A 62501 4 WALKER PHYSICAL CHEMISTRY CHEM. LIB. QD 453 WISH 1922 UNIV MICH e. . kaantudeželaniakasŁUAR 3 MIC HIG ! NON? M MICH HIG CHIG MICHIF 1. AN GA Lis '*'*'* E AIR M Smu INIMIIDEMMIOKT TULIE LIBRARY ARTES HUDSTATI AVA 1837 ATO MATRISTS VERITAS PL UNIVERSITY OF MICHIGAN PLURIOUS UNUM TUEROR HIC SI-QUÆRIS-PENINSULAM-AMĶINAM · CIRCUMSPICE APATATATATATATA UKINU SCIENTIA OF THE KAVOSUJIN 息 ​VET 80 NCHIS VIN AUR Bl vn MIC AN ht MI 34 UNIVERS UNA CHIG MICA LIGH IN" F INDS BRIA "LES MICHIG کتری 16 Get INC 1077 A UNIVER UN asi? ACT UN 3+ ARK/ UNIVE Pa 251 A MIC પરંતુ AN AICA RSITY OLES AN MICHIG. Jonat 11G AN AN MIC MINA FWITH NIVERS " and 1 کی موریار 廠 ​BESE HIG 3 A May PATSY XXIV URIK MIC HIGAN NER 136 AL MIC LA ALEXA SA B M AIN AN A+ 燦​盛 ​LORD MAKE UNIVER ERS, MI MIC NIVERS 22.02 A ERS MILA RS کیا Be AV AIN AN MIC HIG MICHI 14 YER MIC UNIV ARE qkaf 131 That's the win the start of and 140 AN SITY UNI IN HIG AR M MIC CHIG ERSITY IN MIC NIVERS ERS Hai Kaly UN BETTER AN AN SITY! 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NIV $4 ERSITY IVERS CENTRISING In ។ ERSIT AN MICHIG GAN MICHIG MIC SITY MIC SAIND NIVER MIC NIVERS UNI ક AN NIVERS VER IVERS HIG RSITY P ENLIS OF HIG MICHIG MIC NIVE ERS ANI CHIG PAMAEX AIN era M MI MI MI AN Chem Rib Q ]] 453 W181 1922 INTRODUCTION ΤΟ PHYSICAL CHEMISTRY &AM MACMILLAN AND CO., LIMITED LONDON BOMBAY・ CALCUTTA MADRAS • MELBOURNE THE MACMILLAN COMPANY NEW YORK BOSTON CHICAGO . DALLAS SAN FRANCISCO • • • · THE MACMILLAN CO. OF CANADA, LTD. TORONTO INTRODUCTION ΤΟ PHYSICAL CHEMISTRY BY SIR JAMES WALKER LL.D., F.R.S. PROFESSOR OF CHEMISTRY IN THE UNIVERSITY OF EDINBURGH NINTH EDITION * } MACMILLAN AND CO., LIMITED ST. MARTIN'S STREET, LONDON ……………… i 1922 rec COPYRIGHT First Edition, 1899. Third Edition, 1903. Fifth Edition, 1909. Sixth Edition, 1910. Second Edition, 1901 Fourth Edition, 1907 Reprinted 1910 Seventh Edition, 1913 Eighth Edition, 1919. Ninth Edition, 1922 PRINTED IN GREAT BRITAIN chemistry teacher CORA EXTRACT FROM THE PREFACE TO THE FIRST EDITION THIS book makes no pretension to give a complete or even systematic survey of Physical Chemistry; its main object is to be explanatory. I have found, in the course of experience in teaching the subject, that the average student derives little real benefit from reading the larger works which have hitherto been at his disposal, owing chiefly to his inability to effect a connection between the ordinary chemical knowledge he possesses and the new material placed before him. This state of affairs I have endeavoured to remedy in the present volume by selecting certain chapters of Physical Chemistry and treating the subjects contained in them at some length, with a constant view to their practical application. As I have assumed that the student who uses this book has already taken ordinary courses in chemistry and physics, I have devoted little or no space to the explanation of terms or elementary notions which are adequately treated in the text-books on those subjects. I have throughout avoided the use of any but the most elementary mathematics, the only portion of the book requiring a rudimentary knowledge of the calculus being the last chapter, which contains the thermodynamical proofs of greatest value to the chemist. Since it is of the utmost importance that even beginners in physical chemistry should become acquainted at first hand with original work on the subject, I have given a few references to papers generally accessible to English-speaking students. August 1899. J. WALKER. PREFACE TO THE NINTH EDITION NOTWITHSTANDING the shortness of the period which has elapsed since the appearance of the last edition, it has been found necessary to revise many chapters, and practically to rewrite those dealing with the intimate nature of the atom and the system of the elements. Reference is made in the text to volumes dealing with special branches of the subject: to the reader who wishes to pursue its study as a whole may be recommended N. C. M. Lewis's System of Physical Chemistry, 3 vols. (1920). August 1921. J. W. vii ' UNITS AND STANDARDS OF MEASUREMENT THE SIMPLE GAS LAWS. SPECIFIC HEATS CONTENTS THE ATOMI THEORY AND ATOMIC WEIGHTS THE PERIODIC LAW SOLUBILITY CHAPTER I CHAPTER II CHAPTER III CHAPTER IV CHAPTER V CHAPTER VI FUSION AND SOLIDIFICATION CHAPTER VII PAGE 1 8 22 25 33 44 55 ix X INTRODUCTION TO PHYSICAL CHEMISTRY VAPORISATION AND CONDENSATION THE PHASE RULE ALLOYS. CHAPTER IX THE KINETIC THEORY AND VAN DER WAALS'S EQUATION HYDRATES ز CHAPTER VIII THERMOCHEMICAL CHANGE CHAPTER X CHAPTER XI CHAPTER XII CHAPTER XIII CHAPTER XIV VARIATION OF PHYSICAL PROPERTIES IN HOMOLOGOUS SERIES CHAPTER XV RELATION OF PHYSICAL PROPERTIES TO COMPOSITION AND CONSTITUTION CHAPTER XVI THE PROPERTIES OF DISSOLVED SUBSTANCES CHAPTER XVII OSMOTIC PRESSURE AND THE GAS LAWS FOR DILUTE SOLUTIONS PAGE 69 81 96 111 120 128 138 48 62 172 1 ป CONTENTS xi CHAPTER XVIII DEDUCTIONS FROM THE GAS LAWS FOR DILUTE SOLUTIONS MOLECULAR COMPLEXITY METHODS OF MOLECULAR WEIGHT DETERMINATION COLLOIDAL SOLUTIONS CHAPTER XIX BALANCED ACTIONS 1 CHAPTER XX CHAPTER XXI ELECTROLYTES AND ELECTROLYSIS CHAPTER XXII ELECTROLYTIC DISSOCIATION CHAPTER XXIII · CHAPTER XXIV CHAPTER XXV RATE OF CHEMICAL TRANSFORMATION CHAPTER XXVI REL TIVE STRENGTHS OF ACIDS AND OF BASES. CHAPTER XXVII EQUILIBRIUM BETWEEN ELECTROLYTES { PAGE 184 191 212 221 230 247 265 286 297 310 xii INTRODUCTION TO PHYSICAL CHEMISTRY NEUTRALITY AND SALT-HYDROLYSIS CHAPTER XXVIII ELECTROMOTIVE FORCE APPLICATIONS OF THE DISSOCIATION THEORY CHAPTER XXIX CHAPTER XXX CHAPTER XXXI POLARISATION AND ELECTROLYSIS INDEX CHAPTER XXXII DIMENSIONS OF ATOMS AND MOLECULES CHAPTER XXXIII RADIO-ACTIVE TRANSFORMATIONS CHAPTER XXXIV ATOMIC NUMBER AND ISOTOPY CHAPTER XXXV THE STRUCTURE OF ATOMS CHAPTER XXXVI THERMODYNAMICAL PROOFS PAGE 324 336 352 365 1 371 381 391 $398 405 433 1 1 CHAPTER I UNITS AND STANDARDS OF MEASUREMENT To express the magnitude of anything, we use in general a number and a name. Thus we speak of a length of 3 feet, a temperature difference of 18 degrees, and so forth. The name is the name of the unit in terms of which the magnitude is measured, and the number indicates the number of times this unit is contained in the given magnitude. The selection of the unit in each case is arbitrary, and regulated solely by our convenience. In different countries different units of length are in vogue, and even in the same country it is found convenient to adopt sometimes one unit, sometimes another. Lengths, for example, when very great are expressed in miles; when small, in inches; and we also find in use the foot and yard as units for intermediate lengths. Such units as these British measures of length were fixed by custom and convention, and for the purposes of everyday life are convenient enough. When we come, however, to the discussion of scientific problems, we find that they are un- suitable and inconvenient, leading to clumsy calculations the greater part of which could be dispensed with if the units of the various magnitudes were properly selected. There is, for instance, no simple relation between any of the British units of length and the usual British standard of capacity-the gallon. Now lengths, volumes, and weights often enter in such a way into scientific calcula- tions that the existence of simple relations between the units of these magnitudes enables us to perform a calculation mentally which would necessitate a tedious arithmetical operation were the units not thus simply related. } The first requirement of a convenient system of measurement is that all multiples and subdivisions of the unit chosen should be decimal, in order to be in harmony with our decimal system of numeration. For scientific purposes the decimal principle of measure- ment is uniformly accepted, save in circular measure and in the measurement of time, where the ancient sexagesimal system (of 1 B 2 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. counting by sixties) still in part prevails. The unit of length is subdivided into tenths, hundredths, and thousandths if we wish to use the smaller derived units; if we desire a larger derived unit we take it ten, a hundred, or a thousand times greater than the funda- mental unit. The choice of this fundamental unit is, as has been already stated, quite arbitrary. At the time of the French Revolution, when a decimal system was first adopted in a thoroughgoing manner, the unit of length, the metre, was selected because it was of a convenient length for practical measurement, and in particular because it was supposed to have a natural relation to the size of the earth, ten million metres measuring exactly the quadrant of a circle through the poles. The value of the fundamental length therefore depended theoretically on the determination of the length of the earth's meridional quadrant. But the degree of accuracy with which this determination can be made is far inferior to that obtainable in comparing two short lengths (say a metre) together. If the metre then were strictly defined by the earth's dimensions, the fundamental unit of length would change with every fresh determination of the polar circumference. For practical purposes the metre is legally defined as the distance, under certain conditions, between two marks on a rod of platinum-iridium preserved in Paris, and the supposed exact relation to the earth's circumference has been given up.1 Copies of this standard have been made and distributed, and the relations between them and similar standards of length, such as the yard, have been determined with great exactness. For many scientific purposes, the hundredth part of the metre, the centimetre, is a convenient unit, and in what follows we shall generally make use of it in calculations. The millimetre is also much used, and for microscopical and submicroscopical lengths we have μ µ = 1 micron = 10−4 cm. = 10-4 cm. 10-3 mm. 1 micromillimetre = 10~7 cm. = uu 10-6 mm. The Ångström unit (A.U.) of spectroscopy is 10-8 cm. or 10-7 mm. The unit of mass (or weight),2 the gram, was primarily defined as the mass (or weight) of 1 cubic centimetre of water at the Here we temperature at which its density is greatest, viz. 4° C. depend on the constancy of the properties of an arbitrary substance (pure water) to establish a relation between the units of weight and of cubical capacity, or volume. The original standard kilogram was constructed in accordance with this relation, being made equal to 1000 grams as above defined, i.e. equal in weight to a cubic - 1 According to recent measurement, the ten-millionth part of the earth's quadrant is nearly 0.1 millimetre longer than the standard metre. 2 When a chemist uses an ordinary balance he directly compares weights, but, since at any one place weight is proportional to mass, he indirectly compares the masses of the substances on the two pans. So far, then, as measurement of quantity of material by means of the balance is concerned, the terms are interchangeable. 1 4 I UNITS AND STANDARDS OF MEASUREMENT 3 3 decimetre of water at its maximum density point. Since it is possible, however, to compare weights with each other with much greater accuracy than is attainable in the measurement of volumes, the exact relationship between the two units has been allowed to drop, and for exact purposes the kilogram is defined as the weight of the platinum standard kilogram kept in Paris. For the purposes of the chemist the relation between the units of weight and volume as originally defined may be looked upon as exact, since the error is not greater than 0·01 per cent, a degree of accuracy to which the chemist attains only in exceptional circumstances. For measuring the volume of liquids, the litre is defined, not as a cubic decimetre, but as the volume occupied by a quantity of water which will balance the standard kilogram in vacuo at 4° C. The unit of time seldom enters into chemical calculations. The standard for the unit is derived from the length of time necessary for the performance of some cosmical process-for example, the time taken by the earth to perform a complete revolution on its axis. The minute, as measured by a good-going clock or watch, is the unit generally employed in determinations of the velocity of chemical actions. The units of length, weight, and time being once fixed, a great many derived units may be fixed in their turn. The C.G.S. system, which takes, as the letters indicate, the centimetre, the gram, and the second for the fundamental units, is very frequently employed, especially in theoretical calculations, and we shall often have occasion to use it. For example, instead of expressing the average atmospheric pressure as that of a column of mercury 76 cm. high, it is expedient in many calculations to give the pressure in grams per square centimetre. The conversion may easily be performed as follows. Suppose the cross-section of the mercury column to be 1 sq. cm., then if the height of the column is 76 cm., the total volume of mercury is 76 cc., and the pressure per square centi- metre is the weight of 76 cc. of mercury, viz. 1033 g. 2 The average pressure of the atmosphere, then, is equal to 1033 g. per square centimetre. 1 To give an idea of the difficulty of accurate measurement when volumes are con- cerned, the following example may suffice. By Act of Parliament a gallon was defined as equal to 4543458 litres, this number being derived from the weight of a cubic inch of water in grains, and the relation between the inch and the decimetre, a litre being supposed equal to a cubic decimetre. The original definition of the gallon is "the volume at 62° F. of a quantity of water which balances 10 brass pound weights (true in vacuo, and of specific gravity 8·143) in air at 62° F. and 30 inches pressure (barometer reduced to the freezing point) and two-thirds saturated with moisture.' From this definition, and the relation of the pound to the kilogram, Dittmar calculated that 1 gallon is equal to 4.54585 litres, a value which differs from this statutory relation by 1 part in 2000. By an Order in Council of May 1898, the gallon was declared equivalent to 4.54596 litres. 2 This value is obtained by multiplying 76 by the specific gravity of mercury, viz. 13.59. 11 4 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. The specific volume of a substance may be defined as the number of units of volume which are occupied by unit weight of the substance. In the above system it is therefore the number of cubic centimetres occupied by one gram. The specific weight, or density, of a substance is the number of units of weight which occupy unit volume: in the above system, the number of grams which occupy one cubic centi- metre. These definitions are directly derived from the units of weight and volume, and are independent of the properties of any substance save that considered. For purposes of measurement in the case of solids and liquids, however, it is much more convenient and accurate to com- pare the density or specific volume of one substance with that of another arbitrarily chosen as standard, than to effect an absolute deter- mination by measuring both weight and volume of one and the same substance. The substance usually chosen as the standard of com- parison is water, and that for two reasons. In the first place, water is easily obtained and easily purified; in the second place, water is a standard substance in other respects; in particular it is the substance which was used to fix the relation between the units of weight and of volume. From this relation the density of water in our units is 1 (cp. p. 2), so that if we take water as our standard, and refer all densities to it as the unit, we have what are really relative densities, although if measured at 4° the numbers obtained do not differ greatly from the absolute densities of the substances. The actual comparison is made in the case of liquids by weighing the same tared vessel filled with water, and filled with the liquid whose density is to be determined. Although the actual volume is unknown, it is known to be the same in both cases, so that the quotient of the weight of liquid by the weight of water gives the relative density of the liquid. In order to specify with exactness the density of a substance, it is necessary to indicate at what temperature the determination or com- parison has been made. As substances change in temperature they invariably change in volume, so that the absolute density varies with the temperature. In specifying the relative density, or specific gravity, it is not only necessary to give the temperature at which the substance is weighed, but also the temperature at which the equal volume of water is weighed. Thus we meet with such data as the following, 1.0653, which indicates that the weight of the substance at 15.5° is 1.0653 times as much as the weight of the same volume of water at 4°. As a general rule, the weights are both for the same temperature, say 15°; or the substance is weighed at this tempera- ture and referred either directly or by calculation to water at 4°, its maximum density point. In the latter case the density given is very nearly equal to the absolute density of the substance at the specified temperature. 15'5 4 I UNITS AND STANDARDS OF MEASUREMENT 5 Quantities of matter may always be expressed in terms of the unit. of weight irrespective of the form in which the matter may exist. When we come to measure energy we find no such common unit in which its amount may be expressed. Each form of energy, such as heat, work, electrical energy, has its own special unit, but according to the Law of Conservation of Energy, a given amount of any one form of energy is under all circumstances equivalent to constant amounts of the other kinds of energy; we have therefore merely to ascertain the relation between the different special energy units in order to express a given amount of energy in terms of any one of these units. Thus electrical energy is usually expressed in volt-coulombs, but it may easily be expressed in terms of the mechanical or heat units, the numbers then indicating the amounts of mechanical or thermal energy into which the electrical energy may be converted. The "absolute " unit of mechanical energy, the erg, is the product of the unit of length into the "absolute" unit of force, or dyne, i.e. the force necessary to impart to a mass of 1 g. in a second a velocity of 1 cm. per second. For our purposes, however, it will generally be more convenient to use gravitation units, in the definition of which the value of the earth's gravitational attraction for bodies on its surface is involved. The unit of force thus defined is the weight of 1 g., and is equal to 981 dynes. The unit of mechanical energy is therefore the product of this unit into the unit of length, i.e. the gram-centimetre. To obtain the unit of heat and a scale of temperature it is again necessary to refer to the properties of some arbitrarily-chosen substance or substances. In the construction of a scale of temperature it is customary to fix two points by means of well-defined and presumably constant properties of some standard substance, and divide the range between these two points into equal arbitrary units as measured by the change in property of some substance caused by change of temperature. Thus in the centigrade scale the two points which are fixed are, first, the freezing point of the standard substance water, and second, its boiling point under a pressure of 76 cm. of mercury. When We use a mercury thermometer it is assumed that equal changes in volume of the mercury correspond to equal changes of temperature,¹ so by dividing the whole change of volume of the mercury between the freezing and boiling points of water into 100 equal portions, we obtain an ordinary centigrade thermometer. Similarly, if we use a hydrogen thermometer, we assume that equal increments of volume correspond to equal increments of temperature. In each case we depend for our measurement of temperature on the properties of some particular substance, and it does not at all follow that the temperature as measured by one substance will exactly 1 In mercury and other liquid thermometers we really deal with the difference of the volume change of the liquid and of the vessel containing it. 6 CHAP. INTRODUCTION TO PHYSICAL CHEMISTRY coincide with the temperature as measured by another. In point of fact, the temperatures registered by the mercury and hydrogen thermometers never exactly agree except where the two thermometers were originally made to correspond when their fixed points were determined. For accurate comparative work temperatures are referred to an international hydrogen scale. The "international " hydrogen thermometer is a constant-volume instrument, that is, instead of measuring the change of volume caused by change of temperature, we measure the change of pressure, when the volume is kept constant. The scale is defined as having "fixed points at the temperature of melting ice (0°) and that of the vapour of distilled. water in ebullition (100°), under the normal atmospheric pressure; the hydrogen being taken under the manometric initial pressure of 1 metre of mercury." For temperatures between 0° and 100° the maximum difference between the international and the Kew standard mercury thermometers only amounts to a few hundredths of a degree centigrade. In theoretical calculations absolute temperatures are frequently employed. The absolute scale of temperature has degrees of the same size as centigrade degrees, but starts from a point 273 degrees below zero centigrade. The absolute temperature is there- fore obtained by adding 273 to the temperature on the centigrade scale. The usual unit of heat for chemical purposes is the calorie. It is roughly defined as the quantity of heat required to raise the temperature of one gram of water through one degree centigrade. This quantity is not exactly the same at all temperatures, e.g. the amount of heat necessary to warm a gram of water from 0° to 1° is somewhat different from that required to heat it from 99° to 100°. It is therefore necessary to specify the temperature at which the heating takes place. Practically the calorie measured at the ordinary temperature, say from 15° to 16°, is the most convenient. The great or kilogram calorie is 1000 times the ordinary calorie. * It is a matter of importance to fix the relation of the mechanical unit of energy to the heat unit, i.e. to determine the mechanical equivalent of heat. Mechanical energy, e.g. the energy of a falling body, is converted by friction or otherwise into heat, the amount of mechanical energy and the amount of heat resulting from it being both measured. In this way it has been found that 42,720 gram- centimetres are equivalent to 1 gram calorie at 15°, i.e. 42,720 grams falling through a centimetre will generate enough heat in a friction apparatus to raise the temperature of a gram of water one degree at 15°. In the sequel we shall denote this value by J. Another unit of heat, the joule, is sometimes used for thermo- chemical purposes. It is simply related to the unit of mechanical energy, being equivalent to 10,000,000 ergs, and is denoted by the I 4 1 I UNITS AND STANDARDS OF MEASUREMENT 7 letter j. The relation between the joule and the calorie for 15° is given by the equations 1 cal. = 4·189 j. 1 j=0·2387 cal. The relations between the units employed in electrical measure- ments have been the subject of many accurate experimental investiga- tions. The units have been chosen theoretically so as to give a unit of electrical energy which bears a simple relation to the absolute unit of mechanical energy. The theoretical unit of resistance, the ohm, may be practically defined as the resistance at 0° of a column of mercury 1 sq. mm. in section, and 1.0630 metres long. Specific conductivity is expressed in terms of the conductivity of a substance, of which a cube having 1 cm. edge shows a resistance of 1 ohm between a pair of opposite faces. Thus the specific conductivity of mercury at 0° is 10630 in terms of this unit, as a reference to the practical definition of the ohm will show. The symbol is usually employed to denote the specific conductivity as thus defined. κ The unit quantity of electricity, the coulomb, is defined as the quantity of electricity which will deposit 1.1180 mg. of silver from a solution of a silver salt under properly chosen conditions. The electrochemical unit, or faraday, which is equal to 96,540 coulombs, is the charge of electricity carried by one gram-equivalent of an ion, e.g. the charge carried by 107.9 grams of silver as ion. The unit of current, the ampère, is the current of which the flow is 1 coulomb per second. The unit of potential difference or electromotive force, the volt, is so related to the previous units that when the potential difference between the ends of a resistance of 1 ohm is 1 volt, a current of 1 ampère will pass through the resistance. For purposes of com- parison the difference of potential between the electrodes of a standard galvanic cell is used. Thus a Clark's cell, which is frequently employed for this purpose, has a potential difference between its electrodes equal to 1·433 volts at 15°, and the international Weston or cadmium cell a potential difference of 1.0183 volts at 20°. The unit of electrical energy, the volt-coulomb, is the product of the volt into the coulomb, and is equivalent to 1 joule, 107 ergs, or 10,197 gram-centimetres. In connection with the subject-of units, the student may read CLERK- MAXWELL, Theory of Heat, Chaps. I.-IV. CHAPTER II THE ATOMIC THEORY AND ATOMIC WEIGHTS "" CHEMISTS have for long been in the habit of expressing their experi- mental results in terms of the Atomic Theory propounded by Dalton at the beginning of last century. At the time of its inception the facts which this theory had to explain were comparatively few in number and simple in character. Nowadays, the experimental data are numberless, and often of great complexity, yet the atomic theory is still capable of affording an easy and convenient mode of formulation for them. It is true that in the course of years the notions associated with the term "atom have undergone many changes, but Dalton's fundamental idea is unaltered, and will persist for many years to come. The results of chemical analysis show that most substances can be split up into something simpler than themselves, but that a few, some. eighty, resist all attempts at decomposition, and are incapable of being built up by the union of any other substances. These unde- composed bodies we agree to call elements, and to express the com- position of all other bodies in terms of them. It does not, of course, follow that because we have hitherto been unable to decompose these elements, methods may not at some later time be found for effect- ing their decomposition; indeed, it is known that the radioactive elements are subject to a constant process of spontaneous disintegra- tion into other elements (cp. Chap. XXXIII.). However this may be, we shall see in the sequel that the elements form a group of sub- stances, singular not only with respect to the resistance which they offer to decomposition, but also with respect to certain regularities displayed by them and not shared by the substances which go under the name of compounds. The atomic theory affords a simple explanation of the manner in which elements unite to form compounds. Each element is assumed to be not infinitely divisible, but to consist rather of minute particles- the atoms-indivisible with regard to all ordinary physical and chemical processes and all alike amongst themselves, especially with respect to their weight, although this last assumption requires qualification (cp. 8 CH. II THE ATOMIC THEORY AND ATOMIC WEIGHTS 9 1 11 Chap. XXXIV.). The atoms of different elements are supposed to differ in weight and in other properties. Compounds must, on our assumption, be built up of these atoms, the mode of their hypo- thetical union being chosen in accordance with the observed facts. In the first place, the atoms must be supposed to retain their weight unchanged, no matter in what state they may exist; for all our chemical experience shows that chemical change is unaccompanied by any change in the total weight (or the total mass) of the reacting substances. Again, the same kinds of atoms in the same proportions go to the formation of any particular compound. This must be assumed to bring the theory into harmony with the constant com- position exhibited by all compounds. Further, as the number of compounds actually existing is vanishingly small compared with the possible combinations of the different atoms, various rules have had to be devised regarding the nature and relative number of atoms which may unite with each other, in order that under these restrictions the number of possible combinations may agree more closely with the number of existing compounds. In these rules of " In these rules of "valency," etc., other properties of atoms besides their weight are indicated, and it has been found necessary with regard to the carbon atom in particular to make very specific assumptions as to its nature, and its mode of combination with other atoms. From what has just been said, it is evident that our conception of the atom is made to accommodate itself to the facts it has to explain. Beyond the constancy of weight of each atom, nothing had to be assumed in Dalton's time except that the atoms united in proportions expressible by the simple whole numbers, and these assumptions were made in order that the theory might be in harmony with the laws of constant and of multiple proportions. It should be noted that at the present time we can scarcely speak of a law of simple multiple proportions existing. We know well-defined substances whose com- position we are obliged to express by means of such formulæ as 20H23N, C₁3H1106—to choose two examples at random from organic chemistry. Here there is no approximation to simple multiples. 5 When Dalton's theory had led to the adoption of a simple system for expressing the numerical proportions by weight in which elements combine and substances in general enter into chemical action, the conception of atom was practically allowed to drop, and its place was taken by the purely numerical conception of equivalent weight or combining proportion. The necessity of attaching any other significance to the atom than a constant weight was not generally felt, as the facts were not sufficiently well known to require any extension of the idea. Although the name "atom" continued to be used, the conception was practically that of a number. Laurent, for instance, in his Chemical Method (1853) says: "By the term 'atoms,' I understand the equivalents of Gerhardt, or, what comes (C 10 CHAP. INTRODUCTION TO PHYSICAL CHEMISTRY to the same thing, the atoms of Berzelius." Again, "In order to avoid all hypotheses, I shall not attach to the term 'atom' any other sense than that which is included by the term 'proportional number.' For each element there existed a number which expressed its combining weight, and the composition of any compound was ex- pressed in terms of these weights as standards. It was therefore of extreme importance to fix the standard weights relatively to each other, in order that no ambiguity should exist in the formulation of the compounds. Had there been no law of multiple proportions the task would have been easy. The weight of one element would have been chosen as ultimate standard to which all others would be referred, and as in that case each element would have united with each other element in one proportion only, no ambiguity could possibly exist. But the law of multiple proportions expresses the fact that a given weight of one element may unite with more than one weight of another element to form more than one compound, and that then the ratios of the weights of the second element are expressible by means of the simple whole numbers. Taking the combining weight of hydrogen as standard and unit, the combining weight of oxygen might be either 8 or 16, for 1 g. of hydrogen unites with 8 g. of oxygen to form water, and with 16 g. to form hydrogen peroxide. So it is with most of the other elements. The choice becomes still more complex when the element to be considered forms no compound with the standard element hydrogen (as is often the case) and various compounds with the element oxygen, itself of doubtful combining weight. Some principle had therefore to be adopted for obtaining a consistent system of combining weights, only one for each element, to which all other weights might be referred. The extraordinary insight and acumen of Berzelius enabled him to arrive at a system of numbers not greatly differing from those in use at the present day, although he had little to guide him but A better the principles of analogy and simplicity of formulation. guide was gradually recognised in the uniformity displayed by sub- stances in the gaseous state. When the atomic theory was first propounded, no distinction was made between the ultimate particles of elements and the ultimate particles of compounds; both were alike called atoms. By the atom of a compound was meant the smallest particle of it that could exist, any further subdivision resulting in the splitting of the compound into. its elements, or into simpler compounds. It referred, therefore, to an ultimate particle which might be decomposed chemically, but not mechanically. On the other hand, the ultimate particle of an element was an atom which could not be decomposed either mechanically or chemically. There was thus a slight difference in the sense in which the term "atom" was used in regard to the two classes of substances; II THE ATOMIC THEORY AND ATOMIC WEIGHTS 11 and until this difference was recognised, little help was afforded to It had been observed systematic chemistry from the study of gases. by Gay-Lussac that gases entered into chemical actions in simple proportions by volume, the volumes of the different gases being of course measured under the same external conditions of tempera- ture and pressure. Now, according to Dalton, gases (like all other substances) also entered into chemical action in simple pro- portions by atoms. There was thus necessarily a simple connection between the atoms of gases and the volumes they occupied. Dalton himself, led by other speculations, tried the assumption that equal volumes of different gases contained the same number of atoms, i.e. that the weights of equal volumes of different gases were proportional to the weights of their atoms. He rejected this supposition, however, as not being in harmony with the facts. Two measures of nitric oxide give on decomposition one measure of nitrogen and one measure of oxygen, i.e. on Dalton's assumption, two atoms of nitric oxide give one atom of nitrogen and one atom of oxygen. But there must be at least two atoms of nitrogen in two atoms of nitric oxide if the elementary atoms are really indivisible, which contradicts the above hypothesis. Avogadro showed how this difficulty might be overcome by distinguishing in the case of the elements between the different senses of indivisibility above referred to. The gist of his reasoning is as follows: The particles of a gas may be supposed to be the smallest particles obtainable by mechanical division, whether they are particles of elements or of compounds. But it does not follow, even for the elements, that this limit of mechanical or physical divisibility is also the limit of chemical divisibility. The gaseous particles of a compound. can be chemically decomposed into something simpler; so may the gaseous particles of an element, only in the latter case the products of the decomposition of the gaseous particle must be atoms of the same kind, and not atoms of different kinds as with the compounds. He made the distinction, therefore, between atoms and molecules as these terms are used at the present day. The molecules are the mechanically indivisible gaseous particles, which may consist each of more than one elementary atom-of different kinds in the case of compounds, of the same kind in the case of elements. The hypothesis which Avogadro now made to account for the relations between combining weights and volumes was the following. Equal volumes of different gases under the same physical conditions contain the same number of molecules; or, the weights of equal volumes of There is different gases are proportional to the weights of their molecules. now no longer any difficulty encountered in the volume relations of nitric oxide and its decomposition products. According to Avogadro, two molecules of nitric oxide give one molecule of nitrogen and one molecule of oxygen. But as two molecules of nitric oxide must 12 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. contain at least two atoms of nitrogen and two atoms of oxygen, the molecule of nitrogen must contain at least two atoms of nitrogen, and the molecule of oxygen two atoms of oxygen. The general adoption of this one principle has practically proved sufficient to fix the combining weights of the elements, and provide us with our present system of atomic weights. Although Avogadro formulated his hypothesis in 1811, the times were not ripe for it; and it was only after 1858, when Cannizzaro published a sketch of its con- sistent application to organic and inorganic chemistry, that its adoption. became at all general. In the middle of the century the greatest confusion prevailed, different writers using different systems of com- bining weights, so that each compound had as many different formula as there were competing tables of equivalents. So long as weights only were considered, one system was as good as another; but when volume relations, physical properties, and the nature of the substances produced in chemical actions are taken into account, the present system of atomic weights is the only one which gives a simple and consistent expression of the results. The hypothesis of Avogadro often goes under the name of Avogadro's Law, but it must be borne in mind that it is a purely hypothetical statement, and not to be confounded with a generalised expression of fact such as the Law of Constant Proportions. This latter is independent of any theory, while the former is entirely theoretical—a fact which of course in no way impairs its usefulness. From another point of view Avogadro's principle may be looked upon. as giving a definition of the magnitude we know as the gram-molecular weight, or molar weight (p. 14). In the following pages a sketch is given of the mode employed in fixing our present system of atomic weights with the help of Avogadro's principle. We are not for this purpose concerned with the absolute weight of atoms and molecules-all that we require is to express their weights relatively to each other. To begin with, a standard must be chosen to which all chemical weights are to be referred. The standard is purely arbitrary, and in choosing it we only consult our own convenience. Two elements, hydrogen and oxygen, have been selected on practical grounds for this purpose. Dalton chose hydrogen because it had the smallest equivalent of the elements; Berzelius chose oxygen because it was easy to compare the equivalents of the other clements with that of oxygen directly. Hydrogen does not form many well-defined and easily-analysed com- pounds with the other elements, so that comparison with the equiva- lent of hydrogen has usually to be indirect, frequently through the medium of oxygen. Oxygen has now been definitely chosen by inter- national agreement as the standard element, and its atomic weight has been fixed as 16 exactly. The actual number selected as standard magnitude is purely arbitrary; Dalton adopted 1 as the atomic weight II THE ATOMIC THEORY AND ATOMIC WEIGHTS 13 of hydrogen, Berzelius made the equivalent of oxygen equal to 100, and referred the other numbers to this. The reason why the number 16 has been chosen to express the atomic weight of oxygen when that element is taken as standard, is that when hydrogen, as was formerly the case, is made to have an atomic weight equal to unity, the number which then expresses the atomic weight of oxygen is very nearly equal to 16. Thus, whether the old or the new standard is adopted, we obtain practically the same set of numbers for the atomic weights of the other elements, if round numbers only are required. From recent researches carried out by independent investigators the ratio HO has been determined with accuracy to be 1: 15·88, or 1·0076: 16. With O= 16 then, we have a set of atomic weights all 1.0076 times greater than when H = 1. For accurate purposes this difference between the two systems must be considered, but for the everyday purposes of chemistry the system is practically one of round numbers based on O=16, although we still often look on H = 1 as the theoretical standard, and use it in calculations not requiring the highest degree of accuracy. 1 In fixing our system of atomic weights on Avogadro's principle, the weights of equal volumes of gases are compared. Now the weights of equal volumes of gases are proportional to their relative densities, and for the present purpose these densities are most conveniently referred to that of hydrogen = 2, or, for accurate purposes, of oxygen = 32, or of pure air 28.979. The reason for choosing the density of hydrogen equal to 2 instead of equal to 1, as is customary, is that on the former assumption the density of a gas is expressed by the same number as its molecular weight, instead of by half that number. The selection of the unit is quite arbitrary, and we may as well choose that which leads to the greatest simplicity. The atomic weight of hydrogen is 1, and on Avogadro's principle we must argue that the molecule of hydrogen contains two atoms, in order to satisfy, for example, the volume relations for the formation of hydrochloric acid gas from hydrogen and chlorine. Two volumes of hydrochloric acid gas are formed from one volume of hydrogen and one volume of chlorine, so that by reasoning similar to that employed above in the case of nitric oxide, each hydrogen molecule. and each chlorine molecule must contain at least two atoms. The molecular weight of hydrogen is therefore at least 2 if its atomic weight is 1. It is convenient thus to express molecular weights in the same unit as atomic weights, for then the molecular weight is simply the sum of the atomic weights contained in the molecule. The determination of the approximate molecular weight of a substance therefore resolves itself into finding the weight of its vapour in grams which will occupy the same volume as 2 g. of 1 The volumes must of course be measured under the same conditions of temperature and pressure. 14 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. hydrogen measured under the same conditions of temperature and pressure. This weight is called the gram-molecular weight or molar weight of the substance, and the volume which it occupies is called the molar volume. It is practically the same for all gases, and at 0° and 76 cm. may be taken equal to 22·4 litres. Only rough values of the molecular weight are in this way obtained, for the determination of the density of gases and vapours does not under ordinary circumstances admit of any great degree of accuracy; besides which Avogadro's principle can be applied in strictness only to ideal gases, i.e. to those which obey the simple gas laws with absolute exactness. For ideal gases the molar volume is 22:412 litres. It is easy, however, to arrive at the true molecular weight from the approximate value obtained from vapour-density determinations by taking account of the results of analysis, which are susceptible of great accuracy.¹ It is evident that the true molecular weight must contain quantities of the elements which are exact multiples or submultiples of the combining proportions of these elements, and the combining proportions are determined by analysis alone. We therefore select as true molecular weight the number fulfilling this condition which is nearest the approximate molecular weight. For example, the molecular weight of sulphuretted hydrogen, as determined from the vapour density, is 344, i.e. 34'4 g. of sulphuretted hydrogen occupy the same volume as 2 g. of hydrogen under the same external conditions. But we know that for 1 g. of hydrogen in the sulphide there are 160 of sulphur. The true molecular weight therefore must contain a multiple of 10 g. of hydrogen, and the same multiple of 16.0 of sulphur. The number 34.0 evidently does this, for it is equal to 2 (10+160). We consequently take 340 as the molecular- weight of hydrogen sulphide instead of the number 34-4 obtained from the determination of the vapour density. The atomic weight of an element is now deduced from the molecular weights of its gaseous compounds as follows. A list of the molecular weights of the gaseous compounds is prepared, and the quantities of the element in these weights of the compounds are noted alongside. The figures in this second column are the results of analysis alone, and their greatest common measure is in general equal to the atomic weight of the element. Examples are given in the following tables. The first column of numbers contains the gram-molecular weights of the compounds; the second column contains the number of grams of the element in question in the gram-molecular weight; the third column shows the relation of these numbers to their greatest common measure. 1 A method for determining accurate molecular weights of gases from their physical properties without considering the results of analysis will be found in Chap. IX. -- II THE ATOMIC THEORY AND ATOMIC WEIGHTS 15 Compound. Hydrochloric acid Hydrobromic acid Hydriodic acid Water Hydrogen sulphide Hydrogen Ammonia Hydrogen phosphide Methane. Ethane Water Carbon monoxide Phosphorus oxychloride Nitric oxide Oxygen Carbon dioxide Sulphur dioxide Chlorine peroxide Sulphur trioxide Methyl nitrate Osmium tetroxide · Hydrochloric acid Chlorine peroxide Nitrosyl chloride Cyanogen chloride Chlorine. · Ammonia Nitric oxide Nitrogen peroxide Methyl nitrate ` Cyanogen chloride Nitrogen. Nitrous oxide Cyanogen • • • • • Chlorine monoxide Thionyl chloride Sulphuryl chloride Phosphorus trichloride Phosphorus oxychloride Chloroform • Boron trichloride Carbon tetrachloride • •· • · * · • • • · • HYDROGEN OXYGEN • • • ❤ " CHLORINE • • · • · NITROGEN · I. 36.5 81 128 18 34 2 17 34 16 30 18 28 153.5 30 32 44 64 67.5 80 77 255 36.5 67.5 65.5 61.5 71 87 119 135 137.5 153.5 119.5 117.5 154 17 30 46 77 61.5 28 44 52 II. 1 HAT∞∞∞∞∞∞ 1 1 2 2 2 3 3 4 6 16 16 16 16 32 32 32 32 48 48 64 G.C.M. G.C.M. 35.5 35.5 35.5 35.5 71 71 71 71 106.5 106.5 106.5 106.5 142 14 14 14 14 14 28 28 28 G.C.M. III. 1 1 G.C.M. 1 2 x 1 2 x 1 2 x 1 3 x 1 3 x 1 4 x 1 6 x 1 1 x 16 1 x 16 1 x 16 1 x 16 2 x 16 2 x 16 2 x 16 2 × 16 3 x 16 3 x 16 4 x 16 1 16 1 x 35.5 1 x 35·5 1 x 35.5 1 x 35.5 2 x 35.5 2 x 35.5 2 x 35.5 2 x 35.5 3 x 35.5 3 × 35.5 3 x 35·5 3 x 35.5 4 x 35.5 35.5 1 x 14 1 x 14 1 x 14 1 x 14 1 x 14 2 x 14 2 x 14 2 x 14 14 16 CHAP. INTRODUCTION TO PHYSICAL CHEMISTRY Compound. Methane Chloroform Carbon monoxide Carbon dioxide Cyanogen chloride Ethylene Ethane Cyanogen Acetylene Propane Butane Pentane Hexane Benzene • · • • • • Hydrogen sulphide. Sulphur dioxide Sulphur trioxide Sulphuryl chloride Sulphur Carbon disulphide • • • • • CARBON · SULPHUR • I. 16 119.5 28 44 61.5 28 30 52 II. 12 12 12 12 III. 1 x 12 1 x 12 1 × 12 1 x 12 1 x 12 THI 12 24 24 2 × 12 2 x 12 24 24 26 44 2 x 12 2 x 12 3 x 12 36 48 58 72 86 78 60 72 72 4 x 12 5 × 12 6 x 12 6 × 12 2880 34 64 80 135 64 76 32 32 32 32 64 64 Hydrogen=1 Oxygen =16 Chlorine 35.5 Nitrogen =14 Carbon =12 Sulphur =32 G.C.M. 12 1 x 32 1 x 32 1 x 32 1 x 32 2 x 32 2 × 32 G.C.M. 32 Proceeding on the principle explained above, we should now make the following table of the atomic weights of these elements:— These numbers are the generally accepted values for the atomic weights of the elements above considered, and indeed if any element has a large number of volatile compounds whose molecular weights can be determined from their vapour densities, the principle we have adopted leads to practically certain conclusions. The weights of any element contained in the molecular weights of its compounds must be equal to the atomic weight or must be multiples of it, so that if we take the greatest common factor of these multiples, it must either be a simple multiple of the atomic weight or the atomic weight itself. From this it appears, then, that the method is one which only gives values which the atomic weights cannot exceed. But if a great many volatile compounds of the element are known, the chance is very slight that the greatest common factor is still a multiple of the atomic weight, i.e. that another substance may be discovered which will not. contain this greatest common factor in the weight of the element which enters into its molecule; and so the atomic weight fixed in the II THE ATOMIC THEORY AND ATOMIC WEIGHTS 17 manner indicated above has a great degree of probability. If, on the other hand, only a few volatile compounds of the element are available for vapour-density determinations, the method may conceivably fail to give the correct result. Nowadays, however, we are in possession of other means of determining molecular weights besides the method of vapour densities (cp. Chapter XIX.), so that for each element a great many compounds of known molecular weight can be tabulated, even though the volatile compounds of some may be few in number. If we make use of these new methods, the atomic weights of the elements as determined from the molecular weights of their compounds are practically fixed with certainty. The inert gases, argon, helium, etc., enter into no combinations, so that no analysis is possible, and the only method for ascertaining their characteristic weights is the determination of their vapour densities. The atomic weights must in the meantime be assumed equal to their densities compared with hydrogen equal to 2, i.e. their molecules must be assumed to consist of one atom until we find reason to the contrary. That this assumption is justified will be seen in the sequel (Chaps. IV. and V.). Other methods have been used for fixing the atomic weights of the elements, but they must now be looked on rather as checks on the method from molecular weights than as methods of independent applicability. Amongst these the methods depending on Dulong and Petit's Law (Chap. IV.), and on the Periodic Law (Chap. V.) will be referred to in the sections treating of these regularities. From what has been said, the fixing of exact values of the atomic weights consists of two problems-the determination of a set. of equivalents from accurate quantitative experiments, and then the selection of one of these according to some definite principle such as that referred to in the preceding pages. The nature of the quantitative experiments performed depends on the character of the element whose equivalent is to be determined, but it usually consists in the exact comparison of the weights of two substances which contain the same quantity of the element. As far as possible, the experiments are chosen of a simple kind, and should involve the smallest possible number of other elements. Berzelius, for example, dissolved 25.000 g. of lead in nitric acid, evaporated the lead nitrate to dryness and ignited the residue carefully, thereby obtaining 26.925 g. of lead oxide. Twenty-five g. of lead combine therefore with 1925 g. of oxygen, so that if we take 0=16, we obtain the value 207·8 as a possible equivalent of lead. Taking the mean of four similar experiments, we obtain the value 207-3. These experiments may, however, all be affected by a systematic error, i.e. by an error which is the result of this particular method of experiment. The equivalent must therefore be determined by some other method as a check. Berzelius converted lead directly into lead sulphide, and obtained с 18 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. 207.0 as the combining weight of lead if that of sulphur was taken as 32. By converting weighed quantities of lead oxide into lead sulphate, he obtained Pb = 207-0, on the assumption 0 = 16 and S = 32. Other observers using different methods have all got numbers approximating to 2070, the value of 207 2 obtained by Baxter and Grover from analysis of the chloride being now accepted as the most probable. The error in a well-conducted quantitative transformation under ordinary laboratory conditions is about 0.1 per cent of the amount transformed. To secure an error as low as 0.01 per cent extraordinary precautions have to be taken, and it is only in the hands of skilled operators that this degree of accuracy can be attained. In order, too, that this accurate work may be of value, it is plainly essential that the substances operated on should be of a degree of purity corre- sponding to the excellence of the quantitative determinations. Now chemically pure" substances usually contain an amount of impurity greatly exceeding 0.01 per cent of their weight, so that special skill and trouble must be devoted to the purification of the substances employed in the determination of the gravimetric ratio. Even after an experimental accuracy of 0.01 per cent has been secured with pure substances the error in the value of the equivalent is generally much larger than this. Thus, supposing that Berzelius made an error of only this amount in the conversion of lead into lead oxide, 25 g. of lead would give 26.925 g. of oxide, with a possibility of error of 0.0025 g. This means, however, that the quantity of oxygen with which the lead has combined is 1·925, with a possible error of 0·0025 g., i.e. an error of 0.12 per cent on the amount of oxygen. Now this amount is required in fixing the equivalent of lead, so that the error in the equivalent is not less than 0.12 per cent, and the value 207 8 may therefore be two units out in the first decimal place, even on the assumption of this very great experimental accuracy. It may be taken for granted that only for comparatively few elements. has an accuracy of 0.1 per cent on the atomic weight been attained. The elements whose atomic weights were determined by Stas with great ex- perimental refinement are generally accepted as being the most accurate of the older determinations.1 A systematic revision of the atomic weights of the more important elements has been for some years in progress at Harvard University, a revision which aims at determining the ratios of these atomic weights with the utmost attainable accuracy. As an example of modern work carried out by many different in- vestigators, we may take the determination of the ratio in which oxygen and hydrogen combine to form water. Here gases have to be weighed a fact which greatly enhances the experimental difficulty, so that the highest degree of accuracy can scarcely be expected. (6 • 1 These are silver, the halogens, potassium, sodium, lead, sulphur, with less accurate determinations for lithium and nitrogen. II THE ATOMIC THEORY AND ATOMIC WEIGHTS 19 Dittmar and Henderson Scott and others Cooke and Richards Keiser. Morley Berthelot Observers. • Leduc Noyes Burt and Edgar · · • Atomic Weight of Hydrogen for O=16. 1.0084 1.0082 1.0082 1·0075 1.0076 1·0074 1.0075 1·0079 1.0077 Here the extreme difference between the values of different observers is 0.10 per cent an exceedingly good agreement. The value generally accepted at present for the atomic weight of hydrogen is the mean 1.0076, although in the laboratory the approximation 10 is usually sufficient. = The table of atomic weights on p. 21 gives the most probable values for practical use, each figure being significant. Thus, the atomic. weight of carbon, C 12:00, indicates that the atomic weight probably lies between 11·995 and 12.005. It is possible that the atomic weights of some elements have been determined with greater accuracy than that given, but the reverse is probably more often the case. Occasionally the last figure is printed in small type. This indicates either that the value is the mean of several not very concordant determinations, or that, for some other reason, a considerable amount of uncertainty attaches to the figure. An inspection of this table shows that there are some forty elements whose atomic weights have been determined with such accuracy that the first figure after the decimal point may be supposed to possess a real significance. Now, according to the rules of prob- ability, we might expect three-tenths of these, namely thirteen, to have values of the first decimal figure lying within 0·15 of a whole number. Instead of three-tenths, however, we find that more than one-half of this number of elements, namely twenty-four, possess atomic weights diverging by 0.15 or less from the whole numbers. If we only consider elements whose atomic weights are less than 100, the chances of accuracy of the first decimal place are increased, since a unit in that place forms a higher proportion of the whole; yet here also the pro- portion of elements diverging by 0.15 at most from whole numbers is considerably more than half; and if we consider only the elements whose atomic weights were determined by Stas, we find that two- thirds of them have atomic weights diverging by less than 0.1 from whole numbers. It is difficult to believe that this approximation to whole numbers is fortuitous, and we shall see in Chapter XXXIV. how it may be accounted for. 20 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. For the origin of the conceptions atom and molecule, and for the system atisation of the atomic weights, the student should consult-Alembic Club Reprints, No. 2, Foundations of the Atomic Theory; Alembic Club Reprints, No. 4, Foundations of the Molecular Theory; Alembic Club Reprints, No. 18, CANNIZZARO, Sketch of a Course of Chemical Philosophy. An interesting account of the work of Stas, and a general discussion of methods of atomic-weight determination, will be found in the "Stas Memorial Lecture" by J. W. MALLET (Journal of the Chemical Society, 1893, 63, p. 1). Some work by Stas on pure chemicals is reprinted in the Chemical News, 1895, 72. The student may also be referred to the following papers dealing with modern determinations of equivalents :- T. W. RICHARDS (Publications of the Carnegie Institution of Washington, 1910, 125), “Determination of Atomic Weights.' "" E. W. MORLEY (Smithsonian Contributions, 29, and Zeitschrift für physi- kalische Chemie, 1896, 20), "Combining Ratio of Oxygen and Hydrogen." P. A. GUYE (Journal de chimie physique, 1916, 14), "The Sources of Error in Determinations of Atomic Weights." R. J. STRUTT (Philosophical Magazine, 1901 [6th series], 1, p. 311), "Tendency of the Atomic Weights to approximate to Whole Numbers"; also E. FEILMANN (Proc. Chem. Soc., 1912, 28, p. 283). II THE ATOMIC THEORY AND ATOMIC WEIGHTS 21 Aluminium Antimony Argon Arsenic Barium Bismuth Boron Bromine Cadmium Cæsium Calcium Carbon Cerium Chlorine Chromium Cobalt Columbium Copper Dysprosium Erbium Europium Fluorine Gadolinium Gallium Germanium Glucinum Gold Helium Holmium Hydrogen Indium Iodine Iridium Iron Krypton Lanthanum . Lead ¤§‹Ã£ ¤µÂ˜88.8☎ 66 8 ¯ à Á à ¤˜ 8 8 ¯ à ŠŠ: H In I Ir Fe Kr La 323522 Pb Lithium Li Lu Lutecium Magnesium Mg Manganese Mn Mercury Hg ATOMIC WEIGHTS 27.1 120.2 39.9 75.0 137.4 208.0 10.9 79.92 112.4 132.8 40.07 12.00 140 2 35.46 52.0 58.97 93.2 63.56 162.5 1674 152 19.0 157.3 70.1 72.5 9.1 197.2 4.00 163.5 1.0076 114's 126.92 193.1 55.85 82.9 139.0 207.2 7·00 175 24.32 54.93 2004 Molybdenum Mo 96'0 Nd 1443 Neodymium Neon Ne Nickel Ni Niton Nitrogen Osmium Oxygen Palladium Radium Rhodium Rubidium COOZZZZ Ruthenium Samarium Scandium Selenium Silicon Silver Sodium Strontium Sulphur Tantalum Tellurium Terbium Thallium Thorium Thulium Tin Titanium Tungsten Uranium Vanadium Xenon Ytterbium Yttrium Zinc Zirconium 20.2 58.6s BREEDER AND Z 2224 Os 190·9 Pd Phosphorus P Platinum Pt 39.10 Potassium K Praseodymium Pr 140.9 Ra 226'4 Rh 102'9 Rb 85.45 Ru 1017 Sm 1504 Se 44.1 Se 79.2 14.01 > > £ £ × Ñ & 16.000 106.7 31.0 195.2 28.3 Ag 107.88 23.00 87.63 32.07 1812 Te 127.5 Tb 1592 204'0 Th 232.2 Tm 1685 Sn 118.7 Ti 48.1 W 1840 U 238.2 512 130.2 173'8 89.3 Zn 65.38 90'6 CHAPTER III THE SIMPLE GAS LAWS THE student is doubtless familiar with the fact that the laws regu- lating the physical condition of gases are of an extremely simple character and of universal applicability. Pressure and temperature affect the volume of all gases to nearly the same degree, no matter what the chemical or other physical properties of the gases may be, so that we can state for gases the following general laws :- 1. Boyle's Law. The volume of a given mass of any gas varies inversely as the pressure upon it if the temperature of the gas remains constant. 2. Gay-Lussac's Law.-The volume of a given mass of any gas varies directly as the absolute temperature of the gas if the pressure upon it remains constant. 3. The pressure of a given mass of any gas varies directly as the absolute temperature if the volume of the gas remains constant. Any one of these laws may be deduced from the other two, and as an example we may take the deduction of the third law from the laws of Boyle and Gay-Lussac. Let the original pressure, volume, and temperature of the gas be Po, vo, and To. Let the gas be heated at the constant pressure po to the temperature T₁. Then, according to Gay-Lussac's Law, 1' To = V T if V is the volume assumed at 7, 1' Now let the gas be compressed to the original volume at the constant temperature 7. According to Boyle's Law, the product. of the pressure and volume of a gas always remains the same if the temperature is constant. We have therefore at T₁ 1 PoV = P10 if P₁ represents the value of the pressure after compression to the volume v。. But from the first equation we have 22 CHAP. III 23 THE SIMPLE GAS LAWS T 1 Ꮴ To Substituting this value in the second equation, we obtain Po = P1 which is an expression of the third law for the variation of pressure with temperature at constant volume. whence པར To Ti These three separate laws may be put into the form of a single equation as follows. Let the pressure, volume, and temperature of a mass of gas all change, the original values being P。, vo, To, and the final values ₁, 1, T₁. Suppose first the volume to remain constant at Tj , and the temperature to assume its final value T₁. The gas will then have a pressure P, given by the equation Po To P Ti v. 1 Let the gas now expand from 2 to v₁, at constant temperature T₁; the new pressure ₁ will then be given by the equation 1 Pv₂ = P1º1 P = P11 20 Po% To P11 Ti 1 Substituting this value of P in the pressure-temperature equation, we have finally i.e. the product of the pressure and volume of a gas is proportional to the absolute temperature. The final result may also be made to assume the form Povo _P11; To Ꭲ, that is, the expression pu/T for a given quantity of gas always remains constant. Since if pressure and temperature are constant the quantity of a gas is proportional to the volume taken, the actual value of the expression po/T is proportional to the quantity or volume considered. Now, since according to Avogadro's principle the gram-molecular weights of all gases occupy the same volume under the same condi- tions of temperature and pressure, it follows that for these quantities the expression will have a constant value, no matter what the nature of the gas is, or the conditions under which the volume is measured. We have therefore in general for all gases 24 INTRODUCTION TO PHYSICAL CHEMISTRY CH. III PV T Ꭱ == where R is a constant, and the volume occupied by the gram mole- cule of the gas. At 0° and 76 cm. the gram-molecular volume or molar volume is 22.4 litres, or, for a perfect gas, 22,412 cc. (cp. p. 14). The pressure p in grams per square centimetre is 1033 (p. 3), and T is 273. To calculate the value of R we have therefore R, 1033 × 22,412 273 Unit of Energy. Joule (Volt-coulomb). Calorie at 15° Volt-faraday Litre-atmosphere = 84,760. The product p is of the dimensions of energy and is expressed in the above calculation in gravitation units or gram-centimetres. For other energy units the gas-constant R assumes other values, the most important of which are as follows:- • Value of Gas-constant R. 8.315 1.985 0.8613 × 10-4 0·08207 The thermal equation with the calorie as unit of energy is fre- quently employed in the approximate form pV = 2T in measuring the work done upon or done by a gas when it changes its volume under external pressure (cp. Chap. IV.). Thus if a gram molecule of a gas is generated by chemical action, say from the interaction of zinc and an acid, an amount of heat will be absorbed in performing the external work of expansion equal to 27 calories, i.e. to 546 calories if the action takes place at 0° C. Other examples of the use of this equation will be found in the chapters dealing with specific heats, the gas laws for solutions, and more especially in the chapter on thermodynamics. It should be borne in mind that these simple gas laws, and the deductions from them, apply in strictness only to ideal or "perfect" gases, no actually existing gas obeying them exactly. For most purposes of calculation, however, they may be applied to ordinary gases without any very serious error being committed, if care be taken when applying them to vapours near the point of condensation or to gases under very great pressures (cp. Chapters VIII. and IX.). ▼ CHAPTER IV SPECIFIC HEATS As early as 1819, Dulong and Petit noticed that the products of the atomic weights of the elements into their specific heats were approxi- mately constant. Of course at that time the atomic weights of the elements had been determined with very little certainty, but taking - the atomic weights even as they stood there was an unmistakable appearance of regularity. In cases where there was doubt as to which equivalent of an element ought to receive the preference, the rule given by Dulong and Petit was helpful. It was evidently more rational to choose that equivalent which conformed to the regularity displayed by the other elements than an equivalent which would make the element under discussion appear exceptional. This means of selecting a standard number from amongst the different possible equivalents of an element to represent its atomic weight proved of great use during the period when Avogadro's principle was not recognised as a safe guide in fixing molecular and atomic quantities; and even after the recognition of this principle, the Law of Dulong and Petit rendered much service in precisely those cases where the application of Avogadro's Law failed from want of data. It must be borne in mind that the Law of Dulong and Petit is a law on precisely the same footing as Avogadro's Law. According to Avogadro's principle, we take as molecular weights those quantities of gases which occupy a certain volume under given conditions, and from these molecular weights we deduce by a consistent process the atomic weights of the elements. According to Dulong and Petit's rule, we take as the atomic weights of the solid elements those quantities - which have a certain capacity for heat. As it happens, the atomic weights deduced by the one method coincide with the atomic weights deduced by the other. Since, then, the two rules are not contradictory but lead to the same result, considerable confidence may be placed in the system of atomic weights arrived at by their aid, and the periodic regularities referred to in Chapter V. seem further to justify this trust. • 25 26 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. The rule of Dulong and Petit may be stated in the form that the capacity of atoms for heat is approximately the same for all solid elements. The atomic heat, as it is called, is obtained by multiplying the specific heat of the element into its atomic weight, the product being nearly equal to 6·4 when we measure specific heats and atomic weights in the customary units. The following table gives the atomic weights and the specific heats of the solid elements, and also the corresponding products, the atomic heats:- Lithium Glucinum Boron (amorphous) Carbon (diamond) Sodium Magnesium Aluminium Element. Calcium Scandium Chromium Manganese Iron Nickel Cobalt. Copper. Zinc • Silicon (crystalline) · Phosphorus (yellow) Sulphur (rhombic) Potassium • 1 Rhodium Palladium • Silver . Cadmium Indium Tin • Antimony Iodine. Tellurium Lanthanum • • Molybdenum Ruthenium • •· Gallium Arsenic (crystalline) Selenium (crystalline) Bromine (solid) Strontium Zirconium • · • ▸ • • Cerium Tungsten Osmium Iridium Platinum Gold Mercury (solid) • 4 • · • • • • • • 4 A Atomic Weight. 7 9 11 12 23 24 27 28 TABLE I 31 32 39 40 44 52 55 56 59 59 63 65 70 75 79 80 87 91 96 102 103 106 108 112 114 119 120 127 127 139 140 184 191 193 195 197 200 S Specific Heat. •94 •41 25 •14 *29 •245 •20 •16 *19 •18 *166 •170 •153 •121 •122 •112 •108 •107 *093 ·093 *079 ⚫082 '080 *084 ·074 ⚫066 *072 •061 *058 *059 *056 •054 *057 ·054 *052 ·054 *047 *045 *045 ·033 ·031 ·0324 ⚫032 ⚫032 *032 AXS Atomic Heat. 6.6 3.7 2.8 1.7 6.7 5.9 5.4 4.5 5.9 5.7 6.5 6.8 6.7 6.3 6.7 6.3 6.4 6.3 5.9 6.1 5.5 6.2 6.3 6.7 6.5 6.0 6.9 6.2 6.0 6.3 6.0 6.0 6.5 6·5 6.2 6.8 6.0 6.2 6.3 6.1 6.2 6.3 6.3 6.3 6.4 AXS at 50° abs 1.35 0.12 0.24 0.03 3.5 1.7 1.1 0.77 2.4 1.7 5.0 2.8 0.7 1.3 1.0 1.2 1.2 1.6 2.5 1 1.9 2.9 3.6 4.8 2.4 1.4 1.1 1.4 2.0 2.6 3.5 { 3.4 2.9 4.6 3.7 4.6 4.6 1.7 1.5 1.9 2.6 3.2 4.6 IV 27 SPECIFIC HEATS Thallium Lead Bismuth Thorium Uranium ⚫9 •8 .7 .6 .5 4 It will be observed that though the values of the atomic heats fluctuate to some extent, they are generally not far from the mean 6·4. The curve (Fig. 1) shows the regularity in graphic form. Atomic weights are plotted on the horizontal axis and specific heats on the .3 .2 Li Element. GI Specific Heats O • Na Mg Al P Si' S K Ca ⚫Sc Cr S Specific Heat. AXS Atomic Heat. ITTI ⚫0335 ⚫031 ·030 *0276 ⚫0276 6.8 6.4 6.3 6.4 6.6 Atomic Weights 20 40 Fe Cu A Atomic Weight. 204 207 209 232 239 60 As Sr Mo 80 Ag Sb 100 Cc W 140 I 1 180 160 Hg Pb 200 AXS at 50° abs. 4.8 5.0 4.5 4.6 3.3 120 FIG. 1. vertical axis. If the product were exactly equal to 64, the graph would be the smooth curve shown in the figure. The actual values are indicated near the symbols, and it will be seen they seldom fall far away from the curve. The only elements which show exceptionally small values are elements of low atomic weight. 220 Th Ur 240 The values of the specific heat used in the table and curve were all obtained at temperatures above the atmospheric temperature. In the case of some elements, however, the specific heat varies greatly with the temperature, and the question arises: At what temperature should the comparison of the specific heats be made? Thus for carbon we have the values given in Table II. 28 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. TABLE II DIAMOND Temperature. Specific Heat. Atomic Heat. 0.0635 0.76 - 50 +10 0.1128 1.35 85 0.1765 2.12 206 0.2733 3.28 607 0.4408 5.3 806 0.4489 985 0.4589 5.4 5.5 GRAPHITE Temperature. Specific Heat. Atomic Heat. - 50 +10 61 202 642 822 978 0.1138 0.1604 0.1990 0.2966 0.4454 0.4539 0.4670 1.37 1.93 2.39 3.56 5.35 5.45 5.5 We see from the table that the two crystalline varieties of carbon differ very much in their specific heat at low temperatures, but have identical values at high temperatures. The specific heat of wood char- coal is considerably smaller than that of the crystalline modifications of carbon. It may be noted in general that the denser variety of each element has the smaller specific heat. At low temperatures there is no approximation to the value of 64 for the atomic heat, though at high temperatures this value is approached. Dewar has shown from measurements of the mean specific heats of the elements at 50° abs. or - 223° C. that the atomic heat at this low temperature is not constant, but rises and falls periodically with. increase of atomic weight and is generally much lower than the value of 6.4 which prevails at the ordinary temperature. His values for the atomic heat at this temperature are given in the last column of Table I. Theoretical considerations lead to the conclusion that at the absolute zero the specific heat, and therefore the atomic heat, of solid elements is nil. Experiment seems to confirm this. Thus the atomic heat of copper 16 at 50° abs. falls to 0.05 at 15° abs., whilst the atomic heat of diamond is practically nil at all temperatures below 40° abs. An approximate rule for compounds, of like nature to Dulong and Petit's principle for elements, was stated by Neumann in 1831. On comparing compounds of similar chemical character, he found that the products of their specific heats and molecular weights were constant. This may be stated in the form that the molecular heats of similarly constituted compounds in the solid state are equal. This rule is a special case of a regularity of wider scope. Kopp showed that very frequently the molecular heat of a solid compound was equal to the sum of the atomic heats of its component atoms. The atoms in com- bining together to form molecules thus retain their heat capacity practically unchanged. By making use of this general principle, the specific heats of elements in the solid form have been calculated even when the elements themselves were not known to exist in the solid state. Thus the specific heat of solid oxygen could not be determined directly, but by taking the molecular heats of a great number of solid oxygen compounds and subtracting from them the heat capacities of the atoms of the other elements present, the remainder (which was IV SPECIFIC HEATS 29 ľ supposed to be the heat capacity of the oxygen atoms in the compound) always gave an atomic heat for oxygen in round numbers equal to 4. This value was therefore taken as the atomic heat of oxygen in the solid state, whence on dividing by 16 we get 0.25 as the specific heat of solid oxygen. The foregoing rules apply only to solids. The specific heat of liquids usually varies very much with the temperature, but definite relations have been discovered in certain classes of compounds. Thus Schiff found that all the ethereal salts of the fatty acids examined by him had the same specific heat, no matter what their molecular weight was, provided that the specific heats were all measured at the same temperature. In other groups of liquid substances less. simple relations exist, but there is generally some approach to regularity. The specific heat of gases presents a different aspect from the specific heats of solids or of liquids. If we compress a gas, we warm it, although we supply no heat to it; if we allow a gas to expand under atmospheric (or other) pressure, it cools, although no heat has been abstracted from it. Now specific heat may be defined as the number of heat units which must be supplied to 1 gram of a substance in order that its temperature may be raised 1 degree centigrade at the ordinary temperature. But we can raise the temperature of 1 gram of a gas through 1 degree centigrade, without supplying any heat at all, by simply compressing it. We should therefore have to say that its specific heat under these conditions is zero, as its temperature has been raised without any heat being supplied. On the other hand, if we allow the gas to expand sufficiently against pressure while heat is being supplied to it, its temperature may be kept absolutely constant, the heat supplied merely counterbalancing the cooling due to the expansion. The specific heat of the gas in this case is infinite, for a finite amount of heat has been supplied and has only produced an infinitely small rise of temperature. The specific heat of a gas there- fore depends on its volume relations. It may have any positive specific heat between zero and infinity, or even any negative specific heat, according to the manner in which its volume changes during the process of heating. We must always specify, therefore, under what conditions we measure the specific heat of a gas. There are two con- ditions which are usually regarded as standard conditions. In the one case the volume of the gas is not allowed to change during the change of temperature. This gives the specific heat at constant volume. In the other case the volume is allowed to change during the change of temperature and the gas is permitted to expand or contract under a constant pressure. This gives the specific heat at constant pressure, which for practical measurements is the most convenient, the gas being allowed to change its volume at the atmospheric pressure. 30 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. The relation between these two specific heats is easily determined. Suppose we raise the temperature of 1 g. of gas from 0° to 1°, maintaining the volume constant; the quantity of heat required is the specific heat of the gas at constant volume. Let the temperature of the same gas be raised now from 0° to 1°, the gas being allowed to expand against the atmospheric pressure according to Gay-Lussac's Law; the quantity of heat required is the specific heat at constant pressure. The quantity of heat required in the second case is greater than in the first, owing to the fact that the gas on expanding against pressure cools by doing work. Heat must be supplied to make up for this cooling, and to find the amount of this heat we measure the amount of work the gas does in expanding against the atmospheric pressure. As we have already seen in Chapter III., the work done is the product of the pressure into the change of volume, and if a gram molecule of the gas is considered, this work, in terms of the gram calorie, is approximately 2 T for the whole volume occupied by the gas. Now at 0°, T= 273, and a gas when heated from 0° to 1° 1 expands of its volume. 273 In terms of the gram calorie, then, the 1 273 work performed on the expansion is heat the gram molecule of gas from 0° to 1° requires 2 cal. more at constant pressure than at constant volume, the change of volume in itself having no direct effect on the specific heat of the gas (cp. p. 36). It is usual to call the amount of heat required to raise the temperature of the gram molecule one degree, the molecular heat, this being the product of the molecular weight into the specific heat. The molecular heat of a gas at constant pressure is thus 2 cal. greater than the molecular heat at constant volume. × 2 × 273 = 2 cal. To It is a matter of great difficulty to determine the specific heat of a gas at constant volume, owing to a comparatively small amount of the heat supplied going to heat the gas, the greater part being absorbed by the vessel which contains the gas. It is much easier to determine the specific heat at constant pressure, for then we can pass large quantities of the gas through a worm contained in a calorimeter, and The molecular heats measure the amount of heat it gives or takes. at constant volume may then be obtained by subtracting 2 cal. (the value of the work of expansion per gram molecule) from the molecular heat at constant pressure. The ratio of the specific heats at constant pressure and constant volume can be ascertained with great accuracy by Kundt's method from a determination of the velocity of sound in the gas under consideration. From the kinetic theory of gases it follows that if the molecules of a gas are monatomic, so that all the heat they receive from without goes only to increase their rectilinear velocity, and not to perform IV SPECIFIC HEATS 31 any kind of change within the molecule, the molecular heat at constant pressure will be 5, and at constant volume 3, i.e. the ratio will be 1.67. If a molecule consists of more atoms than one, we should expect that a larger quantity of heat would be absorbed for the same rise of temperature, a portion of the heat being employed within the molecule in changing the relations of the constituent atoms to each other. The molecular heat at constant volume would therefore be greater than 3, say 3+ m. But the molecular heat at constant pressure is obtained by adding 2 to that at constant volume; 5 + m it is therefore 5+ m, and the ratio 5 theoretical ratio 3* Cp Cn is consequently less than the 3 + m Accordingly, when the molecule of a gas consists of a single atom, we expect the ratio of the specific heats to be 1.67; for here no heat can be expended in rearrangements of atoms within the molecule. Where the molecule of a gas consists of a number of atoms, we expect on the other hand that a portion of the heat will be expended in effecting new relations between the constituent atoms, and will not go entirely towards increasing the rectilineal velocity of the molecule as a whole, and therefore that the ratio of the specific heats will be less than 167. Until comparatively recently one elementary gas only was known for which the vapour- density results led to the conclusion that there was only one atom in the molecule. This gas was mercury vapour, and its density corresponded to the molecular weight 200, which is identical with the atomic weight. Now Kundt and Warburg, from determination of the speed of sound in mercury vapour, found that the ratio of its specific heats is 1·67, a figure in accordance with the theoretical value deduced for monatomic gases. The inert gases, argon, helium, etc., resemble mercury in this respect. The ratio of their specific heats is approximately 1.67, and we may therefore conclude that they are elementary substances having only one atom in the molecule. If they were compound substances they would necessarily contain more than one atom in the molecule, and one would expect the ratio to have a lower value. Gases with two atoms in the molecule should on certain theoretical grounds have a ratio of specific heats not exceeding 14; those with more than two a ratio not exceeding 1.33. A reference to the following table affords justification of this conclusion : [TABLE 32 INTRODUCTION TO PHYSICAL CHEMISTRY CH. IV RATIO OF SPECIFIC HEATS Cp/Co Diatomic Gases. Monatomic Gases. 1.65 1.65 1.64 Helium, He Argon, Ar Neon, Ne Krypton, Kr Xenon, Xe 1.67 Mercury, Hg. 1.67 1.69 • • • 1.41 1.40 1.40 Hydrogen, H₂ Nitrogen, N2 Oxygen, O Chlorine, Cl₂ Nitric oxide, NO 1.40 Carbon monoxide, CO 1·40 1.35 • • • 1.32 Polyatomic Gases. Nitrous oxide, N₂O Carbon dioxide, CO2 Sulphur dioxide, SO, 1.31 1.28 Hydrogen sulphide, HS 1.34 Ammonia, NH, Methane, CH4 1.32 1.31 • At low temperatures the ratio of the specific heats of the per- manent gases assumes values greater than those at the ordinary tem- perature. Thus at 180° the ratios in the case of nitrogen and hydrogen are 1·47 and 1.60 respectively, instead of the values 1.40 and 1·41 given in the above table. It should be noted by the student that the specific heat at constant volume of a "perfect" gas is independent of the volume which the gas occupies. No known gas is in this sense "perfect,"-indeed the deviation from perfection of existing gases has important practical applications, e.g. in machines for producing liquid air (Chap. VIII.),— but the ideal conception is of value as simplifying many thermo- dynamical calculations. J. DEWAR, Proc. Roy. Soc., 1914, A 89, p. 158: Atomic heats at 50° abs. CHAPTER V THE PERIODIC LAW WE have learned in what has preceded that there are about eighty distinct chemical substances which have resisted all our attempts to decompose them into anything simpler. The question now arises : Are our elements in truth undecomposable, or are they merely com- pounds which we are unable to decompose? It is inexpedient at the present stage to attempt a definite answer to this question, and it may be left open without affecting our views on practical chemical problems at all. We may be tolerably certain, however, that if our well-known elements are really compounds, they are all compounds of the same type, that type being different from that of compounds of the elements among themselves. The reason for this statement is not far to seek. We have seen that the product of the atomic weight and specific heat of solid elements is nearly constant, and equal to 64 if the ordinary units for atomic weight and specific heat are adopted. Here, then, we have a regularity amongst the elements-a regularity which is not shared by the compounds (cp. Neumann's Law, p. 28). It is evident, then, from this alone, that whatever the ultimate nature of our elements may be, they form a class of substances apart from all others. The same conclusion is reached when we tabulate properties of the elements other than the specific heat. Let us take, for example, the specific gravity of the solid elements, and as before tabulate the values on the vertical axis, atomic weights being tabulated on the horizontal axis. The points thus obtained do not lie, as with the specific heat, approximately on a continuous curve, but, as the lines in Fig. 2 indicate, on a series of somewhat irregular curves, which resemble each other closely in general appearance. If the whole series is looked upon as a single curve, that curve is what is termed a periodic curve, i.e. one which after a certain interval repeats itself. Each repetition is called a period, and the periods have been marked in the figure by Roman numerals. The elements alone fall naturally into their places on this curve; compounds, if we take their formula weight as representing the atomic weight, fall far away from it, so 33 D 34 CH. V INTRODUCTION TO PHYSICAL CHEMISTRY that we are again justified in assuming that our elements, if they are composite, are not compounds in the ordinary sense. The periodic character of the curve shows the existence of many relations which do not appear in the curve of specific heats. There are evidently two sorts of periods in the figure, the first two being very much smaller than the others. These are called the short periods; the others are long periods. We see at once from the curve that elements which occupy similar positions in the separate periods. are similar in their chemical character. Thus the first element in each complete period, long or short, is an alkali metal, the second is a metal of the alkaline earths, and the last element a halogen. It will be noticed, too, that similar elements very often in this diagram lie on or near straight lines. Thus if we join the points for lithium and cæsium, the first and the last alkali metals, we find that the straight line passes close by the points for sodium, potassium, and rubidium, the three intermediate alkali metals. Similarly, the points. for the metals of the alkaline earths, together with those of the closely related metals glucinum and magnesium, lie very nearly in one straight line. The halogens, chlorine, bromine, and iodine, may also be connected by a straight line, and so too may arsenic, anti- mony, and bismuth; zinc, cadmium, and mercury; sulphur, selenium, tellurium. The platinum metals (osmium, iridium, platinum; ruthen- ium, rhodium, palladium) lie at the summits of the last complete long periods, the related metals iron, nickel, and cobalt occupying the corresponding position in the first long period. These relations have been indicated in the diagram by means of dotted lines. As in the case of the specific heat, it is here again somewhat difficult to select the values of the specific gravity which are to be used for comparative purposes in such a table as the above. The specific gravity varies with the temperature and with the chemical and physical states of the substances considered. For example, carbon in the form of diamond has the specific gravity 3.3, whilst in the form of graphite it has only the value 2-15. Similar differences are found with the various modifications of sulphur and phosphorus. The metals, too, according to their mode of preparation and physical treatment, have often very different specific gravities-that of cobalt, for instance, varying between 8.2 and 9.5. In the diagram the highest value has been entered in such cases as probably being the most appropriate for comparison. There is a function of the specific gravity which is very frequently tabulated for the purpose of exhibiting the periodic character of the properties of the elements, namely the atomic volume. The atomic volume is the product of the atomic weight into the specific volume, or, what is the same thing, the quotient of the atomic weight by the specific gravity. The curve, first drawn by Lothar Meyer, is shown in Fig. 3 (p. 37). The periodic character of the curve is again well marked, 28 24- Specific Gravities 20 16 12 8 4 Al MgSi B GU S !!! I. Na I-CI Ca 20 40 Mn Cr V Co Ni Cú 60 Zn- Ga As Ge -III- Se Br 80 Mo Nb Zr Sr --Rb 100 Rh Pd Ag In IV Cd Sn Sb 120 Te FIG. 2. Cs La Ba Atomic Weights 140 Sa 160 Ta V W 180 Os Pt Au Hg TI Pb Bi 200 220 Th U VI 240 35 36 INTRODUCTION TO PHYSICAL CHEMISTRY CH. V only now the shape of the periods is different, the maxima corre- sponding to the minima of the specific-gravity curve, and vice versa. The difference between the short and long periods is once more evident, and similar elements once more occupy corresponding positions on the curve. In general, we may say that elements on steep portions of the curve have very pronounced chemical characters: as we proceed from left to right, the elements on descending portions of the curve are base-forming, and those on ascending portions acid-forming. Elements at or before the minima cannot be said to be decidedly acid- forming or decidedly base-forming; in different stages of oxidation they may be either. Most well-defined properties of the elements are periodic, e.g. melting point and magnetic power. When the numerical values are tabulated against the atomic weights, the curve obtained is broken up into periods, the shape of the period varying according to the property tabulated. Even the curve for specific heat is periodic at low temperatures (cp. p. 28). Not only are the properties of the elements themselves periodic, but the properties of their corresponding com- pounds are also frequently periodic. Thus if we tabulate the heat of formation of the chlorides of the elements against the atomic weights of the elements which combine with the chlorine, a periodic curve. results. The same periodicity is observed in the molecular volumes of the oxides, and in the melting and boiling points of various series. of corresponding compounds. We may tabulate the elements in another way without reference to any particular property such as the specific gravity, but in a manner which brings out the general periodic character of the properties of the elements with reference to the atomic weights. Newlands observed in 1864 that if the elements were arranged in the order of their atomic weights, similar elements occurred in the series at approximately equal intervals. He stated this regularity in the form of a "Law of Octaves," according to which every eighth element in the series belonged to a natural group, all the members of which resembled each other more than they did the other elements. Thus the alkali metals fell into one of these groups, the metals of the alkaline earths into another, etc. This rule, however, is not very accurate, and had to be relinquished when better data for the atomic weights accumulated. It is to Mendeléeff and to Lothar Meyer that we owe the periodic arrangement of the elements in its present form. In Table I., p. 38, the elements are ordered according to their atomic weights, beginning at the top of the left-hand column, pro- ceeding downwards to the bottom, then passing to the top of the next. column on the right, and so on. The vertical columns represent periods, which are denoted by the same Roman numerals as in Figs. 2 and 3. A complete short period contains eight elements-a complete long period, eighteen, which Atomic Volumes 60 50 40 30 20 10 Gl I Li B C Na Mg 20 CI II AI S P Si K Ca 40 III Fe Mn Co Ni Cr 60 As Zn Cu Br Se Ga 80 Rb Sr Zr Nb Mo IV Pd Ag Ru Rh 100 Te Sn Sb In Cd Atomic 1 FIG. 3. 1 120 ∞ Ba Weights La Ce 140 Sa 160 Ꮩ W 180 Pt Au Os Tr ΤΙ Hg 200 Bi Pb 220 Th VI 240 y ! 37 38 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. i. 1. (H) 2. 3. 4. 5. 6. 7. 8. 1. 2. 3. 4. 5. 6. 7. 0. He i. I. Li Gl B C N 0 F Ne I. II. Na Mg Al Si Ar II. TABLE I III. K Ca Sc Ti V Cr Mn Fe Co 8 2 3 3 Cu Ru Rh Ni Pd Zn Ga Ge P As S Se Cl Br IV. Rb Sr Kr III. Y Zr Cb Mo ... Ag Cd In Sn Sb Te I Xe IV. V. Cs Ba Rare metals Ta W Os Ir Pt Au Hg TI Pb Bi Po Nt V. VI. Ra Th ... U Even Series. Transition Elements. Odd Series. are made up of two series connected by a transition group of three elements. Within any one period, whether short or long, there is no sudden change in general chemical properties as we pass from one element to the next in order. Descending any series, we proceed from base-forming elements, through elements whose oxides are neither strongly acidic nor strongly basic, to acid-forming elements. In the short periods, which consist of only one series, we start from elements which form powerful bases, and end with elements which form powerful acids. In the long periods the uppermost or even series start with strong base-forming elements, and end with elements which form both acidic and basic oxides. The lower or odd¹ series, 1 The terms "odd" and "even" refer to the numbers of the series in the original arrangement of Mendeléeff. V 39 THE PERIODIC LAW on the other hand, begin with only moderately strong base-forming elements, pass to elements which are very markedly acid-forming, and terminate with the inert elements of the argon group. The elements connecting the odd and even series in the long periods are intermediate in their behaviour between the elements they connect ; those of the first row (iron, ruthenium, osmium) forming acids as well as bases, the others forming only bases. When we pass from one period to the next, there is a sudden change in the properties of consecutive elements. The second-last element of Period I., F = 19, is in complete contrast in its chemical nature to the first element of Period II., Na = 23. Between these elements, the one very powerfully negative (or acid-forming), the other powerfully positive (or base-forming), we have the absolutely neutral or inert element neon, Ne 20. Similar abrupt transitions occur in the passage from Period II. to Period III., and from Period III. to Period IV. These sudden changes in the chemical properties correspond in Figs. 2 and 3 to sudden changes in the direction of the curves. If we consider the elements in a horizontal row, denoted by an Arabic numeral in the table, we find that they are on the whole such as would naturally fall together in a classification of the elements. according to their general chemical characters. Thus in the first row we have lithium, sodium, potassium, rubidium, and cæsium, metals of the alkalies; in the second row, calcium, strontium, barium, and radium, metals of the alkaline earths; and so on. It will be observed that the numbers of the rows in the table are repeated, running in order from 1 to 8, and then beginning again at 1. There are therefore two rows, one in the even and one in the odd series, which are represented by the same number. Between the members of these two rows there is a general similarity of properties, although it is not so great as that between members of the same row. Thus the compounds of vanadium, columbium, etc., in the even series, bear considerable resemblance to those of phosphorus, arsenic, bismuth, etc., in the odd series. The most striking property which exemplifies this is the combining capacity or valency of the elements. All the elements of two rows indicated by the same number, form, in general, compounds with precisely similar formulæ. Take, for example, the 'highest" oxides, i.e. the oxides containing most oxygen, of the rows marked 5, which form the two subdivisions of one natural family. We find their formula to be :- (C Group 5 Even series Odd series V205 V.Os Ch₂O Ta05 NO PO As2O5 Sb2O5 Bi₂05. 5 5 Two atoms of all the elements of Group 5, then, combine with five atoms of oxygen. The same thing holds good for other groups, each 40 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. group having its own combining capacity, which increases regularly from Group 1 to Group 8, as may be seen in Table II., where instead of the elements of Table I. the oxides of these elements have been tabulated. In the case of elements forming more than one oxide, the highest base-forming or acid-forming oxide is usually taken. The oxides enclosed in curved brackets are not known, but are the anhydrides of well-characterised series of salts. GROUP 1. 2. Qic to con 3. 4. 5. 6. 7. 8. ció Lá con 2. 3. 5. 6. 7. I. II. Ligo Na₂O Mg202 Gl₂02 O B203 Al2O3 C204 Si204 N2O5 ... ... I. TABLE II III. K₂O Ca₂O2 Sc₂O3 Ti₂O4 V205 Cr₂06 Mn207 (Fe2O6) Co₂O Ni₂03 Go Zn202 Ga₂03 Ge₂O4 P205 As₂06 S206 (Se₂O6) Cl₂07 II. III. IV. Rb₂O Sr202 Y203 Zr₂O Cb2O5 Mo₂06 .. Ru₂08 Rh₂O Pd₂O Ag₂0 Cd202 In203 Sn204 Sb2O5 Te₂O 1207 IV. 6 V. Cs₂O Ba202 [ M 2 O 3 ] * [M₂O] Ta05 W206 OS 208 Ir₂04 Pt204 Au₂O Hg₂O2 T1203 Pb204 BigOp V. * M stands here for an atom of a metal of the rare earths. VI. Ra202 Th₂04 U₂06 VI. In order that the amounts of oxygen in the various oxides may be readily compared, the formulæ have been written throughout so as to contain two atoms of the element combined with oxygen, and are consequently not in every case the usual formula. The number of the group, with few exceptions, corresponds to the number of oxygen atoms united to two atoms of the elements of the group. It must be conceded, however, that in order to get this regularity the oxides tabulated have been selected somewhat arbitrarily. Thus for copper we have selected the lower oxide CuO instead of the equally characteristic higher oxide CuO. The selection in the case of sulphur, too, is quite arbitrary. Sulphuric anhydride, SO, is not the highest acid-forming oxide, as we have the oxide S,Ŏ, corre- sponding to the well-characterised persulphates. Thus, while there is undoubted periodic regularity in the formulæ of the oxides of the elements, the regularity is not so definite as at first sight appears from a study of a table such as that given above. The same sort of regularity appears when we tabulate the highest chlorides, bromides, etc., instead of the oxides; only in these cases V 41 THE PERIODIC LAW there are still more exceptions. The highest chlorides of the elements of the even series are given in the following table as examples: GROUP 1. 2. 3. 4. 5. 6. 7. HNSTON 5. 7. Group. 1. Si có vị có 1 2. I. II. LiCl NaCl 3. TABLE III 4. 5. 6. 7. III. KCI CaCl2 ScCl3 TiCl VCI CrCl3 MnCl₁? We here perceive that there is a general correspondence between the group number and the number of chlorine atoms with which one atom of the elements can combine, but that there is a tendency of the elements in the higher groups to unite with fewer atoms than are given by the group number. When we study the compounds of the elements with hydrogen or the alcohol radicals, we find that the combining capacity of the different groups is somewhat different from before. In the first place, it is chiefly elements of the odd series that unite with hydrogen or the alcohol radicals to form definite compounds. As we proceed downwards in the periods, we find that the combining capacity does not now steadily increase, but reaches a maximum at Group 4, there- after to decrease regularly. Thus in the first period we have :- Group Methyl Compound. Li(CH3) G1(CH3)2 B(CH3)3 C(CH3) N(CH3)3 O(CH3)2 F(CH3) IV. RbCl SrCl₂ YC13 ZrCl CbCl5 MoCl Sulphuric acid Perchloric acid V. CsCl BaCl2 [MC1₂] Compound. Sodium hydroxide Magnesium hydroxide Aluminium hydroxide Silicic acid Orthophosphoric acid [MC] TaCl5 WCI6 ... The power of combining with hydroxyl groups seems to be deter- mined by the number of hydrogen atoms with which the element can unite. This may be seen from the following table, which gives for the elements of the second period those compounds which contain most hydroxyl groups :- VI. LiH ... RaCl2 Th Cl UCI, Hydrogen Compound. BHS CHA NH3 OH₂ FH Formula. Na(OH) Mg(OH)2 AI(OH)3 Si(OH) PO(OH)3 SO2(OH)2 CIO (OH) 42 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. 5 Corresponding to the oxide P₂O, there should be the hydroxide P(OH), if for all the oxygen the equivalent amount of hydroxyl were introduced. But this compound does not exist, the acid PO(OH)3, orthophosphoric acid, being the compound with the greatest number of hydroxyl groups in the molecule, apparently in connection with the fact that there is a hydrogen compound PH3, but no hydrogen compound PH, It has been pointed out that not only are elements in the same horizontal row similar, but that there is no sudden change in the properties of consecutive elements in the vertical series. We should therefore expect to find a general resemblance amongst the elements gathered together in one part of the table, and this resemblance does in fact appear. The upper rectangle formed by the dotted lines in Table I. contains the metals of the rare earths, which occur in groups in a few minerals found only in certain localities, and which, from their similarity of chemical properties, are extremely difficult to separate from one another. The lower rectangle contains all the "metallurgical " elements, i.e. the heavy metals which are prepared on the large scale. from their ores, although all the elements enclosed within this rectangle. are not of technical importance. Bordering on the rectangle are the light metals magnesium and aluminium, and the half-metals arsenic, antimony, and bismuth, all of which are commercially important. When we look at the position of the elements composing these groups on the curve of specific gravities, we find that the metals of the rare earths occupy the middle of the ascending portions of the curve in the long periods, and that the "metallurgical" metals lie at or close after the maxima. On the curve of atomic volumes, the metallurgical metals are found at, or immediately after, the minima. The distinction between metals and non-metals is not a very sharp one, for the properties of the one class gradually merge into those of the other. There is, however, a classification which is generally adopted in practice, and this classification is in conformity with the periodic table. The elements to the left of the dotted line at the lower left- hand corner of Table I. are the non-metals as usually understood. With the exception of hydrogen all the other elements tabulated are metals. As to the blanks in Table I., we may reasonably expect that some of them at least will be filled up by elements hitherto undiscovered, and that the new elements will be of the same kind as those we already know. When the table was first constructed, the number of blanks in it was greater than now, and Mendeléeff was bold enough to predict the existence of elements to fill the gaps. His predictions have been more than once fulfilled. The three well-defined elements, gallium, germanium, and scandium, have all been discovered since his formulation of the Periodic Law, and have fallen naturally into their places in the third period. Mendeléeff not only predicted the existence. of these elements, but also their chief chemical and physical properties. V THE PERIODIC LAW 43 from a consideration of the properties of their nearest neighbours in the table. By consulting the specific-gravity curve (p. 35), it is easy to see, for instance, that an element of atomic weight 70 will probably have a specific gravity intermediate between the specific gravities of zinc and arsenic, the nearest well-known elements on either side. The value will therefore lie between 57 and 7.2, say about 6.4. The actual specific gravity of gallium, at wt. 70, is 5.93. Similarly the melting point and general chemical properties might be roughly pre- dicted. It should be said that experiments on the newly discovered elements entirely justified Mendeléeff's prediction of their properties. The non-valent elements of the helium-argon group, unsuspected when the periodic table was first constructed, fall naturally into their places between the univalent negative elements on the one hand and the univalent positive elements on the other. It must be admitted that hydrogen finds no satisfactory place in the periodic system. If we regard its valency as decisive, then it must be placed either at the head of the univalent positive elements (the alkali metals), or of the univalent negative elements (the halogens). In chemical properties it is different from either group; although in acids it is easily replaced by the metals, and in organic compounds by the halogens. In physical properties it resembles fluorine and chlorine, rather than lithium or sodium. Since it is an electro-positive element itself, it is usually classed with the positive univalent metals, but its properties are in most respects such that it has no close analogue among the elements, and it is owing to this singularity that the difficulty in classifying it arises (cp. Chap. XXXV.). In recent years much attention has been devoted to the study of radioactive substances. Of well-defined radioactive elements there are four, radium, niton, thorium, and uranium, of which the first two are immensely more active than the others. It will be observed that these are all elements of high atomic weight, and fall into the last vertical column of the table on p. 38, and at the end of the preceding column. In a few instances the natural grouping of the elements is out of harmony with the strict sequence of atomic weight. Thus iodine, at wt. 127, should precede tellurium, at wt. 1276, instead of following it; and the same is true of argon, at wt. 39-9, and potassium at wt. 39. Such exceptions and the light thrown on the system of the ele- ments by the study of radioactive substances and by X-ray research will be dealt with in the sequel (Chapter XXXIV.). CHAPTER VI SOLUBILITY THE solutions we most frequently meet with in inorganic chemistry are solutions in which water is the solvent. The substances dissolved may be gaseous (e.g. ammonia, hydrochloric acid) or liquid (e.g. bromine), but in the vast majority of cases they are solid. When excess of a solid, such as common salt, is brought into contact with water at a given temperature, it dissolves until the solution reaches a certain concentration, after which the solution may be left for an indefinite time in contact with the solid without either undergoing any change, provided that the temperature is kept constant, and that evaporation is prevented. Such a solution is said to be saturated with respect to the salt at the given temperature, and the solubility of salt in water under these conditions is merely an expression. for the strength of the saturated solution. The concentration may be ex- pressed in various ways, e.g. 100 parts of water are said to dissolve at the given temperature so many parts of salt, or the saturated solution contains so much per cent of the solid, or so many grams of salt are contained in 100 cc. or in a litre of the solution. All these methods of expressing the strength of solutions are in common use, the last especially for volumetric work, and it is a matter of convenience which to choose in any particular case. When a solution is capable of dissolving more salt at the given temperature than it already contains, it is said to be unsaturated; when it is made by some device to contain more salt than it would be capable of taking up directly under the given. circumstances, it is said to be supersaturated. The one and only test to find whether a solution is saturated, unsaturated, or super- saturated, with respect to a given solid, is to bring it into contact with that solid. If it is saturated, it will remain unchanged; if unsaturated, it will dissolve more solid; if supersaturated, it will deposit the excess of dissolved substance till the concentration diminishes to that of the saturated solution. The solubility of all substances varies with the temperature. With most solids it increases as the temperature rises. Thus 100 g. 44 CHAP. VI SOLUBILITY 45 of water will dissolve 356 g. of sodium chloride at 0°, and 40 g. at 100°; 50 g. of sodium thiosulphate¹ at 0°, and 102 g. at 40°; 13 g. of potassium nitrate at 0°, and 247 g. at 100°. In some cases the solubility diminishes as the temperature rises. For ex- ample, calcium citrate is more soluble in cold water than in boiling water, and many other calcium salts of organic acids exhibit a similar behaviour. In the case of calcium acetate (Fig. 4) there is first a slight diminution of solubility with rise of temperature and then a slight increase. It will be seen from the curves of Fig. 4, which represent the 100 80 60 40 20 o! 20 O Na 2 S 203,5H 20 KNO3 Na Cl Ca Ac₂,2H20 40° FIG. 4. 60 KBr 80° 100° number of grams of anhydrous salt dissolved in 100 grams of water, that the variation of solubility of different salts with water is very different. The solubility of sodium chloride or of calcium acetate varies very little with the temperature, whereas the solubility of potassium nitrate or of sodium thiosulphate increases rapidly. The degree of curvature also varies greatly. The curves for sodium chloride and potassium bromide are almost straight lines. Those for the other substances of the diagram are perceptibly curved. Solubility curves are almost invariably, like curves of vapour pressure, convex to the temperature axis. Calculated as anhydrous salt. 46 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. The solubility curves for sodium sulphate are represented in Fig. 5. Here instead of one curve we have three, one for the anhydrous salt, one for the unstable heptahydrate, and one for the decahydrate, which is the common crystalline variety. Below 33° this hydrate is stable in presence of water, and its solubility may be determined in the ordinary way. Above this temperature it splits into the anhydrous salt and water, so that the solubility measured above 33° is the solubility of the anhydrous salt, this being the solid with which the solution is actually in contact. It will be observed from the diagram that the two curves run independently, and cross at the temperature of 33°. The dotted prolongation of the curve of the anhydrous form 60 40 20 0° Na₂ 804 7H40 20° a Ná₂SO43 10H 20 FIG. 5. Na 2 SO 4 40° 60° may be realised on account of the union of the anhydrous salt and water just below 33° being so slow as to permit the measurement of its solubility before its transformation into the decahydrate. If we take a point in the diagram lying between the dotted line and the lower curve, that marked a, for example, we find that it represents the concentration of a solution which is unsaturated with respect to the anhydrous salt, and supersaturated with respect to the hydrate Na₂SO4, 10H₂O. We can prove that this is so in reality, for if we bring the solution into contact with the anhydrous sulphate it will dissolve some of the salt; whilst if we bring it into contact with the hydrated salt it will deposit a fresh quantity of the hydrate. It is very necessary, then, to specify exactly the solid with respect to which a solution is said to be saturated or not. To speak of a solution being a saturated solution of sodium sulphate is ambiguous, VI SOLUBILITY 47 į for while the solution may be saturated with regard to one form of the solid salt, it may be unsaturated or supersaturated with regard to another. When a saturated solution of hydrated sodium sulphate, Na₂SO4, 10H₂O, is made in warm water so that none of the solid remains, and is then cooled to the ordinary temperature, care being taken that there is no separation of the salt, the solution may be kept unaltered in the supersaturated state for a great length of time. It frequently happens that the solution on standing deposits crystals of another hydrate, Na2SO4, 7H2O, which at low temperatures is more soluble and less stable than the decahydrate. The solution then becomes saturated with respect to this heptahydrate, but is still supersaturated with respect to the decahydrate, as may be seen by dropping in a crystal of the latter, when further crystallisation at once commences. The curve for this unstable heptahydrate is re- presented in the figure by a dotted line, which at all temperatures lies above that of the decahydrate. It is mostly in the case of salts which crystallise with water of crystallisation that we can easily form supersaturated solutions. Salts which separate only in the anhydrous state do not often form aqueous supersaturated solutions, the excess being usually deposited as the cooling progresses. Sodium chlorate, however, is a good example of an anhydrous salt which does form supersaturated solutions. The question of the transformation of different hydrates into each other will be discussed in Chap. XII. It has been frequently observed that the state of division of a solid has a considerable influence on its solubility when the fineness of the powder is such that a comparatively great diminution of surface results from the dissolution of a small quantity of solid. Thus, whilst the saturated solution of CaSO4, 2H,O at 25° is 0.0153 normal with ordinary particles, the solubility increases to 0.0182 normal with particles averaging 03 μ in diameter (p. 2). By reducing the size of the particles of barium sulphate from 1.8 μ to 0.1 μ, the solubility is almost doubled. Liquids sometimes mix with each other in all proportions, e.g. water and alcohol; sometimes not at all, e.g. water and mercury. In some instances there is partial miscibility. When we shake up water and ether together in about equal proportions, the water dissolves a little of the ether, and the ether a little of the water, the two saturated solutions then separating. The lower layer is a saturated aqueous solution of ether, the upper layer is a saturated ethereal solution of water. Suppose that with two partially miscible liquids, A and B (say aniline and water), the solubility of each liquid in the other increases with rise of temperature. On raising the temperature we should find that the composition of the two saturated layers, or phases, as they are called, would tend to approximate to the 48 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. same value. We may represent this diagrammatically, as in Fig. 6. Temperatures are plotted on the vertical axis, and points on the hori- zontal axis represent the percentage composition of the liquids. As the temperature rises, the solubility of B in A and of A in B increases, so that the corresponding points on the two curves approximate. At a certain temperature the two curves will meet. This means that at that particular temperature the solution of A in B and the solution of B in A have the same composition, i.e. are identical. At this temperature the liquids are miscible in all propor- tions. There is therefore no funda- 100 mental distinction distinction between wholly miscible and partially miscible liquids ; for liquids which only mix partially at one temperature may mix in any pro- portion at another. We have here, likewise, an illustration of the arbitrary manner in which we employ the terms "solvent" and "dis- solved substance." We have spoken of the solubility of A in the solvent B for one part of the curve, and the solubility of B in A for the other. At the temperature where the saturated solutions become identical it is obviously impossible to make any distinction between. solvent and dissolved substance; and in general, it may be said that, though for the sake of convenience we often thus discriminate between the two constituents of the solution, there is in the solution itself no actual distinction of this kind. If it suits our purpose we may call dissolved substance what is usually termed solvent, and vice versa, with- out fear of committing any theoretical error.¹ The solubility of some pairs of miscible liquids in each other increases with rise of temperature, e.g. aniline and water (Fig. 6); the mutual solubility of other pairs diminishes, e.g. dimethylamine and 165 Temperature 80 40 Per cent Aniline FIG. 6. (C 1 The following table gives the mutual solubility at 22° of water and some common organic solvents :- Chloroform Ligroin Carbon disulphide Ether Benzene Amyl alcohol Aniline Vols. Substance in 100 Vols. Water. 0.42 0.34 0.17 8.11 0.08 3.28 3.48 Vols. Water in 100 Vols. Substance. 0.15 0.33 0.96 2.93 0.22 2.21 5.22 It has been pointed out that, as far at least as water and organic liquids are concerned, partial miscibility involves the presence of an "associated" liquid (cp. Chapter XX.), non-associated" liquids being completely miscible. Rowers VI 49 SOLUBILITY 170 water. The diminution of solubility by rise of temperature is very easily seen in the case of water and paraldehyde. If these liquids. are shaken up together in a test-tube at the ordinary temperature, they separate on standing into two clear layers. If now the test- tube is plunged into warm water, the aqueous layer at once becomes turbid from separation of the excess of paraldehyde. Many lactones exhibit a peculiar behaviour with water. The aqueous solution prepared by shaking up the two substances at the ordinary temperature becomes turbid at about 40° owing to the lessened mutual solubility, but at 80° again becomes clear. Here there is first diminution of the solubility of the lactone in water by 210° 130° 90° 50° 20 40 60 Percentage of Nicotine Fio. 7. 80 100 rise of temperature to a minimum value, after which we have increas- ing solubility with further rise of temperature. The peculiarities of all these cases of partial miscibility are best studied from Fig. 7, which shows the mutual solubility of the liquids. nicotine and water. At temperatures below 61° and above 210° the liquids are completely miscible, at intermediate temperatures they are only partially miscible. The curve of solubilities is therefore a closed curve, the upper portion corresponding to the case of aniline and water, and the lower to dimethylamine and water. As the tempera- ture rises from 60° the solubility of each liquid in the other diminishes until it reaches in both cases a minimum at about 130°, after which further heating augments the solubility. Gases vary very much with respect to their solubility in liquids. E 50 CHAP. INTRODUCTION TO PHYSICAL CHEMISTRY For example, one volume of water at 0° and atmospheric pressure dissolves- 1050 505 80 Temp. 0° 4.4 1.8 0.25 0.06 0.04 0.02 0.02 10 20 25 30 40 50 vols. Ammonia. 0.28 0.13 0.07 "" "" "" " "" }} "" "1 Again, one volume of alcohol under the same conditions dissolves— 17.9 vols. Sulphuretted hydrogen. 4.3 3.6 "" "" "} ور "> Hydrochloric acid. Sulphur dioxide. Sulphuretted hydrogen. Carbon dioxide. Ethylene. Argon. Oxygen. Nitrogen. Hydrogen. "" Under ordinary conditions the quantity of a gas dissolved by water almost invariably decreases with rise of temperature. Helium, how- ever, is an exception, its absorption-coefficient in water diminishing from 0° to 25°, thereafter to increase as the temperature is raised. 1 The following table of absorption-coefficients of argon and helium shows the influence of temperature on the solubility of these gases in water- Carbon dioxide. Ethylene. Oxygen. Nitrogen. Hydrogen. Argon. 0.0578 0.0453 0-0379 0.0347 0.0326 0.0286 0.0257 Helium. 0.0150 0.0144 0.0139 0.0137 0.0138 0.0139 0.0140 When the gases are only moderately soluble in the solvent chosen, they obey a simple law with regard to pressure known as Henry's Law. One mode of stating this law is: A given quantity of the liquid solvent will always dissolve the same volume of a given gas, no matter what the pressure may be, so long as the temperature is constant. Another equivalent form is: A given quantity of the liquid will dissolve at constant temperature quantities (by weight) of the gas which are proportional to the pressure of the gas. Thus, if a quantity of water will dissolve 1 g. of oxygen at the atmospheric pressure, it will dissolve 2 g. of oxygen at a pressure of 2 atm. The equivalence of the two forms of the law is evident from the 1 The absorption-coefficient of a gas in a liquid is not the actual solubility, i.e. the volume of the gas dissolved by 1 vol. of liquid at the experimental temperature, but this volume reduced to 0°, VI SOLUBILITY 51 consideration that doubling the pressure of a gas halves its volume, so that the 2 g. of oxygen at a pressure of 2 atm. occupy the same volume as 1 g. of oxygen at a pressure of 1 atm. When a mixture of two or more gases is dealt with, each dissolves in the liquid as if all the others were absent. This rule may be also stated in the form, that when a mixture of gases dissolves in a liquid, each component dissolves according to its own partial pressure, and in this form it is called Dalton's Law. These two laws only hold good with accuracy when the gases are comparatively slightly soluble in water, and when the pressures do not exceed a few atmospheres. With very soluble gases and at great pressures there are great divergences, at least from Henry's Law. This is probably owing in many cases to chemical or quasi-chemical changes which take place when the gas dissolves in water. The gases which dissolve freely in water are all of a more or less pronouncedly acidic or basic nature. Thus amongst the most soluble gases are ammonia and the hydrogen compounds of the halogens. Neutral gases, which do not by union with water form acidic or basic substances, are only sparingly soluble in water, e.g. the chief gases of the atmosphere, the gaseous hydrocarbons, etc. There is also a general connection between easy compressibility to the liquid state at atmospheric temperatures and easy solubility in water. The so-called "permanent gases," nitrogen, oxygen, argon, helium, carbon monoxide, nitric oxide, methane, are all very slightly soluble; the simple gases which are easily condensed to liquids are much more soluble. In general, we may state that when a substance exhibits chemical relationship with a liquid it is more or less soluble in that liquid. Organic compounds containing hydroxyl are very frequently soluble in water, and the more hydroxyl they contain proportionally in the molecule the more soluble they are. Thus the higher monohydric alcohols of the series CnH2n+1OH, say CH3OH, are scarcely soluble, whilst the polyhydric alcohols, such as mannitol, CH(OH), are freely soluble. Hydrocarbons are, as a rule, very sparingly soluble, in simple compounds containing hydroxyl, e.g. water, alcohol; and substances with proportionately much hydroxyl are very slightly soluble in hydrocarbons. The solid hydrocarbons are, however, freely soluble in the liquid hydrocarbons, and that the more the solvent and dissolved substances approach each other more nearly in character. Sulphur and the metals are insoluble in water; the former, however, dissolves freely in liquid sulphur compounds, e.g. carbon disulphide and sulphur chloride, whilst the latter are often soluble in the liquid metal mercury, or in other metals in the fused state, whence the possibility of forming alloys and amalgams. as A process of purification very frequently made use of is that of recrystallisation. This consists in dissolving the impure substance and allowing a portion of it to separate out of the solution, either by 52 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. a change of temperature that will lower the solubility, or by evapora- tion of the solvent. Suppose the substance to contain 90 per cent of pure material and 10 per cent of an equally soluble impurity. The mixture is treated with as much solvent as will dissolve the whole. If we make the provisional assumption that the substances have no influence on each other's solubility, the solution is now saturated with respect to the pure material, but far below the saturation point of the impurity. By removing part of the solvent, say by evaporation, we make the solution supersaturated with regard to the pure material, so that a portion of the latter is deposited. The solution still remains unsaturated, however, with regard to the impurity, so that none of this component falls out. We can therefore go on removing the solvent, and so obtain crystals of the pure material, until we reach the saturation point of the impurity, when the operation must be stopped, for then pure substance and impurity would deposit in equal proportions. In this way ths of the original substance would be obtained in the pure state by the sacrifice of th, which would be more highly contaminated with the original impurity. Purification by crystallising is always attended by loss of material, but if the process of recrystallisation is carried out systematically, this loss in the majority of cases need only be slight. In no case is the process so simple as in the scheme given above, and it is almost always necessary to repeat the crystallisation several times, the separa- tion being even then by no means perfect. In organic chemistry the success of purification by crystallisation largely depends on the choice of solvent, a proper choice leading to a rapid and perfect separation with a good yield of pure substance, where an unsuitable. selection would give a small yield of still impure material. Where, as in the case of the Stassfurt salts, complex recrystallisations of aqueous solutions are carried out, the temperatures of crystallisation are carefully chosen, and the nature of the solvent for any particular salt is varied by using solutions of other salts of varying concentration. In this way, and in successive operations, it is found possible to obtain pure potassium chloride from the very complex salt deposits. A method of separation often practised in organic chemistry is extraction from aqueous solution by means of ether. The principle. of the process is as follows. If a substance is soluble in two liquids which are not themselves miscible, it will distribute itself between the two solvents, when shaken up with them in the same vessel, in a manner depending on its solubility in each of the solvents separately. The simplest case is when the solvents are perfectly immiscible, and when the dissolved substance has the same molecular weight in both. The rule for this case is that the substance, on shaking up, distributes itself between the two solvents in such a proportion that the ratio of the concentrations of the two solutions is equal to the ratio of the solubilities in the separate solvents. VI SOLUBILITY 53 Suppose, for example, that the immiscible substances are water and benzene, that the molecular weight of the substance in these two liquids. is the same, and that the substance is twice as soluble in benzene as in an equal volume of water. On shaking up the substance with benzene and water in quantity insufficient to saturate both these solvents, we shall find that the substance distributes itself so that the concentration of the benzene solution is twice that of the aqueous solution. It will be seen that nothing is here said as to the absolute or relative volumes of the solvents-it is entirely a question of final concentrations. Using the above rule, we might consider the practical question : Whether is it more economical in the above instance to extract the aqueous solution with an equal volume of benzene in one operation, or to apply the same quantity of benzene in several portions, shaking up and separating each time? If the amount of substance in one volume of the aqueous solution is A, then on shaking up with one volume of benzene, we shall have one-third of A remaining in the aqueous. solution, and two-thirds of A in the benzene. We thus with one volume of benzene extract 0.67A if we perform the extraction in one operation. Let us now suppose the extraction to be performed in two stages, using the same total quantity of benzene as before. Το one volume of the aqueous solution we add half a volume of benzene. Let a be the amount extracted by the benzene. A will be the - z amount left in one volume of water; a is the amount in 0·5 volume The concentrations are therefore as A − x to X of benzene. -2.x. - 0.5 But by the above rule the concentration in the benzene is twice that in the water, so we have 2x=2(4x) x=0·54. We have therefore by using half a volume of benzene extracted 0·54, leaving 0.5A behind in the water. Extracting this aqueous solution again with half its volume of benzene, we shall again remove half of what it contains, viz. 0.254; so that by extracting in two successive operations we obtain 0·5A + 0·25A=0·754 instead of 0.674, the result obtained by using the same quantity of benzene in one opera- tion. If we are therefore desirous of getting the maximum amount extracted with a given quantity of the extracting solvent, and time is not in consideration, it is better to apply the extracting liquid in suc- cessive small portions than to use all in one operation. In the actual extraction with ether economy of time has more frequently to be considered than economy of ether, and larger quanti- ties in fewer operations are as a rule employed. This may the more readily be done as the easy volatility enables one portion to be distilled off and made ready for a second operation while extraction with another portion is in progress. The case of ether and water is not the * 54 INTRODUCTION TO PHYSICAL CHEMISTRY CH. VI ideal case for which the solubility rule applies accurately. Ether and water are partially miscible, and the partition coefficient of a substance between them is on account of this not the ratio of the solubilities of the substance in pure ether and pure water separately. It very frequently happens, too, that the molecular weight of the substance. in water is not the same as the molecular weight of the substance dissolved in ether. There is then no constant "partition coefficient ” at all, and the ratio of the concentrations of the two solutions depends on the relative proportion of dissolved substance and solvents present. The effect of a difference in molecular weight in the two solvents will be discussed in the chapter on Molecular Complexity (Chapter XX.). The resemblance between the above problem and that of shaking a gas and liquid up together in a closed vessel is evident. If we regard "vacuum" as a solvent, we may state Henry's Law in the form that there is always a definite "partition coefficient" between the vacuum and the liquid solvent, this partition coefficient being the solubility of the gas in the liquid as defined on p. 50. Let equal volumes of water and nitrogen be shaken up in a stoppered bottle. The solubility of nitrogen is n, i.e. one volume of water dissolves n volumes of nitrogen. The final concentration of the gas in the water is thus n times the final concentration of the gas in the vacuum; or the partition coefficient between water and vacuum is n : 1 = n. The original amount of nitrogen is, say, N. Let aN be the amount dis- solved. We have N-xN for the amount remaining, and as the volumes of vacuum and water are equal, the final concentrations are N-xN and xN respectively; but these are in the ratio of 1 to n, so that we have whence - N-xN 1 − x xN X XC n 1 + n 1 n Since according to Dalton's Law the presence of one gas does not affect the solubility of another, we can easily solve problems concerning the solubility of mixtures of gases in the same solvent. The student will find it a useful exercise to calculate from the tables of solubilities given above, the pressure and composition of the residual gaseous mixture of oxygen, nitrogen, and argon obtained by shaking up 1 vol. of purified air with 1 vol. water at 0° and 760 mm. in a closed vessel. If a gas has not the same molecular weight in a given solvent as it has in the gaseous state, there is no fixed "partition coefficient" and Henry's Law is not followed (ep. Chap. XX.). CHAPTER VII FUSION AND SOLIDIFICATION WHEN a solid such as glass is heated, it gradually loses its rigidity, and by degrees, as the temperature is raised, assumes the fluid form, passing through the stages of a tough, inelastic solid and of a viscid, pasty liquid. It cannot be said, therefore, to have a definite melting point. A crystalline substance such as sulphur, on the other hand, when heated remains solid up to a certain temperature, after which the further application of heat liquefies it, the temperature remaining con- stant during liquefaction. This constant temperature is the melting point of the solid, and is a characteristic property of each particular crystalline substance. Not only is the melting point used as a means of identification, but sharpness of melting point, i.e. constancy of tem- perature during liquefaction, is often employed as a test of purity of crystalline substances, especially in organic chemistry. Should a sub- stance crystallise in more than one form, each crystalline variety has its own melting point. The melting point of rhombic sulphur, for example, is 114.5°, while the melting point of monoclinic sulphur is 119.2°. Of When a fused substance is cooled below the point at which the solid melted, it may or may not solidify according to circumstances. Once the solidification does begin, however, the temperature adjusts itself to the melting point of the solid, for that is the only temperature at which the solid and the liquid can permanently exist in contact. course we may put a piece of ice into water at 15°, and the two may coexist for a time though the temperature of the ice is 0°. But here there is no permanent coexistence, for the ice is constantly melting, and the temperature of the water is being lowered. If we wish to ascertain the exact temperature at which the solid and liquid coexist, we must ensure their intimate mixture, otherwise the solid might be at one temperature and the bulk of the liquid at another. When a crystallised solid is heated, it is apparently impossible to warm it above its melting point without its assuming the liquid state. It is easy, however, in the case of most liquids to lower their tempera- 55 56 CHAP. INTRODUCTION TO PHYSICAL CHEMISTRY ture below the melting point of the solid without actual solidification taking place. The freezing point of a liquid, therefore, if defined as the temperature at which the liquid begins to freeze, is not a definite temperature, and as a matter of practice only melting points of pure substances are determined. Although ice when heated melts at 0°, water may be cooled to - 4° or lower without any separation of ice taking place. The liquid is then said to be superfused. The introduction of a crystal of the solid into the superfused liquid at once determines crystallisation, so that a superfused liquid in this respect resembles a supersaturated solution (p. 44). Fusion is always attended by absorption of heat, and solidification` by evolution of heat. When a superfused liquid begins to solidify, therefore, heat is disengaged and the temperature of the liquid rises. This rise of temperature accompanying the separation of the solid goes on until the melting point of the solid is reached, when no further change occurs, for at this temperature the solid and liquid are in equilibrium, i.e. can exist together in any proportion without affecting each other. If there is no heat exchange with the exterior, the solid and liquid remain in contact at the melting point without alteration. If heat is added from an external source, part of the solid liquefies, thereby absorbing the heat supplied. If heat is with- drawn from the mixture, part of the liquid solidifies, thereby producing sufficient heat to allow the temperature to remain constant at the melting point. It has been said above that the introduction of a crystal of the substance into a superfused liquid is in all cases sufficient to determine its crystallisation. This would appear to be true even when the crystalline particle is so minute as to be unrecognisable by ordinary means, for experiments have shown that as small a quantity as one-hundred-thousandth part of a milligram is effective in produc- ing solidification. Experimental researches have also been made to investigate the tendency of a superfused liquid to crystallise by itself, the possibility of the introduction of any crystallised particle from without being excluded. In most superfused liquids crystalline nuclei soon make their appearance at different parts of the liquid, and then begin to grow until all the liquid has solidified, or until the heat disengaged on solidification has raised the temperature of the whole to the melting point. It is obvious that the chance of such a nucleus appearing increases with the quantity of liquid taken, and we might therefore expect to keep a small portion of a superfused liquid for a longer time uncrystallised than we should a larger quantity. This is found to be in accordance with fact, and it is a comparatively easy matter to count the number of nuclei formed in a given time if the superfused liquid is enclosed in a capillary tube, and to determine by measurement the rate at which these nuclei grow. It has been ascertained that the rate of growth is approxi- VII 57 FUSION AND SOLIDIFICATION mately proportional to the degree of superfusion when that degree is not very great, and that the number of nuclei formed in a given volume in a given time at first increases with the degree of super- fusion, but afterwards reaches a maximum, and begins to diminish as the liquid becomes highly superfused. It is possible, therefore, by suddenly cooling a fused liquid far below the proper freezing point, to diminish the tendency to nucleus formation so greatly that the substance may be kept for days, weeks, or even months, without crystallisation taking place. Such a highly superfused substance is no longer a liquid but a solid of the same nature as glass. It has no definite melting point, but would, if crystallisation were prevented, like glass become gradually softer and less viscous until it might be said to have assumed the ordinary liquid form, there being at no time a sudden passage from the solid to the liquid state. If such a glassy solid is brought into contact with the corresponding crystalline substance, it crystallises, but the velocity of the crystallisation is extremely small, the rate being measurable in millimetres per hour. - We have thus no sharp line of demarcation between amorphous solids and liquids, but a gradual passage from the one state to the other. The line of demarcation is rather between crystalline substances and non-crystalline substances, i.e. between substances which have their particles arranged in a regular manner, and those in which the arrange- ment is confused and irregular. It was for long supposed that no regular arrangement of particles could subsist in liquids, the particles. of which have a certain freedom of motion not possessed by solids, but crystalline liquids are now known which possess properties usually encountered only in solid crystals. Thus when para-azoxyanisole is heated, it melts at 114°, but the liquid obtained exhibits a some- what turbid appearance and strong double refraction. This double refraction points to a regular arrangement of particles in the sub- stance, which is yet undoubtedly a liquid so far as its mechanical properties are concerned. It flows easily, and rises in a capillary tube, and from the capillary rise its molecular weight may be calculated by the method of Ramsay and Shields (Chap. XIX.). At 134.1° it suddenly loses its turbidity and double refraction, and becomes in all respects similar to an ordinary liquid. The transition from the state of "crystalline" liquid to ordinary liquid is accom- panied by a diminution in density, but by no change in the molecular weight. On cooling, the reverse changes occur: the ordinary isotropic liquid passes into the doubly refracting liquid at 1341°, and this into the crystalline solid at 114°. The opalescence of the crystalline liquid. would indicate that it is not absolutely homogeneous, and its optical examination with the ultramicroscope points to its being a colloidal solution of the suspensoid type (see Chapter XXI.). The phenomena of nucleus formation, nuclear growth, and extreme 58 CHAP. INTRODUCTION TO PHYSICAL CHEMISTRY superfusion can be very easily shown with hippuric acid, a substance melting at 188°. A small quantity of it is melted at a temperature not exceeding 195°, in order to avoid decomposition, and a portion of the liquid is sucked up into a thin-walled capillary (melting-point tube) about 6 inches long and 1 mm. or less in diameter, the lower end of which is drawn out to a fine jet but not sealed. The liquid is manipu- lated above a flame until it settles in the middle of the tube, and the narrow end of the capillary is then carefully sealed off. The tube is held horizontally by the open end, and moved above the flame until every crystalline particle has disappeared, after which it is suddenly chilled by dipping into a beaker of cold water. The contents have then assumed the state of a transparent glass. If the tube is broken, it is found that the acid is undoubtedly solid in the ordinary sense, though not crystalline. When left to itself it may remain unchanged for days, but if it is dipped into water at 100°, nuclei will appear and grow along the tube at a moderate rate. The growth may be inter- rupted at any time by chilling the tube in cold water. If a small flame be passed rapidly across the tube, nuclei will be found to form at the heated portion and not elsewhere. When a foreign substance is dissolved in a liquid, the freezing point of the solution is lower than that of the pure solvent. It is, strictly speaking, necessary here to specify what substance separates out, but unless it is otherwise expressly indicated we shall always assume that it is the solidified solvent. A solution of a substance in water must be cooled below the freezing point of water before ice separates out, and for small concentrations the lowering of the freezing point is proportional to the amount of the substance in solution (Blagden's Law). The freezing point of a solution is best defined as the temperature at which the solution is in equilibrium with the solid solvent. At this temperature it may be mixed with the solid solvent in any proportion without undergoing change; it neither melts the solid nor deposits any solvent in the solid form. Practical every- day applications of the lowering of the freezing point by substances in solution are to be found in the melting of snow or ice by salt, and in the addition of alcohol or glycerine to wet gas-meters to prevent freezing in cold weather. In the latter case the alcohol or glycerine. added to the water lowers its freezing point greatly; a mixture con- taining, for instance, 25 per cent of alcohol freezes at -13°, so that the external temperature must fall far below the freezing point before. the gas-meter ceases to act. Glycerine has a certain advantage over When alcohol alcohol for this purpose, inasmuch as it is non-volatile. is used it slowly evaporates, and the solution, owing to this cause, becomes both smaller in bulk and weaker. Glycerine not only does not evaporate itself, but also greatly reduces the vapour pressure of the water (Chap. VIII.), so that it both diminishes the chances of freezing and prevents rapid evaporation. VII 59 FUSION AND SOLIDIFICATION The action of salt in melting snow or ice may be seen from the following consideration. A solution containing a little salt is in equilibrium with ice at a temperature somewhat below 0°. If we cool the solution below this particular temperature, ice will separate, and the remaining solution will therefore become more con- centrated. Its freezing point will consequently be lower than that of the original solution. If we continue the cooling process, more and more ice will fall out and the solution will become still more concen- trated and have a still lower freezing point. This process might be 10° -10 - 20 - 30 10 Ice 20 ¡FIG. 8. 30 Na CI, 2H₂0 Na Cl 40 conceived to go on without limit if salt were infinitely soluble in water. But we know that there is a limit to the solubility of salt in water, i.e. at a given temperature a certain quantity of water will only take up a definite amount of salt and no more. This finite solubility of salt in water accordingly sets a limit to the lowering of the freezing point. We may best understand this by making use of a combined solubility and freezing-point diagram (Fig. 8). The horizontal axis gives the concentration, i.e. parts of salt to 100 parts of water. Temperatures are plotted on the vertical axis. If we lower the temperature of a solution containing a little salt, ice will begin to separate out and the freezing point will fall as the concentration of the remaining solution increases. The relation between the concentra- tion of the solution and its freezing temperature (i.e. the temperature at which it is in equilibrium with ice) is represented by the curve 60 CHAP. INTRODUCTION TO PHYSICAL CHEMISTRY If on the on the left, which we shall call the freezing-point curve. other hand we begin with a saturated solution of salt above 0° and cool it, salt will separate out and the solution will become less concentrated, for the solubility of the salt diminishes with falling temperature. The relation between the concentration and temperature at which the solution is in equilibrium with the solid salt is given by the curve on the right, which is therefore an ordinary solubility curve. This solubility curve is not that of anhydrous sodium chloride, but of the dihydrate NaCl, 2H2O which is formed at temperatures below 0·15° C. The two curves evidently tend to approximate with fall of temperature, the concentration of the solution which is in equilibrium with ice increasing, and the concentration of the solution which is in equilibrium with salt diminishing; and at a certain temperature they meet. We shall now consider what occurs when we follow a salt solution down either of the curves to the point C where the curves intersect (about 21° and 29 per cent concentration). Starting with a weak solution, we find that as the temperature falls, ice separates and the solution becomes more concentrated. This goes on until the freezing-point curve cuts the solubility curve, at which point the solution remaining behind becomes saturated. If we attempt to lower the temperature still further, ice separates out; but the solution remaining is now supersaturated, and therefore deposits salt until the saturation point is again reached. But the solution which was cooled was a saturated solution, so that the ice and salt must have separated out in the proportions in which they existed in this saturated solution, i.e. the solution solidifies like a single substance. The same result is arrived at if we follow the curve for the cooling of a strong solution of salt instead of a weak one. When the solu- tion is cooled a point is at last reached at which the solution is saturated. The result of further cooling is that some of the salt separates out, the solution being now saturated for the lower tempera- ture. As we cool the solution, therefore, it becomes less and less concentrated, the concentrations following the solubility or saturation curve on the right-hand side of the diagram. At last a point is reached where the saturation curve cuts the freezing-point curve. If we could follow the solubility curve below this point, further cooling would cause more salt to separate, but the weaker solution resulting would now be below its freezing point, and consequently ice would begin to separate out, and this would continue until the concentration of the solution was restored to the original value it possessed where the two curves intersect. If we take a solution of the particular con- centration given by the point of intersection of the freezing-point and saturation curves, we may cool it without the separation of either salt or ice until we come to the temperature corresponding to the intersection. If we continue to abstract heat, the solution solidifies V VII 61 FUSION AND SOLIDIFICATION as a whole, both ice and salt separating in the proportions in which they exist in the solution, without any change of temperature taking place until all is solid. The solution then behaves like a pure liquid, and has a definite freezing point. For this reason the separated substance was supposed to be a definite compound of salt and water, and was called a cryohydrate. It is now known, however, that the salt (in the form of a dihydrate) and ice separate out independently of each other and not as a definite compound. Cryohydric solutions, in fact, are quite analogous to the constant-boiling mixtures referred to in Chapter VIII.; they are constant-freezing mixtures. It can be easily seen from a study of the diagram that no solution of salt in water can exist in a stable state below the cryohydric tem- perature, and that this temperature is the lowest that can be produced by mixing ice and salt, or snow and salt together. If we consider a hori- zontal line cutting the diagram at the temperature 0°, we see that there is only a certain range of concentrations, viz. from 0 per cent to 36 per cent, possible for salt solutions, for above the latter concentration we should have supersaturated solutions, and salt would separate. A similar line at 10° cuts both curves, and it is only the part between the two curves, viz. from 15 per cent to 32 per cent, that represents possible concentrations for salt solutions. If we had a stronger solution than 32 per cent, it would be supersaturated at that temperature, and salt would fall out; if we had a weaker solution at that temperature than 15 per cent, it would be below its freezing point, and ice would separate. Taking horizontal lines at still lower temperatures, the parts enclosed between the two curves, i.e. the range of possible concentrations, become smaller and smaller, until at the cryohydric temperature it has shrunk to a point. At that temperature, therefore, there is only one salt solution possible, and below that temperature none is possible at all, unless indeed it be an unstable supersaturated or superfused solution, which when brought into contact with the corresponding solid will immediately assume the temperature and concentration of the cryohydric solution. At and below the cryohydric temperature, ice and salt can exist together without affecting each other; and at the cryohydric tem- perature they can also exist together with the cryohydric solution. Above the cryohydric point, ice can only exist in contact with a solution having one definite concentration, and salt with a solution having another definite concentration. If we mix, therefore, ice, salt, and water at a temperature above the cryohydric point, and keep them well mixed, there can be no real equilibrium. The water will tend to dissolve salt until it becomes saturated, but this saturated solution will not be in equilibrium with ice, being too concentrated (compare the diagram at -10° say). Ice will therefore melt and the solution will become more dilute, and again dissolve more salt, with the result that more ice will melt. Now a certain amount of heat ―― * 62 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. must be supplied in order to melt ice. If this heat is not supplied from an external source it must be derived from the mixture itself. The temperature of the mixture will therefore fall, and if we continue to mix, the same causes will effect a further lowering of temperature until the cryohydric point is reached, at which one and the same solution can be in equilibrium with both ice and salt. The solution adjusts itself to this concentration, and ice and salt now melt (if heat from external sources reaches the mixture) in the proportions of the cryohydric solution, and the temperature will remain constant at the cryohydric point until all the ice or all the salt has disappeared. It may happen, of course, that the original proportions of ice, salt, and water may have been such that one of the solids will disappear before the cryohydric point is reached. No further cooling can occur after this disappearance takes place, and the mixture is not then a satisfactory freezing mixture. In order that a freezing mixture may be effective and reach the cryohydric point, thorough mixing of the ingredients is also necessary, for on this depends the attainment of the final equilibrium. For this reason snow and salt form a better freezing mixture than pounded ice and salt, for in the latter case the mixing can scarcely be so thorough. When ice and salt are brought into contact at a constant temperature below the freezing point of water but above the cryo- hydric point, they form a liquid solution at the points at which they come in contact, and this solution strives to adjust itself so as to be in equilibrium with both solids, with the result that one of the solids entirely disappears. If, as is the case in salting snow or ice covered streets, the solution assumes the general ground temperature, both solids disappear when enough salt has been added. The quantity of salt necessary to melt a given quantity of ice depends on the temperature of the ice, which is practically the temperature of the ground immediately below it. This quantity can easily be cal- culated from the above diagram. Let the temperature of the ice be 2° C. From the diagram we find that the solution which freezes at this temperature contains about 3 per cent of salt. If we add less salt than amounts to 3 per cent of the weight of ice to be removed, the whole of the ice will not be melted, but only enough for the formation of a solution of the above strength, this solution then being in equilibrium. with the remaining ice. If we add 3 per cent of the weight of the ice, all the latter will melt. If we add more than 3 per cent of salt, the ice will all melt and the solution will dissolve the excess of salt, unless the excess is so great that the solution becomes saturated before the whole excess is dissolved. If the temperature of the ground (and the ice) is below the cryohydric point, no amount of salt, however great, will melt the ice. It should be noted that the circumstances of the case are here somewhat different from those under which a freezing mixture is usually formed. In the latter case conduction is avoided as far as VII FUSION AND SOLIDIFICATION 63 possible, as the object is to keep the temperature down. If, on the other hand, excess of salt is thrown on the streets, although the temperature may for a short time sink below the temperature of the street, conduction from below restores it to its original value. In the above discussion it will be observed that there is no practical difference between the behaviour of the solutions towards the ice and towards the salt. The solution which can remain unchanged in contact with ice may be said to be saturated with regard to ice, just as the solution which can remain unchanged in contact with common salt is said to be saturated with regard to the salt. This serves to illustrate the conventional nature of the distinction between solvent and dissolved body. As soon as we reach a temperature at which both substances are solid, the solution bears for all practical purposes the same relation to both. What has been said with regard to salt and water holds equally well for other substances, where we have two concentration curves meeting. The hydrates of many inorganic salts afford excellent instances 100 in point. As an example we may take the hy- so 100% 60% drates of ferric chloride. In the diagram temper- atures are tabulated as before on the vertical axis and concentrations 40 on the horizontal axis. The concentrations, 20 however, in this case are expressed not in parts of salt dissolved in 100 parts of water, but as the number of molecules of ferric chlor- ide (Fe,Cl) to 100 mole- -40 cules of water. Ferric chloride is usually met -60 with as the yellow hydrate Fe Cl, 12H₂O or as the black metallic- looking scales of the anhydrous salt. It also exists, however, in the form of hydrates, having the formulæ Fe,Cl, 7H₂O; FeCl¸, 5H₂O; and FeCl¸, 4H₂O. The diagram shows the solubility of all these hydrates. Each hydrate has its own solubility curve, and these curves cut each other at various points. Towards the left of the diagram is given the freezing-point -20 Ice 5 12 aq. A 7 aq. 5 aq. 4 aq. 10 15 20 25 Mols. Fe, Cl to 100 Mols. H₂O FIG. 9. Fe₂ Clo 30 35 64 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. 2 curve of dilute ferric chloride solutions, water being considered the solvent. The curve cutting this is the solubility curve of the hydrate Fe,Cl, 12H₂O. On the left of this curve we have precisely the kind of diagram already given on p. 59, with the difference that the solid salt is now a hydrate of ferric chloride instead of anhydrous sodium chloride. The point where the curves cut is the cryohydric point, and gives the lowest temperature that can be got by mixing ice and Fe, Cl, 12H₂O. To the right of the cryohydric point it will be observed that the curve of the dodecahydrate culminates at a point A, declining from this maximum as the concentration increases. If we draw a horizontal line at a temperature a little below 40°, it will cut the curve of the dodecahydrate in two places, i.e. at one and the same temperature this hydrate has apparently two solubilities, or can be in equilibrium with two ferric chloride solutions of different concentrations. This is not an uncommon occurrence with hydrates, and may be most easily understood by a reference to the maximum point. The concentration here is the same as the composition of the solid hydrate, i.e. 1 mole- cule FeCl to 12 molecules of water. The solution then may be looked upon as the melted hydrate, and the maximum temperature as the melting point of the hydrate. Now we have seen that ice is in equilibrium in contact with water plus a foreign substance at a lower temperature than when in contact with water (i.e. melted ice) alone. The melting point of ice therefore is highest when it is in contact with a liquid of its own composition: it melts at a lower temperature when the liquid with which it is in contact contains a foreign substance. This holds true generally-a single substance always melts at the highest temperature when in contact with the liquid produced by its own complete fusion. Now in the above case the maximum melting point of the hydrate Fe,Cl, 12H₂O occurs when the concentration of the solution corresponds to this formula. If the solution contains either ferric chloride or water in excess of this ratio, the dodecahydrate will melt at a lower temperature. The curve therefore falls both to the right and the left of the concentration given by the ratio of Fe,Cle to water in the dodecahydrate, with the result that there are two different solutions with which the hydrate can be in equilibrium at one and the same temperature. One of these solutions contains more water than the solid with which it is in equilibrium-the other contains more ferric chloride. The branch of the curve to the left of the maximum may evidently either be looked on as a solubility curve of the dodecahydrate in water, or as a melting-point curve of the dodecahydrate in contact with solutions of varying concentration. The branch to the right we usually regard as a melting-point curve; but it also may be looked on as a solubility curve. 6 If we follow now the right descending branch of the curve of the dodecahydrate, we find that we reach a point C' where it cuts the curve VII FUSION AND SOLIDIFICATION 65 of another hydrate, the heptahydrate Fe,Cl, 7H2O. From inspection of the diagram, the point of intersection is evidently of the same nature as the cryohydric point, where the curve for the melting point of ice intersects the curve for the melting point of the dodecahydrate. At the cryohydric point proper, ice and the solid dodecahydrate can exist together with each other and with a solution of a certain concentration: at the other point of intersection, the dodecahydrate and the heptahydrate can exist together and in contact with another solution of definite concentration. If a solution of this particular con- centration is cooled, it solidifies as a whole, and the freezing point remains constant. As before, however, the solid which separates out is not a pure substance, but a mixture of solids-the two hydrates. Proceeding farther to the right, we again reach a maximum and again dip to a "cryohydric" point, viz. the point where the curves of the heptahydrate and the pentahydrate intersect. In still more concentrated solutions. the same phenomena are repeated, until we finally come to a cryo- hydric point where the curve of the tetrahydrate cuts the solubility curve of the anhydrous salt, after which we have a solubility curve of the ordinary type. Every maximum corresponds in composition to a definite hydrate, and in temperature to the melting point of that hydrate, the solid melting at a definite and constant temperature. At every intersection we again have a composition corresponding to a constant melting point and freezing point; here, however, it is not a definite solid that melts or separates out, but a mixture of two solids. It should be noted that whenever we have a constant freezing point, the composition of the liquid and of the solid is the same. If the composition of the solid which separates is not the same as that of the liquid from which it separates, the temperature will fall as the freezing goes on. It is often possible to follow the curves of hydrates below their points of intersection, as the dotted lines in the figure indicate. The phenomena here are of the same nature as those already referred to on page 60. The solution which is saturated with respect to the dotted hydrate is unstable; or rather "metastable" (cp. Chap. X.), and may suddenly deposit another hydrate. It occasionally happens that when two substances are melted together, the liquid when cooled does not deposit one of the substances only, but both at once. This we have seen to be the case with salt solutions having the cryohydric composition, but it is only when the liquid has this peculiar concentration that the deposition of both solids occurs. With other substances, however, both solids may be deposited simultaneously, no matter what the composition of the liquid is. This generally occurs when the two solids crystallise in the same form and are capable of forming isomorphous mixtures. Such homo- geneous solid mixtures are often called "solid solutions," as they F 66 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. present many points of analogy to liquid solutions. If the solids are always deposited in the proportions in which they exist in the liquid, the liquid will have a constant freezing point. An instance closely approaching this exists in the case of two substances which are nearly related chemically, one being the hexachlor-derivative of a cyclo- pentenone, and the other the corresponding pentachlor-monobrom- derivative. The diagram Fig. 10 gives temperatures (melting points) on the vertical axis and compositions (molecular percentages) on the horizontal axis. The curve is nearly a straight line joining the melting points of the two substances. No matter what the composition of the liquid may be, it freezes as a whole and the solid has a practically constant melting point. 100+ 95% 90t 85 87.5° 80L 20 40 60 Percentage Hexachloro-compound. FIG. 10. 80 +100° O 97.7 95° 90° 86° 80° 100 If the solid mixture which separates out has not the same composition as the liquid, but a different one, which varies with the varying composition of the residual liquid, the temperature will fall as solidification progresses. Should a point be reached where the com- position of the solid separating is the same as that of the residual liquid, the liquid will then freeze as a whole. An instance of this kind is afforded by mercuric bromide and mercuric iodide, which are capable of forming isomorphous mixtures. in all proportions. These substances melt at 236.5° and 255.4° respectively, but, as will be seen from the diagram (Fig. 11), the melting points of the isomorphous mixtures by no means lie on the straight line joining the melting points of the pure substances. The lower curve is the curve of "melting points" of solid solutions of different compositions; the higher curve is the curve of "freezing VII FUSION AND SOLIDIFICATION 67 points" of liquid solutions of different compositions. The com- positions of the solid and liquid which are in equilibrium at any one temperature are given by the points on the adjacent curves which are at the same horizontal level; thus the points and s represent the compositions of the liquid and solid which are in equilibrium with each other at 235°. If we consider what occurs when a liquid having the composition 7 begins to solidify, we see that as the solid which separates out contains a larger proportion of iodide, the concentration of the liquid with respect to iodide falls off, and, as the curve shows, the freezing point falls. There is thus no constant freezing point, but, just as in the case of salt solution, a gradual lowering of the tempera- ture of equilibrium as the solidification progresses. Conversely, if we 255 245 235 225 215 IS 20 40 60 80 Molecular Percentage Mercuric lodide. FIG. 11. 255 -245 235° 225 215° 100 consider what occurs when the solid solution slowly liquefies we shall find that there is no fixed melting point, but a gradual rise in tempera- ture as the solid melts. It should be noted that for each composition of liquid there is only one solid which can be in equilibrium with the liquid, and since. the composition of the liquid changes as the freezing progresses, the composition of the solid also undergoes a change. Change in com- position throughout the mass of a solid solution is a comparatively slow process, so that if equilibrium is always to be maintained, the cooling must not be sudden but very gradual. It will be observed that at 216° the two curves of Fig. 11 touch. Here the solid and liquid in equilibrium have the same composition, namely, 60 molecules of bromide to 40 of iodide. If the liquid. of this composition therefore begins to solidify, it suffers no 68 INTRODUCTION TO PHYSICAL CHEMISTRY CH. VII change in concentration, and consequently the solidification pro- gresses at a constant temperature. Here, then, we have a constant minimum freezing or melting point, which corresponds to the cryo- hydric points previously referred to. It must be noted, however, that in this case the two curves touch, instead of intersecting, and that a single homogeneous solid solution separates, instead of an intimate mixture of two different solids as is the case with cryohydrates (cp. Chap. XI.). In the ordinary method of melting-point determination practised by organic chemists, it is well known that pure substances give sharp melting points, whilst impure substances begin to soften and melt at a temperature several degrees lower than the temperature at which they are completely liquefied. This corresponds with the general rule that the presence of a foreign substance lowers the freezing point, the melting point being similarly lowered. In many cases when solids. are brought into contact with each other the melting point of one of them will be depressed. Thus it is possible to liquefy a mixture of filings. of cadmium, zinc, lead, and bismuth by merely leaving them in contact at 100°, although all of them have melting points much above 100°. If a solid, therefore, is mixed with another solid as an impurity, its melting point will very likely be lowered, especially when, as usually happens, the impurity is a chemically similar substance. In this case the two will probably be mutually soluble, when liquid, and mutual solubility is a necessary condition for lowering of the melting point. If the substances are not mutually soluble, e.g. sand and sulphur, the melting point will not be affected. Occasionally it happens in the practice of organic chemistry that a substance, whose nature is unknown, is suspected, on account of similarity of melting point, to be identical with a known substance. In such a case the known and unknown substances are mixed, and a "mixed melting point" is taken. If the substances are identical, the melting point remains unchanged; if they are different, the melting. point is lower and no longer sharp, as each of the mutually soluble organic substances lowers the melting point of the other. Helium Hydrogen Nitrogen Argon Oxygen Methane WHEN a gas is subjected to great pressures it occupies a small volume compared to that which it occupies at the ordinary pressure, although perhaps a larger volume than it would occupy if it contracted exactly according to Boyle's Law (cp. Chap. IX.). Every gas, above a certain temperature which is characteristic for it, exhibits this behaviour, but if the gas be below the characteristic temperature, constantly increased pressure will at length convert it into a liquid. The temperature below which the substance can be condensed into a liquid, and above which it undergoes compression without liquefaction, is called the critical temperature of the substance, and the pressure which at the critical temperature just suffices to condense the gas to the liquid form, is called the critical pressure. Under these conditions the substance has a certain definite critical density, the reciprocal of which is the critical volume. The following table gives the critical temperature and pressure of various substances, the temperatures being in degrees centigrade and the pressures in atmospheres:- • • A VAPORISATION AND CONDENSATION Carbon dioxide Nitrous oxide Sulphur dioxide Ethyl ether Ethyl alcohol Water CHAPTER VIII · → · • * Critical Temperature. - 268 - 241 - 146 - 122 - 118.8 81.8 + 31.1 36.5 157·0 193.8 243.0 374 Critical Pressure. 2.3 11 35 48 50.8 54.9 73.0 71.9 78.3 35.6 63 217 It will be observed that the critical pressure, which is the pressure necessary to liquefy the gas at the highest temperature at which the ܀ 69 70 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. liquefaction of the gas by pressure is possible, in no case except that of water exceeds 100 atmospheres, being in general very much less than this. At lower temperatures than the critical temperature, a smaller pressure than the critical pressure effects the liquefaction. We may indeed say, as a general rule, that if a gas under any given conditions is not liquefied by a pressure of 100 atmospheres, it will not under these conditions be liquefied by any pressure, however great. For the liquefaction of gases a low temperature is the necessary condition, not a great pressure. If the gas is not cooled to its critical point, no pressure which can be applied is capable of lique- fying it, while at temperatures not far below this point the gas may be liquefied at the atmospheric pressure. All known gases have now been reduced to the liquid state, and the liquefaction in all cases can be effected by cooling alone. One of the methods by which oxygen and similar gases were first successfully liquefied was to cool the gas to as low a temperature as possible while it was subjected to great pressure, and then suddenly to release the pressure. The gas expanded, and in doing so performed work. An amount of heat equivalent to this work must therefore have been supplied. This heat was partially abstracted from the gas. itself, with the result that a portion of it liquefied owing to the fall of temperature. An apparatus constructed by Claude for continuous work on this principle is now used for the production of liquid air on the large scale. Another method which has been worked with conspicuous success to liquefy the least condensible gases also depends on the expansion of the gases by diminution of pressure, but the principle involved is entirely different from that just referred to. When any actually exist- ing gas is made to pass, at a suitable temperature, through a porous plug or valve from a high pressure to a low pressure, its temperature is slightly diminished; and if the process is repeated, the gas gets colder and colder, until finally its point of liquefaction is reached. Now while this is true for existing gases, it is not true for an ideally perfect gas. For such a gas there would be no change of temperature (Joule Thomson effect), and liquefaction in this way would be im- possible. The cooling here observed is due to deviations from the simple gas laws, and must not be confused with the cooling produced by the expansion of a gas under suddenly released pressure, which depends chiefly on the external work done, and holds good for perfect. and imperfect gases alike. A diagram illustrating the principle of such an apparatus is given in Fig. 12. The gas before entering on the left of the diagram is compressed by a pump from the atmospheric pressure P to the pressure P' of 100 atmospheres or more; then, following the direction of the arrows, it passes down the central tube of the double-walled spiral coil or heat interchanger (represented in the VIII 71 VAPORISATION AND CONDENSATION diagram as being cut vertically through the middle) to the throttle valve T, where the pressure falls again to the original pressure P. In passing through this valve the gas becomes colder, and circulating back- wards through the annular space between the tubes of the inter- changer, it cools the next portion of gas coming downwards through the centre tube to the valve. This next portion of gas, after it has passed through the valve, is cooled to a lower temperature than the first portion, and serves in its turn to cool the fresh gas arriving at T. The temperature in the neighbour- hood of the valve thus constantly sinks, and finally the point of liquefaction of the gas is reached. Liquid appears at the nozzle N and collects at L, the place of the liquefied gas being supplied by fresh quantities admitted to the pump. a T 07202 FIG. 12. If hydrogen is passed through an apparatus of this kind at the ordinary temperature, it becomes hotter instead of colder on expan- sion through the valve. It can, however, be liquefied successfully if before entering the apparatus it is cooled by means of liquid air to a temperature below the Joule-Thomson inversion temperature (about - 90° C. for 100 atm.), at which neither heat nor cold is produced by its expansion. In order that helium may be liquefied, it must be pre- viously cooled by means of liquid hydrogen boiling under reduced pres- sure, as its inversion temperature is very low. Under atmospheric pres sure the boiling point of helium is 4.3° abs., and under reduced pressure the temperature may be lowered to only 1.5° above absolute zero. All liquids tend to assume the gaseous state, and the measure of this tendency is what we call the vapour tension of the liquid. The tendency is exhibited in very different degree by different liquids. Water if left in an open vessel evaporates slowly; alcohol under the same conditions evaporates much more rapidly, and ether more rapidly still. Mercury, on the other hand, scarcely evaporates at all at the ordinary temperature. That it does so slowly, however, may be proved by suspending a piece of gold leaf a little distance above a mercury surface. After some time the gold leaf will be found to be amalgamated, indicating that the mercury has reached 72 CHAP. INTRODUCTION TO PHYSICAL CHEMISTRY the gold in a state of vapour and combined with it to form a gold amalgam. For a given liquid there corresponds to each temperature a certain definite pressure of its vapour, at which the two will remain in contact unchanged. At that pressure and temperature none of the liquid will pass into vapour; nor will any of the vapour condense to form liquid. The gas pressure of the vapour balances the vapour tension of the liquid, and this gas pressure is said to be the vapour pressure of the liquid at that temperature, and the vapour itself is said to be saturated. For all liquids the vapour pressure increases with rise of temperature, and consequently all liquids evaporate more readily as the temperature is raised. When the vapour tension of the liquid just exceeds the external pressure (usually the atmospheric pressure) on the liquid surface, the liquid passes freely into vapour, and is said to boil. If we heat a liquid in a closed space of appropriate dimensions, the pressure within the space rises, owing to the increase of the vapour pressure of the liquid, and the properties of the liquid and its saturated vapour gradually approximate to each other, until at last, when the critical temperature is reached, the liquid and the vapour become identical, and all distinction between them disappears.1 The pressure registered is then the critical pressure of the substance. Having considered above the effect of raising the temperature of a substance at constant volume, we may now pass to the considera- tion of the effect of varying pressure at constant temperature. Let us imagine the vapour of a liquid enclosed in a cylinder with a movable piston, and let the pressure on the piston be less than the vapour pressure of the liquid at the constant temperature considered. If the pressure on the piston is gradually increased, the gas will be compressed into smaller bulk at a rate greater than would accord with Boyle's Law, till the moment at which the vapour pressure is reached, when the liquid will begin to make its appearance. If the piston is still pressed home the gas will now pass into the liquid state without any further increase of pressure being necessary. The pressure necessary to liquefy a gas at any given temperature need therefore be only slightly greater than the vapour pressure of the liquid at that temperature, which must be always less than the critical pressure, for this is the upper limit of the series of vapour pressures (cp. Chap. X.). If on two axes at right angles to each other we plot the corre- sponding pressures and volumes of a quantity of gas which obeys Boyle's Law, we should get a rectangular hyperbola as isothermal 1 The student may compare this behaviour with what happens when aniline and water are heated together in a closed space (p. 48). At first there are two layers, but as the temperature is raised, these layers approximate to each other in composition (Fig. 6), density, etc., until at 165° they become identical, the whole being then homogeneous. The lowest temperature of complete miscibility has been called the "critical solution temperature" for the pair of liquids. VIII VAPORISATION AND CONDENSATION 73 1 : curve, or curve of constant temperature. Now all gases diverge more or less from this law, and the isothermal curves obtained only approximate to rectangular hyperbolas when the gas is not near the critical condition. A consideration of Andrews' pv diagram for carbonic acid will be instructive (Fig. 13). The critical temperature of carbon dioxide is about 31.1°. At a temperature of 48° the pv curve runs without any flexure, and though it is not a rectangular hyperbola the volume diminishes regularly with increase of pressure. At temperatures nearer the critical temperature the curves show less 120- 100- Atmospheres 80 60 40 1 O 13.1 32.5° 31° 21.5° 4 48° 8 Volume FIG. 13. 12 16 regularity, exhibiting at a certain pressure (about 80 atmospheres) contrary flexure, although at lower and higher pressures they are quite regular. At the critical temperature the curve runs for a very short distance parallel to the axis, when the pressure is 73 atmospheres. At a still lower temperature, when the pressure is increased slowly, the volume diminishes somewhat rapidly, until at a certain pressure the curve suddenly breaks and runs horizontally, so that no further rise of pressure is required to effect a diminution of volume. This takes place when the liquid and gas coexist, and the pressure is then the vapour pressure of the liquid. When that state is reached, the liquid appears and continues to increase in quantity by 74 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. the liquefaction of the gas without further rise of pressure until the gas has entirely disappeared, when the curve suddenly bends upwards from the horizontal. In the liquid a very small change of volume now follows from a considerable rise of pressure. CC At a lower temperature still, the same phenomena are observable: the formation of liquid now takes place at lower pressures, for, as we have already seen, the vapour pressures of all liquids diminish with fall of temperature. In each curve there are two breaks: first when the curve becomes horizontal, and second when it ceases to be horizontal. The value of the pressure for the horizontal portion is the vapour pressure at the particular temperature for which the curve is con- structed. If all the points where the breaks occur are connected, we get a border curve" which is shown by the dotted line in the figure. The region within this border curve gives the pressures and volumes at which the gas and liquid can coexist. On the right of the border curve and above it the substance is a gas: on the left it is a liquid. If we so change the temperature, pressure, and volume, that the corresponding points in the diagram keep entirely outside the border curve, liquid and gas never exist together, and there is no discon- tinuity in the passage from the liquid to the gaseous state, or vice versa. Let us, for example, take liquid carbon dioxide at the temperature 215°, and at the pressure and volume indicated by the point A. Let us raise the pressure of this liquid at constant temperature until it passes the critical pressure, and then, keeping the pressure constant, let us increase the temperature until it is greater than the critical temperature, say 48°. At no point during the rise of temperature does the liquid change its state suddenly; it passes with perfect con- tinuity into the state of gas. By expanding now at constant tempera- ture along the isothermal of 48°, and then cooling at constant volume, we finally come to the same temperature curve as that on which we started, viz. 21.5°, and so have converted the liquid into gas at the same temperature without sudden change of state. This gradual passage from the gaseous to the liquid state, and vice versa, shows us that these states are not so sharply defined from each other as we are in general disposed to imagine, but are rather united by a continuous series of intermediate states. The sharp definition that we are accustomed to perceive is a consequence of the ordinary conditions of temperature and pressure at which we work, rather than of any inherent properties of the substance operated on. In Chapter IX. reference will be made to the theory of van der Waals, which deals with the continuity of the liquid and gascous states in a systematic manner. When a liquid is made to evaporate rapidly, enough heat may be absorbed by the evaporation to cool a portion of the liquid below its freezing point. Thus in the Carré freezing machine ice is made by the rapid evaporation of water under diminished pressure. Again, VIII VAPORISATION AND CONDENSATION 75 } when liquid carbon dioxide escapes into the air from a cylinder in which it has been kept compressed, the sudden reduction of pressure brings about such rapid evaporation that a snow of solid carbon dioxide. is produced. Even hydrogen has been solidified by Dewar by the rapid vaporisation of liquid hydrogen under reduced pressure. On Solids have often, like liquids, a measurable vapour pressure. a windy day, snow and ice may be seen to disappear by evaporation, although the temperature may be far below the freezing point. The solid here passes into vapour directly without first assuming the liquid state, and the vapour is removed by the wind as it is formed. A piece of camphor when left in the open air gradually diminishes in bulk and finally disappears, owing to evaporation without melting. Since, below the melting point, the vapour pressure of the solid is always less : than the vapour pressure of the liquid, the vaporisation in the former case takes place more slowly. Just as small solid particles have a greater solubility than larger particles (p. 47), so it has been observed that small crystals have a greater vapour pressure than larger crystals. Thus particles of menthol 1 to 2p in diameter pass into a smaller number of larger crystals on standing for a time at constant tempera- ture. The conversion is due to the greater vapour pressure of the small particles, which disappear by vaporisation, the vapour con- densing on the larger crystals and further increasing their size. Solids are said to sublime when on heating they pass directly into a vapour, which on being cooled does not condense to a liquid but directly to a solid. Sublimation takes place with ease under ordinary conditions when the solid has at its melting point a vapour pressure not far removed from the external pressure, or, what practically comes to the same thing, when the melting and boiling points of the substance are comparatively close together. Arsenic trioxide, when heated gradually at the ordinary pressure, passes directly into vapour without assuming the liquid state, and can only be made to melt if heated under increased external pressure. On cooling, the vapours pass at once into the solid form. By sufficiently reducing the external pressure, the boiling point of any substance can always be lowered to the neighbourhood of its melting point, and sublimation can take place. Ice, for instance, can be easily but slowly sub- limed at a temperature below the freezing point from one part of an exhausted vessel to another, the ice passing directly into water vapour and the water vapour into ice. In the formation of snow the water vapour assumes the crystalline state directly. Sublimation can in any case be made to occur if the substance is brought to a temperature just below the melting point of the solid, and the vapours derived from it are rapidly cooled. It has been observed that substances volatilise to a greater extent in a gas than they do in a perfectly exhausted space. It is usually assumed for rough purposes that gases exert no influence on each other, 76 CHAP. INTRODUCTION TO PHYSICAL CHEMISTRY F the sum of the values of a physical property for two separate gases remaining the same when the gases are mixed. This cannot be accurately the case, since we are obliged to conclude that the mole- cules even of the same gas influence each other (see Chap. IX.). The reciprocal influence of the gas molecules will evidently be greatest when the molecules are closely packed together, i.e. at great pressures, but it may be noticeable even at ordinary pressures with sufficiently refined means of measurement.¹ If we consider that a substance like water must at temperatures just below the critical temperature exert a solvent action on many other substances as long as it remains a liquid, and if we further consider that saturated water vapour at such temperature does not sensibly differ in its properties from liquid water, seeing that the properties of the saturated vapour and the liquid become identical at the critical point itself, we shall not find it surprising that water vapour under these conditions exerts a solvent action even on solids. It is a well-ascertained fact that gases under high pressure exert a solvent action on solids, and the solvent action of gases under ordinary pressure can only differ from this in degree. It is quite in accordance with our theoretical knowledge, then, that a substance such as iodine should be more volatile in air than in a very perfect vacuum, as Dewar has ascertained. The air here acts as "solvent," and indeed we can always look on a mixture of two gases as a solution of one in the other, although the influence of the "solvent" is not marked as it is in most liquid solutions. When a substance is dissolved in a liquid, the vapour pressure of the latter is lowered at all temperatures, and the lowering for small amounts is approximately proportional to the quantity of substance dis- solved in a given amount of the liquid (Wüllner's Law). If we take, for example, a fairly dilute aqueous solution of a very soluble substance such as sugar or calcium chloride (which are not themselves volatile), and evaporate it on a water bath at 100°, the evaporation will at first proceed. rapidly, for the vapour pressure of the dilute solution is not much less than that of pure water, being therefore at the beginning nearly equal to one atmosphere. As the evaporation proceeds, however, the solution becomes more concentrated, and the vapour pressure for 100° becomes continually less. The evaporation therefore progresses more slowly, and towards the end the vapour pressure of the water in the solution is so low that practically no vapour comes off at all and the solution cannot be further concentrated. If the solid, on the other hand, is not very soluble and begins to separate out on evaporation, the solution 1 According to Dalton's law of partial pressures, the total pressure of a mixture of gases is equal to the sum of the pressures which the different gases would exert if each separately occupied the whole space afforded to the mixture. Even for moderate pres- sures this law is not absolutely exact. A better approximation to the experimental data is given by the following rule :-If we measure the volumes of the separate gases and of the mixture at the same pressure, the sum of the volumes of the component gases is equal to the volume of the mixture. VIII VAPORISATION AND CONDENSATION 77 1 soon reaches its maximum concentration, and the vapour pressure no longer diminishes, so that the evaporation proceeds at a uniform rate until all the water has been driven off. To get a given vapour pressure of solvent from a solution, the solution must be heated to a higher temperature than would be required for the pure solvent, since at a given temperature the vapour pressure of the solution is lower than that of the solvent. A con- sequence of the diminution of the vapour pressure of a liquid by the dissolution in it of some foreign substance is therefore that the boiling point of the solution is higher than that of the pure solvent. We may thus obtain an aqueous liquid of higher boiling point than 100° by dissolving some fairly soluble non-volatile substance in water. By employing common salt we obtain a brine bath, which is sometimes used instead of a water bath for heating substances to temperatures slightly above 100°. It should be noted that in this case the vessel to be heated must be immersed in the brine, for the steam from the boiling solution will only heat it to 100°, although the temperature of the salt solution may be as high as 110°. As the boiling point of a saturated solution of calcium chloride is 180°, this substance may be used when still higher temperatures are required. Distillation of Mixtures.-We supposed in the preceding para- graph that the dissolved substance was non-volatile. When the dis- solved body is volatile as well as the solvent, each substance lowers the vapour pressure of the other, and the boiling point of the mixture may be higher or lower than the boiling point of either, depending on the sum of the vapour-pressure values of the two substances. If, for example, we take a mixture of methyl alcohol and water, the vapour given off consists of a mixture of the vapours of these liquids, and the mixture boils when the temperature is such that the sum of the two partial vapour pressures is equal to the atmospheric pressure. In general, the composition of the remaining mixture will not be the same as the composition of the vapour evolved, so that as the evaporation or distillation goes on, the temperature of ebullition changes, since for each mixture a different temperature is required in order that the sum of the vapour pressures may reach the constant atmospheric pressure. It is evident that in the process of distillation as ordinarily conducted the boiling point of the mixture must gradually rise, since the more volatile portions evaporate first. If the original mixture contains much water and little alcohol, the boiling point will gradually rise as the more volatile alcohol (boiling point 66°) distils off, and finally pure water will remain behind when the boiling point reaches 100°. We can therefore, as a rule, free an aqueous liquid from alcohol by heating on a steam bath. If, on the other hand, the solution contains much methyl alcohol and little water, it is practically impossible to get rid of all the water by distillation. At the boiling point of the alcohol, pure water has, it is true, a considerable vapour pressure, but when the 78 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. mixture consists almost wholly of alcohol, this vapour pressure is lowered very much, and it is increasingly difficult to get perfect separa- tion of the two constituents as the quantity of water in the liquid gets less and less. 100° O The mixtures of some liquids differ from the example given above inasmuch as they tend to separate on distillation, not into the two components, but into one of the components and a constant boiling mixture of both, which can be distilled unchanged, the com- position of the dis- tillate being the same as the composition of 97 the residual mixture. The constant boiling mixture is that which has either a greater vapour pressure than that of any other mixture, or a less vapour pressure than that of any other mixture. The mix- ture of formic acid and water containing 75 per cent of the acid has a lower vapour pressure than any other mixture, i.e. it is the mixture with the highest boiling point. If, then, we distil any given mixture of formic acid and water, the composition of the residue will always approximate more closely to this mixture of maximum boiling point than will the composition of the distillate. 0 ข 20 40 Percentage Propyl Alcohol FIG. 14. 60 80 21 100 On the other hand, a mixture of propyl alcohol and water con- taining about 75 per cent of the former has, under ordinary con- ditions, a greater vapour pressure than any other mixture of the two substances. It is therefore the mixture of minimum boiling point at atmospheric pressure, and if we distil any mixture what- ever of the two substances, the distillate will always approximate more closely in composition to this mixture than will the residue in the distilling flask. Fig. 14 represents diagrammatically the relation between concentration and temperature for mixtures of propyl alcohol and water at atmospheric pressure. The resemblance to the curves of Fig. 11 is obvious. The upper curve here gives the composition of the vapour, and the lower curve gives the composition of the liquid. The two curves touch at a composition of about 75 per cent propyl VIII VAPORISATION AND CONDENSATION 79 alcohol, i.e. at this concentration both liquid and vapour have the same composition and the liquid boils unchanged. At lower con- centrations, say at lv, it will be observed that the vapour contains more propyl alcohol than the corresponding liquid, whilst at higher concentrations, say at v'l', it contains more water. Thus if we distil a liquid containing 80 per cent of the alcohol, the alcoholic concentra- tion of the distillate will be less than that of the residue. In the same way the mixture of ethyl alcohol and water containing 96 per cent of alcohol has a minimum boiling point at atmospheric pressure, so that it is impossible by distillation alone to concentrate. ethyl alcohol beyond this strength. Certain constant boiling mixtures (mixtures of maximum boiling point) were for long looked upon as true chemical compounds, for their composition was not affected by distillation. Nitric acid, for example (boiling point 86°), forms with water a constant boiling mixture which contains 68. per cent of acid and boils at 126°. If a weaker acid than this is distilled, water comes over in excess, and the residue eventually attains the above composition and boiling point. If a stronger acid than this is distilled, the distillate contains excess of nitric acid, whilst the residue grows weaker and the boiling point rises until the values for the constant boiling mixture are attained. The dilution of the residue is assisted in the case of concentrated nitric acid by the partial decomposition of the acid into oxygen, nitrogen peroxide, and water, the water remaining for the most part in the distilling apparatus. Hydrochloric acid forms with water a similar mixture of constant boiling point. This mixture contains 20.2 per cent of acid, and distils unchanged at 110° under atmospheric pressure. That such constant boiling liquids are mixtures and not chemical compounds is proved by the composition of the liquid which distils unchanged at any one pressure, varying with the pressure at which the distillation is con- ducted. Thus the mixture which distils unchanged at 2 atm. contains 190 per cent of hydrochloric acid, differing therefore in composition from the constant boiling mixture under ordinary pressure. The same error of assigning definite chemical union to mixtures which pass unchanged through some physical process has often been committed, and instances of this nature have already been referred to (cp. p. 61). When liquids that are only partially miscible are distilled together, a distillate of definite composition is obtained as long as two separate layers are present, for, as can be easily shown, the same vapour is given off by each of the two layers, and the only result of distilla- tion is to change the relative quantities of the two layers, the boiling point remaining constant until one of the layers vanishes, after which the distillation proceeds as in the case of two completely miscible liquids. When two liquids that are completely immiscible are subjected to distillation from the same vessel, neither influences the vapour pressure of the other; and when the sum of their vapour pressures is 80 INTRODUCTION TO PHYSICAL CHEMISTRY CH. VIII equal to the external pressure, distillation goes on without change in the composition of the distillate until one of the liquids disappears. Nitrobenzene and water may serve as an instance of a pair of nearly immiscible liquids. The mixture boils at 99° under a pressure of 760 mm. Now water at this temperature has a vapour pressure of 733 mm.; the remaining 27 mm. is therefore due to the vapour pressure of the nitrobenzene. Although the pressure exerted by the nitrobenzene is thus relatively small, the weight of nitrobenzene which distils over with the water is considerable; and it is this circumstance, apart from convenience of boiling point, which renders distillation with steam, an operation so often employed in organic chemistry, a practical success. The weights may be calculated from the vapour pressure by means of Avogadro's Law. The molecular weight of water is 18; the molecular weight of nitrobenzene is 123. If we consider the weight of the gram-molecular volume at 99°, viz. 22.4 (273 +99) litres, the mixed vapour will be found to consist of 273 18 × 733 760 123 × 27 g. of water vapour and g. of nitrobenzene vapour. 760 The ratio of the weight of water to nitrobenzene in the vapour is then 18 × 733 to 123 × 27, or roughly 4 to 1; and this is the ratio of the weights in the distillate. Thus, although nitrobenzene has only 1/27 of the vapour pressure of water at the boiling point of the mixture, one-fifth of the liquid collected is nitrobenzene. This, of course, is due to the molecular weight of the nitrobenzene being so much greater than that of the water. If an organic substance is unaffected by water and has a vapour pressure of even 10 mm. at 100°, distillation with steam for purposes of purification will in general be repaid. For although its vapour pressure may be only an insignificant fraction of that of water, the higher molecular weight makes up for this, and appreciable quantities come over and condense with the steam. It is thus the low molecular weight of water which renders it so specially suited for vapour distillation. CHAPTER IX THE KINETIC THEORY AND VAN DER WAALS'S EQUATION In the preceding chapters we have seen that a consideration of the composition and properties of substances, and the changes which they undergo, has led to the conception of atoms and molecules; but as yet we have not dealt with the mechanical constitution of these substances; in other words, we have not considered how the molecules go to build up the whole-whether they are at rest or in motion, or whether in the different states of matter there is a difference in the state of the molecules. It is plain that the kind of matter most suitable for study from this point of view is matter in the gaseous state, for in this form substances obey laws which in point of simplicity and extensive application are not approached by substances in either of the other states of aggregation. We have the simple laws of Boyle, Gay- Lussac, and Avogadro, which connect in a perfectly definite manner the pressure, temperature, volume, and number of molecules in all gaseous substances, whatever their chemical nature or other physical properties may be. These laws point to great simplicity in the mechanical structure of gases, and to the sameness of this structure for all gases. Various hypotheses have from time to time been put forward to explain the behaviour of gases, but only one has been found to be at all satisfactory, and to some extent applicable to the other states of matter. This hypothesis is called the kinetic theory of gases, and is, in its present form, chiefly due to the labours of Clausius and Maxwell. According to this theory, the particles of a gas-which are identical with the chemical molecules-are practically independent of each other, and are briskly moving in all directions in straight lines. It frequently happens that the particles encounter each other, and also the walls of the vessel containing them; but as both they and the walls are supposed to behave like perfectly elastic bodies, there is no loss of their energy of motion in such encounters, merely their directions and relative velocities being changed by the collision. 81 G 82 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. The total pressure exerted by a gas on the walls of the vessel con- taining it is due to the impacts of the gas molecules on these walls, and is measured by the change of momentum experienced by the particles on striking the walls. Suppose a particle of mass m to be moving with the velocity c, and let it impinge on the wall at right angles. The particle will rebound in its line of approach with a velocity equal to its original velocity, but of course with the opposite sign. The original momentum was mc, the momentum after collision is - mc, so that the change of momentum is 2mc. If we calculate now the change of momentum suffered by all the particles of a quantity of gas in a given time by collision with the walls, we are in a position to give the total effect on the walls, and thus the pressure. For the sake of simplicity, we imagine that the vessel is a cube, the length of whose side is s, and that all the molecules have the same mass m, and the same velocity c. Let the total number of molecules in the gas be n. The molecules, according to the original assumption, are moving in all directions, but the velocity of each may be resolved into three components parallel to the edges of the cube, the components being related to the actual velocity through the equation x² + y²+ z² = c². Consider a single molecule with respect to its motion between two opposite sides of the cube. Its velocity component in this direction is a, and the number of impacts on the sides in unit time will be The change of momentum X S on each impact is 2ma, so that the total change of momentum of the molecule in unit time, i.e. the force it exerts on the walls considered, X is 2mx. The action of the molecule on these walls, therefore, is S 2mx2 S Thus the action of the molecule on all the walls is > and on the other two pairs of walls it will be 2m(x² + y² + ≈2) 2mc2 S S 2my2 2mz2 $ S and 2nmc2 Now in the whole quantity of gas there are n`molecules, so that the total force exerted by the gas on the walls of the cube is S The surface of the six sides of the cube is 6s2, and the quotient nmc² therefore gives the force per unit surface or pressure. But s³ is 3.$3 nmc2 3v = or pv nmc². All the magnitudes on the right of this equation are constant at constant temperature, hence the product of the pressure and volume of the gas is constant, and thus from the assumptions of equal to ", the volume of the cube, so that we have finally p . . . IX KINETIC THEORY AND WAALS'S EQUATION 83 the kinetic theory we deduce Boyle's Law. In the above deduction the vessel was supposed to have the cubical form; but any space may be considered as made up of a large number of small cubes, and as the impacts on the opposite sides of all faces common to two cubes would exactly neutralise each other, the presence of these internal partitions does not affect the impacts on the outer faces of the external cubes, which in the limit constitute the walls of the containing vessel. Since mc² is the kinetic energy of a single molecule, the expres- sion nmc² or . nmc2 may be read as two-thirds of the total kinetic energy of the gas, and we may say that the product of the pressure and volume of a gas is equal to two-thirds of the total kinetic energy of its molecules. We learn that systems of moving particles, such as gases are imagined to be on the assumptions of the kinetic theory, are in equilibrium with each other when the mean kinetic energies of their particles are equal;¹ and we know that gases are actually in physical equilibrium when their pressures and temperatures are equal, i.e. they may then be mixed without the pressure or temperature undergoing alteration. Let us consider a number of gases at the same pressure and at the same temperature. If the temperature of the gases is in every case altered in the same degree, the pressure remaining constant, the gases are still in physical equilibrium, and consequently the kinetic energies of their particles must be altered in an equal degree. But the product of pressure and volume of a gas is proportional to the kinetic energy of its particles, and this product has therefore been altered to the same extent for each gas. Since, however, the pressure remained constant throughout, the volume of each gas has thus undergone the same relative change. Thus the kinetic theory enables. us to deduce that the volume of different gases is affected equally by the same change of temperature if the pressure remains constant, i.e. that all gases have the same coefficient of expansion. Avogadro's Law may also be deduced from the kinetic theory by making use of considerations similar to the above. Take equal volumes. of two gases at the same temperature and pressure. Since p = p', and v = v′, pv=p'v', and consequently {n · {mc² = {n' • §m'c'². But the mean kinetic energies of the particles of the two gases must also be equal, since they are in mechanical equilibrium, ¿.e. {me² = {m'c²², whence, dividing the first equation by the second, n=n'. Equal volumes of different gases, therefore, at the same temperature and pressure contain the same number of molecules. Again from pv=3nme and from the gas law pv 2T, where R is expressed in thermal units (p. 24), we may deduce the relation 1 The validity of this statement is disputed by some physicists, : 84 INTRODUCTION CHAP. • TO PHYSICAL CHEMISTRY - 2T = }· ±nmc² or e = 37, where e = {nmc², the kinetic energy of a gram molecule of the gas. Now if we heat a gram molecule of a mon atomic gas from T to T + 1, all the heat goes to raise the energy of motion of the particles provided the volume is kept constant. We have therefore e₁ − c = 3(T + 1) − 37' = 3. − But the difference of kinetic energy 1 e is here the molecular heat of the gas, so we have as final result that the molecular heat of a monatomic gas at constant volume is numerically equal to 3 cal. We have seen (p. 30) that the molecular heat at constant pressure exceeds that at constant volume by 2 cal.; the molecular heat at constant pressure is thus 5 cal., and the ratio of the molecular and specific heats is P = 1.67 in the case of a monatomic gas (cp. p. 31). 5 Cv Imm 3pv กากาง = We may write the equation pv nmc² in the form c= But from Avogadro's Law and the simple gas laws we deduce that pv the quotient is constant for all gases at constant temperature, N = consequently c is proportional to Here m is the mass of the single molecule, which for different gases is proportional to their relative densities. Therefore the molecular velocities of different gases at the same temperature are inversely proportional to the square root of their relative densities, a result which is in harmony with the experimental fact that the velocity of diffusion (and the) velocity of transpiration) of a gas is inversely proportional to the square root of its density. It is possible to obtain a value for the speed of the molecules of 13 pv a gas by substituting the known values in the equation c= กาน = = 46,100 centi- Thus for 32 g. of oxygen under standard conditions we have p= 1,013,000 dynes per square centimetre (see pp. 3, 5), mn = 32 g., and 3 x 1,013,000 × 22,400 v = 22,400 cc., so that со 32 metres per second. The molecule of oxygen therefore at 0° C. moves at the rate of 46,100 cm. per second, or nearly 18 miles per minute. The speed of the molecules of any other gas at any temperature may d.T be got from the formula c₁ = co €1 in which d₁ is the density d₁. 273' of the gas, do the density of oxygen, and 7 the temperature in the absolute scale. In the foregoing we have spoken of the velocity of the particles of a gas as if all the particles had the same velocity. This, however, cannot be the case, for even though the particles had the same velocity ทาง KINETIC THEORY AND WAALS'S EQUATION 85 at the beginning of any time considered, the velocities of the individual. particles would speedily assume different values owing to their en- counters. It must be understood, therefore, that the velocity in the above formulæ means a certain average velocity, some of the particles having a greater and some a smaller speed than corresponds to this value. The bulk of the particles have velocities in the neighbourhood of this mean velocity, and the farther we diverge from the mean, the fewer particles we find possessing the divergent values. If two different gases are brought together, their particles in virtue of their rapid motion in straight lines will soon leave their fellows and mix with the particles of the other gas. This process of intermixture we are acquainted with practically as gaseous diffusion. Two gases, no matter how different their densities may be, will mix uniformly if brought into the same space, but the rate of intermixture is very much slower than what we should expect from the rate at which the particles move. This discrepancy, however, may be easily explained. The particles in a gas at ordinary pressure are comparatively close together, and consequently encounter each other frequently; so that, though their rate of motion between individual encounters is very great, their path between points any distance apart is, owing to these encounters, a very long and irregular one, and the rate of mixing is therefore com- paratively small. ▪ After the mixture of two gases has attained the same composition in every part, there is no further apparent change; but the motion of the particles, and thus the mixing process, is supposed to go on as before, only now the further mixing does not alter the composition. The kinetic theory may be applied in a general way to the study of the processes of evaporation and condensation. In a liquid the particles have not the same independence and free path as gas particles, although they may be assumed to be identical with the gaseous molecules and have equal velocities. A gaseous substance, in virtue of the freedom of its molecules, can expand so as to fill any space offered to it. A liquid does not do so at low pressures, but retains its own proper volume, although its molecules still possess sufficient independence to move easily between collisions, and thus enable the liquid under the influence of gravitation to accommodate itself to the shape of the vessel containing it. In spite of the clinging together of the liquid molecules, it happens that some of them near the surface have sufficient motion to free themselves from their neighbours, and leaving the liquid altogether, to become free gas molecules. If these gas molecules move away unhindered, other molecules from the liquid will take their place; and so the liquid will go on giving off gas molecules until all has evaporated. If, however, the liquid is kept in a closed space, the gas molecules which leave its surface will be able to proceed no farther than the walls of this space, and must eventually return in the direction of the 86 CHA INTRODUCTION TO PHYSICAL CHEMISTRY liquid. It will consequently happen that some of them will strike the surface of the liquid again and be retained by it. But the liquid molecules still continue as before to become gas molecules and leave the surface of the liquid, so that at one and the same time there are molecules entering and molecules leaving this surface. When in a given time as many molecules leave the liquid as are reabsorbed by it, no further apparent change takes place the relative quantities of the liquid and the vapour remain the same. A stationary state of balance. or equilibrium has thus set in, and we may now look at what determines this state. The number of molecules leaving the liquid depends on the temperature, for it is only those molecules which attain a certain velocity that will succeed in freeing themselves; and the motion of the molecules of a liquid, like those of a gas, increases with the temperature. The number of the molecules reabsorbed by the liquid depends on the number of gas molecules striking the surface in a given time, i.e. on the number of molecules contained in a given space, and on their speed. As we have seen, this number and the speed together determine the pressure exerted by a gas, so the number of molecules reabsorbed depends on the pressure. Temperature thus regulates the number of molecules freed, and gaseous pressure the number of molecules bound in a given time; consequently for each state of equilibrium when these two numbers are equal, a definite temperature will correspond to a definite gaseous pressure of the vapour in contact with the liquid-or vapour pressure of the liquid, as it is shortly termed. Every liquid, therefore, has at each tempera- ture a definite vapour pressure; and this vapour pressure increases as the temperature rises, for more molecules at the high temperature will have the speed necessary to free them. It may be noted that as it is the molecules with the greatest speed, i.e. with the highest tempera- ture, that first leave the liquid, the average temperature of the liquid. must sink as evaporation goes on, unless heat is supplied from an external source. The kinetic theory not only gives us the ordinary gas laws, which are, strictly speaking, obeyed only by ideal gases and not by any actual gas, but also when properly applied affords us a probable explanation of the deviations from the gas laws which are experi- mentally found. So far, we have considered the gas molecules as mere physical points occupying no volume whatever; but certainly if gaseous particles are supposed to exist materially at all, they must be supposed to possess finite though small dimensions. It is evident that the volume in which these particles have to move is not the volume occupied by the whole gas, but this volume minus at least the volume of the particles. So long as the volume occupied by the gas is great and the pressure small, the volume of the particles vanishes in comparison with the total volume, and the gas laws are Jak le a at sa pa IX KINETIC THEORY AND WAALS'S EQUATION 87 closely followed; but when the pressure is great and the total volume small, the volume of the particles themselves bears a considerable proportion to this whole, with the consequence that the divergence from the gas laws is great. Owing to this cause, the pressure would increase in a greater ratio than the volume would diminish, as the following reasoning will show. Suppose a molecule to be oscillating between two parallel walls in a direction at right angles to them, and suppose the distance between. the walls to be equal to 100 times the diameter of the molecule. It is evident that the molecule from its contact with one wall has to travel, not 100 diameters before it comes in contact with the other, as it would have to do if it were a point without sensible dimensions, but only 99. It will therefore hit the walls oftener in a given time than if it were without sensible dimensions, and that in the ratio of 100 to 99. Now suppose the distance between the walls to be reduced to 10 molecular diameters. The particle has now only to travel 9 times its own diameter in order to pass from contact with the one wall to contact with the other. It will therefore in a given time hit the walls oftener in the ratio of 10 to 9, or 100 to 90, than if it were a mere point. By diminishing the distance of the walls to one-tenth, therefore, we have increased the pressure not to ten times the original. value, but to this value multiplied by 99: 90, i.e. the pressure has increased to eleven times its former magnitude. We might now write the gas equation pv = RT in the form p(v − b) = RT, where b is a constant for each gas theoretically equal to four times the volume of the gas particles. But there is still another influence at work which interferes with simple obedience to the ideal gas laws. The particles of a liquid undoubtedly exercise a certain attraction on each other, and this attraction must still persist when the liquid particles have become particles of vapour, only in the latter case the particles are in general so far apart that the effect of the attraction is inconsiderable. If the gas is compressed into a smaller volume, however, the influence becomes felt more decidedly owing to the comparative proximity of the particles. Van der Waals has assumed that this attraction is proportional to the square of the concentration of the gas, or reciprocally proportional to the square of the volume. The effect of the mutual attraction of the particles is the same as if an additional pressure were put upon the gas, so that the correction is applied by adding it to the value of the external pressure. If a denotes the coefficient of attraction, i.e. the value of the attraction when the gas occupies unit volume, then the correction for any other a volume is The equation for the behaviour of a gas under all conditions is, therefore, according to van der Waals, 88 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. { This equation not only gives the behaviour of the so-called permanent gases very accurately up to high pressures, but even that of a com- paratively compressible gas like ethylene. The following table gives the values of pv for ethylene at 20° actually found by Amagat, and those calculated from the equation (v – 0·0024) 1·005441, p being expressed in atmospheres, and v being made equal to 1 when p = 1. + p al (p + 1 ) (v − b) = RT. . · 0.00786 v2 1 45.8 84.2 110.5 176.0 282.2 398.7 1000 pv (observed). 1000 781 399 454 643 941 1248 ! 1000 pv (calculated). 1000 782 392 446 642 940 1254 The agreement between the observed and calculated values is very satisfactory. If the gas were an ideal gas the values of pv would remain the same for all pressures. We see, however, that this con- stancy is far from being attained. The gas is at first more compressible than corresponds to Boyle's Law, and then at higher pressures less compressible, the minimum value of the product occurring when the pressure is about 80 atmospheres. From the form of the equation it may be seen that the two corrections act in opposite ways, the value of the product pv being diminished by the attraction, and in- creased by the finite dimensions of the molecules. At low pressures the effect of the attraction greatly overweighs the volume correction, which in its turn becomes preponderant when the pressure reaches a high value and the total volume becomes small. With ethylene at the temperature considered, the two corrections balance each other at about 80 atmospheres, and here the gas within narrow limits of pressure obeys Boyle's Law, for the product pv then remains sensibly constant. All gases hitherto investigated, with the exception of hydrogen and helium, give similar deviations from Boyle's Law: the product of pressure and volume at first diminishes, afterwards to increase as the pressure rises. At higher temperatures the deviations are of the same kind, but not so marked. This may be seen directly from the formula, the constants a and b being theoretically independent of the tempera- ture, and the value of the expression on the right-hand side increasing in direct proportionality with the absolute temperature. In the case IX KINETIC THEORY AND WAALS'S EQUATION 89 of hydrogen and helium there is no preliminary diminution of the value of pv, on account of the constant of attraction a being so small that its effect is counterbalanced from the first by the effect of the constant b. Daniel Berthelot has developed a method whereby accurate molec- ular weights of gases may be ascertained from their densities without recourse being had to the results of analysis (cp. p. 14). He assumes that for perfect gases Avogadro's rule is strictly accurate, i.e. that the molecular weights of perfect gases would be exactly proportional to their densities, instead of being only approximately proportional as is the case for ordinary gases under ordinary conditions. The following development of Berthelot's method used by Guye will serve to illustrate one mode of procedure. Berthelot established by his method that the gram-molecular volume of an ideal gas is 22.412 litres at normal temperature and pressure (p. 14). For unit volume (1 litre) of any actual gas at normal pressure (1 atmo- sphere), van der Waals's equation may be made to assume the form L (1 + a)(1 − b) = 22.412) 'M' where L is the weight of 1 litre under normal conditions, and M the true molecular weight of the gas. Now, as we shall see, a and b can be determined from the critical data of the gas. By substituting the values of a and b and the weight of the normal litre the molecular weight M can be calculated. Guye has pointed out, however, that the corrections a and b determined directly for the critical condition cannot be used at 0°, because van der Waals's equation is after all only approximate, and the coefficients a, b vary with the temperature. By using the constants a, bo corrected for this temperature variation, instead of a, be given directly by the critical data, results in perfect accordance with the analytical data are obtained. In the following table M, indicates the crude molecular weight as calculated from the density L and the simple gas laws, M, the molecular weight deduced by Guye, and M. the molecular weight found from analytical data :- 1 2 3 Gas. Hydrogen, H₂ Nitrogen, No Carbon monoxide, CO Oxygen, O2 (standard) Argon, Ar Carbon dioxide, CO₂ Nitrous oxide, N₂O Hydrochloric acid, HCl Sulphur dioxide, SO2 Acetylene, CH₂ M1. 2.0125 28.007 28.001 32 39.864 44.267 44.284 36.741 65.536 26.216 Mo. 2.0153 28.013 28·003 32 39.866 44.003 44.000 36.484 64.065 26.018 M3. 2.0152 28.02 28.00 32 44.00 44.02 36.45S 64.06 26.016 The difference between M, and M, for the permanent gases (with 3 90 CHAP. INTRODUCTION TO PHYSICAL CHEMISTRY small a and b) is slight; for the often amounts to nearly 1 per cent. P γ ∞ a 6 FIG. 15. 5 This cubic equation has in general three solutions, so that for each value of p we have in general three corresponding values of v. The graph of the equation for constant values of a and b, and for various values of 7', is given in Fig. 15. The isothermal curves thus ob- tained would represent the behaviour of a substance at various temperatures. The curves for the lower temperatures are sinuous, and are cut by horizontal lines of constant pressure, sometimes in three points and sometimes in one. When the curve is cut by the horizontal line only once, the point of intersection gives the real solution of the equation, the other two solutions being imaginary. The resemblance of these curves to the curves on p. 73, which roughly express the result of actual experiment, is at once evident. In the case of the theoretical curves. we have no sudden breaks such as we have in the actual discontinuous passage of vapour into liquid by increasing pressure. Van der Waals's equation assumes a continuous passage from the liquid to the vaporous state, and vice versa, such as we find when we take the temperature and pressure above their critical values for the substance under con- sideration. When the passage between the two states is discontinu- ous, as it usually is, we proceed along a horizontal line from one part of the theoretical curve to another, this line of constant pressure cutting the curve in three points. The volume of the substance as liquid is the 4 3 compressible gases the difference The differences between M, and M, are throughout trifling. 2 The equation of van der Waals is especially interest- ing in its application to the continuous passage from the gaseous to the liquid state, as it holds good not only for gases, but also in many ways for liquids. If we rearrange the equation. so as to give coefficients of the powers of v, we obtain ab 2 23 A RT - (b + 11 )² + 10. 102 P a -v p p 0. IX KINETIC THEORY AND WAALS'S EQUATION 91 least of the volumes corresponding to the constant pressure; another volume, the greatest, is the volume of the vapour derived from the liquid; the substance in a homogeneous state occupying the third intermediate volume is unknown. By studying supersaturated vapours and superheated liquids, we can advance along the theoretical curve for short distances beyond B and C without discontinuity, but the substance in these states is comparatively unstable. In the neigh- bourhood of the third volume the state of the substance is essentially unstable, increase of pressure being followed by increase of volume, and so we cannot hope to realise it. Van der Waals has pointed out, however, that in the surface layer of a liquid, where we have the peculiar phenomena of surface tension, it is possible that such un- stable states exist, and that the passage from liquid to vapour may after all in the surface layer be really a continuous one. It will be noticed that, as the temperature increases, the sinuous portion of the curve gets continually smaller, and the three volumes. get closer and closer together, finally to coalesce in a single point. Here the three solutions of the equation become identical, the volume of the liquid becomes equal to the volume of the substance as gas, and there is no longer any discontinuity or distinction between the liquid and gaseous states. In short, the substance at this point is in the critical condition: the curve is the curve of the critical temperature, the pressure is the critical pressure, and the volume is the critical volume. When the three roots of a cubic equation become equal, certain relations exist between this triple root and the coefficients of the powers of the variable. If the equation is x³ – Ax² + Bx − C = 0, and if the triple root is represented by §, the following relations hold good:- K 3A; 32 B; &C. In van der Waals's equation there are only the pressure and volume of the gas, the constants a and b, and the gas constant R. Now we can express the constant R in terms of the constants a and b as follows. Under normal conditions, let po = 1, = 1, and T, = 273. Then the equation 0 becomes whence a (+)(-b)-RT = (1 + a)(1 − b) = 273R (1 + a)(1 − b ) ; R= 273 92 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. or, if we make ẞ, the coefficient of expansion, we have 3 R=ẞ(1+a)(1 - b). Van der Waals's equation then becomes (cp. p. 90) ab b)) 02 p whence 2.3 1 Ø3 π in which -(0+ b+ If we denote now by p, π, and 0 the critical values of v, p, and T respectively, we obtain for the critical equation, in which the three roots become identical, 36=b+ 362= Ꮎ ; a > π ab π BT(1+a)(1b) P ; B0(1 + a)(1 – b) π 36 (critical volume), a 2762 (critical pressure), Sa 27ẞb(1 + a)(1 − b) a 2² + -v p b φ 3' Here we have general expressions for the critical values of a substance in terms of the constants which express the deviation of the substance from the laws for ideal gases; and conversely we can give the numerical values of these constants from the values of the critical data, viz.- www.d α = a = 0. (critical temperature). 3п ф2. These relations have been tested in several cases, and a fair approxi- mation of the calculated to the observed values has been found. If in van der Waals's equation we express the values of the pressure, temperature, and volume as fractions of the corresponding critical values, and at the same time express these latter in terms of the deviation constants, the equation becomes (e + 23 ) (3n − 1) = 8m, − − T <= 2; n = 2; m = 7 € φ Ꮎ Here everything connected with the individual nature of the substance Ex 93 KINETIC THEORY AND WAALS'S EQUATION has disappeared, and we have an equation which applies under certain restrictions to all substances in the liquid or gaseous state, just as the gas equation holds good for all gases, independently of their specific nature. The chief point to be noted here is that whereas for gases the temperature, pressure, and volume may be measured in the ordinary units without impairing the validity of the comparison of different gases, it is necessary in the case of liquids to effect the comparison under "corresponding" conditions, the temperatures of the two liquids to be compared being, for instance, not equal on the thermometric scale, but being equal fractions of the critical temperatures of the two substances. It is one of the tasks of physical chemistry to compare the physical properties of different substances, and to trace, if possible, some connection between their magnitude and the chemical constitution of the substances considered. Now we know that most physical properties of substances vary with the temperature and pressure. The question therefore arises: At what temperature and pressure are we to compare the properties of different substances? It is evidently purely arbitrary to make the comparison under the so-called normal conditions of 0° and 760 mm., for these conditions have no relation whatever to the properties of the substances themselves, and are merely chosen for convenience' sake as being easily attainable in the circumstances in which we work. Van der Waals answers the question by saying that the properties ought to be compared at corresponding temperatures and pressures, meaning thereby at temperatures and pressures which are equal fractions of the critical values on the absolute scale. Suppose, for example, we were to compare ether and alcohol. with respect to some particular property. The critical temperature of ether is 194° C., or 467 absolute; that of alcohol is 243° C., or 516 absolute. Let the property for the alcohol be measured at 60° C., then we shall have as the corresponding temperature r for ether 273+x 273 +60 or x = 28° C. The pressure when small has not, as 467 516 a rule, a great effect on the properties of liquids, so that in general we may make the comparison at the atmospheric pressure without com- mitting any serious error. It should, however, be stated at once that the data for the comparison of different substances under corre- sponding conditions are for the most part still wanting, so that it is not known whether the theoretical conditions would lead to sensibly greater regularities than those observed among the properties when measured under more usual conditions. Various equations have been proposed to take the place of van der Waals's equation. Amongst these the equation of Dieterici is worthy of mention. It may be written in the form 1 p(-b) RT, e 1 ᎡᎢ n 94 CHAP INTRODUCTION TO PHYSICAL CHEMISTRY in which e is the base of the natural logarithms and A a constant depending on the mutual attraction of the gas-molecules. It differs from the equation of van der Waals in that the correction for attrac- tion is applied as a factor and not as a term added to p. When v is great the equation reduces to the simple gas equation, and the pressure-volume graphs for various temperatures closely resemble those given by van der Waals's equation (Fig. 15). In some respects Dieterici's equation (especially when a power 7" is used instead of T in the correcting factor) represents experimental results more closely than that of van der Waals, but under other conditions the latter is the more satisfactory. In solids there is no rectilinear motion of molecules: indeed the conception of molecule in a pure crystalline solid seems to be in- applicable (cp. Chap. XXXII.). Instead there is a definite arrangement of atoms which vibrate about fixed centres. The effect of rise of temperature is to increase the amplitude of vibration of the atoms, until, at the melting point of the solid, they disengage from their fixed associations and regroup themselves as mobile molecules of a liquid. The kinetic theory affords also some account of the phenomena of solution. If we take the case, for example, of the solution of a gas in a liquid, we can easily see that the gas molecules impinging on the surface of the liquid may be held there by the attraction of the molecules of the solvent. When, however, a number of the gas molecules have accumulated in the liquid, some of them, in virtue of their motion, will fly out from the surface of the solution, and this will happen the more frequently the more molecules there are dissolved in the liquid. But as the number of gas molecules striking the surface of the liquid remains constant at constant pressure, it will at last come to pass that the number of molecules entering and leaving the liquid will be the same. There is then equilibrium, and the liquid is saturated with the gas. As the number of gas molecules striking the liquid surface is proportional to the pressure, the number of molecules leaving that surface when the liquid is saturated, and consequently the number of molecules dissolved in the liquid, is likewise proportional to the pressure. This is Henry's Law, and Dalton's Law also follows at once; for in a gaseous mixture the number of molecules of each gas striking the surface is proportional to its partial pressure in the mixture, and independent of the other components. It will be seen from this explanation that there is a great similarity between the solution of a gas in a liquid and the phenomena of evaporation and condensation. The same analogy appears when we consider the solution of a solid. When a soluble crystalline substance is introduced into a solvent, some of its particles become detached and enter the solvent. After a time certain of these detached particles come into contact IX 95 KINETIC THEORY AND WAALS'S EQUATION with the solid again, and are retained by it. This give-and-take process goes on until the same number of particles leave the solid and return to it in a given time. No further apparent change then takes. place, and the solution is saturated. The number of particles which return to the solid evidently depends on the number of them in unit volume of the solution, i.e. on the strength or concentration of the solution. If the solid is brought into contact with a stronger solution than the above, more particles will enter the crystal than will leave it, and so the crystal will increase in size. Such a solution is super- saturated with regard to the solid. In a weaker solution, fewer particles will come into contact with the solid and be retained by it than will leave it, i.c. the solution is unsaturated and the crystal will dissolve, in part at least. In the chapter on evaporation and condensation we had occasion to refer to the vapour tension of liquids, meaning thereby the tendency of the liquids to pass into vapour under the specified conditions. There is equilibrium when the vapour tension of the liquid is balanced by the gaseous pressure of the vapour above the liquid. A similar term has been employed to express the tendency of a substance to pass into solution, the substance having a definite solution tension for each solvent with which it is brought into contact. When the pressure of the dissolved substance in the solution is equal to the solution tension of the solid there is equilibrium. Here a new conception is introduced, namely, the pressure of a substance in solution. What this pressure is, and how it may be measured, will be seen in Chapter XVII. A brief account of the Kinetic Theory will be found in CLERK-MAXWELL, Theory of Heat, chap. xxii. The student is also recommended to read, in connection with this chapter, J. P. KUENEN, on Condensation and Critical Phenomena," Science Pro- gress, New Series, 1897, vol. 1, p. 202 and p. 258. DANIEL BERTHELOT gives an account of his method for molecular weights of gases in several papers in Comptes rendus, 1898, vol. 126. P. A. GUYE, "Departure of Gases from Avogadro's Law," Journal de chimie physique, 1908, vol. 6, p. 769. (6 CHAPTER X THE PHASE RULE A SUBSTANCE is in general capable of existing in more than one modification. For example, water may exist as ice, as liquid water, or as water vapour. Sulphur may exist as vapour, liquid, and as two distinct solids, namely, as monoclinic and as rhombic sulphur. Para- azoxyanisole, as we have seen, forms not only solid and gaseous modi- fications, but can also exist as a crystalline liquid distinct from the ordinary non-crystalline liquid. All such modifications, when they exist together, are mechanically separable from each other, and are in this connection called phases. A single substance may assume the form of many different phases, but these phases cannot in general all exist together in stable equilibrium, being subject to certain restric- tions as to their coexistence, which may be stated in the form of definite rules. As a familiar example we shall take the substance water in the three phases-ice, water, and vapour. The physical conditions deter- mining the equilibrium of these phases are temperature and pressure. We know that at the pressure of 1 atmosphere, water is in equilibrium with ice at the temperature of zero centigrade, and with water vapour at the temperature of 100° centigrade. For a given pressure, then, there is a definite temperature of equilibrium between such a pair of phases; and we shall also find that for a definite temperature there is a definite equilibrium pressure. Consider the two phases, water and water vapour. To each temperature there corresponds a fixed vapour pressure, which is the pressure of water vapour, or gaseous phase, which is in equilibrium with the water, or liquid phase. By drawing the pressure-temperature diagram, therefore, of a substance, we are enabled to study conveniently the equilibrium between its phases. In Fig. 16 the line OA represents the vapour-pressure curve of water, each point on the line corresponding to a certain pressure measured on the vertical axis, and to a certain temperature measured on the horizontal axis. Ice, like water, has a vapour-pressure curve of its own, and this has been represented in the diagram by the 96 CHAP. X 97 THE PHASE RULE line OB. It will be observed that the two vapour-pressure curves have not been represented as one continuous line, but as two lines intersecting at a point O. If we inquire into the meaning of this intersection, as interpreted from the diagram, we find that at a certain temperature t ice and water have the same vapour pressure, for the point O corresponding to this temperature belongs both to the vapour- pressure curve of water and the vapour-pressure curve of ice. It is easy now to show that there is in fact a temperature at which the vapour pressures of ice and water are identical. Water at its freezing point is in equilibrium with ice, i.e. ice and water can coexist at this temperature in any proportions whatever, and these proportions will Pressure B с A.-- t M A, 1 才 ​L N Temperature FIG. 16. P remain unchanged if the mixture is surrounded only by objects of this same temperature. Now let the water and ice coexist, not, as is usual, under the pressure of one atmosphere, but under the pressure of their own vapour. The temperature of coexistence will no longer be exactly 0°, but a temperature very slightly higher; otherwise the conditions are unchanged. If the ice, at this equilibrium temperature between ice and water, have a vapour pressure higher than that of water at the same temperature, diffusion will tend to bring about equalisation of pressure, i.e. the pressure of vapour over the ice will become less than its own vapour pressure, so that the ice will evaporate; and the pressure of vapour over the water will become greater than its own vapour pressure, so that water will be formed by condensation. Ice will have H 98 CHAP. INTRODUCTION TO PHYSICAL CHEMISTRY therefore been converted into water, which is contrary to our original postulate that the proportions of water and ice present are not under the circumstances subject to alteration. Similarly, if water at the equilibrium temperature had a greater vapour pressure than ice, ice would be formed indirectly through the vapour phase at the expense of the liquid water, so that our postulate in this case also would be contradicted. There only remains, then, the alternative that the vapour pressures of ice and water are equal when the ice and water are in equilibrium, which is in accordance with the representation of the diagram. At any point on the line OA water and water vapour can coexist in equilibrium, at any point on the line OB ice and water vapour can coexist. At the point O, where these two lines intersect, all three phases can exist together in equilibrium, and such a point is therefore called a triple point. When a substance can only exist in three phases, only one triple point is possible. The triple point in the case of water is not quite identical with the melting point of ice, because the melting point is, strictly speaking, defined as the temperature at which the solid and liquid are in equilibrium when the pressure upon them is equal to one atmosphere. At the triple point the pressure is not equal to the atmospheric pressure of 760 mm., but to the vapour pressure of ice or water, which is only 4 mm. Now, it has been shown, both theoretic- ally (cp. Chap. XXXVI.) and experimentally, that pressure lowers the equilibrium temperature of ice and water by about 0·007° per atmo- sphere, so that the freezing point under atmospheric pressure is about 0.007° lower than the triple point. The effect of pressure on the melting point of ice may be represented in the diagram by the line OC, slightly inclined from the triple point towards the pressure axis. At any point on this line, ice and water are in equilibrium with each other, the temperature of equilibrium falling with increase of pressure. The diagram for equilibrium of the three phases of water consists therefore of three curves meeting in a point, the triple point. At any other point on the curves, two phases can coexist in equilibrium :— (a) Water and water vapour on OA ; (b) Ice and water vapour on OB; (c) Water and ice on OC. At O, the common point of intersection, all three phases are in equilibrium together. The three lines divide the whole field of the diagram into three regions. At pressures and temperatures represented. by any point in the region AOB water can only exist permanently in the state of vapour. At any point in the region AOC it can only exist as liquid water, and at any point in the region BOC it only exists as ice. The curve OA separates the region of liquid from the region of vapour, but the separation is not complete. The curve is the curve of vapour pressures, and, as we have already seen, there is X THE PHASE RULE 99 a limiting pressure beyond which the vapour pressure of a liquid. cannot rise. This is the critical pressure of the substance, and it is attained at the critical temperature. The curve OA, therefore, ceases abruptly at a point A, the values of the pressure and temperature at which are the critical values. Beyond A there is no distinction between liquid and vapour; the two phases have become identical. It is possible to pass from a point M in the liquid region to a point N in the gaseous region in an infinite number of ways, which may be represented on the diagram by straight or curved lines. If these lines. cut the line OA, there is discontinuity in the passage, for at the pressure and temperature represented by the point of intersection the two phases will coexist. For example, we may pass from M to N, by means of lines parallel to the axes, along the path MLN. The line ML represents increase of temperature at constant pressure; the line LN diminution of pressure at constant temperature. The pressure at L is greater than the vapour pressure of the liquid at the constant temperature considered, and so the substance exists at this point as liquid only. As the pressure is gradually reduced, a point is at last reached at which it is exactly equal to the vapour pressure of the liquid. The liquid now begins to evaporate, and the two phases exist together, the point at which this occurs being the point of intersection of LN with OA. No further reduction of pressure can be effected until all the liquid has been converted into vapour, after which the pressure may be diminished until it attains the value represented by the point N. If, on the other hand, we follow the line MPQN, which does not cut the line OA, we can pass from the state of liquid at M to the state of vapour at N without any discontinuity whatever. We first increase the temperature, following the line MLP, to a value above the critical temperature, the pressure being all the time above the critical value. This takes us into the region where there is no distinction between liquid and vapour, so that by first reducing the pressure and then lowering the temperature we pass without any break to a sub- stance in the truly vaporous state at N, the substance at no time having been in the state of two distinct phases. In what has been said above as to the condition of the substance in the various regions of the diagram, it has been assumed that stable states only are under consideration. If we disregard this restriction, then we may have, for example, liquid water in the region BOC, for water may easily be cooled below its freezing point without actually freezing, and exist as liquid at points to the left of O. Such super- cooled water has a vapour-pressure curve which is the continuation of the curve OA, and has been represented in the diagram by the dotted line OA'. This curve lies above the vapour-pressure curve for ice, so that at any given temperature below the freezing point the vapour pressure of the supercooled liquid is greater than the vapour pressure of the solid. This rule holds good for all substances, and we find in 100 CHAP. INTRODUCTION TO PHYSICAL CHEMISTRY general that the vapour pressure of the stable phase is less than the vapour pressure of the unstable phase. It should be noted that the instability in such examples is only relative. A supercooled liquid may be kept for a very long time without any solid appearing (cp. Chap. VII.), but as soon as the smallest particle of the substance in the more stable solid phase is introduced, the less stable, or, as it has been called, the metastable phase is transformed into it. That the metastable substance should have a higher vapour pressure than the stable substance is not surprising, if we consider that the phase of higher vapour pressure will always tend to pass into the phase of lower vapour pressure when the two substances are allowed to evaporate into the same space, although they are not themselves in contact. The vapour of the substance of higher vapour pressure will on account of that higher pressure diffuse towards the substance of lower vapour pressure and there condense. More of the metastable phase will then evaporate in order to restore equilibrium between itself and the vapour, with the result that there will again be diffusion and condensation on the stable phase until all the metastable phase has been thus indirectly converted by evaporation into the stable phase. The vapour at pressures and temperatures represented by points on the line OA' is in an unstable state with regard to the solid ice, being supersaturated, although it is only saturated with regard to the supercooled liquid. It is likewise possible to supersaturate vapour at temperatures above the freezing point, i.e. to have the substance in the state of vapour in the region COA; and also to have a liquid substance in the region AOt by superheating. Water, for example, if free from dissolved gases, may be heated to 200° or over at the atmospheric pressure without boiling. It has always been found impossible, on the other hand, to heat a solid above its melting point. Water in the form of ice has never been observed in the region COA. We shall next proceed to the consideration of a substance capable of existence in more than three phases, taking sulphur as our ex- ample. Here we consider not only the liquid and vaporous phases, but the two solid phases of rhombic and monoclinic sulphur. Rhombic sulphur is the crystalline modification usually met with, and this on heating rapidly melts at 115°. If we keep it at a temperature of 100°, however, for a considerable time, we find that it becomes converted into the other modification, monoclinic sulphur. This latter on heating does not melt at 115° but at about 120°, in accordance with the general rule that each crystalline modification of a substance has its own melting point. If the monoclinic sulphur be cooled to the ordinary temperature, it gradually passes again into the rhombic modification. We should be inclined, therefore, to say that at the ordinary temperature rhombic sulphur is in a stable state, while monoclinic sulphur is in a metastable condition. At 100° the reverse X 101 THE PHASE RULE : is the case here monoclinic sulphur is the stable variety, and rhombic sulphur the metastable variety. We have seen that in the case of solid and liquid there is a temperature at which both phases are stable together, namely, the melting point. Above or below this temperature only one of the phases is stable. We should therefore expect by analogy that there is a temperature at which the two solid phases of sulphur should be equally stable, i.e. should be able to coexist without any tendency. of the one to be converted into the other. Careful experiment has revealed such a temperature. At 95.6°, the transition or inversion temperature, both rhombic and monoclinic sulphur are stable, and can exist either separately or mixed together in any proportions. Below this temperature the monoclinic phase gradually passes into the rhombic phase; above it, the rhombic phase gradually passes into the monoclinic. A transition temperature of this sort is then quite comparable to a melting point, the chief difference being that while a solid can never be heated above its melting point without actually fusing, a substance like rhombic sulphur may be heated above its transition point without undergoing transformation. It is thus possible to investigate the properties of rhombic sulphur up to its melting point, 115°, although between 95.6° and that temperature it is in a metastable condition, and is apt to suffer transformation into the stable monoclinic modification. The transition point, like the melting point, is affected by pressure, and in the case of sulphur increase of pressure has the effect of raising the transition temperature. The transition point of sulphur is most easily determined by means of the change in density which occurs when the sulphur passes from one crystalline modification to the other. The instrument employed is termed a dilatometer, and resembles in construction a thermometer with a large bulb, which is made to contain a mixture of the two kinds of crystals together with a suitable liquid (best one in which the substance is slightly soluble). If the dilatometer is im- mersed in a bath whose temperature is constant and slightly above the inversion point, the liquid in the capillary will rise owing to the monoclinic sulphur which is formed having a larger volume than the rhombic sulphur from which it was produced. If, on the other hand, the dilatometer is immersed in a bath whose temperature is slightly below the inversion point, the level of the liquid will slowly fall owing to the reverse transformation. Thus at 96·1° a slow rise was observed in the case of sulphur, and at 95.1° a slow fall. It was therefore concluded that the transition point is intermediate between these two temperatures, say 95.6°. With some substances the transition takes place so slowly that many hours, or even days, must elapse before it can be determined with certainty if expansion or contraction is taking place. This dilatometric method is very generally available, but may be supplemented by vapour pressure or solubility determinations, the 102 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. temperature at which the same two phases have the vapour pressure or the same solubility being the transition point (p. 106). The transition phenomena may be represented diagrammatically by means of temperature-pressure curves (Fig. 17). The line OB in the figure represents the vapour-pressure curve of rhombic sulphur; OA is the vapour-pressure curve of monoclinic sulphur. These vapour- pressure curves must meet at the transition point, for at that tempera- ture both modifications are equally stable and must have the same vapour pressure. Below that temperature the vapour pressure of the Pressure Á B Rhombic 95.6° B 1159- 151° 1288 atm.C - Monoclinic E 120° A Vapour Temperature FIG. 17. Liquid - metastable monoclinic phase must be greater than that of the stable rhombic phase. The line OA', therefore, which is the prolongation of the line OA, represents the vapour-pressure curve of monoclinic sulphur below the transition point. Above 95.6° rhombic sulphur is the metastable phase, and consequently has the greater vapour pressure. This is represented in the diagram by the dotted line OB', which is the continuation of the line OB. The line OC gives the effect of pressure on the transition point, sloping upwards away from the pressure axis in order to represent rise of transition point with rise of pressure. The point O thus corresponds very closely to the point O of Fig. 16, being like it a triple point at which there is stable X 103 THE PHASE RULE equilibrium of three phases, viz. rhombic, monoclinic, and gaseous. The lines OA, OB, and OC diverging from O represent as before the conditions of equilibrium between pairs of phases, and the dotted lines OA' and OB' similar conditions in metastable regions. It has been already stated that monoclinic sulphur melts at about 120°. At this point also a triple point must exist, for here monoclinic sulphur, liquid sulphur, and sulphur vapour are in equilibrium. The melting point of monoclinic sulphur is raised by pressure, instead of being lowered, as is the case with water. We have therefore three curves intersecting at A, namely, OA, representing the vapour pressure of monoclinic sulphur; AD, representing the vapour pressure of liquid sulphur; and AC, representing the influence of pressure on the melting point. It happens that the lines OC and AC representing the effect of pressure on the transition point and on the melting point of monoclinic sulphur, although both sloping upwards from the pressure axis, meet at a point C, corresponding to a temperature of about 151°. This point C is also a triple point, for at the pressure and temperature which it represents, three phases-rhombic, monoclinic, and liquid sulphur-can coexist in equilibrium. At pressures above this, monoclinic sulphur has no stable existence at any temperature whatever. We have seen that rhombic sulphur may be heated above its transition point to a temperature at which it melts, viz. 115°. The rhombic is here a metastable phase, and so also is the liquid formed by its fusion. We are therefore now dealing with a triple point in a metastable region, represented in the diagram by the point B', which is the intersection of the prolonged vapour-pressure curves OB and DA for rhombic sulphur and liquid sulphur respectively. The dotted curve from B' to C represents the effect of pressure on the "metastable" melting point of rhombic sulphur, this curve being continued in the curve CE for the effect of pressure on the "stable" melting point of rhombic sulphur, the possibility of transition into monoclinic sulphur ceasing at C. The chief features of the diagram may thus be represented as follows, metastable conditions being enclosed in brackets:- BOCE ECAD BOAD OCA BO, (OB') OA, (OA') AD, (AB') OC AC CE, (CB') REGIONS-DIVARIANT SYSTEMS. Rhombic Liquid Vapour Monoclinic CURVES-MONOVARIANT SYSTEMS. • + Rhombic, vapour Monoclinic, vapour Liquid, vapour Rhombic, monoclinic Liquid, monoclinic Rhombic, liquid 104 CHAP. INTRODUCTION TO PHYSICAL CHEMISTRY A с (B') TRIPLE POINTS-NONVARIANT SYSTEMS. • • • · Rhombic, monoclinic, vapour Monoclinic, liquid, vapour Rhombic, monoclinic, liquid (Rhombic, liquid, vapour) The actual behaviour of sulphur is more complicated than the diagram indicates. Sulphur differs from other elements by the liquid phase containing several distinct varieties in equilibrium with each other. Thus monoclinic sulphur melts at 119.25°, and the liquid freezes at the same point immediately after melting. If the liquid, however, is kept for some hours at a temperature just above 120° the freezing point is found to have dropped several degrees. This lower- ing of the freezing point would indicate (p. 58) that a new substance had appeared in the liquid. This substance is a variety of sulphur known as Sπ, and like the other crystalline forms of sulphur is soluble in carbon disulphide. The well-known darkening and increase of viscosity of liquid sulphur on heating to 250°-300° indicates the presence of still another variety, which is known as Su, and yields insoluble plastic sulphur when quickly cooled. Diagrams similar to the sulphur diagram can be drawn for most other substances that exist in more than one crystalline modification. For example, para-azoxyanisole yields a similar figure, although one of the crystalline modifications is in this case a liquid. The two chief triple points are here the points at which the solid crystal passes into the liquid crystal, and where the liquid crystal passes into an ordinary liquid. The first point is usually spoken of as the "melting point of the substance, and the second as its transition point. In the case of a single substance, there is only one point, the triple point, at which any three phases can exist together. On this account, a system consisting of three phases of a single substance is called a nonvariant system, for if we change any of the conditions — here temperature or pressure-one or more of the phases will cease to exist. When the system consists of two phases, it is said to be monovariant, there being for each temperature one pressure, and for each pressure one temperature, at which there is equilibrium. When the system. consists of only one phase, it is said to be divariant, for within certain limits both the temperature and the pressure may be changed arbitrarily and independently. The regions in the diagram therefore correspond to divariant systems; the curves to monovariant systems; and the triple points to nonvariant systems. When the systems considered contain two distinct components, say salt and water, and not one, as in the preceding instances, the phenomena become more complicated; for here, besides temperature and pressure, we have a third condition, viz. concentration, entering into the determination of phases. The liquid phase, for example, may be pure water, or it may be a salt solution of any concentration up >> X 105 THE PHASE RULE to saturation. The phase rule developed by Willard Gibbs furnishes us, however, with general methods for treating such systems theoretic- ally. It states, for instance, that if the number of phases exceeds the number of components by 2, the system is nonvariant. As we have seen, this is true for one component, and it is equally true for two components. With the components salt and water we have a nonvariant system when the four phases, salt, ice, saturated solution, and vapour, coexist. There is only one temperature, one pressure, and one concentration at which the equilibrium of these four phases can take place; the point at which these particular values are assumed is called a quadruple point, and it coincides practically with what we have hitherto called the cryohydric point (cp. p. 60). Again, the phase rule states that if the number of phases exceeds the number of components by 1, the system is monovariant. If, therefore, there are three phases with the components salt and water, a monovariant system will result. Suppose the phases are salt, solution, and water vapour. If we fix one of the conditions, say the temperature, the other conditions adjust themselves to certain definite values. At the given temperature, the salt solution assumes a definite concentration, viz. that of the saturated solution. This solution of definite concentration has a definite vapour pressure, less than that of pure water. By fixing the temperature, therefore, we also fix the concentration and the pressure. Suppose, again, that the three phases are ice, salt solution, and water vapour. Let the concentration of the solution be fixed, and it will be seen that the temperature and pressure adjust themselves to definite values. First, a solution of the given concentration can only be in equilibrium with ice at a certain temperature fixed by the rule for the lowering of the freezing point in salt solutions (cp. p. 58). At this temperature the solution being of a fixed concentration will have a vapour pressure defined by the law of the lowering of vapour pressure in solutions. By fixing the concentration, therefore, we likewise fix the temperature and pressure of equilibrium. If the number of phases is equal to the number of components, the phase rule states that the system is divariant. Let the two phases in our example with two components be salt and solution, and let the temperature be fixed. The pressure and concentration are no longer fixed as in the last case, but may vary in such a way that a given variation in the pressure produces a concomitant variation in the concentration of the saturated solution. It is necessary, of course, that the pressure should be above a certain limiting value in order that the third phase, of vapour, should not appear. That the concentration of the saturated solution, i.e. the solubility of the salt, changes with the pressure has been experimentally ascertained in a number of cases. Sometimes the solubility increases with pressure, sometimes it diminishes, according as the volume of the solution is 106 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. less or greater than the volume of the solvent and dissolved sub- stances separately. If the two phases considered be salt solution and vapour, it is obvious that although the pressure is fixed, the con- centration and the temperature are not thereby defined. An increase of concentration will counteract the effect of a rise in the temperature, so that concentration and temperature may be made to undergo con- comitant variations even though the pressure remains constant. If the number of phases is less than the number of components by 1, the system is then, according to the phase rule, trivariant. In the case of two components, the trivariant system has only one phase, and with our example of salt and water, the salt solution may be taken as the most representative phase, since it contains both com- ponents. If the temperature and pressure are both fixed, we are still at liberty to vary the concentration as we choose, i.e. a change of pressure at the fixed temperature causes no concomitant change in the concentration. Here, then, we meet with the greatest degree of freedom in varying the conditions in the case of two components, as we cannot reduce the number of phases further. An instructive exemplification of the phase rule is afforded by the distillation of two partially miscible liquids. It has been stated on p. 79 that when a mixture of such liquids is distilled, a distillate of definite composition is obtained as long as two separate layers are present, the composition of the layers themselves remaining unaltered. When one of the layers vanishes, the composition of both residue and distillate varies continuously. In the first case, where we have two liquid phases and one vaporous phase, the number of phases exceeds the number of components by 1, so that the system is monovariant. Consequently, if we fix the pressure at the atmospheric value, the temperature of the equilibrium and the concentrations of the three phases are also fixed. The constant temperature is the constant boiling point of the mixture, and the vapour has a constant com- position, as have also the two layers of liquid. As the distillation progresses the only change that occurs is in the proportions in which the three phases are present. If one of the layers of liquid disappears before the other, which will in general happen if the distillation is continued, the number of phases becomes equal to the number of components, and the system is divariant. Fixing the pressure there- fore no longer involves a fixed temperature and fixed concentrations : these are now subject to variation. The variations of the different magnitudes are however not independent, but concomitant, definite compositions of vapour and liquid corresponding to each temperature. With two components we sometimes get diagrams for melting and transition points which closely resemble those obtained for one component, when instead of pressure we substitute concentration and neglect pressure altogether. Thus with the two components paratoluidine and water, we may draw the following diagram X THE PHASE RULE 107 (Fig. 18). On the vertical axis solubilities or concentrations are measured instead of pressures, and on the horizontal axis temperatures are plotted as before. The line BO represents the concentrations of the solutions in equilibrium with solid paratoluidine at different temperatures, i.e. it is the solubility curve of solid paratoluidine in water.¹ If we heat the system above 43°, the solid phase disappears, and a liquid phase takes its place. Now, the liquid phase has its own solubility curve LO, and this must cut the solubility curve of the solid at the point at which the solid melts. This can be 2.60 2.20 1.80 _Solubility__ 1.40 1.00- 0.60 20° B 30° Solid Paratoluidine 40° Liquid Paratoluidine 50° Temperature Fio. 18. 60° 70° shown in the same way as that adopted to prove that water and ice have the same vapour pressure at the melting point, by substituting in this case solubility for vapour pressure. In general, we may say that if two phases are in equilibrium with each other, and one of them is in equilibrium with a third phase, then the second of the original pair will also be in equilibrium with the third phase. We see that there is thus considerable resemblance between the melting point of a substance under its own saturated vapour and the melting point of a substance under its own saturated solution. If we bear in mind that concentration of a solution corresponds to pressure of a gas 1 The solubility curve for the solid is really that of the monohydrate which is formed. when the solid is brought into contact with water. 108 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. (Chap. XVII.), the reason for the resemblance of the solubility and pressure diagrams becomes apparent. The similarity is also observable in the case of transition points. If we consider the two components, sulphur and an organic liquid capable of dissolving it, the general rule referred to in the preceding paragraph teaches us that at the transition point of rhombic and monoclinic sulphur the solubility of rhombic and monoclinic modifica- tions in the solvent must be the same. For if a certain solution is in equilibrium with one of the modifications, it must be in equilibrium with the second also, since at the transition point the two modifications are in equilibrium with each other. We may say shortly, therefore, that the vapour-pressure curves and the solubility curves of two modifications of the same substance cut at the transition point. This actually gives us in some cases a practical method of determining the transition point. Sometimes the transition of one modification into the other proceeds with such extreme slowness that it is almost impossible to observe the transition temperature directly. If, how- ever, we investigate carefully the vapour pressures or the solubilities of the two modifications at different temperatures, we can construct curves of vapour pressure or solubility; these curves will be found to intersect, and the point of intersection may be taken as the point of transition. Formation of New Phases.-Under conditions where a new phase may appear, it does not necessarily follow that it must appear. When a crystalline solid is heated to its melting point, it invariably melts if any further heating is attempted. Here we have the new phase, the liquid, making its appearance as soon as the conditions are such that its stable existence becomes possible. If we cool the liquid, on the other hand, we may easily reach temperatures considerably below its freezing point without any solidification actually taking place. Here the new phase, the crystalline solid, does not appear when its existence becomes possible, but may remain unformed for an indefinite period. We meet with the same reluctance to form new phases at transition points. Rhombic sulphur can exist at temperatures above 95.6°, the transition point into monoclinic sulphur, and the latter may remain for a long time unchanged even at the ordinary temperature, which is far below the transition point into the rhombic modification. New hydrates of well-known substances are constantly being dis- covered since systematic investigations have been directed to their formation, although it is practically certain that conditions compatible with their existence must have previously been encountered in actual work with these substances. Although sodium sulphate in the form of the decahydrate is efflor- escent under ordinary atmospheric conditions, its vapour pressure being greater than the pressure of water vapour in the atmosphere, yet it may, if perfectly pure, remain for a long time in the air without a trace X THE PHASE RULE 109 of efflorescence being observable. On the other hand, there are prob- ably salts which are under the atmospheric conditions capable of taking up moisture to form a higher hydrate, and yet remain unaffected in the air (Chap. XII.). In each case, removal or absorption of water would result in the formation of a new phase, but the tendency to the formation of the new phases is so small that their formation may be delayed or never occur at all. It must be borne in mind that the reluctance to the production of new phases only applies to the first appearance of the new phase. As soon as the smallest particle of it appears, or is introduced from without, its formation goes on steadily and in many cases very rapidly. Supersaturated solutions of sodium thiosulphate, for example, may be kept for years without showing any tendency to crystallise, but if the merest trace of the solid crystalline phase is introduced, the whole mass becomes solid in the course of a few seconds. Similarly water, if perfectly air-free, may be heated to a temperature much above its boiling point, but in such a case, when the smallest bubble of vapour is formed in the interior of the liquid, the whole passes into the new vaporous phase with explosive violence. When ice and a salt (or other substance soluble in water) are brought together at a temperature below the freezing point, there is the possibility of the formation of a new phase-the solution-and this phase generally forms, the temperature then under favourable. conditions falling to the cryohydric point. It is questionable, how- ever, if this is invariably the case; and it seems quite possible that two substances in the solid state might be brought together at a temperature above the cryohydric point without liquefaction taking place. It is evident from what has been said in this chapter that it is not always the phase most stable under the given conditions which actually exists. A metastable phase may exist for an indefinite time without passing into the most stable phase, provided that this latter phase is entirely absent. As soon as the stable and metastable phases are brought into contact, however, the former begins to be produced at the expense of the latter. The transition from the metastable to the stable phase takes place as a rule fairly rapidly, but in some instances the transformation is so slow as to be practically unobserv- able. Strongly overcooled liquids, for example, crystallise with extreme slowness, even after they have been brought together with the stable crystalline phase (cp. p. 57). The crystalline modifications of silica (quartz and tridymite) are at ordinary temperatures so far beneath their temperature of transformation, if such exists, that they show no tendency to reciprocal transformation, and both forms must. be accounted stable. The same holds good for the two forms of calcium carbonate, aragonite and calespar. Yellow phosphorus is at ordinary temperatures metastable with regard to red phosphorus, which is the stable form, yet in the dark it keeps for an indefinite time. 110 INTRODUCTION TO PHYSICAL CHEMISTRY CH. X without undergoing much alteration, even although it may be in contact with red phosphorus. When we inquire what phase will be formed when there is the possibility of formation of several different phases, we find that it is not, as we might be inclined to expect, always the most stable phase that is formed, but rather a metastable phase, which may there- after pass into the stable phase. A substance, then, in passing from an unstable to the most stable phase very frequently goes through phases of intermediate degrees of stability, so that the transformation does not occur directly, but in a series of steps. Liquid phosphorus, for instance, which is itself metastable with regard to red phosphorus, does not on cooling pass into the latter, most stable, modification, but into the metastable yellow phosphorus. Molten sulphur, again, when quickly cooled by pouring into cold water, does not pass directly into the stable rhombic sulphur, but into the comparatively unstable plastic sulphur, which then in its turn undergoes transformation into more stable varieties. It is a common experience in organic chemistry to obtain substances first in the form of oils which afterwards crystallise, sometimes only after long standing. The alkali salts of organic acids, for example, on acidification with a mineral acid in aqueous solution, very frequently do not yield the free acid in the most stable solid form, but as an oily liquid, which crystallises with more or less rapidity. Thus if solid paranitrophenol is dissolved in caustic soda solution, and the solution is then acidified with hydrochloric acid, the paranitrophenol is liberated as an oil which crystallises after a few minutes. If we consider that the least stable phase of a substance has always the greatest vapour pressure, or, if we are dealing with solutions, the greatest solubility, the formation of intermediate metastable phases is not perhaps so remarkable as might at first sight appear. In the above instance of paranitrophenol, the system before acidification con- sists of a liquid phase only, since for our present purpose we may neglect the vapour phase altogether. On acidification, the new phase may not. make its appearance for some moments, owing to the general reluctance exhibited in the formation of new phases, and the new phase which eventually does make its appearance is that which entails least altera- tion in the system, i.e. that which leaves most in the solution. In other words, the more soluble and less stable phase is formed first, the less soluble and more stable phase only appearing as a product of the transformation of the former. For further information concerning the subjects dealt with in this chapter the student may consult : A. FINDLAY, The Phase Rule and its Applications (1918). D. A. CLIBBENS, The Principles of the Phase Theory (1920), CHAPTER XI ALLOYS i THE nature and behaviour of alloys under varying conditions may be taken as a useful application of the phase rule to systems of more than one component with liquid and solid phases only. It is well known that the physical properties of alloys are by no means those which would be calculated from their composition and the corresponding properties of their components. It is therefore natural to infer that alloys in general consist of metals not merely in juxtaposition, forming a more or less uniform mixture, but of metals combined with each other chemically, or forming with each other solid solutions. In all cases alloys are formed of aggregates of crystals, which may be of the same kind when the alloy is truly homogeneous, or which may be mixtures of different kinds, some of them pure metals, some compounds, and some solid solutions. The mechanical properties of alloys, like those of pure metals, depend greatly on the size and orientation of these component crystals. The methods available for the intimate investigation of alloys are microscopical, chemical, and thermal. Quantitative chemical analysis of the ordinary type is used to give the empirical composition of the alloy, but this affords no information as to the nature of the small constituent crystals. These, however, may be detected and differentiated quali- tatively by the use of appropriate chemical agents. Thus a piece of hard-tempered steel will dissolve without residue in dilute acid, whilst the same steel, with the temper drawn by heating and slow cooling, will on treatment with the same acid leave a crystalline residue having approximately the composition represented by the formula Fe,C. The untempered steel therefore contains a crystalline constituent which is not to be found in the tempered steel. Again, grey cast iron when dissolved in concentrated acids leaves a crystalline residue of graphite. A much more effective method of qualitative analysis is found in the microscopic examination of the etched surface of the alloy. A small piece of the alloy is ground flat and finely polished. After treatment with an etching agent for a period ranging from a few seconds to a few minutes, the alloy no longer presents a uniform 111 112 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. surface to microscopic examination by reflected light, but an irregular surface, showing distinctly the outline of the various constituent crystals, if the etching has been properly carried out. Amongst etching solutions commonly used in the microscopic examination of alloys are dilute acids, iodine solution, ammoniacal cuprous chloride, ferric chloride, and in general chemical agents which readily attack metals. These etching materials attack the different crystals to different extents and thus render their presence evident. Even in the case of a pure metal the crystalline structure may often be made apparent by etching the polished surface. The chief thermal method employed in the investigation of alloys is that based on the rate of heating or cooling. If a substance suffers no change of state or of chemical condition on cooling, then if the temperature of the substance is noted at equal intervals of time as it cools, the cooling curve obtained will be perfectly regular. If, on the other hand, any change, whether physical or chemical, occurs accompanied by evolution or absorption of heat, the curve will no longer be regular but will exhibit more or less marked irregularities, for evolution of heat will diminish the rate of cooling, and absorption of heat will increase it. The results of such chemical, micrographic, and thermal investiga- tions are usually expressed in temperature-composition diagrams, the interpretation of which for binary alloys will now be given in a few typical examples. First we shall consider the case of two metals which are completely miscible in the fused state, but form neither compounds nor solid solu- tions on solidification. In general characteristics this case corresponds to that of a salt and water (p. 104), and it may conveniently be studied in the instance of the alloys of lead with antimony, used for accumulator plates, as antifriction metal, etc. On the vertical axis of the diagram (Fig. 19) temperatures are registered, and on the horizontal axis the percentage of antimony in the system. The region marked I is that of liquid mixtures of lead and antimony. The curve AB is the freezing-point curve, or solubility curve, of lead, the curve BC is the freezing-point curve, or solubility curve, of antimony (compare Fig. 8). These two curves cut at B, which corresponds to a cryohydric point in the water-salt diagram, and is in the case of alloys called a eutectic point or point of maximum fusibility. It will be noted. that it represents the lowest temperature, 245°, at which the liquid can exist in a stable condition, and the composition, 13 per cent of antimony, represented by the point is that of the eutectic alloy, i.e. that which melts completely at this minimum temperature. Solidification of any liquid alloy will commence when the temperature falls to such a point that the composition ordinate cuts the curve AB, or BC. If the liquid contains less than 13 per cent of antimony the solid which first separates is lead, if it contains more than this amount the solid which first | 113 XI ALLOYS crystallises out is antimony. If the liquid has originally the eutectic composition, or if by previous crystallisation of excess of either metal it has reached it, lead and antimony crystallise out side by side in a state of very intimate admixture, and the liquid then solidifies as a whole at constant temperature. The curves AB, BC of incipient crystallisation are called the liquidus curves of the alloys. Curves of incipient fusion on the 700 600 500 400 A 326 300 a 200 100 0 II M' n' p'B p IV P 13 20 M 772 M" III V 40 60 Percentage of Antimony FIG. 19.-LEAD-ANTIMONY DIAGRAM. 245 80 C632 C 100 other hand are called solidus curves. A study of the diagram will show what these curves are. Every solidified alloy consists either of the pure eutectic mixture, or of this eutectic mixed with crystals of either lead or antimony. If the solid has the eutectic composi- tion it will when heated liquefy as a whole at the constant tempera- ture of 245°. If the solid has another composition than this, the eutectic portion will still liquefy at this temperature when heated, the lead and antimony melting together in the eutectic proportions at constant temperature until one of these metals is completely liquefied, I 114 CHAP. INTRODUCTION TO PHYSICAL CHEMISTRY when we shall have the liquid eutectic at the "eutectic" temperature mixed with one or other of the solid metals. This corresponds exactly to what would happen if a mixture of ice and salt were heated from a very low temperature to the cryohydric point. When this point is reached the ice and salt melt together in the cryohydric proportions to form the cryohydric solution, and this process of melting will go on at constant temperature until one of the solids has disappeared, when the cryohydric solution will be left in contact with the excess of the other solid, and the temperature may again be raised. The solidus curve of the alloys of lead and antimony therefore consists of a straight line of the constant eutectic temperature parallel to the composition axis. A point in region I of the diagram consists of one phase only. A point in any other region consists of two phases. In II the phases are solid lead and solution, in III solid antimony and solution. In both IV and V the phases are solid lead and solid antimony. It is con- venient, however, to distinguish these regions and conceive them as consisting of eutectic and lead in IV, eutectic and antimony in V. From the diagram the quantities of each phase for a given composi- tion and temperature may easily be calculated. Suppose we are dealing with the point M in region III. This system consists of solid antimony and a liquid having the composition represented by the line ap. If x is the weight of liquid, the total weight ac being taken as unity, then x (ap) is the quantity of antimony in the liquid state, and 1 - x the quantity of antimony in the solid state. But the total quantity of antimony in the system is represented by the line am, so that 1 − x + x (ap) = am, whence x (ap − 1) = am – 1. But since ac=1 we have x (pc)= mc or x= Similarly if we considered a point M' in mc pc am' region II we should have x = As m or m' approaches B it is seen ap' that the proportion of the system in the liquid state becomes greater am' mc until, at B, or becomes unity, and the system is completely ap liquid. pc If we wish to know for the regions IV and V, which are separated from each other by the vertical of the eutectic composition, how much eutectic mixture and how much of lead or antimony in addition to this is contained in the system, we find as above, for all temperatures beneath am' aB the eutectic temperature, x= for the fraction as eutectic if lead is in excess of the eutectic composition, and x = if antimony is in excess. mc Bc The important series of alloys of lead and tin was thought to be similar in type to the series of lead and antimony, but it has been since shown that solid solutions of tin in lead exist (and therefore XI 115 ALLOYS in all probability also of lead in tin), so that the diagram for this pair of metals is somewhat more complicated. The case of solid solutions may be simply studied with the alloys of antimony and bismuth. These metals are chemically similar and iso- morphous, and apparently mix in all proportions both in the liquid and the solid state. The alloys therefore are perfectly homogeneous and each consists of only one kind of crystal which contains both metals. The diagram is given in Fig. 20. For each temperature there is a liquid solution in equilibrium with 700 600 500 400 300 268 200 A 20 II III 772 40 Percentage of Antimony 60 L M 80 S S FIG. 20.-ANTIMONY-BISMUTH DIAGRAM. 632° 100 In a solid solution. Thus at 560° the liquid with the composition AL is in equilibrium with the solid solution AS. A system represented by a point in the region I is entirely liquid, in region III entirely solid. region II, say at m, it is partly liquid of the composition AL and partly solid of the composition AS. The fraction by weight as liquid can be æ calculated as before: xAL + (1 − x)AS = AM, x(AL – AS) = AM – AS MS LS - MS ML. whence aLS = MS or a = the solid being 1 x or LS' LS That is, the quantities as liquid and solid respectively are as the lengths MS to ML, or ms to ml. Similar reasoning may be applied to Figs. 11 and 14 (Chaps. VII. and VIII.). Alloys of gold LS 116 CHAP. INTRODUCTION TO PHYSICAL CHEMISTRY and silver are of this same type, which differs essentially from that previously considered, as there can be no solidification or liquefaction at constant temperature. At the eutectic temperature in the former type three phases coexist, and this fixes the temperature. Here there are never more than two phases in coexistence and these have always different compositions, so that on fusion or solidification there is always change of temperature. It may happen, however, in the case of completely miscible metals that solidus and liquidus show a maximum or minimum as was found for mercuric bromide and iodide (Fig. 11). The alloys of gold and copper afford an instance in point. These metals form a continuous series of 1200' 1100 1063 1000 900 800° A 1772 M 20 Liquidus Solidus 40 60 Percentage of Copper FIG. 21.-GOLD-COPPER DIAGRAM. 80 · 1084 100 homogeneous solid alloys with a minimum at which the liquid and solid have the same composition. The diagram is given in Fig. 21. When the system has the composition given by AM it solidifies at constant temperature, although there is no eutectic in the sense in which the term is used above. On heating a solid mixture, the temperature of incipient. fusion is not constant as is the case with a eutectic consisting of two phases, but varies with the composition. The characteristic form of solidus for a eutectic mixture is a straight line parallel to the composi- tion axis, which is here wanting, the solidus being at the minimum only momentarily parallel to this axis. A third case essentially different from either of those preceding is when the metals form a true chemical compound. This may be exemplified in the alloys of magnesium and zinc, which do not form XI 117 ALLOYS 650 A solid solutions of any sort, but combine to produce the compound MgZn. The diagram of these alloys is given in Fig. 22, and it will be noticed that it closely resembles the solubility diagram of a hydrated salt. Besides the pure metals melting at 419° and 650° we have the definite compound melting at 575°. The liquidus curve consists. of four portions: AB, representing the lowering of the freezing point of magnesium by the addition of zine; BC the lowering of the freezing point of the compound MgZn, by the addition of magnesium; CD the 700° 600 500 400 A 300 200 0 II V 20 I I. Liquid. II. Mg + liquid. III. MgZn, + liquid. IV. Zn + liquid. I B 344° 40 52 60 Percentage of Zinc VI III FIG. 22.-MAGNESIUM-ZINC DIAGRAM. C 575° III 368° D C" VII 80 84.3 E 419° V. Mg + eutectic B. VI. MgZn, + eutectic B. VII. MgZn, + eutectic D. VIII. Zn + eutectic D. 2 IV Ε΄ 97 100 VIII lowering of the freezing point of the compound by the addition of zinc; and DE the lowering of the freezing point of zinc by the addition of magnesium. There are two eutectic points, B and D, at which the compound separates with excess of magnesium on the one hand and with excess of zinc on the other. The following table gives the substances found in the various regions of the diagram:- 118 CHAP. INTRODUCTION TO PHYSICAL CHEMISTRY The following point should be noted with regard to the curve BCD culminating at C, the melting point of the compound MgZn,. In the figure it is shown as a continuous curve with a maximum at C. Now if the compound melted without decomposition, the curves BC and CD should be two independent freezing-point, or solubility, curves cutting each other at a sharp angle at C, just as the curves AB and BC do at the eutectic point. The reason why the two curves merge into each other and form a continuous curve with a maximum at C, the tangent at that point being parallel to the composition axis, is that the compound on melting is partially decomposed, so that the addition of a small quantity of either of the component metals is not the addition of a new substance altogether, but only supplements the quantity of that metal already existing in the fused mass from decomposition. Indeed the bluntness of the apex may be taken as an indication of the extent to which decomposition has taken place on fusion. The hydrates of Fig. 23 must thus be largely decomposed when in the fused state. The solidus here consists of the two eutectic lines A'C", C'E' and the three points A,E,C, corresponding to the metals and the compound MgZn₂. If we take a solid of the composition MgZn,, its point of incipient fusion C is much higher than that for any other composition. A determination of the temperatures of incipient fusion of a number of alloys of different composition is therefore of great value in indicating the presence of a compound by the sudden rise in temperature at which the phenomenon takes place when the composition of the compound is approached. Of course in order to be visible (say in a melting-point tube such as that used in organic chemistry) the fusion must have proceeded to some considerable extent, and therefore the observed temperatures will not lie truly on the solidus lines and points, but somewhat as indicated by the dotted lines of the diagram. These examples indicate the general character of the diagrams for binary alloys where the conditions are comparatively simple. The diagrams may be complicated by the existence of partial miscibility of both liquid and solid solutions, the existence of several compounds of the constituent metals, the instability of certain solutions or compounds above or below a fixed temperature, the existence of metals in more than one modification, etc. Thus the diagrams for the important tin- copper alloys (ordinary bronzes) and for the still more important iron- carbon alloys (irons and steels) are excessively complicated and require. special investigation and study. One circumstance may be noted with respect to certain bronzes and steels they may be tempered, that is, their physical and mechanical properties may be greatly altered according as they are cooled slowly or quickly from high temperatures. If between the ordinary tempera- ture and the high temperature to which the alloy has been heated there exists a transition temperature, above which one condition of the alloy is stable, and below it another, then if the alloy is cooled slowly, XI ALLOYS 119 the transformation during the passage through the transition tempera- ture has time to occur, and the alloy reaches the ordinary temperature in a stable state. If, on the other hand, the alloy is suddenly chilled, the passage through the transition temperature is so rapid that the transformation does not take place, and the alloy reaches the ordinary temperature in the unstable condition which is now practically fixed and permanent. The condition of such a chilled alloy resembles that of the unstable glasses referred to on p. 57. Thus a steel containing more than 0.85 per cent of carbon when heated to a temperature of considerably over 700° and suddenly cooled, is extremely hard and brittle compared with the same steel slowly cooled from the same temperature. The reason is that at temperatures above 700° solid solutions of carbon in iron can exist, but at lower temperatures these solutions are unstable, the whole of the carbon either combining with iron to form the compound cementite, Fe,C, or separating as graphite from the alloys containing more than about 2 per cent of carbon. The steel hardened by cooling still retains its carbon for the most The soft annealed part in the metastable state of solid solution. steel has its carbon in the form of a carbide. The tempering of the hardened steel consists in reheating it to temperatures between 200° and 300°. At these temperatures the transformation from the hard meta- stable state to the soft stable state takes place at such a rate that the extent of the transformation can be easily regulated, and the hardness thus reduced to any required extent. C. H. DESCH, Metallography (1918). G. H. GULLIVER, Metallic Alloys (1919). CHAPTER XII HYDRATES HYDRATED salts present many interesting aspects when viewed from the standpoint of the phase rule. The components here are the anhydrous salt and water, and the number of phases which they form may be very great, each solid hydrate being a phase distinct from the others. Let us take as our first example sodium sulphate in the form of the decahydrate Na2SO4, 10H₂O, and the anhydrous salt Na₂SO4· The solubility curves of these solids have been already given on p. 46, the concentrations being referred to the two components, anhydrous salt and water. The two solubility curves intersect at 33°, i.e. the two solids are at that temperature in equilibrium with the same solution. They must, therefore, according to the rule already given, be in equilibrium with each other, i.e. 33° is the temperature of transition of the decahydrate phase into the anhydrous phase. This may be confirmed directly by heating the decahydrate alone. At 33° it melts, but the fusion is not complete, for besides the liquid phase, a new solid phase, the anhydrous salt, comes into existence. We have therefore at 33° the four phases of decahydrate, anhydrous salt, saturated solution, and water vapour, all in equilibrium. This point is thus a quadruple point, and as the system consists of two com- ponents and four phases, it is nonvariant. Consequently, if we alter the temperature, pressure of vapour, or the concentration of the solution, the equilibrium will be disturbed. If the alteration is only slight and temporary, the equilibrium will re-establish itself; if the alteration is permanent, some of the phases will disappear. Many instances like the above are known. The essential feature is that one hydrate loses water, forming a solution and a lower hydrate or anhydrous salt. When this is the case there is a definite transition temperature from one hydrate to the other, the higher hydrate, i.e. that with the greater amount of water of crystallisation, existing below the transition temperature, and the lower hydrate above this point. Sometimes a hydrate on being heated melts without separation of a new solid phase. An example of this kind is to be found in the 120 CH. XII 121 HYDRATES ordinary yellow hydrate of ferric chloride, Fe₂Cl, 12H2O, already referred to on page 63. This hydrate on heating melts completely at 37°, the liquid having the same composition as the solid. Here, then, we are dealing with a melting point in the ordinary sense, the three phases of solid, liquid, and vapour existing together, but, as in the case of the compound of magnesium and zinc (p. 117), the fusion is here accompanied by partial decomposition, as the flattened maximum indicates. If the system consisted of only one component, the number of phases at the melting point would exceed the number of components by 2, and the system would be nonvariant. But the number of components is 2, and the number of phases at the melting point exceeds the number of components by 1 only, so that the system is monovariant according to the phase rule. This is as much as to say that we are not fixed down to absolutely definite values of temperature, pressure, and concentration for the equilibrium of the solid, liquid, and vaporous phases, but may alter any one of these conditions within limits, the alteration of one being attended by concomitant alterations in the two other factors. For example, we may change the concentration of the liquid by adding one or other of the components to it. Such a change in the composi- tion of the liquid phase will bring about a certain definite change in the temperature of equili- brium and in the vapour 100 pressure. If, on the other hand, we alter the temperature, it will be 80 found that the vapour pressure and the con- centration of the liquid phase will undergo cor- responding variations. The curves of temper- ature and concentration for the hydrates of ferric chloride are given in Fig. 23. In this dia- gram pressure is not 20 considered, and the curves represent equili- -40° brium curves between solid and liquid phases. The line on the left re- presents the temper- atures at which ice is in equilibrium with solu- tions of ferric chloride of various concentrations; it is, in freezing-point curve of ferric chloride solutions (cp. p. 63). 60+ 40 20 Ice 12 aq. A 5 7 aq. J 5 aq. 1 4 aq. 10 15 20 25 Mols. Fe, Cl to 100 Mols. H₂0 FIG. 23. Fe₂ Clo 30 35 short, the The curve 122 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. CAC' gives the equilibrium of solid dodecahydrate with ferric chloride solution; it is the solubility curve of the dodecahydrate. The point C, where it intersects the ice curve, is the cryohydric point, and lies at - 55°. At a temperature of 37° the liquid with which the dodeca- hydrate is in equilibrium has the same composition as the dodecahydrate itself. This temperature may therefore be called the melting point. of the dodecahydrate, and it is the maximum temperature at which the hydrate can exist either by itself or in contact with any solution of ferric chloride. Addition of either water or ferric chloride to the liquid will lower the temperature at which the dodecahydrate will separate from the solution. If we follow the curve for the dodecahydrate to greater concentra- tions, we find that another hydrate may make its appearance at C', which lies at about 27°. It will be seen that this point resembles the cryohydric point C, inasmuch as it is a quadruple point, the four phases being the dodecahydrate, the new heptahydrate Fe,Cl, 7H₂O, the saturated solution, and aqueous vapour. The only difference be- tween the equilibrium here and at the cryohydric point is that the two solid phases are both hydrates, while at the cryohydric point one of the solid phases is ice. The point C'corresponds to the point of intersection in the sodium sulphate diagram (Fig. 5, p. 46). It is the intersection of the solubility curves of the dodecahydrate and the heptahydrate, and therefore represents the transition point of these two phases. An investigation of the curve of the heptahydrate shows that it is of the same nature as the curve of the dodecahydrate. It reaches a maximum as before, the concentration of the solution and the temperature there being the composition and melting point of the hydrate. This curve for the heptahydrate finally cuts the curve of a lower hydrate, and similar curves are repeated until at last the solubility curve of the anhydrous salt is reached. The curve for each hydrate reaches a maximum temperature, which is the melting point of the hydrate, and cuts the curves for other hydrates at temperatures which are transition temperatures. In the preceding instances we have seen how water may be removed from hydrates by continually raising the temperature, solu- tion being at the same time present. Now, it is possible in many cases to remove the water of crystallisation from a hydrate without any solution being formed at all. This can be done most conveniently by placing the hydrate in an evacuated desiccator over a substance such as sulphuric acid or phosphorus pentoxide. The hydrate at a given temperature generates a small, and in most cases measurable, pressure of water vapour. If the pressure of water vapour above the hydrate, is kept beneath this value, the hydrate will lose water and be con- verted into a lower hydrate or the anhydrous salt. In an evacuated desiccator containing phosphorus pentoxide the pressure of water vapour is practically kept at zero, so that the loss of water by the XII HYDRATES 123 hydrate goes on continuously. Copper sulphate in the form of the pentahydrate CuSO4, 5H₂O, for example, gradually loses water under these conditions, and is converted into the greenish-white monohydrate CuSO4, H₂O. The vapour pressure of this hydrate is so small at the ordinary temperature that the salt remains practically unchanged in the desiccator. In this mode of dehydration of a hydrate, we have only three phases coexisting, viz. the higher hydrate, the lower hydrate, and aqueous vapour. The liquid phase is entirely wanting. Now, the system consists of two components, the anhydrous salt and water, so that the number of phases exceeds the number of components only by one. The system is therefore monovariant, i.e. we can change one of the conditions without destroying the equilibrium altogether, the other conditions at the same time undergoing con- comitant alterations. It should be noted that the condition of con- centration is here practically absent, for there is no phase present in which the concentration varies continuously as it does in a solution. The effect of passing from one hydrate to another at constant temperature is seen in the accompanying diagram (Fig. 24), which represents the dehydration of copper sulphate pentahydrate at 50°. The dehydration does not proceed in one step from the pentahydrate to the anhydrous salt, but in three stages, two intermediate hydrates. being formed. Each of these hydrates has its own vapour pressure, which is the minimum pres- sure of water vapour with which the hydrate can re- main in equilibrium at the given temperature; and where two hydrates co- exist, the observed vapour pressure is the vapour pres- sure of the higher hydrate. Pressures have been tabu- lated on the vertical axis, and composition in mole- cules on the horizontal axis. Until the molecule of copper sulphate has lost two molecules of water, the vapour pressure remains constant at 47 mm., after which there is a sudden drop to 30 mm. The first of these values is the pressure of the pentahydrate; the second is the pressure of the trihydrate formed as the first step in the dehydration. This value of the pressure is retained until two more molecules of water Pressure Б Н,О 47 mm. 3 H₂O 30 mm. 4.5 mm. 1 H₂O Composition FIG. 24. O H₂O 124 CHAP. INTRODUCTION TO PHYSICAL CHEMISTRY have been lost, when it sinks suddenly to 45 mm., indicating that the trihydrate has been completely transformed into the monohydrate with a vapour pressure of 4·5 mm. Further dehydration produces no diminution of the pressure until all the water has been lost,. when, of course, the pressure altogether disappears. This method of systematic measurement of vapour pressure during the dehydration. of a hydrate at constant temperature can be used to ascertain the existence of intermediate hydrates, which may not be easily prepared in other ways. If the dehydration were conducted at another temperature than 50°, a similar diagram would result; the values of the pressures for the Pressure B S Solutions Pentahydrate Trihydrate Monohydrate Anhydrous Salt Temperature FIG. 25. A P T M different temperatures would, however, be all higher or all lower than before. As the system with three phases is a monovariant one, the temperature and the pressure may be altered without the equilibrium being destroyed, but to a given alteration of the one there corre- sponds a definite alteration of the other. Each hydrate, therefore, has a vapour-pressure curve precisely like that of liquid water. These curves are represented in the temperature-pressure diagram of Fig. 25 by the lines BM, BT, BP, for the monohydrate, trihydrate, and pentahydrate respectively. The vapour-pressure curve of ice is repre- sented by the line OB, and that of water by the line OA, the point O where these curves intersect being the freezing point, or more correctly the triple point. SP is the curve of vapour pressures of solutions saturated with the pentahydrate at different temperatures. 1 II 125 HYDRATES This curve has a smaller vapour pressure than that of pure water, and consequently cuts the curve for ice at a temperature below the freezing point. Since the solutions become more concentrated as the temperature rises, the lowering of the vapour pressure becomes more marked, and the curve recedes from that of pure water (cp. p. 76). The line SP is the curve for the equilibrium of the three phases, pentahydrate, solution, and vapour. At the point S, where it cuts the ice curve, the three phases are also in equilibrium with ice, so that S represents the cryohydric point for copper sulphate. As the lower hydrates do not seem to exist in contact with ice or an aqueous solution in stable equilibrium, these curves probably do not cut the lines SP or OB at all, unless, indeed, we represent them as all meeting OB at B, the point at which the pressure becomes zero, as has been indicated in the diagram. If the pressure of water vapour does not reach the vapour pressure of the monohydrate, copper sulphate will exist as the anhydrous salt. It can exist then as anhydrous copper sulphate in contact with water vapour at any point in the region beneath MB. If the pressure of water vapour is equal to the vapour pressure of the monohydrate, this salt can exist in presence of the anhydrous salt and water vapour at points on the curve MB. If the pressure is greater than the vapour pressure of the monohydrate, the anhydrous salt ceases to exist, and passes into the monohydrate. The region of existence of the monohydrate is between the lines MB and TB which indicate the pressures at which it can coexist with the anhydrous salt and the next higher hydrate respectively. Similarly, TBP is the region of the trihydrate, BP giving the pressures at which it can coexist with the pentahydrate. The region of this, the highest, hydrate is BPS. The form of this region is different from that of the previous regions, because the penta- hydrate phase can at certain pressures and temperatures coexist with ice. If the pressure of water vapour is increased to values above those given by the vapour-pressure curve PS, some of the vapour may con- dense with formation of a new phase, viz. solution. The region of the existence of solutions is PSOA, bounded by the vapour-pressure curve of the saturated solutions, of ice, and of pure liquid water respectively. The curve SP for the vapour pressure of saturated solutions meets the curve of the "vapour pressure" of the pentahydrate at the transition point P (about 106°), when the pentahydrate is transformed into the trihydrate and saturated solution. The diagram throws some light on the behaviour of hydrated salts when exposed to an atmosphere containing the ordinary amount of moisture. The pressure of water vapour in a well-ventilated labora- tory in this country is about 8 to 10 mm. on the average at 15°, i.e. the atmosphere is about two-thirds saturated with moisture. If the vapour pressure of a hydrate is greater than this amount at the atmospheric temperature, the hydrate will lose water, i.e. will 126 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. 2 effloresce. This is the case, for example, with common washing soda, Na₂CO₂, 10H₂O, which, when exposed to the atmosphere, loses water in the form of vapour, with production of a lower hydrate. If, on the other hand, the pressure of water vapour in the atmosphere is greater than the vapour pressure of the hydrate, the water vapour may condense, and a higher hydrate or a solution may be formed. Thus, if anhydrous copper sulphate, or one of the lower hydrates of this salt, be exposed to the atmosphere, the water vapour will be slowly absorbed with ultimate formation of the pentahydrate, for all the hydrates of copper sulphate have a lower vapour pressure than the pressure of the water vapour usually found in the atmosphere. If calcium chloride or its common hydrate, CaCl2, 6H,O, is exposed to the air, it deliquesces, i.e. forms a liquid phase. Here the vapour pressures of the hydrate and of the saturated solution amount to only 2 or 3 mm. at the ordinary temperature, and are lower than the pressure of water vapour commonly in the atmosphere. The result is that a solution is formed which will become more and more dilute by absorption of water vapour, the process coming to an end when the vapour pressure of the solution is equal to the pressure of water vapour in the atmosphere. All deliquescent substances are very soluble in water. This great solubility is essential, for otherwise the solution could not be of sufficient concentration to have a vapour pressure lower than about two-thirds of the vapour pressure of water at the same temperature, i.e. lower than the actual pressure of water vapour in the surrounding atmosphere. It must be noted that there is an important difference between the "vapour pressure" of a hydrate and the vapour pressure of a single sub- stance which vaporises as a whole, i.e. a system in which the composition of the vapour and of the solid or liquid phase is the same. There is also on the other hand a difference between the "vapour pressure" of a hydrate and that of a solution of a non-volatile substance, inasmuch as the composition of a solution is variable, whilst that of a hydrate is not. In the case of a single substance (one component) or of a solution (two components), if the pressure of aqueous vapour is raised above the vapour pressure value, vapour will condense, and in the second case the composition of the solution will alter. In the case of a hydrate the pressure of water vapour over it may be increased beyond the vapour pressure value without any change occurring. The difference between the hydrate and the single substance is that with the same number of phases there is one component more in the first case, and therefore one more degree of freedom-here freedom to vary the pressure at constant temperature. The difference between the solid hydrate and a solution of the same substance is that, although the number of phases and components in both cases is the same, the composition of the two phases in the case of the hydrate is invariable, so that pressure alone can be varied at constant XII HYDRATES 127 temperature, whereas in the case of solution, if an attempt is made to keep up a constant pressure of water vapour slightly greater than that of the vapour pressure of the original solution, the solution adjusts its composition to the new value of the vapour pressure, i.e. it becomes more dilute and the vapour pressure rises until it corresponds to the new pressure of aqueous vapour, or until the vapour phase disappears altogether by condensation. Here then the pressure is variable at constant temperature, but the variation is always accom- panied by a change in composition of the liquid phase. If the pressure of aqueous vapour over a hydrate is greatly increased above the vapour pressure value, a higher hydrate or a solution may be formed, in which case a new phase is introduced, and a degree of freedom is lost, i.e. at constant temperature the pressure can no longer be varied. If, on the other hand, it is attempted to reduce the pressure below the value of the vapour pressure, the hydrate gives up water and is converted into the anhydrous salt or into a lower hydrate. Here again a new phase is introduced, and to constant temperature corresponds a constant pressure. The vapour pressure of a hydrate is therefore not any pressure of aqueous vapour with which it may be in equilibrium, but the pressure with which it is in equilibrium in presence of its next lower hydrate (or of anhydrous salt in the case of the lowest hydrate). With two components there can only be invariable vapour pressure at constant temperature if two phases are present besides the vapour. Thus, in Fig. 25 BP is the "vapour pressure" curve of the pentahydrate, BT of the trihydrate, and so on, SP and all the lines below it representing the pressure values for the contact of two phases. It was stated on p. 100 that the least stable form of a substance, at a given temperature, has always the greatest vapour pressure. This statement, it should be borne in mind, was applied only to systems of one component, i.e. to substances which vaporise as a whole, and not to two-component systems such as the hydrates of non- volatile salts, which afford the vapour of only one of the components. The vapour pressure of a stable hydrate may be greater than the vapour pressure of an unstable hydrate at the same temperature. Thus in the example of sodium sulphate, the "vapour pressure" of the unstable heptahydrate at 15° is less than the "vapour pressure of the stable decahydrate at the same temperature. Here each hydrate is in contact with the anhydrous salt as well as with water vapour. If the hydrates are in contact with their saturated solutions, then the vapour pressure of the unstable system is always less than the vapour pressure of the stable system, for these pressures are the vapour pressures of the solutions, and the more concentrated solution in equilibrium with the unstable form (cp. Fig. 5) has always the lower vapour pressure. "} CHAPTER XIII THERMOCHEMICAL CHANGE A CHEMICAL change is almost invariably attended by a heat change, the latter being generally of such a nature that heat is given out during the progress of the action. Vigorous reactions are accompanied by con- siderable evolution of heat; in feeble reactions, on the other hand, the heat evolution is comparatively small as a rule, and in some cases gives place to heat absorption. In special circumstances there may be neither evolution nor absorption of heat, but instances of this kind are practically confined to the reciprocal transformation of optical isomerides. From the fact that vigour of chemical action frequently goes hand in hand with heat evolution, it was at one time thought that measuring the amount of heat evolved in any given action was tanta- mount to measuring the chemical affinity of the substances taking part in the action; but this point of view has of late years been given up owing to practical difficulties in reconciling it with the facts, and to a general advance in our theoretical knowledge of the subject. If heat evolution were to be taken as an accurate measure of chemical affinity, there is an obvious difficulty in explaining why certain changes take place with absorption of heat, since this would correspond to a negative chemical affinity, and there would therefore be no reason, chemically speaking, why the action should take place at all. By introducing the heats of attendant physical changes in a somewhat arbitrary way, it was found possible to explain away the exceptions, but the explanations were in many instances so laboured that it became expedient to drop the rule altogether, in the strict sense, and be content with the recognition of a general parallelism between the amount of heat evolved in an action and the readiness with which it takes place. The amount of heat change attendant on a chemical change is perfectly definite in ordinary circumstances, and is easily susceptible of exact measurement. A gram of zinc when dissolved in sulphuric acid will always occasion the same heat development if the conditions of the chemical action are the same. If the conditions are different, the 128 CH. XIII THERMOCHEMICAL CHANGE 129 thermal effect will also be different. Thus it is necessary in the first place to ensure that in each case exactly the same chemical action occurs. The action of zinc on sulphuric acid differs according as the acid is concentrated or dilute. In the former case, zinc sulphate, sulphuretted hydrogen and sulphur dioxide are the chief products, in the latter case zinc sulphate and hydrogen. These are essentially different chemical actions, and evolve different amounts of heat for a given quantity of zinc dissolved. But even if we ensure that the only products are zinc sulphate and hydrogen, there will still be a difference in the heat development if the sulphuric acid in two cases is at different degrees of dilution. The difference here, however, will be slight, and may for most purposes be neglected. Again, a difference in the temperature at which the action takes place will occasion a difference in the heat evolution; but in this case also the difference is comparatively slight, and negligible for small variations of temperature. Lastly, if the zinc and sulphuric acid form part of a voltaic circuit, as in a Daniell or a Grove cell, the heat evolution is then very different from what it is if the chemical action is not accompanied by the generation of an electric. current. From the standpoint of the conservation of energy, these phenomena are easily understood. Each substance, under given conditions, pos- sesses a certain definite amount of intrinsic energy, so that if we are dealing with a system of substances, a definite amount of energy is associated with that system as long as it remains unchanged. If it now changes into another group of substances, each of these will have its own intrinsic energy, and the new system will in general have a dif- ferent amount of energy from that of the original system. Suppose the second system has less energy than the first. From the law of con- servation, it is plain that the difference of energy between the two systems cannot be lost, but must be transformed into some other kind of energy. Now the energy difference between two systems is usually accounted for as heat, and in our example the heat evolved during the solution of zinc in dilute sulphuric acid measures the difference of the intrinsic energy of the zinc and dilute sulphuric on the one hand, and hydrogen and dilute zinc sulphate on the other. If pure sulphuric is taken instead of a mixture of sulphuric acid and water, the second. system is now sulphur dioxide and zinc sulphate, mostly in the solid anhydrous state-a system which has quite a different amount of intrinsic energy associated with it from hydrogen and dilute aqueous solution of zinc sulphate, so that the energy differences (and therefore the heat evolved) are widely divergent in the two cases. When the zinc and sulphuric acid form part of a galvanic cell, the initial and final systems are the same as above, so that there is the same energy difference as before; but now all the energy does not pass into heat, some of it being transformed into electric energy, which takes the shape. of an electric current passing outside the system. The consequence K 130 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. is that much less heat is obtained by the solution of the zinc in this case than was obtained when no electric current was generated. Dealing now with smaller heat effects, we find that the intrinsic energy of a system is not the same at one temperature as it is at another; for if we wish to raise the temperature we must supply energy in the form of heat to the system, the quantity supplied depend- ing on the heat capacity of the substances which compose the system. In changing from one system to another, therefore, at different tempera- tures, different amounts of heat will be evolved, for in general the heat capacities of the two systems will be different. If we dilute a solution of sulphuric acid or of zinc sulphate, we find that a heat change accompanies the process, and as this heat of dilution is not as a rule the same for two substances, the total thermal effect depends on the concentration of the solutions employed. From the principle of the conservation of energy it follows that if we have in a chemical change the same initial system and the same final system, the same thermal effect will always be produced no matter how we pass from the first system to the second, provided that no other form of energy than heat is concerned in the transformation. Hess, who originally worked this out experimentally, gives the follow- ing numerical example. Pure sulphuric acid was in one experiment neutralised with ammonia in dilute aqueous solution; in other experi- ments it was first of all diluted with varying amounts of water before neutralisation, the heats of dilution and the heats of neutralisation being noted in each case. The experiment resulted as follows:- Mols. Water. Heat of Neutralisation. 0 1 25 5 Heat of Dilution. 0 77.8 116.7 155.6 595.8 518.9 480.5 446.2 Sum. 595.8 596.7 597.2 601.8 The first column gives the number of molecules of water which were added to one molecule of sulphuric acid; the second gives the number of heat units evolved on the addition of the water; and the third gives the number of heat units evolved on the neutralisation of the resulting solution by dilute ammonia. It will be noticed that the sum of the two heats is very nearly the same in the four cases, for in each the starting-point is from pure sulphuric acid and dilute ammonia, and the product is dilute ammonium sulphate. This constancy of the total heat evolved is frequently made use of in thermochemistry for the determination of heat changes not easily accessible to direct measurement. Yellow phosphorus, for example, on conversion into red phosphorus is known to give out a considerable amount of heat, but the direct determination of this amount is a matter of some difficulty. An indirect determination, on the other hand, may be made with the greatest ease. Favre found that when a gram atom of yellow phosphorus is oxidised to an aqueous solution of phosphoric XIII THERMOCHEMICAL CHANGE 131 acid by means of hypochlorous acid, the oxidation is attended by the disengagement of 2386 heat units. A gram atom of red phosphorus in similar circumstances yields 2113 heat units. If now a gram atom of yellow phosphorus were first converted into red phosphorus, and this then oxidised to phosphoric acid, the total heat evolution would be 2386 heat units, since the sum must be equal to the heat evolved in the direct oxidation. But the second part of the action, viz. the oxidation of the red phosphorus, yields 2113 units, so the first stage of the action, viz. the transformation of the yellow into the red phosphorus, must yield 23862113-273 units. We have no means of determining the amount of intrinsic energy in any substance; we can only measure differences between the intrinsic energies of certain substances, or systems of substances. If all substances were mutually convertible, directly or indirectly, we might take one substance as standard, and refer all intrinsic energies to it by means of numbers stating the quantity of energy possessed by the substance in excess of the standard. But chemical substances are not mutually convertible without restriction. In particular, the elements cannot be converted into each other by any means in our power. We are therefore unable to compare the intrinsic energies of the elements together, and so for purposes of calculation we may adopt any value for them that we please. The easiest system is to make the intrinsic energies of all the elements equal to 0, and refer all other intrinsic energies to this value for the elements. If in the equation. 1 Pb+ I2 = Pbl₂, we take the ordinary chemical symbols of the elements and compounds. as signifying the amounts of intrinsic energy in the substances, as well as the quantities of the substances themselves, the equation does not balance, for in the conversion of lead and iodine into lead iodide there is heat evolution, viz. 39,800 cal. for one gram atom of lead, so that the equation to be an accurate energy equation should read Pb+ I2 = PbI₂+ 39,800 cal. 2 The intrinsic energy of a gram molecule of lead iodide is 39,800 cal. less than the sum of the intrinsic energies of the atoms from which it is formed, and is therefore equal to - 39,800 cal., since the sum of the intrinsic energies of the elements is zero. If we actually write the 1 Observations on the radioactive elements, radium, thorium, etc., and their emanations show that a continuous process of disintegration of their atoms is in progress. This process of disintegration, however, is of a very different order from the common chemical actions, the heat evolved for a given material change being millions of times greater than that of the most vigorous chemical action at present known. Such processes, which are entirely beyond our control, will not be considered in this chapter. . 132 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. amounts of energy associated with the various substances, we have the equation Pb + I2 0 + 0 Pbl₂2 - 39,800 cal. + 39,800 cal. Now, the heat given out on the production of lead iodide, or any other substance, from its elements, is called the heat of formation of the substance, and we see from the above instance that this must be equal to the intrinsic energy of the substance with the sign reversed; for on the left-hand side of the equation the sum of the energies is always equal to zero, being the intrinsic energy of elements alone, so that the sum of the energies on the right-hand side must also be zero, and the intrinsic energy of the compound thus equal to its heat of formation with the sign reversed. The heats of formation of compounds from their elements are for this reason very important in thermochemical calculations, their practical use being as follows. If from the sum of the heats of formation on the right hand of an ordinary chemical equation we subtract the sum of the heats of formation on the left hand, we obtain the heat given out or absorbed during the reaction. If the difference has the positive sign, the heat is evolved; if it has the negative sign, the heat is absorbed. If we reverse the signs of the heats of formation, i.e. if we write the values of the intrinsic energies, and subtract the sum on the right hand from the sum on the left, we arrive at the same result. As an example, we may take the displacement of copper from copper sulphate by metallic iron according to the equation Fe + CuSO4, Aq = Cu + FeSO4, Aq; where Aq indicates that the substance to whose formula it is attached is in aqueous solution. The heat of formation of copper sulphate in solution is 198,400 cal. per gram molecule, and of ferrous sulphate under the same conditions 235,600 cal. The two metals have of course no heats of formation. If we subtract, therefore, the heat of formation of copper sulphate from that of ferrous sulphate, we get the heat of reaction required, viz. 37,200 cal. Writing the equation with the values of the intrinsic energies, we have Fe + CuSO = Cu + FeSO 4 0 - 198,400 cal. =0 - 235,600 cal. + 37,200 cal. If we know the heat of a reaction and the heats of formation of all the substances but one concerned in the action, we can calculate the heat of formation of that substance directly from the energy equation. For example, the heat of neutralisation of hydrochloric acid by caustic soda, when both substances are in aqueous solution, is 13,700 cal., the heats of formation of dissolved hydrochloric acid, dissolved caustic soda, and liquid water respectively being 39,300 cal., 111,800 cal., and XIII THERMOCHEMICAL CHANGE 133 I 68,300 cal. If we let z represent the unknown heat of formation of sodium chloride in aqueous solution, we obtain the following equation :- HCl, Aq + NaOH, Aq = NaCl, Aq + H₂O - 39,300 cal. – 111,800 cal. X CH+2O,= = -x+ 0 M whence x = 96,500 cal. When an element like sulphur exists in more than one modification, it is necessary to specify which modification we assume to have zero intrinsic energy, as there is always a heat change in passing from one modification to another. As a rule, the commonest or the most stable variety is taken as the standard, as convenience dictates. Heats of formation of sulphur compounds are generally referred to rhombic sulphur; those of phosphorus compounds to yellow phosphorus. In the case of carbon compounds we seldom deal directly with heats of formation, but rather with heats of combustion, on account of their practical importance, and also on account of the ease with which they can be determined. The heat of formation, however, can easily be calculated from the heat of combustion. We find, for example, that methane has a heat of combustion equal to 213,800 cal., the products. of combustion being carbon dioxide and water. Now, the heat of formation of carbon dioxide from carbon in the form of diamond is 94,300 cal., and of water 68,300 cal. For the heat of formation of methane we have therefore the following equation:- ― CO, + 2H₂O 0 = 94,300 cal.2 x 68,300 cal. +213,800 cal., - 68,300 cal. + 13,700 cal., whence a = 17,100 cal. We see from this that the combustion of methane gives out less heat than we should get by burning the same quantity of carbon and hydrogen as the free elements, and this is true of most carbon compounds. The difference between the heat of combustion of a hydrocarbon and that of the carbon and hydrogen composing it is not as a rule very great, so that a calculation of the latter gives an approximate value for the former. Thus the heat of combustion of the carbon and hydrogen in amylene, C.H10, would be 5 × 94,300 cal. + 5 × 68,300 cal. = 813,000 cal. The heat of combus- tion of amylene vapour was found by direct experiment to be 807,600 cal., a number differing only slightly from the preceding one. When a carbon compound contains oxygen as well as hydrogen, its heat of combustion may be calculated roughly by means of "Welter's rule." According to this rule, the oxygen is subtracted from the molecular formula together with as much hydrogen as will suffice to convert it completely into water, the heat of combustion of the carbon and hydrogen in the residue then giving an approximate value of the heat of combustion of the whole compound. As an 134 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. example we may take propionic acid, CHO₂, whose heat of combustion. has been found by direct experiment to be 386,500 cal. If we sub- tract 2H2O from the molecular formula, we are left with the residue CH₂, the elements of which have the heat of combustion 3 x 94,300 cal.+ 68,300 cal. = 351,200 cal. 2 62 6 It is evident that the approximation is here by no means close, the error in this case being about 10 per cent. A better result may usually be obtained by subtracting the oxygen, not with the corre- sponding quantity of hydrogen, but with the corresponding quantity of carbon, and then estimating the heat of combustion of the elements in the residue. In the above example we subtract CO, from the formula CH₂O₂, and have C₂H, left as residue. This gives the heat of combustion 2 × 94,300 cal. + 3 × 68,300 cal. = 393,500 cal., a much better approximation to the experimental value. As another instance of the two methods of calculation, we may take cane sugar, C12H22O11 By Welter's rule we subtract 11H₂O from the molecule, and for the residue C₁₂ get the heat of combustion 12 × 94,300=1,131,600 cal. By the other method we subtract 5.5CO, in order to dispose of the 11 atoms of oxygen, and obtain the residue 6.5C and 11H₂. The heat of combustion of these quantities of the elements is 6.5 × 94,300 + 11 × 68,300 1,364,200 cal. The value experimentally found is 1,354,000 cal., a number much closer to the second calculated value than to the first. 2 12 = Some hydrocarbons have a greater heat of combustion than that of the carbon and hydrogen contained in them. The heat of combustion of acetylene, for example, is 310,000 cal.; the heat of combustion of the two atoms of carbon and the two atoms of hydrogen contained in its molecule being 2 x 94,300 cal.+ 68,300 cal. = 256,900 cal. This corresponds to a heat of formation of - 53,100 cal., i.e. this amount of heat is absorbed on formation of acetylene from its elements. We have here, then, an example of an endothermic compound formed from its elements with heat absorption, in contradistinction to the bulk of compounds, which are exothermic, i.e. are formed from their ele- ments with evolution of heat. Other common examples of endothermic compounds are carbon disulphide, which is formed with a heat absorption of 28,700 cal., and gaseous hydriodic acid, which is formed with a heat absorption of 6100 cal. Endothermic compounds like these are comparatively unstable, and give out heat on their decom- position. Hydriodic acid gas, for instance, is decomposed by gentle heating; carbon disulphide can be split up into its elements by mechanical shock; and carbon and hydrogen may be regenerated from acetylene by the passage of electric sparks through the gas. The substance hydrazoic acid, or azoimide, N,H, has a large negative heat of formation, which is no doubt closely associated with its extremely explosive properties. XIII THERMOCHEMICAL CHANGE 135 Such endothermic compounds are formed directly from their elements with difficulty, if they can be formed at all. At ordinary temperatures their direct formation does not take place, but if the elements are brought into contact at a very high temperature, then combination may occur. Thus carbon disulphide is formed by passing sulphur vapour over red-hot carbon. Acetylene is produced when carbon and hydrogen are brought into contact at the very high temperature of the electric arc. This behaviour is exactly opposite to what we find with the common exothermic compounds, which are stable enough at ordinary temperatures, but are frequently decomposed by high temperatures. It has already been stated that for thermochemical purposes it is. necessary to specify exactly the condition of each of the substances concerned in the action under discussion. This is not only true for the chemical condition, but also for the physical state of the sub- stances. We must know whether the substances are in the solid, liquid, or gaseous state, or, if they are in the state of solution, in what solvent, and at what dilution. This is so because change of physical state is accompanied by heat change, which must be taken into account in thermochemical investigations. Liquid sulphur, on combining with oxygen to give sulphur dioxide, will not give out the same amount of heat as rhombic sulphur, for the latter on melting absorbs about 300 cal., which must therefore be added to the heat of combustion of the rhombic sulphur. The correction for the difference between the solid and liquid states is often small, and never amounts to more than about 5000 cal. If the substance is in the state of vapour, the heat of vaporisation must be added to the thermochemical data for the liquid. This correction is often considerable, amounting approximately to twenty-five times the boiling point of the substance on the absolute scale (Trouton's rule). Thus the correction for water according to this rule would be 25 × 373=9325 cal., the actual heat of vaporisation at 100° being 9700 cal. The heat of formation of liquid water at the ordinary temperature from oxygen and hydrogen is 68,300 cal., the number we have used throughout in the above calculations. At 100° the heat of formation of liquid water is somewhat less, viz. 67,600 cal. If, now, we want to find the heat of formation of gaseous water at 100°, we must subtract the heat of vaporisation of the liquid, viz. 9700 cal., and thus obtain 57,900 cal. as the heat of formation of water vapour. There is still another circumstance which must be taken into con- sideration when a chemical action is accompanied by the disappearance or formation of gases; or, in general, when the action is accompanied by a great change of volume. Each gram molecule of gas generated performs an amount of work equal to 27 cal., for, as we have seen, the equation pv = RT becomes pv=27 for the gram molecule, and R has the value 2 in calories. This amount of heat, then, is 136 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. absorbed on production of the gas. If, on the other hand, a gram molecule of gas disappears, a corresponding amount of heat is pro- duced in the action. At 27° the actual amount per gram molecule is 2 × (27 + 273) = 600 cal., the value being the same for all gases. This correction is of importance in the case of carbon compounds, which, under ordinary circumstances, are burned at the atmospheric pressure, the volume increasing considerably during the combustion. The actual thermochemical measurement is usually made, on the other hand, in a closed "calorimetric bomb," the volume thus remaining constant. If we consider the combustion of toluene, for instance, we have the following volume relations:- C₁₂Hg + 90₂ = 7CO₂ + 4H₂O ; 8 2 2 9 vols. 7 vols. or, if the substances are all in the gaseous state, C₁Hg + 902=7CO₂ + 4H₂O. -8 1 vol. 9 vols. 7 vols. 4 vols. Each volume in the above equations is the gaseous gram-molecular volume, the volume of the liquid substances being negligible. If both the toluene and the water formed by its combustion are in the liquid state, there is a shrinkage of two volumes on completion of the combustion. If all the substances are in the gaseous state, there is an expansion of one volume. On the supposition that the combustion takes place in the calorimetric bomb at 27° with evolution of m calories, then if the liquid toluene is burned at constant pressure, we shall have a heat evolution of m + 1200 cal. Suppose now that the toluene is burned as vapour at 27° by passing a stream of air or oxygen through the liquid and igniting the mixture at a jet, the water vapour being carried off without condensing; and suppose further that the heat of vaporisation of toluene and water at this temperature are t and w respectively, then we can calculate the heat of combustion under these conditions as follows. To vaporise the gram molecule of toluene, t heat units are absorbed, and this heat is given out again when the toluene ceases to exist as such, so to the heat of combustion of the liquid we must add this heat of vaporisation. But the water obtained in the previous case. was liquid water, in the formation of which from vapour there were evolved w heat units per gram molecule. This amount of heat is not given out if the water remains in the gaseous state, so that from the heat of combustion given above we must now subtract 4w. Finally, there is now an expansion of one volume, so that 600 cal. must be subtracted from the heat of combustion at constant volume. The heat of combustion under the circumstances is therefore m + t - 4w - 600 cal. The apparatus used to measure heats of chemical change is XIII THERMOCHEMICAL CHANGE 137 essentially the same as that used in physics for measuring heat quantities, and in particular the water calorimeter is universally employed. The chemical action is allowed to take place in a chamber immersed in a known amount of water of known temperature, and the change of temperature brought about in this water by the chemical action is noted. As the apparatus itself, viz. vessels, thermometers, stirrers, etc., is heated along with the water it contains, its water equivalent, i.e. the quantity of water which has the same heat capacity as the apparatus, must be determined and added to the quantity of water actually employed in the experiment. This can be done by adding a known quantity of heat to the apparatus and ascertaining the resultant change of temperature in the water of the calorimeter. The chief source of error in such experiments lies in heat exchange with external objects by conduction and radiation. To reduce this error to a minimum, the chemical action must be made to go as fast as possible, and the temperature of the calorimeter must never be allowed to depart greatly from the temperature of the room in which the experiment is made. Conduction is avoided by having the calorimeter surrounded by one or two vessels with stagnant air spaces between them, contact between the vessels being made by a bad heat conductor, such as cork, and reduced to as few points as possible. If the calorimeter is constructed to contain half a litre of water, the heat capacity of the apparatus is small in comparison, and by the use of a thermometer which can measure differences of temperature to a thousandth of a degree, very accurate results can be obtained with a relatively small expenditure of material. The most convenient form for the calorimeter is that of a cylinder, whose height is one and a half times to twice its diameter, so that in many cases an ordinary beaker serves the purpose very well, a larger beaker with a cover being the surrounding vessel. For many thermochemical deter- minations, especially those involving solutions, Dewar tubes and bulbs may be used with advantage as calorimetric vessels. For further information concerning the methods and results of thermo- chemistry, the student may consult― MUIR AND WILSON, Elements of Thermal Chemistry. JULIUS THOMSEN, Thermochemistry. CHAPTER XIV VARIATION OF PHYSICAL PROPERTIES IN HOMOLOGOUS SERIES IN the homologous series of organic chemistry, for example the series of the saturated alcohols, there is a close resemblance in chemical properties amongst the members, so that it is possible to give general methods for the preparation of the substances and general types of action into which they enter. The actual readiness with which the substances are formed or are acted on by other substances may, and usually does, differ from case to case, there being a gradation in chemical activity as successive members of the series are considered. Sodium, for instance, acts on the alcohols with formation of sodium alkyl oxides and hydrogen according to the equation 2ROH + 2Na = 2RONa + H,, • but the vigour of the action is very different according as the alcohol is one high or low in the series. With methyl alcohol CH,. OH and with ethyl alcohol C,H,. OH the action is brisk; with amyl alcohol CH₁₁. OH it is sluggish at the ordinary temperature. 11 Corresponding to this gradation of chemical activity within the series we have a gradation in physical properties, and here, on account of the accuracy with which these physical properties can be measured, the differences are more readily observed and more readily brought under general rules. We may take first for consideration the specific gravities in the series of normal primary saturated alcohols, which are exhibited in the following table (p. 139). The values of the specific. gravity are for 0°, and are referred to the specific gravity of water at 0°. It will be seen that as the molecular weight of the alcohol increases, the specific gravity increases likewise. The difference in composition from step to step is one atom of carbon and two atoms of hydrogen; and to this constant difference, CH2, there corresponds a continually diminishing difference in the values of the specific gravities as the series is ascended. • 138 CH. XIV 139 HOMOLOGOUS SERIES Methyl alcohol Ethyl Propyl Butyl Amyl Hexyl Heptyl Octyl Nonyl of v = ď and V رد در >> 2) }} }} "" "" "" 2) "" Methyl alcohol Ethyl Propyl Butyl Amyl Hexyl Heptyl Octyl Nonyl }} "" >> "} An exception is found in the first member of the series. Reasoning by analogy from the other members, we should expect methyl alcohol to have a considerably lower specific gravity than ethyl alcohol, but instead of this it has a higher specific gravity, the value being inter- mediate between those for the second and third members of the series. The exceptional behaviour of the first member of a series is not confined to this series, or to this property, being of frequent occurrence amongst organic compounds. If instead of considering the specific gravities of the compounds (i.e. the weights which occupy unit volume) we consider the molecular volumes (ie. the volumes occupied by the molecular weights), we are enabled to bring greater regularities to light. The specific volume v is the reciprocal of the density d, and the molecular volume is the product of the specific volume and the molecular weight, or the molecular weight divided by the density. The values 1 M are contained in the following table :— d >> = CH3.OH C₂H. OH CH7. OH CH₂. OH CH₁₁. OH 11 C&H13. OH C₂H15. OH • CH17. OH C9H19. OH Specific Gravity. Difference. 0.812 0.806 0.817 0.823 0.829 0.833 0.836 0.839 0.842 M CH₂OH 32 C₂H₂OH 46 1.241 C₂H₂OH 60 1.224 C₁H₂OH 74 1.215 C&H₂OH 88 1.206 C&H13OH 102 1.201 C₂H15OH 116 1·196 CH₂OH 130 1.192 17 C9H19OH 144 1.188 หน 1.231 Difference. +10 - 17 – 9 9 - 5 10 T I 100 5 هر 4 -·006 +011 +'006 pofiti + ·006 +·004 + '003 + '003 + .003 I Difference. 39.4 57.1 73.4 89.9 106.1 122.5 138-7 154.9 171.1 +17.7 +16.3 +16.5 +16·2 +16.4 +16.2 +16.2 +16.2 140 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. 2 The regularity exhibited by the molecular volumes is much more striking than that displayed by the specific volume or by the specific gravity. Here the difference between neighbouring members, instead of continuously diminishing, remains practically constant throughout the series. The volume occupied by the molecular weight of the alcohol is increased by 162 units for every addition of CH₂ to the molecule of the alcohol. The value 16.2 may therefore be looked upon as the "molecular" volume of CH, under the given conditions, and in this particular homologous series. Under other conditions the value for CH, may be, and is, different. The volume of organic compounds is usually affected greatly by temperature; the coefficient of expansion of ethyl alcohol being, for example, some twenty times. as great as that of water at the ordinary temperature. It is therefore of importance to determine under what conditions the volumes of different compounds are to be compared, more especially when they belong to different series. In the above instances the specific gravities were measured at 0° (and compared with water at 0°). This choice of temperature is evidently arbitrary, bearing no relation to the properties of the compounds themselves, but as far as we have seen. it has the merit of leading to regular results. 2 Kopp found, by studying a great many liquid substances, that if the molecular volume of each was determined at its own boiling point, not only were the regularities within each series preserved, but the same regularity held good for practically all series. No matter what homologous series was studied, Kopp found that a difference of com- position of CH, corresponded to a constant difference in the molecular volume, the value being in his units 22. It must be noted that this volume is not absolutely constant, but is merely an average, the actual differences being liable to slight fluctuations about the mean. The value is greater than that obtained when the homologous compounds are all measured at the same temperature, because, as we shall see, the Thus boiling points in homologous series rise as the series is ascended. the molecular volumes of two neighbouring compounds measured at their respective boiling points will show a greater difference than if they were measured at the same temperature, for the molecular volume of the compound with greater molecular weight is ascertained at a higher temperature than the molecular volume of the substance with lower molecular weight, and there is therefore the expansion between the two temperatures to be added to the value that would be obtained if both were measured at the boiling point of the lower compound. Besides this regularity others come to light. It was found by Kopp that the densities of isomeric compounds (measured at their boiling points) were equal, and consequently that their molecular volumes were also equal under these conditions. For example, he found the following numbers for compounds having the formula C&H1202:- XIV 141 HOMOLOGOUS SERIES Methyl valerate Ethyl butyrate Butyl acetate Amyl formate. Normal primary Iso-primary Tertiary • Acid. Formic Acetic Propionic Butyric Valeric Caproic • C₂H¸. CH₂. CH₂OH (CH3)2CH. CH₂OH (CH3)3C.OH • CH202 C₂H,02 C3H6O2 C4H802 C5H1002 C6H12O2 When the measurements are made at the same temperature, this con- stancy is not displayed if the boiling points of the isomeric substances are widely different. Thus for the butyl alcohols we have, when the densities are all measured at 20° (against water at 4°)— Molecular Volume. 59,000 cal. 213,300 367,900, 522,700, 676,700,, 831,200 · Boiling Point. 117° 107° 83° 15 • The alcohol with highest boiling point has the greatest density, i.e. the smallest volume when the measurements are all made at one temperature. Its higher boiling point, however, allows of greater expansion, so that when the determinations are made at the boiling points, the increase of volume due to the higher temperature to some extent compensates for the smaller original volume at 20°. The choice of the boiling points of the compounds as the temperatures at which the comparisons are to be made, although it involves the properties of the compounds themselves, is still to a certain extent arbitrary, inasmuch as they are the temperatures at which all the substances have an arbitrary vapour pressure, viz. 76 cm. The justification of this choice lies in the fact that the observed regularities under these conditions are great. Another mode of working up the volume data will be found in Chapter XIX. The heat evolved by the complete combustion of an organic com- pound (the carbon becoming carbon dioxide, and the hydrogen becoming water) is an example of a property exhibiting constant differences between neighbouring members of a homologous series when molecular quantities are compared. The following table contains the heats of combustion of gram-molecular weights of the fatty acids expressed in calories: ?) 149.2 149.3 149.3 149.8 Density. 0.810 0.806 0.786 Difference. 154,300 154,600 154,800 154,000 154,500 For each difference in composition of CH, there is a difference in the molecular heat of combustion amounting on the average to 154,300 cal. 142 CHAP. INTRODUCTION TO PHYSICAL CHEMISTRY This difference is found to be practically the same in all homologous series. Thus for the alcohols we have-- Alcohol. Methyl Ethyl Propyl Butyl Amyl Hydrocarbon. C₂H16 CgH18 C9H 20 C10H22 C11H24 C12H26 C13H28 C14H30 C18H32 C18H34 16 t 100.5 125.5 149.5 173.0 194.5 214.5 CH₂O C₂H₂O C3H8O C4H100 C6H12O 234.0 252.5 270.5 287.5 168,500 cal. 324,600,, 481,100 637,600, 793,400,, A property which in general varies regularly in homologous series is the boiling point. As we ascend a simple series the boiling point. invariably rises, but the rise at each succeeding step in general grows smaller and smaller as the molecular weight increases. The following table gives the boiling points of some of the normal saturated hydrocarbons. In the second column under t is the boiling point at 76 cm. in the centigrade scale; under T we have the boiling point in the absolute scale, i.e. t + 273. t (calculated). 100.8 126.1 149.9 172.5 193.8 214.2 234.3 253.0 271.1 288.9 "" "" T 373.5 398.5 422.5 Difference. 446 0 467.5 487.5 507.0 525.5 543.5 560.5 156,100 cal. 156,500,, 156,500,, 155,800,, Difference. 25.0 24.0 23.5 21.5 20.0 19.5 18.5 18.0 17.0 The boiling points of most series exhibit a regularity similar to the above, but the differences are not usually so great as is the case with the hydrocarbons. The boiling points of most members of a homo- logous series can be expressed by a fairly simple empirical formula. M is the molecular weight of the compound, T its boiling point in the absolute scale, and a and b constants for the series, then in general If TaMb. The values under "t calculated" in the above table were obtained by means of a formula of this kind, the constants for the series being XIV HOMOLOGOUS SERIES 143 a = 37.38 and b = 0·5. In some series (e.g. the alcohols, the alkyl bromides, and the alkyl iodides) a formula of this type cannot success- fully be applied, but in most cases it gives accurate results. The student must again be reminded that the boiling point of a series of compounds is a magnitude arbitrarily selected in so far as the pressure under which the compound boils is itself quite arbitrarily chosen equal to the average pressure of the atmosphere. It is found, however, that a formula of the above type is capable of expressing the relation between boiling points and molecular weights under any pressure. In the same series the constant a has different values for different pressures, whilst the constant b retains the same value for all the pressures. Thus the boiling points of the above series of hydrocarbons under 3 cm. pressure may be expressed by the formula T29.68M05, instead of T37.38M0·5 which is valid for 76 cm. The constancy of b for different pressures leads to the following results. If we take two substances belonging to the same series, we have for their boiling points at a certain pressure p T= aM³, and T₁ = aM₂°. 1 1 At another pressure p' we have the boiling points T=a'M', and T₁ = a'M. - 1 M, M, and b remain the same throughout; so, by dividing each equation of the first pair by the corresponding equation of the second pair, we obtain T T a 7 a and T T T a 1 a "} whence That is, if b remains constant for different pressures, the ratio of the boiling points (expressed in the absolute scale) at any two given pressures will remain the same for all members of the homologous series. Transposing the last equation, we have Τ T TT₁ 1 Τ T," i.e. the ratio of the absolute boiling points of two substances belonging to the same homologous series is independent of the pressure. S. Young has shown that the rise of boiling point for an increment of CH, is mainly a function of the absolute temperature of ebullition, and may be expressed for any series with considerable accuracy by the formula A=144-86/700148√T 144 1 INTRODUCTION TO PHYSICAL CHEMISTRY CHAI in which ▲ is the difference between the boiling point T of a substance on the absolute scale and that of the next higher member of the series. The boiling points of isomeric substances are not in general identical, as may be seen, for example, in the case of the butyl alcohols given in the table on p. 141. We usually find, as here, that the isomers containing the longest carbon chain boil at a higher tempera- ture than those with branched carbon chains. Malonic nitrile As a rule, the boiling point of the first member of a homologous series is considerably higher than that calculated from the formula which includes the other members of the series. This abnormally high boiling point is displayed still more markedly when, instead of one characteristic group, the first member of the series has two. Thus, for example, in the simplest series of the dicyano-derivatives, the first member has a boiling point which is actually higher than those of the three succeeding members:- Methyl-malonic nitrile Ethyl-malonic nitrile Propyl-malonic nitrile The glycols behave similarly Ethylene glycol Methyl-ethylene glycol Ethyl-ethylene glycol (CN)2CH2 (CN)2CH.CH3 (CN)2CH.CH,CH, (CN),CH. CH₂CH₂CH¸ Acid (Even). Caproic C6H12O2 Caprylic C8H1602 Capric C10H 2002 Lauric C12H2402 Myristic CH2302 Palmitic C16H22O2 Stearic C18H3602 Arachic C20 H4002 CH₂(OH). CH₂(OH) CH₂(OH). CH(OH). CH¸ CH(OH). CH(OH). CH₂CH, Melting Point. 1.5 +16.5 +31·4 +44 +54 +62 +68 +75 218° 197° 206° - 10.5 +12.5 +28 +40·5 +51 +60 +66.5 216° 197° 188°. 192° The melting points in homologous series often show the pecul- iarity that the substances with an even number of carbon atoms form a regular series by themselves, and those with an odd number of carbon atoms form a regular series by themselves. The following table contains the melting points of the higher fatty acids :- Acid (Odd). 2 2 Difference. - 21 + 9 +10 - 9 + 4 CH₁402 Enanthylic C9H1802 Pelargonic C11H22O2 Undecylic C13H26O2 Tridecylic C15H300 Pentadecylic C17H3402 Margaric C19H38O₂ Nondecylic XIV HOMOLOGOUS SERIES 145 If we take the successive members of the series, we find an alternate rise and fall in the melting point as we pass from one acid to the next. If, however, we separate the acids into those with an even and those with an odd number of carbon atoms, we have a rise in the melting point in each series. It will be observed here again that the differences decrease as we ascend the series. In some homologous series the difference between the homologues with even and those with uneven numbers of carbon atoms is so marked that in the one case we may have the melting point rise, and in the other case fall, as we ascend. The normal saturated dibasic acids afford an instance in point:- Succinic Adipic Suberic Sebacic Acid (Even). Decane-dicarboxylic Dodecane-dicarboxylic C₁H₂O 4 C6H1004 CgH1404 C10H1804 C12H22O4 C₁H2604 Melting Point. 181° 149° 141° 133° 127° 123° 98° 103° 107° S 110° 113° Acid (Odd). C,H₂O₁ Glutaric C₂H₁₂O₁ Pimelic Azelaic 4 C9H1604 C11H2004 Nonane-dicarboxylic C13H04 Brassylic In the even series the melting point falls, in the odd series the melting point rises, as the molecular weight increases. Once more the rise or fall diminishes in magnitude step by step as we ascend the series. 2 4 The student may have observed that in these melting-point tables the lowest members of the series have been omitted. This is so because they do not fall under the general scheme which includes the higher members of the series. For example, the first member of the normal dibasic acids, oxalic acid, C,H,O,, melts at 189°, and the second member, malonic acid, CHO, melts at 133°. It is evident that the melting points are not included in the general scheme which suffices for the other members of the series. In the series of the fatty acids the same irregularity is displayed. Acetic acid, C₂HO₂, which, if the fall observable as we descend the series were maintained to the end, should melt many degrees below zero, in reality melts at + 16.5°. In general, we may say that the lowest member (or members) of a series departs from the regular behaviour exhibited by the higher members amongst them- selves. Exceptions to this rule occur, but they are comparatively rare. 2 4 The separation of the members of a series into those with even and those with odd numbers of carbon atoms sometimes appears in other properties besides the melting points. Thus in the same series of the normal dibasic acids we have the solubility of the odd members in water considerably greater than the solubility of the even members, as the following table shows:- L 146 CHAP. INTRODUCTION TO PHYSICAL CHEMISTRY Oxalic Succinic Adipic Suberic Sebacic Acid (Even). Solubility. C₂H₂O4 8.8 C₁H₂O4 6.9 6 C&H1004 CgH140 0.142 C10H1804 0.01 Decane-dicarboxylic C12H2204 0.003 1.5 140 100 4.5 0.12 0.014 0.004 Acid (Odd). C₁H₁₁ Malonic 4 C3H8O4 Glutaric C₂H120₁ Pimelic C9H1604 Azelaic C11H2004 Nonane-dicarboxylic C13H2404 Brassylic 4 The solubilities are given as parts of acid dissolved by 100 parts of water at the ordinary temperature (15° – 20°). It will be seen that in the separate series the solubility in water falls off very rapidly as the number of carbon atoms in the molecule increases. This is in accordance with what was stated in the chapter on solubility. The solubility of an acid in water is probably connected with the presence in it of the hydroxyl (-OH) or carboxyl (–COOH) group (cp. p. 51). Other things being equal, the greater the proportion of hydroxyl (or carboxyl) in the molecule, the greater will be its solu- bility. As we go up the series the proportion which the hydroxyl bears towards the rest of the molecule diminishes, and along with this goes on diminution of the solubility in water. It is evident from the above tables that in this particular series there is some fundamental difference between the homologues with an even and thos; with an odd number of carbon atoms. This difference is probably to be sought for in some property affecting the substance in the solid state, for both the melting point and the solubility are properties of the solids, i.e. it is the presence of the solid that deter- mines both. The liquid may be supercooled when not in contact with the solid, and the solution may be supersaturated if the solid is not present. But the solid cannot be heated above its point of fusion without melting, and the solution in contact with it is always exactly saturated. It is in this sense that we say that it is the solid that determines the melting point and the solubility—not the liquid or the solution. The analogy that we see between solubility and fusing point in the above tables is an example of a rule of fairly general applicability. We usually find that, when we compare similar substances, fusibility and solubility go together. If we consider a set of isomeric substances, for instance, we find that the order of solubility is usually the same as the order of fusibility, i.e. the most soluble isomer has the lowest melting point. The order of the solubility of isomers is frequently independent of the nature of the solvent. Thus, if one of two isomers is more soluble in water than the other, it will still be the more soluble if XIV HOMOLOGOUS SERIES 147 alcohol, ether, benzene, etc., be the solvents employed instead of water. In some special cases, not only the order but even the ratio of the two isomers remains nearly the same for all solvents. For example, it has been found that meta-nitraniline is, on the average, 1.3 times more soluble than para-nitraniline in 13 different solvents, the ratio of solubility only varying from 1.15 to 1:48. It has also been found that the order of solubility of corresponding salts of isomeric acids is also very frequently the order of solubility of the acids themselves. It must be borne in mind, however, that these rules are all liable to well-marked exceptions. . CHAPTER XV • RELATION OF PHYSICAL PROPERTIES TO COMPOSITION AND CONSTITUTION THE properties of substances, when studied in relation to their com- position and structure, have been divided into three classes. In the first class we have those properties which are possessed by the atoms unchanged, no matter in what physical or chemical state these atoms may exist. Such properties are called additive, and the best in- stance of an additive property is found in weight (or mass). Each atom retains its weight unaltered, whether it exists in the free state or whether it is combined with other atoms. When atoms combine, the weight of the compound is the sum of the weights of the com- ponent atoms. This is only another way of stating one of the funda- mental assumptions of the atomic theory (Chap. II.), the assump- tion, namely, which takes account of the indestructibility of matter. Radioactivity must also be regarded as a purely additive (i.e. atomic) property, since all the evidence collected shows that the radioactivity of a given element is absolutely independent of its mode of com- bination (cp. Chap. XXXII.). No other property is additive in the stri sense, although in the case of some properties there is an app ximation to the additive character. For example the molecular heats of solid compounds are approximately equal to the sum of the atomic heats of their component atoms (p. 28). We We have seen in homologous series that there exists between neighbouring members a difference in molecular volume which is Fractically constant. For a difference in composition of CH, there is the constant difference of 22 in the molecular volume, and this dif- ference retains nearly the same value for all homologous series. may therefore attribute the value 22 to the group CH2, for whenever the methylene group enters a molecule the molecular volume is increased by this amount. Here we are evidently dealing with an additive property, but the additive character is modified by other influences, for the difference 22 is not absolutely constant, but fluctuates slightly about this value. It should be noted, also, that this number only holds good for the liquid state, for the measure- 148 CH. XV PHYSICAL PROPERTIES 149 ments from which it is derived were all made at the boiling points of the liquid substances. By comparing the molecular volumes of liquids differing in composition in various definite ways, Kopp was able to establish a set of approximate rules such as the following: (a) When two atoms of hydrogen are replaced by one atom of oxygen, there is a very slight increase in the molecular volume. (b) One atom of carbon may replace two atoms of hydrogen with- out sensible alteration of the molecular volume. From these rules, in conjunction with the preceding one, we may draw the following deductions :-If the increase of molecular volume for CH, is 22, and if one atom of carbon is equivalent to two atoms of hydrogen, we may assign the value 11 to an atom of carbon, and the value 11÷255 to the atom of hydrogen. These values, then, are assumed to be the atomic volumes of carbon and hydrogen. Since there is a slight increase in the molecular volume when two atoms of hydrogen are replaced by one atom of oxygen which is attached to the same carbon atom, the atomic volume of oxygen must be somewhat greater than 11. On the average it is 12.2. It is found, however, that when hydrogen is replaced by hydroxyl, the increase of molecular volume is not 12.2, corresponding to the addition of one oxygen atom, but only about 7·8. When oxygen, therefore, is attached to one carbon atom, it contributes more to the molecular volume than when it is partially attached to carbon and partially to hydrogen. Here we come across an influence which modifies the additive character of all properties except weight and radioactivity, namely the influence of structure or constitution. The molecular volume is not a purely additive property-it is in part constitutive, i.e. is dependent not merely on the number and kind of atoms in the molecule but also on their arrangement. We must therefore attribute to oxygen two atomic volumes-12.2 when it is attached to carbon so as to form the carbonyl group (CO), and 7.8 when it is part of the hydroxyl group, or is attached to two different carbon atoms, as in the ethers. We are now in a position to deduce the molecular volume of a compound containing only carbon, hydrogen, and oxygen, by adding together the volume values of its constituent atoms. Thus, a com- pound of the formula CɑH₂O.O'a, where O' denotes oxygen in a hydroxyl group, has the molecular volume V = 11a + 5·5b + 12·2c + 7·8d, if the molecular volume is determined at the boiling point of the liquid. For example, in valeric acid, CH,. CO. OH, we have a = 5, b = 10, c = 1, d= 1, so that V = 55+ 55 + 12·2 + 7·8 = 130. The molecular volume as experimentally ascertained is 130-5. agreement in the majority of cases is not so good, e.g. the molecular The 150 CHAP. INTRODUCTION TO PHYSICAL CHEMISTRY volume of ethyl oxalate found from the formula is 161, and that found by experiment 167. In this connection it must be remembered that Kopp's original rules are not strictly accurate, so that the deductions from them are always liable to slight error. From the consideration of compounds containing other elements than those already referred to, values have been deduced for the atomic volumes of nitrogen, the halogens, sulphur, phosphorus, etc. Sulphur and nitrogen, like oxygen, have different values for the atomic volume according to the mode in which they are combined with other atoms. When an element or radical exists in the liquid state, the molecular volume deduced from its compounds in general agrees with that of the free element or radical. Thus the volume of Br, deduced from bromine compounds would be 534, and the molecular volume of free bromine is actually 53.6. The value of NO, deduced from compounds contain- ing it as a radical is 31·5, the value for the free oxide being 32′0. 2 No purely additive property can throw any light on the size of the molecule or its constitution, as each atom retains the numerical value of the property unchanged, whether it exists in the free state or whether it is combined with other atoms in any way whatever. As the value is therefore unaffected by the kind and extent of the combination, it can clearly give no indication as to what that mode or extent of combination may be (cp. Chap. XIX. § 7). When the property is modified by constitutive influences, as is the case with the molecular volume of liquids, it may throw light on the constitution of a substance. Thus, if of two isomeric substances it were suspected that one contained an atom of carbonyl oxygen, whilst the other contained an atom of hydroxyl oxygen, that with the larger molecular volume would be the compound containing the carbonyl oxygen. The refractive power of liquids is a property which, like the molecular volume, is in general additive in character, although modi- fied by constitutional influences. The refractive index itself cannot be taken as a measure of the refractive power when its relation to the chemical nature of the substance is under investigation, for the index varies greatly with temperature, etc. A better measure is found in the specific refractive constant 171 N d or (n - 1)v, where n is the refractive index, d the density, and v the specific volume. This expression varies very slightly with the temperature, and is little influenced by the presence of other substances, so that it has frequently been used in the comparison of different liquids. Another n2 - 1 1 specific refractive constant is given by the expression n² + 2° d' n² - 1 When the n² + 2° values obtained by its aid are compared with the values of (n − 1)v, it is found that they have an advantage over the latter, inasmuch as or v, which was arrived at on theoretical grounds. 2 XV 151 PHYSICAL PROPERTIES they are not only independent of the temperature, but also of the state of aggregation. With the empirical refraction constant there is considerable divergence between the values in the liquid and gaseous states, whilst the numbers obtained for the theoretical constant are the same in both cases. Thus for water at 10° we have Liquid Gaseous (n − 1)v. 0·3338 0.3101 n2-1 n2+2 0.2061 0.2068 · α T 1001' บ. The molecular refractive power is the product of the molec- ular weight into the specific refractive power as measured by either of these expressions. Thus we have the "empirical " molecular refrac- tion, Mv(n − 1), or (n - 1), and the "theoretical" molecular refraction, n² - 1 n² 1 Mv, or V. It is found that the molecular refraction of n² + 2 n² + 2 liquids as measured by either of the formulæ is essentially an additive property modified by constitutive influences, so that an atomic refrac- tion for carbon, hydrogen, oxygen, chlorine, etc., may be calculated. If the refractive powers of each atom in the molecule be added together, the sum gives the molecular refraction of the compound. As is the case with the atomic volumes, different values have to be attributed to the atomic refraction of oxygen, according as it is carbonyl, or hydroxyl, or ether oxygen, a distinction in this case being necessary between the last two kinds which is not required for atomic volumes. When a substance contains an ethylene linkage, its atomic refraction is considerably higher than would be reckoned from the atomic refractions of the elements composing it; and when an acetylene linkage is known to exist in the molecule, the excess is even greater. "Atomic" refractions have therefore been attributed to "double bonds and "triple bonds," which must be added to the atomic refractions of the elements themselves when the total molecular refraction is calculated. }) Some liquids exhibit the phenomenon of optical activity—that is, when placed in the path of a polarised ray they rotate the plane of polarisation in one sense or the other. Substances which rotate the plane of polarisation in the direction of the hands of a watch are said to be dextrorotatory; substances which rotate it in the opposite direction are said to be lævorotatory. The specific rotatory power is usually denoted by the symbol [a], which is obtained by dividing the actual rotation a observed in the polarimeter by the length of the layer of liquid through which the light passes, and by the density of the liquid at the temperature of observation. The length of the tube is usually given in decimetres. The molecular rotation is the product of the specific rotation into the molecular weight, or more usually the hundredth part of this value, i.e. [m] = Ma 100dl 152 CHAP. INTRODUCTION TO PHYSICAL CHEMISTRY 3 where a is the observed rotation, l the length of liquid, and d its density. It has been suggested that a better expression would be [8]=, since we are obviously here concerned not so much with the molecular volume of the liquid as with the average distance between the centres of the molecules, the ray of light passing in a straight line through the liquid, and meeting a number of molecules proportional to the cube root of the molecular volume. Since the rotatory power varies. with the temperature and with the wave length of the light employed, it is necessary to specify both of these in stating the value of the specific or molecular rotation. When we inquire into the nature of liquids and dissolved sub- stances which show optical activity, we find that they possess in almost every case one or more asymmetric atoms of carbon, silicon, tin, sulphur, nitrogen, etc., i.e. atoms which are directly united to elements or radicals all different from each other. If we imagine the four radicals attached to a carbon atom to be at the four corners of a tetrahedron, α d с FIG. 26. V a ♡ C FIG. 27. we find that we can arrange them in two essentially different ways, as is shown in the accompanying figures. Here the tetrahedra are sup- posed to be resting with one face on the paper, the summits being towards the reader. If we call the four different groups a, b, c, d, and place the group d at the summit, then in Fig. 26 the order abc is in the direction of the hands of a watch, and in Fig. 27 in the reverse direction. It will be observed that each of these arrangements is the mirror-image of the other, and that they are not superposable, differing from each other as a right hand differs from a left hand. This want of superposability of object and mirror-image, or enantiomorphism, is a convenient test of asymmetry and in consequence of the possibility of optical activity. If two of the groups in the figures are made the same, the asymmetry vanishes, as we can see if we make a=b, for then the two figures become identical. No altogether satisfactory answer has yet been obtained to the question of what determines the value of the molecular rotation in any particular case. As we have seen, the rotation should vanish if two of the groups become identical, and it may also be perceived from the figures that if we interchange the positions of any two XV 153 PHYSICAL PROPERTIES of the groups, the sign of the rotation will thereby be changed. If we suppose each of the groups to be endowed with some definite property causing the rotation, and denote the value of this function by a, B, y, 8 for the groups a, b, c, d respectively, the rotation of the asymmetric carbon atom must be determined by an expression of the following or similar type:- (a – B)(B − y)(y – 8)(a − y)(a – 8)(B – §). If the function becomes the same for two of the groups, the expression becomes zero, i.e. the rotation vanishes, and if we interchange any two throughout, the sign of the whole expression is changed, i.e. the rota- tion from dextrorotatory becomes lævorotatory, or vice versa. What the function is we do not know. It may be connected with the weight of the radicals, but cannot be the weight itself, since substances are known to be optically active which have two groups of equal weight attached to the asymmetric carbon atom. Certain regularities have been observed among the molecular rotations of members of homologous series, but numerous exceptions occur. Many of the ap- parent divergencies, however, may be accounted for by the assumption that different members of the same series may have different degrees of molecular complexity when in the liquid state (cp. Chap. XX.). Some carbon compounds are known to be optically active although they do not possess an asymmetric atom of carbon or of any other element. In such cases the molecule as a whole (as represented by means of carbon tetrahedra) possesses an asymmetric structure, so that the existence of two opposite modifications is indicated by the stereo- chemical formula. Indeed, it is this molecular asymmetry which is always the determining factor in the appearance of optical activity, the presence of so-called asymmetric atoms being merely special cases of the general principle. The following example indicates the nature of this molecular, as distinguished from atomic, asymmetry. In Fig. 28 is represented a tetrahedral carbon atom, one edge xy being in the plane of the paper, and another ab at right angles to the plane of the paper. If a, b, x, y represent different groups, the carbon atom is asymmetric. Now, consider a compound of the A / type C=C=C X Y In it there is no asym- ቲ W a FIG. 28. 7 B metric carbon atom, in the ordinary sense, as each carbon atom is connected by a double bond with one other, i.e. two of the valencies are satisfied by the same group and are thus equivalent. The stereochemical representation of a compound of this type is given in Fig. 29 and Fig. 30. It is evident from inspection that each of these 154 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. two figures is the mirror-image of the other, and that the two mirror- images are not superposable. There are thus two essentially different ways of arranging the group A, B, X, Y on the three carbon atoms, so that although there is no asymmetric carbon atom in the molecule, the molecule as a whole exhibits enantiomorphism and consequently optical activity. The relationship of this molecular asymmetry to atomic asymmetry may be seen most easily by considering Fig. 29 from a different point of view. Let the carbon tetrahedra represented in the figures be viewed in the direction of a vertical line passing through the middle points of ab and xy. Then A will appear to coincide with a, B will fall on b, X on x, and Y on y, i.e. the system appears to project" into the asymmetric carbon atom abxy. CC The asymmetry, however, is different from that of an asymmetric X X X a A FIG. 29. b B Y رو so that the formula is y ·CH₂ – CH₂- -CH2-CH₂- B a A FIG. 30. t carbon atom, for if we make, say, B=X, the mirror-images will still be unsuperposable and the substances optically active, although the system would "project" in the above manner into a symmetric carbon atom. Optically active compounds of this simple type have not yet been prepared, but a a well- authenticated substance has been investigated in which the double bonds between the middle and the external carbon atoms are represented by the system-CH2-CH2- A B c. The substance, an acid, B Y has no asymmetric carbon atom in the ordinary sense but yet exists in two modifications of opposite optical activity. Some crystalline substances, such as quartz, are optically active, but the activity here is not due to the arrangement of atoms within XV PHYSICAL PROPERTIES 155 the molecule, but rather to a certain arrangement of the crystalline particles. A consequence of this is that the activity disappears when the crystalline structure is destroyed, i.e. when the substance passes into the fused or dissolved state. As a general rule, substances which are active in the liquid or dissolved state are not active when crystalline, but a few substances are known which exhibit optical activity both as crystals and as liquids. The reason why the activity generally disappears on crystallisation is that the crystals are almost invariably optically biaxial, so that there is no direction in the crystal which shows simple refraction, the phenomena of optical activity being obscured by the phenomena of double refraction. A solid "glass," e.g. uncrystallised sugar candy, shows the same optical activity as the fused mass. All liquids when placed between the poles of, a magnet or in the core of an electromagnet become optically active. The character of the magnetic optical activity, however, is essentially different from that exhibited by liquids which are naturally active. If we place a tube. containing a naturally active liquid between the prisms of a polarising apparatus, and so adjust the prisms that no light passes through the system when we place a light-source at one end and the eye at the other; and if we then reverse the positions of the light-source and The the eye, we find that the passage of the light is still obstructed. sense and magnitude of the activity, then, are independent of the direction of the light. A naturally inactive liquid in a magnetic field behaves quite differently. If we first adjust the prisms so that no light passes, and then reverse the positions of eye and light-source, we find that light now traverses the system quite freely. We find, in fact, on readjusting the prisms to darkness, that a substance which appeared originally dextrorotatory is now to an equal extent lævorotatory. The difference in the nature of the activity may best be made clear by analogy. The action of a naturally active liquid resembles the action of a screw. If a screw is right handed (dextro- rotatory) when viewed from one end of its axis, it is right handed when viewed from the other. A naturally active liquid if dextro- rotatory when viewed from one end remains dextrorotatory though the direction of the light passing through it is reversed. The polarity of a magnetically active liquid resembles that of a muff. If the lie of the hair in a muff when viewed from one end of its axis is in the direction of the hands of a watch, it will be in the opposite direction if we view it from the other end of the axis. A magnetically active liquid is similarly dextro- or lævo-rotatory according to the direction in which the light passes through it, the magnetic field being supposed constant. The specific magnetic rotation is usually given as the ratio of the specific rotation of the substance (determined as in naturally active liquids) to that of water under the same conditions: the molecular magnetic rotation is this value multiplied by the molecular weight of the substance and divided by 18, the supposed molecular weight 156 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. of water. We find once more that when the molecular magnitudes for homologous substances are compared, there is a constant difference. for the group CH₂, so that to this extent the property is an additive The constitutive influence, however, is much more marked here than in any of the previous instances; and the property is therefore valuable when applied to solving problems of constitution. one. When a relation has been established between the constitution of well-known substances and the values of a certain property possessed by them, it is legitimate to draw inferences regarding the unknown constitution of other substances from the known value of this property in their case, the same rules being applied as to substances of known constitution. Of course, the worth of the deduction depends entirely on the number and variety of compounds which have been investigated, and on the exactness of the empirical rules established from the in- vestigation. Melting points and boiling points, for example, are occa- sionally useful in indicating the probable structure of a compound, but it is seldom that the evidence based on them alone is to be treated with any degree of confidence. To take an instance in point, it was assumed that the decane-dicarboxylic acid given in the table of melting points on page 145 has the normal structure, entirely on the strength of its melting point falling into the regular scheme displayed by the even members of the homologous series which are known to have the normal structure; for if its structure were not normal, it would in all prob- ability have a melting point diverging widely from the scheme which included the members of the series which were known to be normal in reality. This evidence is slight, but for want of anything better it had a certain validity, although any well-established fact to the con- trary would have been conclusive against it. It is now definitely known that the acid has the normal structure. When the connection between the constitution and the value of a physical property is better marked and susceptible of being laid down in more definite rules, as is the case with the molecular refraction or magnetic rotation, greater confidence can be placed in the conclusions drawn from the value of the property in a particular compound as to the constitution of that compound. Great care, however, must be exercised in certain cases, for we are liable to draw conclusions which are not warranted by the facts. Thus, for example, from a considera- tion of the molecular refraction of aromatic compounds, it has been concluded that benzene has three ethylene linkings in the molecule, i.e. that Kekule's formula HC HC H с с H CH is the correct one. Against this I XV PHYSICAL PROPERTIES 157 we must place the fact that benzene does not behave chemically as if it had a molecule containing three ethylene linkings, and at present the chemical evidence must be held to outweigh the evidence of the molecular refraction. The real difficulty in a case like this is that we can generally express the chemical behaviour of the fatty compounds by means of three different kinds of carbon linking-simple, ethylenic, and acetylenic, all perfectly well defined chemically; whilst in the aromatic compounds we meet with something entirely new, and not readily brought into any of the above classes of carbon linking. How we are best to represent this new kind of linking we do not at present know. Various attempts have been made, mostly based on the supposed properties of the tetrahedral carbon atom-some of them yielding ordinary static formulæ, some of them kinetic formulæ in which the carbon atoms are assumed to be in a state of constant vibration in a specified manner. These formulæ have all their peculiar merits and demerits. In view of the conflict of evidence, none can be held to be entirely satisfactory, but the following formulæ, HC HC H C CH CH HC HC H C H CH CH с H I. Centric Formula. II. Conjugated Formula. which are perfectly symmetrical, best represent in a simple way the properties of benzene. The first formula makes no very definite assumptions as to the mode of union of some of the carbon valencies; the second formula indicates that in benzene there are three pairs of conjugated double bonds, which are different from both simple and ethylenic linkings, the dotted lines being taken to represent the saturation of partial (half) valencies. As we have seen in Chapter XIII., the molecular heat of combustion is essentially an additive property, though subject to modifications conditioned by differences in constitution. The modifications occa- sioned by changes of constitution are often very slight, so that as long as we are dealing with saturated compounds, it is found that isomeric substances have approximately the same heats of combustion. Thus for propyl alcohol we have 498,600 cal., and for isopropyl alcohol 493,300 cal.; for anthracene and phenanthrene, which both have the formula C14H10, we have 1,694,300 cal. and 1,693,500 cal. respectively. In other cases the differences are greater, but when substances of nearly the same character are considered, the differences are not of much importance. Unsaturated compounds exhibit great divergence from saturated compounds, and values have been attributed to the ethylene linkage and to the acetylene linkage. Conclusions, too, have 158 CHAP. INTRODUCTION TO PHYSICAL CHEMISTRY l been drawn as to the constitution of benzene from the heat of combustion. This amounts for benzene, CH, to 787,800 cal., the value for the isomeric substances, dipropargyl, CH: C. CH₂. CH₂. C: CH, and dimethyl- diacetylene, CH,. CC.CC.CH,, being 882,900 cal. and 847,400 cal. respectively. Here the differences for the isomeric substances are great, owing to undoubted differences in the constitution-especially in the mode of linking of the carbon atoms. The actual heat of combustion of benzene agrees well with the assumptions of the conjugated formula. Some properties are exceedingly valuable in certain special cases. for giving us an insight into the constitution of chemical compounds. For instance, if a new hydrocarbon displays optical activity, we know that in all likelihood it contains one or more asymmetrical carbon atoms, i.e. carbon atoms which are combined with four different kinds of atoms or groups of atoms; for practically all well-investigated optically active hydrocarbons have been proved to contain such asymmetric carbon atoms. Again, for organic acids we have the molecular conductivity in aqueous solution (cp. Chap. XXIII.). This property, as we shall see, is closely related to the strengths of the acids, and also to their constitution. The molecular conductivity of acids varies with the concentration of the solution in which they exist, and the actual variation is different according to the acid which is dissolved. There is a general rule, however, to which the variations for nearly all acids conform, and this enables us to calculate for each acid a constant independent of the concentration of the solution whose conductivity is measured. It is this dissociation constant which has proved useful in the discussion of questions regarding the constitution of organic acids. In it there is scarcely a trace of an additive nature in the sense in which the term has hitherto been used. For example, the dissociation constants for the fatty acids are- Formic Acetic Propionic Butyric Isobutyric Valeric Caproic Heptylic H.COOH CH,. COOH C₂H. COOH CзH. COOH C₂H. COOH Oxalic Malonic Succinic Glutaric Adipic CH,. COOH CH₁₁. COOH C8H11 · CH13. COOH 100 k. 0.0214 0.00180 0.00145 0.00175 0.00159 0.00156 0.00147 0.00146 If we except the first number of the series, which as usual diverges from the others, we see that the constants have all approximately the same values, although constant additions are made to the molecule. In the case of the normal dibasic acids of the oxalic series we have a very different table :- (COOH)2 CH₂(COOH)2 C₂H(COOH)2 C₂HG(COOH)2 C₁Hg(COOH)2 100 k. 10.0 (?) 0.163 0.00665 0.00475 0.0037 XV 159 PHYSICAL PROPERTIES Here the values of the constant sink steadily as the molecular weight increases, although the fall becomes less and less as we proceed. In the normal acids the carbon atoms are supposed to be linked to each other in a continuous chain. The two carboxyl groups are there- fore at opposite ends of the chain. Now, in the fatty monobasic acids, in which there is only one carboxyl group, the addition of CH, has little influence on the dissociation constant. We therefore attribute the great diminution of the value in the series of dibasic acids to the increasing distance of the carboxyl groups from each other. In oxalic acid the carboxyl groups are directly attached, and their proximity is assumed to increase the dissociation constant. Each separate group has acid properties, and the two groups reinforce each other's acidity. In a higher member of the series, e.g. adipic acid, it is true that we have still two acid groups, but they are so far apart that we suppose them to have little reciprocal influence in increasing the acid properties. This mode of viewing the relation between constitution and dissocia- tion constant leads on the whole to consistent results where the constitution has been well ascertained. The influence of constitution on dissociation constants may be seen in the hydroxypropionic acids. The presence of the hydroxyl group increases the value of the dissociation constant, and that the more the nearer it is to the carboxyl group. Acid. Propionic a-Hydroxypropionic B-Hydroxypropionic Glyceric Formula. CH3. CH. COOH CH3. CH(OH). COOH CH₂(OH). CH₂. COOH CH₂(OH). CH(OH). COOH H-C-COOH || H-C-COOH The replacement of a second hydrogen atom by a hydroxyl group again raises the constant. Generally speaking the replacement of a hydrogen atom by any of the following groups raises the value of the dissociation constant: OH, SH, Cl, Br, COOH, CN, SCN. Maleïc acid. 100 k=1.17 100 k. ·00145 •0138 ·00311 *0288 2 The so-called geometrical or space isomerism is often accompanied with striking differences in the values of the dissociation constants. From the saturated dibasic acid, succinic acid, COOH. CH2. CH₂. COOH, are derived two unsaturated acids, maleïc acid and fumaric acid, which both in all probability have the structural formula COOH. CH: CH. COOH. We often express the difference between these two acids by writing their formulæ thus: COOH-C-H II H-C-COOH Fumaric acid. 100 k=0·093 the difference corresponding to a supposed difference in the arrange- ment of the hydrogen and carboxyl groups on the carbon tetrahedra. 160 CHAP. INTRODUCTION TO PHYSICAL CHEMISTRY The dissociation constants are much greater than that of succinic acid, and the constant of maleïc acid is twelve times that of fumaric acid. The above mode of formulation might lead us to expect such a difference, for in maleïc acid the two carboxyl groups are on the same side of the central plane, and in fumaric acid on different sides, and consequently more remote from each other. The study of absorption spectra, especially in the ultra-violet region, has shown that the nature of the absorption is closely connected with the chemical structure of organic compounds. The method of work adopted is to photograph the spectra obtained from an iron or tungsten arc (which are rich in lines in the ultra-violet region) through varying thicknesses of a solution of the substance under investigation, and to measure on the photographs so obtained the wave-lengths of the edges of any absorption bands which are superimposed on a general absorp- tion. The spectrum obtained through a very thin layer of the solu- tion may show a faint narrow band, which increases in width as the thickness of the layer increases, until at last with very thick layers it may be completely merged in the general absorption. Absorption curves are drawn by plotting the frequency of the edges of the bands (i.e. the reciprocal of the wave-length) on a horizontal axis and the logarithm of the thickness of layer on a vertical axis. These curves are characteristic, and for substances of the same chemical constitu- tion, closely similar. As an example of the method in its simplest form we may take its application to determining the constitution of isatin. Isatin is capable of reacting as if it possessed one or other of the following formulæ :- NH C&H4 CO = C,H, со Lactam formula. N CO and CH CO Lactim formula. Probably these two substances are in a solution of isatin in equilibrium with each other and capable of reciprocal transformation, but the point of equilibrium may lie very near one end or the other, as is the case with many similar "tautomeric" substances (cp. Chap. XXIV.). Now two methyl derivatives are known, namely NCH 3 C6H4 C.OCH3 со Co which do not pass one into the other. On a comparison of the absorp- tion curve of isatin with those of its methyl derivatives, it is found that there is a close resemblance between it and the curve for the N-methyl derivative, the curve for the O-methyl derivative being altogether different. It is therefore concluded that isatin in solution is, preponderantly at least, the lactam and not the lactim. The occurrence of an absorption band in the spectrum of a com- C.OH N I XV PHYSICAL PROPERTIES 161 pound is often attributed to a manifestation of "residual affinity" existing in the molecule. Fully saturated organic compounds show no banded absorption, but compounds containing "double bonds," or atoms such as those of oxygen or nitrogen, which are capable of assuming a higher valency than is associated with them in the ordin- ary formulation, frequently display well-marked absorption spectra. Here "residual affinities" and "partial valencies and "partial valencies" come into play which are not expressed in the ordinary organic constitutional formulæ. From the purely chemical point of view, the investigation of the rela- tionship between the value of physical properties and the constitution. of chemical compounds is chiefly important as affording a means of ascertaining the otherwise unknown constitution of certain compounds by a determination of their physical constants. Too much reliance, however, should not be placed on this mode of settling chemical constitutions. The interpretation of the results is often doubtful, and that most frequently in cases where the determination of the constitution by chemical methods presents special difficulty. Το sum up, we may say that the physical method usually offers valuable indications of the direction in which a chemical solution of the problem is to be sought, rather than a final solution of the problem itself. It should be borne in mind, however, that the formulæ usually employed to express constitution are felt to be too rigid, and that there is a tendency to replace them by less definite and more flexible formula which aim at a better correspondence with the diverse behaviour of compounds under different conditions. Besides additive and constitutive properties, we have what have been called colligative properties. The numerical value of these properties depends only on the number of molecules concerned, not on their nature or magnitude. For example, if we take equal numbers. of gaseous molecules of any kinds whatever, they will always occupy the same volume when under the same conditions (Avogadro's Law). The gaseous volume then is a colligative property. We make use of these properties in the determination of molecular weights, and they will therefore be further referred to under that heading. S. SMILES, The Relations between Chemical Constitution and Physical Pro- perties (London, 1919). E. C. C. BALY, “Absorption Spectra of Organic Compounds," British Association Report, 1920, p. 222. M CHAPTER XVI THE PROPERTIES OF DISSOLVED SUBSTANCES If we ask ourselves the question: "To which state of aggregation is the state of a substance in solution comparable?" we find that there are only two answers admissible, viz. the liquid state or the gaseous state. It is obvious that when a solid is dissolved in a liquid it at once loses the properties which are characteristic of the solid state. Its particles become mobile, and all properties which depend on regular arrangement of particles disappear. Thus the solid may be a double- refracting crystal; its solution exhibits none of the phenomena of double refraction. It may be an optically active solid, and yet its solution may show no signs of optical activity. In such cases the passage of the substance into solution exhibits considerable analogy to the passage of the substance into the liquid state. A double-refracting crystal almost invariably loses its double refraction when it melts, and most substances which are optically active in the crystalline state are inactive after fusion. The analogy which here holds good between the dissolved and the liquid states might, however, be equally well applied to the dissolved and the gaseous states. It is true that the solution of a substance in a liquid solvent is itself a liquid, but it by no means follows that the state of the substance within the solution is accurately comparable to that of a liquid. Indeed, if we look a little more closely into the matter we find that in the case of dilute solutions, at least, there is far more likelihood of the dissolved substance being in some respects in a condition comparable with that of a gas. One of the characteristic properties of a gas is its power of diffusion. If its pressure (or what is proportional to its pressure, its density or its concentration) is greater at one part of the space con- taining it than at another, the gas will move from the region of higher to the region of lower pressure or concentration, until by this process of diffusion the pressure or concentration is everywhere equalised. This process of diffusion goes on independently of the presence of another gas. A coloured gas such as bromine vapour may be seen to diffuse against gravitation into another gas, say air, until 162 CH. XVI PROPERTIES OF DISSOLVED SUBSTANCES 163 the colour of the contents of the cylinder is everywhere of the same depth. The rate at which the diffusion takes place is, however, greatly influenced by the presence of another gas, the rate becoming rapidly less as the concentration of the other gas increases. This may be easily rendered evident by taking two tall cylinders, one evacuated, and one containing air, and breaking a bulb containing liquid bromine at the bottom of each. In the cylinder containing air the diffusion takes place slowly, an hour perhaps elapsing before the bromine vapour reaches the top of the vessel in quantity. In the evacuated cylinder the diffusion is apparently instantaneous. The particles of the foreign gas thus obstruct the movement of the particles of bromine, and render the process of diffusion slower. Now, in the case of a substance in solution we have the same process of diffusion as we have with gases. If we take a solution of a coloured substance, such as bromine itself, or preferably a coloured salt like copper sulphate, and place it in the bottom of a cylinder, afterwards covering it with a layer of pure water, we find that the colour of the copper sulphate solution gradually rises in the cylinder, proving that the copper sulphate is moving from a region of greater concentration to a region of less or no concentration. In this case also the diffusion is against gravity, for the solution of copper sulphate if concentrated has a much higher specific gravity than water, so that the centre of gravity of the liquid will be raised as the copper sulphate becomes more uniformly distributed throughout it. It is true that the diffusion of a dissolved substance is very much slower than the diffusion of a gas, months or even years elapsing before uniform concentration is attained in a cylinder not more than a foot high. But the difference is only a difference in degree, for it has been shown that gases under great pressures mix with extreme slowness against the action of gravity, many of the characteristic phenomena of the critical point, where the pressure is high, being obscured owing to this cause, unless the substances are mechanically mixed by stirring. Even solids have the property of mixing by diffusion. Thus Roberts-Austen, who kept surfaces of gold and lead in contact during long periods at the ordinary temperature, was able after four years to detect the presence of gold in the layer of lead 7 mm. distant from the surface of contact. This phenomenon is quite in accordance with the conception of solid solutions to which reference has already been made (p. 65). When we come to consider the arrangement of the particles of a substance in dilute solution relatively to each other, we again find a resemblance to the gaseous state, as in the case of chlorine water, for instance. Water at the ordinary temperature takes up about 2.2 times its volume of chlorine. As the dissolved chlorine is uniformly distributed in the solution, the average distance between the chlorine 164 CHAP. INTRODUCTION TO PHYSICAL CHEMISTRY particles in the chlorine water is not very much less than the average distance between the particles in the chlorine gas; and if the chlorine water is only half saturated with chlorine, the distance between the particles is practically the same as the distance between the particles of the gas. So far, then, as the relative position of the particles of a gas and of the same substance in dilute solution is concerned, there is great similarity between the two states; and as the particles of a gas at low pressure have very little influence on each other, so we may suppose that the particles of a substance in dilute solution are mutually independent. If we ask what concentration of a solution is comparable to the concentration of a gas under ordinary conditions, we find that although the concentration is comparatively small, it is still such as may be frequently met with in ordinary laboratory work. A gram molecule of a gas at 0° and 760 mm. occupies 22.4 litres; a solution then containing a gram molecule of dissolved substance in 22.4 litres at 0° will be equally concentrated with a gas under standard con- ditions. This solution in the terminology of volumetric analysis is about twenty-second normal, and solutions twentieth or even fiftieth normal are by no means uncommon in volumetric work. There is therefore an a priori probability that the state of a substance in dilute solution resembles in some respects the state of a gas, and it would not be surprising, therefore, to find that dissolved substances obey laws comparable to the gas laws. In the sequel we shall see that this is actually true. In order to get some idea of the effect of the solvent on a dissolved substance, we shall consider briefly some properties of sub- stances which are easily measurable both for the substances themselves and for their solutions. It is evident that the properties most suited for study in this respect are those which are possessed by the dissolved substance alone and not by the solvent. We get a good example of such a property in the case of optically active liquids dissolved in optically inactive solvents. We can easily measure the specific rotation given by oil of turpentine, say, in the pure state and in various inactive solvents. If the solvent has no influence on the dissolved body, the specific rotation ought to remain the same whether the substance is in solution or whether it is in a state of purity. The specific rotation of lævorotatory oil of turpentine is 3701°. Solutions containing 10 per cent of this substance dissolved in various solvents gave the following specific rotations, as calculated by the formula [a] = where p is the lp' number of grams of active substance in v cubic centimetres of solution (comp. p. 151) :— av Solvent. Alcohol Benzene Acetic acid Specific Rotation. 38.49° 39.45° 40.22° 1 XVI 165 PROPERTIES OF DISSOLVED SUBSTANCES There is evidently here an action of the solvent, for these numbers are all different from the number obtained for the pure substance, and not only are they thus divergent, but they differ also from each other. The optically active liquid alkaloid, nicotine, shows the same behaviour in still more striking fashion. The specific rotation of the pure substance is 168.5°; the specific rotation of a 15 per cent solution in alcohol is 141°, and that of a 15 per cent solution in water is only 81°. Here the specific rotation of the nicotine sinks to less than half its original value when the substance is dissolved in about six times its own weight of water, so that we cannot avoid 25 20% Specific Rotation 15 10t LO 5 100 Ethylic Tartrate 80 in Water in Methyl Alcohol in Ethyl Alcohol 60 40 Percentage of Tartrate FIG. 31. 20 the inference that the water exercises some powerful influence on the nicotine dissolved in it. In the other instances given above the solvents also play an important part in modifying the properties of the substances dissolved in them, although to a less extent than is the case with nicotine; and it will be observed that each solvent exerts its own peculiar influence. The extent to which the rotatory power is affected by the solvent depends on the concentration of the solution. The less of the active substance there is in the solution, the more does its specific rotation diverge from the value for the pure substance. This may be seen in the accompanying diagram (Fig. 31), which gives the specific rotation of ethyl tartrate alone and in solutions of varying concentrations. The effect of water is again specially great, 166 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. the specific rotation in 14 per cent aqueous solution being three times as great as the rotation for the pure ester. It has been suggested that the value of the specific rotation of the same substance in different solvents varies owing to a variation in the molecular complexity (Chap. XX.) of the dissolved substances. Whilst this may be true in some instances, the explanation is not sufficiently wide to cover all the observed facts. A definite regularity appears when we consider the rotations of aqueous solutions of different salts of optically active acids and bases. In order to obtain comparable numbers for the different salts, it is Molecular Rotation 60 55 45ㅏㅏ ​40 35 50 K Να Ba Am 40 Li Camphorates in Water Mg 30 20 Percentage of Camphorate FIG. 32. 10 Ca 0 necessary to work, not with specific rotations, but with molecular rotations, i.e. the products of the specific rotation and the molecular weight. If we take, for example, the more soluble salts of camphoric acid, it appears that the value of the molecular rotation tends towards the same limit in each case as the dilution increases. This is clearly shown in Fig. 32, where the percentage strength of the solution is plotted on the horizontal axis, and molecular rotations on the vertical axis. The values for the molecular rotations are obtained by multiply- ing the specific rotations by the molecular weights, and dividing by 100 in order to get convenient numbers. The curves, which are wide apart at the higher concentrations, come closer and closer together as the strength of solution diminishes, and might ultimately meet in the XVI PROPERTIES OF DISSOLVED SUBSTANCES 167 same point at zero concentration. This final concurrence, however, cannot be established directly owing to the difficulty of obtaining readings of sufficient exactness with very dilute solutions.¹ The same regularity appears with the soluble salts of other optically active acids, such as malic acid, tartaric acid, and quinic acid; and also with the soluble salts of optically active bases, such as the alkaloids quinine, cinchonine, etc. We may say then, in general, that the molecular rotation of the salts of an optically active acid or base always tends to a definite limiting value as the concentration of the solution diminishes. This regularity is known as Oudemans' Law, and we may now attempt its interpretation. Taking the camphorates as our example, we note in the first place that the activity of the salts is due to the acid-the bases, potash, soda, etc., with which the acid is combined, being optically inactive. Indeed, it is only when we have such a combination of active acid with inactive base, or active base with inactive acid, that Oudemans' Law holds good. Now the salts of camphoric acid, to judge from the general run of the curves in Fig. 32, would in very strong solution have quite divergent molecular rotations; and this we know definitely to be the case with the salts of malic acid, some of which in concentrated solution have a positive and some a negative rotation, although their rotations at extreme dilution are all negative. Where the influence of the solvent is least, therefore, the salts have different molecular rotations; where the influence of the solvent is The water greatest, they have nearly the same molecular rotation. therefore tends to bring the acidic, or active, portion of the salts in some way into the same state, for from the similarity of rotation we must argue similarity of condition, as this is no chance agreement, but a general law. This might come about in several ways, but here we shall consider only one possibility. Suppose that the water decomposed the salts into free acid and free base progressively as more and more water was added to the solution. In the concentrated solutions only a relatively small proportion of the gram-molecular weight of the salt will be de- composed, so that the total molecular rotation will be made up of a relatively small rotation for the free acid, and a relatively large rotation for the acid still bound up with the base in the form of salt. As the different salts have different rotations, their concentrated solu- tions will differ considerably in their molecular rotations, as the total rotation is here due mostly to the salt. It is quite different with dilute solutions in which, on our assumption, the salt is almost entirely decomposed into free acid and free base. Here the total rotation is due almost wholly to the acid in each case, the undecomposed salt 1 The salts of a-bromosulphocamphoric acid have a very high rotatory power, and it is therefore possible to investigate them in centinormal aqueous solutions. At this dilution the molecular rotations of the soluble salts are identical. 168 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. contributing very little to the total; so that no matter what the salt is, the value of the molecular rotation will be the same. The assumption, then, of the progressive decomposition of the salt into acid and base under the influence of the solvent water would thus account for the phenomena of molecular rotation in dilute solutions; but it is on other grounds an improbable one, and fails to explain the numerical value of the limiting molecular rotation for the following reason. A consideration of the curves of Fig. 32 shows that if they intersect on the line of zero concentration at all, it will be at a value of about 39°. This value, therefore, would be on the above assumption the molecular rotation of camphoric acid in dilute solution; but the value actually found for a solution of camphoric acid of 0.6 per cent strength is 93°, an altogether different number. We must therefore conclude that the supposition of a decomposition into acid and base by the water is untenable, and endeavour to find another explanation. It will be seen that the essence of the explanation attempted above is the assumption. of the independence of the inactive basic and the active acidic portion of the salt in dilute solution. If on any hypothesis of the action of the water we can suppose that the active negative part of the salt is removed from the influence of the positive part of the salt as the dilution increases, we have a sufficient explanation of the constancy of the molecular rotation in weak solutions, and it only remains to select such a hypothesis as shall conform with the other properties of dilute solutions, and give a rational account of the numerical value of the limiting rotation. We shall meet with such a hypothesis in a later chapter. When we deal with properties which are possessed by solvent and dissolved substance alike, it is more difficult to ascertain definitely if the property of the dissolved substance has been changed by the fact of its having passed into solution. As a rule we find that the sum of the values of the property for substance and solvent is not the value for the solution. For example, the volume of solvent and solute is never exactly equal to the volume of the resulting solution, the pro- cess of solution being usually accompanied by contraction in volume. When water and alcohol are mixed in equal measures there is a con- traction of about 3 per cent of the total volume. This shrinkage may be due to a change in the specific volume of the water or the alcohol, or both, so that it is by no means easy to apportion the total volume change between the two substances. Even when we take a solution and dilute it by adding more of the solvent, we generally find that a change of volume occurs, a contraction on dilution being the usual result (cp. Chap. XIX. § 7). There is an undoubted regularity in the density of aqueous salt solutions. If we consider, for example, the density of normal solutions. of a number of salts, we find that the difference in density between a chloride and the corresponding bromide is constant; that the difference XVI 169 PROPERTIES OF DISSOLVED SUBSTANCES between a chloride and the corresponding sulphate is constant; in short, that the difference between corresponding salts of two acids is approximately constant, no matter what the base is with which the acids are combined. On the other hand, we find that the difference in the densities of equivalent solutions of corresponding salts of two bases is always the same, and independent of the acid with which they are united. Examples are given in the following table, where the densities are those of normal solutions :- Cl K 1.0444 1.0157 NH Difference NO3 Cl Difference 0.0287 K 1·0591 1·0444 0.0147 NH, K Na Ba ¿Ca AA4940 Mg YZn Br 1.0800 1.0520 Cu Ag 0.0280 0.0288 NH4 1·0307 1·0157 0.0150 Na 1·0540 1.0396 0.0144 I 1.1135 1.0847 From a consideration of this table, it is evident that we can obtain the density of the normal solution of any salt by adding to the density of a salt chosen as standard two numbers, or moduli, one of which is characteristic of the base of the salt, and the other characteristic of the acidic portion of the salt. This regularity is known as Valson's Law of Moduli. Valson chose ammonium chloride as his standard, because its normal solution had the smallest density of any of the salts he investigated. The moduli for the principal series of salts are given below :- 0.0000 0.0296 0.0235 0.0739 0.0282 0.0221 0.0410 0.0413 0.1069 S04 1·0662 1.0378 Cl Br I NO 3 50+ 0.0284 Sr 1.0811 1·0667 0.0144 NO 3 1·0591 1.0307 0.0284 Ba 1·1028 1·0887 0.0141 0.0000 0.0370 0·0733 0.0160 0.0200 If these moduli are added to the density of normal ammonium chloride solution, viz. 10153, the densities of the other normal salt solutions are obtained, the values holding good for 18°. Thus, if we wish to know the density of an equivalent normal solution of copper sulphate, we add to 10153 the modulus of copper plus the modulus of the sulphates, viz. 0.0413+00200, and obtain 10766 as the result, in good accordance with the experimental number. Not only is the law of moduli valid for normal solutions, but also for solutions of other (moderate) concentrations, the moduli being multiplied by the concentration of the solution expressed in terms of a normal solution as unity, and then added to the density of the 170 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. ammonium chloride solution of corresponding strength. We have the following values for the densities of various ammonium chloride solutions, which can be used as standards: Normality. 1 2 3 4 Density. 1.0153 1.0299 1.0438 1.0577 Should we wish to know the density of a thrice normal solution of calcium bromide, we add to 1.0438 three times the sum of 0·0282 and 0.0370, viz. 0.1956, and obtain 1.2394, the value actually found by experiment being 1·2395. It is found that other properties of salt solutions besides the density are susceptible of similar treatment by the method of moduli. One salt solution is taken as standard, and from the value for it the values of the others may be obtained by adding the modulus for the acidic portion and the modulus for the basic portion of the salt under consideration. It will be observed that, strictly speaking, the optical rotations of dilute salt solutions are treated by means of moduli. If we take an inactive salt as standard, we can get the value of the rotation of any salt by adding together a modulus for the acidic and a modulus for the basic portion of the salt. Thus if we compare in the case of the camphorates dilute solutions of equivalent strengths, we find that their rotations are all practically the same, and equal to what we may term the modulus for the acidic portion of the salt, the moduli for the different inactive basic portions being all equal to zero. From what has been said above, it will appear already that the properties of a substance in general undergo an alteration when the substance is dissolved in a liquid. The question as to whether the process of solution is to be regarded as chemical or physical has been much discussed, and this alteration in the value of well-defined properties has led many chemists to the conclusion that solution must be classed amongst the chemical processes. The readiness with which many substances crystallise from aqueous solution in the form of hydrates naturally indicates the existence of compounds of those substances and water in the aqueous solution itself. It is extremely difficult, however, to ascertain to what extent such hydrates exist in solution, or what their composition actually is. There can be no reason- able doubt that the dissolved substance and the solvent frequently react on each other so as to influence each other's properties, but at present. we are without any general theory as to the origin or nature of this influence, and even without empirical regularities, except in a few special cases, to enable us to say in any given instance how the influence is most likely to become apparent. On the other hand, if we look upon dilute solutions with respect to volume, pressure, and XVI PROPERTIES OF DISSOLVED SUBSTANCES 171 temperature relations as affecting the dissolved substance, we find we can often neglect the influence of the solvent altogether, and obtain simple laws of the most general applicability, as will be shown in the succeeding chapters. T. S. PATTERSON, "Influence of Solvents on the Rotation of Optically Active Compounds," Journal of the Chemical Society (1901), 79, 167, 477; (1902), 81, 1097, 1134; and subsequent years. E. W. WASHBURN, "Hydrates in Solution," Technological Quarterly (1908), 21, 360. CHAPTER XVII OSMOTIC PRESSURE AND THE GAS LAWS FOR DILUTE SOLUTIONS WE have seen in the preceding chapter that in some respects there is considerable analogy between the state of a substance existing as gas and the state of a substance in dilute solution. In the present chapter it will be shown that the analogy is more than superficial, and that laws exist for substances in dilute solution which are quite comparable with the simple laws for gases. These gas laws (cp. Chap. III.) connect together the volume, pressure, and temperature of the gaseous substances to which they apply, so that we have first to consider what we are to understand by the volume, pressure, and temperature of a substance in solution. The temperature of a substance in solution is evidently the temperature of the solution itself. The volume of a gas we take to be the volume in which the substance as gas is uniformly distributed. Now the volume in which a dissolved. substance is uniformly distributed is the volume of the solution, so that this volume corresponds to gaseous volume. There still remains to find for solutions the analogue of gaseous pressure. In the case of gases, the pressure we consider is that on the walls of the containing vessel, and may be measured directly. With substances in solution it is different here the pressure on the walls of the containing vessel is not the pressure of the dissolved substance, but is the gravitational pressure of solvent and solute combined. If we could exactly counteract the force of gravitation, there would be no pressure of the liquid on the walls of the vessel at all. There is therefore for substances in dilute solution no obvious magnitude corresponding to gaseous pressure; and yet, until this magnitude was discovered, no progress was made in the theory of dilute solutions. An experiment with gases will serve to show in what direction the analogue of gaseous pressure might be sought. In the case of the liquid solution we have two substances, and we wish to estimate the pressure of one of them. Can we in the case of a mixture of two gases find a method for measuring directly not only the total pressure of the two gases, but the partial pressure contributed by one of them? 172 CH. XVII OSMOTIC PRESSURE AND THE GAS LAWS 173 a There is a theoretical method by means of which we can do this, and experiments have been made which go far to confirm the theory. Suppose that of two gases, A and B, one of them, B, can pass through a certain diaphragm, whilst A cannot. Let the gas A be enclosed in a vessel made of the material through which A cannot pass, and let the vessel be connected with a manometer which will measure the pressure of the gas within it. Suppose that the original pressure of A within the vessel is half an atmosphere, and let the vessel and its contents be immersed in the gas B, whose pressure is maintained steadily at one atmosphere. The gas B by supposition can pass freely through the material of which the vessel enclosing A is constructed, and it will do so until its pressure inside the vessel is equal to its pressure outside the vessel, viz. one atmosphere. For, if there is to be equilibrium between the gas inside the vessel and the same gas outside the vessel, there must be no difference of pressure of B throughout the whole space which it occupies, for the gas A exerts no appreciable influence on B. Inside the vessel there is now total pressure of one and a half atmospheres, one atmosphere being the partial pressure of B, and half an atmosphere being the partial pressure of A, which has remained constant at its original value, owing to the impermeability of the vessel to this gas. Outside the vessel we still have the pressure of one atmosphere, so that the internal pressure registered when equilibrium occurs is half an atmo- sphere greater than the pressure outside the vessel. The excess of pressure inside is evidently due to the gas A which cannot pass through the diaphragm, so that, by taking the difference in pressure on the two sides of the diaphragm, we obtain the partial pressure of the substance to which the diaphragm is impermeable. It is not an easy matter to get a diaphragm which is quite permeable to one gas, and quite imper- meable to another; but palladium at a moderately high temperature fulfils the conditions fairly well. Palladium has the property of absorbing hydrogen at ordinary temperatures, and parting with the absorbed gas again when heated in a vacuum to temperatures above 100°. It exhibits this behaviour with regard to no other gas, so that at temperatures of about 200° it forms a diaphragm permeable to hydrogen, but impermeable to gases such as nitrogen, carbon monoxide, or carbon dioxide. Experiments have been made with one of these gases inside a palladium tube, and an atmosphere of hydrogen at known pressure outside the tube. Theoretically, one would expect the internal pressure to increase by an amount equal to the external pressure of hydrogen. This was found to be nearly but not quite the case, the actual increase amounting in different experiments to from 90. to 97 per cent of the theoretical increase. Still the result is close enough to show that this method of measuring the pressure due to one substance in a mixture might be applied in other cases with success. Let us now deal with a liquid solution, say a solution of cane 174 CHAP. INTRODUCTION TO PHYSICAL CHEMISTRY sugar in water. If we could procure a semipermeable diaphragm of the proper kind, i.e. one which would be permeable to water and impermeable to sugar, we might be in a position to ascertain the pressure in the solution due to the presence of the sugar, and find how it varied with the concentration of the solution, the temperature, etc. The botanist Pfeffer, while working at the osmotic pheno- mena in vegetable cells, prepared various membranes which proved to be perfectly permeable to water, and impermeable to some sub- stances dissolved by the water. Such membranes had been pre- viously discovered by Moritz Traube, who, however, did not give them such a form as to permit of accurate work being done with them. They are essentially precipitation membranes, and their formation can. be easily studied on a small scale. If a solution of copper acetate is added to a solution of potassium ferrocyanide, a chocolate-brown precipitate of copper ferrocyanide is produced; and if the two solu- tions are brought together very carefully, mechanical mixing being avoided, the precipitate assumes the form of a fine film or membrane separating the two, and impermeable to both the dissolved substances. The experiment may be carried out in the following fashion, which was proposed by Traube. A piece of narrow glass tubing about 6 inches in length is left open at one end, and closed at the other by means of a piece of rubber tubing provided with a clip. Into this tube a few drops of a 2.8 per cent solution of copper acetate are sucked up, by compressing the rubber beneath the clip with the fingers and then releasing it. The tube is now lowered into a test- tube containing a few cubic centimetres of a 2.4 per cent solution of potassium ferrocyanide. If the liquid in the inner tube forms a plane surface at its mouth, which can be secured by a slight movement of the rubber tubing, the copper ferrocyanide is deposited as a fine transparent film which closes the opening of the tube. That diffusion of the dissolved salts is prevented by this membrane is evident from the fact that the membrane remains transparent and of excessive tenuity for a very considerable period, showing that the copper and potassium salts no longer come into contact. A substance such as barium chloride, which is easily recognisable in small quantities, may be added to one of the membrane-forming solutions, best to that which. is to be placed in the inner tube, and it will be found that even when gravitation would aid the mixing, none of the barium chloride passes through the septum. Whilst these experiments show the possibility of existence of membranes permeable to a liquid solvent and not to certain substances which might be dissolved in it, they are of no use for an investi- gation into the pressure exercised by dissolved substances, for the films are so delicate as to be ruptured by very slight pressure or mechanical disturbance. Pfeffer solved the problem by depositing such films in the pores of unglazed vessels of fine earthenware, such as those XVII OSMOTIC PRESSURE AND THE GAS LAWS 175 employed in experiments on the diffusion of gases. This he did by placing one of the membrane-forming solutions in the inside of the porous pot and the other solution outside. The two solutions gradually penetrating the wall from opposite directions, at last meet in the interior of the wall, and there deposit a semipermeable film across the pores in which they meet. The film, although as delicate in this as in the former case, is now capable of withstanding a much higher pressure, on account of the support which the material of the If the film is to be porous cell affords it (cp. Chap. XIX. § 2). exposed to high pressures, great precautions have to be taken in its preparation, so as to avoid all possibility of rupture of the membrane; but if it is merely desired to show the phenomena qualitatively, it may be done simply as follows. The most convenient form of porous vessel to use is that of a bulb provided with a neck into which a rubber stopper may be inserted. These bulbs are used in gas diffusion experiments, and need no further preparation than washing and soaking for a day in running water. The neck of the bulb is dried and coated inside and outside with melted paraffin wax, which is allowed to solidify. A solution of copper sulphate (2.5 grams per litre) is introduced into the bulb up to a level above the bottom of the paraffin coating, and the bulb is then placed in a beaker, into which is poured a solution of potassium ferrocyanide (2·1 grams per litre) until the bulb is immersed up to the neck. After standing for some hours the bulb is taken out of the solution, emptied, and rinsed with water. If now the bulb is filled with a solution of sugar, and placed in pure water, the water will pass into the interior through the semipermeable membrane. This is best seen by inserting into the neck a well- fitting stopper through which passes a length of narrow glass tubing open at both ends. As the water passes into the interior of the bulb, the solution rises in the narrow glass tube, and the pressure on the inner surface of the semipermeable diaphragm increases. Owing to the resistance the water experiences in passing through the fine pores of the bulb the process is a slow one, but in the course of an hour a rise of several inches may be noted, and in twenty-four hours the solution may have risen from six to ten feet in the tube. As a rule, the membrane prepared in this way without any special precautions being taken, breaks down when the pressure much exceeds ten feet of water, and the level of the liquid no longer rises. į With a perfect membrane capable of resisting high pressures it becomes a question when the increase of pressure within the bulb will come to an end. Pfeffer devoted his attention to this question, and attained the following results. For a given solution the pressure rises slowly until a certain maximum value is reached, after which the pressure remains constant. This maximum pressure, called by Pfeffer the osmotic pressure, varies with the nature of the dissolved ! I 176 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. substance. The following table contains the maximum pressures in centimetres of mercury which are observed for one per cent solutions of the undernoted substances: Cane sugar. Dextrine Potassium nitrate Potassium sulphate Gum. Concentration. 1 2 2.74 4 • 6 Concentration. 0.80 1.43 3.3 · • The pressure was found to be very nearly proportional to the concen- tration of the solution, as may be seen from the following table for cane sugar, concentrations being given in percentages, and pressures in centimetres of mercury :- ... 101.6 151.8 208.2 307.5 For potassium nitrate the ratios are less constant : Pressure. 130'4 218.5 436.8 Temperature. 6.8° • Pressure. 53.5 13.2 14.2 22.0 47.1 16.6 178 193 7.2 Pressure. 50.5 52.1 53.1 54.8 Ratio. 53.5 50.8 55.4 52.1 51.3 As the concentration increases, the ratio of pressure to concentration here diminishes, but Pfeffer showed that this was really due to the membrane not being perfectly impermeable to potassium nitrate, a small quantity of the salt escaping, especially at the higher pressures, so that the proper maximum was never reached. Ratio. 163 153 133 Temperature also influences the maximum pressure, as the follow- ing results with a one per cent sugar solution indicate :- There is obviously here a regular increase of pressure with rise of temperature. All these experiments were made by Pfeffer in 1877, but it was not until 1887 that van 't Hoff published a theory of dilute solutions which takes them as its experimental basis. The semipermeable membrane furnishes us with a means of directly measuring a pressure due to the presence of the dissolved substance, so that we may take it as the analogue in dilute solutions of gaseous pressure in gases. We are now in possession of all the magnitudes necessary to enable us to investigate the pressure, volume, and temperature relations of substances in dilute solutions, and to compare the numerical results of the investigation with the corresponding relations for gases. The pressure we consider is the osmotic pressure; the XVII OSMOTIC PRESSURE AND THE GAS LAWS 177 temperature is the temperature of the solution; and the volume is the volume occupied by the solution. Pfeffer's results show, in the first place, that the osmotic pressure at constant temperature is proportional to the concentration of the solution, i.e. to the amount of substance in a given volume. In other words, the osmotic pressure of a given quantity of substance is inversely proportional to the volume of the solution which contains it. Here, then, is a law in perfect analogy to Boyle's law for gases: the volume varies inversely as the pressure. The following table for mannitol from recent exact experimental data of Morse and his collaborators shows the accuracy with which in solution Boyle's law may be followed. Grams mannitol in 100 grams water. 1.82 3.64 5.46 7.28 9.10 Weight-normal concentration. 0.1 0.2 0.3 0.4 0.5 Osmotic Pressure at 10°. 2.314 atm. 4.609 6.940 9.209 11.613 "" رر }} >> Osmotic Pressure at 40°. 2.557 atm. 5.107 7.664 10.216 12.804 >> }} "" >> From these numbers it is at once apparent that the osmotic pressure is proportional to the concentration as expressed above. A weight- normal aqueous solution is one in which a gram-molecular weight of solute is dissolved in 1000 grams or 1 litre of water: the customary volume-normal solution is one in which a gram-molecular weight is contained in 1 litre of solution. For very dilute solutions the two definitions practically coincide, but for concentrated solutions they differ considerably. Weight-normal solutions are generally used in calculations for the determination of molecular weights of substances. in solution (cp. Chap. XIX.). Van 't Hoff showed from Pfeffer's data that the osmotic pressure of a given solution is approximately proportional to its absolute tem- perature. A consideration of Morse's data for mannitol shows that the conclusion is in this case strictly valid. If we sum the osmotic pressures at 10° in the table given above we obtain 34.685, and at 40° 38-348. The ratio 38.348/34 685 is 1-106, and the ratio of the corresponding temperatures expressed in the absolute scale, i.e. 313/283, is likewise 1.106. Here, then, we have another law exactly comparable to the law for gaseous substances: If the volume is kept constant the pressure is proportional to the absolute temperature (cp. p. 22, Law 3). From this and Boyle's law may be deduced Gay-Lussac's law for dilute solutions, namely that for constant osmotic pressure the volume of the solution must be proportional to the absolute temperature. Combining these laws we may now say that for dilute solutions (for which the two definitions of normality coincide) the product of osmotic pressure and volume is proportional to the absolute tempera- ture, i.c. we may write the equation pv = RT, N 178 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. where p is now the osmotic pressure, for substances in dilute solution as well as for gases, and it only remains to find how the constant R is related to the corresponding constant for substances in the gaseous state. On p. 24 we calculated the value of this constant for a gram molecule of a gas, and we shall now proceed to evaluate it from Pfeffer's data for a gram molecule of sugar dissolved in water. For a one per cent solution of cane sugar at 0° Pfeffer observed that the osmotic pressure was 49.3 centimetres of mercury. This corresponds to a pressure of 49.3 × 13.59 gram centimetres (cp. p. 3). The gram- molecular weight of cane sugar is 342, and consequently the volume of a one per cent solution containing the gram-molecular weight is nearly 34,200 cubic centimetres. The absolute temperature of the solution is 273, so that we have for the constant- 49.3 × 13.59 × 34,200 273 R: 83,932, a value practically identical with the value obtained for the gas constant, which in the same units we found to be 84,760 (p. 24). We can make the corresponding calculation from Morse's results. for mannitol as follows. By extrapolating the results to 0.01 normal solution (a concentration for which the definitions of normality practi- cally coincide) we obtain the osmotic pressure 0·231 atm. at 10° C. The osmotic pressure in gram centimetres is thus 0.231 × 1033 (cp. p. 3); the volume containing 1 gram molecule of solute is 100,000 cubic centimetres as a close approximation; the absolute temperature is 283. Thus we have R = = 0.231 × 1033 × 100,000 283 = 84,300, again practically identical with the gas constant. In the case of cane-sugar solutions of greater concentration than that of Pfeffer, Morse found that whilst at low temperatures the value of the solution constant is from 6 to 10 per cent higher than the gas constant, at higher temperatures the two constants are identical. Glucose solutions, like those of mannitol, conform closely to the gas laws. So far, then, as pressure, temperature, and volume relations are concerned, the analogy between gases and substances in dilute solution is complete. The identity of the constant in the two cases shows that the osmotic pressure of a substance in dilute solution is numerically equal to the gaseous pressure which the substance would exert were it contained as a gas in the same volume as is occupied by the solution. In fact, if we imagine that the solvent is suddenly annihilated, we should have the osmotic pressure on the semipermeable membrane replaced by a gaseous pressure of equal magnitude. This similarity between gaseous and dissolved substances is of the utmost importance, XVII OSMOTIC PRESSURE AND THE GAS LAWS 179 * for it enables us to transfer to dissolved substances conclusions arrived at from the consideration of the temperature, volume, and pressure relations of gases. For example, it at once enables us to determine the molecular weights of dissolved substances from simultaneous measurements of the temperature, volume, and osmotic pressure of the solution, just as the molecular weights of gases and vapours are deter- mined from similar magnitudes. The accurate measurement of osmotic pressure is an experimental task of the utmost difficulty, and has been attempted by only one or two investigators, so that molecular weights are seldom determined directly in this way. There are, however, other magnitudes, susceptible of easy and exact measurement, which are known to be proportional to the osmotic pressure, and these are now made use of for molecular-weight determinations, as will be shown in a subsequent chapter. A consideration of the diffusion of substances in solution shows that just as in gases we have movement from regions of higher to regions of lower pressure, so in solutions we have movement from regions of higher osmotic pressure to regions of lower osmotic pressure. Osmotic pressure, then, we may take to be the driving force in solu- tions, and if we calculate its value, we find it to be very considerable. Thus the osmotic pressure of a normal solution is over 22 atmospheres (cp. p. 24), or 330 lbs. per square inch. In spite of this high driving power, the process of diffusion in solution is, as we have seen, a very slow one. Nernst has calculated that the force necessary to drive a gram of dissolved urea through water at the rate of 1 cm. per second is equal to a weight of forty thousand tons. The resistance, then, which the water offers to the movement of the dissolved substance is enormous. This we must take to be due to the smallness of the dis- solved particles. A substance in the state of fine dust may take many days to settle, even in a perfectly still atmosphere, while the same weight of substance in the compact state would fall to the ground in as many seconds. The driving force in the two cases is the same, namely, the gravitational attraction of the earth for the given substance, but the resistance which the air offers to the small particles is incom- parably greater than that offered to the compact mass. Osmotic pressure in a solution may conveniently be regarded as present, whether a semipermeable membrane renders it evident or not. The osmotic pressure in the ordinary reagent bottles of the laboratory is of the dimensions of 100 atmospheres. This pressure cannot of course be borne by the walls of the bottle, nor is it apparent at the free surface of the liquid. Where the liquid comes in contact with the enclosing vessel, there we find a liquid surface, and a consideration of the magnitude of the forces at work in the phenomena of surface. tension leads us to believe that the pressure at right angles to the free surface of a liquid, and directed towards the interior of the liquid, is measurable in hundreds and even thousands of atmospheres. Osmotic 180 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. pressures, then, large as they are in ordinary solutions, are small compared to the surface pressures in liquids, and could not be evident. at the free surface of liquids. Various hypotheses have been put forward to explain the nature of osmotic pressure, but none of them can be accounted satisfactory. They are based upon more or less probable suppositions as to the nature of the movement of the dissolved molecules, the degree of attraction between them and the particles of the solvent, the capillary phenomena in the "pores" of the semipermeable partition, and the like. Our ignorance of such matters is, however, so great that no profitable conclusions have hitherto been arrived at. One thing may be said about osmotic pressure which is independent of any supposition as to its nature, and has indeed been already indicated. in what has preceded. The osmotic pressure of any solution is inde- pendent of the nature of the semipermeable membrane. Pfeffer A tested several mem- branes besides mem- branes of copper ferrocyanide. For ex- ample, membranes of Prussian blue or of tannate of gelatine can be deposited in a por- ous cell in the manner described for copper ferrocyanide, and have a similar action with regard to water and substances held by it in solution. Pfeffer found that in general the osmotic pressure of any one solution varied with the nature of the membrane he employed, but this was in reality due to the membranes not being perfectly impermeable to the dissolved substance, so that the maximum pressures recorded did not correspond to the actual osmotic pressure, but fell short of it in a degree dependent on the extent to which the membrane leaked. A theoretical proof may be given that the nature of the membrane does not influence the value of the osmotic pressure provided that it is perfectly impermeable to the dissolved substance and permeable to the solvent. Suppose two membranes to exist, one of which, A, generates with a given solution and solvent a higher osmotic pressure than the other membrane B. If the hydrostatic pressure in the vessel containing the solution is less than the osmotic pressure, liquid will flow through the membrane from solvent to solution; if it is greater than the osmotic pressure, liquid will flow from the solution to the solvent. Let the solution, solvent, and diaphragms be combined. into one working system, as in the figure. If the solution is originally at a greater hydrostatic pressure than corresponds to the value of the osmotic pressure generated by B, solvent will flow out through the dia- phragm, and the pressure inside will diminish and tend to reach this value. Solvent Solution FIG. 33. B Solvent XVII OSMOTIC PRESSURE AND THE GAS LAWS. 181 But as soon as the pressure diminishes to a value less than the osmotic pressure generated by A, solvent will flow through A into the cell. The pressure inside will still be too great for B, and solvent will therefore continue to flow out. There will thus be a continuous flow of solvent through the cell from left to right, and as the conditions are not changed by this transference, the flow might go on indefinitely and the current be made use of to perform work, i.e. in this way we could obtain a perpetual motion (cp. Chap. XXXVI.). Since this is impos- sible, our assumption that the osmotic pressures generated by the two diaphragms are different must be incorrect, and we are forced to conclude that the osmotic pressure of a solution is independent of the diaphragm used in measuring it, provided that the diaphragm is completely impermeable to the dissolved substance. Although the direct measurement of osmotic pressure is surrounded by so many difficulties, it is often possible to tell whether a solution has an osmotic pressure greater than, equal to, or less than another solution of the same or a different substance. This may be done either with the aid of a precipitation membrane such as Traube employed, or by means of a natural semipermeable membrane. When a precipita- tion membrane is formed at the end of a tube as described above, the two solutions which form the precipitate have in general different osmotic pressures. But the solvent water moves through the membrane from the solution with less, to the solution with greater, osmotic pressure. If the solution outside the cell has the greater concentration it gains water, becomes more dilute in the immediate neighbourhood of the membrane, and rises in the external liquid owing to its lesser specific gravity. This is easily rendered visible by means of a Töpler apparatus, which detects very small differences in the refractive power of liquids. If the external solution has a smaller osmotic pressure than the internal solution, water will be transferred inwards through the membrane, and the external solution will become more concentrated in the neighbourhood of the membrane, the change betraying itself by differences in density and refractive power as before. If, finally, the two solutions have equal osmotic pressures, no trans- ference of water takes place, and there is consequently no change in the density or refraction of the solutions. We have here, then, a method for determining when two membrane-forming solutions are isotonic, a term frequently applied to solutions having the same osmotic pressure. The following method makes use of natural semipermeable mem- branes. It is known that the protoplasm of vegetable cells has a sort of skin which serves to a certain extent as a semipermeable membrane, for it keeps dissolved substances in the cell sap from passing outwards, while it admits of the free passage of water. If the proto- plasm of the cell then is brought into contact with pure water or a solution of smaller osmotic pressure than the cell contents, water will 182 CHAP. INTRODUCTION TO PHYSICAL CHEMISTRY pass through the skin inwards to the protoplasm. If, on the other hand, the cell conte it has a smaller osmotic pressure than the solution with which the protoplasm is brought into contact, water will pass outwards through the skin. Should the external solution finally have the same osmotic pressure as the solution within the protoplasmic skin, there will be no transference of water between the cell and the external solution. In the case of some cells, the passage of water to or from the protoplasm is easily visible on account of the apparent increase or diminution of the volume. Thus when a suitable vegetable cell is brought into contact with a solution of higher osmotic pressure than that of the solution within the protoplasm, the granular or coloured cell contents are seen to shrink away from the cell wall owing to the loss of water and contraction in volume which they experience. By diluting the external solution, it is easy to find a con- centration which just ceases to produce this contraction; then the cell contents are isotonic with this solution. In the same way, a solution of another substance may be found which is isotonic with the same cell. These two solutions are then isotonic with each other. This last state- ment is proved by experiments which show that two solutions which have been found to be isotonic with respect to one kind of cell are also isotonic with regard to other kinds of cell. In this again we have an indication that the nature of the membrane has no influence on the osmotic pressure, if only it is impermeable to the dissolved substance. In what follows it will practically always be assumed that the osmotic pressure of a solution is proportional to its concentration. This is by no means always a strict relation, and holds good only for somewhat dilute solutions; indeed, this is only what might be expected even if there were exact similarity of the state of a substance in solution and as a gas. As has been already stated above, some of the ordinary laboratory solutions have osmotic pressures of nearly 100 atmospheres. Now, concentration of a solution corresponds to absolute density in the case of a gas. Boyle's law for gases states that the pressure of a gas is proportional to its absolute density, or inversely proportional to its volume, which is the same thing. But this by no means necessarily holds good for pressures as high as 100 atmospheres. We could not expect, then, that the corresponding law for solutions—namely, that the osmotic pressure is proportional to the concentration-should be exactly true at similar high osmotic pressures. In general, we can expect no exact proportionality between osmotic pressure and concentration at strengths above normal, and we very often find that much more dilute solutions have to be con- sidered in order to get the simple laws to apply with any strictness. An account of the preparation and use of osmotic cells will be found in the following papers :— XVII OSMOTIC PRESSURE AND THE GAS LAWS 183 H. N. MORSE and collaborators, Amer. Chem. Journ. (1911), 45, pp. 91, 237, 383, 517, 554; (1912), 48, p. 29. H. N. MORSE, "The Osmotic Pressure of Aqueous Solutions," Carnegie Institute of Washington Publications, No. 198, 1914. E. FOUARD, Journal de physique (1911), 5. sér., 1, p. 627. EARL OF BERKELEY and E. G. J. HARTLEY, Philosophical Transactions, 1906, 206 A, 481. For a general account of osmotic pressure, see A. FINDLAY, Osmotic Pres- sure (London, 1919); 'Osmotic Pressure: a General Discussion," Trans. Faraday Society, 1917, 13, pp. 119-189. DEDUCTIONS FROM THE GAS LAWS FOR DILUTE SOLUTIONS IN discussing the evaporation and solidification of solutions in Chapters VII. and VIII., we have met with empirical laws which find a theoretical basis in the conception of osmotic pressure. Such are the law that the vapour pressure of a solution is less than the vapour pressure of the pure solvent by an amount proportional to the strength of the solution (p. 76); that the boiling point of a solution is higher than the boiling point of the solvent by an amount proportional to the strength of the solution; and that the freezing point of a solution is lower than the freez- ing point of the solvent by an amount proportional to the strength of the solu- tion (p. 58). MARKE CHAPTER XVIII The more strict thermodynamical deduction of these relations from the gas laws for dilute solutions is given in Chapter XXXVI., but in this chapter we can show, at least approximately, the relations of the various magnitudes. In the first place, we shall consider the connection between osmotic pressure and the relative lowering of the vapour pressure. In the figure (Fig. 34) B represents a porous bulb with a semipermeable mem- brane deposited within the wall (p. 174). It is filled with a known solution and immersed in the pure solvent. The solvent will enter the bulb until the solution in the tube rises to a height where the difference of level of the liquids inside and outside the bulb causes a pressure equal to the osmotic pressure of the solution. Let this equilibrium be reached in an atmosphere which consists only of the vapour of the solvent, and let the difference in level be represented by h. The B h FIG. 34. 184. CH. XVIII DEDUCTIONS FROM THE GAS LAWS 185 temperature throughout is supposed to remain at the constant value T on the absolute scale. In the first place, we note that the pressure of vapour at the level l of the surface of the solution must be the same inside and outside the tube. If it were not, being, let us assume, greater inside than outside, vapour would pass from the region of higher to the region of lower pressure. This would lower the pressure of vapour immediately over the surface of the solution, and some of the solvent would therefore evaporate in order that the pressure should regain its former value, which is the value in equilibrium with the solution. The original value outside the tube is the value corresponding to the vapour pressure of the pure solvent, so that the accession of vapour from the inner tube would increase the pressure of vapour above its equilibrium value, and consequently some of the vapour would condense at the surface of the pure solvent. The process would be then, in short, that some of the liquid inside the tube would distil over and condense to liquid outside the tube. This would evidently increase the concentra- tion of the solution within the bulb, and so we should no longer have To restore osmotic equilibrium between the solution and the solvent. the osmotic equilibrium, some of the solvent would pass inwards through the membrane and dilute the solution to its original concen- tration. The liquid in the tube would then regain its original height and vapour pressure, and the whole process of distillation would recommence. On the assumption, then, that the pressure of vapour at is greater inside the pressure tube than outside at the same level, we have a continuous circulation of solvent from the solution to the solvent as vapour, and from the solvent to the solution as liquid through the semipermeable membrane. The current thus generated could theoretically be used to perform work, and we should there- fore have a form of the perpetual motion, which is impossible. Similarly, if the pressure at the level is supposed to be greater outside the tube than inside, we should have a continuous circulation of solvent in the opposite direction. The conclusion which we must adopt, then, is that the pressure within and without the tube at the same level is the same. If now ƒ is the vapour pressure of the solvent at the given tempera- ture, and f' that of the solution, the difference f-f' is evidently the difference of pressure between the levels at the two liquid surfaces, i.e. at the top and bottom of the height h. This difference in pressure is due to the weight of the column of vapour between the two levels on a surface of one square centimetre, and is equal to the product of the height and absolute density of the column of vapour, i.e. to hd, if d is the density expressed in grams per cubic centimetre. Let us now consider a gram-molecular weight of this vapour. For it we have pv = RT, 186 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. where R is the ordinary gas constant. But the density d is the weight divided by the volume, i.e. d = M/v, where M is the molecular weight of the solvent in the gaseous state. The pressure of the gaseous solvent is ƒ, so that we obtain v = RT|ƒ, and fM RT d = 1 We have seen above that f-f' = hd, or, substituting the value of d here found, ƒ – f' = h. fM whence RT ƒ -ƒ' f h M ᎡᎢ f Now ff' is the lowering of the vapour pressure of the solvent, and ƒ -ƒ' is therefore the proportional or relative lowering, with which we are alone concerned. We have thus obtained an expression for the relative lowering of the vapour pressure in terms of the "osmotic height" h, and constants for the gaseous solvent. It is now an easy matter to express h in terms of the osmotic pressure of the solution, and another constant for the solvent. The osmotic pressure, i.e. the excess of pressure inside the cell over that outside, is equal to the height of the column h into the absolute density of the liquid. This we may denote by s, which is the absolute density of the pure solvent, for if the solution considered is very dilute, its density will not greatly differ from the density of the solvent. We have, then, if p represents the osmotic pressure, p=hs, or h=2. Substituting this value of h in the previous equation, S we obtain ƒ -ƒ' M f =PSRT (1) Here the relative lowering is expressed in terms of the osmotic pressure of the solution and magnitudes referring to the solvent, which for constant temperature are constant. It appears, then, that the relative lowering of the vapour pressure of a liquid by the solution in it of some foreign substance is at any one temperature proportional to the osmotic pressure of the solution and independent of the nature of the dissolved substance. By making use of the gas laws for solutions we can eliminate temperature from the above expression, and put it in a simpler form. If we express the concentration of the solution in the form that ʼn gram molecules of the solute are contained in W grams of the solvent, then we have for n gram molecules the equation pv = nRT. XVIII DEDUCTIONS FROM THE GAS LAWS 187 Now the volume in which these n gram molecules are contained is equal to II, the weight of solvent, divided by s, the density of the solvent, i.e. v W's, so that p Substituting this value in the former equation, we obtain f-f' nsRT M f W´SRT or ƒ_ƒ' f (2) The relative lowering is now expressed in terms of the concentration of the solution and the molecular weight of the solvent in the gaseous state, which is constant. The temperature has, according to this result, no influence on the value of the relative lowering-a conclusion. which is in accordance with the experimental data. For a given. weight of any one solvent, the relative lowering is proportional to the number of molecules of dissolved substance in solution. Consequently, if we find that for certain quantities of different substances, dissolved in the same weight of the same solvent, the relative lowering of the vapour pressure of the solvent is the same, we conclude that these quantities contain the same number of dissolved molecules, and thus we obtain a method for determining the molecular weights of substances. in solution. The molecular weight of a dissolved substance might also be calculated in terms of the relative lowering of the vapour pressure by finding the numerical value of p from equation (1), and introducing the osmotic pressure thus obtained into the gas equation. For a given amount of the same substance dissolved in the same weight of different solvents, we find from equation (2) that the relative lowering is proportional to the molecular weight of the solvent in the gaseous state, provided that the molecular weight of the dis- solved substance remains the same in the different solvents. or ns RT W f - f The expression for the relative lowering receives a still simpler form if, instead of the actual weight of the solvent, we introduce the number of gram molecules of the solvent. The weight of solvent may be expressed as the product of the number of molecules into the gram-molecular weight of the substance in the gaseous state, i.e. as MN, if N represents the number of gram molecules of the solvent as gas. We have therefore Фотодерматити в N IV f M. N M, MN ƒ-j f (3) The relative lowering is here given as the ratio of the number of N N • 188 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. dissolved molecules to the number of molecules which the solvent would produce if it were converted into vapour. It must be emphasised that the number of molecules N in the above equation does not denote the number of liquid molecules in the solvent, but only the number of gaseous molecules derivable from the liquid. This caution is necessary, because it has frequently been supposed that the equation enables us to determine the molecular weight of the liquid solvent, which is not the case. As we shall see, methods exist for determining the molecular weights of liquids, but this is not one of them. In deriving the above equations, we have assumed that the specific gravities of the solutions considered are the same as the specific gravities of the respective solvents. This assumption only holds good for very dilute solutions, but it gives serviceable approxi- mations in practical work, as the following numerical examples will show: A solution of 2.47 g. of ethyl benzoate in 100 g. of benzene showed a relative lowering of vapour pressure equal to 0.0123, i.e. if the vapour pressure of pure benzene is 1, the vapour pressure of the solu- tion is 1 0·0123. The temperature at which the determination was made was 80° C., and at this temperature the density of benzene is 0.812. The molecular weight of gaseous benzene is 78. If we sub- stitute these values in equation (1), we obtain 78 0·0123 = P・ 0·812 × 84,700 × (273 + 80)' = whence p 3830 g. per square centimetre, or over 3-7 atmospheres. If we wish to calculate the molecular weight of ethyl benzoate from these data by means of equation (2), we get by substitution 0.0123 Ն 100 78, whence n = 0·0158, i.e. in 2:47 g. of benzoate of ethyl there are 0·0158 gram molecules. There is therefore one gram molecule of dissolved ethyl benzoate in 2.47/0.0158 = 156 g., or 156 is the molecular weight of ethyl benzoate when it is dissolved in benzene. It must be re- membered that this number is only approximate, but still it is sufficient to show that the molecular weight of the dissolved sub- stance is practically that of the gaseous substance, which is 150. Equation (3) leads to the same result, for in order to get the number of molecules N we have to divide the number of grams taken, viz. 100, by the molecular weight, viz. 78. The determination of the lowering of vapour pressure is somewhat too difficult and tedious to be much used in fixing the molecular weights of dissolved substances, and it is therefore preferable to XVIII DEDUCTIONS FROM THE GAS LAWS 189 ascertain in its stead the elevation of the boiling point, which is for dilute solutions nearly proportional to it, and susceptible of easy and rapid determination. As a solution of a non-volatile substance at a given temperature has a lower vapour pressure than the pure solvent, it is evi- dent that in order that the solution and solvent may have the same vapour pres- sure, e.g. equal to the standard atmospheric pres- sure, it is necessary to heat the solution to a higher temperature than the solv- The boiling point of the solution solution is therefore always higher than the boil- ing point of the solvent. A consideration of the accom- panying diagram (Fig. 35) will show that for small changes, the lowering of the vapour pressure and the elevation of the boiling point are nearly proportional. ent. In the figure the three curved lines represent the vapour pressure curves of a pure solvent and of two solutions of different concentrations. At the temperature t, the intercept aa, represents the actual lowering of the vapour pressure of the solution 1, and the ratio aa, ta the relative lowering. For the temperature T we have the corresponding magnitudes cc, and cc, Te. Since for any one solution the relative lowering is independent of the temperature, we have aa, tacc₁: Te. Similarly for the second solution aa: ta = cc: Te, so that aa: aª² CC1 : cc₁cc2, or αа₁: α₁α₂ = СС₁ : C1C2• 12 CC1 If the curves were straight lines, they would in virtue of these proportions meet in one point, but if we only consider the very small lowerings observed in dilute solutions, their directions are so nearly the same that they may in small intervals be treated as parallel straight lines. Consider the line ab, parallel to the temperature axis. This is a line of constant vapour pressure, and cuts the curves at points corresponding to the temperature t₁ and t The intercepts ab, and ab, represent the elevations of the boiling point, if the boiling point at the atmospheric pressure is t. Now if the curves were parallel straight lines we should have aа₁:αª¸ ab₁: ab₂, i.e. the elevation of the boiling point would be proportional to the lowering of the vapour pressure, and thus also to the osmotic pressure. 1 Pressure C C2 a T ແດ t Temperature FIG. 35. ཆ3 Another easily determinable magnitude which is proportional to the osmotic pressure is the depression of the freezing point in dilute. solutions. By means of a diagram similar to the above we can show 190 INTRODUCTION TO PHYSICAL CHEMISTRY CH. XVIII that this depression is approximately proportional to the lowering of the vapour pressure, and thus indirectly establish the connection with osmotic pressure. In the figure (Fig. 36) a0 represents the vapour- pressure curve of the liquid solvent, say water, and a0' that of the solid solvent, ice. The tem- perature t, where these two curves intersect, is the freezing point of the pure solvent (cp. p. 98). The curves 1 and 2 re- present as before the vapour-pressure curves of two solutions. These cut the ice curve at two points, b₁ and b₂, the cor- responding temperatures t, and to representing the freezing points of the two solutions, i.e. the temperatures at which ice and the solutions are in equilibrium, and at which, therefore, they have the same vapour pressure. Now as before we may treat the curves 0, 1, and 2 as three parallel straight lines if we only consider small concentrations. We have then aa₁: aa₂ = ab₁ : ab₂ = tt₁: tt2. But tt₁ is the depression of the freezing point for the solution 1, and tt, the depression for the solu- tion 2. Consequently, we have the depressions of the freezing point. proportional to the lowerings of the vapour pressure, aa, and aa2. The freezing-point depressions are thus proportional in dilute solutions to the osmotic pressures of the solutions, and can therefore be sub- stituted for the latter in ascertaining molecular weights. 2 1 2 It has now been shown on approximate assumptions that Pressure | ხი. b₁ " Ja₁ az to t, t Temperature FIG. 36. 1 2 Elevation of boiling point = cP, Depression of freezing point = c'P, and These where P is the osmotic pressure, and c and c' constants. constants remain in each case the same for a given solvent, and are valid for all dissolved substances. They correspond in their nature to the constant factor on the right-hand side of equation (1) in this chapter. They depend on the properties of the solvent, and their derivation from these properties will be shown in Chapter XXXVI. CHAPTER XIX METHODS OF MOLECULAR WEIGHT DETERMINATION 1. Gaseous Substances-Vapour Density WHEN we can obtain a substance in the gaseous state, the determination of its molecular weight resolves itself, as we have seen in Chapter II., into ascertaining what weight of the vapour in grams will occupy 22.4 litres at 0° and 760 mm., or, if we deal with smaller quantities, what weight in milligrams will occupy 22·4 cc. In general, we cannot weigh these volumes of the vapour under the standard conditions. In the case of water, for example, it is impossible to get a pressure of 760 mm. of vapour at 0°, the vapour pressure of water at that temperature being only a few millimetres of mercury. We can, however, make the actual determination under any conditions we please, and then reduce to the standard conditions by means of the gas laws. The practical problem to be solved, then, in vapour-density determinations for the purpose of finding molecular weights is to measure the weight, volume, temperature, and pressure of a given amount of substance in the gaseous state. This may be done in various ways, as the following short description of the principal methods will show. Dumas's Method. In this method the weight of a known volume of gas or vapour is determined, a globe of known capacity being filled with the gas at atmospheric pressure and known temperature, sealed off, cooled, and weighed. The volume of the globe is ascertained by weighing it when empty and when filled with water. Its weight when filled with air minus the weight of air contained in it (which can be calculated from the volume and the known density of air) gives the weight of the empty globe. This, when subtracted from the weight of the globe filled with the gas under investigation, gives the weight of that gas which fills the given volume at the given pressure and temperature. The method, when applied to the vapours. of substances liquid at the ordinary temperature, usually assumes 191 192 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. the following form: The bulb is of 50 to 100 cc. capacity and has the shape shown in the figure (Fig. 37). Several grams of the liquid substance are placed in the bulb, which is then immersed in a bath of constant temperature about 20° higher than the boiling point of the liquid. The liquid in the bulb boils and expels the air with which the bulb was originally filled. After the vapour ceases to escape, the narrow neck of the bulb is sealed off near the end with a small blowpipe flame. There is then in the bulb a known vol- ume of vapour at a known tem- perature and pressure, so that all FIG. 37. that has now to be done is to ascertain the weight of the vapour. This method is somewhat troublesome, and is usu- ally applied to substances which are only volatile with difficulty. Hofmann's Method. Here, instead of taking a known volume of vapour and measuring its weight, we take a known weight of substance and measure the volume which it occupies as vapour, and the pressure which the vapour exerts. The apparatus employed for the purpose is shown in Fig. 38. It consists of an inner tube about a metre long and 12 millimetres in bore, which is graduated in cubic centi- metres. This is filled with mercury, and inverted in a mercury trough so that at the top of the tube a Torricellian vacuum is formed. Outside this tube is a wider tube which acts as a vapour jacket, the vapour of a boiling liquid passing in through the narrow tube d, and issuing, together with the con- densed liquid, through the side tube just above the mercury. A weighed quantity of the liquid, the density of whose vapour is to be determined, is introduced into the inner tube in a bulb or very small stoppered bottle made for the purpose, and containing only about a tenth of a cubic centimetre. The inner tube is then heated by the vapour from a liquid boiling in a suitable vessel attached to d, the boiling being continued until the vapour issues freely from the lower tube and the level of the mercury in the inner tube no longer alters. The weighed quantity of substance has now been converted into vapour, and occupies the volume FIG. 38. b A d XIX DETERMINATION OF MOLECULAR WEIGHTS 193 above the mercury at the top of the graduated tube, at the boiling point of the liquid used for heating, and under a pressure equal to the atmospheric pressure minus the height of the column of mercury ab. The external liquid used for heating the inner tube to a constant temperature is usually water. This evidently will vaporise any liquid with a boiling point under 100°, but in reality it can be used for liquids. with much higher boiling points, for the vaporisation takes place in the inner tube under reduced pressure. Thus a liquid such as aniline, with a boiling point of 180°, is easily vaporised in the Hofmann apparatus by boiling water, if the quantity taken is such that the vapour only occupies a small proportion of the total volume of the graduated tube. Variable ய www Pressure Method. In this method a known weight of substance is vaporised at known temperature in a constant volume, and the rise in pressure due to the conversion of the substance into vapour is noted. The method has been proposed from time to time by differ- ent investigators, but has only recently received the attention it deserves. The form of apparatus here described is that employed by Lumsden. The vaporising bulb A (Fig. 39), of about 100 cc. capacity, is blown in one piece with the surrounding vapour jacket B, in which a liquid of suitable boiling point is kept n a state of ebullition. A side tube of arrow bore makes connection through a three-way tap with the manometer CE, the mercury of which is always adjusted to he mark E before a reading is taken. The neck of the vaporising bulb has a diameter of 8 mm., and is furnished with short side-tube for the insertion of a glass rod to support the small bottle containing the weighed substance. The neck is protected from undue heating by a shield of asbestos paper (not shown in the figure) which rests on the top of the vapour jacket. FIG. 39. To perform an experiment, the liquid in the jacket is boiled for ifteen minutes by means of a bunsen flame, which plays on the wire- gauze bottom of the asbestos box, in a hole in the top of which the apour jacket rests. The temperature is then constant and the mercury 0-1 A B 0 194 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. is adjusted to the mark E, all parts of the apparatus having free con- nection with the air through the three-way tap. The capsule con- taining the substance is next placed on the support at D, a rubber stopper inserted into the neck of the flask, and the tap turned so as to retain connection with the manometer but to shut off connection with the air. By turning the support the capsule is released and falls into the bulb, where vaporisation at once begins. While the substance is being vaporised the movable graduated limb of the manometer is gradually raised so as to keep the mercury in the other limb at the mark E. After a minute the pressure has become constant, the mercury is accurately adjusted to the mark, and the difference in level read off. This difference is due to the pressure of the vaporised substance. If we therefore know the volume of the apparatus, the boiling point of the liquid in the jacket, the weight of the substance, and the pressure due to it, we have all the data for calculating its molecular weight. It is true that the vapour is not distributed uniformly through the bulb, but since no change in pressure would result if diffusion proceeded until uniformity of composition had been attained, the increase in pressure immediately after vaporisation is the pressure which the vapour would exert if it uniformly filled the whole apparatus. A principle may be here adopted which is often available for simplifying molecular weight and other similar calculations. If in this case the temperature and volume are kept constant, i.e. if the same apparatus and the same liquid in the jacket are used, the molecular weights of two substances are proportional to the weights. of them taken, and inversely proportional to the pressures produced, i.e. M₁ w₁. Po 1 201 Mo wo P1 0 w₁ 21. P1 P1 If therefore we take a substance of known molecular weight M。 and determine the pressure po caused by the vaporisation of a weight w。 of it, we can then by means of the above formula find a "constant" c for the apparatus which includes both the volume and the temperature, neither of which need be known, if only both be kept at the former values when the measurements for the substance of unknown molecular weight M₁ are made. This mode of working is especially convenient. for high temperatures, where it is much easier to keep a temperature constant for a time than to determine exactly what the temperature is. It also eliminates volume corrections for expansion of the bulb by rise of temperature and for the difference in temperature between. the bulb proper and its connections. 1 Victor Meyer's Method. This method is akin to Hofmann's inasmuch as a weighed amount of liquid is vaporised, but it differs or M₁ = MoPo 1 го о XIX DETERMINATION OF MOLECULAR WEIGHTS 195 from the other methods on account of the gas whose volume, tempera- ture, and pressure are actually determined not being the vapour under investigation, but an equal volume of air which has been displaced displaced by the vapour. On account of the simplicity of the ap- paratus and the conven- ience in manipulation, this method is widely adopted in practical work where great accuracy is not re- quired. { St S CHHURTING TO The apparatus consists of a cylindrical vessel of about 100 cc. capacity with a long narrow neck, having a top piece furnished with two side tubes. One of these side tubes acts as a delivery tube, and is con- nected by means of a piece of thick-walled, narrow- bored rubber tubing with the gas-measuring tube g, FIG. 40. which at the beginning of the experiment is filled with water. Through the other side tube a glass rod projects into the neck, con- nection being made by means of a short piece of rubber tube, permitting a good deal of play. A weighed quantity of the liquid whose vapour density is to be determined is contained in the small bulb s, which is held in place by the rod t. closed at the top by a cork, and contains a little in order to protect the glass from injury by the fall of the bulb. The wide cylindrical portion and a large part of the neck are heated by means of a liquid boiling in an external cylinder with a bulb-shaped end. To perform an experiment, the level of water in the measuring tube is adjusted to zero by moving the reservoir n, the bulb is put into position, and the heating started while the tube is still open at the top. When the temperature of the whole apparatus has become steady, the tube is corked, and the level of the water in the measuring tube is observed for some minutes. If it has altered slightly, it is re- adjusted to zero, and the bulb is let drop by drawing back the rod t for a moment. The liquid at once begins to vaporise, and air passes over into the measuring tube. To keep the gases in the apparatus at the atmospheric pressure, and thus prevent leakage, the water in the reservoir is kept at the same level as the water in the measuring tube g T #k 几 ​The principal tube is asbestos at the bottom, 196 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. * by continually lowering the reservoir. The vaporisation is complete in about a minute, as a rule, and if after two or three minutes the level of water in the measuring tube does not change, the volume is read off, the barometric pressure and the temperature on a thermo- meter near the measuring tube being noted at the same time. If a substance is taken, such as bromine, which gives rise to a coloured vapour, the distribution of the vaporised substance within the appa- ratus may be readily seen. The distribution of temperature throughout the whole apparatus is the same before and after the volatilisation of the substance, and therefore the volume of air collected is the same as the volume of vapour formed, after reduction to the temperature and pressure at which the collected air is measured. The temperature is the temperature of the water, the pressure that of the atmosphere minus the vapour pressure of water; for the vapour pressure of the water over which the gas is collected and the pressure of the gas itself together make up the total pressure, which is equal to that registered by the barometer. In the simplest form of apparatus the gas is collected in a graduated tube over water contained in a shallow dish, the side tube being in this case long, of narrow bore, and bent to the appropriate form for delivery. It will be seen that this method of vapour-density determination does not involve a knowledge of the temperature at which the vaporisation takes place, for since all gases are equally affected by changes of temperature, the contraction in volume of the hot air on cooling is the same as the contraction which the vapour itself would experience. We thus, instead of measuring the volume at the temperature of vaporisation, measure the reduced volume at the atmospheric temperature. The liquid used for heating should have in general a boiling point at least as high as that of the experimental substance, and the ebullition of the heating liquid should be so brisk as to make the vapour condense two-thirds of the way up the outer tube. The mode of calculation of a molecular weight from the observed data for the vapour density may be seen from the following example. The bulb contained 0·1008 g. of chloroform, boiling point 61°, and was dropped into a tube heated by the vapour from boiling water. The air collected measured 22.0 cc., the temperature being 16.5°, and the height of the barometer 707 mm. Now the vapour pressure of water at 16.5° is 14 mm., so that the actual pressure of the gas was 707 - 14 = 693 mm.. We have only now to solve the following proportion: If 100.8 mg. of chloroform vapour occupies 22:0 cc. at 16.5° and 693 mm., what number of milligrams will occupy 22-4 cc. at 0° and 760 mm. ?—i.e. to evaluate the expression 100·8 × 22·4 × (273 + 16·5) × 760 22·0 × 273 × 693 XIX DETERMINATION OF MOLECULAR WEIGHTS 197 The result we obtain is 119, the actual molecular weight of chloroform as calculated from its formula being a little over 118. It should be borne in mind that in the determination of molecular weights from vapour densities only approximate results are obtained, for we assume that the vapours obey the simple gas laws exactly, which is by no means the case when the vapour is at a temperature only a little removed from the boiling point of the liquid from which it is produced. As, however, the vapour density is used in conjunction with the results of analysis in fixing the accurate value of the molecular weight, an error amounting to 5 or even 10 per cent of the value is unimportant, the number obtained from the vapour density merely determining the choice between the simplest formula weight and a multiple of it. The molecular weight of chloroform in the above example can from the formula be only 118 or a multiple of 118; and the vapour-density estimation shows conclusively that the simplest formula is here the molecular formula. 2. Dissolved Substances-Osmotic Pressure Assuming the complete similarity in pressure, temperature, and volume relations of substances in dilute solution and of gases, we can evidently determine the molecular weight of a dissolved substance by simultaneous observations of its weight, temperature, volume, and osmotic pressure. Pfeffer found, for example, that a one per cent solution of cane sugar at 32° had an osmotic pressure of 544 mm. One gram, or 1000 mg., of cane sugar here occupied approximately 100 cc. We have then as before the proportion: If 1000 mg. of sugar occupy 100 cc. at 32° and 544 mm., what number of milligrams will occupy 22.4 cc. at 0° and 760 mm.? The answer is 1000 × 22·4 × (273 + 32) × 760 100 × 273 × 544 = 349. The molecular weight of cane sugar calculated from the formula C12H22O11 is 342. It is evident, then, that the molecular formula of cane sugar in aqueous solution is the simplest that will express the results of analysis. Were it not for the extreme difficulty of obtaining a membrane perfectly impermeable to the dissolved substance, this method would be the most suitable and the most accurate for determining molecular weights of substances in very dilute solution. Rapid and accurate determinations of osmotic pressure have been made by means of semipermeable membranes deposited in collodion films which are supported on a cylinder of metal gauze. The osmotic pressure of the solution to be investigated is balanced against that of an isotonic sugar solution. If the concentrations of the two solu- 198 CHAP. INTRODUCTION TO PHYSICAL CHEMISTRY , tions in osmotic equilibrium are known, the ratio of the molecular weight to that of cane sugar (342) can be calculated. It was found, for example, that the concentrations of isotonic solutions of mannitol and cane sugar were 1.5 and 2·8 respectively. The molecular weight 342 × 1.5 of mannitol was therefore = 183. The weight correspond- ing to the formula of mannitol, C6H14O6, is 182. This method is applicable to very dilute solutions. - 2.8 3. Dissolved Substances-Lowering of Vapour Pressure An example of how a molecular weight of a dissolved substance may be estimated by this method has been given in the preceding chapter (p. 188). The method has as yet little practical importance, and is scarcely ever employed. M Barger's method. In this method the principle is used that equimolar solutions have the same vapour pressure, and that a solution of higher molar concentration will absorb the vapour of the solvent from one of lower molar concentration. A solution of known concentration of the substance whose molecular weight is to be determined is tested against successive standard solutions of known molar concentration. The two solutions to be compared are intro- duced as alternate drops into a horizontal glass capillary, the drops. being separated by air spaces. The lengths of the short columns of liquid in the capillary are now measured by means of a microscope provided with an eyepiece scale. If the drops are of equal molar concentration their length will remain unchanged. If they are of different concentrations the drops of the more concentrated solution will slowly increase in length at the expense of the less concentrated solution. By repeated trials two standard solutions of neighbouring concentration may be found, one of which absorbs solvent from the unknown solution and the other loses solvent to it. The molar concentration of the unknown solution must then fall between the concentration of these standard solutions, and thus the molecular weight of the dissolved substance may approximately be determined. 4. Dissolved Substances Elevation of Boiling Point This is a practical method for determining the molecular weights of substances in solution, and is coming more and more into general use. An essential condition for its success is that the dissolved substance should not itself give off an appreciable amount of vapour at the boiling point of the solvent. It is only applicable, therefore, to substances of comparatively high boiling point, say over 200°, and cannot be employed with success for volatile substances such as alcohol, benzene, or water. Two forms of apparatus may be described, which differ principally in the mode of heating. • XIX DETERMINATION OF MOLECULAR WEIGHTS 199 Beckmann's Apparatus.-In this form of apparatus the solution is raised to its boiling point by the indirect heat from a burner. Now in ascertaining the boiling point of a liquid, it is customary to place the thermometer, not in the boiling liquid itself, but in the vapour coming from it. In this way superheating is avoided. The liquid itself may be at a temperature considerably above its true boiling point, but a thermometer placed in the vapour will show very little sign of this superheating. The plan, however, cannot be adopted in ascertaining the boiling point of a solution. The vapour which comes from the solution of a non-volatile substance is the vapour of the solvent, a part of which condenses to liquid on the bulb of the thermometer. Now the temperature at which the condensed and vaporous solvents are in equilibrium on the bulb of the thermometer is the boiling point of the solvent and not that of the solution, so that the temperature registered by a thermometer placed in the vapour from a boiling solution is the boiling point of the solvent, slightly raised perhaps by radiation from the hotter solution. It is necessary then to immerse the bulb of the thermometer directly in the boiling solution if the boiling point of the latter is to be determined, i.e. the temperature at which the solution and the vapour of the solvent are in equilibrium. When, therefore, the source of heat is external and necessarily of a higher temperature than the boiling point which has to be measured, precautions of the most rigorous kind have to be adopted in order to prevent superheating of the liquid whose boiling point is in question. In Beckmann's apparatus this is done as follows. The boiling tube A (Fig. 41) is about 2.5 cm. in diameter, provided with a stout platinum wire fused through its end, and filled for about 4 cm. with glass beads. The platinum wire is for the purpose of conducting the external heat into the solution so as to get the bubbles of vapour to form chiefly at one place and prevent superheating. The glass beads are used in order to split up the large bubbles of vapour into smaller bubbles, so that more intimate mixture of the solution and the vapour of the solvent may be secured. A more effective method of preventing superheating is to use instead of beads Beckmann's platinum tetrahedra. These are made by tightly rolling up thin platinum foil, and cutting off with shears pieces of the cylinder thus produced, at distances roughly equal to the diameter of the roll, the cylinder being rotated through 90° after each cut. The pieces of platinum cut off in this way are in the form of tetrahedra, which, on account of the large surface, sharp edges, and good conducting power, greatly facilitate boiling. When neither solvent nor solute attacks aluminium, tetrahedra of aluminium foil may be substituted. The thermometer, which is divided into hundredths of a degree, and may be read to thousandths, is placed so that its bulb dips partly into the glass beads or tetra- hedra. A device for rendering such a thermometer available for 200 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. solvents of widely different boiling points is described in section 5 of this chapter. The inner vessel is surrounded by a vapour jacket, charged with about 20 cc. of the solvent, which is kept boiling during the experi- ment. This jacket reduces radiation towards the exterior to a minimum, and consequently only a comparatively small amount of heat has to be supplied to the solution in order to make it boil, whereby the risk of superheating is greatly reduced. Both vessels are provided with reflux condensers, which may either be air condensers, Kas in the figure, or water condensers of the ordinary type, according to the volatility of the solvent. - www 12:1 K₂ B PORN PICS ORKESTERASEK 19 COMO UMANA CERERETT ANNAPEE BEHRENAE LIZZARELLESLIE PEZZANAKLIZEDE h₂ hi FIG. 41. שר B The small asbestos heating chamber C has two asbestos rings, h, and he, which protect the boiling vessel from the direct action of the flame of the burner, and two asbestos funnels, ss, which carry off the products of combustion. The heat of the burners reaches the liquid in the vapour jacket through the ring of wire gauze visible in section at d as a dotted line. To perform the experiment, a weighed quantity of the solvent (15 or 20 g.) is brought into the boiling tube, the heating begun, and the thermometer read off when it has become steady, which may not be before an hour has elapsed. The condensed solvent should only drop back very slowly from the condenser, and so the boiling must not be hastened by using a large flame. The condenser K, is then removed, and a weighed quantity of the experimental substance added, best in the form of a pastille if a solid, or from a specially shaped pipette if a liquid. The boiling point will now be found to rise, and the thermometer will after a short time again become stationary. The difference between the first and second readings of the thermometer is the elevation of the boiling point. Another weighed quantity of the substance may now be added, and the temperature of equilibrium again read off. This will give a second value for the molecular weight. XIX 201 DETERMINATION OF MOLECULAR WEIGHTS In some modern forms of Beckmann's apparatus the heat is sup- plied, not externally by burners, but internally by means of a spiral of wire, immersed in the liquid below the thermometer and heated electrically. A Landsberger's Apparatus. -Since the boiling point of a solution of a non-volatile substance is the temperature at which the solution is in equilibrium with the vapour of the solvent, we can bring a solution to its boiling point by continually passing into it a stream of vapour from the boiling solvent. As long as the solution is under its boiling point, some of the vapour will condense, and the latent heat of condensation will go to heat the solution until finally the boiling point is reached, when the vapour will pass through the solution without further condensation if no heat is lost to the exterior. Here there is little risk of superheating, since the vapour which heats the solution is originally at a lower temperature than the solution itself; so that if we surround the solution with a jacket of the vaporous solvent, we have all the conditions for real equilibrium, at least so far as the determination of molecular weights is concerned. Landsberger's apparatus secures these conditions in a very simple manner, and a slight modification of it is shown in Fig. 42. The apparatus consists of a flask F, a bulbed inner tube N, which contains the solution, and a wider tube E, which is connected with a Liebig's condenser C either through a side tube as in the figure, or through a tube sealed into the bottom of E. The vapour is generated in F (which contains the boiling solvent), passes through the solution. in N, from which it issues through the hole H, to form a vapour jacket between the two tubes, and finally passes into the condenser. The lower end of the delivery tube R, where the vapour passes into the solution, is perforated with a rose of small holes, so that the vapour is well distributed through the liquid. The bulb prevents portions of the liquid being projected through H if the boiling is vigorous. The boiling point of the solvent is first determined by placing enough of the pure solvent in N to ensure that the bulb of the thermometer is just covered by the liquid when equilibrium has been attained. This quantity usually amounts to from 5 to 7 cc. The parts of the apparatus are then put together, and the boiling of the solvent in F begun. To ensure regular ebullition, it is necessary to place in F a few fragments of porous tile, which must be renewed every time the boiling is interrupted. If the ebullition is brisk, the vapour heats the liquid in N to the boiling point in the course of a few minutes, as is evidenced by the reading of the thermometer rapidly becoming constant. The boiling is now interrupted, the tube emptied, and the whole process repeated, with the addition of a weighed quantity of the substance under investigation to 5-7 cc. of solvent in N. The difference between the temperature now observed and the former temperature is the elevation of the boiling point, and it only remains. 202 CHAP. INTRODUCTION TO PHYSICAL CHEMISTRY to determine the weight of solvent employed. This is done by detaching the inner tube N, with thermometer and delivery tube, and weighing to centigrams. If from this weight we subtract the weight of the substance taken, and the tare of the tubes, etc., we obtain the weight of the solvent present when the temperature of equilibrium was reached. If great accuracy is not desired, several successive determinations IN TAT VOUCHLESSKABEÇE פרייזס יייסדי -------➖➖KSER- T R H FIO. 42. ヨ ​Centimetres B F S with the same quantity of substance may be made by replacing the tube with its charge and continuing the passage of vapour,. interrupt- ing the boiling from time to time to ascertain the amount of solvent present at the moment of reading the temperature. In this case, instead of determining the weight of solution, it is more convenient to read off its volume in cubic centimetres by having the tube N appropriately graduated, as shown in the figure, and removing the thermometer and delivery tube at each interruption in order to read the volume. The successive determinations are less accurate on account XIX DETERMINATION OF MOLECULAR WEIGHTS 203 1 'of the boiling points being observed under somewhat different con- ditions, the pressure of the column of solution increasing, for example, as the solvent condenses in N.¹ For ordinary rough laboratory work a thermometer graduated into fifths of a degree is sufficiently accurate. The calculation of the molecular weight is carried out as follows. For each solvent we have a constant k, which is the elevation produced if a gram-molecular weight of any substance were dissolved in a gram of the solvent. Of course such an elevation is purely fictitious as it stands, but it has a real physical meaning if we take it to be a thousand times the elevation which would be produced if a gram-molecular The weight of the substance were dissolved in 1000 g. of the solvent. hundredth part of this constant, i.e. the elevation caused by dissolving 1 g. molecular weight in 100 g. of solvent, is often spoken of as the molecular elevation. For the solvents ordinarily employed the constants are as follows: Solvent. Alcohol Ether Water Acetone Chloroform Benzene h 1150 2110 520 1670 3660 2670 0.032 0.022 0.018 S k' Γ΄ Δ’ The constants in the first column, k, refer to 1 g. of solvent; those in the second column, K, refer to 1 cc. of solvent at its own boiling point, and are useful if we measure the volume of the solution instead of ascertaining its weight. In the calculation we assume exact proportionality between the concentration of the solution and the elevation of the boiling point. We thus obtain for the molecular weight the expression S k M LA' where A is the elevation, s the weight of substance, and L the weight of solvent, both expressed in grams; or M where is the volume of solution in cubic centimetres. As an example of the calculation we may take the elevation produced by camphor in acetone. An elevation of 109° was produced by 0.674 g. camphor dissolved in 681 g. acetone. We have, therefore, 0.674 x 1670 M = 6.81 x 1·09 * = 151. K 1560 3030 540 2220 2600 3280 1 The rise of boiling point due to the pressure of the liquid column alone is given in the following table in fractions of a degree for a columu 1 cm. high- Chloroform Water Acetone Benzene Ether Alcohol 0.017 0.016 0.013 204 CHAP. INTRODUCTION TO PHYSICAL CHEMISTRY 16 The molecular weight of camphor, according to the formula C10H100, is 152. An estimation by volume resulted as follows. An elevation of 1.47° was found for 8.1 cc. of an acetone solution containing 0.829 g. camphor. This gives M - 0.829 × 2220 8.1 × 1.47 = 154. The molecular weight of camphor in acetone solution is thus in accordance with the simplest formula that expresses its composition. 5. Dissolved Substances-Depression of Freezing Point-Raoult's Method The apparatus chiefly used for determining molecular weights by the freezing-point, or cryoscopic, method is that devised by Beckmann, and figured in the following illustration (Fig. 43). It consists of a stout test-tube A, provided with a side tube, and sunk into a wider test-tube B, so as to be surrounded by an air space. The whole is fixed in the cover of a strong glass cylinder, which is filled with a substance at a temperature of several degrees below the freezing point of the solvent. The inner tube is closed by a cork, through which pass a stirrer and a thermometer of the Beckmann type. This thermometer has a scale comprising 6° and divided into hundredths of a degree, but the quantity of mercury in the bulb may be varied by means of the mercury in the small reservoir at the top of the scale, and thus the instrument can be adjusted for use with solvents having widely different freezing points. To perform an experiment, a weighed quantity (15 to 20 g.) of the solvent is placed in A, and the external bath is regulated to a few degrees below the freezing point of the solvent. Thus if the solvent. is water, a freezing mixture at about -5° should be placed in the external cylinder C. The temperature of A is lowered by taking it out of the air jacket and immersing it directly in the freezing mixture until a little ice appears. It is then replaced in the air jacket, and the liquid in it is stirred vigorously. As there is invariably overcooling before the ice and water are thoroughly mixed, the thermometer rises during the stirring until it reaches the freezing point, after which it remains constant. This constant temperature is then read off. The tube A is now taken out of the cooling mixture, and a weighed quantity of the substance under investigation is introduced. and dissolved by stirring, all the ice being allowed to melt save a small residue. The tube is replaced in the air jacket and the temperature allowed to fall in order that the liquid may become slightly overcooled. Stirring is then recommenced. The thermometer rises, remains constant for a very short time, and then slowly sinks. The maximum temperature is read off, and taken as the freezing point of the solution. The reason for the subsequent sinking of the XIX DETERMINATION OF MOLECULAR WEIGHTS 205 temperature of equilibrium is plain. As long as the solution remains in the cooling mixture, ice continues to separate. This results in making the remaining solution more concentrated than the original solution, so that its temperature of equi- librium with the solidified solvent will sink (ep. p. 58). The highest temperature registered corresponds therefore most closely to the freezing point of the solution whose concentration is expressed by the weights of substance and solvent taken, although even this is evidently not high enough, for some of the solvent has necessarily separated out as ice before equilibrium can be attained at all. The calculation is precisely the same as that for the elevation of the boiling point. Each solvent has a constant K of its own, representing the fictitious lowering of the freezing point caused by dissolving one gram molecule of substance in one gram of solvent. The hundredth part of this constant, i.e. the depression caused by dissolving 1 gram molecule of substance in 100 grams of solvent, is usually termed the molecular depression. The constants for the most common solvents are as follows:- Solvent. Water Acetic acid Benzene Phenol The formula for calculation is s K ΙΔ’ ΜΕ = ΚΙ 1870 3880 4900 7500 1.458 x 4900 100 x 1.220 The formula C₂HO requires 58. 6 C- 58.6. A where M is the molecular weight of the dissolved substance, s its weight in grams, L the weight of the solvent in grams, and ▲ the observed depression. A solution of 1458 g. acetone in 100 g. benzene showed a depression of 1·220 degrees, whence we have the molecular weight B D WEAR BADNAALILIMUNDOOR TERRITZ!!! FIG. 43. An indispensable requirement, of this method is that the solvent should separate out in the pure state without admixture of the 206 CHAP. INTRODUCTION TO PHYSICAL CHEMISTRY solid dissolved substance. If this does not occur, the method is worth- less as a practical means of ascertaining molecular weights. 6. Pure Liquids-Surface Tension A method for determining the molecular weight of pure liquids as distinguished from substances in liquid solution, was indicated by the Hungarian physicist Eötvös in 1886, but received no attention until it was taken up in a practical manner by Ramsay and Shields in 1893. From theoretical considerations Eötvös reasoned that the expression y (Mv) 3 Ύ where y is the surface tension, M the molecular weight, and v the specific volume, would in the case of all normal liquids be affected equally by the same change of temperature. According to the simple gas laws, the expression p(Mv) is affected equally by temperature for all gases. There is an obvious similarity between these two expressions. For the pressure p in the one we have the surface tension y in the other. For Mv, the molecular volume in the one, we have (Mv), the molecular surface in the other. In the case of gases we might calculate the molecular weight from the relation as follows. The ratio of the change in the expression to the corresponding change of temperature is c, whence M : P。(Mv。) — p₁(Mv₁) to - t₁ Yo(Mv₁) — Y₁(Mv₁)i to - t₁ If we where c is a constant having the same value for all gases. determine this constant once for all in the case of one gas taken as standard, we can calculate the molecular weights of other gases in terms of the molecular weight of the standard gas. Similarly for liquids we have = c(to − t₁) Povo - P1v1 k, whence M [_k(to t₁) 12 Yoo* - 711 where k is a constant having the same value for all liquids. If then we determine the numerical value of this constant for one standard liquid, we can calculate the molecular weight of other liquids in terms of the molecular weight of this standard liquid. It should be mentioned that both the above expressions hold good only when the' molecular weight does not change with the temperature, and are not applicable to gases like nitrogen peroxide in the one case, or liquids like water in the other, where there is such a change. The method for determining the surface tension adopted by Ramsay and Shields was to measure the capillary rise of the liquid XIX DETERMINATION OF MOLECULAR WEIGHTS 207 in a narrow tube. The simplest form of apparatus they used is shown in Fig. 44. FG is the capillary tube, open at the top and blown out to a small bulb at the bottom, in which there is a minute opening to admit the liquid contained in the wider tube A. D is a closed cylinder of very thin glass, which contains a spiral of iron wire and is connected with the capillary by means of a fine glass rod. The capillary tube and liquid under investiga- tion are introduced into A through the tube C before it is drawn out and sealed. After being drawn out at I, the open end of C is connected with the air pump, and the liquid within the tube boiled under diminished pressure, with application of heat if necessary. While the vapour is still issuing from the tube the narrow portion is rapidly sealed off at I. The tube now contains nothing but the liquid and its vapour, and is ready for the experiment. In order to maintain the liquid at a constant temperature, the tube is surrounded from L upwards by a mantle. through which flows a stream of water heated to the desired point. HH represents the section of a magnet which by its attraction for the iron spiral is made to adjust the level of the capillary so that the liquid within it is always at the same place G, a few milli- metres from the end, where the bore of the capillary has been previously determined by means of a micro- scope and micrometer scale. The difference of level between the liquid within and without the capillary is read off by a telescope and scale attached to the apparatus. The temperature is then changed and a fresh reading made, in order to ascertain the temperature-variation. To obtain the surface tension y from the observed capillary rise, we have the approximate formula H Radius of capillary Temperature Capillary height Density 0.0129 cm. 19.4° to 4·20 cm 1.264 L OF 46.1° t 3.80 cm. 1.223 ܝ ееееееее Agrdh = y, FIG. 44. where g is 981, the gravitational acceleration in cm. sec.², h the capillary height in centimetres, r the radius of the capillary tube at G in centimetres, and d the density of the liquid at the temperature of observation. The value of y is then obtained in dynes per centimetre. B The following values were observed by Ramsay and Shields for carbon bisulphide : A H 208 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. From these numbers we have the surface tensions Yo=0·5 × 981 × 0·0129 × 1·264 × 4·20 = 33.58 at 19.4° 0.5 × 981 × 0.0129 x 1.223 × 3·80 29.41 at 46.1°. Y1 = M The mean value of k for various liquids is 2.12, so that for the molecular weight we have, remembering that v = 1/d, - 2.12(19446.1) 33.58/1.264-29-41/1.223 J بتدا = 81.5. The molecular weight of carbon bisulphide corresponding to the formula CS2 is 76. The divergence here found between the theoretical molecular weight and that calculated from the variation of the surface tension with temperature is considerable, and is connected with the fact that the constant k has not precisely the same value for all liquids, but varies nearly 20 per cent, although most liquids give constants not far removed from the mean value - 2:12. If we estimated the molecular weights of gases from the change in the product of pressure and molecular volume with the temperature as suggested above we should find similar variations, as it is only for the gases which are liquefied with difficulty that we have nearly the same temperature coefficient. Thus there is a difference of 6 per cent between the coefficients of expansion of hydrogen and sulphur dioxide, and this would lead to a corresponding error in the molecular weight of the one referred to that of the other if we used the method analogous to that here used for liquids. As we see, the surface-tension method only gives us the molecular weight of one liquid as compared with that of another liquid, and does not in itself afford us evidence of the relation between the molecular weight of a substance in the liquid state compared with that of the same substance in the gaseous state. A discussion of this point is reserved for the next chapter. 7. Traube's Volume Method Isidor Traube suggested a method for determining molecular weights which differs in principle from any of those previously men- tioned. As in the former cases the method depends on the numerical determination of the value of a physical property, namely, the density, but in addition Traube's method presupposes a knowledge of the com- position and constitution of the substance whose molecular weight is to be ascertained. No such knowledge is required for the application of any of the other methods. Instead of considering with Kopp the molecular volumes of liquids at their respective boiling points (cp. p. 140), Traube worked up the density data of liquids at constant temperature and arrived at the following system. As before, the molecular volume of a liquid is ļ XIX DETERMINATION OF MOLECULAR WEIGHTS 209 constituted of the sum of the atomic volumes of the atoms contained in it, but (and herein consists the peculiarity of Traube's method) there is always to be added to the sum of the atomic volumes a constant magnitude termed the molecular co-volume. The following table contains the values of some of Traube's atomic volumes at 15°, Kopp's numbers for the temperatures of ebullition being added for the sake of comparison:- OE >r с H O (in first hydroxyl group) (in subsequent hydroxyls) O (in CO group) O (attached to different C's) S (not attached to 0) Molecular co-volume. 158n 0.957 nE d • N · ▸ • Atomic Volumes. n\v + 25·9, 25.9 Eld-Zv Traube at 15°. It is apparent that there is little parallelism between the two sets of numbers, and that a knowledge of the constitution is even more necessary for Traube's treatment than for Kopp's. From Kopp's equation for the molecular volume it is impossible to calculate a molecular weight, since all the terms depend on the values for atoms only. It is possible to make the calculation from Traube's equation, because here, in addition to the atomic values, we have a term-the co-volume-which depends, not on the atoms, but on the molecule, having the constant value of 25.9 cc. for each molecular quantity in grams. Traube's mode of calculation may best be shown by means of a concrete example. A liquid is known to have the empirical formula CH₁403=158, and it is also known that all the oxygen is combined with carbon only. Its density at 15° is 0.957. The molecular formula must be (CHO)n and the molecular weight 158n; the problem is to find n. According to Traube we have the equation n{(8 × 9·9) + (14 × 3·1) + (3 × 5·5)} + 25·9, whence n = 0·995. Since n is here practically equal to 1, the molecular weight of the liquid is the same as the empirical formula weight. In general if E is the empirical formula weight of a liquid, & the density at 15°, and Ev the sum of the atomic volumes in the empirical ormula, then 9.9 3.1 2.3 0.4 5.5 5.5 15.5 25.9 Kopp at b. p. 11.0 5.5 7.8 7.8 12.2 7.8 22.6 0 Traube applies his method to solutions as follows. A quantity S of the solution is considered which contains the empirical formula P 210 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. weight E of the dissolved substance in grams. If d is the density of the solution, then S/d is the volume which it occupies. If, further, D is the density of the solvent, then V: = be occupied by the quantity of solvent which is contained in S of the solution. The difference between the volume of the solution and of the solvent which it contains Ꮪ ՊՆ = S - E D d S-E D n = may be called the "formula solution volume" of the dissolved sub- stance, and corresponds to E/d, the "formula volume" in the equation for a pure liquid. The calculation is then made exactly as before. For non-aqueous solutions at 15° we have 25.9 V - Zv is the volume which would When water is the solvent, the value of the molecular co-volume in dilute solution is not 25.9 but 12.4, so that for dilute aqueous solutions 12.4 V - Ev Traube's method gives results both for solutions and liquids which generally accord fairly well with the molecular weights obtained by other methods, but sometimes the results are quite discordant. It must be borne in mind that in Traube's equation the essential term is the co-volume, which is a comparatively small residue derived from a complex system of atomic volumes more or less arbitrarily fixed. When the method, therefore, gives an abnormal value of n, the abnormality may be real, or it may arise from some constitutional influence not being taken into account, and is, therefore, correspond- ingly uncertain. 8. Other Methods Besides the methods already mentioned, there are other methods of determining molecular weights of substances in solution based on the osmotic-pressure theory. These methods are not in general use, but occasionally they afford valuable information where the usual means are unavailable. If we take a liquid such as ether, which is only partially miscible with water, and determine its solubility in the water, we find that this solubility is diminished if we dissolve in the ether a substance which is at the same time insoluble in the water. Just as the solution of a substance in ether diminishes the vapour pressure of the ether, so also it diminishes its solubility in any liquic XIX DETERMINATION OF MOLECULAR WEIGHTS 211 I with which it is partially miscible; and the diminution of solubility gives us a means of ascertaining the molecular weight of the substance dissolved in the ether just as the diminution of the vapour pressure does. In the case of liquids the method is of no special value, but in the case of solid solutions the results obtained are of some importance. The so-called solid solutions with which we have chiefly to deal are isomorphous mixtures, i.e. crystals which are uniformly composed of two crystalline substances which present similarity in crystalline form as well as of chemical composition (cp. p. 65). Such mixed crystals are in a sense comparable with liquid solutions, and one crystalline substance may be said to be dissolved in the other. The method of diminished solubility can be applied to find the molecular weight of the dissolved substance. If the mixed crystal consists of a large quantity of the substance A, in which a small quantity of the substance B is dissolved, the molecular weight of B may be determined by finding the diminution of the solubility of A in any solvent which it occasions. In the practical investigation the difficulty is encountered that both A and B are usually soluble in the same solvents on account of their chemical analogy, without which there can be no isomorphous mixture. The calculation in such a case becomes more complex, and the results are of doubtful significance. For a more detailed description of the methods the student may consult- A. FINDLAY, Practical Physical Chemistry, London (1917). RAMSAY AND SHIELDS, Journal of the Chemical Society, 63 (1893), p. 1089, "The Molecular Complexity of Liquids." WALKER AND LUMSDEN, ibid. 73 (1898), p. 502, "Determination of Molecular Weights-Modification of Landsberger's Boiling-Point Method"; also H. N. MAcCoy, American Chemical Journal, 23 (1900), 353. J. S. LUMSDEN, Journal of the Chemical Society, 83 (1903), p. 342, “A New Vapour Density Apparatus." I. TRAUBE, Raum der Atome, Stuttgart, 1899. E. FOUARD, Journal de physique, 5. séries, 1 (1911), 627, "Osmotic Pressure Method with Collodion Films." P. BLACKMAN, Chemical News, 101 (1910), p. 121, "A Convenient Modification of Hofmann's Method." G. BARGER, Journal of the Chemical Society, 85 (1904), p. 286, "A Microscopical Method of determining Molecular Weights." CHAPTER XX MOLECULAR COMPLEXITY THE molecular weights as determined by any of the methods of the preceding chapter are average molecular weights. Indeed, the methods. do not directly give molecular weights at all, but rather the number of gram molecules in a given number of grams of the substance con- sidered (cp. calculation, p. 188). As a rule, the molecules of any one substance under given conditions are all of the same size, so that in most cases the molecular weight as determined is perfectly definite. But even with gases we have sometimes to deal with molecules of a single substance which are not of the same magnitude, with the result that the molecular weight determined from observation is not the weight of any one kind of molecule, but a weight intermediate between real extreme values. Thus the molecular weight of nitrogen peroxide deduced from its vapour density under atmospheric pressure at 4° is 74.8, while under atmospheric pressure at 98° it is 52. Now these molecular weights cannot belong to molecules all of one kind, for the molecular weight corresponding to the simplest formula NO, is 46, while for the next simplest, N,O,, it is double this, or 92. It is evident, therefore, that under the given conditions we must be dealing with a mixture of simple and complex molecules, and that the observed molec- ular weights are merely average molecular weights of all the molecules present. We have here, then, a case of a substance existing in at least two different states of molecular complexity under the same con- ditions. Experiment shows that raising the temperature or lowering the pressure favours the existence of the simple molecules, whilst lowering the temperature or raising the pressure favours the existence. of the complex molecules. 2 The formation of complex, usually double, molecules is frequently encountered with vapours at temperatures near the boiling points of the liquids from which they are derived. The vapours of the fatty acids, for example, have in the neighbourhood of the boiling points of the liquids, molecular weights considerably above those found when the density of the vapour is taken at a higher temperature. It should be remarked, however, that this is by no means the case for all 212 CH. XX 213 MOLECULAR COMPLEXITY liquids, and is on the whole exceptional, although it is true that we very seldom get numbers for the vapour density exactly equal to the theoretical value when the vapour is near its point of condensation. The thorough-going analogy between substances in the gaseous and dissolved states furnishes us with a means of comparing their molecular weights in these two conditions. As a rule, we may say that the molecular weight in dilute solution is the same as the molec- ular weight of the substance in the state of vapour. Since, however, even for vaporous substances we find variations in the molecular weight under different conditions, we cannot expect in every case that there should be identity of the gaseous and dissolved molecules. In general, we may say that substances which tend to form complex molecules in the gaseous state exhibit the same tendency in solutions, the extent to which the formation of complex molecules proceeds depending largely on the nature of the solvent, as well as on the temperature and osmotic pressure (concentration) of the solution. The numbers in the following table give the percentage of nitrogen peroxide existing as double molecules in various solvents, the concen- tration in each case corresponding to an osmotic pressure of about seven atmospheres. At this pressure, and at the temperatures given. in the table, the gaseous substance would exist almost entirely as double molecules; as a matter of fact, the substance is liquid under these conditions, with the molecular formula NO, as judged by the method of Ramsay and Shields. Solvent. Acetic acid Ethylene chloride Chloroform Carbon bisulphide Silicon tetrachloride. Double Molecules at 20°. 97.7 95.8 92.3 87.8 84.3 Double Molecules at 90°. 95.4 91.3 85.5 77.5 74.0 This question of the influence of the solvent on the molecular weight of the dissolved substance is one of practical importance in the selection of a solvent in which to determine the molecular weight of a given substance by means of the freezing- or boiling-point method. As a rule, what we wish to obtain is the smallest molecular weight, and it is therefore expedient to select a solvent in which the tendency to the formation of complex molecules is as little marked as possible. Of the ordinary solvents, water and alcohol are those in which the for- mation of complex molecules is least apparent; so that the former would be preferably chosen for the cryoscopic method, and the latter for the boiling-point method. Acetone comes next in order, and is suitable for determining the elevation of the boiling point; acetic acid is a similar solvent for the cryoscopic method. In ether, chloro- form and benzene, the tendency to association of the simple molecules. is often considerable, so these solvents should not be used if it is suspected that the substance tends to form complex molecules. Of the substances which show a tendency to the formation of 214 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. complex molecules, organic bodies containing the groups hydroxyl (OH) and cyanogen (CN) are of the most frequent occurrence. Thus the alcohols and the carboxyl acids almost invariably tend to form complex molecules in a solvent such as benzene. The subjoined tables serve to show this behaviour, the freezing-point method with benzene as solvent being employed. In the first column is given the number of grams. of substance dissolved in 100 g. of benzene. ETHYL ALCOHOL, C₂H(OH) = 46 Molecular Weight. Concentration. 0.494 1.088 2.290 3.483 8.843 14.63 0.465 1·195 2.321 4.470 8.159 50 61 82 100 159 208 ACETIC ACID, CH3(COOH)=60 Molecular Weight. Concentration. 110 115 117 122 129 10.85 16.55 23.30 PHENOL, CH(OH)=94 Concentration. 0.651 2.589 7.255 Concentration. 0.337 1.199 2.481 3.970 7.980 17.29 PHENETOL, CH¿(OC₂H¿)=122 BENZOIC ACID, СН(COOH)=122 Concentration. Molecular Weight. 0.567 1·444 2.603 4.725 It will be noticed that except in the case of ethyl alcohol the molec- ular weight does not even in the strongest solutions rise much above twice the value for the simplest molecule corresponding to the gener- ally accepted formula. With ethyl alcohol the formation of complex molecules proceeds much further than this, if we assume the simple gas laws to hold good for the solutions of the strengths investigated. To give an idea of the behaviour of a substance which shows no tendency to molecular complexity, the numbers for phenetol solutions are subjoined. This substance is derived from phenol, and has the formula CH¿(OC₂H) and the molecular weight 122. The solvent was again benzene, and with the removal of the hydroxyl group the power to form associated molecules is lost. 5 2 Molecular Weight. 120 119 121 Molecular Weight. 144 153 161 168 188 223 122 125 128 223 228 232 236 Even in what must be accounted a very concentrated solution the molecular weight does not here greatly depart from the normal value corresponding to the formula. Soluble salts, strong acids, and strong bases in dilute aqueous solu- tion invariably exhibit too small a molecular weight, whether this is determined by the osmotic-pressure, vapour-pressure, freezing-point, or XX 215 MOLECULAR COMPLEXITY boiling-point methods. The normal molecular weight of sodium chlor- ide, corresponding to the formula NaCl, should be 58.5, but all the above methods give for its molecular weight when dissolved in water numbers approximating to 30, i.e. about half the normal value. Here we are dealing with a dissociation instead of an association of the simplest molecules. It is evident that unless we halve the atomic weights of sodium and chlorine, and write the formula na cl, where na and cl represent the half atomic weights, the two molecules into which the normal molecule of sodium chloride dissociates cannot be the same. The most probable assumption to make is that the atomic weights retain their ordinary value, and that the normal molecule splits up into two different molecules, Na and Cl. The average molecular weight of the sodium chloride in solution would then, if the dissociation were complete, be half the normal molecular weight corresponding to the simplest formula. It is preferable to make this assumption of dissocia- tion into atoms rather than the assumption that our ordinary atomic weights are twice what they should be, because we find that in other cases the last assumption would not account for the molecular weight observed. Thus in very dilute solutions of sulphuric acid or sodium sulphate, the atomic weight deduced from the boiling-point or freezing- point methods is considerably less than half the normal molecular weight, so that halving the atomic weights of the constituent atoms would be insufficient to produce the low molecular weight actually observed. The assumption that the dissociation is into products of different kinds receives support from what we learned of the properties of salt solutions. It will be remembered that very frequently the properties of salt solutions were such that the positive and negative radicals appeared to be independent of each other (cp. p. 169). Now if we assume that the independence is caused by the salt actually split- ting up into these radicals, each of which then acts, as far as osmotic pressure and the derived magnitudes are concerned, as a separate molecule, we have an explanation both of the peculiarities of the physical properties of aqueous salt solutions and of the low molecular weights exhibited by the dissolved salts. In a subsequent chapter the subject of dissociation in salt solutions will be treated in greater detail. This dissociation is not confined to aqueous solutions, although it is exhibited by them to the greatest extent, but is found also, for example, in alcoholic and acetone solutions. The following numbers were obtained by the boiling-point method, and show the abnormally small molecular weight in aqueous and alcoholic salt solutions :— SODIUM ACETATE IN WATER, C₂H₂O₂Na=82 Concentration. Molecular Weight. 1.01 4.19 6.32 10.69 46 48 46 45 SODIUM IODIDE IN ALCOHOL, NaI 150 Molecular Weight. Concentration. 1.56 5.00 8.22 14.35 109 118 115 108 216 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. In the chapter on solutions we found that a substance distributed itself between two immiscible solvents in such a way that there was a constant ratio of the concentrations of the substance in the two solu- tions, depending on the solubility of the substance in the solvents separately. This constant partition coefficient is observed, however, only when the dissolved substance has the same molecular weight in both solvents. If, for example, we shake up acetic acid in small quantity with benzene and water, we do not get a constant partition coefficient of the acid between the two solvents independent of the quantity of acetic acid present, as we should if the molecular weight were the same in both solvents; but we get a ratio of concentrations which varies as the values of the concentrations themselves change. This is due to acetic acid in benzene solution consisting practically of double molecules, whereas in aqueous solution it consists of practically single molecules. In such a case we have the following rule for the concentrations in the two solvents. Let the solvents be A and B, and let the concentrations of the dissolved substance in these two solvents be C and C, respectively. Then if the dissolved substance in the 1 B solvent A has a molecular weight n times greater than the molecular weight of the substance when dissolved in B, the ratios • "JC/C or C/C B CB 0.043 0.071 0.094 0.149 ጎ are constant. It will be seen that when the molecular weight in both solvents is the same, we get the partition coefficient as a particular case of the general rule. In the above-mentioned instance of the distribution of acetic acid between benzene and water, where the molecular weight in benzene is approximately twice that in water, the ratio C2/C, should be approximately constant. The following experiments show that this is indeed the case. In the first column we have the concentration of acetic acid in the benzene, in the second the concentration in the water, in the third the ratio of these concentrations, and in the fourth the expression which should be. nearly constant. W B Cw 0.245 0.314 0.375 0.500 B CW/CB 5.7 4.4 4.0 3.4 w/CB 1.40 1.39 1·49 1.69 As the third column shows, there is no constant partition co- efficient. It has already been pointed out (p. 54) that there is a great analogy between the partition coefficient of a substance between two solvents and the solubility coefficient (or, shortly, solubility) of a gas in a XX MOLECULAR COMPLEXITY 217 same. liquid according to Henry's Law, and the mere existence of this law is sufficient to indicate that for the substances to which it applies the molecular complexity in the dissolved and gaseous states is the If the molecular complexity of a substance in the gaseous state is different from what it is when dissolved in a given solvent, Henry's Law, that the amount of gas dissolved is proportional to the pressure, i.e. that the ratio of the concentrations of the substance in the two states is constant, no longer holds good. Thus the solubility of carbon dioxide in water is not exactly proportional to the pressure, varying by as much as a hundred per cent, between 1 atm. and 30 atm. This is no doubt due to the formation of gaseous C2O4 molecules under great pressures, as may be deduced from vapour- density determinations. If we make allowance for this increase of molecular complexity of the gaseous phase, as compared with the dis- solved phase, on increase of pressure, we find that the theoretical expression, where n now varies with the pressure, gives a fair approxi- mation to constancy. When we come to compare many properties of substances in the liquid state with each other, we find that liquids containing the hydroxyl group, such as the alcohols, water, and the fatty acids, are quite exceptional in their behaviour. In the first place, it must be noted that the boiling points of such compounds are exceptionally high, and this alone might lead us to suspect an exceptionally high molecular weight in the liquid state. As a rule, we find that on comparing the boiling points of similar compounds, the compound with highest molecular weight has the highest boiling point. If we substitute a CH, group for a C₂H, group we have, in accordance with the general rule, a fall in the boiling point, and similarly if we substitute H for CH. But if these groups are attached to an oxygen atom, we find that though the substitution of CH, for C¸H lowers the boiling point, the substitution of H for CH, raises it greatly. The following substances afford examples of this be- haviour:- 5 5 Ethyl methyl ether Dimethyl ether Methyl alcohol Water Propyl acetate Ethyl acetate Methyl acetate Hydrogen acetate C₂H. O. CH, 5 CHg. O. CH. CHg. O. H H.O.H CH. COOCH, CH.COOCH CHS. COOCH3 CH₂.COOH Boiling Point. 11° - 24° +66° 100° 102° 77° 57° 118° Difference. - 35 +90 +34 - 25 - 20 +61 218 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. Another point in which organic hydroxyl compounds differ from most other liquids is the following. If a substance obeyed the gas laws exactly, its density at the critical temperature and pressure could easily be calculated from Avogadro's Law. Now all liquids have a much greater critical density than the calculated value, and for most of them the actual critical density is 3.77 times the theoretical. With hydroxyl compounds, however, the factor is greater than this normal value, varying from 4 to 5. This points to association of simple molecules under the critical conditions. Again, the vapour-pressure curves of most liquids do not cut each other when tabulated on the same diagram, but the curves of the hydroxyl compounds often cut each other and sometimes those of the "normal" liquids. This exceptional behaviour once more points to the existence of complex molecules in the liquids, which are progress- ively decomposed as the temperature rises. As we have seen in the preceding chapter, the surface-tension method for determining molecular weight gives a constant approxi- mately equal to - 2.1 for most liquids. The constants for the follow- ing substances are much lower than this mean value, and vary with the temperature-alcohols, fatty acids, water, acetone, propionitrile, and nitroethane. It is impossible to calculate the molecular weight of such substances by means of the formula given on p. 206, for that formula assumes that the molecular weight remains constant through the range of temperature examined. By suitably altering the formula, however, probable values may be obtained for substances whose molecular weight changes with the temperature, and a few of these are exhibited in the subjoined table, in which t is the temperature and n the associa- tion factor, i.e. the number of times the molecular weight of the liquid is greater than that corresponding to the ordinary formula :- t 0° 20 60 100 140 t METHYL ALCOHOL ጎ 2.65 2.32 2.06 1.86 1.75 90° WATER + 20 110 180 220 N 1.71 1.64 1.52 1.40 1.29 ACETIC ACID t 20° 60 100 140 280 - 90° ETHYL ALCOHOL t +- 20 N 2.13 1.99 100 180 220 1.86 1.72 1.30 N 2.03 1.65 1.39 1·15 1.03 In each case a diminution of n with increasing temperature is observable, indicating that the associated molecules decompose as the temperature is raised. In all the above instances, except water, the XX MOLECULAR COMPLEXITY 219 association factor is greater than 2, so that we are probably dealing with molecules more complex than double molecules. Traube's method also leads to the conclusion that the molecules of these substances are complex, although his values of n are generally lower than those obtained by the method of Ramsay and Shields. Reviewing the facts regarding the associative tendency of the common solvents, we find first that hydroxylic solvents such as water and alcohol of themselves tend to form complex molecules, whilst as solvents they are characterised by their dissociative power—that is, they not only prevent the association of molecules dissolved in them, but even in the case of salts dissociate the normal molecules into still simpler molecules. Hydrocarbon solvents such as benzene, on the other hand, which themselves form no complex molecules, promote the association of molecules dissolved in them, and in no case exhibit a dissociative action comparable with that shown by the other class of solvents. These facts may be harmonised to some extent by the following general consideration. A self-associating liquid has mole- cules which possess a certain power of combination with each other; it is not improbable, therefore, that this power of combination may be exerted on the molecules of substances dissolved in the liquid, counter- acting any power of self-association which these molecules may them- selves possess. This would be in harmony, for example, with the normal molecular weight which alcohol shows when dissolved in water. On the other hand the molecule of a solvent such as benzene, which shows no tendency to unite with other like molecules, may be supposed to be without action on the molecules of substances dissolved in the liquid, thus permitting these substances still to display any self-asso- The ciative tendency they may exhibit in the undissolved state. behaviour of alcohol and phenol in benzene solution would in this way be intelligible. We may thus distinguish between inert solvents, without self-association, and without action on dissolved substances; and active solvents, themselves self-associative, which tend to inter- act with substances dissolved in them. The saturated hydrocarbons are typical examples of the first class; water is typical of the second class. A rough but instructive illustration of the effect of a solvent on a The vapour dissolved substance may be found in the case of iodine. of iodine is of a violet colour, and one might therefore expect that if iodine were dissolved in an absolutely inert solvent, the colour of the solution would likewise be violet. This is approximately the case for the saturated hydrocarbons, chloroform, and carbon disulphide, as is well known. These solvents, therefore, may be supposed to have. little influence on the iodine dissolved in them. Water and alcohol, on the other hand, yield brown solutions, and may therefore be reasonably supposed to form some species of compound with the If iodine, as there is no evidence of dissociation of iodine molecules. 220 CH. XX INTRODUCTION TO PHYSICAL CHEMISTRY iodine is dissolved in glacial acetic acid at the ordinary temperature the solution is brown, but when the solution is heated to the boiling point of the acid it assumes a distinctly pink colour. This behaviour may be held to indicate that the compound of iodine with acetic acid, stable at the ordinary temperature, decomposes when heated to the vicinity of 100°. Mixtures of inert liquids which show no tendency to self-associa- tion have in general properties which may be calculated from the properties and proportions of the components. Thus the volume of the mixture is equal to the sum of the volumes of the components ; the energy content of the mixture is equal to the sum of the energy contents of the components, i.e. there is no heat change on mixing; the heat capacity of the mixture is equal to the sum of the heat capacities of the components, etc. In this respect mixtures of non- associative liquids resemble mixtures of gases, which throughout obey Dalton's Law (p. 76) not only with reference to pressure or volume but to most other properties. It might be objected that as we compare the molecular weights of liquids only amongst themselves, we have no right to compare the molecular weight of a substance in the liquid state with that of the same substance in the dissolved or gaseous state. The mere fact of the continuity of the gaseous and liquid states (Chap. VIII.) is not in itself sufficient to indicate that the molecular condition is the same in the two cases; but where the surface-tension method shows the liquid to have a molecular weight corresponding to a complex molecule, there we have behaviour amenable to much less simple laws than those which hold good for the bulk of liquids. We should be disposed, therefore, to assume that the origin of the comparatively simple laws is to be sought in the molecular condition in the liquid and vaporous states being the same. For if the molecular conditions in the two states were different, even in the case of normal liquids, it would be difficult to explain the abnormalities shown by liquids such as alcohol. That the normal molecular weight in solution is identical with the normal molecular weight in the gaseous state is practically certain from the existence of Henry's Law, and from the perfect analogy in pressure, volume, and temperature relations exhibited by dissolved and gaseous substances. CHAPTER XXI COLLOIDAL SOLUTIONS OF recent years colloidal solutions have formed the subject of numerous investigations, for they not only present points of much theoretical interest, but are also of great practical importance in both biological and industrial chemistry. These solutions, sometimes known as pseudo-solutions, are characterised by their very small osmotic pressure and derived magnitudes, in some cases by their power of gelatinising or coagulating, and by the want of any definite saturation point. The dissolved particles in true solutions are, as we have seen in the preceding chapters, presumed to be molecules of the solute. itself or these in combination with each other or some molecules of the solvent. The particles in colloidal solutions are on the whole larger aggregates than those in true solutions, but it is not unlikely that the two kinds of solution gradually merge one into the other and The that no sharp line of demarcation can be drawn between them. term sol is frequently employed to indicate a colloidal solution as distinguished from a true solution. We have so far looked upon true solutions as being perfectly homogeneous and consisting of one phase only. Colloidal solu- tions are regarded as consisting of two phases, one the solvent and the other the dissolved or, better, the dispersed phase. In a true solution there is no question of a surface existing between the two constituents. In a sol some of the dispersed particles are of sufficient magnitude to present in the ordinary sense a surface of separation with respect to the rest of the system, with corresponding development of surface forces, to which many of the properties of colloidal solutions must be attributed. The presence of the relatively coarse dispersed phase may be shown by means of Tyndall's experiment. If a beam of light from an arc-lamp is passed through a dust-free solution, and observed from a direction at right angles to its path against a dark ground, it is practically invisible, whereas if it is passed through a colloidal sol the beam is plainly visible (as it would be in a pure liquid containing dust-particles in suspension), and the light when viewed through a Nicol's prism is seen to be polarised. This test is 221 222 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. exceedingly delicate, but cannot be regarded as affording an absolutely sharp distinction between colloidal and true solutions. A convenient division of sols into two classes may be made according as the dispersed phase is solid or liquid. If the dispersed phase is solid, the sol resembles a suspension with the suspended particles in a very fine state of division, and is therefore called a suspensoid. If on the other hand the dispersed phase is liquid, the solution resembles an emulsion of very fine liquid particles, and is therefore called an emulsoid. The colloidal solutions of the metals in water and organic solvents yield, on account of their simplicity, the best examples for the study of suspensoid solutions. They may be prepared either by reduction of the salt of a metal in aqueous solution, e.g. gold from gold chloride. by alkali and formaldehyde, or by electrical dispersion of the metal, which is of very general application. For example, if an electric arc is formed under pure water between two stout platinum wires, the ends of which are brought near enough for the dis- charge to pass, fine particles of platinum are shot off and dispersed through the water so as to produce a brown colloidal solution of A platinum. The solution. SO obtained is stable, shows no visible particles under the microscope, and passes unchanged through ordinary filtering materials. That the dissolved substance is not a compound of platinum, formed, for example, by the action of platinum on the water or dissolved gases, is proved by the fact that the platinum in colloidal solution retains all the catalytic activity of metallic platinum, which is absent from platinum compounds. This colloidal solution differs from a true solution inasmuch as the platinum it contains is readily precipitated by the addition of a small quantity of a good electrolyte. Similar precipitation by electrolytes is frequently met with in colloidal solutions. A C FIG. 45. M B The size of the platinum particles in such a sol may be estimated by means of an instrument known as an ultramicroscope. This instrument consists of a powerful microscope applied to the examina- tion of the solution through which a narrow but intense beam of light passes horizontally. Although with ordinary illumination the platinum particles are invisible in the microscope on account of their minute dimensions, yet when they are intensely illuminated by being XXI COLLOIDAL SOLUTIONS 223 placed in the path of a powerful beam of light they become visible as points by the light which they reflect and scatter. In the same way the stars, of too small angular dimensions to be themselves visible objects, become visible as light centres in a dark field; and just as the stars are invisible by daylight, so the metallic particles in the beam which passes through the solution are invisible unless light from other sources is carefully excluded. The nature of the instrument is diagrammatically illustrated in Fig. 45. The very shallow horizontal beam of light AA is passed through the solution in the cell C, the microscope objective in B serving as condenser to concentrate the light in the cell. The path of the ray in the cell is observed by the vertical microscope M. The appearance of the beam in the cell as observed by the microscope is shown in Fig. 46, where the circle represents the field of vision in the cell C. If pure water or a true solution is in the cell, the path of the beam is practically invisible, as there is no reflection or scattering of the light. When the cell is filled with the colloidal sol, however, the particles of the colloid appear as small specks of light in brisk motion in all directions. By means of the micrometer scale of the microscope indicated in the figure, the number of particles in a definite volume of the solution may be counted, and the concentration of the solution being known, the average weight of the particles may be determined. Assuming that the particles have the same density as the massive metal, it is thus possible to estimate the volume and average diameter of the particles. Estimated in this way, the diameter of the particles in a colloidal platinum solution was found to be about 45 µµ, or 45 x 10-6 mm. When a dilute solution of gold chloride is made alkaline with potassium carbonate and reduced by means of formaldehyde, or by hydra- zine, or an ethereal solution of phosphorus, colloidal gold solutions are produced, the fineness of the particles varying with the concentration of the solutions employed and with the conditions of reduction. The colour of the colloidal gold solutions by transmitted light is subject to great variation, all shades from blue, through brown, green, ruby-red, yellowish-red to reddish-yellow being observed. The diameter of the particles varies from 300 up to less than 10 µp, which is the limit of resolving power of the ultramicroscope. The instrument, however, still shows by a diffuse illumination of the path of the beam the presence of particles smaller than those which can be individually detected. This corresponds to the nebulous appearance of a star-cluster in a telescope of insufficient power to resolve it into the individual points of light. As we shall see in Chapter XXXII., the diameter of a molecule is of the order of 1 μu, so that in the gold particles of solutions which FIG. 46. 224 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. cannot be resolved by means of the ultramicroscope, we at least approximate to particles of molecular dimensions. The correlation between size of colloidal gold particles and colour of solution is not definite. Red gold sols and blue gold sols may be obtained which have particles of the same size. The change from red to blue which accompanies incipient coagulation is generally attributed to the massing together of the individual particles to form clusters, the clusters of a blue solution, however, not being necessarily larger than the individual particles of a red solution. It has been said above that the metallic particles observed by the ultramicroscope in a colloidal solution are in a state of brisk agitation, known as Brownian movement. This Brownian movement is shown by all dispersed particles of less than 3000 μu diameter, and becomes livelier as the particles are smaller, and as the temperature is higher. On the assumptions of the kinetic molecular theory, it is possible to calculate the motion of minute suspended particles due to the impact. on them of the molecules by which they are surrounded. The agree- ment between the calculated and observed motions is satisfactory, so that we can with confidence attribute the Brownian movements to molecular bombardment. The actual path of individual particles has been recorded photographically by means of the ultramicroscope and cinematographic apparatus. Many metals have now been produced as suspensoids, not only in water but also in organic solvents. A general method for their pro- duction in the latter, say in ether, is to cover the metal, cut into small pieces, with pure ether, and then employ a high-tension oscillatory discharge, instead of a direct arc. The discharge is produced by immersing in the ether the terminals of an induction coil, with several Leyden jars in parallel with the coil. The oscillatory discharge passes as an arc between the pieces of metal, rapid dispersion of which takes place. Most of the metallic solutions obtained in this way are brown, but those of the alkali metals in ether are brightly coloured. It has already been stated that suspensoids are coagulated and precipitated by the addition of electrolytes to the solution. This precipitation is irreversible, that is, the coagulum or precipitate once formed cannot be brought back again into the finely dispersed con- dition. Colloids of this nature, which do not readily redissolve after separation, are said to be lyophobe. The precipitating action of well- ionised electrolytes is doubtless connected with the fact that the metallic particles in the suspensoid solution are electrically charged with regard to the solvent. That they are so charged may be proved by immersing two electrodes in the solution, when it is found that the particles travel to the anode, indicating that they carry a negative charge. Most good electrolytes effect the precipitation, and the concentrations required are usually small. In the case of particles carrying a negative charge, the precipitation appears to be due chiefly to the positive XXI 225 COLLOIDAL SOLUTIONS ion, and takes place the more readily as the valency of the positive ion is greater. Thus a solution of calcium chloride is more effective as a precipitant than an equivalent solution of sodium chloride. In pure water nearly all suspensoids are charged negatively, the chief exceptions amongst inorganic substances being the metallic hydroxides, which have a positive charge. In the latter case the valency of the negative ion determines the readiness with which an electrolyte effects the precipitation. Thus a ferric hydroxide sol is precipitated much more readily by a solution of sodium sulphate than by an equivalent solution of sodium chloride. When two sus- pensoid solutions of opposite electrical charges are mixed, the particles at once precipitate each other. A typical reversible emulsoid solution, e.g. one of gelatine, or agar- agar, shows heterogeneity under the ultramicroscope, but is very different in its properties from a suspensoid solution. The suspensoid solution is mobile, whilst the emulsoid solution is viscous even at small concentrations, and at greater concentrations sets to a stiff jelly on cooling. The emulsoid solution, again, is not coagulated by the addition of small quantities of electrolytes. The addition of large quantities of neutral salts, however, may effect the coagulation, but in this case the action is reversible, that is, the coagulum can again be brought into emulsoid solution in pure water. Colloids of this type are said to be lyophile. When a warm solution of gelatine is cooled, it sets to a jelly or gel if the original sol is sufficiently concentrated. Ultramicroscopic particles are most clearly visible in solutions of about 0.5 to 1 per cent. concentration, which in the cold assume the form of soft jellies. At higher and at lower concentrations the visibility of the particles diminishes. The setting of a sol to a gel may be likened either to a process of incipient crystallisation or else with more probability to the phenomena observed when a solution of aniline in water is cooled with separation of two liquid phases (p. 48). Under the ultramicroscope a 0.5 per cent sol of gelatine, which appears almost homogeneous at higher temperatures, shows when a certain degree. of cooling has been reached ultramicroscopic particles in a state of confused motion, which on further cooling coalesce into comparatively large flocks, the motion of the particles within which gradually ceases. The separation of two liquid phases exhibits similar phenomena under the ultramicroscope, the main difference being that the small ultramicroscopic particles which appear near the point of separation fuse together to definite liquid drops. In the gel the flocks form a sort of fine meshwork in which the remainder is entangled much as water is in a sponge. A dilute aqueous gel offers little more resistance to the diffusion or electrolytic conduction of salts dissolved in it than does pure water, proving that there is free liquid communication throughout. 226 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. At first sight it is somewhat surprising that two liquid phases may be supposed to give rise to a stiff jelly with many of the mechanical properties of a solid. The phenomenon, however, is familiar enough when an additional gaseous phase is present, which in itself would not lead towards rigidity. Thus in foams such as whipped cream, or mayonnaise (egg and oil), we can by sufficiently fine mechanical division of the liquid phases, even though a gaseous phase is enclosed, obtain a more or less rigid mass, very different in mechanical properties from the phases from which it was derived. Practically solid lubricants may be made by intimate mixing of mineral oil with small quantities (about 1 per cent) of soap solution. It should be borne in mind that in dealing with ultramicroscopic particles there is more than the usual difficulty in distinguishing between solid and liquid (p. 57) when the particles are not crystalline. Soap solutions form an interesting and important class of rever- sible colloids. The soaps are salts and therefore their solutions show electrical conductivity owing to ionisation (Chap. XXII.), and since the acids from which they are derived are weak acids, they are parti- ally resolved in aqueous solution into acid and base (Chap. XXVIII.), the extent of hydrolysis being, however, comparatively slight. If we take sodium oleate as an example of a soap salt we find that one and the same solution at the same temperature may be brought to exist in the form of a fluid sol, a transparent elastic gel, or a white opaque curd. Apart from the difference in mechanical rigidity the sol and the gel have the same physico-chemical properties, e.g. their electrical conductivity, their refractive index, and their osmotic activity, as measured by the lowering of vapour pressure, are identical. The curd on the other hand shows a much smaller electrical conductivity, part of the soap having separated in the form of white fibres which hold enmeshed a solution more dilute than the original sol or gel. Emulsoids very generally exert a protective action on suspensoids with regard to their precipitation by electrolytes. Thus a very small quantity of gelatine added to a colloidal solution of gold will prevent the coagulation and precipitation of the gold particles by the addition of an electrolyte. The explanation generally given of this action is that the liquid emulsoid particles envelop the solid suspensoid particles with which they come in contact, and so prevent their coalescence to larger aggregates. The degree to which protective action is exercised by a colloid is often given in terms of the so-called gold number, i.e. the weight in milligrams of the substance which just fails to prevent the change from red to violet, characteristic of incipient aggregation, when 1 cc. of a 10 per cent sodium chloride solution is added to 10 cc. of gold sol of a concentration of about 0·0055 per cent. The smaller the gold number of the substance the greater is its protective action. Gold numbers of ordinary colloids vary from 0.005 for gelatine to 5 for starch. XXI COLLOIDAL SOLUTIONS 227 The capability of substances to form gels, including gelatinous precipitates, is much more general than was at one time imagined. The chief requisite for the formation of gels seems to be that the concentration of the substance if formed by precipitation should be very much greater than its true solubility in the liquid solvent. Thus while it is impossible to obtain a gel of a soluble crystalline substance like sodium chloride in water, it is easy to obtain gelatinous sodium chloride in an organic liquid like benzene in which the sodium chloride is insoluble. Again, barium sulphate when pre- cipitated from aqueous solutions of ordinary concentration is not gelatinous, but if we mix very concentrated solutions of a barium salt and a sulphate, the barium sulphate appears in the gelatinous form. For example, if we shake up nearly saturated solutions of manganese sulphate and barium thiocyanate together, we obtain a stiff jelly like that formed when silicic acid is precipitated from a soluble silicate. Such gels, unlike those of gelatine or agar, are irreversible and cannot by heating or dilution be made to revert to the state of sols. It is characteristic of both suspensoids and emulsoids that the diffusibility of the particles in both is small compared with that of molecules in true solution. Graham, to whom the first investigations of the subject are due, divided dissolved substances into two classes, crystalloids and colloids, according as they diffused rapidly or slowly in water. The following table gives the diffusion coefficients of typical examples :- Crystalloids Nitric acid Sodium chloride Urea. Cane-sugar 2.1 1.0 0.9 0.4 Colloids Pepsin Egg-albumin Caramel Emulsin • • • · 0.07 0.06 0.05 0.04 Crystalloids may also be distinguished from colloids by their power of passing through certain animal and vegetable membranes, and on this distinction Graham based a process for the separation of the two classes, which he called dialysis. A crystalloid in solution encounters little resistance in passing through a moist membrane of animal or vegetable parchment, but a colloid only passes through such a membrane very slowly, if at all. If therefore a mixture of a crystalloid and a colloid in solution be placed on one side of the membrane, the other side being washed by pure water, the crystalloid will gradually pass through the membrane and leave the colloidal solution behind. A method, known as ultrafiltration, has been applied to colloidal solutions, and is analogous to dialysis. An ordinary filter-paper is impregnated with gelatine which is afterwards hardened in formalde- hyde solution (formalin). This yields a membrane which is, like parchment, practically impermeable to colloids, and if the membrane. is properly supported, and pressure applied to the colloidal solution, the solvent with any dissolved crystalloid may be expressed, and so a 228 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. 4 process resembling filtration may be effected. For a given colloid a certain concentration of the gelatine used in preparing the membrane may be found, which under the pressure applied (about 5 atm.) will just suffice to hold back the colloid. A more dilute gelatine will permit the colloid to pass. Thus a 2 per cent gelatine solution will hold back colloidal platinum prepared according to Bredig's method of electrical dispersion, but permits the passage of silicic acid, which requires a paper impregnated with gelatine solution of 45 per cent to stop it. The conclusion is that the particles of silicic acid in the colloidal solution of that substance are smaller than the particles of colloidal platinum. By this method we can therefore roughly arrange colloidal solutions in the order of the average size of their particles, and the results obtained agree in the main with those derived from ultramicro- scopic study and determination of molecular weight. The ordinary methods for the determination of molecular weight in solution may be applied to colloidal solution of both types. As we might imagine, from the slowness of diffusion displayed in these solutions the molecular weights found are very high. Not only are they extremely great, but in many cases they are very variable and uncertain owing to the unavoidable presence of small quantities of dissolved crystalloids to which the observed magnitude (say freezing-point depression) used in the determination of the molecular weight may be largely due. The molecular weight of dextrin which is not completely retained by an ultrafilter made with 10 per cent gelatine is estimated at about 1000, and that of silicic acid at about 50,000. How far we are justified in interpreting the experimental result in the usual way in order to draw deductions as to molecular weights is doubtful, for no definite relation- ship has been established between concentration of colloid and the corresponding lowering of the freezing point, or any similar magnitude. Considerable light has been thrown on the structure and magnitude of colloidal particles by means of X-ray analysis. It will be shown. (Chap. XXXII.) that a homogeneous beam of X-rays may be applied to elucidate the internal structure of a crystal. By an ingenious modification the method may be applied, not to a single definitely oriented crystal, but to a rod of compressed crystalline particles of every possible orientation. By measurement of the lines obtained on a photographic film, conclusions may be drawn both as to the structure of the crystals, and as to the size of the crystalline particles. Liquids and truly amorphous solids, e.g. glass, give none of the characteristic lines of crystals. Investigation showed that the particles of colloid gold protected by gelatine and evaporated to dryness are in reality. crystalline, and that they possess the same structure as that of massive gold crystals. The values for the magnitudes of the colloidal particles. of gold agreed closely with those obtained by other methods. Colloidal silicic acid appeared to be mostly amorphous but contained many crystalline particles. Gelatine was completely amorphous. XXI COLLOIDAL SOLUTIONS 229 W. W. TAYLOR, The Chemistry of Colloids, 1921. E. HATSCHEK, Introduction to the Physics and Chemistry of Colloids, 1919. WOLFGANG OSTWALD, Handbook of Colloid Chemistry, 1919. R. ZSIGMONDY, Kolloidchemie, 1920. This work contains an appendix by P. SCHERRER on the determination of the structure and size of colloidal particles by means of X-rays. British Association Reports, 1917, 1919, 1920: "Colloid Chemistry and its General and Industrial Applications." First, Second, and Third Reports. These reports on Colloid Chemistry are also published separately by H.M. Stationery Office. M. E. LAING and J. W. MCBAIN, "Sodium-Oleate Solutions," Journal of the Chemical Society, 117 (1920), p. 1506. \ CHAPTER XXII ELECTROLYTES AND ELECTROLYSIS IF we take platinum wires from the terminals of a battery and join their free ends by another metallic wire, we find that a current of electricity flows through the system without being accompanied by any motion of ponderable matter. If we dip the free ends of the wires from the terminals into water acidulated with sulphuric acid, we find that an electric current again flows through the system, but that now the passage of the current is accompanied by chemical phenomena and motion of matter. Oxygen appears at one wire where it dips into the solution, and hydrogen at the other; and if we continue. the passage of the current, taking measures to prevent mechanical mixing in the solution, we shall find that the sulphuric acid will accumulate round the wire at which oxygen is evolved. We distinguish, therefore, between two kinds of electrical conduc- tion, viz. metallic conduction, which is unaccompanied by material. change, and electrolytic conduction, which is essentially bound up with movement, and usually chemical change, in matter. In this chapter we are concerned with electrolytic conduction and the accompanying phenomena, and have first to ascertain what substances are conductors in this sense. Comparatively few pure substances act as electrolytic conductors, the chief exceptions being fused salts and bases. Fused silver chloride conducts electricity freely, and is itself decomposed during the process, and there is even a perceptible electrolytic conduction in the substance in the solid state at temperatures not far removed from its melting point. It was by the electrolysis of fused salts that many of the metals were first prepared. Lithium and magnesium, for example, may be easily obtained by the passage of an electric current through their fused anhydrous chlorides; and Davy first discovered the metals of the alkalies by electrolysing the fused bases, potassium and sodium hydroxides. At present, aluminium is manufactured on the large scale by the electrolysis of fused aluminium oxide, and many other technical applications of similar processes are being developed. 230 CH. XXII ELECTROLYTES AND ELECTROLYSIS 231 Electrolysis of such autolytic conductors has as yet offered little of a regular character adapted to theoretical treatment from the chemical point of view, and the subject has not hitherto been systematically worked out. The result is that our systematic knowledge of electro- lysis is confined almost entirely to the second class of electrolytes, namely solutions, and, in particular, aqueous solutions. Pure water can scarcely be called an electrolyte, its conductivity being extremely small. Dry liquid hydrochloric acid in the same way cannot be called an electrolyte; yet if we bring these two substances together, the resulting solution of hydrochloric acid is an excellent conductor of electricity, and undergoes decomposition when electrolysed. The conductivity, therefore, is not a property of either constituent of the solution, but of the aqueous solution itself. It is not every solvent that confers the conducting property when a substance such as hydrogen chloride is dissolved in it. Chloroform, for example, does not conduct electricity itself, neither does a chloro- form solution of hydrochloric acid. The nature of the solvent, there- fore, plays an important part in determining whether the resulting solution will conduct or not. If a substance is such that its aqueous solution is an electrolyte, then its solution in ethyl and methyl alcohol will also conduct electricity, but not so well as the aqueous solution. Acetone ranks with the alcohols in this respect. Ether follows next, and solutions in chloroform or benzene and other hydrocarbons scarcely conduct at all. Inert solvents, then (cp. p. 219), do not yield con- ducting solutions, whilst active solvents form solutions which conduct to a greater or less extent. That a relationship exists between the magnitude of the dielectric. constant of a solvent and its general ionising power has been pointed out by Nernst and Thomson. The relationship is based on the circumstance that the higher the dielectric constant of a medium, the feebler is the mutual attraction of oppositely charged particles immersed in it. The following table gives the dielectric constants of some of the common solvents: Water Methyl alcohol Ethyl alcohol Acetone Ether. Benzene • • ∞ co 81 33 25 17 4.5 2.2 From this table it is apparent that the dielectric constant has a great influence in determining the ionising power of a solvent (cp. p. 263). The conductive property does not depend only on the nature of the solvent, however, but also on the nature of the dissolved substance. In general, it may be said that the only substances which exhibit con- ductivity in aqueous solution in any marked degree are salts, acids, and bases, of which the salts and bases also conduct autolytically at high 232 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. temperatures when fused. An aqueous solution of sugar or alcohol, for example, does not conduct much better than water itself, and cannot in the ordinary sense be called an electrolyte. It should be noted that the conducting solution is, properly speaking, the electrolyte, but by a convenient transference the term is often applied to the dissolved substance, the solvent in such a case being usually understood to be water. We therefore speak of acids, bases, and salts as electrolytes, meaning thereby that their aqueous solutions conduct electricity. It is often expedient to make a distinction between electrolytes, half-electrolytes, and non-electrolytes. In the first class are included practically all salts, together with the strong acids and bases, e.g. hydrochloric and sulphuric acids, potassium and sodium hydroxides. The half-electrolytes comprise the weak acids and bases, e.g. acetic and benzoic acids, ammonia and hydrazine. Non-electrolytes are neutral substances which are not salts, e.g. sugar, alcohol, urea. There is, strictly speaking, no sharp line of demarcation between these classes, intermediate substances existing which cannot be definitely classified within any one set. The distinction is based on degree of conductivity, in which there is no sudden break; but we may say that normal aqueous solutions of the electrolytes conduct electricity well, those of half-electrolytes conduct rather poorly, and those of non-electrolytes very feebly, or practically not at all. Thus normal hydrochloric acid has a conductivity two hundred times greater than normal acetic acid, and this again a conductivity many hundred times greater than a normal aqueous solution of alcohol. It should be noted at once by the student that although weak acids and bases are only half-electrolytes, their salts are good electrolytes. Normal potassium acetate, for example, has a conductivity fifty times that of acetic acid; and normal ammonium chloride a conductivity more than a hundred times as great as the conductivity of normal ammonia. Neglect or forgetfulness of this relation has often led to serious error, and it should therefore be impressed firmly on the memory. When a solution of sulphuric acid in water is electrolysed, the electrodes being of platinum or other resistant material, oxygen, as we have said, comes off at one of the electrodes and hydrogen at the other. The electrode at which the oxygen appears is called the positive electrode or anode, and is connected with the positive pole of the battery which generates the current; that at which the hydrogen is evolved is termed the negative electrode or cathode, and is con- nected with the negative or zinc pole of the battery. It was observed by Faraday that the amount of decomposition in such an electrolyte is proportional to the amount of electricity which flows through it. We have here, then, a direct proportionality between quantity of matter and quantity of electricity. For example, each gram of hydrogen liberated by an electric current corresponds to the passage through the electrolyte of 96,540 coulombs or 1 faraday. It is of no XXII ELECTROLYTES AND ELECTROLYSIS 233 moment whether the current which liberates the hydrogen is strong or weak, whether much or little time is occupied in the decomposition, whether the sulphuric acid solution is more or less concentrated, so long as hydrogen alone is evolved; the result is always the same-a given quantity of electricity liberates in each case the same amount of hydrogen. What here holds good for hydrogen also holds good for other elements or groups of elements. A given quantity of electricity passed through a solution of copper sulphate always deposits the same quantity of copper on the cathode. On this depends the use of the hydrogen or copper voltameter, by means of which a quantity of electricity is measured by finding the amount of hydrogen or copper which it has liberated. Not only is the quantity of hydrogen liberated by a given amount of electricity unaffected by the concentration, temperature, etc., of the electrolytic solution: it is even unaffected by the nature of the dissolved substance, provided that this substance is of such a kind as to permit of the evolution of hydrogen at all. Thus if the same current is passed successively through dilute solutions of sulphuric acid, hydro- chloric acid, and sodium sulphate, it will be found that the same amount of hydrogen is liberated by the current in each solution. When we compare the volumes of oxygen and hydrogen evolved simultaneously at the anode and cathode from a solution of sulphuric acid, we find that if the solution is dilute and the current has been passed for some time before the measurement is begun, in order to get rid of the effect of initial subsidiary reactions, the volume of hydrogen is double that of the oxygen. The gases are thus liberated in the proportions in which they combine, i.e. in chemically equivalent proportions. The same thing may be observed if we take other solutions. The quantity of copper deposited on the cathode from a solution of copper sulphate is exactly equivalent to the oxygen liberated at the anode by the same current. An indirect consequence of this is that if we send the same amount of electricity through solutions of sulphuric acid and copper sulphate, the amount of copper deposited by the current in the one solution will be equivalent to the amount of hydrogen liberated by the same current in the other. A quantity of electricity equal to 1 faraday will therefore deposit 317 g. of copper from the solution of a cupric salt, as this is the amount equivalent in these salts to 1 g. of hydrogen. In general, we may say that the electrochemical equivalents of substances are identical with their chemical equivalents, if we define electrochemical equivalent as the amount of substance liberated by the same quantity of electricity as liberates 10076 g. of hydrogen. This relation, together with the proportionality established between the amount of electricity and the amount of chemical action, gives the most general expression of Faraday's Law. If we inquire as to what happens within the electrolytic solution 234 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. during electrolysis, we must assume that matter travels along with electricity, in order to explain the changes of concentration that occur round the electrodes. Faraday introduced the term ion to denote the matter which travels in the electrolyte, and as in each solution matter travels towards both electrodes, the term anion was used to denote the matter travelling towards the anode, and cation the matter travelling towards the cathode. It is not an easy matter to determine in any given case what the ions really are, and the views now held are not entirely in accordance with those of Faraday. The following system, however, is self-consistent and involves no contradiction, while it affords a satisfactory account of most of the phenomena. In the aqueous solution of an acid the cation is hydrogen, and the anion the acid radical. In the solution of a base, the cation is the metal or metallic radical, e.g. ammonium, NH, and the anion hydroxyl, OH. In the solution of a salt the cation is the metal or metallic radical and the anion the acid radical. The cations carry the positive electricity, and therefore move towards the negative electrode or cathode. The anions carry the negative electricity, and therefore move towards the positive electrode or anode. Many facts lead us to believe that the ions in aqueous solution are hydrated, but for simplicity we may look upon the ions as being the simple positive and negative radicals. The quantitative phenomena of electrolysis are accounted for if we assume that for monobasic acids, monacid bases, and their salts, each gram ion is charged with a faraday of electricity, which it loses when it reaches the oppositely-charged electrode. Take, for example, a dilute aqueous solution of hydrochloric acid. The positive ion in this case is assumed to be hydrogen, and the negative ion chlorine, the water being supposed to play no part in the conductivity. Each gram of hydrogen ion is charged with a faraday of positive electricity, and moves towards the negative electrode. There it is discharged and becomes ordinary hydrogen, which is liberated at the electrode. Now while this is going on at the negative electrode, an equal quantity of negative electricity must be neutralised at the positive electrode, as the same current flows through the whole circuit. This quantity of negative electricity is supplied by the gram equivalent of the negative radical, viz. 35.5 g. of chlorine. The chlorine when discharged of its electricity does not in general appear at the positive pole as such entirely. If the solution of hydrochloric acid is con- centrated, the bulk of the discharged chlorine is liberated, but in dilute. solutions it attacks the water of the solvent, combining with the hydrogen and liberating an equivalent quantity of oxygen. As a rule both oxygen and chlorine are produced, but if both be accurately estimated they are found to be together equivalent to the hydrogen evolved at the negative pole. If the solution electrolysed is one of sodium sulphate, the positive XXII ELECTROLYTES AND ELECTROLYSIS 235 : 4 ion is sodium, and the negative ion SO. From the formula of sodium sulphate it is evident that one SO ion is equivalent to two sodium ions. Thus for every 23 g. of sodium ion which loses a faraday of positive electricity, half of 96 g. of sulphate ion will lose a faraday of negative electricity. In dealing with electrolytes we shall often find it convenient to give the charges of the ions in the formulæ. Each charge of 1 faraday of positive electricity will be indicated by a dot attached to the gram-symbol of the positive ion, and each charge of 1 faraday of negative electricity will be indicated by a dash attached to the gram-symbol of the negative ion. Thus sodium sulphate will be written Na',SO,", and sodium chloride Na Cl'. Neither sodium nor the sulphate radical is capable of independent existence in presence of water, so that they are not obtained as products of the electrolysis, the products of their action on water appearing in their stead. The sodium acts on water with production of hydrogen and sodium hydroxide, according to the equation 2Na + 2H₂O = 2NaOH + H2, while the sulphate radical acts on water with production of sulphuric acid and oxygen :— SO4 + H2O = H₂SO4 + 0. The equations represent the action on water of equivalent quantities of the discharged ions, the amounts of hydrogen and oxygen formed by the action being therefore also equivalent. According to this view, the liquid round the anode should become acid, and the solution round the cathode alkaline. This can easily be shown to be the case, and if proper precautions be taken to prevent diffusion within the liquid, the quantity of sulphuric acid formed at the anode is found to be exactly equivalent to the quantity of caustic soda formed at the cathode. It is convenient to have a system of names for the ions derived from acids, bases, and salts, which shall represent not so much the ions. as particles, but rather the ionic substances. The following system has been proposed, in which the names are derived directly from the names of the ionised salts. The positive ions receive their names from the names of the positive radicals of the salts, acids, or bases by replacement of the terminations by the suffix -ion, e.g. hydrion H', sodion (or natrion) Na', calcion Ca¨, argention Agʻ, ammonion NH, etc. When one radical, e.g. iron, Fe, exists in two sets of salts, the positive ions. of these salts may be distinguished from each other by a prefix indicating the electro-valency, thus diferrion Fe", triferrion Fe. The names of all negative radicals terminate in ate, ite, or ide. Correspond- ing to these we have the terminations for the negative ions -anion, -osion, and -idion respectively. Thus we obtain the names sulphanion SO", sulphosion SO,″, sulphidion S", hydrosulphidion HS', carbanion Co¸”, 3 236 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. hydroxidion OH', etc. Using these names we can say that a solution of sodium chloride contains so many grams per litre of non-ionised sodium chloride, so many grams of natrion, and so many grams of chloridion, thus treating the products of ionisation as ordinary substances—a method of treatment which has many advantages. It has been supposed in what has been said above that the material of the electrodes is not attacked by the discharged ions, a condi- tion which is practically secured if the electrodes are constructed of platinum, or, as often occurs in practice, of dense, conducting carbon. If we electrolyse the solution of a silver salt, say silver nitrate, between two silver electrodes, we find that the argention is discharged and deposited as silver on the negative pole. At the same time, an equivalent quantity of the negative nitranion NO,' is discharged at the positive silver electrode. The nitrate radical in this case is neither liberated as such, nor does it attack the water. It combines with the silver to form silver nitrate, so that the whole electrolytic process has here consisted in the transference of silver from the anode to the cathode, and a change in the concentration of silver nitrate round the electrodes. Such a process is made use of in electroplating, the anode consisting of silver and the cathode of the object to be silver- plated. The silver salt is chosen of such a type as to ensure a coherent film of silver on the surface of the plated object, and is usually a double cyanide of potassium and silver. In the formation of hydrogen gas from the hydrogen ions of hydrochloric acid it is evident that we have union of the discharged atoms, as each molecule of hydrochloric acid can contribute only one atom of hydrogen. Here then there is, strictly speaking, action of the discharged ions on each other. This is not uncommon, and is most evident in the actions of the discharged negative ions of carboxylic acids. If we take, for instance, potassium propionate, and subject it to electrolysis, the positive ion K' goes to the cathode, and the negative ion CH,. CH2. COO', propanion, to the anode. The dis- charged potassion as usual acts on the water with formation of potassium hydroxide and evolution of hydrogen. The discharged negative ion acts in a variety of ways. A portion of it acts on the water with production of the acid and liberation of oxygen, 2CH3. CH₂. CO0 + H₂O = 2CH¸. CH₂. COOH + O. 2 Under favourable conditions, however, the discharged anions react with each other according to the following equations:- Butane 2CH. CH2. COO = CH₂. CH2. CH₂. CH3 + 2CO₂, 2CH. CH₂. COO = CH,. CH2. COOH + CH₂: CH₂ + CO2, 2CH. CH₂. COO = 2 2 2 Ethylene CH, . CH, . COO.CH, . CHg + CO 2 3 2 3 Ethyl propionate XXII ELECTROLYTES AND ELECTROLYSIS 237 The amount of ethyl propionate produced is not great, but in other cases the corresponding compound is formed in considerable quantity. Butane, also, is only a subsidiary product, the chief substances formed being carbon dioxide and ethylene. With other acids the compounds. corresponding to butane may form the bulk of the product of the interaction of the anions. The laws concerning the migration of the ions in opposite direc- tions towards the electrodes were ascertained experimentally by Hittorf. It has been said that in a solution of silver nitrate electrolysed between silver electrodes, the only change is a transference of silver from the anode to the cathode, and a change in the concentration of the silver salt round the two electrodes. By properly constructing the apparatus so that mechanical convection of the dissolved salt between the two electrodes is prevented, the exact change in concentration in the neighbourhood of the electrodes can be measured, and from this change the relative speeds of the two ions can be calculated. It might be thought on a superficial consideration that the anion and cation must move at the same rate since they are liberated in equivalent proportions at the opposite electrodes. In silver nitrate solution, for instance, there is one nitrate ion discharged at the anode for each silver ion discharged at the cathode. The equivalence of dis- charge would, however, be retained for any relative rate of motion of the two ions, as the following scheme will show. In it the positive ions are represented by + and the negative ions by P is the positive electrode or anode, N is the negative electrode or cathode. D is a porous diaphragm to prevent convection currents. To begin with, let there be 6 molecules on each side of the diaphragm, as represented in Scheme I. I N | ++++ + +:++++++ N P D The concentration in each compartment before any current has passed may thus be represented by 6. Let a current now be passed, and let the positive ions alone be capable of movement, the negative ions. remaining in their original compartments. The state after 3 cations have passed through the partition from the anodic to the cathodic compartment is represented in II. II +++++++++:+++ P 238 CHAP. INTRODUCTION TO PHYSICAL CHEMISTRY Each ion without a partner is supposed to be discharged and liberated, and it will be seen that although the negative ion has not moved at all, the number of liberated positive and negative ions is the same. The number of complete molecules in the cathodic compartment has not altered; the number of complete molecules in the anodic compart- ment has been reduced to 3. Let now both ions move at the same rate, i.e. let one negative ion cross the diaphragm to the right for each positive ion that crosses it to the left. If four ions of each kind are discharged, we shall have the state shown in III. III N + ++ + + + + + + + + + -/+ N Here the concentration in both compartments has been reduced to the same extent, namely from 6 to 4. Let, finally, the positive ion move at twice the rate of the negative ion, ¿.e. let one negative ion cross the partition to the right in the same time as two positive ions cross it to the left. After three ions of each kind have been discharged, we have the scheme— IV P ++++++++++++ P The concentration has here fallen off from 6 to 5 in the cathodic com- partment, and from 6 to 4 in the anodic compartment. It is obvious from the diagrams that the loss of concentration in any of the compartments is proportional to the speed of the ion leaving the compartment. Thus in the last example the anodic compartment loses cations twice as fast as the cathodic compartment loses anions, so that the fall of concentration round the anode is twice as great as the fall of concentration round the cathode in the same time. If both ions move at the same rate, the concentrations in the two compartments fall off at the same rate, as in III. We therefore get the ratio of the speeds of the two ions from the observed falls in concentration round the two electrodes as follows: Fall round anode Fall round cathode Speed of cation Speed of anion It must be borne in mind that it is the cation which leaves the anode, and produces the fall of concentration round that electrode, XXII ELECTROLYTES AND ELECTROLYSIS 239 In actual practice the conditions are usually somewhat different from those indicated above. For example, the relative speeds of the ions of silver nitrate may be conveniently determined in an apparatus of the form shown in Fig. 47. In this form a diaphragm is dispensed with, the construction of the vessel itself preventing mixing to a sufficient extent. The vessel has two limbs connected by a short, wide tube, the longer limb being furnished at the lower end with a tap. The cathode C in the short limb consists of a piece of silver foil connected to the battery wire by a piece of silver wire. The anode A consists of silver wire which is bent in the form of a flat spiral. The portion which forms the stem is protected by being enclosed in a narrow glass capillary throughout the entire length immersed in the silver nitrate solution. A feeble current is passed between the electrodes through the silver nitrate solution with which the vessel is charged, the amount of electricity passed being registered by means of a voltameter or other suitable in- strument. At the expiration of some hours, half the solution is carefully withdrawn by opening the tap at the anode, and the amount of silver held in solution determined by analysis, the original concentration of the silver nitrate solu- tion having been previously ascertained. In this case the concentration round the anode does not fall off at all; for although silver ions leave the anodic limb, each nitrate ion that arrives at the anode dissolves from it one ion of silver so as to form silver nitrate. On the sup- position that the ions move at the same rate, the state of the solution may be represented by Scheme V., which is comparable with Scheme III. V + +++ +++ +i+ +|+ +++++++ FIG. 47 The concentration at the cathode has fallen off as before from 6 to 4, but the concentration at the anode has increased from 6 to 8, the total number of complete molecules remaining the same before and after the passage of the current. It is easy, however, to ascertain how many cations have crossed the diaphragm to the left, and thus 240 CHAP. INTRODUCTION TO PHYSICAL CHEMISTRY find what the fall in concentration round the anode would have been had no silver been dissolved from the electrode. The reading of the voltameter tells us the quantity of electricity which has passed through the solution, i.e. the number of negative ions which have been dis- charged at the anode. In the above case it is 4. Now a molecule of silver nitrate appears at the anode for each negative ion dis- charged. If no cations had left the anodic compartment, there would thus be an increase in the concentration equal to 4; but the actual increase is only 2: therefore had no silver been dissolved from the anode there would have been a fall of concentration equal to 2, or 2 cations have crossed the diaphragm to the left in the same time as 2 anions have crossed it to the right. In an actual experiment a current which deposited 32.2 mg. of silver in a silver voltameter was passed through a solution of silver nitrate contained in an apparatus similar to that of Fig. 47. The fall of concentration at the cathode corresponded to 16.8 mg. of silver as silver nitrate, and the rise in concentration round the anode to the same, since the total quantity of silver nitrate in solution necessarily remained unaltered. Had no silver ions migrated from the anode, the rise in concentration would have been 32.2, so that the fall due to migration of the cations is 32.2 - 16·8 = 15·4. We have therefore Speed of cation, Ag Speed of anion, NO, 3 Fall round anode Fall round cathode. 212 From the speed ratio for any substance it is easy to calculate what Hittorf called the transport numbers of the ions of the substance. If only the positive ion of a substance moves, as in Scheme II., this ion is responsible for the total electricity carried, the negative ion having no share in the transport. If both ions move, they share the transport between them, and as each equivalent of the ions has the same charge, the share of each in the transport is evidently proportional to the speed at which it moves. If u and v are the speeds of migration of the positive and negative ions respectively, represents the share V 15.4 16.8 U U + V V W + v taken by the cation in the transport, and the share taken by the anion. These are the transport numbers of Hittorf. It is cus- tomary to denote the transport number of the anion by n. The transport number of the cation is therefore 1-n, since the sum of the two transport numbers. and is equal to 1. As has been W V W + v W + v indicated above, the ratio of the transport numbers of the ions is the ratio of their speeds, so that we have 1 n 0.917. N XXII ELECTROLYTES AND ELECTROLYSIS 241 If we wish to express n in terms of the ratio of the speeds u/v=r, we have, from the above equation, N N = n = 1 1 + 2 As a numerical example, we may again take silver nitrate. For this salt we have 1 1 +0.917 = 0.522. = This may also, of course, be got directly from the fall of concentration round the electrodes, which are proportional to the speeds of the ions. We have V 16.8 U + v 15.4 + 16.8 0.522. K From his researches on the conductivity of dilute salt solutions, Kohlrausch established a very simple relation connecting the transport numbers and the molecular conductivity of the dissolved substance. By the molecular conductivity μ of a solution is meant its specific conductivity (see p. 7) multiplied by the volume of the solution in cubic centimetres which contains one gram - molecular weight of the dissolved substance. In very dilute salt solutions the value of the molecular conductivity is independent of the concentration of the solution, and Kohlrausch found that this constant value was for different salts additively made up of two terms, one depending on the positive and the other on the negative radical, i.e. on the positive and negative ions. On considering any one salt, he found that the ratio of the terms for the two ions was the ratio of the speeds of migration of the ions. By properly choosing the units it was therefore possible to state for very dilute salt solutions, which exhibit a molec- ular conductivity independent of further dilution, the simple relation μ = 1 U + 2, μ when is the molecular conductivity, and u and v numbers propor- tional to the speeds of the positive and negative ions. The numbers u and v do not, of course, represent the actual speeds of the ions, if the ordinary units are employed for the molecular conductivity. The absolute values of the velocities may, however, be determined in certain cases, and from them the other values may be calculated. The rate of progression of a coloured ion, either in water or in a jelly, due to a given difference of potential, can, under suitable conditions, be observed and measured, and even the speed of colour- less ions may be measured by observations of the change of refractive index, as the ions migrate from one part of the solution to another. The relative values so obtained agree well with those derived from Kohlrausch's observations on molecular conductivity. R 242 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. F The following table contains the speeds of the principal univalent ions in dilute aqueous solution at 18°, when the difference of potential between the electrodes 1 cm. apart is 1 volt. The velocities of migration are given in centimetres per hour. H K NH₁ Na 4 Ag Cations. 10.8 I 2.05 1.98 1.26 1.66 он Cl I Anions. NO3 C2H3O2 5.6 2.12 2.19 1.91 1.04 The movement of the ions through practically pure water is seen, from the above table, to be a very slow one. If we calculate the force required to drive 1 g. of hydrion through water at the rate of 1 cm. per second, it is found to be equal to about 320,000 tons weight. On reference to p. 179, it will be found that this number is of the same order of magnitude as the corresponding number calculated from the rate of diffusion of urea by means of the osmotic-pressure theory, viz. 40,000 tons. It is somewhat curious that when the rates of migration of a series of ions such as those afforded by the metals of the alkalies are com- pared with each other, the metals with the greatest atomic weights move with the greatest speed. Thus potassion has a greater velocity of migration than sodion, and this again than lithion, whilst cæsion, the heaviest ion of the alkali metals, moves at the greatest rate of all. This is usually accounted for by the assumption that the degree of hydration of lithion is greater than that of sodion, which is in its turn greater than that of potassion, and so on. On this view the positive ion of lithium salts is not lithium itself, but lithium plus a comparatively large amount of water which travels with it, and there- fore reduces its speed. The coincidence in the magnitudes of the driving forces would lead us to suspect that the resistance offered to the diffusion of sub- stances in water and to the passage of ions through water under the influence of electric forces is of the same kind, and further inquiry serves to bear out the supposition. The resistance is connected with the viscosity or internal friction of the liquid, which may be measured by the time the liquid takes to flow through a narrow tube under given conditions. When the fluid friction increases, the resistance to the passage of substances through the liquid increases, and the rate of diffusion and rate of ionic migration diminish in consequence of the increased resistance which the particles in motion through the fluid experience. The addition of a small quantity of a substance Corre- such as alcohol to water increases the viscosity of the water. sponding to this increase we find that the rate of diffusion is less when a substance is dissolved in water containing a little alcohol than the rate of diffusion when water alone is the solvent, no matter what the XXII ELECTROLYTES AND ELECTROLYSIS 243 dissolved substance may be. Similarly the speed of ions in water containing alcohol is less than their speed in pure water. Again, when the temperature of water is raised, its fluidity increases. Corresponding to this we have increased rate of diffusion and ionic migration as the temperature increases. There is even a rough proportionality between the different magnitudes. Thus at the ordinary temperature the fluidity of water increases at the rate of about 2 per cent per degree. In close accordance with this, the rate of migration of ions through water increases about 2 per cent at 15° for a degree rise in temperature. Indeed, if slow-moving ions alone are considered, the influence of temperature on the conductivity of aqueous solutions and on the fluidity of water is almost exactly the same within wide limits, both conductivity and fluidity apparently dis- appearing at a temperature of about 35°. The assumption that the ions are hydrated is again of service in enabling us to understand this exact accordance, for the sluggish ions are supposed to be most highly hydrated, so that the resistance they experience in passing through water is practically the resistance of their "water atmo- sphere" against water, i.e. precisely what determines the viscosity of pure water. The actual determination of the molecular conductivity as usually carried out in chemical laboratories proceeds as follows:-The principle adopted is to determine the resist- ance of the given solution by the ar- rangement known as Wheatstone's bridge, M shown in the diagram, Fig. 48. R is a resist- ance box, by means of which resistances of known value can be introduced. W is the resistance to be measured-that of a piece of wire, for example. G is a galvanometer, and B a battery to produce current. MM' is a platinum wire of uniform resistance stretched along a metre scale subdivided into millimetres. Connection is made between the galvanometer and any point on this wire by means of the sliding contact C. In general, a current flows through the galvanometer, but in the special case when the resistance of R is to that of W as the resistance of a is to that of b, no current flows through G, and the galvanometer exhibits no deflection. To determine the resistance of W we place a known resistance in the box R, and move the contact C along the platinum wire until the galvanometer shows no deflection. We know then that R: W = a:b. R a G C || B FIG. 45. ŏ M' 244 CHAP. INTRODUCTION TO PHYSICAL CHEMISTRY But the platinum wire is of uniform resistance, so that the resistance. of a is to the resistance of b as the length of a is to the length of b, and these lengths may be read off directly on the scale along which the wire is stretched. When the resistance to be measured is that of an electrolyte, it is generally impossible to use a galvanometer owing to the polarisa- tion at the electrodes when a steady current flows through the electrolyte. This polarisation may be M' avoided by the use of an alternating current instead of a direct current, but then the galvanometer is use- less to indicate the alternating current. Its place, however, may be taken, as Kohlrausch has shown, by the telephone, which is silent when no current passes through it, but sounds when it is traversed by alternating currents. The alternating current is got from the secondary terminals of a small induction coil, worked by the battery B, which is usually a single accumulator, or a bi- chromate cell. The modified arrangement is shown in Fig. 49, where S is the electrolytic solution, T the telephone, and I the induction coil. M a R с B FIG. 49. T D b S The vessel which contains the solution is generally of the type proposed by Arrhenius, and shown in Fig. 50 in natural size. The electrodes are made of stout platinum discs, fitting closely to the cylindrical vessel. A short platinum stem from each is sealed into a glass tube, held firmly in the ebonite cover C. The connecting wires are passed down the glass tubes till they make contact with the platinum wires by means of a drop of mercury introduced into each tube. This form is most suitable for very small conductivities. For solutions which have greater conductivity, a modified type narrowed at the lower end may be employed, which has the advantage of being considerably cheaper on account of the much smaller size of the platinum electrodes used in its construction. If the platinum electrodes are bright, there is no sharp minimum of sound in the telephone at any position of the sliding contact. It is therefore necessary to cover the surface of the platinum with a coating of finely-divided platinum, and the success of the method largely depends on how the platinisation of the electrodes is effected. If a fine velvety coating of platinum. black does not entirely cover the inner surfaces of the electrodes, the sound minimum is more or less indistinct owing to the polarisations XXII ELECTROLYTES AND ELECTROLYSIS 245 of the make and break induction currents not exactly neutralising each other, and the measurements of resistance in consequence are more or less doubtful. The platinisation is carried out by electrolysing a solution of platinum chloride (chloroplatinic acid) between the electrodes by means of a direct current. The best solution to employ is one containing 30 parts water, 1 platinum chloride, and 0.008 lead acetate. The electrolysing current is so regulated that there is a feeble gas evolu- tion from the anode and a fairly brisk evolution from the cathode. The direc- tion of the current is occasionally reversed, and the platinisation is complete when each electrode has served as cathode for about fifteen minutes. A large induction coil is not necessary for the success of the method; in fact, a small toy coil with a high rate of alterna- tion works best in the chemical laboratory. It should be some six feet distant from the measuring wire, and enclosed in a box so that the sound from the make and break of the coil itself does not interfere with the sound in the telephone. If the vibrating tongue found on such a coil is replaced by a piece of watch-spring (pro- tected where the spark passes by a layer of thin platinum foil wrapped round it), the instrument may be made to work practically without noise, especially if the current is suitably reduced by interposing a resistance between the cell and coil. FIG. 50. Another, and for most purposes very convenient, method permits the use of a galvanometer, instead of a telephone, as zero-instrument. A rotating commutator reverses the direction of the current from an ordinary battery many times a minute in the electrolytic cell, and thus prevents polarisation. On the same spindle is another reversing apparatus which undoes the work of the first so far as the galvanometer is concerned, so that though an alternating current passes through the electrolytic cell, the current through the galvanometer is always in the same direction and thus gives a steady deflection. The cell containing the electrolytic solution is immersed in a bath of constant temperature, usually 25°, fluctuations of more than a tenth of a degree from the mean being inadmissible, owing to the great varia- tion of the conductivity with the temperature. For chemical purposes the dilutions generally are made to increase 246 CH. XX. INTRODUCTION TO PHYSICAL CHEMISTRY in powers of 2. If the substance under investigation is sufficientl soluble, a solution containing a gram equivalent, or a gram molecule in 16 litres is first prepared, and 20 cc. of this solution introduce into the electrolytic cell by means of a 10 cc. pipette twice filled Another 10 cc. pipette is marked by direct experiment so as to remov exactly as much water as the first pipette delivers. After the firs reading of the conductivity has been taken, 10 cc. of the solution ar removed by the second pipette and 10 cc. of water added by means o the first pipette, after which the diluted solution is well mixed by motion of the electrodes. The solution is now twice as dilute a formerly, i.e. its dilution is 32. The reading is repeated after th solution has attained the temperature of the bath, and the dilution process is gone through anew. When the dilution has reached 102. the process is usually stopped, except in the case of strongly-dissociated salts, for the conductivity caused by impurities in the distilled wate renders the values uncertain. For further information concerning methods for the determination o electric conductivity in solutions, see FINDLAY, Practical Physical Chemistry (1917); KOHLRAUSCH AND HOLBORN, Leitvermögen der Elektrolyte (1916) W. C. D. WHETHAM, "Rotating Commutator Method," Proc. Roy. Soc. 6€ (1900), p. 192; EDGAR NEWBERY, "A New Method for the Determina tion of Conductivity," Trans. Chem. Soc. 113 (1918), p. 701. F. KOHLRAUsch, "The Resistance of the Ions and the Mechanica Friction of the Solvent," Proc. Roy. Soc. 71 (1903), p. 338. A CHAPTER XXIII IN the preceding chapter we have become acquainted with some of the fundamental facts of electrolysis; in the present chapter we pro- ceed to the consideration of a mechanical scheme which affords a simple mode of representing them, as well as many other peculiarities of electrolytic solutions. Before entering on the discussion of this scheme, however, it is necessary to draw attention to two further facts which must be accounted for by any satisfactory theory of electrolysis. In the first place, it has been ascertained by careful experiment GUARA ELECTROLYTIC DISSOCIATION R FIG. 51. R that electrolytic solutions obey Ohm's Law as strictly as do metallic conductors, the current being proportional to the electromotive force for all values of the force. A direct consequence of this is that no electrical energy is expended in splitting up the dissolved salt molecules into their constituent ions, as was first indicated by Clausius. In the second place, when the circuit is completed between two electrodes, the products of electrolysis appear simultaneously at both, no matter how far apart they may be. Thus if we have, as in Fig. 51, a narrow glass tube RR, 1 cm. in bore and 40 cm. long, connected at the ends with wide tubes containing the two electrodes A and C, and 247 248 CHAP. INTRODUCTION TO PHYSICAL CHEMISTRY filled with sulphuric acid, the products of the electrolysis of the sulphuric acid will make their appearance at the electrodes as soon as connection is made through a powerful battery. If the anode A is of copper and the cathode a piece of platinum wire, the blue colour of copper sulphate and the bubbles of hydrogen at C will be observed simultaneously. Now the first hydrogen ions cannot therefore come from the same molecules of sulphuric acid as the first sulphate ions, which unite with the copper to form copper sulphate, for the ions, in whatever way they might be supposed to move, could not traverse a distance of 40 cm. in a few seconds, as may be seen from the table of rates of migration given in the preceding chapter (p. 242). In the schemes for the representation of the migration of the ions, we have assumed that there is a constant change of partners going on as the ions travel to the opposite electrodes. This idea was intro- duced by Grotthus, who conceived that the molecules at the two electrodes were split up into their positive and negative constituents under the influence of the electric charges on the electrodes, and that the intermediate molecules changed partners according to a scheme like the following, the first action of the electric charges being to direct all the positive ends of the molecules towards the negative electrode and the negative ends towards the positive electrode:- N K jā KXCI + K මගෙ (CI) K + K (CI) + kā KCI CIK මගෙ + මම CIK + (CI)(K + điền đồn K)(CI KXCI + + ōā jā K)(Ci K CI CIK බල් + මගෙ CIK + điền K CI + මග මර්ග + (K)(CI) CI P Ը. CI This, however, does not get over the difficulty indicated by Clausius, an account of whose views may be given in the words of Clerk Maxwell. "Clausius has pointed out that on the old theory of electrolysis, according to which the electromotive force was supposed to be the sole agent in tearing asunder the components of the molecules of the electrolyte, there ought to be no decomposition and no current as long as the electromotive force is below a certain value, but that as soon as it has reached this value a vigorous decomposition ought to commence, accompanied by a strong current. This, however, is by no means the case, for the current is strictly proportional to the electromotive force for all values of that force. "Clausius explains this in the following way :-According to the theory of molecular motion, of which he has himself been the chief XXIII ELECTROLYTIC DISSOCIATION 249 founder, every molecule of the fluid is moving in an exceedingly irregular manner, being driven first one way and then another by the impacts of other molecules which are also in a state of agitation. "This molecular agitation goes on at all times independently of the action of electromotive force. The diffusion of one fluid through another is brought about by this molecular agitation, which increases in velocity as the temperature rises. The agitation being exceedingly irregular, the encounters of the molecules take place with various degrees of violence, and it is probable that even at low temperatures some of the encounters are so violent that one or both of the com- pound molecules are split up into their constituents. Each of these constituent molecules then knocks about among the rest till it meets with another molecule of the opposite kind, and unites with it to form a new molecule of the compound. In every compound, there- fore, a certain proportion of the molecules at any instant are broken up into their constituent atoms. At high temperatures the proportion becomes so large as to produce the phenomenon of dissociation studied by M. Ste. Claire Deville. "Now Clausius supposes that it is on the constituent molecules in their intervals of freedom that the electromotive force acts, deflecting them slightly from the paths they would otherwise have followed, and causing the positive constituents to travel, on the whole, more in the positive than in the negative direction, and the negative constituents more in the negative direction than in the positive. The electromotive force, therefore, does not produce the disruptions and reunions of the molecules, but, finding these disruptions and reunions already going on, it influences the motion of the constituents during their intervals of freedom." The constituent molecules referred to in the above passage are, of course, the positive and negative radicals of the dissolved salt, i.e. the cation and anion of which it is assumed to be composed. At any one time then we have, on the hypothesis of Clausius, some proportion of the salt molecules split up into their constituent ions, which, with their electric charges, move towards the appropriate electrodes. It must be observed that this state of partial dissociation of the dis- solved substance is the normal condition of the liquid, and exists whether there is an electric current passing through the solution or not. All that the electric forces do is to direct the dissociated charged products to the electrodes and there discharge them. Nothing has been said as to the proportion of dissolved substance which is thus dissociated into ions. For the purpose of accounting for the validity of Ohm's Law in electrolytic solutions, a very small proportion will suffice, provided that the small quantity is always regenerated by the action of the molecules themselves without any interference of the electrical forces. In proportion as the free ions are removed from the solution at the electrodes, Clausius supposes them to be 250 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. regenerated by the collisions of the undissociated molecules, so that the process of conduction and electrolysis goes on. If we are to give the hypothesis definiteness and precision, however, we must take account of the relative quantities of the electrolyte in the dissociated and undissociated states. The manner of doing this was first pointed out by Arrhenius, and it is to his hypothesis of electrolytic dis- sociation that we must resort if we wish to explain quantitatively the phenomena exhibited by electrolytic solutions, whether during electrolysis or in their ordinary state. Arrhenius supposes substances which give solutions that conduct electricity freely to be almost entirely split up into their constituent ions, while substances which yield solutions of feeble conductivity are supposed by him to be split up only to a very small extent. In fact, he proposes to measure the degree of dissociation of a substance by the conductivity of its solutions. On his hypothesis, only those molecules which are split up into their constituent ions play any part in the conduction of electricity, the undissociated molecules remaining idle. It is obvious, therefore, that the conductivity of any given solution. depends on two factors-the number of ions in the solution, and the rate at which these ions move. To simplify matters we will, in what follows, only consider univalent ions, i.e. those derived from monacid bases, monobasic acids, and the salts which they form by mutual neutralisation. Every ion derived from these substances has the same charge of electricity, i.e. 1 faraday per gram-ion. Since each carrier of electricity has the same load, the quantity carried can depend only on the number of carriers and the speed at which they move. Now the rate at which the ions move may, as we have seen, be determined from the work of Hittorf and Kohlrausch. It only remains, therefore, to find the number or proportion of ions in any given solution. Kohlrausch ascertained experimentally that the molecular conduc- tivity of a salt increases as the dilution of the solution increases, and that the rate of increase of conductivity gets smaller and smaller as the dilution gets greater, until finally the molecular conductivity remains constant, although the addition of water to the solution is continued. This may best be figured as follows. Consider a cell of practically infinite height and of rectangular horizontal section, two parallel sides. of which are of platinum, and are placed at a distance of 1 cm. from each other. These platinum sides may be used as electrodes. Let there now be introduced into the cell one cubic centimetre of a solution containing a gram-molecular weight of common salt (58.5 g.) dissolved in a litre of water. If the resistance offered to the passage of the current through the liquid is measured in ohms, the reciprocal of the number obtained represents the specific conductivity of the solution (p. 7). If we add the rest of the litre of normal solution, the conduction between the electrodes will be 1000 times as great as before, and this represents the conductivity of one gram molecule XXIII 251 ELECTROLYTIC DISSOCIATION dissolved in 1000 cc. of water, i.e. is the molecular conductivity (p. 241) at the dilution of 1 litre. If now we add water so as to make up the volume to 2 litres, and again determine the molecular conductivity, we find that the value for the dilution 2 is greater than before, although the value of the specific conductivity has diminished. As we add more water so as to increase the volume in which the gram-molecular weight of the salt is contained, the molecular conduc- tivity will also increase, practically to reach a limit when the dilution amounts to about 10,000 litres. The numbers obtained by Kohlrausch for sodium chloride at 18° are given in the following table: Dilution=v 10 lit. 20 100 500 1,000 5,000 10,000 ∞ "} "" }" "" >> "} He Mol. Cond. μ 92.5 95.9 102.8 106.7 107.8 109.2 109.7 110.3 Dissociation=m •839 •870 •932 •967 *977 …. M ... From this table it will be seen that the rate of increase of the molecular conductivity is much greater when the dilution is small than when it is great. Doubling the quantity of water when the dilution is 1 adds more than 6 units to the conductivity; doubling the quantity when the dilution is 500 only adds 1 unit to the conductivity. The molecular conductivity, as has been said, depends only on (1) the number of ions, and (2) the rate at which they move. We have therefore to determine to which of these causes the increase of molec- ular conductivity on dilution is due. The rate of the ions depends on the resistance offered by the liquid to their motion. Now at a dilution of 10 we have 58.5 g. of salt dissolved in 10,000 g. of water. So far as viscosity is concerned, this is practically pure water, and further additions of water should have no appreciable effect in changing the resistance offered to the passage of the ions. We may suppose then that the rate at which the ions travel is practically unaltered after a dilution of about 10 is reached, so that the increase of conductivity with further dilution is not due to any increase of speed of the ions, but to an increase in their number. In the imaginary cell considered above we have always the same amount of salt between the electrodes, but evidently as we add water we obtain a greater pro- portion of ions. With increasing dilution the salt then must split up more and more into ions capable of conveying the electricity, if we are to account for the increase of molecular conductivity which the salt exhibits. When all the salt has been split up into its ions the increase of molecular conductivity with dilution must cease, for further dilution can neither increase the speed of the ions nor augment their number. In the case of sodium chloride the salt is almost entirely 252 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. ionised at a dilution of 10,000, and we find that salts in general exhibit this behaviour. The limiting value of the molecular con- ductivity corresponding to complete ionisation is called the molecular conductivity at infinite dilution, and is usually denoted by paso Since in dilute solutions the addition of more water is assumed not to affect the speed of the ions, it is obvious that in a cell such as we considered above, the conductivity of the solution is directly pro- portional to the amount of ionised substance in it. Now we know that at infinite dilution all the sodium chloride is ionised. At finite dilutions, therefore, the degree of dissociation, i.e. the proportion of the whole which exists in the state of ions, is equal to the quotient of the molecular conductivity at the dilution considered by the molecular conductivity at infinite dilution. The degree of dissociation or ionisa- tion is generally denoted by m, so we have the equation m Мо Моо For example, if we wish to ascertain the degree of ionisation of sodium chloride at a dilution of 10 litres, i.e. in decinormal solution, we divide the molecular conductivity, 92.5, by the molecular con- ductivity at infinite dilution, say 110, and obtain as quotient 0.84. In a decinormal solution of sodium chloride then at 18°, a little over eighty per cent of the salt is split up into its constituent ions. It must be very specially emphasised that the molecular con- ductivity itself is no measure of the degree of ionisation of a dissolved substance; the true measure is the ratio of this conductivity to the molecular conductivity at infinite dilution. The degree of ionisation is not proportional to the molecular conductivity unless under certain conditions which must be carefully specified. For example, the molecular conductivity of a decinormal solution of sodium chloride. at 50° is 160. This is much greater than the molecular conductivity at 18°, but the increase cannot come from an increase in the number of ions, as the salt at 18° is already more than four-fifths ionised. The great increase in the molecular conductivity is due to the increase in the other factor, namely, the speed of the ions. The fluid friction of the solution is greatly diminished by the rise in temperature, and consequently the ions move much faster, thus in a given time conveying more electricity. At 50° the molecular con- If we therefore divide 160 ductivity at infinite dilution is about 197. by this number we obtain an ionisation approximately equal to 0.81, which is practically the same value as we got for 18°. In general, we find with salts that rise of temperature, while greatly augmenting the molecular conductivity, has very little effect on the degree of ionisation, the increase in the conductivity being almost wholly due to the increased speed of the ions. In a similar way the addition of non-conducting substances to salt XXIII ELECTROLYTIC DISSOCIATION 253 solutions lowers the conductivity without appreciably altering the ionisation as measured by the ratio of the molecular conductivity at the dilution considered to that at infinite dilution. Thus diethyl- ammonium chloride dissolved in water and in mixtures of water and ethyl alcohol gave the following values at 25° for decinormal solutions, and for infinite dilution Percentage of Alcohol by Volume. 0 10.1 30.7 49.2 72.0 90.3 M10 90.4 69.2 43.2 32.0 25.5 18.1 μα 114.7 89.1 57.5 45.8 42.0 41.4 M10 M = Мо 0.788 0.778 0.751 0·699 0.607 0.438 The column headed μ gives the effect of the alcohol in reducing the speed of the ions, since at infinite dilution the salt is entirely ionised in each case, the ionised amount being therefore the same throughout. In the last column we have the degree of ionisation as measured The by the ratio of the molecular conductivity at 10 litres to μ∞• addition of alcohol diminishes both the speed of the ions and the ionisation, so that the molecular conductivity in decinormal solution is reduced from both these causes. It will be noticed, however, that these two effects of the addition of alcohol do not go hand in hand. The degree of ionisation is scarcely affected by the first substitutions. of alcohol for water (up to 30 per cent), while the speed of the ions, and consequently the molecular conductivity, is reduced to one-half. On the other hand, when nearly all the water has been replaced by alcohol, the effect of further additions is scarcely noticeable on the speed of the ions, but very marked on the degree of ionisation, and consequently on the molecular conductivity, in decinormal solution. On the whole the molecular conductivity in decinormal solution in 90 per cent alcohol is only about a fifth of what it is in pure water if the speed of the ions alone had been affected, the reduction would have been to a value a little more than a third of the value for water; the balance of the reduction is due to diminution in the degree of ionisation. The student is particularly recommended to a close study of the above examples, in order that he may become familiar with the two factors on which the molecular conductivity depends, as beginners almost invariably neglect to take account of the change in speed of the ions under different conditions, and thus from the values of the molecular conductivity draw utterly erroneous conclusions regarding the degree of ionisation. Degree of ionisation is never proportional to molecular conductivity unless the speed of the ions is the same in the two solutions compared. If two dilute solutions contain the same salt dissolved in the same solvent at the same temperature, then the degree of ionisation of the substance in the two solutions is pro- portional to the molecular conductivity, for the maximum molecular 254 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. conductivity is the same in both cases. But if in the solutions compared the dissolved substance is different, if the solvent is different, or if the temperature is different, then the molecular conductivity is no longer a measure of the degree of ionisation, for the maximum molecular conductivity will no longer be the same. If we investigate the influence of dilution on the molecular con- ductivity of dilute solutions, we find that the weak acids and bases which form the group of half-electrolytes obey a law which was deduced by Ostwald from the law of mass-action, as will be shown in a subsequent chapter. Since other conditions are the same, and increase in dilution does not affect the speed of the ions, the change in the molecular conductivity observed is due entirely to change in the degree of ionisation. If we represent the degree of ionisation by m and the dilution by v, the following relation holds good :- V The constant is usually denoted by k, and is called the dissociation constant. In the subjoined tables are given the values obtained at 25° for acetic acid and ammonia respectively. 8 16 32 64 128 256 512 1024 m² (1 − m)v υ 8 16 32 64 128 256 ACETIC ACID, CH3COOH Mcs=387 M 4.63 6.50 9.2 12.9 18.1 25.4 35.3 49.0 constant. μ 3.4 4.8 6.7 9.5 100m 1.193 1.673 2.380 3.33 4.68 6.56 9.14 12.66 AMMONIA, NH₁(OH) Mcs = 252 100m 100% 1.35 0.0023 1.88 0.0023 2.65 0.0023 3.76 0.0023 5.33 0.0023 7.54 0.0024 Mean 0.0023 In the third column of the tables is given the percentage ionisa- tion, i.e. the degree of ionisation multiplied by 100; and in the fourth column is one hundred times the value of the dissociation constant derived from the above formula. This centuple constant is 13.5 19.0 Mean 100% 0.00180 0.00179 0.00182 0.00179 0.00179 0.00180 0.00180 0.00177 0.00180 XXIII ELECTROLYTIC DISSOCIATION 255 often used instead of the smaller number on account of its leading for all substances to more convenient figures. The values of the constant at particular dilutions vary as a rule only 1 or 2 per cent from the mean, and this variation is due to errors of observation, which are greatly magnified in the calculation of the constant. In the first place, it is evident from the tables that acetic acid and ammonia in equivalent solutions are about equally ionised, the former into hydrion, H', and acetanion, CH,COO', the latter into ammonion, NH, and hydroxidion, OH'. If we compare these tables with that given on p. 251 for a good electrolyte, it will be seen that dilution has a far greater influence on the molecular conductivity in the former case than the latter. For sodium chloride an increase of the dilution from 1 to 100,000 only increases the molecular con- ductivity by about half its value; while an increase in the dilution from 8 to 1024 increases the molecular conductivity of acetic acid more than tenfold. Although the molecular conductivity thus increases much more rapidly with the dilution than is the case for good electrolytes, yet the increase is not nearly proportional to the increase in the dilution. When the degree of ionisation is small the molecular conductivity is roughly proportional to the square root of the dilution, as may be seen from a consideration of the general formula m² k. (1 − m)v For small degrees of ionisation, 1 m is not greatly different from 1 so that the formula becomes whence m² = kv, or m=k' √v. If in the general formula we put m=0·5, i.e. if we assume that the electrolyte is half ionised, we obtain a conception of the physical dimensions of the constant k. The expression becomes 0.52 1 magdada 2v (1 -0.5)v 1 = · k, = k, or c=k, if c = c=! V* In words, the dissociation constant k is numerically equal to half the concentration at which the substance is half dissociated, the concen- tration being expressed in gram molecules per litre. Thus for acetic acid we have k = 0·000018, whence c= 0.000036, i.e. acetic acid is half ionised when the concentration of its aqueous solution is 0.000036 normal. It is obvious that if in such a solution, which contains only about 2 parts of acetic acid per million, the acid is only half ionised, the direct determination of the molecular conductivity for solutions in which the acid is wholly ionised is an impossibility. Yet the 256 CHAP. INTRODUCTION TO PHYSICAL CHEMISTRY value of the molecular conductivity must be known in order that the degree of ionisation may be calculated. It has therefore to be determined indirectly by means of Kohlrausch's Law. Although weak acids and weak bases are but half-electrolytes, their salts are good electrolytes, and as much dissociated in solution as the corresponding salts of strong acids and bases. Thus for sodium acetate at 25° we have the following numbers :---- V 32 64 128 256 512 1024 00 บ 10 20 100 500 1,000 H' Η K' For ammonium chloride, Kohlrausch found at 18° numbers much the same as those for sodium chloride, p. 251, viz.— Na Ag મ Cations. 347 80.5 82.7 85.1 87.4 89.3 91.7 94.9 73.8 50.5 62.5 m 0.858 0.882 0.907 0.927 0.949 0.966 μ 110.7 115.2 122.1 126.2 127.3 131.1 It is an easy matter, then, to find numbers for the molecular con- ductivity at infinite dilution in the case of salts of weak acids or bases. Now, according to Kohlrausch's Law, there is a constant difference between the maximum molecular conductivities of all acids and their sodium salts-a difference due to the difference in speed of hydrion and sodion. But strong acids like hydrochloric acid. are at equivalent dilutions quite as much ionised as their sodium. salts, so that their maximum molecular conductivities may be deter- mined experimentally. For these acids we can thus get directly the difference between their maximum molecular conductivity and that of their sodium salts. This difference, amounting at 25° to about 296 in the customary units, when added to the maximum molecular conductivity of the sodium salt of an acid, which can always be directly determined, gives the molecular conductivity of the acid itself at infinite dilution, and this enables us to give the degree of ionisation of the acid at any other dilution. The following table gives the conductivity values for some of the principal ions at 25°: ጎ 0.845 0.879 0·931 0.962 0.970 ... Anions. 196 OH' NO3' 71.0 CIO 65.0 Cl' 75.2 XXIII ELECTROLYTIC DISSOCIATION 257 These numbers are proportional to the speeds of the ions, and the maximum molecular conductivity of the salt, acid, or base may be obtained by adding together the numbers for the corresponding cation and anion. Thus the maximum molecular conductivity of sodium chloride at 25° is 50.5 +75°2125·7. It should be noted that not- withstanding their importance, the values for H and OH' are less accurately known than those for the other ions. Good electrolytes, curiously enough, do not obey Ostwald's dilution law, which holds so accurately for half-electrolytes. Several empirical relations have been suggested connecting the degree of ionisation m as measured by the ratio pop with the dilution ", amongst them van 't Hoff's dilution formula 00 m³ (1 − m)²v = constant. These empirical formulæ may now, however, be discarded in favour of Ghosh's dilution formula 3/v. log m= constant, which has a theoretical basis, and is discussed later in the chapter. According to the foregoing hypothesis of electrolytic dissociation, aqueous solutions of salts, acids, and bases have a certain proportion of the dissolved molecules split up into charged ions, and the amount in any given case is determinable from measurements of electrical conductivity. The ions are supposed to be independent of each other, and ought to act as separate molecules if the independence is complete. We should therefore expect that salt solutions, when investigated by the customary methods of molecular-weight determina- tion, should exhibit exceptional behaviour, and give for the molecular weights of the dissolved substances, values smaller than those which we should deduce from the ordinary molecular formulæ. As has already been indicated, such abnormal values are frequently observed. When the molecular weight of sodium chloride and other similar salts is determined from the freezing or boiling points of their aqueous solutions, the numbers obtained are equal to little more than half the values given by the formula NaCl, i.e. the depressions of the freezing point and the elevations of the boiling point have almost twice the normal value. A normal solution of cane sugar freezes at - 187°; a normal solution of sodium chloride freezes at - 3·46°. Such an abnormally large value for the depression indicates that there are more dissolved molecules in the normal solution of common salt than there are in the normal solution of sugar, although each solution in the usual system of calculation is supposed to contain 1 gram molecule per litre. It should be noted at once that it is only electrolytic solu- tions which show these abnormally high values, the values given by S 258 CHAP. INTRODUCTION TO PHYSICAL CHEMISTRY non-electrolytic solutions being almost invariably equal to, or less than, the normal values. Abnormally small values we have already attributed to molecular association (cp. p. 213), so that we ought, by parity of reasoning, to attribute the abnormally large values for salt solutions to a dissociation of the molecules. Van 't Hoff introduced for salt solutions a coefficient i which repre- sents the number by which the normal value of the freezing point, etc., must be multiplied in order to give the value actually found. Thus for the solution of sodium chloride we have the depression 3:46°, instead of the "normal" value 1.87 shown by solutions of non- electrolytes containing 1 gram molecule per litre. In this case ¿ — 3·46/1·87—1.85, i.e. the normal value 1.87 has to be multiplied by 1.85 in order to bring it up to the observed value 3:46. In the first place, it is to be noticed that for salts, acids, and bases which, according to the theory of electrolytic dissociation, split up into two ions, the coefficient i is never greater than 2. If dissociation into two ions were complete, the value for the depression of freezing point, etc., would be twice the normal value, since each molecule represented by the ordinary chemical formula becomes two independent molecules. by dissociation. Now at common dilutions the ionisation is never complete, so that the value of the depression should be something less. than 2, as is actually found by observation. When the dissociation hypothesis admits of a dissociation into more than two charged molecules, the depressions of the freezing point give values of i greater than 2. Thus strontium chloride, according to the dissocia- tion hypothesis, splits up at infinite dilution into the three ions, Sr., Cl', and Cl', the first of which has two charges of positive electricity, and the others a charge each of negative electricity. At moderate dilutions, therefore, we ought to expect a value of i less than 3, but probably greater than 2, as the dissociation of salts is generally high. We find in accordance with this that strontium chloride in decimolar solution gives the depression 0.489, instead of the value 0·187 obtained for non-electrolytes. The value of i is therefore 0.4890.187 2.6. = It is obvious from the above examples that there is a direct numerical relation between the degree of ionisation in any given case and van 't Hoff's coefficient i, so that if one is given the other can be calculated. If the degree of ionisation of a binary salt in solution is m, then there will be present in the solution 1 - m undissociated molecules and 2m dissociated molecules, in all 1 + m molecules for each molecule represented by the chemical formula. The depression of the freezing point, etc., will therefore have 1 + m times the normal value, i.e. 1 + M. ¿ Should the original molecule split up into n ions when the ionisation is complete, m again representing the ionised proportion, there will be present 1 - m undissociated molecules, and nm dissociated molecules, XXIII ELECTROLYTIC DISSOCIATION 259 in all 1+ (n-1)m molecules for each original molecule, i.e. we shall have i = 1 + (n − 1)m. - Giving m in terms of i, we have m = i − 1 for dissociation into two ions, and กาว i – 1 n - 1 for dissociation into n ions. A comparison of the values of i deduced for the same solution directly from the freezing point, and indirectly from determinations. of m by means of the electric conductivity, shows very fair accordance between the two methods. In the first place, it is found that in the case of non-electrolytes where m = 0, we have i=1, i.e. we obtain the normal value for the freezing-point depression, etc.; and Arrhenius showed from his own freezing-point determinations that the values obtained for with electrolytes differ from the corresponding values calculated from the electric conductivity by not more than 5 per cent on the average. Although this difference is comparatively great, better accordance was scarcely to be expected owing to the difficulty in effecting the comparison. It must be borne in mind that the molecular conductivity can only give accurate figures for the degree of ionisation when the solutions are so dilute that further dilution does not sensibly change the speed of the ions. Now this condition necessi- tates a dilution of at least 10 litres, i.e. the solution must not be more concentrated than decinormal, even in the most favourable case. But the normal depression for a decinormal solution is only 0.187°, so that to obtain an accuracy of 1 per cent on the value of i, as measured by the freezing-point depression, we must be able to determine depressions with an accuracy of about a thousandth of a degree centigrade. It is by no means easy to attain this degree of accuracy, the error in ordinary careful work with Beckmann's apparatus being nearer a hundredth of a degree than a thousandth. For more dilute solutions the relative error on the depression is of course still greater, and extraordinary precautions have to be taken if the freezing points are to be of any value in calculating van 't Hoff's coefficient i, or the degree of ionisation m. The following table gives a comparison of the degree of ionisation of solutions of potassium chloride as calculated from Whetham's determinations of the conductivity at 0°, and from the mean value of the best series of observations on the freezing point by different investigators. In the first column we have the dilution in litres, in the second the percentage dissociation deduced from the freezing points, and in the third the same magni- về 260 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. tude calculated by means of the dissociation hypothesis from the conductivity at 0°:- Dilution. V. 10 20 33.3 100 Percentage Ionisation=100 m. From Freezing Point. From Conductivity. 89.6 86.0 88.8 91.7 90.8 93.2 95.3 96.2 This is a typical example of the numerical agreement attained by the two methods of calculation, which, it must be remembered, are experimentally entirely independent of each other. As a rule the divergencies amount to about 2-3 per cent of the actual values, the numbers deduced from the conductivity being generally the greater, but the parallelism between the series is always close. In stronger solutions the differences are much larger, but this of course arises from the conditions for the comparison not being fulfilled. From what has been stated above, it is obvious that the hypothesis of electrolytic dissociation affords a qualitative explanation of the anomalous behaviour of the solutions of salts, strong acids, and strong bases with regard to freezing point, boiling point, and in general all magnitudes directly derivable from the osmotic pressure. Not only does it do this, but it explains also very directly the additive character of most of the properties of salt solutions. In a previous chapter it has been shown that the properties of salts in aqueous solution are most simply explained when we assume that they are additively composed of two terms, one depending on the basic or positive portion of the salt, and the other on the acidic or negative portion. Accord ing to the theory of electrolytic dissociation, the salt is actually decomposed on dissolution in water into its positive and negative ions, which then lead a mutually independent existence, each conferring therefore on the solution its own properties undisturbed by the properties of the other ion. The total numerical value of any given. property in a salt solution should consequently be made up of the value for the positive ion plus the value for the negative ion, at least when the solutions considered are dilute, and so the additive character of the properties of dilute salt solutions is satisfactorily accounted for. It should be borne in mind, however, that additive properties (p. 148) are also frequently encountered amongst non-electrolytes. The theory of electrolytic dissociation offers, too, a simple explana- tion of a set of facts that are so familiar that we generally accept them without any attempt at accounting for them. It is well known that salts in aqueous solution enter into double decomposition with the greatest readiness. If we add a soluble silver salt to a soluble chloride, a precipitate of silver chloride is at once produced. The positive and negative radicals here exchange partners, and they do so readily because the positive and negative for the most part exist free in the solution XXIII ELECTROLYTIC DISSOCIATION 261 as ions, so that the whole action practically consists of the union of the silver ions with the chloride ions to produce the insoluble silver chloride. If alcohol is used as solvent, the action takes place with equal readiness. If we take now a solution in alcohol of an organic chloride, such as phenyl chloride, we find that it is a non-electrolyte, and corresponding with this, it may be mixed with an alcoholic solution of silver nitrate without any double decomposition taking place. Even after boiling for a consider- able time there is little or no silver chloride produced. In this case there are no free chloride ions in the solution to unite directly with the silver ions, and consequently the action is much slower. Other organic halogen compounds act much more rapidly than phenyl chloride when in alcoholic solution, but it is doubtful if any act with a speed even approximately comparable with that of the inorganic chlorides. Reactions between salts in aqueous solution, which do not proceed by a simple rearrangement of the ions, are much slower than the double decompositions where the ions undergo no change except rearrangement. We have instances of this type of reaction in the oxidations and reductions occurring in aqueous solution, such as the conversion of ferrous into ferric salts, or stannous into stannic salts, and vice versa. Although one may find exceptions to these rules, they are yet of a general character such as we might expect on the theory of electrolytic dissociation. · It has occasionally been urged that the existence of chlorine in a solution of sodium chloride cannot be accepted even hypothetically, as the solution shows none of the properties of a solution of chlorine. This, of course, rests on a misunderstanding. What we suppose to exist in the solution is not chlorine, but chloridion. The molecule of the former consists of two uncharged atoms of chlorine, the molecule of the latter consists of one atom of chlorine, possibly hydrated, and charged with negative electricity according to Faraday's Law, i.e. every 35 46 g. of chlorine as ion has 96,540 coulombs of electricity associated with it. But we know that a charge of electricity pro- foundly affects the chemical properties of substances. The mere fact of the chemical changes accompanying electrolysis is evidence of this, and other instances exist in plenty. Neither aluminium nor mercury decomposes water at the ordinary temperature, but if the aluminium is coated with mercury the amalgam formed has this power, hydrogen being evolved and aluminium hydroxide produced. The action of the copper-zinc couple is similar. These metals separately are unable to perform chemical reactions which are easily brought about by zinc coated with copper. In each of these cases the two metals on coming into contact assume electrical charges which greatly modify their ordinary chemical character. With regard to the electrolytic behaviour of non-aqueous solutions, it may be said of them in general that the relations they exhibit are much less simple in character than those shown by solutions in which 262 CHAP. INTRODUCTION TO PHYSICAL CHEMISTRY water is the solvent. Besides organic solvents such as the alcohols, which have been already mentioned, solvents like liquid ammonia, liquid hydrocyanic acid, and liquid sulphur dioxide have been em- ployed, and many interesting results obtained. For example, bromine, which is scarcely a conductor when dissolved in ordinary solvents, gives a molecular conductivity when dissolved in liquid sulphur dioxide greater than that of many weak acids in aqueous solution. Again, triphenylmethyl chloride (CH),CCl in liquid sulphur dioxide is as good a conductor as methylammonium chloride N(CH3)H,Cl, which is universally accepted as a true salt. Potassium iodide dissolved in liquid iodine yields a solution which is an excellent conductor. What the ions are in such cases is somewhat difficult to determine, but these examples make it plain that our views as to ionogenic substances must be widened so as to include many other substances than the ordinary acids, bases, and salts. If we refer to the table of dielectric constants on p. 231 we observe that those solvents which have a high value of the constant are precisely those which are self-associative, and tend to react with. substances dissolved in them (p. 219). In virtue of their possession. of both of these properties it is probable that prior to ionisation these solvents combine with the dissolved substances, the compound pro- duced then dissociating into electrically charged molecules or ions. A glance at the conducting combinations with non-aqueous solvents. mentioned in the preceding paragraph will show that there is a tendency to combination between solvent and solute. Thus potassium. iodide is known to form a periodide with iodine even in aqueous solution. The cause of the absence of exact agreement between the con- ductivity and freezing-point method of estimating ionisation, and also the want of obedience to the mass-action dilution law of Ostwald (p. 254) in the case of good electrolytes has given rise to much discussion, and many explanations have been offered of the apparent anomaly. The most successful of these is that put forward by J. C. Ghosh, who assumes that salts such as potassium chloride are completely ionised at all dilutions. In the solution, however, all the ions are not free to move, owing to the influence of the electric charges, and only the mobile ions contribute to the conductivity. The proportion of mobile. ions present at any dilution may be calculated from purely electrical data, so that the chemical law of mass-action does not enter into the problem at all. Ghosh's general formula is as follows- A M = С nRT A where n is the number of ions into which a molecule of the electrolyte XXIII ELECTROLYTIC DISSOCIATION 263 : dissociates, and A is the work done when the ions constituting a gram molecule pass beyond one another's sphere of attraction. For a binary univalent salt we have, A where N = Avogadro's number (Chap. XXXII.) = 6·06 × 1023, e = Charge on the ion (Chap. XXXII.)=4·77 × 10-10 electrostatic units, €= Dielectric constant of water =81, V = Dilution in cubic centimetres. 2. 10 20 50 100 200 500 1000 If we evaluate the expression for m at 18°, the gas constant R being given in ergs (p. 24), we obtain, log10m= or, if we express the dilution v in litres as usual, V/v. log₁m= −0·1626. According to this formula the "active" proportion of all salts yielding two univalent ions is the same at the same dilution, and may be measured as before by the ratio / The following table contains the values of m as deduced from Kohlrausch's data for four salts in order to compare these with the values calculated from Ghosh's formula: 3 Ne² 2N € 3 T Ꮴ KCI. *845 .875 ·906 *925 ·941 •956 •964 0·434 × (6·06 × 1023) × (4·77 × 10-10)2 × 2 × (8.32 × 107) × 291 × 81 × 3√ √ NaCl. ⚫836 •867 •905 ⚫929 •947 •964 *972 12·12 × 1023 NaNOg. NHẠC. .838 •836 •871 •875 •904 •908 *923 *929 •947 •940 •955 •964 •964 •972 Average. Theoretical. ·839 •872 •906 •927 944 •960 •968 •841 •872 ·904 *923 •938 •954 •963 } The agreement between the theoretical and experimental values is satisfactory. The slight difference between the values of m for sodium and ammonium chloride in the table above and those on pp. 251 and 256 is due to the use of slightly different values of μ, these being liable to a little uncertainty owing to the extrapolation necessary for fixing them, as they are, of course, inaccessible to direct experiment. Ghosh's general formula applies not only to univalent electrolytes, but also when appropriate values are introduced, to electrolytes of the type of Na¿SO", Ba¨Cl2, Zn¨SO". Non-aqueous solutions also come within its scope. In the case of half-electrolytes the ordinary law of mass-action. applies, modified only to a very slight extent by the electric equilibrium which prevails in the case of the strong electrolytes. 264 INTRODUCTION TO PHYSICAL CHEMISTRY CH. XXIII The student who wishes to pursue the subject of Electrolysis and Electro- chemistry from the dissociation point of view may be referred to S. ARRHENIUS, Electrochemistry (1902); and R. LÜPKE, Electrochemistry (1903). Tables of Conductivity Data are given in KOHLRAUSCH AND HOLBORN'S book, Leitvermögen der Elektrolyte, insbesondere der Lösungen (1916). The fundamental paper of ARRHENIUS "On the Dissociation of Substances dissolved in Water" appeared in Zeitschrift für physikalische Chemie, 1 (1887), p. 631. References to the literature for non-aqueous solutions are given by P. WALDEN in a paper "On Abnormal Electrolytes," ibid. 43 (1903), p. 385. B. D. STEELE, D. M'INTOSH, and E. H. ARCHIBALD, "The Halogen Hydrides as conducting Solvents," Philosophical Transactions, 205 A (1905), p. 99. JAMES KENDALL AND J. E. BOOGE, "The Mechanism of the Ionization Process," Jour. Amer. Chem. Soc. 39 (1917) p. 2323. J. C. GHOSH, "The Abnormality of Strong Electrolytes," Trans. Chem. Soc. 113 (1918), pp. 449, 627, 707, 790. Compare D. L. CHAPMAN and H. J. GEORGE, Philosophical Magazine, 41 (1921), p. 799. 1 CHAPTER XXIV BALANCED ACTIONS MANY of the chemical actions familiar to the student in his laboratory experience are reversible, i.e. under one set of conditions they proceed in one direction, under another set of conditions they proceed in the opposite direction. Thus if we pass a current of hydrogen sulphide into a solution of cadmium chloride, double decomposition occurs, according to the equation. CdCl2 + H2S = CdS + 2HCl. If the precipitate is now filtered off and treated with a solution of hydrochloric acid of the requisite strength, the action proceeds in the reverse direction, the equation being CAS + 2HCl = CdCl2 + H2S. What determines the direction of the action in this case is apparently the relative concentrations of hydrogen sulphide and hydrogen chloride present in the solution. If hydrogen sulphide solution is added to a solution of a cadmium salt which contains a considerable quantity of free hydrochloric acid, a part only of the cadmium will be precipitated as sulphide, part remaining as soluble cadmium salt. We are here dealing with a balanced action, and we shall find it convenient to formulate actions of this type by means of the ordinary chemical equation for the action with oppositely-directed arrows instead of the sign of equality, thus :- CdCl₂+ H₂S CdS + 2HCl. As a rule, in analytical work we do not wish to stop half-way in a chemical action, and therefore choose such conditions for the action that it will proceed practically to an end. Cadmium chloride is completely precipitated as sulphide when there is little hydrochloric acid present, and when a considerable excess of sulphuretted hydrogen has been added. 265 266 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. A balanced action of the same sort, but with the point of balance or equilibrium towards the other end of the reaction, is to be found when a solution of hydrogen sulphide is added to a solution of zinc chloride. A white precipitate of the metallic sulphide is formed, but even though a very large quantity of hydrogen sulphide is present, the precipitation is never complete, and a very moderate quantity of free hydrochloric acid will prevent the precipitation altogether. It is thus possible, by suitably choosing the conditions, to effect a separation of cadmium from zinc by means of sulphuretted hydrogen, although in the two cases we are dealing with balanced actions of precisely the same type. If some dilute hydrochloric acid is added to the solution of the mixed metallic chlorides, hydrogen sulphide will precipitate the cadmium almost completely, and pre- cipitate practically none of the zinc salt, even though it is present in great excess. If ammonium hydroxide solution is added to a solution of a magnesium salt, say magnesium chloride, part of the metal is precipitated as magnesium hydroxide, in accordance with the equa- tion MgCl2 + 2NH₂OH = Mg(OH)2 + 2NH¸C1, but the precipitation is never complete, for the reverse action Mg(OH)2 + 2NH¸C1 = MgCl₂ + 2NH¸OH occurs simultaneously, and a state of balance results. In analytical practice we have usually excess of ammonium chloride present from the beginning, so that the addition of ammonium hydroxide produces no precipitate in the solution of magnesium salt. The reverse action. can be easily studied by shaking up freshly-precipitated magnesium hydroxide with a solution of ammonium chloride, when the magnesium hydroxide will be found to dissolve (cp. Chapter XXIX.). We may consider such cases of chemical equilibrium from the same standpoint as we adopted in Chapter IX. for physical equilibrium. In the last instance given above, viz. the addition of ammonium hydroxide solution to magnesium chloride solution, if we mix definite quantities of solutions of definite strengths, the precipitation of magnesium hydroxide will come to an end when the reaction has proceeded to a certain ascertainable extent. Equilibrium is then reached, and the system undergoes no further apparent change. We may still conceive the opposed reactions to go on as before, but at such a rate that exactly as much magnesium hydroxide is formed by the direct action as is reconverted into magnesium chloride by the reverse action. Now at the beginning of the action there is no magnesium hydroxide or ammonium chloride present at all; these are only formed as the direct action proceeds. There is therefore at the beginning no reverse action. It is obvious that if a state of balance XXIV BALANCED ACTIONS 267 is to be reached, the rate at which the direct action proceeds must fall off, or the rate at which the reverse action proceeds must increase, or else both these changes must occur together. As the action progresses, the relative proportions of the reacting substances and the products of the reactions vary, and so we should be disposed to connect the actual rate at which magnesium hydroxide is formed or decom- posed with the amounts of the different substances in solution. Further information on this point may be got by adding ammonium. chloride to the solution of magnesium chloride from the beginning. If we add a very small quantity of ammonium chloride before we add the ammonium hydroxide, keeping the relative proportions of the other substances the same as before, we shall find that the amount of magnesium hydroxide precipitated is smaller than before, and if we go on increasing the amount of ammonium chloride little by little, we at last reach a point when no magnesium hydroxide is precipitated at all. Evidently, then, the presence of ammonium chloride favours the reverse action, and that in a manner proportionate to the amount added. Increase in the amount of ammonium hydroxide, on the other hand, favours the direct action, so that on the whole we should be inclined to suspect that the rate of a reaction depended on the amount of reacting materials present. >> Guldberg and Waage, from a consideration of many experiments, formulated the connection between rate of action and amount of reacting substances in the following simple way. The rate of chemical action is proportional to the active mass of each of the reacting. substances. This rule must in the first instance be taken to apply to dilute solutions or gases, for it is in their case only that "active mass can be properly defined. By active mass Guldberg and Waage under- stood what we usually term the molecular concentration of a dissolved or gaseous substance, i.e. the number of molecules in a given volume, or in the ordinary chemical units, the number of gram molecules per litre. It is possible, however, as Arrhenius has suggested, that this molecular concentration in solution is not really a measure of the active mass, and that instead of it we ought to substitute the osmotic pressure of the dissolved substance. For our present purpose, we may take the active mass to be proportional to the molecular concentration without risk of committing any serious error, for, as we have already seen, there is, at least in dilute solution, almost exact proportionality between osmotic pressure and molecular concentration. Suppose we are dealing with the following chemical action- A + B = C + D, where the letters represent single molecules of the substances as in ordinary chemical formulæ. Let the molecular concentrations of the original substances A and B be a and b respectively. The rate of the reaction will then, according to Guldberg and Waage, be proportional * 268 CHAP. INTRODUCTION TO PHYSICAL CHEMISTRY to a and also proportional to b, i.e. it will be proportional to the product ab. At the beginning of the action, then, we have the expression Rate = ab × constant, if by rate of reaction we mean the number of gram molecules of each of the reacting substances transformed in the unit of time, usually the minute. The constant, therefore, in the above equation, which is generally denoted by k, represents the rate at which the action would proceed if each of the reacting substances were of the molecular con- centration 1 at the beginning of the reaction, as may easily be seen from the transformed equation. 7c= Rate ab It must be remembered that as the action progresses the concentration of both A and B will fall off, and that the rate at which these substances are transformed into the substances C and D must there- fore progressively diminish. If at the time t the concentration of A has diminished by a gram molecules per litre, the concentration of B will have diminished by a similar amount, and the rate at which these substances are then transformed will be Rate = k(α − x)(b − x). The constant k is still the same as in the preceding equation, and has still the same significance, as may be seen from a consideration of the formula. It is indeed characteristic of the reaction, being inde- pendent of the concentrations of the reacting substances, although variable with temperature, nature of the solvent, etc. It is customary to call it the velocity constant of the action, and the student is recommended to bear in mind that it is the rate at which the reaction would proceed were the reacting substances originally present, and constantly maintained, at unit concentration. If the reaction A + B = C + D is not reversible, the transformation of A and B into C and D will go on at a gradually decreasing rate until at least one of the reacting substances has entirely disappeared. If the action, on the other hand, is reversible, the transformation of C and D into A and B will begin as soon as any of the former substances are formed by the direct action. If we suppose no C and D to be present when the action commences, we have at the beginning of the direct action c=0 and d=0. At the time t, when x of the original products has been transformed, we have c and d = a. If k' is the velocity constant of the reverse action, then at the time t Rate = k'x². Now after the direct action has proceeded for a certain time, which we XXIV BALANCED ACTIONS 269 may call 7, a state of equilibrium sets in. Let the diminution of the molecular concentration of A and B have at that time the value έ, then the rate of the direct transformation is k(a — §)(b − §). The rate of the reverse transformation at the same time is k'ε². But these rates must be equal if the system is to remain in equilibrium, for in a given time as much of A and B must be formed by the reverse action as disappears in the same time by the direct action. The equation k(a – §)(b − §) = k´§² ' હૃ consequently holds good, and this when transformed gives * (a — §)(b − §) - *2 k T = constant. Since k and k' are independent of the concentration, their ratio is also independent of the concentration, so that for equilibrium we have the product of the active masses of the substances on one side of the chemical equation, when divided by the product of the active masses of the substances on the other side of the chemical equation, giving a constant value no matter what the original concentrations may have been. If we wish to make the formula perfectly general, we may suppose that the original concentrations of the products of the direct action were c and d respectively. At the point of equilibrium the concentrations of these substances will then be c + έ and d + §, so that the constant magnitude will be (a – έ) (b − §) (c + §)(α + §) K k When, as very frequently happens, the original substances are taken in equivalent proportions, and none of the products of the direct. action are present at the beginning, the constant quantity has the simple form (a – §)² k हुए k' in which a represents the original molecular concentration of both reacting substances. No good practical instance of the application of the above formula. is known, although approximations to it are in some cases obtainable. The best is perhaps the equilibrium between an ester, water, and the acid and alcohol from which the ester is produced. Thus if we allow acetic acid and ethyl alcohol to remain in contact, they will interact with production of ethyl acetate and water. The action, 270 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. however, will not be complete, because the reverse action will simul- taneously take place, the ethyl acetate being decomposed by the water. into acetic acid and ethyl alcohol. The equation for the reversible action is CH¸ . COOH + C₂Н5. OH⇒ CH¸. COOС₂Н¿ + H₂O. 3 2 5 If the acetic acid and ethyl alcohol are taken in equivalent propor- tions, the action ceases when two-thirds of the substances have been transformed into ethyl acetate and water. Supposing the active. mass of the acid and alcohol to have been originally 1, the active masses at equilibrium will be = 1 − 3/3/3/ - Acetic acid Alcohol Ethyl acetate = 3, Water =3, = 1-2 = }}, 1-号 ​and the constant quantity will be (1 − §)² - = 1 | - ↓ - × X 1. £2 323 323 × 3232350 • = }}, (1 − )(3 — §) €2 }, This constant holds good for any proportions of the reacting substances, being the ratio of the velocity constants of the opposed reactions. We can therefore use it to determine the extent to which a mixture in any proportions of acetic acid and ethyl alcohol will be transformed. Thus, if for 1 equivalent of acid we take 3 equivalents of alcohol, at what point will there be equilibrium? If & represents the amount transformed at equilibrium, we have whence = 0·9, i.e. 90 per cent of the acid originally taken will be converted into ester by 3 equivalents of alcohol against 66.6 converted by 1 equivalent of alcohol. Direct experiment has shown that this amount of the acid is actually transformed. In this instance the substances are merely in solution in each other, but the presence of a neutral solvent, such as ether, in no way affects the equilibrium, although it greatly reduces the speed of the opposed reactions. There is an example of equilibrium in aqueous solution which has been verified with the utmost strictness for a very large number of substances, namely, the equilibrium between the ions of a weak acid or base and the undissociated substance itself. Suppose the substance considered to be acetic acid. If a gram molecule of the acid is dis- solved in water so as to give v litres of solution, its active mass is 1/v in our units. No sooner is the acid dissolved than it begins to split up into hydrion and acetanion. The rate at which this action XXIV BALANCED ACTIONS 271 progresses is, according to the principle of mass action, proportional to the active mass of undissociated acetic acid present at the time considered. Let the proportion of acetic acid dissociated be m. The proportion undissociated will therefore be 1-m, with the active mass 1 - m The rate of dissociation will thus be V 1 m C. or V where c is the velocity constant of the dissociation. But the action is a balanced one, so that undissociated acetic acid will be re-formed from the ions at the same time as the dissociation proceeds by the direct action. When the proportion of the acid transformed into ions is m, each of the ions will have the active mass m/v, and thus we shall have for the rate of reunion C 2 & (M): c if c' is the velocity constant of this reaction. Suppose now, in particular, that m is the proportion decomposed into ions when equilibrium has taken place. The rates of the opposed reactions must then be equal, and we obtain the equation 1 - m ข M² (1 − m)v_c´ ε (m C 2 k. This is Ostwald's dilution formula referred to on p. 254, and it was first deduced by him from the mass-action principle of Guldberg and Waage. The experimental verification has been given for acetic acid and ammonia (p. 254). What holds good for them is equally true. for many hundreds of feeble acids and bases which have been in- vestigated. The principle of mass action, which has been found above to hold good for substances in solution, also holds good for substances in the gaseous state. Nitrogen peroxide, when dissolved in chloroform, dissociates into simpler molecules (p. 213), according to the equation N₂O₁ = NO₂+ NO₂· 24 2 This is quite analogous to the equation for the dissociation of a weak acid or base, the only difference being that the products of dissociation are here of the same kind instead of being of different kinds. If m represents the dissociated proportion, 1m the undissociated propor- 272 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. tion, and v the dilution, we have, as before, the following expression regulating the equilibrium— m² (1 − m)v = constant, and this has been proved to be in accordance with the observed facts. Now nitrogen peroxide is also known in the gaseous state, and dissociates under these conditions precisely in the same manner as when it is dissolved in such a solvent as chloroform. The chemical equation for the dissociation is the same as before, and so is the expression for the amount of dissociation. The dilution v, in this case, is the volume occupied by 1 gram molecule of the gaseous substance. In cases of balanced action into which gaseous substances enter, the amount of dissociation is most readily determined by ascer- taining the pressure and density. With a given amount of substance occupying a certain volume at a known temperature it is easy to calculate what pressure it will exert, according to Avogadro's principle, there is no dissociation of the molecules. If there is dissociation, the pressure will, cæteris paribus, be greater, for in the given space there are more molecules than would be if no dissociation had oc- curred. With nitrogen peroxide, simultaneous determinations of the pressure and density are made at constant temperature, both being varied by changing the volume, and from these the magnitudes entering into the above formula can be calculated. In this case also there is close agreement between the observed numbers and the numbers calculated from the formula. Considering the matter from the kinetic molecular point of view, it is obvious that the position of equilibrium will in some cases depend on the volume of the system. Dilution increases the degree of dissociation of electrolytes in aqueous solution; and dilution increases the dissociation of nitrogen peroxide, whether in the state of solution or in the gaseous state. Each molecule of undissociated material decomposes on its own account, and is independent of the presence of other molecules. The proportion of the undissociated molecules, there- fore, which will decompose in a given time is entirely unaffected by the volume which the gas may be made to occupy. Dilution then does not affect the number of molecules which will dissociate in a given time. But dilution does affect the number of molecules re-formed from the dissociated products in a given time, for here each dissoci- ated molecule must meet with another dissociated molecule if an undissociated molecule is to be reproduced. Now the chance of two molecules meeting depends on the closeness with which the molecules. are packed. If the particles are close together, they will encounter each other frequently; if they are far apart, they will meet only rarely. Suppose that we double the volume in which a certain amount of nitrogen peroxide is contained. Each NO, molecule has XXIV 273 BALANCED ACTIONS 2 now to travel much farther than before in order to meet another NO₂ molecule, and the number of encounters in a given time will therefore be diminished. The rate therefore of the reverse reaction corresponding to the equation N₂O₁ = 2NO₂ 2 4 is considerably reduced by doubling the dilution; the rate of the direct action is unaffected. If then there was equilibrium between the direct and reverse actions before the system was made to occupy a larger volume, this equilibrium will be disturbed and a new equilibrium will be established at a point of greater dissociation. Here there will be proportionately less of the undissociated substance, and proportion- ately more of the products of dissociation, in order that the rates at which the undissociated nitrogen peroxide is decomposed and re-formed may again be the same. In general, we may say that diminution of volume, or increase in pressure, displaces the point of equilibrium towards the system which occupies the smaller gaseous volume. Thus dissociation, accompanied by increase of gaseous volume, is diminished when the pressure is increased, and conversely. In the important technical reaction expressed by the following equation 2SO,+0 = 280, 2SO3 3 vols. 2 vols. the system on the right of the equation occupies the smaller volume and its production will therefore be favoured at constant temperature by increase of pressure. In actual practice the action, by the use of finely divided platinum as catalyst, takes place so readily that it is not necessary to exceed atmospheric pressure. On the other hand, high pressures are employed for the synthetic production of ammonia, for which the equation is 2N₂+ 3H 2 5 vols. 2NH, 2 vols. Here there is a great contraction on combination, and pressure has a correspondingly great effect. At the ordinary pressure and the most favourable temperature that can be practically employed, the yield of ammonia is altogether trifling, but by the use of pressures varying according to the process from 100 atm. to 1000 atm. a sufficiently large proportion of ammonia may be obtained to render the method successful. When the volume on both sides of the equation is the same, pressure has no effect on the point of equilibrium. Thus in the technical synthesis of nitric oxide from nitrogen and oxygen according to the equation. N₂+ 0,2NO 2 2 vols. 2 vols. T 274 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. pressure may be disregarded and temperature alone considered. That change of volume is without effect on such an equilibrium may be seen from the consideration that in each reaction two molecules have to meet in order to change into the other system. The chances of encounter are then equally affected by change of volume, and the position of equilibrium is not disturbed. The above instances of balanced action are all of such a type that the system in which the equilibrium occurs is a homogeneous system. -either a homogeneous mixture of gases, or of substances in solution. We have now to consider cases of heterogeneous equilibrium, the system consisting of more than one phase. As an example, we may take the dissociation of calcium carbonate by heat, according to the equation CaCO, → CaO + CO, Here we are dealing with two solids and one gas. The active mass of the gas may either be measured by its concentration, i.e. density, or its pressure, the two being closely proportional. There is evidently a difficulty in expressing the active mass of a solid in a similar way. The pressure of a solid cannot be measured in the same way as the pressure of a gas, and the active mass could scarcely be expected to be proportional to the density of the solid, which is the only direct meaning we can give to concentration in this connection. The most instructive way of looking at such an equilibrium is to imagine it to take place entirely in the gaseous phase, the solids simply affording continuous supplies. of their own vapour. Every liquid has, as we have seen, a definite vapour pressure for each temperature. At 360° the vapour pressure of mercury is 760 mm. At the ordinary temperature it is not easily measurable, being too small, but the presence of mercury vapour over liquid mercury may easily be rendered evident at temperatures much below the freezing point. Ice, too, has a vapour pressure of a few millimetres at the freezing point, and there is no very good reason to think that this vapour pressure would disappear entirely at any temperature, however low, although it might become vanishingly small. We may then freely admit the possibility of calcium carbonate or calcium oxide vapour over the respective substances, although the pressure of these vapours is so small that they escape our means of measurement, or even of detection. If we are to assume the existence of these minute quantities of vapour, we must assume that the same laws are followed in their case as in those instances which are acces- sible to our measurement. In particular, we must assume that at a given temperature calcium carbonate, for example, will have a perfectly definite vapour pressure, which will remain constant as long as the temperature remains constant. We have thus in the gaseous phase a constant concentration or pressure of the vapour of the solid, the presence of the solid maintaining the pressure at its proper value, XXIV BALANCED ACTIONS 275 although the substance may be continually removed from the gaseous phase by the chemical action. This constant concentration is obviously unaffected by the quantity of solid present, as the vapour pressure of a small quantity is as great as that of a large quantity of the same solid. From the above reasoning, then, we conclude that the active mass of a solid is a constant quantity, being in fact proportional to the vapour pressure of the solid, which is constant for any given tempera- ture. Experiment confirms this conclusion, which was, indeed, arrived at by Guldberg and Waage on purely experimental grounds. For the action CaCOg = CaO + CO, if P, P' and p denote the equilibrium pressures (or active masses) of calcium carbonate, calcium oxide, and carbon dioxide respectively, we have at the point of equilibrium the following equation- kP kP = k'P'p, or p = KP" in which all the quantities are constant except p. At any given temperature, then, the pressure of carbon dioxide over any mixture of calcium carbonate and calcium oxide has a fixed value quite inde- pendent of the proportions in which the solids are present. This particular pressure of carbon dioxide, which is called the dissociation pressure of the calcium carbonate, is the only pressure of carbon dioxide which can be in equilibrium with calcium carbonate, calcium. oxide, or with any mixture of the two. Greater pressures are in equilibrium with calcium carbonate only, smaller pressures are in equilibrium with calcium oxide only. As the temperature increases, the dissociation pressure likewise increases, so that we can get a temperature curve of dissociation pressures resembling that for the vapour pressures of a liquid. Similar phenomena to the above are met with in the dehydration of salts containing water of crystallisation. The hydrated and the anhydrous salts are here the solid substances, and water vapour the gas. For a given temperature each hydrate has a certain dissociation pressure of water vapour over it (cp. p. 124). There is here, however, the complication that a salt usually forms more than one hydrate, in which case the dehydration often proceeds by stages, the hydrate with most water of crystallisation not passing immediately into water vapour and the anhydrous salt, but into water vapour and a lower hydrate. Take, for example, the common form of copper sulphate, the pentahydrate CuSO4, 5H,O. At 50° this hydrate gradually loses water (ep. Fig. 24, p. 123), and becomes converted into the trihydrate CuSO, 3H,O. As long as any pentahydrate remains, a definite dissociation pressure of 47 mm. of water vapour persists over the solid. If the water vapour is removed, the pentahydrate will give up 276 CHAP. INTRODUCTION TO PHYSICAL CHEMISTRY 2 more water, and the equilibrium pressure will be re-established. If the dehydration still goes on, the pressure will remain constant at 47 mm. until all the pentahydrate has been converted into trihydrate, when the pressure will suddenly fall to 30 mm. This new pressure is the dissociation pressure of the trihydrate, which now begins to lose water and to pass into the monohydrate CuSO4, H₂O. As long as there is trihydrate present the pressure of 30 mm. will be maintained, but as soon as all the trihydrate has passed into monohydrate, the pressure again drops suddenly to 4.5 mm., which is the dissociation pressure of the latter substance. The solid dissociation product is now the anhydrous salt, and when all the monohydrate has been converted into this the pressure of water vapour drops to zero. What holds good here for the dehydration of hydrates also applies to the removal of ammonia from such compounds as AgCl, 3NH,, in which the ammonia may be removed in successive stages, with formation of intermediate compounds, e.g. 2AgCl, 3NH,. Sometimes dissociation results in the formation of two gaseous substances derived from one solid, e.g. the dissociation of ammonium salts, the solid chloride furnishing both ammonia and hydrochloric acid. In this case also there is a constant dissociation pressure for each temperature. If P is the constant vapour pressure of the undissociated ammonium chloride, and p the gaseous pressure of the ammonia or of the hydrochloric acid, these being equal since the two gases are produced in molecular proportions by the dissociation of the ammonium chloride, we have kP = k'p², or p² = NH₂HS 4 kP K whence the total dissociation pressure 2p is also constant. Balanced actions of this kind have been studied with ammonium hydrosulphide and similar compounds which dissociate at comparatively low temperatures. From the above equation it is apparent that the product of the pressures of ammonia and the acid is constant, i.e. for the balanced action kP=k'pp', or pp': constant, NH3 + H₂S the constant quantity is the product of the pressures of ammonia and hydrogen sulphide. If these substances are derived entirely from the dissociation of ammonium hydrosulphide, the pressures will be equal, but it is possible to add excess of one or other gas from the beginning, in which case they will no longer be cqual. Their product, however, will retain the same value as before, though their sum, i.c. the total gas pressure, will have changed. This is evident from the equation kP K constant, XXIV BALANCED ACTIONS 277 in which p and p' represent the pressures of the ammonia and the sulphuretted hydrogen respectively. Since p and p' enter into the equation in the same way, an excess of the one will have precisely the same effect on the equilibrium as an excess of the other. This conclusion also is verified by experiment, as may be seen from the following table, which contains some of the results of Isambert with ammonium hydrosulphide. Under p is the pressure of ammonia in centimetres of mercury, under p' the pressure of hydrogen sulphide, the other columns giving the variable total pressure and the constant product of the individual pressures. For this series the temperature of experiment was 17.3°. p 15.0 (10.3 Excess of HS 37.7 Excess of NH3 41.6 5.35 p' 15.0 21.4 41.9 6.43 5.59 p+p' 30.0 31.7 47.2 44.1 47.2 pp' 225 220 224 243 232 It is true that the numbers in the last column do not exhibit very great constancy, but this is due to experimental error, as comparison with other similar series of numbers serves to show. In the previous cases of balanced action we have had gaseous substances on one side of the chemical equation only. We now proceed to deal with a case where gaseous substances occur on both sides. If steam is passed over red-hot iron, a ferroso-ferric oxide of the composition Fe,O, is formed, the water being reduced to hydrogen. For the sake of simplicity we may assume that the product of oxida- tion of the iron is ferrous oxide, FeO, so that we have the equation 4 5 H₂O + Fe = FeO + H₂. If we continue to pass steam over the solid, all the iron is eventually oxidised, notwithstanding which, however, the action is a balanced one. For if we take the oxide produced, and heat it in a current of hydrogen, the hydrogen is oxidised to water, and the iron oxide reduced to metallic iron, the reduction being complete if the current of hydrogen is continued for a sufficient length of time. The reason why we can in each case complete the action is that the passage of the current of gas disturbs the equilibrium, which is never completely established. The nature of the equilibrium may be seen from the equation H₂O+Fe FeO + H₂. 2 On each side we have one solid substance and one gas. Since the gases are now on opposite sides of the equation, the equilibrium is regulated by the quotient of their pressures instead of by their product, as in the preceding case. Let p, P, P', p' be the equilibrium pressures of the reacting substances in the order in which they occur in the chemical equation. We obtain the equilibrium equation. 278 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. p k'P' p kP kpP = k'p'P', or constant. The ratio of the pressures of water vapour and hydrogen thus determines the equilibrium. If, as occurs when we pass water vapour over the iron, the ratio of the pressure of water to the pressure of hydrogen is greater than the equilibrium ratio, the action proceeds until all the iron has been converted into oxide. On the other hand, when we pass a current of hydrogen over the heated oxide, the above ratio is always less than the equilibrium ratio, and the oxide is completely reduced. The ratio changes with temperature, and becomes equal to unity at about 1000°. At this temperature, therefore, if we take equal volumes of water vapour and hydrogen, and pass the mixture over either iron or the oxide Fe̟ O̟, no chemical action will take place, for the pressures are then in the ratio for equilibrium. Corresponding to the preceding cases of equilibrium with gases we have similar instances with substances in solution. In aqueous solution, however, the equilibrium is very often complicated by the occurrence of electrolytic dissociation. Several cases of this kind will be considered in a subsequent chapter. A case of heterogeneous equilibrium in solution which is analogous to the dissociation of calcium carbonate is to be found in the action of water on insoluble salts of very weak insoluble bases combined with soluble acids. An organic base such as diphenylamine is so weak that its salts when treated with water split up almost entirely into the free acid and free base. Diphenylamine itself is practically insoluble in water, and so is the picrate formed from it by its union with picric acid. When the solid picrate is brought into contact with water, it partially dissociates with formation of insoluble base and soluble picric acid. We have, therefore, the equation Diphenylamine picrate Diphenylamine + Picric acid exactly analogous to the equation Calcium carbonate Calcium oxide + Carbon dioxide, with the exception that the picric acid is in the dissolved state, whereas the carbon dioxide is in the gaseous state. Now in the gaseous equilibrium we found that for each temperature a certain pressure, or concentration, of gas was produced. We should expect, therefore, that in the solution-equilibrium for each temperature there should exist a certain osmotic pressure, or concentration, of the dis- solved substance which should be necessary and sufficient to determine the equilibrium, independent of the proportions in which the insoluble solid substances may be present. This has been confirmed by experi ment. At 406° a solution of picric acid containing 13.8 g. per XXIV BALANCED ACTIONS 279 litre is in equilibrium with diphenylamine and its picrate, either singly or mixed in any proportions, and if diphenylamine picrate is brought into contact with water at this temperature it dissociates until the concentration of the picric acid in the water has reached this point. If we bring carbon dioxide at less than the dissociation pressure into contact with calcium oxide, no carbonate is formed. Similarly, if we bring at 406° a solution of picric acid, having a concentration of less than 13.8 g. per litre, into contact with diphenylamine, no diphenylamine picrate will be formed. On the other hand, if the solution has a greater concentration than 13.8 g. per litre, diphenyl- amine picrate will be produced until the concentration has sunk to that necessary for the equilibrium. This may be readily shown experimentally, on account of the different colours of diphenylamine and its picrate. The base itself is colourless, picric acid yields a yellow solution, and diphenylamine picrate has a deep chocolate- brown colour. A solution of picric acid at 40.6° containing 14 g. per litre at once stains diphenylamine deep brown, but a solution at the same temperature containing 13 g. per litre leaves the diphenyl- amine unaffected. If the weak base were soluble instead of insoluble, we should have an equilibrium for solutions corresponding to the dissociation of ammonium hydrosulphide. Urea is such a base, and the picrate of urea is very sparingly soluble as such in cold water, so that we have the equation Picrate of urea Urea + Picric acid. On the left-hand side of the equation we have a solid; on the right- hand side we have two substances in solution. The sparingly soluble urea nitrate and oxalate afford similar instances. Phenanthrene picrate, when dissolved in absolute alcohol, dissociates into phenanthrene and picric acid, both of which are soluble. This case has been investigated, and found to obey the law of mass action. The calculation is, however, complicated on account of the picric acid. being partially dissociated electrolytically, and part of the phenan- threne being associated in solution to larger molecules than that corresponding to the ordinary molecular formula. There are plenty of instances in solution corresponding to the action of steam on metallic iron. If barium sulphate is boiled with a solution of sodium carbonate, it is partially decomposed, according to the equation BaSO4 + Na2CO3 = BaCO3 + Na,SO4 Here both the barium salts are insoluble and the sodium salts soluble, so that there is one solid and one salt in the dissolved state on each side of the equation. The active masses of the barium salts may be accounted constant during the reaction, for although they are generally 280 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. spoken of as "insoluble," they are in reality measurably soluble in water (cp. Chap. XXIX.). The aqueous liquid in contact with them will therefore be and remain saturated with respect to them, i.e. their concentration and active mass in the solution will be constant. The equilibrium will thus be determined by a certain ratio of the con- centrations of the soluble sodium salts, independent of what the actual values of the concentrations may be. Guldberg and Waage found by actual experiment that the concentration of the carbonate should be about five times that of the sulphate if the solution is to be in equilibrium with the insoluble barium salts. Such a solution will convert neither sulphate into carbonate nor carbonate into sulphate. It should be noted, however, that more recent experiments have shown that the ratio of carbonate to sulphate in the equilibrium solutions varies somewhat with the total concentration. This may be due to changes in ionisation, molecular association in concentrated solutions, or the like, but the data at present available do not permit a definite conclusion to be drawn. In general when we are dealing with salts in solution, Guldberg and Waage's Law has only an approximate application if we use concentration in the ordinary sense as the measure of the active mass of the dissolved substances, for by doing so we entirely neglect the effect of electrolytic dissociation. When the reactions are considered more closely, we find that it is usually the ions that are active, so that we ought really to consider ionic concentrations in the majority of cases, instead of the total concentrations of the sub- stances. It happens, however, that in a great many instances the influence of dissociation is such that it affects the two opposed reactions equally, and in such cases the approximate conditions of equilibrium may be arrived at although ionisation is entirely neglected. In the instance given above of the equilibrium between soluble carbonates and sulphates and the corresponding insoluble barium salts, all the substances are highly ionised, and approximately to the same extent on both sides of the equation, so here the application of Guldberg and Waage's Law in its simple form gives results approximately in accordance with experiment. If we increase the active mass of any substance playing a part in a chemical equilibrium, the balance will be disturbed, and the system will adjust itself to a new position of equilibrium, that action taking place in virtue of which the active mass of the added substance will be diminished. Thus if we have carbon dioxide at a pressure such that the gas is in equilibrium with a mixture of calcium oxide and calcium carbonate, and suddenly increase its active mass by increasing the pressure upon it, a new equilibrium will be established by the action CaO + CO,= CaCO 3 taking place, the tendency of which is to diminish the active mass, or XXIV 281 BALANCED ACTIONS pressure, of the carbon dioxide. Again, if we have an aqueous solution containing sodium sulphate and sodium carbonate in such proportions as to be in equilibrium with the corresponding barium salts, and add an extra quantity of sodium sulphate to the solution so as to increase the active mass of this salt, the equilibrium will be disturbed, and will readjust itself by means of the action Na2SO4 + BaCO3 = Na¿CO3 + BaSO4, which will go on until the concentration of the sodium sulphate has fallen to a value which gives the equilibrium ratio with the new concentration of sodium carbonate. The principle here enunciated is of special importance in its For application to dissociation, whether gaseous or electrolytic. convenience' sake it may be stated for this purpose in the follow- or more of ing form. If to a dissociated substance we add one the products of dissociation, the degree of dissociation is diminished. Thus phosphorus pentachloride, which when vaporised dissociates according to the equation PCI PCI, + Cl2, 5 gives a vapour density little more than half that which corresponds to its usual formula. If, however, the pentachloride is vaporised in a space already containing a considerable quantity of phosphorus trichloride, the vapour density is such as would correspond to the formula PC, for the pentachloride. Here one of the products of dissociation has been added, viz. PCI, and the dissociation of the pentachloride has been in consequence diminished to such an extent that the vapour density has practically the normal value. In the dissociation of ammonium hydrosulphide (p. 276), the addition of either ammonia or hydrogen sulphide to the normal dissociation products will diminish the degree of dissociation. Here, however, the undissociated substance exists only to a very small extent in the vaporous state, the consequence being that a smaller quantity of the salt is vaporised. This may be seen by reference to the numbers given on p. 277. At 17.3° the dissociation pressure is 30 cm., half of this pressure being due to ammonia and half to hydrogen sulphide. If, now, we allow the hydrosulphide to dissociate into an atmosphere of hydrogen sulphide having a pressure of 36.6 cm., the increase in pressure will only be 107 cm., i.e. this will be the dissociation pressure under these conditions. If the dissociation is allowed to take place in an atmosphere of ammonia of 36 cm. pressure, we have a similar diminution of the dissociation to about one-third of its normal value. Suppose two substances are dissociating in the same space, and have a common product of dissociation: it is evident from what has been said that the degree of dissociation of each will be lower than -- 282 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. Tw the value it would possess were the substances dissociating into the given space singly. Take, for example, a mixture of ammonium hydrosulphide and ethylammonium hydrosulphide. These substances dissociate according to the equations 2 5 NH₁(C₂H₂)HS NH₂(C₂H¿) + H₂S, NH₂HS ⇒ NH3 + H₂S, and have hydrogen sulphide as a common dissociation product. When they dissociate into the same space, the effect is that as each supplies an atmosphere of hydrogen sulphide for the other to dissociate into, the dissociation pressure of each is diminished below the value it would have were it dissociating alone. Thus at 26.3° ammonium hydrosulphide has a dissociation pressure of 53'6 cm., and ethyl- ammonium hydrosulphide has a dissociation pressure of 13.5 cm. If the two substances did not affect each other when dissociating into the same space, the dissociation pressure of the mixture would be the sum of the dissociation pressures of the components of the mixture, viz. 67.1 cm. The value actually found by experiment is much lower than this, viz. 524 cm., which is even lower than the dissociation pressure of ammonium hydrosulphide itself. It should be stated that this result is not in accordance with the law of mass action, if pres.. sures are taken as the measure of the active mass of the gases. The theoretical result is 55 3 cm., somewhat greater than the dissociation pressure of the ammonium hydrosulphide. When, as in the above instance, two unequally dissociated sub- stances are brought into the same space, the more dissociated substance has a much greater effect on the degree of dissociation of the less dissociated substance than vice versa. This we might expect, for the more dissociated substance increases the concentration of the common dissociation product far above the value which would result from the dissociation of the less dissociated substance; while the latter, even if dissociated to its normal extent, would only slightly increase the concentration of the common dissociation product beyond the value obtained from the more dissociated substance alone. There is thus a great relative increase in the first case and a small relative increase in the second, the effects on the dissociation being in a corresponding degree. It remains now to discuss the effect of temperature on balanced action. A rise of temperature is almost invariably accompanied by acceleration of chemical action. In a balanced action, therefore, both the direct and reversed actions are accelerated when the temperature is raised. The effect on the opposed reactions is, however, not in general equally great, with the result that the point of equilibrium is displaced in one or other sense. This displacement is intimately con- nected with the heat evolved in the reaction. If the direct action gives XXIV BALANCED ACTIONS 283 out a certain number of calories per gram molecule transformed, the reverse reaction will absorb an exactly equal amount of heat. Now rise of temperature always affects the equilibrium in such a manner that the displacement takes place in the direction which will determine absorption of heat. If, therefore, the direct action is accompanied by evolution of heat, the action will not proceed so far at a high as at a low temperature, for if we start with equilibrium at the lower tem- perature, and then heat the system to a higher temperature, the heat- absorbing reverse action occurs, and the point of equilibrium moves backwards. If, on the other hand, the direct action absorbs heat, the action will proceed farther at high than at a low temperature. In gaseous dissociation at moderate temperatures the dissociation is almost always accompanied by absorption of heat, so that the degree of dis- sociation increases as the temperature is raised. Thus the dissociation pressure of calcium carbonate rises with rise of temperature, and so does the dissociation pressure of salts like ammonium hydrosulphide, as the following table shows:- DISSOCIATION PRESSURE OF AMMONIUM HYDROSUlphide 'Mm. of Mercury. 155 216 310 Temperature. 7.7° 12.2 17.6 22.4 27.6 Nitrogen peroxide, again, which at the ordinary temperature is only about 20 per cent dissociated into the simple molecules NO2, is at 130° almost entirely dissociated. In the technical reactions 2SO₂ + 0₂ 2.2SO3 O2 421 573 2N₂ + 3H₂ ✈ 2NH3 2 heat is evolved in the union and therefore rise of temperature tends to diminish the amount of combination. In consequence it is sought to carry out the processes at as low a temperature as possible, the reactions being accelerated by means of appropriate catalysts. On the other hand the production of nitric oxide N₂ + O₂ 2NO 2 ← is accompanied by absorption of heat, and high temperature favours the combination. Consequently the temperature of the electric arc is employed for the synthesis, and the product is cooled as rapidly as possible to a point where the velocity of the reverse action is so small as to render it negligible. The following table illustrates the difference between the effect of ** 284 CHAP. INTRODUCTION TO PHYSICAL CHEMISTRY temperature on the synthesis of ammonia and that of nitric oxide- Percentage of Ammonia in equilibrium mixture at 200 atm. 11.9 5.7 3.0 1.7 1.1 A Temperature. 550° 650° 750° 850° 950° Temperature. 1600° 1750° 1900° 2300° 3000° Percentage of Nitric Oxide from air as original mixture. 0.43 0.67 1.0 2.0 5.0 Since the heat of dissociation into ions is sometimes positive, sometimes negative, a rise of temperature may in some. cases be accompanied by increased ionisation, in other cases by diminished ionisation. A considerable diminution of the degree of ionisation with rise of temperature has been proved for hydrofluoric acid and hypophosphorous acid. The rule which has here been applied to the displacement of chemical equilibrium with change of temperature is equally applicable to physical equilibrium. If we take a quantity of a liquid, and enclose it in a space greater than its own volume, a certain proportion of the liquid will assume the vaporous state, heat being absorbed in the vaporisation. If now we raise the temperature, keeping the volume constant, the equilibrium will be disturbed, and the endothermic action. will occur, i.e. more liquid will be converted into vapour, and the vapour pressure thus become greater. A number of cases of balanced action of an interesting type have been classified under the name of dynamic isomerism. The two substances, ammonium thiocyanate and thiourea, are isomeric, both having the empirical formula CSN,H,, and their isomerism at temperatures below 100° does not differ in any special way from the isomerism of other substances. If either substance is fused, however, it is partially transformed into the other isomeride, a balance being attained at a point where the liquid consists of about 80 per cent of ammonium thiocyanate and 20 per cent of thiourea. In this particular instance the substances have no marked tendency at the ordinary temperature to pass into each other when pure, so that the opposed reactions and the balance between them can be studied experimentally. Other instances have been investigated in which a change of one isomeride into the other in solution can be followed in the polarimeter, owing to a difference in the optical activity of the two substances. It is probable that most of the phenomena of "tautomerism" and desmotropy" met with in organic chemistry are referable to similar causes. Liquids, for example, which are usually written with the group - CH₂. CO -, very frequently act as if they contained the enol group, - CH: C(OH) –, and the liquids themselves often give values for their physical properties which would accord with their being mixtures of the two isomerides. That the liquids may be such mixtures is highly probable, seeing that other instances of isomeric balance are now well authenticated. 66 XXIV BALANCED ACTIONS 285 Cases of balanced action are discussed in the following papers, which may be consulted by the student: J. T. Cundall—“Dissociation of Nitrogen Peroxide," Journal of the Chemical Society, 59 (1891), p. 1076; 67 (1895), p. 794. Compare also W. OSTWALD, ibid. 61 (1892), p. 242. J. WALKER and J. R. APPLEYARD "Picric Acid and Diphenylamine,” ibid. 69 (1896), p. 1341. J. WALKER and J. S. LUMSDEN-" Dissociation of Alkylammonium Hydrosulphides," ibid. 71 (1897), p. 428. T. M. LOWRY—"Dynamic Isomerism," ibid. 75 (1899), p. 235. CHAPTER XXV RATE OF CHEMICAL TRANSFORMATION IN the preceding chapter we have had occasion to use the conception of reaction velocity in order to facilitate the discussion of chemical equilibrium, without entering into the question of how such a magni- tude can be practically determined. It is only in comparatively few cases that an accurate determination is possible at all, for the vast majority of reactions either take place so rapidly, or are complicated to such an extent by subsidiary reactions, that no specific coefficient of velocity for the reaction can be calculated. One of the simplest reactions, and one of the earliest to be studied with success, is the inversion of cane sugar. When this substance is warmed with a mineral acid in aqueous solution, it is gradually con- verted into a mixture of glucose and fructose, and the process of conversion can be accurately followed by means of the polarimeter. The cane-sugar solution has originally a positive rotation, the invert sugar produced by the action of the acid has a negative rotation. If therefore we place the sugar solution in the observing tube of a polari- meter, and read off the angle of rotation from time to time, we can tell how the composition of the solution varies as time progresses without in any way disturbing the reacting system. For example, the original rotation of a cane-sugar solution was found to be 46.75°, whilst the rotation of the same solution after complete inversion was 18.75°. The total change in rotation, therefore, corresponding to complete con- version of the cane sugar into invert sugar was 46.75° + 18.75° 65.5°. After the lapse of an hour from the beginning of the reaction, the rotation was found to be 35.75°. In that time the rotation had thus diminished by 11:00°, so that the fraction of the original sugar transformed was 11.00 65.5 = 0·168. By a similar calculation the quantity of sugar transformed at any other time could be arrived at. Grading The following table gives the rotations actually observed at different times: 286 CH. XXV RATE OF CHEMICAL TRANSFORMATION 287 Time in Minutes. 0 30 60 90 120 150 210 330 510 630 00 Rotation. 46.75° 41.00 35.75 30.75 26.00 22.00 15.00 2.75 - 7:00 - 10:00 - 18.75 or k C12H22O11 + H2O = C6H12O6 + C6H12O6 Glucose Fructose dx dt t From this table it is evident that the rate at which the reaction proceeds falls off as less and less cane sugar remains in solution. In the first two hours the change in rotation is 20 75°; in the two hours from 510' to 630' the change is only 300°. This is in accordance with the principle of Guldberg and Waage that the amount transformed in a given time will fall off as less cane sugar remains to undergo transformation. The chemical equation expressing the reaction is log As the action progresses, water disappears as well as cane sugar; but if we consider that the action takes place in aqueous solution, it is obvious that any change in the active mass of the water can only be very slight, and so, for practical purposes of calculation, the active mass of the water may be taken as constant. The action is not a balanced one, but proceeds until all the cane sugar has been trans- formed. If a is the original concentration of the cane sugar, and a the quantity transformed at the time t, the rate of transformation at that time will be, according to the unimolecular formula, = k(ax), a a where de represents the very small quantity transformed in the very small interval dt starting at the time t, and is the coefficient of velocity of the action. From this equation the integral calculus enables us at once to find a relation between x and t, the corresponding values for any stage of the reaction in terms of the original concentra- tion and the velocity constant. This relation has the form a a 11.00 16.00 20.75 24.75 31.75 44.00 53.75 56.75 a=65.50 20 X ekt a ... 2 2.3 t 1 The values of the expression log t a 5.75 a ܝ.. log10 a Constant. ', - 0.001330 1332 1352 1379 1321 1371 1465 1463 1386 ―――― X are given in the last column. X 288 CHAP. INTRODUCTION TO PHYSICAL CHEMISTRY of the preceding table, and it will be seen that they remain tolerably constant throughout the reaction. It is unnecessary in calculating the value of the constant to convert the rotations into actual sugar concentrations, as we are dealing with the ratio α X For a, which expresses the total concentration, we may take the difference between the initial and final readings, viz. 46·75 − ( − 18·75) = 65·5, and for x the difference between the initial reading and that at the time t. These differences are given in the third column of the table on p. 287. The agreement between experiment and the theory of Guldberg and Waage is therefore in this case quite satisfactory.¹ The part played by the acid in the inversion of cane sugar is not well understood. The acid itself remains unaffected, but the rate of the inversion is nearly proportional to the amount of acid present if we always use the same acid. In such an action as this, the acid is said to act as a catalyst or accelerator, and this behaviour on the part of acids is by no means uncommon, although the different acids vary very much in their accelerative power. As we shall see later, the accelerative power of acids furnishes us with a convenient measure of their relative strengths. a A similar accelerating action of acids is found in the catalysis of esters. If an ester such as methyl acetate is mixed with water, the two substances interact, with formation of acetic acid and methyl alcohol (cp. Chap. XXIV.). This is in reality a balanced action, but if the solution of the ester is very dilute, the transformation is almost complete, and the action becomes of the same simple type as the inversion of cane sugar, the equation being CH¸. COOCH3 + H₂O = CH₂ . COOH + CH₂OH. If no acid is present, the action takes place with extreme slowness, but in presence of the strong mineral acids, such as hydrochloric acid, it progresses with moderate rapidity. Although the progress of the action cannot conveniently be followed by a physical method, as in the sugar inversion, a chemical method may be employed without disturbing the reacting system. At stated intervals a measured portion of the solution is removed and titrated with dilute alkali. As the action progresses, the titre becomes greater, owing to the pro- duction of acetic acid, and from this increase we may deduce the amount of transformation. It is found that at each instant the rate of transformation is proportional to the amount of ester present, and a velocity constant may be calculated by means of the same formula as was used for sugar inversion. From a comparison of the velocity 1 The inversion of cane sugar was studied, both practically and theoretically, by Wilhelmy, with the result arrived at above, before Guldberg and Waage enunciated their principle, and the numbers in the table are taken from Wilhelmy's work. XXV RATE OF CHEMICAL TRANSFORMATION 289 constants obtained with different acids, it appears that the relative accelerating influence of the acids is the same for both actions. Saponification of esters by alkalies affords us an example of a bimolecular reaction. The equation for the saponification of ethyl acetate by caustic potash is CHg . COOC,H + KOH=CH, . COOK + C,HOH, 3 5 and the action proceeds until one or other of the reacting substances entirely disappears. If we take both substances in equivalent propor- tions, and represent the active mass of each by a, the general equation for the rate of transformation will be integration of which leads to the equation 1 k Methyl acetate Ethyl acetate Propyl acetate Isobutyl acetate Isoamyl acetate N • dx dt Experimental work confirms this conclusion, the expression on the right-hand side of the last equation being in reality constant. The course of the saponification can be easily followed by removing measured portions of the solution from time to time, and titrating them with acid. As the action proceeds, the amount of acid required to neutralise the potassium hydroxide in solution falls off proportion- ally. For a given ester, equivalent solutions of the caustic alkalies and the alkaline earths effect the saponification at practically the same rate, the rate of saponification for ammonia being very much smaller. For a given base the rate of saponification is greatly affected by the nature of both the acid radical and the alkyl radical which go to form the ester. Thus at 9.4° the velocity constants of various acetates on saponification by caustic soda are as follows:- = = k(α − x)² Ethyl acetate Ethyl propionate . Ethyl butyrate Ethyl isobutyrate Ethyl isovalerate. Ethyl benzoate • X • t``a(a — x)' - • · · While the corresponding numbers for various ethyl esters with the same base at 14·4° are • · • 3.493 2.307 1.920 1.618 1.645 3.204 2.186 1.702 1·731 0.614 0.830 Other instances of bimolecular reactions which have been investigated are the formation of sodium glycollate from sodium chloracetate and caustic soda, according to the equation U 290 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. CH,C1. COONa + NaOH = CH,(OH). COONa + NaCl, and the formation of such salts as tetra-ethyl-ammonium iodide from tri-ethylamine and ethyl iodide, in accordance with the equation N(C,H,)g + C,HI=N(C,H,)_I. 5/3 Trimolecular reactions are, comparatively speaking, rare, amongst those which have been most thoroughly investigated being the reduction of ferric chloride by stannous chloride, viz. 2FeCl, + SnCl = 2FeC1, + SnCl 2 4' and the reduction of a silver salt in dilute solution by a formate, according to the equation 2AgC₂H₂O₂ + NaCO₂H = 2Ag + CO2 + HC₂H3O2 + NaC₂H₂O, For such reactions we have the following expression for the rate:- dx Ic dt = k(a-x)³, where the reacting substances have each the original active mass a. On integration this becomes 1 x(2a − x) t¨ 2a² (a − x)²' and it has been ascertained experimentally that the expression on the right-hand side of the equation is actually a constant. Few actions have hitherto been investigated which follow the equation expressing the rate for a higher number of molecules than three. At first sight this is surprising, for in our ordinary chemical equations we are familiar with well-known reactions where the number of reacting molecules is much greater than three. It would appear, however, that these complicated reactions take place in stages, so that each is really an action composed of successive simple reactions. When we measure the rate of a complicated action, we are in general measuring an average rate of a series of reactions which may be progressing successively or simultaneously, and it is the rate of the slowest of these actions which plays the principal part in determining the total rate. Should the other actions proceed at an immeasurably faster rate than the slowest action, the whole action will appear to go at a rate and be of a type regulated by this action alone. Hence it is that we so frequently find complex actions pro- ceeding in such a way as to suggest that they are much simpler in type than the total action really is. The saponification of an ester of a bibasic acid affords a good instance of the progress of a reaction in stages. For example, the XXV RATE OF CHEMICAL TRANSFORMATION 291 saponification of diethyl succinate by caustic soda has been shown to proceed, not according to the equation. C,H,(COOC,H,), + 2NaOH = C,H,(COONa),+2C,H,OH, 5/2 but according to the equations C,H,(COOC,H)2 + NaOH = C,H,(COOC,H,)(COONa) + C,H,OH, C₂Н(COOСН¿)(COONa) + NaOH = C₂H(COONa)2 + С₂͸ÓH. 5 On the last assumption, Guldberg and Waage's principle leads to a certain expression for the velocity of the total action involving the two velocity constants of the single actions, and the experimental rate has been found to agree with the theoretical requirements. The fact that the action does in reality proceed in two stages may be easily demonstrated by treating ethyl succinate in alcoholic solution with half the calculated quantity of caustic potash necessary for complete saponification. Instead of half the ester being completely saponified, and half being left untouched, about three-fourths of the original quantity is, under favourable conditions, converted into the potassium ethyl salt C₂H (COOEt) (COOK), the product of the first stage, one- eighth remaining unattacked, and one-eighth being converted into the dipotassium salt. In general, then, it may be accepted as a fact that in actions which are expressed by comparatively complicated chemical equations, involving the interaction of a large number of molecules, we are dealing with a series of simpler actions, not more than three molecules being involved in each of these. The velocity of a given action is usually greatly affected by change of temperature. Almost invariably a rise of temperature is accompanied by a large increase in the rate of the reaction, the speed being very frequently doubled for a rise of five or ten degrees, starting at the ordinary temperature. It is somewhat difficult to account for this very high temperature coefficient. On any of the usual hypotheses regarding the rate of molecular motion and its variation with the temperature, it is impossible to assume that the speed of the molecules increases so greatly that they encounter each other twice as often when the temperature rises from 15° to 20°. Probably only a certain small proportion of the total number of molecules is active at any one temperature, this proportion increasing rapidly as the temperature rises. The rate of disintegration of radio-active substances (Chapter XXXIII.) is completely independent of the temperature at which the disintegration takes place. Very slight changes in the nature of the medium greatly affect the speed of a reaction. Thus if we remove 10 per cent of the water in which the conversion of ammonium cyanate into urea is taking 292 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. | place, and bring the solution up to its original volume by adding acetone, which takes no part in the reaction, the rate of the conversion increases by nearly 50 per cent. In ethyl alcohol the rate of trans- formation of the cyanate into urea is thirty times as great as it is in pure water, other conditions remaining the same. The following table contains the coefficients of velocity observed for the bimolecular reaction N(C₂H5)² + C₂H₂¿I = N(C₂H¿)I in different solvents :- Solvent. Hexane Heptane Xylene Benzene Ethyl acetate Ethyl ether Methyl alcohol Ethyl alcohol Allyl alcohol Benzyl alcohol Acetone • • · · • • ▸ 1000 k 0.180 0.235 2.87 5.84 22.3 0.757 51.6 36.6 43.3 133 60.8 The great range of the speed of one action is very well shown by this table. We cannot imagine that the reacting molecules meet each other in benzyl alcohol 700 times as often in a given time as they do in hexane. Either we must assume that there is a greater proportion of active molecules in the former solvent than in the latter, or that the addition takes place at a larger proportion of the encounters of the reacting molecules, a supposition which is practically identical with the first. We very frequently find that below a certain temperature chemical action apparently will not occur, while above that temperature the action takes place freely. Thus we speak of the temperature of ignition of a mixture of gases, meaning usually thereby the lowest temperature which, when given to one part of the mixture, will pro- duce chemical action in the whole mass. For example, we say that the temperature of ignition of a mixture of air and saturated carbon bisulphide vapour is about 160°, because a glass rod heated to that temperature and applied to any part of the mixture will inflame the whole. If we inquire more closely into such actions, however, we generally find that the action occurs at temperatures below the ignition temperature, but that it will not then propagate itself under ordinary circumstances. The experimental proof of this is given by maintaining the whole mixture at a temperature somewhat below the ignition point, and noting after a time if the action has progressed. The reason for the non-propagation at lower temperatures is that the action then takes place only slowly. The heat evolution consequent on the chemical change is therefore spread over such an interval of time XXV RATE OF CHEMICAL TRANSFORMATION 293 that the temperature of the mixture remains below the ignition tempera- ture, so that the action becomes slower and slower if no external heat is supplied, and eventually ceases. At the ignition temperature the action takes place at such a rate that the heat evolution is sufficiently rapid to keep the temperature of the gaseous mixture up to the ignition point, and even to raise it still higher, with the result that the action proceeds with an ever-increasing speed. We therefore see that the so- called temperature of ignition depends on the generally-observed rapid increase in the rate of chemical action with rise of temperature, and may vary with the original temperature of the whole mixture. The same thing holds good, and may be followed more easily, with certain solids. Solid ammonium cyanate, for instance, may be kept. for many months at the ordinary temperature without undergoing any notable transformation into urea, according to the equation NHCNO = CO(NH2)2⋅ At 60° the action is fairly rapid, if the temperature is kept up externally, but the heat evolution is still too slow to enable the action to go on at an increasing rate. If the external temperature is 80°, however, the action proceeds swiftly, and the rapid heat evolution raises the temperature to such an extent that the whole passes almost instantaneously into fused urea, the melting point of which is 132°. If the heat evolution is comparatively small, as it is with ammonium cyanate, the increasing rapidity of the reaction may be observed through a considerable range of temperature. If, on the other hand, the heat evolution is great, as it is with all explosives, the action when it takes place perceptibly is usually propagated at once, owing to the rapid rise in temperature of the particles in the immediate neighbourhood of the reacting particles. The rate of propagation of explosion in solid explosives when fired is very great, rising in some instances to about five miles per second. The rate of propagation of the explosion wave in gaseous mixtures, such as that of oxygen and hydrogen, is somewhat less than this, averaging about a mile and a half per second, and being therefore of the same dimensions as the average rectilinear velocity of the gaseous molecules at the temperature of the explosion (cp. p. 84). Substances which react vigorously at the ordinary temperature usually lose their chemical activity entirely when cooled to the tem- perature of boiling liquid air. This we must attribute to the lowering of the rate of chemical action by fall of temperature, the speed being so greatly diminished that the action does not perceptibly occur at all in any moderate length of time. That there is not an entire cessation of action is probable, since in certain cases we can follow the gradual slack- ening of the action to extinction as the temperature falls. Thus sodium. and alcohol, which react briskly at the ordinary temperature, with evolu- tion of hydrogen, become less and less active as the temperature is 294 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. lowered, until finally the hydrogen formed by the action is so small in quantity as to escape observation. If we reflect that we often see the reaction velocity halved by a fall of 5° in temperature, we can conceive that a fall of 100° might by successive halving reduce the reaction velocity to a millionth of its original value. We may some- times find it convenient, in accordance with this, to look upon chemical inactivity as being not absolute inactivity, but rather the progress of a chemical action at a rate too slow for detection. We have now to consider briefly the points of resemblance between physical transformation and chemical transformation. In the change from solid to liquid, and vice versa, there is (for a given pressure) a definite temperature of transformation (p. 55). Above this tempera- ture the solid passes into the liquid; below this temperature the liquid passes into the solid. So it is for the transformation of one crystalline modification into another, as in the example of the two modifications of sulphur. Above the inversion temperature, one modification is stable ; below the inversion temperature, the other. If a liquid is cooled to a temperature slightly below its inversion temperature (the freezing point) and brought into contact with the more stable crystalline phase, it will assume the crystalline state, the rate of crystallisation being for small temperature differences proportionate to the degree of overcooling. A maximum rate, however, is attained when the degree of overcooling reaches a certain value, and below the temperature corresponding to this, the rate of crystallisation rapidly falls with further diminution of temperature (cp. p. 57). The same thing has been observed with the reciprocal transformation of crystalline modifications: at first the rate of transformation increases as the temperature falls below the tem- perature of inversion, afterwards, however, to diminish rapidly as the fall of temperature proceeds. No doubt monoclinic sulphur, cooled to a temperature considerably below zero, would exhibit little tendency to pass into the more stable rhombic form, for even at ordinary tempera- tures the process takes a comparatively long time. We have already seen that an overcooled "glass" may remain for a very long time in contact with the more stable crystalline modification if the overcooling is sufficiently great, without crystallisation progressing at a sensible. rate. If we heat the "glass," however, to a higher temperature, the transformation proceeds with ever-increasing rapidity until a tempera- ture a few degrees below the inversion point is reached. This is evidently comparable to the chemical transformation of solid ammonium cyanate into urea. Ammonium cyanate is the less stable form, and although it may be kept for a very long time at 0° in contact with urea without undergoing appreciable transformation, the transition goes on at an increasing rate as the temperature is raised. Here the inversion point is not known, but it must at least be considerably over 80°. The reverse transformation of urea into ammonium cyanate has only as yet. been investigated in aqueous solution. XXV RATE OF CHEMICAL TRANSFORMATION 295 When cyanic acid vapour condenses below 150°, it forms cyamelide; when it condenses above 150°, it forms cyanuric acid; the chemical action being in both cases one of polymerisation, as cyanic acid has the formula CNOH, cyanuric acid the formula (CNOH)2, and cyamelide a still more complicated formula, expressed generally by (CNOH). This would indicate that cyanuric acid had the inversion point 150°, cyamelide being the stable form below that temperature, and cyanuric acid the stable form above that temperature. The conversion of cyamelide into cyanuric acid above 150° has actually been observed, but the reverse action has not hitherto been noticed, probably on account of its slowness. If we construct a pressure- temperature diagram for the three phases, we shall find it exactly analogous to the diagram for the three physical states of aggregation of water. From the foregoing instances it is evident that considerable analogy exists between the effect of temperature on chemical and on physical change, especially when the chemical change is reversible: and it may often be found expedient to look upon apparently irreversible chemical changes as being in reality reversible, but with a temperature of inver- sion so high that the reverse action has never been realised. It will be noted that in all of the above instances there can be no coexistence in equilibrium of two systems which are mutually insoluble and capable of reciprocal transformation, except at the inversion point. Above the inversion point one of the systems is stable; below the in- version point the other is stable; at the inversion temperature itself, both systems are equally stable. This rule holds good both for chemi- cal and physical changes. If, on the other hand, the chemical systems are mutually soluble, there can be equilibrium at any temperature for which they only form one phase, the proportions of each system present changing in this case with the temperature. The fused mixture of ammonium thiocyanate and thiourea forms an example of this one-phase equilibrium of two reciprocally transformable systems. If we fuse thiourea, a certain proportion of it passes into ammonium thiocyanate with a measurable velocity, and we should expect that a different pro- portion would be transformed according to the temperature at which the system was maintained. Another example of one-phase equilibrium of reciprocally transformable systems is to be found in the solutions of dynamic isomerides (p. 284), or in the substances themselves if they be liquid. The actual transformation in this case also has been noted, and its velocity measured. Traces of water vapour have been found to play a very important part in determining the occurrence, or at least the rate, of many chemical actions. Thus ammonia and hydrochloric acid, when both in the gaseous state, unite readily under ordinary circumstances to form ammonium chloride. If precautions are taken, however, to have the gases absolutely dry, they may be mixed without any union taking 296 INTRODUCTION TO PHYSICAL CHEMISTRY CH. XXV place. On the other hand, ammonium chloride, when vaporised, dis- sociates to a very great extent into ammonia and hydrochloric acid, as is rendered evident by the vapour density being only about half the normal value calculated from the molecular formula NHCl by the help of Avogadro's principle. If the ammonium chloride is perfectly dry, however, the vapour density is normal, thus showing that no dissocia- tion has taken place. The reversible action NH + HCI NH Cl is therefore apparently dependent on the presence of traces of water for its occurrence either in the direct or the reverse sense. No satis- factory explanation of the action of the moisture has yet been given. It has not indeed been clearly established whether the action is alto- gether inhibited by the absence of water vapour or whether it still goes on, but at a greatly diminished rate. In the latter case the action of the water might be comparable to the action of acids in accelerating the inversion of cane sugar, the hydrolysis of esters, and the like. J. W. MELLOR-Chemical Statics and Dynamics (1904). RIDEAL and TAYLOR-Catalysis in Theory and Practice (1919). CHAPTER XXVI RELATIVE STRENGTHS OF ACIDS AND OF BASES Ir is customary and correct to speak of sulphuric acid as a strong acid, and of acetic acid as a weak acid, and the statement is the outcome of our general experience of the chemical behaviour of these substances. In comparing two such acids there is no difficulty; they are so different in their properties that no one could mistake their relative strengths. But if we compare two acids which are closer together in the scale of strength, say hydrochloric and nitric acids, it is impossible from our general chemical experience alone to say which is the stronger, and we must resort to a more exact definition of strength and to more accurate experiment. A method which, when properly applied, leads to useful and con- sistent results, is that of the displacement of one acid from its salts by another. If an equivalent of sulphuric acid be added to a given quantity of an acetate in solution, the change in properties of the solution is sufficient to indicate that practically the whole of the sulphuric acid has been neutralised, and the corresponding quantity of acetic acid liberated. There is therefore no doubt that the sulphuric acid is much stronger than the acetic acid, being capable of turning the latter out of its salts. But the method must be applied with caution, or it leads to contradictory and inconsistent results. If, for example, we take sodium silicate in aqueous solution and add hydro- chloric acid, sodium chloride will be formed and silicic acid liberated, but yet at a very high temperature, as in the process of glazing earthen- ware, silicic anhydride in presence of water vapour is capable of decomposing sodium chloride with expulsion of hydrochloric acid. Without any further principle than that of displacement to guide us, the first experiment would show that hydrochloric is stronger than silicic acid, and the second that silicic acid is stronger than hydro- chloric acid. Again, if we pour aqueous acetic acid on sodium carbonate, there is immediate effervescence due to the expulsion of carbonic acid. Yet a solution of potassium acetate in nearly absolute alcohol is decomposed by carbonic acid to a great extent with pre- cipitation of carbonate. From these experiments it is impossible to say whether acetic or carbonic acid is the stronger, since the expulsion 297 298 CHAP. INTRODUCTION TO PHYSICAL CHEMISTRY takes place in different senses according to the conditions of experiment. There is, of course, no doubt among chemists that hydrochloric acid is much stronger than silicic acid, and that acetic acid is much stronger than carbonic acid. We must therefore inquire more closely into the experiments which seem to point to the contrary conclusions. In the first place, it is obvious from the experiments themselves that we are dealing with actions which can, according to circumstances, take place in either sense, i.e. with balanced actions. Carbonic acid, or its anhydride and water, can always displace a little acetic acid from its salts, although it is a much weaker acid. If all the substances remain within the sphere of the reaction this displacement will not go far, as the reverse reaction will set in and soon establish equilibrium. The action expressed by the equation CH₂. COOK + CO2 + H2O = KHCO3 + CH,. COOH would therefore speedily come to an end if the products of the action. were to remain and accumulate in the system. But if one of the products, say potassium hydrogen carbonate, is insoluble, or nearly so, its active mass cannot increase beyond a certain small amount, however much of it may be formed, for it falls out of solution, i.e. the true sphere of action, as soon as it is produced. Whilst, therefore, in aqueous solution, in which all the substances remain dissolved, the carbonic. acid succeeds in displacing only a small proportion of the acetic acid, in alcoholic solution it displaces a much larger amount, owing to the insolubility of the hydrocarbonate. In the action of silica on a chloride, we have again one of the substances removed from the sphere of action as it is produced, viz. hydrochloric acid, which at the high temperature of the experiment escapes as vapour. In solution, only an infinitesimally small proportion of hydrochloric acid would be displaced by silicic acid, owing to the reverse reaction which would at once set in, but at the high tem- perature no reverse action is possible at all, since one of the reacting substances leaves the sphere of action as soon as it is formed. Similarly, we are not in a position to judge of the relative strengths of hydrochloric and hydrosulphuric acids by experiments with sulphides insoluble in water. Sulphuretted hydrogen will at once expel hydro- chloric acid from copper chloride, even if excess of hydrochloric acid is present in solution. Yet sulphuretted hydrogen is a very feeble acid compared to hydrochloric acid. The displacement is due to the insolubility of the copper sulphide, which is removed from the sphere of action as it is produced. That such experiments lead to no definite conclusion may be seen by taking the same acids with another base. Though sulphuretted hydrogen is passed through a solution of ferrous chloride until the solution is saturated with it, a scarcely percep- tible precipitate of ferrous sulphide will be formed, and if a little hydrochloric acid is added to the solution from the beginning, no XXVI RELATIVE STRENGTHS OF ACIDS AND BASES 299 precipitate will be formed at all. Here, then, the same two acids exhibit totally different behaviour relatively to each other, according to the nature of the base for which they are competing. Now, in the above instances the acids have been selected of as widely different natures as possible, so that the fallacy of the reasoning based on the experiments mentioned is obvious. But similar fallacious reasoning is prevalent, and passes without detection, when experiments. are discussed regarding acids where common chemical experience supplies no answer as to their relative strengths. Thus it is almost invariably a settled conviction in the minds of students that sulphuric acid is a stronger acid than hydrochloric acid because it expels the latter from its salts. No doubt this is the fact if we evaporate a solution of a chloride and sulphuric acid to dryness, or nearly so. But, of course, in this case the hydrochloric acid is expelled as vapour, and cannot therefore participate in the reverse reaction. The hydrochloric acid is expelled, not because it is a feebler acid than sulphuric acid, but because it is more volatile. These examples suffice to show that expulsion of one acid from its salts by another cannot be used as a proof that the latter is the stronger acid, unless the two acids are competing under circumstances equally favourable to both. The proper conditions are secured when the reacting systems form only one phase, namely, that of a solution. As soon as one of the components of the reacting systems is removed as a gas or as an insoluble solid or liquid, the system which does not contain that substance as one of its components is unduly favoured at the expense of the other system. If we are to compare the strengths of hydrochloric and sulphuric acids, then, we shall do best to take a soluble sulphate and add to it an equivalent of hydrochloric acid, the base being so chosen that the chloride formed is also soluble. Since all salts which have potash or soda as base are soluble, a potassium or sodium salt is generally selected, and the competing acid added to its aqueous solution. Thus equivalent solutions of hydrochloric acid and sodium sulphate may be mixed, and the composition of the resulting solution investigated, in order to ascertain in what proportion the base distributes itself between the two acids, the assumption being that the stronger acid takes the greater share of the base. In general, it is necessary to use a physical method for determining the composition of the solution, since the application of a chemical method would disturb the equilibrium. That an equilibrium is actually being dealt with is ascertainable from the fact that the solution has exactly the same properties in all respects, whether the base was originally combined with the sulphuric acid or with the hydrochloric acid, as will be presently shown in a numerical example. The two methods which have been most extensively applied are the thermochemical method of Thomsen and the volume method of Ostwald. 300 CHAP. INTRODUCTION TO PHYSICAL CHEMISTRY When a gram molecule of sulphuric acid in fairly dilute solution (about one-half equivalent normal) is neutralised by an equivalent. amount of caustic soda of similar dilution, a production of 31,380 calories is observed. The same amount of soda neutralised by hydro- chloric acid is attended by a heat evolution of 27,480 cal. Now if the addition of hydrochloric acid to a solution of sodium sulphate pro- duced no effect chemically, we should also expect no thermal effect. If, on the other hand, all the sulphuric acid were expelled from combination with the soda, we should expect an absorption of 31,380 cal. – 27,480 cal. = 3900 cal. Now an actual heat absorption of 3360 cal. per gram molecule was observed by Thomsen. This would indicate that the greater proportion of the base was taken by the hydrochloric acid, an equivalent quantity of sulphuric acid being expelled from its combina- tion with the soda. If we assumed that the heat absorption were directly proportional to the amount of chemical action, the proportion of sulphate converted into chloride would be 33603900 = 0·86. This proportion, however, is not the correct one, for the sulphuric acid liberated reacts with the normal sodium sulphate remaining, to produce. a certain quantity of sodium hydrogen sulphate, the formation of which is attended by absorption of heat, so that the total heat absorp- tion observed is too great. Special experiments show that the correc- tion to be applied in order to eliminate the effect of this action is 760 cal., the heat absorption due to the displacement of sulphuric by hydro- chloric acid thus being 3360760 2600 cal. The proportion of sul- phuric acid expelled is therefore 2639, or two-thirds. 39, or two-thirds. When, there- fore, equivalent quantities of sulphuric and hydrochloric acids compete for a quantity of base sufficient to neutralise only one of them, the hydro- chloric acid takes two-thirds of the base and the sulphuric acid one- third. The reverse experiment of adding sulphuric acid to a solution of sodium chloride showed that the final distribution of the base between the acids was the same as above so far as could be judged from the heat effect. Since the hydrochloric acid always takes the larger share of the base, we conclude that it is the stronger acid, at least in aqueous solution. Thomsen, by working in this way, compiled a table of the avidities of different acids, from which it is possible to tell at once how a base will distribute itself between any two of them if all three sub- stances are present in equivalent proportions. The avidities of some of the commoner acids are given below:- Acid. Nitric Hydrochloric Sulphuric . Oxalic Orthophosphoric Monochloracetic Tartaric Acetic • ¿- • M Avidity. 100 100 49 24 13 9 5 3 Co XXVI RELATIVE STRENGTHS OF ACIDS AND BASES 301 In order to find the distribution ratio from this table, we proceed as follows. Let the acids be sulphuric and chloracetic, the avidities being 49 and 9 respectively. If the base and these acids are present in equivalent proportions, the base will share itself between the acids in the ratio of their avidities, i.e. the sulphuric acid will take g and the chloracetic acid. 58 Ostwald's volume method is based on similar principles. Instead of heat changes, the changes of volume accompanying chemical reactions. are measured. The substances used by him were contained in aqueous solutions of such a strength that a kilogram of solution contained 1 gram equivalent of acid, salt, or base. The specific volumes of these solutions were carefully determined so that the change of volume pro- duced by chemical action might be ascertained. Thus the volume of a kilogram of potassium hydroxide solution was found to be 950-668 cc., and of a nitric acid solution 966.623 cc. If, on mixing these solutions, no change of volume occurred, the total volume would be 1917.291 cc. But the volume actually found on mixing the solutions was 1937.338 cc. The neutralisation of the acid and base is thus accompanied by an expansion of 20.047 cc. Similarly, changes of volume accompany other chemical reactions, and the extent to which a given action has occurred can be measured by the volume change. A solution of copper nitrate had a volume equal to 3847·4 cc., and an equivalent solution of copper sulphate 38403 cc. Solutions of nitric and sulphuric acids had the volumes 1933.2 and 1936 8 respectively. If no action occurred on mixing the copper sulphate solution with the nitric acid solution, the total volume would be 3840 3+ 1933·2 = 5773.5; if complete trans- formation into copper nitrate and sulphuric acid took place, the total volume would be 3847 4+ 1936.8=5784.2. The actual volume found by mixing the copper nitrate and sulphuric acid solutions was 5780 8, and by mixing the copper sulphate and nitric acid solutions 5781.3. These two volumes are practically identical, and we may take as their mean 5781·0. We have therefore the numbers- • Difference · All Copper Sulphate. 5773.5 7.5 Actual. 5781.0 3.2 No Copper Sulphate. 5784.2 The actual equilibrium is evidently nearer the system containing no copper as sulphate than the system containing all the copper as sulphate, and if we assume direct proportionality, the base is shared by the nitric and sulphuric acids in the ratio of 7.5 to 3.2, or nitric acid takes 70 per cent of the base, leaving the sulphuric acid 30 per cent. This result is not quite accurate, as allowance has to be made for the slight volume changes consequent on the action of the respective acids on their neutral salts. When this correction is applied, it appears that the nitric acid takes 60 per cent of the base and the sulphuric acid 40 per cent. 302 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. A table of avidities can be constructed for the different acids from similar data, and a comparison with Thomsen's avidities derived from. thermochemical experiments shows that the two methods yield results in harmony with each other, at least so far as relative order of the acids is concerned. The actual avidity numbers differ considerably in many instances, but it has to be borne in mind that the thermochemi- cal measurements are on the whole less accurate than the volume measurements, and the numbers derived from them consequently less trustworthy. In special cases the distribution of a base between two acids may be studied by making use of other physical properties than those already mentioned. For example, measurements of the refractive. index of solutions often lead to satisfactory results, and also measure- ments of the rotatory power when optically active substances are in question. The principle involved is identical with that just described, any differences being merely differences in detail. A method which differs in principle from the distribution of a base between two competing acids, and may also be applied to the deter- mination of the relative strengths of acids, is to ascertain the accelerat- ing influence exerted by different acids on a given chemical action. For example, the inversion of cane sugar has long been known to take place much more rapidly in presence of an admittedly strong acid like sulphuric or hydrochloric acid, than it does in presence of an equiva- lent quantity of an admittedly weak acid like acetic acid. The strong mineral acids have thus a greater accelerating effect than the weak organic acids, and it is natural to infer that an exact determination of the specific accelerating powers of different acids might lead to a knowledge of their relative strengths. There is no obvious connection between this method and the preceding method of relative displace- ment, but a connection exists, as will be shown later, and the results obtained are in general quite in harmony with each other. The method of sugar inversion as practised by Ostwald was performed in the following manner. Normal solutions of the various acids were mixed with an equal volume of 25 per cent sugar solution and placed in a thermostat whose temperature remained constant at 25°. The rotation of each solution was taken from time to time, and a velocity constant calculated according to the formula given on p. 287. The order of these velocity constants is the measure of the accelerating powers of the acids, and presumably a measure of their relative strengths. Another action well adapted to investigating the accelerating power of acids is the catalysis of methyl acetate (cp. p. 288). Ostwald mixed 10 cc. of normal acid with 1 cc. of methyl acetate, and diluted the mixture to 15 cc. This solution was then placed in a thermostat at 26°, and its composition ascertained at appropriate intervals in the manner already indicated. A calculation of the velocity constant by the usual formula for unimolecular reaction gave the required measure XXVI RELATIVE STRENGTHS OF ACIDS AND BASES 303 of the accelerating power. A very sensitive reaction, which may be used for the same purpose, is the catalysis of diazoacetic ester by acids, according to the equation Ng:CH.COOEt + H,O = N, + HỌ.CH, . COOEt. 2 2 This action, which scarcely occurs at all in pure water, is greatly accelerated by the presence of even weak acids in excessively dilute solution, and its progress may be readily followed by measurement of the volume of nitrogen evolved. A comparison of the results obtained by the different methods is given in the following table, the value for hydrochloric acid being made in each case equal to 100, in order to assist the comparison :- Hydrochloric Nitric Sulphuric Oxalic Orthophosphoric Monochloracetic Tartaric Acetic Hydrochloric Nitric Sulphuric Oxalic Orthophosphoric Monochloracetic Avidity. 100 100 49 Tartaric Acetic 24 13 9 5 3 Velocity Constants. Sugar Inversion. Catalysis of Acetate. 100 96 54 100 100 53 18.6 6.2 4.8 It is at once evident that the order in which the acids follow each other is the same in all cases, and in especial it will be seen that the numbers expressing the accelerating powers of the acids are closely similar, although the accelerating influence was exerted on entirely different chemical actions. The avidity numbers differ considerably from the others, but the general parallelism of the results cannot be denied, and we are therefore justified in adopting the acceleration method as a means of measuring the relative strengths of acids, although its theoretical justification is not immediately obvious. It was pointed out by Arrhenius that if we arrange the acids in the order of their relative strengths, they are also arranged in the order of the electrical conductivities of their equivalent solutions. This may be seen in the following table, the first column of which contains the mean value of the velocity constants of sugar inversion and catalysis of methyl acetate, and the second that of the electric conductivities of equivalent solutions, all values being referred to that for hydrochloric acid as 100:- 18 ** Velocity Constants. 6.2 4.5 2.3 0.4 0.4 100 91.5 54.7 17.4 4.3 2.3 0.35 Electric Conductivity. 100 99.6 65.1 19.7 7.3 4.9 2.3 0.4 304 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. The parallelism is here unmistakable, the numerical values in the two columns being often practically identical. At first sight it appears a matter of difficulty to associate the electric conductivity of an acid with its strength, i.e. its chemical activity in so far as it behaves as an acid; but the dissociation hypothesis of Arrhenius furnishes the clue to the nature of the con- nection. All acids in aqueous solution possess certain properties peculiar to themselves which we class together as acid properties. Thus they neutralise bases, change the colour of certain indicators, are sour to the taste, and so on. We are therefore disposed to attribute to them the possession of some common constituent which can account for these common properties. On asking what aqueous solutions of the various acids have in common, we find for answer "hydrion" if we adopt the hypothesis of electrolytic dissociation. Let us suppose the peculiar properties of acids to be due to hydrion. How in that case are we to explain the different strengths of the acids Evidently on the assumption that different acids in equivalent. solution yield different amounts of hydrion. The acid which in a normal solution produces more hydrion will be the more powerful. acid, so that on this hypothesis the degree of ionisation of an acid. furnishes a measure of its strength. But if we compare equivalent solutions of different acids under the same conditions, the electrical conductivity is closely proportional to the degree of ionisation of the dissolved substance. This arises from the fact that the speed of hydrion is much greater than the speed of any negative ion with which it may be associated. The conductivity, then, of any acid solu- tion is due principally to the hydrion it contains, so that if we compare the conductivities of solutions of different acids, the values we obtain are nearly proportional to the relative amounts of hydrion in the solutions, and thus to the relative strengths of the acids. Since it is a very easy matter to measure the conductivity of solutions, this method of determining the relative strengths of acids has practi- cally superseded the other methods, especially in the case of the weaker organic acids. For them it is possible to calculate a dissociation con- stant according to the formula given on p. 254, and this constant is very generally accepted as a measure of their strength, for which reason it is sometimes spoken of as the affinity constant of the acids. It will be remembered that the theoretical dissociation formula only applies to half-electrolytes, the strong and highly dissociated mineral acids giving only constants with empirically modified formulæ such as that of van 't Hoff. These empirical constants might be used as "affinity constants" for the strong acids, since they, like the true disso- ciation constants, give a measure of the relative dissociations of different acids independent of the dilution, but as their significance is doubtful, and the degree of constancy attained is not after all very great for the highly-dissociated acids, they have not so far come into general use. XXVI RELATIVE STRENGTHS OF ACIDS AND BASES 305 İ Since the degree of ionisation cannot go beyond 100 per cent, we have here a natural limit set to the strength of acids. The limit is reached at moderate dilutions for some of the monobasic acids, which are therefore the strongest acids which can be met with in aqueous solution. These are hydrochloric, hydrobromic, hydriodic, nitric, and chloric acids amongst the common inorganic acids; amongst the organic acids we have the alkyl sulphuric acids, such as hydrogen ethyl sulphate, and the sulphonic acids, such as benzene sulphonic acid or ethane sulphonic acid. No dibasic acid is as strong as these monobasic acids, sulphuric acid being the strongest of this type. The fatty acids, such as acetic acid, are very much weaker than these. It should be noted that at great dilutions the differences in strength between acids begin to disappear. At infinite dilution all acids are equally dissociated, so that no one will contribute more hydrion than another, and consequently all acids under these con- ditions will have the same strength. As has already been indicated, no such state can actually be realised, but it is well to remember that, in general, differences in strength of acids are more marked in com- paratively strong solutions than in very dilute solutions. Thus we have the following numbers for acetic acid and its chlorine substitution products, which represent the percentage degree of dissociation or the proportion of available hydrogen existing in the solution as hydrion, i.e. in the active state. Acetic Monochloracetic Dichloracetic Trichloracetic Dilution 32 Litres. 2.4 20 70 90 128 Litres. 4.7 35 88 95 512 Litres. 9.1 57 98 99 At the dilution 32, trichloracetic acid has 37 times as much hydrion as an equivalent solution of acetic acid; at the dilution 512 it has only 11 times as much. It is true that between the dilutions 32 and 512, acetic acid gains only 6.7 per cent, where trichloracetic acid gains 9, but this is on account of the great difference in strength between the acids, and it can be seen that between 128 litres and 512 litres the actual gain of the acetic acid is greater than the gain of the trichloracetic acid, and this would be more and more evident as the dilution proceeded. In the case of the other acids which are more nearly equal in strength, the equalisation of strength as dilution proceeds is much more evident. From 32 litres to 512 litres, trichlor- acetic acid only gains 9 per cent, where dichloracetic acid gains 28, and monochloracetic acid gains 37. At 32 litres trichloracetic acid is nearly 30 per cent stronger than dichloracetic acid; at 512 litres their strengths are almost equal. It now remains to show the connection between the degree of ionisation of two acids and the proportion in which they will share X 306 CHAP. INTRODUCTION TO PHYSICAL CHEMISTRY a base between them, when both are competing for it in the same solution. Let the acids HA and HA', and the base NaOH, be dissolved in water so that 1 gram molecule of each is contained in v litres of the mixed solution. For the sake of simplifying the calculation, we shall also suppose that the acids are weak and obey Ostwald's dilution formula (p. 254), that their degree of ionisation at the dilution considered is very small, and that the degrees of ionisation of the sodium salts produced are equal, which will very nearly be the case. Let x be the amount of the acid HA neutralised by the soda, then the quantity of HA' neutralised will be 1 - x, since the total quantity of soda is 1. If now h represents the quantity of hydrion in the solution, and d the common ionisation factor of the two sodium salts, we shall have the following quantities of the various substances existing in equilibrium with one another :— Na A NaA' HA HA' Hion · • • Na ion A ion. A' ion. x, of which (1-d)x un-ionised. X, (1 − x), of which (1 − d)(1 − x) un-ionised. (1-x), of which practically all un-ionised. a, of which practically all un-ionised. h, a very small amount, from HA and HA'. d(x+1−x)=d, from NaA and NaA'. dx, from NaA almost entirely. d(1 − x), from NaA' almost entirely. - 7:2 (H ion) × (A ion) = k. (un-ionised HA)v Substituting the values given in the above table, we obtain hdx (1 − x)v Similarly for the acid HA' we get Now in order that equilibrium may exist between the un-ionised HA and the ions H and A, the requirements of Ostwald's dilution formula must be fulfilled, and we have hd(1-x) XV · = k. K. Dividing the first of these equations by the second we have k X or 1 − x (1 − x)² ¯¯ k - だ ​i.e. the ratio of the avidities of two acids is equal to the square root of the ratio of their dissociation constants, if all the conditions mentioned above are fulfilled. We can get a direct relation between the avidities and the degree of ionisation at a given dilution by taking account of the dilution XXVI RELATIVE STRENGTHS OF ACIDS AND BASES 307 formulæ for pure solutions of the acids. For the acids HA and HA' let the degrees of ionisation at the dilution v be m and m' respect- ively. We have then m² (1 − m)v ― therefore But we found above = Since the degree of ionisation of the acids is by hypothesis very small, 1 m and 1-m' become very nearly equal to 1, and conse- quently by division we have approximately k, and 1 22 (1 − x)² 2 X ― m² 12 m X ทาง (1m')v k K k 7" m m' = 7) k'. i.e. ratio in which a base distributes itself under the conditions named between two acids, is practically equal to the ratio of the degrees of ionisation of the separate acids at the same concentration, or, in other words, is practically equal to the ratio of the electrical con- ductivities of the acids under similar conditions of dilution. The dissociation theory of Arrhenius, therefore, furnishes us with a satisfactory explanation of the connection between the various methods of measuring the strengths of acids; the fundamental assumption being that the activity of acids, as acids, is due entirely to the presence of hydrion, the various acids differing from each other only in the quantity of hydrion produced when equivalent amounts are dissolved in a given quantity of water. The sour taste, the action on indicators, the accelerating effect on sugar inversion, etc., are all attributed to the hydrion, and increase in intensity as the amount of hydrion increases. It may be noted, however, that in the case of strong acids the non-ionised molecules seem also to exert an accelerative catalytic action. The methods employed for measuring the relative strengths of bases are in all respects similar to the methods adopted for acids. The distribution of an acid between two competing bases, the accelerating effect of different bases on certain chemical actions, and the electrical conductivities of equivalent solutions of the bases, have all been investigated, and have been found to lead to consistent results. If we inquire into what is common to solutions of all bases, we find that the dissociation hypothesis gives us "hydroxidion" for answer. Just as it attributed the essential properties of acids to hydrion, so it attributes the essential properties of alkalies to 308 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. hydroxidion. This is responsible for the alkaline taste, the action. on indicators, and for the power of neutralising acids. Bases differ from one another in strength according as their equivalent solutions produce more or less hydroxidion, that base being the strongest which produces the most. The electrical conductivity of equivalent solutions of the soluble bases gives us a means of judging their relative strengths, but a reference to the table of ionic velocities on p. 242 will show that the value of the conductivity is not such a direct measure of the strength of bases as it is of acids, in view of the fact that the speed of the hydroxide ion does not exceed the speeds of the positive ions in the same proportion as the speed of the hydrogen ion exceeds the speeds of the negative ions. The total conductivity of the base is therefore due. to a less extent to hydroxidion than the conductivity of an acid is due to hydrion. The degree of ionisation, however, if calculated from the conductivities in the usual way (p. 252), gives the correct measure of the strengths of bases when in equivalent solution, and the dissociation constant of bases has the same significance in this respect as the dissociation constants of acids, the stronger base being that with the greater constant. A direct velocity method, which has been applied to determining the strengths of bases, is the rate at which they saponify methyl acetate (cp. p. 288). The saponification is apparently effected by hydroxidion, and equivalent solutions of different bases will give velocity constants (at the beginning of the saponification at least) which will be proportional to the amount of hydroxidion in the solution, i.e. to the strengths of the bases. The following numbers were obtained in fortieth normal solution of the various bases, the value of the constant for lithium hydroxide being made equal to 100 :— Lithium hydroxide Sodium hydroxide Potassium hydroxide Thallium hydroxide Tetraethylammonium hydroxide. Triethylammonium hydroxide Diethylammonium hydroxide Ethylammonium hydroxide Ammonium hydroxide • · • • · • • • 100 98 98 89 75 14 16 12 2 The strong alkalies, lithia, soda, potash, have practically reached the limit of strength, their ionisation at moderate dilutions being almost. complete. The hydroxides of the metals of the alkaline earths are equally strong, as similar experiments have proved. Ammonia, or, as it partially exists in solution, ammonium hydroxide, is a com- paratively feeble base, bearing much the same relation to the strong alkalies as the weak organic acids to the strong mineral acids. The alkylammonium hydroxides are all stronger bases than ammonium hydroxide, the tetra-alkyl hydroxides being nearly comparable in XXVI RELATIVE STRENGTHS OF ACIDS AND BASES 309 1 point of strength to the caustic alkalies, as indeed is evident from their general chemical behaviour. An acceleration method by means of which the relative strengths of bases may be determined is the transformation of the alkaloid hyoscyamine into the isomeric alkaloid atropine, the course of which can be followed with the polarimeter. The amount of the acceleration here seems to be proportional to the concentra- tion of hydroxidion in the solution, but the method cannot be applied with great strictness on account of secondary decompositions of the atropine, which interfere with the calculation of the velocity constant. The bases, potassium hydroxide, sodium hydroxide, and tetramethylammonium hydroxide, were found to have nearly equal accelerating effects, which is in agreement with the results of the saponification methods. Another acceleration method depends on the change of volume. observable when diacetone-alcohol undergoes transformation into acetone according to the equation CH, . CO. CH, .C(OH)CH,),= 2CH, . CO. CH. This unimolecular action takes place in aqueous solution at a rate nearly proportional to the concentration of hydroxidion in the solution, and its progress may be followed by observing the increase of volume from time to time as the action proceeds in a dilatometer (p. 101). Other methods applicable to very weak acids and bases will be dis- cussed in Chapter XXVIII. OSTWALD'S papers on the Affinity Constants of Organic Acids are in the Zeitschrift für physikalische Chemie, 3, pp. 170, 241, 369 (1889). CHAPTER XXVII EQUILIBRIUM BETWEEN ELECTROLYTES IN the chapter on balanced action we have seen that when the active mass of one or more of the products of a dissociation is increased, the degree of dissociation is diminished (p. 281). This rule is especi- ally important when we deal with solutions of salts, acids, and bases, all of which are electrolytically dissociated, and that to very different degrees. In cases of gaseous dissociation we can usually add one of the products of dissociation without adding anything else at the same time. This cannot be done with dissolved electrolytes, for the nature of the dissociation is such that the solution must always remain electrically neutral, although the products of dissociation are electric- ally charged. When, therefore, we add one of the products of dis- sociation to an electrolytically dissociated substance, we are compelled to add at the same time an electrically equivalent quantity of an ion oppositely charged. Thus if we consider a solution of hydrogen acetate, we find that we can only add hydrion by adding an acid, say hydrochloric acid, which not only contributes hydrion but chloridion. as well. Similarly, if we wish to increase the amount of acetanion in the given volume, we can only do so by adding to the solution an acetate, which yields metallion at the same time as it yields acetanion. Notwithstanding this complication, however, the equilibrium equation. for hydrogen acetate still remains the same, namely— act. mass H˚ × act. mass С₂H¸О₂ = const. x act. mass undiss. C₂HO₂ 3 2 42 whether a small quantity of another ion is added or not; so that if we increase the active mass of either the hydrion or the acetanion, the active mass of the un-ionised hydrogen acetate is also increased, i.e. the degree of ionisation is diminished. As has already been indicated, the effect on the degree of ionisation is greater when the substance considered is only slightly ionised. Now acetic acid, even in moderately dilute solution, is only feebly ionised, so that the addition of an equivalent quantity of a strongly ionised acid like hydrochloric acid practically reduces 310 CH. XXVII EQUILIBRIUM BETWEEN ELECTROLYTES 311 the ionisation of the acetic acid to zero. Suppose, for example, that at the given dilution hydrochloric acid is fifty times more ionised than acetic acid, the addition of an equivalent of the former will practically reduce the quantity of acetanion to a fiftieth part of the original value, for it has increased the amount of hydrion about fifty- fold, and the product of the active masses of the ions must remain practically constant, the active mass of the un-ionised hydrogen. acetate suffering but little change from the diminution of the ionisa- tion. The degree of ionisation is thus reduced to about a fiftieth part of its former magnitude. A similar reduction takes place if we add an equivalent quantity of an alkaline acetate to a solution of acetic acid. Although the acid itself is feebly ionised, its salts are as highly ionised as those of strong acids, with the result that the amount of the acetanion is greatly increased, and the amount of hydrion correspondingly diminished. This reduction of the ionisa- tion of weak acids by the addition of their neutral salts to the solution is of great practical importance, as the amount of hydrion which they produce, and consequently their activity as acids, is thereby greatly reduced (Chap. XXVI.). The addition to a strong acid of an equivalent or greater quantity of one of its neutral salts has very little effect on its activity as an acid, as measured by the proportion of hydrion to which it gives rise. This is due to the acid as well as the salt being almost completely ionised. If, for example, we add an equivalent of sodium chloride to a solution of hydrogen chloride, we at most double the amount of chloridion. The increase of the un-ionised amount is, however, also relatively great, so that to fulfil the requirements. of the equilibrium formula a very small diminution of the amount of hydrion is necessary, and consequently the activity of the acid is little affected. What holds good for acids likewise holds good for bases, the strength of which is measured by the proportion of hydroxidion derived from them. If to a feebly ionised solution of ammonium hydroxide we add ammonium chloride, which is highly ionised, we greatly increase the amount of ammonion, and diminish to a corresponding extent the amount of hydroxidion necessary for equilibrium. The base ammonium hydroxide, then, loses much of its activity when accom- panied in solution by ammonium salts. Strong bases like potassium hydroxide are only slightly affected by the addition of a neutral salt yielding the same metallion, for although the relative change on the undissociated proportion may be great, the same actual change has a very slight effect on the ionised proportion, and thus leads to only a slight diminution in the concentration of hydroxidion. The addition of neutral salts to a dibasic acid like sulphuric acid is a much more complicated phenomenon than the addition of neutral salts to monobasic acids. This is chiefly due to the formation of acid 312 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. salts, which dissociate rather as salts than as acids. Thus when we add an equivalent of sodium sulphate to a solution of hydrogen sulphate, a certain proportion of the two original salts remains in the solution, accompanied, however, by the intermediate acid salt, sodium hydrogen sulphate, NaHSO. The sodium sulphate dissociates largely into the ions Na and SO"; the sulphuric acid dissociates into H and SO", but also produces the ion HSO; the sodium hydrogen sulphate, finally, dissociates chiefly into Na' and HSO. There are therefore three un-ionised substances in solution, viz. H₂SO4 Na2SO4, and NaHSO4; and at least four kinds of ions, viz. Na', ɨ, ÍSO, and SO", H', so that the equilibrium is somewhat complex. 4 In this connection it may be stated that dibasic acids very generally dissociate in solution into one hydrogen ion and the residue of the molecule, the second replaceable hydrogen atom not splitting off as an ion until the greater quantity of the first has been removed. Thus sulphuric acid in fairly strong solution in all probability contains com- paratively little of the ion SO4", the dissociation being principally into Hand HSO. At greater dilutions, however, the ion SO," appears in quantity, probably owing to the splitting up of the hydrosulphanion HSO, into H and SO". One result of this is that weak dibasic acids. give a dissociation constant of exactly the same character as that of a monobasic acid, the formula used in deriving which assumes that the ionisation takes place into two ions only. Thus malonic acid does not primarily dissociate according to the equation 4 4 CH CH2 but according to the equation CH, COOH 2 COOH /COOH COOH V 16 32 64 128 256 512 1024 /COO' COO' CH₂ 100m 14.85 20.20 27.15 /COO' 2 COOH The equilibrium is therefore of exactly the same type as the dissocia- tion equilibrium of acetic acid, and follows the same law. 35.9 46.4 58.6 70.8 + 2 H', This is seen in the following table, which gives the dissociation constant for malonic acid (cp. p. 254): + H˚. The constancy of the expression 100k = shows that the primary dissociation is into 1007 0.159 0.159 0.158 0.157 0.157 0.162 0.168 100m² (1 - m)v in the last column two ions and not into XXVII EQUILIBRIUM BETWEEN ELECTROLYTES 313 m3 would be constant. It - three, in which case the expression (1 − m) v² will be noticed that after 50 per cent of the first hydrogen has split off as ion, the constant begins slightly to rise. The rise probably indicates that the second hydrogen atom is now being affected, i.e. that the action /COO' CH, COO 2 CH₂ 2 COOH + H' has commenced to be appreciable. The exact point at which this second action begins varies very much with different acids. Many acids show the secondary dissociation when the primary has proceeded to the extent of about 50 per cent; but some acids-maleïc acid for example-give a good constant up to 90 per cent primary dissociation. For acid salts of comparatively feeble polybasic acids one may say that almost invariably the primary dissociation is into metallion and the rest of the molecule, no hydrion appearing until this primary dissociation is far advanced, although, as indicated above, hydrion may arise from the acid itself produced by the action of the water. Thus sodium hydrogen malonate dissociates primarily according to the equation + Na', CH₂ and the acid character of the solution, judged by the methods of the preceding chapters, is very feebly marked. In general, when we mix two electrolytic solutions, we cannot calculate the conductivity of the mixed solution from those of the components by the simple alligation formula,¹ because each dissolved substance affects the dissociation of the other, and thus alters the number of carrier ions. From the law of mass action, however, as applied to electrolytic equilibrium, we ascertain that there must be certain solutions which can be mixed together without alteration in the amount or nature of the ions, and therefore without change in the average conducting power. Such solutions are called isohydric, and we shall first investigate the conditions for isohydry in the case of two electrolytes giving rise to a common ion, say HA and HA', each of which obeys Ostwald's dilution law. Let the dilutions of the two isohydric solutions be v and respectively, and their degrees of ionisation m and m'. For the acid HA we have the equilibrium equation /COONa COOH /COO' CH₂ COOH 2 m² (1 − m)v ސ li, 1 If two substances when mixed retain their specific values of a property unchanged by the process of mixture, the specific value of the same property for the mixture can be а Å + b B calculated by the alligation formula in which a and b represent the proportions at b of the components, and A and B the specific values of the property for the components. 314 CHAP. INTRODUCTION TO PHYSICAL CHEMISTRY and for the acid HA' the corresponding equation M'2 (1 − m´)v′ whence If now we mix these isohydric solutions, the volume becomes v + v′, and the amount of hydrion m+m'. For the acid HA under the new conditions we have now the equilibrium equation (m + m')m (1 − m)(v + v´) Dividing this equation by the first, we obtain m + m' (m + m²) v (v + v')m M or Now ― = = k'. 1, or m' v ทาวน V m V m' 21 = k. v + v V ; m m' is the concentration of hydrion in the acid HA, and v V concentration of hydrion in the isohydric solution of the acid HA', and these two concentrations prove to be equal. We may say, therefore, that solutions of electrolytes containing a common ion are isohydric when the concentration of the common ion in the different solutions is the same. is the It is often convenient for purposes of calculation to imagine an actual mixed solution to be split up into its component isohydric solutions, which may be afterwards ideally mixed without any change. in the ionisation of the electrolytes occurring. For example, a solution containing equivalent quantities of hydrogen acetate and sodium acetate may be imagined to exist in a rectangular vessel with a movable vertical partition through which water can freely pass, but not the dissolved substances. Let the hydrogen acetate be on one side of the partition and the sodium acetate on the other. The partition is now to be moved until the concentration of the common ion, acetanion, is the same on both sides. Since the sodium acetate is highly ionised, it must receive most of the water in order that the concentration of the acetanion may be as small as that derived from the slightly ionised hydrogen acetate. The position of the partition for isohydry must therefore be as shown in Fig. 52, which represents a horizontal section of the vessel, the liquid rising to the same level on both sides of the diaphragm. It must be borne in mind that con- XXVII EQUILIBRIUM BETWEEN ELECTROLYTES 315 centrating the solution of hydrogen acetate does not involve an equal concentration of the hydrion or acetanion, for as the dilution diminishes, the degree of dissocia tion, and therefore the proportion of ions, diminishes also, although at a smaller rate. Beginners are apt to reason that if the solution of one electrolyte is ten times more ionised than an equiva- lent solution of another, it is only necessary to concentrate the second solution to a tenth of its volume in order that the ionic concentrations of the two solutions may become equal. In view of the diminution of the degree of ionisation as the dilution diminishes, a much greater degree of concentration is necessary. FIG. 52. If we consider the mixing of two salts which have no common ion, say NaCl and KBr, the problem becomes much more complicated. When the two salts are mixed, we have not only the substances origin- ally in solution, i.e. the un-ionised salts NaCl and KBr, and their ions, Na, K, Cl, and Br', but also the new un-ionised substances NaBr and KCl. Let there be prepared isohydric solutions of the different salts, NaCl being made isohydric with NaBr, by getting the natrion of the same concentration in both solutions; KC may then be made isohydric with NaCl by making the chloridion of the same concentrations in the two solutions. KBr may finally be made isohydric with KCI by equalising the concentrations of the kalion. Any two of these solutions then which possess a common ion may be mixed in any proportions without change in the dissociation. If we wish to mix all four, we must take volumes of the solutions such that the products of the volumes of reciprocal pairs are equal. If a, b, c, d be the volumes of the iso- hydric solutions of NaCl, NaBr, KCl, and KBr respectively, such that ad = bc, then the solutions may be mixed in these proportions without change in the ionisation. Using a dia- grammatic representation similar to that adopted for mixtures of pairs of electrolytes with a common ion, we get the diagram Fig. 53, which fulfils the desired condition. The proof of the condition may be given on the supposition that b Na Br d K Br FIG. 53. α NaCl C K CI Na Ac HAc 316 CHAP. INTRODUCTION TO PHYSICAL CHEMISTRY the substances obey Ostwald's dilution law. Let a, b, c, d be the volumes of the isohydric solutions which when mixed will produce no change in the ionisation. Before mixing, the equilibrium of the sodium chloride will, assuming its quantity to be unity, be represented by the formula M M a a 1 - m a m² (1 − m)a the expression where m is the degree of dissociation. Let the solutions be now mixed. The amount of natrion has now increased in the ratio of a + b to a, since b volumes of NaBr have been added to the original a volumes of NaCl, and the concentration of the natrion is the same in both solutions. But the volume in which this quantity is contained is now a+b+c+d, so that the active mass of the natrion is now a + b m x 1 X a m(a + b) Similarly the amount of a+b+c+d a(a+b+c+d) chloridion increases in the ratio of a + c to a, and its active mass m(a + c) becomes The un-ionised proportion of sodium a(a + b + c + d) chloride remains the same as before, viz. 1 – m. We have therefore for the new equilibrium the equation whence, since k is also equal to m(a + b) m(a + c) a(a+b+c+d) u(a+b+c+d) 1 ทาน a + b + c + d m² — (1 m)a k, (a + b)(a + c) a(a + b + c + d) m² (a + b)(a + c) m² (1 − m)a²(a + b + c + d) ¯¯ (1 − m)a' 1, a² + ab + ac + bc = a² + ab + ac + ad, bc = ad, X [NaCl] × [KBr] [NaBr] x [KC1] k, which was to be proved. According to Guldberg and Waage's Law, we should have for equilibrium in the balanced action NaCl + KBr NaBr + KCl constant, XXVII EQUILIBRIUM BETWEEN ELECTROLYTES 317 where the formulæ in square brackets represent the active masses of the respective substances. This equilibrium formula takes no account of electrolytic dissociation of the various salts, and is only valid under certain conditions of dissociation. The correct formula is [diss. NaCl] x [diss. KBr] [diss. NaBr] x [diss. KCl as may be deduced from the above relation_ad = bc. Since all the solutions to which these letters refer are isohydric in pairs, i.e. have the same concentration of ions, the volumes a, b, c, d are proportional to the quantities of the ions. in the various solutions, i.e. to the dissociated quantities of the salts, and not to their total quantities. When all the substances are highly ionised, Guldberg and Waage's Law leads to very nearly the same result as when the ionisation is con- sidered, and the same holds true when two of the four substances are highly ion- ised. When, however, one or three of the substances are highly ionised, there is usually a great discrepancy between the two modes of calculating the equilibrium, Guldberg and Waage's Law being no longer even approximately true, except in special circumstances. As an example of the application of the theory of isohydric solutions to the equilibrium of four ionised substances, we may take the distribution of a base between two acids obeying Ostwald's dilution law and find the relation of the distribution ratio to the ratio of the dissociation constants of the acids. Let, as before (p. 306), one gram-molecular weight of each of the substances HA, HA', and NaOH be dissolved in a certain volume of water, and let the solution thus obtained be ideally split up into isohydric solutions of the same ionic concentration i. We thus get the diagram Fig. 54. The volumes are again represented by a, b, c, d, and the following table gives the data necessary for the calculation, if x is the amount of HA neutralised by the soda :--- h Na A X = 1, d Να Α' 1 - X FIG. 54. α HA 1 - X C HA X + 318 CHAP. INTRODUCTION TO PHYSICAL CHEMISTRY Total quantity Ionised quantity Degree of ionisation (m) Dilution (v) Division then gives 1 whence HA. 1 X ia ia m² (1 − m)v 1 Xx Now for equilibrium we have a 1 − x ia 2 (1+= 1) * 19 X i²a a = ********** X and similarly for the acid HA', we obtain i2c X k. 1 - x = k'. ᏟᏓ c(1 - x)˜¯¯ As the concentration of the ions is the same in all the solutions, the ionised quantities are equal to the volumes of the solutions multi- plied by the common ionic concentration i. The degree of ionisation, m, is the ratio of the ionised to the total quantity, and the volume divided by the quantity contained in it gives the volume which contains. unit quantity measured in gram molecules, i.e. the dilution v. The acids by supposition obey the theoretical dilution law k By a simplification which has been already adopted when the degree. of ionisation is small, we may neglect m in comparison with 1, and m² write the dilution formula = k. Now, substituting the above values. of m and v for the acid HA, we obtain V * k NaA. HA'. XC = k 2218018 ad = bc, or a/c = b/d bx k d(1-x) k 8.3.2/8 0/2 ic ib ic с a NaA'. 1 - x id id 1 d X 1 X XXVII EQUILIBRIUM BETWEEN ELECTROLYTES 319 As before, we may assume that the two sodium salts are ionised to the same extent, so that in their case the ratio of the ionised quantities is the ratio of the total quantities, i.e. ib id We thus obtain finally the relation 1 X -- X x2 k 2 k = = or 1 - - √ √ (1 − x)² ¯¯ ¯”' X k' that is, the ratio of distribution of the base between the two acids is equal to the ratio of the square roots of the dissociation constants of the acids, a result already obtained on p. 306 under the same assumptions. The student who desires to familiarise himself with the equilibrium of electrolytes in solution is advised to study the subject from the point of view of isohydric solutions, in particular when dealing with two electrolytes containing a common ion, or with double decompositions between electrolytes. In this last case the important fact to bear in mind is that the product of the ionised quantities on one side of the equation is equal to the product of the ionised quantities on the other. If, for example, we are dealing with the double decom- position CH¸ . COONa + HCl = CH₂. COOH + NaCl, M.4 M1 Mg M3 and the molecular concentrations of these substances, when equilibrium has been attained, are m₁, m2, mg, m, with the degrees of ionisation d₁, da, dg, d in the mixed solution, we have always the relation m₁d₁ × måd。 = m¸d¸ × måd¸· This relation we can combine with our knowledge of the general nature of the ionisation of the various substances, and its variation with dilution, to ascertain the actual character of the equilibrium. Thus Arrhenius, to whom the theory is due, has shown that the avidities of two monobasic acids at a given dilution are approximately proportional to the degrees of ionisation which they would have if each were dissolved separately in the given volume of solvent. we showed above to be the case for two weak acids, but it is equally true if both acids are strong, or if one is strong and the other weak. In the case of dibasic acids, like sulphuric acid, the theory cannot easily be applied owing to the excessively complicated nature of the equilibrium caused by the presence of acid salts (cp. p. 312). It is easy, too, to prove from the theory that the degree of 320 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. I A ionisation of a weak acid in presence of one of its salts is nearly inversely proportional to the quantity of salt present. If the weak acid should be in presence of several strongly ionised electrolytes, it can also be shown that its degree of ionisation will be the same as if the ionised parts of these electrolytes were the ionised parts of a salt of the given acid. We have now to examine the nature of the equilibrium between the aqueous solution of a salt and the solid salt itself. To each temperature there corresponds a certain solubility of the salt, i.e. a certain osmotic pressure of the dissolved substance in the solution which is in equilibrium with the solid. Now this osmotic pressure is made up of more than one component: it is the sum of the partial pressures of the undissociated salt and of the ions derived from the salt. The question thus arises: Is it the total osmotic pressure in the solution which directly regulates the equilibrium with the solid salt, or the osmotic pressure of the un-ionised dissolved salt, or finally the osmotic pressure of the ions? The reply given by Nernst to this question is that it is most probably the osmotic pressure of the un-ionised substance which directly determines the equili- brium, and there are many facts which support this conclusion. The un-ionised salt here plays the part of intermediary between the ions and the solid: it is in equilibrium with the ions on the one hand and with the un-ionised solid on the other. Considered in this aspect, the constant total solubility of a solid salt at a given temperature in water is due to the constant concentration of un- ionised substance in the solution, which in its turn is in equilibrium with a constant concentration of the ions. It is possible, however, to supply additional quantities of one or other of these ions to the solution, and we have to inquire into the effect this will have on the solubility equilibrium. In order to secure conditions favourable for calculation and for experimental verification of the results deduced from the theory, it is advisable to consider the equilibrium in the case of a sparingly soluble. salt, so that the solutions considered are dilute. As an example we may take silver bromate, AgBrO,, the concentration of the saturated solution of which at 24.5° is 0.0081 normal. The primary equilibrium which determines the solubility is here supposed to be that between. the solid silver bromate and the un-ionised silver bromate in the solution. The concentration of this last will remain constant if the The temperature remains at 24.5° and the solvent remains water. addition of a small quantity of a perfectly neutral substance, such as alcohol, sugar, and the like, does not appreciably affect the solubility of any substance in water, since the nature of the solvent practically remains the same. Besides the un-ionised silver bromate in the solution, we have argention and bromanion. We can increase the concentration of argention by adding a soluble silver salt, and we XXVII EQUILIBRIUM BETWEEN ELECTROLYTES 321 can increase the concentration of bromanion by adding a soluble bromate. Suppose that we add such a quantity of silver nitrate as to double the amount of argention after equilibrium has been attained. The concentration of the un-ionised silver bromate will, by hypothesis, remain the same as before. But the dissociation equilibrium of silver bromate requires that the product of the concentration of the ions should be equal to a constant into the concentration of the un-ionised salt, i.e. should remain constant. If, therefore, the concentration of the argention is doubled, the concentration of the bromanion must be halved, in order that the product of the two may have the same value as before. Bromanion can only fall out of solution along with an equivalent quantity of some positive ion, and since the only kind of positive ion in the solution is argention, bromate of silver must be precipitated in order to re-establish equilibrium. The effect, then, of adding silver nitrate to the silver bromate solution is to diminish the solubility of the silver bromate, and that in a degree depending on the amount of silver nitrate added. The addition of a soluble bromate acts in precisely the same way. The amount of bromanion at equilibrium is increased, and the amount of argention must be pro- portionately diminished in order to secure the constancy of the product of the two ions. The following numerical example will afford an insight into the mode of calculation. As has already been stated, the concentration of a saturated silver bromate solution at 24.5° is 0.0081 normal. If we assume the salt to be entirely ionised, the product of the ionic con- centrations is 0.0081 × 0.0081 = 0·0000656. Now let a quantity of silver nitrate be added, which, when dissolved in the same water as contains the silver bromate, will make the solution 0.0085 normal with respect to silver nitrate. Again we assume that the silver nitrate is entirely ionised. Suppose that the concentra- tion of the silver bromate now remaining in the solution is a, a smaller quantity than before. The concentration of argention will then be 0.0085 +2, and the concentration of bromanion will be a. We have therefore the product of these concentrations equal to the former product, i.e. whence (0·0085+x)x = 0.0000656, == X = 0.0049. We should consequently expect the addition of silver nitrate to reduce the strength of the saturated solution of silver bromate from 0·0081 to 0.0049. An actual determination showed that the solubility was reduced to 0·0051, which is in fair agreement with the theoretical result. It must be noted, however, that the theoretical result was Y 322 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. deduced on the erroneous assumption that the degree of ionisation of the various substances remained the same throughout the experi- ments. This is, of course, not the case, as the degree of ionisation at the dilutions considered is not equal to unity, and is diminished on the addition of the silver nitrate. It is easy, however, to take account of the change in the degree of ionisation of the silver salts by making use of conductivity determinations, although the formula for equilibrium then becomes somewhat complicated. Making the necessary corrections in the above case, the theoretical number comes out equal to 0·00506, which is very nearly the value observed for the solubility. Since silver nitrate and sodium bromate have practically the same effect so far as ionisation is concerned, we should expect equivalent quantities of these two salts to diminish the solubility of silver bromate equally. Experiment shows that an amount of sodium bromate equivalent to the silver nitrate added in the above experiment diminishes the solubility to 0.0052, a value very nearly identical with the former value. When two sparingly soluble salts yielding a common ion are shaken up with the same quantity of water, each diminishes the solubility of the other in a degree which can be calculated as in the previous instance. Thus the saturated solutions of thallium chloride, TICI, and thallium thiocyanate, TISCN, have a concentration of 0.0161 and 0.0149 respectively in gram molecules per litre. For the constant product of ionic concentrations we have, therefore, 0·01612 and 0.01492, if each is fully ionised. If the solubility of the chloride in presence of the thiocyanate is 2, and the solubility of the thiocyanate in presence of the chloride is y, these two numbers give the concentrations of the chloridion and thiocyanion respectively, while their sum, x+y, gives the concentration of the thallion. We thus obtain the simultaneous equations x(x + y) = 0·01612, y(x + y) = 0.01492, = whence a 0.0118, and y = 0.0101, the numbers found by experiment. being in good agreement, viz. 0·0119 and 0·0107. By taking account of the degree of ionisation as deduced from the conductivity the harmony between the experimental and calculated values is even more marked. The lowering of the solubility of an electrolyte by the introduction into the solution of another electrolyte pos- sessing a common ion with the first is a phenomenon of very general occurrence, the only exceptions being when the two substances form In- a double salt or act on each other chemically in the solution. stances of the application of the theoretical results will be given in Chap. XXIX. It has been pointed out that the very considerable accuracy obtained in applying this theory of the constant ionic product to the XXVII EQUILIBRIUM BETWEEN ELECTROLYTES 323 solubility of salts may be partially due to the balancing of opposed errors, as the theory assumes, for example, that the mass-action law is followed by salts, which, as has been indicated above, is very improbable. If we assume, however, that the ions themselves act normally, any abnormality of the non-ionised salt, although it would affect the dilution-law of the salt, would be without effect on the principle of the constant ionic solubility product; for the intermediate non-ionised salt would act abnormally in the same way on both sides, i.e. towards the ions and towards the solid, so that the abnormality would cut out in the final result between ions and solid. ARRHENIUS, "Theory of Isohydric Solutions," Zeitschrift für physikalische Chemie, 2, p. 284 (1888); "Equilibrium between Electrolytes," ibid. 5, p. 1 (1890). CHAPTER XXVIII NEUTRALITY AND SALT-HYDROLYSIS It is a matter of common knowledge that aqueous solutions of certain normal salts are not neutral to indicators. Thus a solution of aluminium sulphate, or of alum, is distinctly acid, and a solution of sodium carbonate is distinctly alkaline. Now acidity is, on the theory of electrolytic dissociation, attributed to the presence of hydrion, and alkalinity to the presence of hydroxidion, in the solution. We have therefore to account for the presence of these ions in some salt solutions in sufficient concentrations to affect indicators. As these ions cannot come from the normal salts them- selves, we must conclude that they are derived from the solvent water, and must therefore devote some attention to the properties of pure water from the point of view of electrolytic dissociation. Ordinary tap-water has a very considerable electrical conductivity, i.e. it contains ions in moderate quantity. The conductivity of distilled water is very much less, so that we are justified in assuming that the conductivity of tap-water is chiefly due to saline impurities dissolved in it. The more carefully water is distilled the less does its conductivity become; but it is difficult to procure and keep water with a conductivity at 18° of much less than κ= 10-6 (cp. p. 7), since the solvent action of the water on glass and on the carbonic acid of the air soon raises the conductivity to this figure. By the use of silica apparatus and air carefully freed from carbon dioxide and other conducting impurities it is possible, however, to prepare and work with water of a conductivity of 0·06 × 10-6 at 18°. Kohlrausch on purifying water by distillation in vacuo, and condensing the vapour directly in the resistance vessel, found that the purest water he could obtain had a conductivity of 0.040 × 10-6 at 18°, and from a con- sideration of the temperature coefficient of the conductivity of this water he calculated that absolutely pure water would have a conduc- tivity of 0.036 × 10-6 at 18°, or 0·054 × 10-6 at 25°. Knowing the ionic velocities of hydrion and hydroxidion, the only ions to which water can give rise, we deduce from these values that at 25° about 324 CH. XXVIII 325 NEUTRALITY AND SALT-HYDROLYSIS X 1.8 mg. per ton of water is ionised. In other words, the concentra- tion of hydrion H' and of hydroxidion OH' in pure water is about 1 × 10-7-normal at 25°. The value generally adopted for the ionic concentration of hydrion and hydroxidion in pure water at 25° is 1.1 x 10-7-normal, and this value we shall use in all subsequent calculations, although recent experiments tend to show that it is somewhat too high. For pure water at 25° we have the equilibrium equation k[H'] × [OH'] = k’[H₂O] [H] × [OH'] = const., since the active mass of the non-ionised water is so great relatively to the active mass of the ions as to be practically invariable. Sub- stituting for the active masses of hydrion and hydroxidion the values for pure water found above, we obtain as the ionisation constant of water at 25° Kw = [1·1 × 10−7] × [1·1 × 10-7] = 1·2 × 10−14. X This ionisation constant is of the utmost importance, for its value is supposed to hold good not only for pure water but for all dilute aqueous solutions, since in them the active mass of the non-ionised water remains practically constant. The product, then, of the con- centrations of hydrion and hydroxidion is the same in all dilute aqueous solutions, whether acid, neutral, or alkaline, and equal to 12 x 10-14 at 25°. By making use of this number we may ascertain, for example, the concentration of hydroxidion OH' in a decinormal solution of acetic acid. At 25° the dissociation constant of acetic acid is 1.8 × 10-5 (p. 254). For v = 10 we therefore deduce m = 0·013 or [H] = 0.0013, whence X [1·3 × 10-³] × [OH'] = 1·2 × 10−1± [OH'] = 0·9 × 10−11. The concentration of hydroxidion in decinormal acetic acid is there- fore about 10-11-normal. According to what has just been stated, all aqueous solutions must contain both the acid ion, hydrion, and the alkaline ion, hydroxidion. It is expedient, therefore, to ascertain in what sense we may use the terms acid, neutral, and alkaline, as applied to aqueous solutions. Pure water is always looked upon as absolutely neutral, and in it the concentrations of hydrion and hydroxidion are equal. We may consequently define absolutely neutral solutions as those in which the concentrations of hydrion and hydroxidion are equal. Since the product of these ionic concentrations remains con- stant, it follows that all absolutely neutral solutions contain hydrion and hydroxidion in the same concentration as pure water at the 326 CHAP. INTRODUCTION TO PHYSICAL CHEMISTRY same temperature. Having thus defined absolutely neutral solutions, we define acid solutions as those in which the concentration of hydrion is greater than that of hydroxidion, and alkaline solutions as those in which the concentration of hydroxidion is greater than that of hydrion. To determine whether a solution is acid, alkaline, or neutral in the ordinary sense we employ indicators, which by means of a change in colour indicate the passage from one concentration of hydrion (or hydroxidion) to another. Pure litmus (azolitmin) is one of the most sensitive of these indicators. If the concentration of hydrion exceeds 10-6, the colour is sensibly red, if it falls beneath about 10-8, i.e. if the concentration of hydroxidion is about 10-6, the colour is sensibly blue; for intermediate concentrations the solution assumes a purple tint, and is said to be neutral. It is obvious that neutrality to litmus does not necessarily coincide with absolute neutrality, but extends over a wide region on both sides of the true neutral point. For the practical purposes of acidimetry or alkalimetry, however, litmus is sufficiently sharp to fulfil all ordinary requirements. Suppose, for example, we are dealing with 100 cc. of a solution just appreciably acid to litmus, i.e. a solution of hydrion concentration equal to 10-6 normal. To bring this solution to the true neutral point we have only to add 100 × 10-60·0001 cc. of a normal solution of alkali, or 0.01 cc. of a centinormal solution of an alkali; and the further addition of a similar amount would render the solution perceptibly alkaline. Lacmoid closely resembles litmus in its properties as an indicator, but is not quite so sharp. Of other indicators, methyl orange under ordinary conditions changes between 10-4-N hydrion and 10-5-N hydrion, and methyl red between. 10-5-N and 10-6-N hydrion; phenol-phthalein changes between 10-5-N hydroxidion and 10-6-N hydroxidion. We are now in a position to resume consideration of the acid or alkaline reaction exhibited by certain normal salts. It is generally said that the divergence from neutrality is due to the normal salt being partly split up by the water into free acid and free base. The process is termed salt-hydrolysis, and may be represented in the case of the sodium salt of carbolic acid (sodium phenolate) by the reversible equation CH¿ONa + HOH⇒CH₂OH + NaOH. We have then to ascertain if the minute ionisation of pure water is competent to explain the production of hydrion or hydroxidion in salt solutions in sufficient quantity to affect indicators and impart the other distinctive properties of acids or alkalies to the salt solutions. It may be stated at once that salt-hydrolysis takes place to a perceptible extent only in solutions of salts derived from a weak acid XXVIII NEUTRALITY AND SALT-HYDROLYSIS 327 • 4 or a weak base. Strong acids combined with strong bases give salts which show no hydrolysis. The normal salts of strong polybasic acids such as orthophosphoric acid may, however, be strongly alkaline, but here the normal salt Na¸PO is really a salt of the very weak acid HNa,PO, so that the exception is only after all apparent. It is only the first replaceable hydrogen atom in orthophosphoric acid that confers strength on the acid, the others being nearly non- ionisable when the first is replaced. The hydrolysis equation is therefore NaPO4 + H₂O Na₂HPO4 + NaOH, an acid salt being formed instead of the polybasic acid proper. X As a weak acid we may choose phenol, for which k= 1.3 × 10-10, and study the effect of water on its sodium salt in decinormal solution. On the supposition that the hydrolysis is slight and that the sodium phenolate remaining is completely ionised, which at the dilution considered is not far from being the case, we obtain as a first approximation to the concentration of the ion CHÃO' the value 0.1. Now this ion is not only in equilibrium with the positive ion Na, but also with the hydrion H' which is in the solution, and the equation 6 5 [H′] × [C₂H₂O] – 1·3 × 10−10 [C&H₂OH] = must be obeyed. We have just found that [CH.O'] is approximately equal to 0-1 so that [H] 13 x 10-9[CH,OH]. Now since caustic soda and phenol are produced in equivalent quantities by the hydrolysis of the sodium salt, and since the caustic soda is almost completely ionised whilst the phenol is practically not ionised at all, we may take the concentration of hydroxidion as equal to that of free phenol, and write [OH'] = [CH₂OH]. We have thus, from the relation given above, [H'] = 1·3 × 10-9[OH'], and from the equilibrium of the ions of water, [H] x [OH]= 12 x 10-14. Eliminating [H] from these equations, we obtain [OH]= 3 x 10-3 approximately, a concentration well within the range of ordinary indicators for alkali, which will therefore show the solution to be distinctly alkaline. = The complete hydrolysis equations for a salt MA derived from the base MOH and the acid HA may be established as follows. In the solution there will be the following seven kinds of molecule MOH, MA, HA, M, A', H, OH' besides the non-ionised water which exists in such preponderant quantity in all solutions that the variation of its active mass need not be considered. To ascertain the concentrations of these seven molecular species we must obtain seven independent relations 328 CHAP. INTRODUCTION TO PHYSICAL CHEMISTRY between them without the introduction of any fresh variable. To begin with we have (1) (2) [MOH] + [MA] + [M'] = n [MA] + [HA] + [A'] = n (4) (5) (6) (7) where η is the concentration of the solution as made up from the salt and water. In the present case we have [MOH] + [MA] + [M'] = [MA] + [HA] + [A] because equivalent quantities of acid and base are taken; otherwise the two expressions will not be equal, although known from the weights used in making up the solution. condition that the solution shall be electrically neutral gives us [M'] + [H'] = [A'] + [OH']. The (3) For the equilibrium of the positive and negative ions in pairs we have [H] x [OH'] = Kw [M] × [OH'] = ¿[MOH] [M'] × [A'] = ks[MA] [H] × [A´'] = a[HA]. The magnitudes Kw, kь, ks, ka are the ionisation constants of water, the base, the salt, and the acid respectively. From the equation MA + H₂O MOH + HA we obtain the equilibrium equation for the non-ionised substances [MOH] × [HA] [MA] = constant. This, the hydrolysis constant, may be derived from equations (4) to (7) by substitution as follows:- [MOH] × [HA] _ [M] × [OH] [H] x [A] x [M][A] X _ X X X ka [MA] × kb [MOH] × [HA] Kulis [MA] h. kika (8) The hydrolysis constant h is expressed in terms of the ionisation constants, but the expression as it stands is of little utility. In the first place we are ignorant of the concentrations of the non-ionised substances on the left of the equation, and in the second place, although Kw is known, the constants ka and k are only available when the acid and base are feeble, and k, the salt constant cannot be determined (p. 257). We can, however, utilise the equation under certain assumptions which are very nearly true. Once more we shall consider the case of sodium phenolate, derived from the strong base soda, and the weak acid phenol. Both salt and base are strongly ionised, and the degree of ionisation in one and the same solution XXVIII NEUTRALITY AND SALT-HYDROLYSIS 329 may be assumed to be the same, without the introduction of any considerable error. This assumption enables us to substitute in the left-hand member of the equation the ratio of the concentrations of total free base and total salt for the ratio of the concentrations of non-ionised base and non-ionised salt, these ratios being equal, and also to make the ratio kg/k 1, since equality of ionisation in the same solution involves equality of ionisation constants. Again, the concentration of the non-ionised acid is equal to the concentration of total free acid, since in presence of highly ionised electrolytes the degree of ionisation of a weak acid is practically zero. Equation (8) then assumes the form [free base] x [free acid]__ Kw ka (9) [unhydrolysed salt] By similar reasoning, when the acid is strong and the base feeble, we have [free base] x [free acid] Kw [unhydrolysed salt] lip When acid and base are originally in equivalent quantities, i.e. when normal salts are considered, we have V 2 4 5 10 [hydrolysed salt]2 Kw [unhydrolysed salt] ' where k is ka or k according as the acid or base is weak. The relation may be put in a form analogous to Ostwald's dilution formula, viz. n² (1 - n)v h, where n is the hydrolysed proportion. For small degrees of hydrolysis, the degree of hydrolysis is proportional to the square root of the dilution (cp. p. 255). As an example of the application of the formula we may take urea hydrochloride at 25°. N 0.684 0.791 0.820 0.893 N2 (1 − n ) 2! 0.74 0.75 0.76 0.75 =h Here the degree of hydrolysis is great, and it will be observed from the equation h= Kw that this involves a small value of the basic kb constant k. If we perform the calculation we obtain Kw 1·2 × 10 −1± kv h 0.75 as the basic constant of urea. = 1.6 x 10-14 330 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. For sodium phenolate at the dilution v = 100, n was found to be 0.0882 0.912 × 100 0.088. This value gives h= whence ka به شما 1.2 × 10-14 0.85 × 10-4 = = 0.85 × 10-4, 1.4 × 10-10. This value for the dissociation constant of phenol agrees well with that determined from the conductivity of phenol solutions, namely, 1.3 × 10-10. If we have a knowledge, then, of the degree of hydrolysis of the alkali salts of weak acids, or the chlorides, nitrates, etc., of weak bases, we can calculate the dissociation constants of these weak acids or bases. The determination of the degree of hydrolysis is often some- what difficult. It might be thought that the free acid could be titrated with an alkali and an indicator, but a little consideration shows that this is impossible. Suppose that the salt were hydrolysed to the extent of 5 per cent. As soon as we have neutralised that 5 per cent of acid by a strong base, the originally undecomposed proportion of the salt is hydrolysed by the action of the water, and more acid. is produced. Although this in turn is neutralised by the addition of a strong base, the process of hydrolysis still goes on, and the solution will not become neutral until all the acid which was originally combined with the weak base has passed into combination with the strong base with which we titrate the solution. A solution of aniline hydrochloride behaves towards caustic alkali and phenol- phthalein exactly like an equivalent solution of hydrochloric acid, owing to the progressive hydrolysis of the salt. In determining the degree of hydrolysis, then, a method must be employed which will not disturb the hydrolytic equilibrium. Measure- ments of optical or other physical properties have been employed with some degree of success, and also reaction velocity methods. For example, we can tell approximately how much free hydrochloric acid there is in a solution of aniline hydrochloride or urea hydrochloride, by ascertaining at what rate they catalyse methyl acetate or invert cane sugar (ep. p. 302). The velocity constant for the catalysis of methyl acetate is approximately proportional to the concentration of free hydrochloric acid in the solution, so that a determination of the one leads to a knowledge of the other. Similarly, we can ascertain how much free caustic soda there is in a solution of sodium carbonate by finding at what rate the solution saponifies ethyl acetate, the initial rate of saponification being nearly proportional to the amount of free alkali in the solution. Besides these velocity methods, others which depend on solubility may be employed. Phenol, for example, dissolves sparingly in water, but in a dilute solution of caustic soda the solubility is greatly t XXVIII NEUTRALITY AND SALT-HYDROLYSIS 331 increased. On the assumption that the solubility of the phenol itself is the same in water and dilute solutions, and the increased amount dissolved in the alkali is entirely due to the formation of sodium phenolate, it is possible to calculate the amounts of phenol, phenolate, and sodium hydroxide in the solution, and then from equation (9) determine the acidic constant of phenol. A similar method depending on coefficients of distribution between two solvents may be used for the same purpose. When acid and base are both weak, the practical form to which equation (8) reduces is quite different from that used above. Since both acid and base are weak we can substitute the concentrations of total free acid and total free base for the concentrations of the non- ionised substances. We thus obtain [free base] × [free acid] Kw ks [non-ionised MA] ka ko But for k, [MA] we may substitute from equation (6), [M'] × [A'], and since in dilute solution we may regard the salt as nearly all ionised, we may put [M] × [A'] = [total salt]2, so that finally or X [free base] × [free acid]__ Kw [unhydrolysed salt] ka ko' 2 liv' 1 N 1 - n N ✓ ― N when the normal salt is considered. Here the degree of hydrolysis does not vary with the dilution. Ammonium acetate affords an instance of this type of salt, the values of the constants being ka=1.8 × 10-5 and 23 × 10−5, so that K10 Vka kip kv 1·2 × 10-14 2·3 × 1·8 x 10-10 At 25°, then, ammonium acetate is hydrolysed to the extent of about one-half per cent at all concentrations. [HA] = ka N la 1 - n' Since by equation (7) we have [H'] = ka the [A] acidity, i.e. the hydrion concentration, does not alter with dilution in the case where both acid and base are weak, for we have just seen is then constant. For ammonium acetate [H]= 1·8 × 10 −5 N that 1 N × 0·0054 = 0·97 x 10-7 at all concentrations. This is very near absolute neutrality, a coincidence which is due, as shown below, to the practical equality of the acidic and basic constants. If we wish to find the condition that a salt solution should be =0·0054. 332 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. absolutely neutral, we may proceed as follows. For this solution [H]=[OH'], and consequently from (3) [M'] = [A'], [MOH]=[HA], [M] × [OH']_[H'] × [A'] [MOH] [HA] whence from (1) and (2) so that or kb = ka. The condition, then, for absolute neutrality of a salt is that the constants of the acid and base from which it is derived should be equal. Conversely, it may be proved that if the acidic and basic constants are equal, the solution will be neutral, no matter how great the hydrolysis may be. The range of apparent neutrality towards indicators is of course considerable, but it may be said that a salt (of a monobasic acid with a monacid base) will always appear neutral to litmus if both acidic and basic constants, however unequal, are above 10-4. ka The amount of salt hydrolysed in a given solution may vary greatly with the temperature. Generally speaking, the temperature coefficient of ionisation is small, and may be sometimes positive, sometimes negative. Since the hydrolysis constant depends on for a weak acid and strong base, it follows that if ka is nearly constant the effect of temperature on the hydrolysis (for a small degree of hydrolysis) is roughly proportional to the effect of tempera- ture on the ionisation of water. Now rise of temperature greatly increases the ionisation of water, the proportion ionised at 50° being seven times as great as at 0°, and 2.5 times as great as at 25°. 100° the ionised amount is about seven times as great as at 25°. At We found on p. 331 that ammonium acetate at 25° should be about 0.5 per cent hydrolysed. On the assumption that ka and kɩ are in this case but slightly affected by temperature, the degree of hydrolysis at 100° should be 7 x 0.5 3.5 per cent. An experimental determination gave 3.2 per cent in normal solution, and 4.1 per cent in 0·04-N solution, in approximate concordance with the calculated value. www Kw The temperature coefficients of the ionisation constants of many very weak organic acids and bases, the latter particularly, approximate to that of water itself. If they were the same, the value of Kw k would not vary with the temperature, and consequently the degree of hydrolysis would be unaffected by change of temperature. Thus the hydrolysis of urea hydrochloride is practically the same at 25° and 40°. XXVIII NEUTRALITY AND SALT-HYDROLYSIS 333 The following table shows the extent of hydrolysis per cent in decinormal solutions of salts of a few weak acids and bases at 25°. WEAK ACIDS. Potassium phenolate Potassium cyanide Borax Sodium acetate • then • • • • 3.1 1·1 0.5 0.01 WEAK BASES. Aniline hydrochloride. p-Toluidine o-Toluidine Urea "} >> =liv "" = · The hydrolysis of aluminium chloride is of the same order as that of aniline. That an insoluble base such as aluminium hydroxide, or an insoluble basic salt, should exist in solution seems at first sight inexplicable. When we consider, however, the tendency of such substance to form colloidal solutions, the difficulty disappears. It is found that salts of insoluble weak bases give aqueous solutions. which exhibit the Tyndall phenomenon, thus proving that they are not in reality homogeneous, but contain an ordinarily insoluble sub- stance in colloidal solution (p. 221). • There exists an interesting class of substances known as amphoteric electrolytes, which are capable of acting both as acids and as bases. A simple example is to be found in glycine or amino-acetic acid NH, CH, COOH. In virtue of the group NH, this substance can act as a base, and form salts with the stronger acids; whilst in virtue of the COOH group it can act as an acid and form salts with the stronger bases. In aqueous solution by itself it must be conceived as giving rise to both hydrion and hydroxidion, i.e. to be ionised both as an acid and as a base. But in all aqueous solutions the product of the hydrion and hydroxidion concentrations must be constant and equal to 1·2 × 10-14 at 25°, and this consideration gives us the clue to the abnormal behaviour of amphoteric substances in water. By the hydrolysis methods described on p. 330 as applied to the salts of amphoteric substances with strong bases and strong acids, it is possible to calculate the value of their acidic constant ka and basic constant k. Let there be dissolved in water an amphoteric electrolyte HXOH which as an acid is ionised into H and XOH' with the constant ka, and as a base into HX and OH' with the constant k¿. If the concentrations of the various molecular types in solution at equilibrium are # 1.4 0.9 1.8 89.3 [H], [XOH'], [OH'], [HX'], [HXOH], [XOH'] + [HX'] + [HXOH]=n, the known total concentration, [H'] + [HX'] = [OH']+[XOH'], for electrical neutrality, [H] × [OH'] = Kw [H] × [XOH'] [HXOH] -ka [HX'] × [OH'] [HXOH] 334 CHAP. INTRODUCTION TO PHYSICAL CHEMISTRY i.e. we have five equations for determining five unknown quantities, so that the values of the ionic concentrations and the total concentra- tion of non-ionised substance, here represented as fully hydrated, can be ascertained. It follows from these equations that an amphoteric electrolyte cannot obey Ostwald's dilution law unless one of the constants becomes very small. For an ordinary acid [H'] × [X']= [H*]²=ka[HX] whereas for an amphoteric acid, i.e. an electrolyte in which the acidic. constant is considerably larger than the basic constant, we obtain [H]² - ka[HXOH] kv Κω 1 + [HXOH] ༦ If in this expression is vanishingly small, it becomes identical with the expression for a simple electrolyte. If on the other hand Kn [HXOH] is appreciable compared with unity, the value of [H], no ་ which chiefly determines the electrical conductivity, diverges from that of a simple electrolyte with the same acid constant, and that to a different degree at different dilutions, owing to the change in the value of [HXOH], so that there can be no strict obedience to the simple dilution law. This want of harmony has long been known in the case of amino-acids such as anthranilic acid o-C¸Н¸(NH₂)(COOH), which behaves in most respects as an ordinary acid, owing to the value of ka 8.2 × 10-6 being much greater than that of k¿ = 2 × 10-12. Yet this comparatively small basic constant is sufficient to affect the electrical conductivity to an appreciable degree. In the solution of such an amphoteric acid there is always a certain quantity of the ion HX, as well as the ion H', and this is more marked as the solution is more concentrated. The substance in short ionises not only as an acid but also as a kind of internal salt, the basic neutralising the acidic portion, and this type of ionisation is the greater as the basic constant k approaches the acid constant ka. When the constants ka and k for an amphoteric electrolyte are equal, the substance is neutral, i.e. the concentrations of the ions H' and OH' are the same as in pure water. Notwithstanding this neutrality, the solution of the substance would be an electrolyte owing to its ionisation as an internal salt, and it would still be capable of acting as a base towards other acids, and as an acid towards other bases. Such a neutral amphoteric electrolyte would differ from a neutral salt in that its degree of ionisation (and therefore molecular conductivity) would be the same at all dilutions, the actual value depending on the magnitude of the constants ka and k XXVIII NEUTRALITY AND SALT-HYDROLYSIS 335 The theory of salt-hydrolysis was originally given in a paper by ARRHENIUS, Zeitschrift für physikalische Chemie (1890), 5, 16. J. WALKER and J. K. WOOD, "Hydrolysis of Urea Hydrochloride," Journ. Chem. Soc. (1903), 83, 484. R. C. FARMER, "Determination of Hydrolytic Dissociation," ibid. (1901), 79, 863; also ibid. (1904), 85, 1713. J. WALKER, "Theory of Amphoteric Electrolytes," Proc. Roy. Soc. (1904), 73, 155; (1904), 74, 271. } APPLICATIONS OF THE DISSOCIATION THEORY WHEN many of the ordinary chemical reactions are looked upon from the standpoint of the theory of electrolytic dissociation, they present an aspect very different from that to which we are accustomed. The neutralisation of strong acids by strong bases in dilute solution is a typical example. If the acid is hydrochloric acid, and the base sodium hydroxide, we have the equation HCl + NaOH = NaCl + HOH. CHAPTER XXIX Now on the dissociation theory all the substances concerned in this action are highly ionised in aqueous solution, the water itself being the only exception. Writing, then, the equation for the ions, we obtain H' + Cl' + Na + OH' = Na˚ + Cl′ + HOH, or, eliminating what is common to both members of the equation, H' + OH' = HOH. The neutralisation of a strong acid by a strong base consists then essentially in the union of hydrion and hydroxidion to form water. So long as the base, acid, and salt are fully ionised, their nature. makes no difference whatever on the character of the chemical act of neutralisation. This we find to be in conformity with many experimental facts. For example, the heat of neutralisation of one equivalent of a strong acid in dilute solution by a corresponding quantity of a strong base is very nearly 13,700 cal., as the following table shows, the base used being caustic soda :- Acid. Hydrochloric Hydrobromic Hydriodic Chloric. Bromic. Iodic Nitric ► • ▸ • • • • Heat of Neutralisation. 13,700 cal. 13,700 cal. 13,700 cal. 13,800 cal. 13,800 cal. 13,800 cal. 13,700 cal. • 336 CH. XXIX APPLICATIONS OF DISSOCIATION THEORY 337 A similar table for the heats of neutralisation of bases by an equivalent of hydrochloric acid shows that the heat of neutralisation is independent of the base, as long as it is fully ionised. Base. Lithium hydroxide Sodium hydroxide Potassium hydroxide Thallium hydroxide Barium hydroxide Strontium hydroxide Calcium hydroxide Tetramethylammonium hydroxide. · • but Metaphosphoric acid Hypophosphorous acid Hydrofluoric acid Acetic acid Monochloracetic acid Dichloracetic acid. Base. When we deal with weak acids or weak bases the heats of neutralisa- tion often diverge greatly from the mean value for the highly ionised substances. Thus the heats of neutralisation of some comparatively feebly ionised acids by sodium hydroxide are given in the following table, the acids not being so feeble, however, as to have their sodium salts sensibly hydrolysed in aqueous solution :- Acid. · · • Corresponding numbers for weak bases neutralised by hydrochloric acid are:- Ammonium hydroxide Methylammonium hydroxide Dimethylammonium hydroxide Trimethylammonium hydroxide · Heat of Neutralisation. 13,800 cal. 13,700 cal. 13,700 cal. 13,800 cal. 13,900 cal. 13,800 cal. 13,900 cal. 13,700 cal. • • Heat of Neutralisation. 14,300 cal. 15,100 cal. 16,300 cal. 13,400 cal. 14,300 cal. 14,800 cal. Heat of Neutralisation. 12,200 cal. 13,100 cal. 11,800 cal. 8,700 cal. The explanation of these divergencies from the value for highly dis- sociated substances is simple. Ammonium hydroxide is only feebly ionised at the dilution considered; the chemical action is not chiefly H* + OH' = H₂O, H² + NH₂OH = H₂O + NH¸¨‚ 4) i.e. from the "normal" heat of neutralisation must be subtracted the heat necessary to decompose NH,OH into the ions NH and OH'. The heat of ionic dissociation for acids and bases is not as a rule great, so that in the majority of cases the heat of neutralisation even of weak acids and bases does not greatly diverge from the value 13,700 cal. The divergence may be in the one direction or the other, according as heat is absorbed or developed by the ionic dissociation (cp. p. 284). When acids or bases are so weak that their salts undergo extensive hydrolytic dissociation in aqueous solution, i.e. are partially split up Z 338 CHAP. INTRODUCTION TO PHYSICAL CHEMISTRY into free acid and base, the heat of neutralisation is very small. This is owing to the fact that the neutralisation is incomplete, free hydrion. or free hydroxidion remaining in the solution. If we consider the displacement of a weak acid from its salts by a strong acid in the light of the dissociation hypothesis, we find that the resistance of the weak acid to ionisation is the determining circumstance in the reaction. Thus if the salt is sodium acetate and the strong acid is hydrochloric acid, the customary equation becomes Na' + C₂H₂O₂' + H* + Cl′ = Na' + Cl′ + HC₂H₂O„, 2 2 or C,H,O, + H = HC,H,O, The action is essentially a union of hydrion with acetanion, and the nature of the acetate or of the strong acid is a matter of indifference, provided that they are both almost dissociated at the dilution under consideration. Similarly, the displacement of a weak base from its salts by a highly ionised base consists essentially in the union of a positive ion with hydroxidion, thus or NH₁‍ + Cl′ + Na + OH′ = Na˚ + Cl′ + NH₂OH 4 4 NH₁* + OH′ = NH₂OH. 4 The solution of a metal in an aqueous and strongly dissociated acid is principally transference of a positive electric charge from hydrogen to the metal: thus if zinc is the metal and hydrochloric acid the highly dissociated acid, we have Zn + 2H + 2C1' = Zn" + 2C1' + H₂, or Zn + 2H = Zn + H₂. Similarly, the displacement of bromine from a soluble bromide by chlorine is chiefly a transference of an electric charge from bromine to chlorine :- Cl₂ + 2K* + 2Br′ = Br₂ + 2K˚ + 2C1', 2 Cl2 + 2Br′ = Br₂ + 2C1′. When two highly dissociated salts are brought together in solution, we have an ionic equation such as the following, if double decomposi- tion is supposed to occur :— Na + CI' + K' + Br' = Na + Br' + K' + Cl'. Both sides of this equation are the same, i.e. no chemical change has taken place at all. This of course only holds good as long as all the substances remain in the solution. If the solution is evaporated, that salt which is the least soluble will in general fall out first. XXIX APPLICATIONS OF DISSOCIATION THEORY 339 We obtain a similar equation for the action of a strong acid on the salt of an equally strong acid, for example- H' + Cl' + K' + Br′ = H' + Br' + K' + Cl'. Here again both sides of the equation are the same, and thus no action has taken place. It is usual to say that in this case the base is equally divided between the two acids, but it may be seen from the above equation that the ions are free to combine in any way according to circumstances. If one of the acids is weaker than the other, then more of it will exist in the un-ionised state, so that the other acid is said to have taken the greater share of the base. From the examples already given it is evident that the equations. involving ions are usually of a much more general character than the ordinary chemical equations, and more frequently bring out the essential phenomenon common to a number of actions of the same type, as, for instance, neutralisation, and the displacement of a weak acid or base by a stronger. The special actions employed in testing for metallic and acid radicals are practically always reactions of the ions, so that our ordinary tests are tests for ions. Copper, for example, gives a black precipi- tate with hydrogen sulphide, but not under all conditions. As long as the copper to be tested for is in the form of cuprion, whether Cu˚ or Cu¨, the black precipitate is formed when sulphuretted hydrogen is intro- duced into the solution. But if the copper ceases to be cuprion, and becomes part of a more complex ion, the precipitation will not take place. This is the case if we add potassium cyanide to the solution of a cupric salt until the original cyanide precipitate is dissolved. The copper in the solution is then in the state of a complex salt, prob- ably 3KCN, Cu(CN). The formula of this salt should be written. K,Cu(CN), for on solution in water it is ionised into kalion K and the complex negative ion cuprocyanidion Cu(CN),"". The copper is no longer in the form of cuprion, but exists merely as a part of the complex ion, and has in this state no reactions of its own. Sulphuretted hydrogen added to such a solution produces no precipi- tate, and use is made of this fact to separate copper from cadmium. Cadmium, it is true, when its salts are treated with excess of potassium cyanide, ceases for the most part, like copper, to be a positive ion, and enters into the composition of the complex negative ion Cd(CN),", but this ion is not by any means so stable as the corresponding ion containing copper. All such compound ions have the tendency to split up into simple ions, according to equations like the following:- Cd(CN),″ = Cd(CN), + 2CN', Cd(CN)2 = Cd¨ + 2CN', and this tendency exists to different extents with different ions. With the complex ion containing cadmium it is very pronounced; with the 340 CHAP. INTRODUCTION TO PHYSICAL CHEMISTRY complex ion containing copper it is much less evident. In the solu tion of K,Cu(CN), there is thus a scarcely appreciable quantity of cuprion Cu', while in the solution of K₂Cd(CN), the ion cadmion Cd" exists in moderate proportions. The former solution then is scarcely affected by hydrogen sulphide, while the latter is freely precipitated. A solution of potassium ferrocyanide, K,Fe(CN)6, is almost entirely free from diferrion Fe¨, and exhibits none of the ordinary reactions of ferrous salts. 4 The commonest reaction for silver is the production of a precipi- tate of silver chloride by the addition to the silver solution of a soluble chloride. This reaction is for argention, Ag, and not for silver in general. If we add potassium cyanide to a silver solution until the original precipitate of silver cyanide redissolves, the addition of a soluble chloride fails to produce a further precipitate. Here again the silver has become part of a fairly stable complex negative ion argenti- cyanidion, Ag(CN),', which gives off very little argention by dissocia- tion, so that the ordinary reagents for argention may fail to detect its presence. The same holds good for the solution of a silver salt in presence of sodium thiosulphate. When we add this salt to a solution of silver nitrate, we obtain first a white precipitate of silver thio- sulphate, Ag₂SO, which on further addition of sodium thiosulphate dissolves with formation of the double salt NaAgS,O,. This salt dissociates chiefly into the ion Na and the complex negative ion argenti-thiosulphanion, AgS,Og, so that very little argention is at any one time in the solution, and the ordinary reagents produce none of the characteristic silver reactions. 4 A change in the quantity of electricity associated with a positive or negative radical is accompanied by an entire change in the properties. of the radical. Thus the reactions of diferrion, Fe¨, are entirely different from the reactions of triferrion, Fe; and the reactions of permanganion, the ion of the permanganates, MnO4', differ greatly from the reactions of manganion, the ion of the manganates, MnO, In connection with such changes in the electric charges of ions, the student will find it useful to remember that addition of a positive charge or removal of a negative charge corresponds to what is generally known as oxidation in solution; and that removal of a positive charge or addition of a negative charge corresponds to reduction. Thus we are said to oxidise a ferrous salt to a ferric salt when we convert the ion Fe" into Fe, or reduce a permanganate to a manganate when we convert the ion MnO into the ion MnO". In the first instance a positive charge is added; in the second a negative charge. 4 If we write the equation. 2FeSO4 + H2SO4 + Cl2 = Fe2(SO4)3 + 2HCl, on the supposition that all the electrolytes are fully ionised, we obtain XXIX 341 APPLICATIONS OF DISSOCIATION THEORY or 2Fe¨ + 2H + 3SO" + Cl₂ = 2Fe¨¨ + 2H + 3SO," + 2C1', 2Fe¨ + Cl₂ = 2Fe¨¨ + 2C1'. 2 The whole action, from this point of view, reduces itself to the simul- taneous appearance of positive and negative charges. The diferrion assumes a positive charge and is "oxidised" to triferrion. The uncharged chlorine assumes a negative charge and is "reduced" to chloridion. In this instance no oxygen has been transferred, so that it is only by analogy that we can call such a process one of oxidation. It is precisely in these cases, however, that the above mode of viewing the action is sometimes of service. When actual transference of oxygen takes place, the composition alters as well as the charge of the ions, and the ionic conception of the process can only be applied to groups of substances which do not change in composition. Thus if we consider the action or to take place at such an extreme degree of dilution that all the electro- lytes are fully dissociated, the equation becomes or 10FeSO4 + 8H2SO4 + 2KMnO4 = K2SO4 + 2MnSO4 + 5Fe2(SO4)3 + SH₂O 8H₂SO 10Fe¨ + 16H˚ + 2K° + 18SO¸″ + 2MnO' = 10Fe¨¨ + 2K' + 2Mn¨ + 1850″ + 8H₂O, 4 • 5 Fe" + 8H + MnO₁ = 5Fe¨¨ + Mn¨ + 4H₂O. 4 The 4 Here again we have the "oxidation" of diferrion to triferrion. group 8H+ MnO has lost six positive and one negative charge in becoming the group Mn" + 4H,O, i.e. has lost on the whole five positive charges, and has therefore been "reduced" by this amount. When a metal passes into solution as an ion, it gains one or more positive charges, and thus acts as a reducing agent. Thus in the action Zn + H2SO4 = ZnSO4 + H2, Zn + 2H = Zn + H₂, the zinc has been "oxidised" to the state of zincion, while the hydrogen has been "reduced" from the ionic condition to the state of free hydrogen. It is scarcely customary to apply the terms oxidation and reduction to the passage of hydrogen or a metal from the free state to the state of combination in an acid or salt, but the application is obviously justifiable. Free zinc and free hydrogen are undoubtedly reducing agents under proper conditions, while the ionic zinc or hydrogen, in zinc sulphate or sulphuric acid, can in no sense be looked on as reducing substances in dilute solution. Many applications are made in the laboratory and in the operations 342 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. of technical chemistry of the diminution in solubility suffered by a salt, acid, or base when there is added to the solution another electro- lyte having one ion in common with the original electrolyte (p. 320). If we wish to prepare a pure specimen of sodium chloride, we make a strong solution of the impure salt, and pass hydrochloric acid gas into it, or add to it concentrated hydrochloric acid solution. The bulk of the sodium chloride is precipitated, and in a higher state of purity than the salt originally dissolved. Suppose the sodium chloride solution to be saturated, or nearly saturated. The addition of the highly ionised and extremely soluble hydrogen chloride greatly increases the amount of chloridion, and the quantity of natrion must conse- quently be diminished by separation of sodium chloride, in order that the product of the ionic concentrations may be maintained nearly constant. Even if the solution is not nearly saturated to begin with, the addition of chloridion in sufficient quantity from the hydrogen chloride may bring the ionic product up to the constant value, and thus determine the precipitation of a portion of the sodium chloride. If the above process of purification is to be effective, it is essential. that the added substance should be considerably more soluble than the original substance, especially if the two are about equally ionised, as is the case with most salts, strong acids, and strong bases. If we attempted to precipitate a saturated solution of the soluble sodium chloride by the addition of the comparatively sparingly soluble barium. chloride, we should find very little sodium chloride thrown down. This is because the ionic solubility product of barium chloride is much smaller than the corresponding product for sodium chloride, owing to the smaller solubility and also to some extent to the smaller degree of ionisation. The addition of barium chloride contributes therefore very little chloridion to the salt solution, and consequently the natrion is not greatly reduced in quantity by precipitation. The effect is all the smaller, because the great concentration of chloridion. from the sodium chloride necessitates a very small concentration of barion, in order that the ionic solubility product of barium chloride may not be excecded, i.e. the solubility of barium chloride in saturated salt solution is much lower than in water, so that very little of the salt passes into solution to displace the sodium chloride. The sodium salts of aromatic sulphonic acids are often obtained pure from solution by the addition of sodium chloride or caustic soda. These salts are, as a rule, much less soluble than either sodium chloride or sodium hydroxide, and are therefore thrown out of solution in great part when natrion from strong brine or solid caustic soda is added. Organic acids may often be readily precipitated from aqueous solution by the addition of hydrogen chloride, the hydrion being here the active substance. Even a comparatively soluble acid like sulphocamphylic acid may be thrown out of its aqueous solution almost entirely by saturating with gaseous hydrogen chloride. XXIX 343 APPLICATIONS OF DISSOCIATION THEORY The salting out or graining of soap is one of the oldest applications of the principle under discussion. When a fat is saponified with a moderately dilute solution of caustic soda, the whole gradually passes into solution, and to produce a good soap it is necessary to separate the sodium salts of the fatty acids which constitute it from the glycerol formed during the saponification and the excess of caustic alkali that was employed. The separation can be easily effected by the addition of common salt or a strong brine. The sodium salts of the higher fatty acids are comparatively slightly soluble in water, and thus the addition of a soluble salt like sodium chloride throws them almost completely out of solution as a curdy mass. If the fat is saponified by a strong solution of caustic soda, there may be enough sodium in the form of natrion in this solution to throw the soap out as it is produced. The precipitation of soaps in this way is not wholly caused by the ionic effect. Since the soap is partly in colloidal solution there is to some extent precipitation of colloid by means of the added electrolyte (Chap. XXI.). }} If a fat is saponified with potash instead of with soda, a solution. of the potassium salts of the fatty acids is obtained. This solution, if strong enough, assumes on cooling a consistency expressed by the name of "soft soap. Soft soaps are not in general salted out as such, but are used while still mixed with glycerol and excess of alkali. If we wish to obtain the potassium salts free from these admixtures, we may do so by adding to the solution a strong solution of potassium chloride. The potassium salts of the fatty acids then separate out as a somewhat gelatinous mass, containing a consider- able quantity of water and still retaining the characteristics of a soft soap. In the old process of manufacturing hard or soda soaps, the fat was saponified with potash, as that alkali was most readily obtainable from wood ashes. To the solution obtained on saponification sodium chloride was then added. The effect of this was to throw out, not a soft potash soap, but a hard soda soap, in consequence of double decom- position taking place between the sodium chloride and the potassium salts, whereby potassium chloride and the sodium salts were produced. These sodium salts were then thrown out by the excess of sodium chloride. The addition of sodium chloride to a solution of the potas- sium salts could not to any considerable extent throw them out as such from solution, for these substances have no common ion. If we consider that kalion, natrion, chloridion, and the anions of the fatty acids exist simultaneously in the solution, it is evident that these free ions may combine in pairs in two ways, the natrion which was originally associated with chloridion having now the choice of combining with the anions of the fatty acids. This actually occurs, because the ionic solubility product of the sodium salts of these acids is much less than the corresponding magnitude for the potassium 344 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. salts, the sodium salts being much less soluble. Under the given conditions the actual ionic product of the natrion and the anions exceeds the solubility value, and thus a portion of the sodium salts falls out of the solution, the potassium remaining behind mostly as potassium chloride. If we consider the saturated aqueous solution of any salt, the addition of a non-electrolyte will in general affect the solubility to a much smaller extent than the addition of an electrolyte containing one ion in common with the substance originally dissolved. There will, of course, always be some effect, for the solvent is changed by the addition of the foreign substance, and any change in the nature of the solvent has its effect on the solubility of the substance considered. If the substance added is not itself a solvent for the original substance the general effect will be slight precipitation from the saturated aqueous solution. If we add a small quantity of an electrolyte which contains no ion in common with the original electrolyte dissolved, the effect is generally to diminish the actual value of the ionic product of the latter, and thus increase its solubility. This comes about because the addition of the second electrolyte produces double decomposition, so that some of the original ions go to form part of un-ionised molecules. If double salts or acid salts may be formed, of course the equilibrium is thereby greatly complicated, and it is impossible to tell without further information how the addition of the new substance may affect solubilities. The addition of a salt of a weak acid to the acid itself is frequently carried out in the operations of analytical chemistry in order to reduce the strength of the acid, i.e. in order to reduce the con- centration of hydrion (p. 310). If we add to a solution of ferrous sulphate a large excess of sodium acetate, it is comparatively easy to precipitate the iron as ferrous sulphide by means of sulphuretted hydrogen. If no sodium acetate is added, the precipitation does not take place, a dark coloration at most being effected. The old ex- planation of this difference was that by the addition of sodium acetate to the ferrous sulphate solution the acid liberated by the hydrogen sulphide according to the equation FeSO4 + H₂S=FeS + H2SO4 would at once act on the sodium acetate, producing sodium sulphate and acetic acid :- H₂SO4 + 2NаC₂H₂O₂= 2HC₂H¸O2 + Na2SO4. 3 The ferrous sulphide was supposed to be easily soluble in sulphuric acid, but insoluble in the acetic acid. This explanation is insufficient, however, for if we take ferrous sulphate and add to it no more sodium acetate than is necessary for double decomposition, the precipitation XXIX APPLICATIONS OF DISSOCIATION THEORY 345 takes place to a slight extent only; and if we now add acetic acid, the precipitate originally formed dissolves up. Acetic acid, therefore, in these circumstances dissolves ferrous sulphide. If, however, we now add more sodium acetate to the same solution, the ferrous sulphide is reprecipitated. The reprecipitation is consequently due to the reduc- tion of the degree of ionisation of the acetic acid by the addition of sodium acetate, and not to double decomposition alone, as was formerly supposed. Ammonium chloride acts in a similar way on a solution of ammonium hydroxide, greatly reducing the ionisation and therefore the strength of the base. This fact is taken advantage of in the analytical separation of iron, chromium, and aluminium from zinc, manganese, magnesium, etc. The trihydroxides Fe(OH)3, Cr(OH)3) Al(OH), are extremely weak bases, whilst the dihydroxides Zn(OH)2, Mn(OH)2 Mg(OH)2, etc., are comparatively strong. If ammonia solution is added to any salt of these metals, the hydroxide is in each case precipitated, although the precipitation may sometimes only be partial. If, however, excess of ammonium chloride (or other ammonium salt) is present at the same time, the strength of the ammonium hydroxide as a base is so greatly reduced that although it is still capable of completely precipitating the very feeble trihydroxides, the salts of the stronger dihydroxides remain undecomposed. It must be borne in mind, however, that this is not the only mode of action of ammonium chloride, for it has also the property of favouring the formation of complex ammoniacal ions, and thus preventing pre- cipitations which would otherwise occur on the addition of ammonium hydroxide. It was stated on page 320 that the addition of any highly ionised electrolyte, say sodium chloride, would have practically the same effect on the ionisation of acetic acid as the addition of a highly ionised acetate. Whilst this is true, it must not be supposed that the addition of sodium chloride to a solution of a ferrous salt, which has enough acetic acid in it just to prevent precipitation by sulphuretted hydrogen, will have the same ultimate effect as an equivalent quantity of sodium acetate. The degree of ionisation of the acetic acid is indeed diminished in the same ratio as before, but hydrochloric acid is simultaneously formed by double decomposition, and as this is highly ionised, the amount of hydrion in the solution, after the addition of the sodium chloride, is rather greater than before, so that there is an actual increase in the total active acidity of the solution. Precipitation of the sulphide therefore does not occur. In analytical chemistry it is a common practice to wash a precipi- tate with the fluid precipitant, especially in quantitative operations. The theoretical basis of this is, of course, that the sparingly soluble. precipitate has an ion in common with the precipitant, so that it is less soluble in the solution of the latter than in pure water. If other 1 346 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. circumstances permit, therefore, it is advisable to wash with a diluted solution of the precipitant rather than with water alone. When solutions of two electrolytes are brought together, and it is. theoretically possible by double decomposition to obtain a substance which is insoluble, then, in general, the double decomposition actually takes place. Thus when any sulphate is added to any barium salt, double decomposition invariably takes place, barium sulphate being deposited. From the point of view of the dissociation theory the explanation is at once apparent. The solubility product of the ions of barium sulphate is very small, and as soon, therefore, as barion and sulphanion are brought together in quantity to exceed this solubility limit, barium sulphate falls out. Now all barium salts yield ions freely in solution, and all soluble sulphates behave in like manner. Precipitation of barium sulphate therefore invariably occurs, unless, indeed, the solu- tions are so extremely dilute that the solubility product is not reached. Exceptions to this rule are only encountered when there is the possibility of the existence or formation of complex ions, or when a feeble acid or base is one of the pair of substances. If we add a solution of silver sodium thiosulphate to a solution of sodium chloride, there is the theoretical possibility of the action. NaAgS2O3 + NaCl = Na2S2O3 + AgCl, but this action does not occur because there is so little argention yielded by the sodium silver thiosulphate that the solubility product of silver chloride is not exceeded, practically all the silver in solution being in the form of the complex ion AgS,O,' (p. 340). For the same reason, when cyanides or ammonia are present, there is frequently no pre- cipitation where such might be expected by double decomposition. If sulphuric acid itself is added to a barium salt, barium sulphate is precipitated as readily as if a neutral sulphate had been employed. It is different in the case of tartrates. If calcium chloride is added to sodium tartrate solution, a precipitate of calcium tartrate is formed. Should tartaric acid, however, be used instead of sodium tartrate, no precipitate is produced. The reason for this difference in behaviour is not far to seek. When sodium tartrate is employed, tartranion is abundantly present in the solution, the sodium salt being highly ionised, and the solubility product of calcium tartrate is far surpassed. When hydrogen tartrate is employed, we have comparatively little tartranion, for tartaric acid is not highly ionised. The ionic product of calcion and tartranion therefore falls short of the solubility product. for calcium tartrate, and there is no precipitation. For the same reason, some metallic solutions are easily precipitated by an alkaline sulphide and not at all by sulphuretted hydrogen. In general, it may be said that when all the substances concerned are highly ionised in solution, and complex ions are excluded, the precipitation occurs when there is the theoretical possibility of it; while if one of the reacting XXIX 347 APPLICATIONS OF DISSOCIATION THEORY + substances is feebly ionised, the precipitation may only take place to a limited extent or not at all. The precipitation may be almost perfect, even when a feebly ionised substance is concerned, if the solubility product is vanishingly small, i.e. if the substance is scarcely at all soluble. This is the case, for example, with silver sulphide, so that sulphuretted hydrogen, although very sparingly dissociated, easily precipitates this substance from a solution of silver nitrate. In intimate connection with the formation of precipitates on mixing electrolytic solutions, there is the solubility of precipitates in solu- tions of electrolytes. It must be borne in mind that the so-called insoluble substances are merely sparingly soluble substances, and that the presence of electrolytes in the solvent water only alters the solubility by affecting the concentration of the ions which by their union might form the precipitate. Kohlrausch, from measurements of the electric conductivity of the saturated solutions, determined the solubility of some of the commoner "insoluble " substances, and his results are given in the following table, the solubility being expressed in parts per million, i.e. milligrams per litre, at 18° :— Silver chloride Silver bromide Silver iodide • · Mercurous chloride Mercuric iodide Calcium fluoride Barium sulphate Strontium sulphate Calcium sulphate Lead sulphate Barium oxalate Strontium oxalate Calcium oxalate · Barium carbonate Strontium carbonate Calcium carbonate Lead carbonate Silver chromate Barium chromate Lead chromate • · • • Magnesium hydroxide. ་ · • 1 • ་ • Solubility. 1.34 0.107 0.0035 (2) 1 0.3 16 2-3 114 2023 41 $6 46 5.6 24 11 13 3 CO 25 3.5 0.1 9 The solubility product of silver chloride in water is very small, corresponding to the very slight solubility of the salt. If we have the solid salt in presence of water, and add nitric acid to the solution, we disturb the equilibrium very little. The dissolved silver chloride is almost entirely ionised, and the silver nitrate and hydrogen chloride, which might be formed from it by the action of the nitric acid, would 1 This value is probably too high owing to hydrolytic dissociation. Another method gives 0.8 mg. per litre. 348 CHAP. INTRODUCTION TO PHYSICAL CHEMISTRY ; be likewise almost entirely ionised. The addition of nitric acid, therefore, does not appreciably affect the argention and chloridion in the solution, and therefore is without influence on the solubility of the silver chloride. Consider, on the other hand, the effect of the addition of hydrochloric acid on the solubility of calcium tartrate. The calcium tartrate in the aqueous solution is highly ionised, but the addition of hydrochloric acid at once liberates tartaric acid, which is only slightly ionised in the presence of the highly ionised calcium chloride, etc. The concentration of the tartranion is therefore much reduced, and the ionic product falls below the solubility product, i.e. the solution becomes unsaturated with respect to calcium tartrate. More of this salt, therefore, dissolves up, and the process of solution goes on until the ionic product once more reaches the solubility pro- duct. It is evident from these considerations that a given quantity of hydrochloric acid will not dissolve an unlimited amount of calcium tartrate. If, however, we take a sufficient excess of acid, a given amount of calcium tartrate can always be entirely dissolved. In general we may say that strong acids in aqueous solution will not appreciably dissolve salts of equally strong acids, or even of acids nearly as strong, for any double decomposition that takes place in solution does not then appreciably affect the concentration of the ions which regulate the solubility. On the other hand, strong acids will in general easily dissolve insoluble salts of weak acids, for then the weak acid is liberated, which being slightly ionised, reduces the ionic. product, with the result that the solid dissolves to restore the solution equilibrium. Sometimes, when the solubility product of the salt of the weak acid is excessively small, as it is in the case of silver sulphide, even an equivalent of a strong acid like nitric acid will not dissolve an appreciable quantity, for the solubility product is soon reached when silver nitrate and hydrogen sulphide begin to accumulate. in the solution. Weak acids, as we might expect, do not dissolve the "insoluble" salts of stronger acids. Whilst calcium oxalate is freely soluble in hydrochloric acid, it is almost insoluble in acetic acid. In the first case the concentration of oxalanion in the aqueous solution is diminished by the addition of the hydrochloric acid, whereas in the second it is not sensibly affected. Calcium oxalate, therefore, must dissolve in hydrochloric acid solution to restore the solubility product. When acetic acid solution is used, the ionic product scarcely departs. from the solubility value, so that little or no calcium oxalate need pass into solution. As has already been indicated, the number of ions of a given kind. in a solution may be greatly altered by the formation of complex ions. If to a saturated solution of silver chloride in contact with the solid there is added a quantity of potassium cyanide, the free cyanidion unites with a portion of the argention to form the complex ion AgC,N,'. The ionic product of silver and chloride ions, therefore, falls below the 2 2 XXIX APPLICATIONS OF DISSOCIATION THEORY 349 equilibrium value, i.e. below the solubility product, with the result that silver chloride must dissolve in order to restore equilibrium. If the quantity of cyanide added is small, the silver chloride need not dissolve wholly, but if a sufficient excess of cyanide is employed, the solution of the silver chloride will be complete. Should the solubility product be extremely small, as is the case with silver sulphide, potas- sium cyanide has only a slight solvent action, and the silver sulphide dissolved is easily reprecipitated on addition of potassium sulphide. The solvent action of sodium thiosulphate or ammonia on “insoluble silver compounds is similar in origin, the increased solubility being due to the formation of complex ions with corresponding disappear- ance of argention. It will be noticed that the solubility of silver chloride, bromide, and iodide in water (given in the table on p. 347) is the same as the order of their solubility in ammonia, as an applica- tion of the above theory would lead us to expect. "} The theory of electrolytic dissociation affords some assistance in understanding the action of the indicators used in acidimetry and alkalimetry. The indicators are themselves acid or alkaline in nature, but are necessarily very feeble compared with the acids or alkalies whose presence they indicate. Their action depends on a change of colour which they undergo on neutralisation. Phenol-phthalein, for example, is a substance of very weak acid character, being in aqueous solution almost entirely un-ionised and colourless. If, however, we add a strong alkali such as sodium hydroxide to it, the sodium salt is formed and imparts an intense pink colour to the solution. All the soluble salts of phenol-phthalein are thus coloured,¹ and the colour is of the same intensity in equivalent solutions if these solutions are very dilute, so that we are justified in concluding in terms of the dissociation hypothesis, that the colour is due to the anion from phenol-phthalein, as this is the substance common to all dilute solu- tions of salts of phenol-phthalein. The undissociated substance has no colour. Let us consider what happens as we titrate a solution of an acid, using phenol-phthalein as an indicator. In presence of the acid, the indicator, being a very feeble acid, is even less ionised than it would be in pure water, and consequently no colour is per- ceptible. As soon, however, as the acid originally present in the solution is neutralised and a drop of alkali in excess is added, the corresponding alkaline salt of phenol-phthalein is formed, i.e. the anion from the phenol-phthalein is produced, and the solution at once assumes the pink tint characteristic of it. In order to have a sharp indication of the neutral point, it is necessary first that the acid to be titrated should be considerably 1 Phenol-phthalein is an example of a "pseudo-acid," i.e. an acid the salts formed from which have not the same chemical constitution as the acid itself. Thus when soda is added to a solution of phenol-phthalein the pink salt obtained is not, strictly speaking, a salt of the colourless phenol-phthalein at all, but a salt of an isomeric acid. This circumstance, however, in no way affects the explanation given in the text. 350 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. stronger than phenol-phthalein itself, and that the alkali should be a strong alkali. If the acid to be titrated is so feeble that its salts even with strong bases suffer hydrolytic dissociation in aqueous solution, it is obvious that phenol-phthalein is incapable of sharply indicating the neutral point, for long before sufficient alkali for complete neutralisa- tion has been added, the salt formed will begin to split up into free acid and free base, part of which will neutralise the phenol-phthalein, and thus produce a faint pink colour which will gradually deepen in intensity as the addition of alkali progresses. Carbolic acid, and other phenols, therefore, cannot be titrated with alkali and phenol-phthalein, on account of the hydrolysis which their salts suffer (cp. p. 330). Polybasic acids, too, very frequently form normal salts which are partially hydrolysed in aqueous solution, and with them also no definite indication of the neutral point can be obtained. Thus the ordinary sodium phosphate, i.e. di-sodium hydrogen phosphate, although formally an acid salt, has an alkaline reaction to phenol- phthalein on account of its slight hydrolysis. Phenol-phthalein roughly indicates neutrality in the case of carbonic acid when sodium. hydrogen carbonate exists in the solution, and gives a strong pink colour with the normal sodium carbonate. Since carbonic acid thus behaves as an acid to phenol-phthalein, i.e. since carbonic acid is a stronger acid than phenol-phthalein, it must be excluded from the alkali with which acids are titrated. This is best done by using baryta as the alkali, and keeping it protected from the carbonic acid of the atmosphere. Any carbonic acid which may have been originally present settles down as barium carbonate, and the clear liquid is therefore free from this source of disturbance. If the base employed is not a strong base, the indication in this case also is uncertain owing to hydrolysis. Thus ammonia should never be used in titrations with phenol-phthalein as indicator, for the ammonium salt of phenol-phthalein, being the product of the union of a weak base with a very weak acid, is hydrolysed in aqueous solution, and thus the neutrality point is not sharply indicated, even though the salt of ammonia with the acid originally in the solution undergoes no hydrolysis. Looked at from another point of view, the ammonium salt formed by the neutralisation of the acid so diminishes the strength of a small amount of ammonia added in excess that the ammonia is unable to ionise the phenol-phthalein sufficiently to give the deep pink colour requisite for a sharp indication, and thus more ammonia must be added, the colour gradually deepening in intensity. Phenol-phthalein then forms a good indicator when weak acids are titrated with strong bases. The application of the dissociation theory to explain the action of other indicators is not so simple as in the case of phenol-phthalein, because these indicators are usually amphoteric, i.e. acidic or basic according to circumstances, and thus form two series of salts, one XXIX APPLICATIONS OF DISSOCIATION THEORY 351 series with strong acids and another with strong bases (cp. p. 333). There may, therefore, be more than one kind of coloured ion in solution, besides, perhaps, coloured un-ionised indicator, so that the interpretation of the actual colours of the different solutions on the dissociation theory is sometimes difficult. Many applications of the electrolytic dissociation theory to ordinary laboratory work will be found in W. OSTWALD, The Scientific Foundations of Analytical Chemistry. J. STIEGLITZ, Qualitative Chemical Analysis, vol. i., Theoretical Principles and their Application (New York, 1912). CHAPTER XXX ELECTROMOTIVE FORCE IN the chapter on the kinetic theory (p. 94) it was indicated that the equilibrium of a solid and its saturated solution might be conceived ast the balancing of two opposed forces-the solution tension or tendency of the solid to pass into the dissolved state on the one hand, and the osmotic pressure exerted by the dissolved substance on the other. A similar conception has been applied by Nernst to the equilibrium existing between a metal such as zinc, and a solution containing the same metal as a positively charged ion, e.g. a solution of zinc sulphate. To the metallic zinc may be attributed a certain electrolytic solution- tension, or tendency to pass into solution in the ionic state. This tendency is balanced by the osmotic pressure of the positively charged zinc, or zincion, in the solution, and a certain osmotic pressure of zincion will exist at which metallic zinc will neither enter the solution nor be deposited from it. This definite osmotic pressure which just balances. the solution tension of the zinc is spoken of as the electrolytic solution pressure of zinc. It is obvious, however, that there must be a profound difference. between this equilibrium of a metal and its electrolytic solution, and the equilibrium between a solid salt and its solution. In the latter case, if the solubility value of the osmotic pressure is not reached, salt is seen to dissolve, and if the solubility value is exceeded, salt is seen to be deposited. Now in the case of zinc and zinc sulphate, the metal may be brought into contact with an aqueous solution of the salt of any concentration whatever, i.e. of any osmotic pressure of zincion, and yet there is neither perceptible passage of zinc into solution. nor deposition of zinc from solution. The reason is not far to seek. When the salt passes into solution, positively charged ions are pro- duced, but at the same time negatively charged ions in equivalent quantity are also formed, and pass into solution with them. When metallic zinc, on the other hand, passes into solution as positively charged zincion, no negative ion is formed at the same time, so that the solution becomes electrostatically charged with positive electricity, 352 CH, XXX ELECTROMOTIVE FORCE 353 and the metallic zinc in consequence negatively charged. The electrical charges prevent the uniform diffusion throughout the solution of the zincion produced from the metallic zinc, and there is formed at the surface of contact of the metal and the solution an electrical double layer (negative on the metal, positive in the solution) resembling a charged condenser. The electrical stress resulting from this double layer serves to compensate the difference between the actual osmotic pressure of the zincion and the electrolytic solution-pressure of the zinc. Should the osmotic pressure of the metallion exceed the electrolytic solution pressure of the metal, as is the case, for example, with copper in a solution of copper sulphate, metallic copper will be deposited on the surface of the electrode. The electrode therefore becomes posi- tively charged, and the solution negatively charged, the negative charge being carried by the sulphanion which forms one side of the electrical double layer at the surface of separation of metal and solution. Here again we have an electrical stress compensating the difference be- tween the actual osmotic pressure and the electrolytic solution pressure. When a metal then is brought into contact with one of its salts, an electromotive force is set up at the surface of contact, which, when the ionic osmotic pressure is less than the solution pressure, would cause a current (positive electricity) to flow from metal to solution, and when the ionic osmotic pressure exceeds the solution pressure, would cause a positive current to flow from solution to metal, provided the conditions are such as to permit the establishment of a current in the system. Since the electrical double layer is only of molecular thickness, the solution or deposition of an extremely small quantity of metal is sufficient to establish equilibrium, so that no noticeable chemical change occurs when a metal is immersed in an aqueous solution of one of its salts, unless indeed the metal, like sodium, decomposes water. The magnitudes representing the electrolytic solution pressures of metals are generally inconveniently large or inconveniently small. Thus, for the calculated solution pressure of zinc we have a magnitude of the order of 1018 atmospheres, and for the solution pressure of silver 10-15 atmospheres. Instead of using them, therefore, we shall deal with the differences of potential between metals and solutions containing the corresponding metallions, i.e. with the electromotive force (due to the difference between the solution pressure of the metal and the actual osmotic pressure of the metallion) which resides at the surface of contact between metal and solution. These electrode potentials are susceptible of direct measurement and are of great practical importance, as the electromotive force of all galvanic combinations is principally due to them. In galvanic cells two metallic conductors are immersed in one or more electrolytic conductors, and it is at the surface of separation of the metallic and electrolytic conductors that the chief electromotive force resides. The other sources of electromotive 2 A 354 CHAP. INTRODUCTION TO PHYSICAL CHEMISTRY force are at the junction of the two metallic conductors, and at the junction of the electrolytic conductors. The electromotive force at the metallic junction does not, however, exceed a few millivolts for most metals, and that at the junction of the electrolytes is also small, if salt solutions alone are used, for these give rise to ions which do not differ greatly in velocity, and it is in general to difference in the speeds of the ions that the electromotive force at the contact of two electrolytic liquids is due. These sources of electromotive force then may be frequently neglected for practical purposes, and the metal- solution contact regarded as the sole seat of the electromotive force of the galvanic combination. With regard to the practical determination of electrode-potentials, it is evident that we cannot construct a system connected with a measuring instrument, whether galvanometer or electrometer, with E a G FIG. 55. π less than two metallic electrodes in contact with the solution or solutions. In actual experiment we therefore of necessity deal with two electrode-potentials, and measure their difference. The method of measurement of electromotive force generally adopted in physico-chemical laboratories is as follows:-The wire ab (Fig. 55) is stretched along a divided scale and provided with a sliding contact c. The two ends of the wire are connected with the opposite poles of a battery E of constant electromotive force (say one or two accumulators) greater than the electromotive force to be measured. The combination whose electromotive force is to be determined is connected with the sliding contact c on the one hand, and on the other with one end of the measuring wire through the galvanometer (or electrometer) G. The contact is moved along the wire until the Then a cell of known electro- measuring instrument indicates zero. motive force, По is substituted for the unknown combination, and the sliding contact again moved until zero is indicated. The lengths of the wire read off are now proportional to the electromotive forces of the unknown and known cells, i.e. ac : ac = π :π0· C XXX ELECTROMOTIVE FORCE 355 As cell of known electromotive force the Clark element (E.M.F. = 1·433 volt at 15°) was formerly almost exclusively employed, but this has been largely superseded by the standard Weston cadmium cell, the international value for which is 10183 volt at 20°. Its temperature coefficient is only + 0·00004 (20° – t°). The electrometer most frequently employed in physico-chemical laboratories is the form of Lippmann's capillary electrometer shown in Fig. 56. The surface tension of a liquid metal such as mercury in contact with an electrolytic solution depends on the difference of potential between the mercury and the solution. If there- fore the mercury is in contact with dilute sulphuric acid as at a in the figure, there is a movement of the mercury in the capillary tube if the mercury is made to assume a different potential from that which it would normally possess when in contact with the electrolyte. Suppose that the mercury in the bulb is connected with one terminal of a galvanic combination, and the mercury in ab with the other terminal, then if there is a potential difference between these terminals the meniscus in the capillary will change its position. The sliding contact c (Fig. 55) is therefore moved until no deflection of the mercury surface takes place on closing the circuit. If the terminal which is connected with the mercury in the bulb passes downwards through the tube of the bulb it must be protected from contact with the sulphuric acid by being sealed within a glass tube throughout the whole length immersed in the acid. Instead of an electrometer, a sensitive moving-coil galvanometer with mirror and scale may be conveniently employed as a zero- instrument. By suitably combining the various electrodes and electrolytic solutions, and measuring the electromotive force of the combinations, it is possible to obtain the relative values of electrode potentials such as those given in the table on p. 359. The absolute values of the electrode potentials are somewhat uncertain owing to the difficulty of obtaining an electrode which has the same potential as the solution in which it is immersed. It is generally assumed that this condition is fulfilled with the mercury dropping electrode. Here the mercury issues from a jet in a fine stream, which breaks up into drops just after it comes into contact with a solution of sulphuric acid. The charge taken up by the mercury is carried off by the rapid succession of drops, so that the still unbroken stream of mercury may be assumed to have the same potential as the solution with which it is in contact. If then we combine this dropping electrode (of assumed zero electrode e FIG. 56. ееее 356 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. potential) with another metallic electrode immersed in the same solu- tion, the electromotive force of the combination is equal to the difference of potential of the second metal and the acid. Knowing now this difference of potential, we may proceed to combine this second electrode with other electrodes and obtain the electrode potentials of these. As has been indicated, these absolute potentials are of little practical importance, since we are usually concerned in measurements with the difference between two of them, so that we only need accurate knowledge of their relative values to obtain this difference. PA The calomel electrode is largely made use of as a standard normal electrode of known potential difference with which other electrodes may be combined for measurement. One form is constructed as shown in Fig. 57. The bottom of the tube is covered by a layer of mercury above which is placed a layer of calomel and then a normal solution of potassium chloride. Metallic connection with the mercury is made by a platinum wire sealed through the end of a glass tube. The side tube with the stopper is filled with normal potassium chloride solution and dipped into the appropriate liquid, after which the stopper may be opened. The absolute potential of this calomel electrode is generally given as 0.560 volt, although recent experiments would appear to show that a value of −0·520 is more probable. FIG. 57. Its chief advantage lies in the fact that it can be combined with very many other metallic electrodes in neutral solutions of their salts. The speeds of the ions of neutral salts do not differ very much from each other, and in consequence the electromotive force developed at the junction of solutions of different salts is comparatively small. On the other hand, if the liquids are acid or alkaline they contain hydrion or hydroxidion, both of which move at much greater rates than the "neutral" ions (ep. p. 256), and thus develop considerable electromotive forces when brought into contact with other solutions. The reason why there is an electromotive force at the junction of solutions with ions of different velocities, or even of different solutions of the same substance whose anion and cation move at different rates, was pointed out by Nernst, to whom the application of the osmotic and ionisation theories to electromotive force is due. Consider two solutions of hydrochloric acid of different concentrations in contact with each other. The relative speeds of migration of the ions may be taken as their relative rates of diffusion in. solution (ep. p. 242). The hydrion of the stronger solution will therefore tend to diffuse into the weaker solution at a greater rate than the chloridion. But XXX ELECTROMOTIVE FORCE 357 if the diffusion actually takes place, the dilute solution will become positively charged from the arrival of hydrion, and the concentrated solution will become negatively charged from the excess of chloridion remaining. There will therefore be set up a difference of potential between the two solutions, which would not exist if anion and cation had the same rate of migration, for then the ionised substance would diffuse as a whole without disturbance of electric neutrality. A practical method of eliminating this diffusion-potential, to a great extent at least, is to connect the two solutions by a concentrated solution of a salt whose anion and cation have as nearly as possible the same rate of migration. The salts used for this purpose are usually potassium chloride, or ammonium nitrate (cp. table, p. 242), the latter having the advantage of very great solubility. In order that an electrode may be used as a normal electrode, it must be of the reversible or non-polarisable type; that is, its electromotive force must be unaffected by the passage of a slight current through the electrode in either direction. This condition is fulfilled with many metals immersed in solutions containing the corre- sponding ion. Thus if we consider a copper electrode in a solution of copper sulphate, the passage of a current from the solution to the metal merely deposits metallic copper, whilst the passage of a current from the metal to the solution is accompanied by the solution of metallic copper as dicuprion. The only change that takes place then is a slight variation in the concentration of the electrolytic solution (in particular of the dicuprion with which the metallic copper is in equilibrium), and this has practically no effect on the potential of the electrode. An electrode of this kind is said to be reversible with regard to the cation. The calomel electrode, on the other hand, is reversible with regard to the anion, i.e. the only effect caused by the passage of a current one way or other is to make a slight variation in the concentration due in the first instance to the chloridion. Thus if a current passes from solution to metal, metallic mercury is deposited from the mercurous chloride, the chloride radical originally united with the metal passing into solution as chloridion to neutralise the cation brought by the positive current. Similarly if the current passes from metal to solution, metallic mercury is converted into mercurous chloride, the necessary chloride radical being derived from the chlorid- ion of the solution. The following schemes may serve to elucidate the process: 1. Hg Hg 2. Hg Hg Hg 1. Hg Hg 2. Hg 1 FIRST CASE 1 HgCl HgCl Cl Hg SECOND CASE CIHg CIHg HgCl HgCl HgCl K' CV'K' Cl′ K' Cl CI' K CIK' CI' K' CI' K' | Cl K CV K' Cl' K' → K' Cl′ K' Cl' . . Κ ← K 358 СНАГ. INTRODUCTION TO PHYSICAL CHEMISTRY The normal Clark and cadmium cells are combinations of two such reversible electrodes, and thus have constant electromotive force. The Clark cell consists of two poles or electrodes, one of mercury, the other of zinc. The zinc rod dips into a magma of zinc sulphate and water, which rests on a layer of a paste of the practically insoluble mercurous sulphate with saturated zinc sulphate solution, beneath which again is a layer of mercury. The mercury which is the other pole of the combination is connected with its terminal by means of a platinum wire immersed in it. The cell may be represented thus: Hg | Hg₂SO4 | ZnSO4 (saturated) | Zn A frequent type of cadmium cell is represented in Fig. 58. One Sealing Wax- Cork Paraffin- Air Hg + Solution+Cd SO4, H2O Paste FIG. 58. Cd + Hg O limb of the H-shaped tube contains mercury, and a paste of mercurous sulphate, mercury and cadmium sulphate. The other limb contains cadmium amalgam (usually of 13 per cent); and the rest of the tube, including the horizontal connecting portion, contains a magma of crystals of the hydrate CdSO, 8/3 H2O and their saturated solution. The cell may be represented as follows:- Hg | Hg₂SO4 | CdSO (saturated) | Cd amalgam. 4 From measurements of the electromotive force of suitable galvanic combinations the following table of electrode potentials has been con- structed. The numerical values are the potential differences in volts between the metals and normal solutions of the corresponding metal- lion referred to the normal calomel electrode as -0.56 volt. In the case of metals which decompose water at the ordinary temperature the direct measurement is of course impossible, but for the sake of completeness numbers for these derived indirectly by calculation from thermochemical data have been given within brackets. The calculation is somewhat uncertain, but the numbers probably give at least the correct relative positions of these metals in the table. The values for magnesium and aluminium are also very doubtful. XXX 359 ELECTROMOTIVE FORCE ELECTRODE POTENTIAL IN VOLTS FOR NORMAL SOLUTIONS OF CATIONS (+2·9) Nickel (+2·5) Tin (+2·4) Lead (+2·3) (+1·9) +1.5? +1.0? +0·80 +0.50 +0·15 +0·07 +0·045 -0.045 Potassium Sodium Barium Strontium Calcium Magnesium Aluminium Manganese Zinc Cadmium Iron Thallium Cobalt K' Na Ba Sr** Ca Mg** Al Mn Zn** Ca** Fe** TI Co" 40 Hydrogen Copper Arsenic Bismuth Antimony Mercury Silver 1 Palladium Platinum Gold Ni" Sn** Pb H' Cu** As Bi Sb Hg Ag Pa" Pt Au*** -0.05 -0.1? -0.12 -0.27 -0.60 -0.6? -0.7? -0.8? - 1.02 - 1.04 - 1.1? -1.2? -1.4? The application of the preceding table may be seen from the following instance. If a rod of zinc is immersed in a normal solution of zincion, an electromotive force of 0.50 volt is produced at the surface of contact which tends to drive a current from metal to solution. Similarly if a rod of copper is immersed in a normal solution of dicuprion, the potential difference produced tends to drive a current from metal to solution with a force of 0·60 volt, or, in other words, tends to drive a current from solution to metal with a force of 0·60 volt. If now we combine the two systems into a closed circuit by connecting the metal rods by a wire and bringing the solutions into electrical contact through the walls of a porous pot, which gives free passage to the ions but prevents mechanical mixing of the two solutions, a current will pass from metal to solution in the zinc compartment, from solution to metal in the copper compartment, and back from copper to zinc by the metallic connecting wire. The electromotive force in this circuit due to the contact of metals and electrolytic solutions is +0.50 +0.60 1·10 volt. But the passage of the current does not alter the general nature of the system, the only changes that take place being deposition of copper, solution. of zinc, and corresponding changes in the concentration of the electrolytes. The system will therefore continue to afford a current at a practically constant electromotive force, and may be used as a galvanic cell. It is practically the ordinary Daniell cell, the electro- motive force of which by direct measurement is 1.09 volt, in agreement with the foregoing value. If a cell is constructed with both metallic poles of the same metal, and each immersed in a solution of the same salt of the same con- centration, say zinc poles in zinc sulphate solution, the electromotive force at the two surfaces of contact will be of the same magnitude but oppositely directed, so that no current will pass through the circuit. Since, however, the electromotive force at the contact of metal and solution depends on the ionic concentration of the solution, if the 1 The cation of the mercurous salts is probably the dyad Hg" and not the monad Hg.. 360 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. system consists of two zinc rods immersed in zinc sulphate solutions of different concentrations, the oppositely directed electromotive forces will not exactly neutralise each other, and there will be a small differential electromotive force in the circuit due to the difference of concentration of the two zinc sulphate solutions. Such a cell is termed a concentration cell, and the electromotive force in the case of an electrolyte with univalent ions of equal velocities of migration is E = 0.057 log¹ volts at 15° (cp. Chap. XXXVI.). If C1 is ten Ca C2 times as great as c, log1 becomes equal to 1, and the electromotive force is therefore 0.057 volts. At 25° the value is 0.059 when one solution has ten times the ionic concentration of the other. The direction of the current is from the dilute to the concentrated solution, and the transference of the dissolved substance is such that the concentra- tions tend to become equal. The electrode potential at 15° between a monad metal and the normal solution of the corresponding metallion is thus 0.057 volt smaller than the electrode potential between the same metal and a decinormal solution of the metallion. The table of electrode potentials can therefore be easily converted from normal solutions to solutions of any other concentration. For cations of greater valency the increase of electrode potential for a tenfold increase 0.057 of dilution is where n is the electrovalency. Thus the electrode , n potential of zinc in a decinormal solution of zincion is 0.50 + = 0.53 volt. Since the electrode potentials of two metals of the same electro- valency, say zinc and copper, are affected to the same extent by change in concentration of the solutions with which they are in contact, it follows that the electromotive force of a Daniell cell, so far as it is derived from the contact of metal with solution, remains unchanged if the solutions of copper and zinc sulphates are of the same ionic concentration, no matter how that concentration may vary. The arrangement of the metals in the table of electrode potentials is usually called the electro-chemical list of the metals, and gives, with some exceptions, the order in which they replace one another from solutions of their salts. Thus magnesium turns out zinc from a solution of a zinc salt, zinc expels copper from a solution of a copper salt, copper displaces silver, and silver displaces gold. The position of hydrogen in the table should be noted. All the metals which precede it are capable of displacing hydrogen from a hydrogen salt, say hydrochloric acid, whilst those which follow it in the list can only displace the hydrogen from an acid under exceptional circumstances. It frequently happens that at the potential theoretically necessary 0.057 2 XXX 361 ELECTROMOTIVE FORCE to liberate hydrogen in a given solution at a given metallic electrode, no hydrogen gas is evolved, and even at potentials much higher than the theoretical value the actual production of hydrogen in quantity does not take place. The excess of the actual over the theoretical value of the potential is known as the overvoltage of the cathode and varies from metal to metal. Aluminium readily dissolves in dilute hydrochloric acid, but is practically unaffected by dilute oxygen acids such as sulphuric and nitric acids, even at the boiling point. A similar anomaly is met with in the action of aluminium on solutions of copper salts. Copper nitrate and copper sulphate may be brought into contact with aluminium without deposition of copper occurring, but if copper chloride is used, or if sodium chloride is added to the sulphate or nitrate solutions, precipitation of copper immediately follows. This anomalous behaviour is probably connected with the circumstance that metallic aluminium under many conditions becomes readily coated with a protective film of hydroxide or basic salt. The displacement of metals from their salts by hydrogen is not readily effected under ordinary conditions. A current of hydrogen bubbled through a solution of platinum chloride (chloroplatinic acid) will precipitate platinum, but similar treatment has no effect on copper sulphate, for example, although copper follows hydrogen in the list. If, however, the copper sulphate solution is exposed at 90° to the action of hydrogen at 100 atmospheres pressure, the copper is com- pletely precipitated in the metallic state. When a metal which precedes hydrogen in the electro-chemical list is brought into contact with the aqueous solution of a salt of another metal which also precedes hydrogen, two things may evidently occur. Take, for example, magnesium in a solution of zinc sulphate. Magnesium, even at the ordinary temperature, decomposes water with evolution of hydrogen. When brought into a solution of zinc sulphate therefore it may displace hydrogen from the water; but it can also displace zinc from the zinc sulphate. As a matter of fact both of these actions proceed simultaneously. Hydrogen is slowly evolved from the beginning, and zinc is seen to be deposited on the magnesium. As the action progresses the evolution of hydrogen becomes much brisker. This is apparently due to the zine and magnesium forming a couple (like the zinc-copper couple) which is much more active in decomposing water than either metal separately. Since the electrode potentials in the table are applicable to metals in normal solutions of the corresponding metallion, and become greater as the concentration of the metallion diminishes, it is evident that for a given difference in the table of electrode potentials the most favourable conditions for a metal A displacing a metal B from solutions. of its salts are that the concentration of the metallion A should be as small as possible and that the concentration of the metallion B should 362 CHAP. INTRODUCTION TO PHYSICAL CHEMISTRY be as great as possible. Magnesium, for example, decomposes water very slowly, the concentration of the hydrion in water being extremely small. If now the water is acidified, the action becomes much more vigorous owing to the increased concentration of the hydrion. The mag- nesium hydroxide formed by the action of magnesium on water is very sparingly soluble, yet owing to its comparatively great ionisation it not only furnishes magnesion in the solution but also greatly diminishes the concentration of hydrion by increasing the concentration of hydrox- idion, for the product of the concentration of hydrion and hydroxidion must remain constant (p. 325). By the addition of ammonium chloride to the water it is possible to counteract both of these retarding influences. In the first place the highly ionised magnesium hydroxide is replaced by ammonium hydroxide, which in presence of the large excess of ammonium chloride is scarcely ionised at all. Thus the concentration of hydroxidion is kept down, and the concentration of hydrion in a proportionate degree increased. In the second place a complex ammoniacal ion containing magnesium is produced, and the concentration of magnesion thus lowered. Magnesium accordingly gives a brisk evolution of hydrogen with a moderately concentrated solution of ammonium chloride. Both zinc and aluminium evolve hydrogen from solutions of the caustic alkalies. Here the concentration of hydrion must be extremely small, but this is more than compensated for by the practical absence of zincion and aluminion from the solution, the dissolved metals pass- ing into the anions of the zincate and aluminate which are produced. Aluminium will not expel zinc from an aqueous solution of zinc chloride, but if caustic soda is added to the solution of zinc chloride till the precipitate of hydroxide at first produced dissolves as sodium zincate, aluminium will then freely precipitate metallic zinc from the solution. This we may attribute to the circumstance that the concentration of aluminion in a solution of sodium aluminate is much smaller than the concentration of the zincion in a solution of sodium zincate, though both are vanishingly small. By removing a metallion from solution as a complex ion, it is possible to invert the position of two elements widely apart in the electro-chemical list. Thus metallic copper (in contact with platinum) will readily displace hydrogen from a solution of potassium cyanide, on account of the copper which passes into solution becoming a constituent of a complex cuprocyanidion (cp. p. 339), so that practically no cuprion exists in the solution at all. Similarly, and for the same reason, it is possible to throw out zinc from a zinc solution con- taining excess of potassium cyanide, by means of metallic copper. It is thus apparent that the displacement of one metal by another is often determined by the ratios of ionic concentrations, which may be themselves so small in absolute magnitude as to defy our ordinary means of detection. XXX ELECTROMOTIVE FORCE 363 ! The variation of the electromotive force with change in concentra- tion of the ions, is frequently utilised to determine very small ionic concentrations, otherwise inaccessible to measurement. Thus by measuring the potential of copper in a solution of a cuprocyanide, it is possible to estimate the vanishingly small concentration of the cuprion present in such a solution. As an example of the application of the measurement of electro- motive force to the determination of small ionic concentration, we may take the estimation of the solubility of silver chloride in decinormal potassium chloride solution. The following combination is set up- :- Ag | 0·1n AgNO3 | 0·1n KNO。 | 0·1n KCl saturated with AgCl | Ag 3 i.e. as one pole we have a silver electrode in decinormal silver nitrate, and as the other a silver wire coated with silver chloride (and thus analogous to the calomel electrode), in a decinormal solution of potassium chloride which has been saturated with silver chloride by the addition of a few drops of silver nitrate. To prevent the two solutions precipitating each other, a decinormal solution of potassium nitrate is interposed in a U-tube. The observed electromotive force is 0.45 volt at 25°. This is composed of the electrode potentials, and the diffusion potentials at the junctions of the various electrolytes. In this case the latter may be neglected as they fall within the limits of experimental error. We have thus for the difference of the electrode potentials at the two silver electrodes 0:45 volt, and this is due to the difference in the concentration of the silver ion in the silver nitrate solution on the one hand, and the silver chloride solution on the other. From the formula (p. 360) we obtain at 25° 0·45=0·059 log¹ Now C1 the concentration of Ag in decinormal silver nitrate is about 0.084 (cp. p. 263), so that 0·45=0·059 (log 0·084 - log c₂) c₂ = 1·95 × 10-9 normal. whence To derive from this solubility of silver chloride in decinormal potassium chloride the solubility of silver chloride in pure water, we may apply the rule of the constant solubility product. [Ag] x [Cl'] = constant. The value of this constant according to the above determination is [1·95 × 10-9] × [8·4 × 10-2]=1·64 × 10−10. For the solubility in pure water, the concentration of the two ions is the same and equal to the square root of this number, viz. 1·28 × 10-5. The saturated solution of silver chloride at 25° is therefore 1.28 x 10-5 364 INTRODUCTION TO PHYSICAL CHEMISTRY CH. XXX normal, and contains 1.82 mg. per litre, a value agreeing sufficiently with that derived by Kohlrausch from conductivity measurements at 18° (cp. p. 347). M. LE BLANC, Text-book of Electro-chemistry (1920). R. A. Lehfeldt, Electro-chemistry (1918). N. T. M. WILSMORE, "Electrode Potentials," Zeitschrift für physikalische Chemie, 35 (1900), p. 291. W. OSTWALD, ibid. p. 333. ¦ CHAPTER XXXI POLARISATION AND ELECTROLYSIS As long as we are dealing with reversible combinations, the passage of a feeble electric current through the system in no way alters the nature of the electrode or of the liquid surrounding the electrode, except in so far as it may effect a change in the concentration of the electrolyte, which has little effect on electromotive force. The electromotive force therefore of the combination is practically constant, and if we apply to the system an opposed electromotive force of a magnitude greater than the direct electromotive force of the cell, a current will pass through the system in the reverse direction to that which the cell itself would generate. Since a metal immersed in the solution of one of its own salts is in general a reversible electrode, we can, as we have already seen, get a reversible combination of zero electromotive force by immersing two strips of the same metal, say copper, in a solution of copper sulphate. The slightest electromotive force applied to this system will effect the passage of a current one way or other, copper being deposited on one electrode and dissolved from the other, and the current will continue to pass indefinitely if the concentration of the copper sulphate round the electrodes is maintained at the same value, say the saturation value at the experimental temperature. Suppose now we immerse platinum electrodes instead of copper electrodes in the copper sulphate solution. The electromotive force of the com- bination is still zero, but we find that we can get no current of any magnitude to pass until the electromotive force applied to the system exceeds a certain value. The reason for this is obvious. As soon as a current passes through the system, copper is deposited on one electrode and sulphanion is discharged at the other with production of sulphuric acid and oxygen. The electrodes and solutions round the electrodes are no longer the same as they were originally, for in place of two platinum plates immersed in copper sulphate solution we have now one platinum plate (the cathode) covered with a thin coating of copper, and the other platinum plate (the anode) partially charged with oxygen and surrounded by a mixture of copper sulphate and 365 366 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. sulphuric acid. A combination of this sort possesses an electromotive force of its own, which acts against the direct electromotive force and neutralises it if the direct force is of insufficient magnitude. The reverse electromotive force is called the electromotive force of polarisation, and is always encountered where the electrodes are non- reversible. When the direct electromotive force is of sufficient strength to overcome the electromotive force of polarisation, a steady current may be made to pass through the system. Oxygen will then be evolved at the anode, and sulphuric acid will accumulate around it. If now the direct current is interrupted, the electromotive force of polarisation will produce a polarisation current in the opposite direction to the original current, and this current will flow, at a rapidly diminishing rate, until equilibrium is attained. In the case just considered, one of the products of the continued action of the direct current, namely oxygen, escapes from the system, all but a small residue absorbed by the platinum, so that the polarisation current almost instantaneously falls off owing to the consumption of this small quantity of oxygen. Secondary cells, or accumulators, depend for their action on a polarisation current. In the ordinary type, the electrodes of the uncharged cell may be said to consist of lead and lead sulphate immersed in diluted sulphuric acid, the metallic lead acting as a conductor and a support for the lead sulphate. Such a combination of similar electrodes and one liquid possesses no electromotive force. When a current is passed through the system, however, the following action takes place:- Cathode Anode 2PьSO4 + 2H₂O = Pb + PbO2 + 2H2SO4 The lead sulphate on the cathode is converted into lead sponge and on the anode into lead dioxide, the concentration of the sulphuric acid increasing. Here all the products of the direct current, the charging current, are retained within the cell, and a very large amount of lead sulphate can in this way be converted into metal and peroxide. When the charging is complete, and the two electrodes are connected through an external conductor, the polarisation electromotive force of about two volts immediately starts a current, the discharge current, in the direction opposite to that of the charging current, and this goes on with very slowly diminishing electromotive force until a large proportion of the lead and lead peroxide have again become converted into lead sulphate, according to an equation which is the reverse of that given above, namely, Pb + PbO2 + 2H₂SO₁ = 2PьSO + 2H2O. 4 4 Here the polarisation current is maintained because the nature of the electrodes undergoes no important change while a very large quantity of material is being transformed. XXXI POLARISATION AND ELECTROLYSIS 367 3 The polarisation electromotive force of a combination may be investigated by closing the polarisation circuit through a measuring instrument simultaneously with the breaking of the primary circuit. The operation may be repeated as frequently as we please by the vibra- tion of a tuning-fork, which alternately makes and breaks the primary circuit at the same time as it breaks and makes the polarisation circuit. If the rate of vibration is sufficiently great this is tantamount to observing the electromotive force of a steady current of polarisation generated by a known steady primary current. It is occasionally of importance to investigate the anodic and cathodic polarisation separately. This may be done by introducing a third electrode, a non-polarisable calomel electrode being very generally employed, which is combined in a polarisation circuit with the electrode under investigation. The electromotive force of the non-polarisable electrode being known, the polarisation of the other electrode can be easily calculated from the total electromotive force of the circuit. The electrode potentials given in the table on p. 359 may be looked upon from the point of view of electrolysis as well as of current generation. In order to deposit zinc in quantity on a cathode from a normal solution of zincion it is obviously necessary that the difference of potential of 0.5 volt between zinc and normal zincion should be compensated by the electrolysing current, for as soon as more than a very minute quantity of zinc separates out on the electrode, which is assumed not to act on the zinc, that electrode behaves as a zinc electrode. The electrode potentials then may be considered as the discharging potentials for cations. Since in order to study the electrolysis of a solution between unattacked electrodes it is neces- sary to know the discharging potential of the anion of the electrolyte as well as that of the cation, a similar table for the commoner negative radicals may be constructed. The ionic concentration is once more normal, except in the case of oxidion and hydroxidion, where the values are for solutions in which the concentration of hydrion is normal. DISCHARGING POTENTIALS OF ANIONS I' Br' O" Cl' Iodidion Bromidion [Oxidion Chloridion [Hydroxidion Sulphanion Hydrosulphanion OH' SO" HSO, -0.80 -1.22 - 1·49] - 1.59 - 1.94] -2.2 - 2.9 It is of course impossible to obtain some of these discharged ions in the free state, and the values of the discharging potentials depend both on the nature of the electrode and of the substances produced. The numbers given are mostly derived from observations of the mag- nitude of direct currents and the corresponding polarisation currents 368 CHAP. INTRODUCTION TO PHYSICAL CHEMISTRY which they produce. The values given in the table are all negative, and for the elements express approximately their electro-affinity or tendency to combine with negative electricity. Thus chlorine will displace bromine from a bromide in virtue of its greater electro-affinity for negative electricity, just as zinc will displace copper from a copper salt in virtue of its greater electro-affinity for positive electricity. As before (p. 360), a diminution of the concentration to one-tenth of its value will augment the discharging potential of a univalent anion by 0·057 volt. From the known concentration of hydroxidion in water it may be calculated that a potential difference of about 1.94081.14 volts is required to discharge it from normal hydroxidion solution, as against 1.94 volts in normal hydrion solution. The decomposing current, i.e. the minimum current which will effect continued electrolysis, may be calculated from the discharging potentials as follows:-Since the cathodic and anodic electrode potentials. act against each other, the electromotive force of the current through the system is equal to the difference of the electrode potentials of cathode and anode. The cathodic potential is therefore taken with the sign given in the table on p. 359, and the electrode potential of the anions on p. 367 with the sign reversed, i.e. with the positive sign. Thus in a solution of hydrochloric acid of normal ionic concen- tration the decomposing current will be Cathode Anode -0.27 (159) 1:32 volt. — = In a decinormal ionic solution it will be ▬▬▬▬▬▬▬▬▬▬▬ – 0·27 + 0·06 − (−1·59 – 0·06) = 1·44 volt. To effect the decomposition of water with discharge of hydrox- idion, a practically constant electromotive force is required, which may be calculated from the values given for normal hydrion solution, viz. -0.27 (194) 1.67. = Since the product of the hydrion and hydroxidion concentrations is constant in aqueous solutions and equal to the product for pure water, and since the increase or diminution of electrode potentials depends on the ratios of concentrations, it follows that whatever addition may be made to the cathodic potential for hydrogen will be compensated by a corresponding diminution in the anodic potential for hydroxyl, so that the decomposing value for water should remain constant at 1.67 volts whether the solution is neutral, strongly acid, or strongly alkaline. An instructive instance of discharging potentials is afforded by the electrolysis of sulphuric acid in approximately normal solution, and may here be given in rough outline. The only positive ion to be XXXI POLARISATION AND ELECTROLYSIS 369 נ ? considered in this case is hydrion, but in the solution the following anions must be supposed to exist - hydrosulphanion, sulphanion, hydroxidion, and oxidion, the last two in vanishingly small amounts in comparison with the first two. The electromotive force of the corresponding decomposing currents would be Cathode. Hydrion -0.27 -0.27 -0.27 -0.27 "" "" "" Anode. Oxidion +1.49 Hydroxidion +1.94 = 1.67 Sulphanion +2.2 = 1.93 Hydrosulphanion +2.9 = 2.63 = - 1.22 volts "" در >> If the electrolysing current has an electromotive force of 1.22 volts, continuous electrolysis might be expected to take place, as oxidion would then be discharged at the anode. This decomposition has actually been observed by taking a large platinised platinum plate as anode and the point of a thin platinum wire as cathode. With an electromotive force of 1.1 volts, hydrogen may be seen to come off at the platinum point, though no visible change occurs at the anode, the small amount of oxygen being absorbed by the platinum. This value is lower than the number given above, and the occurrence of electrolysis is probably due to the formation of an oxide of platinum. Reversing now the electrodes and saturating the platinised cathode with hydrogen, oxygen may be seen to come off at the point which serves as anode when the electromotive force of the direct current reaches about 1.67 volts, i.e. attains the decomposing value for hydroxidion. Even here, however, the extent of electrolysis is slight. It is only when the electromotive force of the current is raised to such an extent as to discharge freely the sulphanion and hydrosulphanion, which exist in plenty in the solution, that the electrolysis becomes really brisk. For the preparation of detonating mixture by the electrolysis of water acidulated with sulphuric acid an electromotive force of at least 2-3 volts is always used, corresponding to the discharging values for sulphanion and hydrosulphanion. Advantage is frequently taken in electrolytic analysis of the different discharging potentials of different metals. To deposit zinc from a normal solution of zincion as sulphate requires a voltage of + 0·50 + 2·2 = 2.7, whilst to deposit copper from a similar solution only requires -0.60 +2.2 1.6 volts. If then we use a source of electrical energy with a voltage of about 1.8, e.g. one lead accumulator, we can deposit the copper from solution without depositing the zinc. By the use of a single accumulator, copper may be deposited from a solution of copper sulphate strongly acidulated with sulphuric acid without the simultaneous evolution of hydrogen which would tend to lessen the coherence of the deposit, and render it less suitable for weighing. To get an appreciable liberation of hydrogen under ordinary conditions from sulphuric acid, requires, as has been indicated. 2 B 370 INTRODUCTION TO PHYSICAL CHEMISTRY CH. XXXI above, 1.93 volts, which is above the voltage of an accumulator which has been running for a little time, viz. 18. To deposit copper from normal dicuprion and sulphanion with platinum electrodes requires only 16 volt. The accumulator will then deposit copper without liberating hydrogen. It must be noted, however, that as the solution retains less and less copper, the voltage required to deposit what remains increases, the concentration of the sulphanion being supposed constant. Thus from decinormal solution of dicuprion, the copper 0.06 2 can only be deposited by 16+ 1.63 volts, and from a millionth normal solution by 1.78 volts, which is still below the voltage of the accumulator. Now as a millionth molecular-normal solution of copper only contains 0·065 mg. per litre, it is plain that the continued action of the accumulator will, for analytical purposes, entirely exhaust the copper solution, without hydrogen being at any time liberated in appreciable quantity. one When two platinised electrodes, one saturated with oxygen and the other with hydrogen at atmospheric pressure, are immersed in a conducting solution, they exhibit a difference of potential of 1·14 volts, and when connected by a conductor form the oxyhydrogen gas-cell. The departure from the theoretical value 1.22 is accounted for as above by the formation of an oxide of platinum. If, now, electrode is placed in the solution of an oxidising agent, this acts as a potential source of oxygen at a different " pressure" from the atmospheric value, and similarly the other electrode placed in a solution of a reducing agent acts as a hydrogen electrode at a "pressure depending on the power of the reducing agent. By measuring the electromotive forces of platinum electrodes immersed in oxidising and reducing solutions it is thus possible to range these solutions in the order of their oxidising or reducing power. When tested in this way permanganate acidified with sulphuric acid is one of the most powerful oxidisers, and stannous hydroxide in alkaline solution one of the most powerful reducers. "" A considerable amount of discussion has taken place as to whether the oxygen and hydrogen which appear at the electrodes when a solution of a salt like potassium sulphate is electrolysed, are primary or secondary products of the electrolysis. There is little doubt that hydrogen and oxygen do appear as primary products of the electrolysis of such a salt solution when the electromotive force employed is just beyond the decomposing value for water, but when large electromotive forces are used and the electrolysis proceeds briskly, the oxygen and hydrogen are in all probability secondary products formed by the action of the discharged ions on water (cp. p. 235). I ! CHAPTER XXXII DIMENSIONS OF ATOMS AND MOLECULES THE conceptions of molar weight and of combining weight are, as we have seen, independent of any hypothesis as to the structure of matter, and for most purposes of elementary inorganic chemistry these numerical conceptions suffice. When, however, we consider the facts of organic chemistry, and particularly the phenomena of isomerism, we find that hypotheses as to structure become indispensable for the co-ordination of the facts. For example, two isomeric substances, say ethyl alcohol and methyl ether, have the same composition and the same molar weight, and yet possess absolutely different properties, both chemical and physical. The simplest way in which we can account for this difference is to assume a difference in structure, and a difference in structure of the molecules at once involves the conception of structural units or particles. Now we have ready to hand in the atomic and molecular theories of Dalton and Avogadro the necessary apparatus for the simple formulation of chemical structure. The ultimate structural units are the atoms of the various elements; the simplest structure is the gas particle or molecule. In terms of the theory our system of combining weights becomes a system of atomic weights, and molar weights become the relative weights of the molecules, or molecular weights. We account for the difference in properties between ethyl alcohol and methyl ether, by assuming that in ethyl alcohol the two carbon atoms which the molecule contains are directly united, whilst in methyl ether they are kept apart by an intermediate atom of oxygen. A reference to Chapter XV. will give numerous instances of the interpretation of specific properties in terms of structure. If we thus assume a discrete structure of matter for chemical purposes, and as we have seen in Chapter IX. the same assumption must be made to account for the physical properties of gases, it is legitimate to inquire as to the absolute size of the grains or particles of which matter is supposed to be constituted. Approximate measures of the size of the ultimate particles of 371 372 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. matter have been obtained in many different ways. Thus some dyes exhibit colour when diluted 100,000,000 times, so that the smallest weighable quantity of the dye must be divisible into at least 100,000,000 parts. Again, the fluorescence of an eosin solution is visible, with sufficiently powerful illumination, in solutions containing only 10-18 milligrams per cubic millimetre, and the ultramicroscope now enables us to assert that particles of colloidal gold exist which have a considerably smaller diameter than 3 μp (p. 223). Faraday prepared films of gold whose thickness he estimated as about one-hundredth of that of a wave-length of light. As every continuous film must have a thickness of at least one particle, this result would give 5 µµ or less as the diameter of the ultimate particle of gold. Observations on soap-bubbles and oil films point to the existence of continuous films only 0.6 μu in thickness. We may deduce then from these observations that no discontinuity appears till we descend to magnitudes of about one-millionth of a millimetre, which may be taken as a rough approximation to the molecular dimensions. It is possible on the assumptions of the kinetic theory of gases to form an estimate of the absolute size of the gas particles which that theory postulates. In such calculations it is generally assumed that the particles are spherical, and the size referred to is the diameter of the molecular sphere. We have seen (p. 84) that the speed of air particles in their free path is of the order of magnitude of 20 miles per minute under ordinary conditions. Now if we observe a layer of bromine vapour diffusing upwards in a vessel filled with air, we see that the process of diffusion is very slow indeed, to be measured rather in millimetres than in miles per minute. In terms of the theory we must attribute this slow diffusion to the innumerable collisions of the particles, the mean free path between two collisions being extremely short. The number of collisions between a given number of particles in a given volume and moving with a given mean velocity is plainly dependent on the dimensions of the particles. Had the particles no physical dimensions there could be no collisions at all, and the larger the particles are, the more frequently must they encounter one another. If we consider the chance which a single foreign molecule, say one of bromine moving upwards, has of penetrating a single horizontal layer of air, we see that it is determined by the ratio which the sum of the cross-sections of all the air particles in the layer bears to the total area of the layer. From experiments on gaseous viscosity, and on the velocity of dif- fusion and of heat conduction in gases, it is possible to determine the mean free path of the moving particles, and the sum of the cross- sections of the particles in a given volume of the gas. The mean free path of an air particle at normal temperature and pressure is calculated as 0.0001 mm., and the sum of the cross-sections of the XXXII DIMENSIONS OF ATOMS AND MOLECULES 373 At particles in 1 cc. of this air is calculated to be 18,400 sq. cm. first sight this area seems enormous compared with the small quantity of matter considered, but a moment's reflection shows that the greatness of the area must be due to the smallness of the individual particles. If a cube of 1 cm. edge is divided up into 1000 cubes of 1 mm. edge, the sum of the cross-sections of the smaller cubes will be 10 times the cross-section of the original cube, for it is plain that when spread in one layer the small cubes will occupy ten times the area and form a layer one-tenth the thickness of the original cube. In general terms, the more finely we divide a given substance, the greater is the surface over which we can spread it, i.e. the greater is the sum of the cross-sections of its particles. Knowing now the sum of the cross-sections of the air particles, we can calculate the radius of the individual particles if we can only ascertain the sum of the volumes of the particles. We have already seen how an estimate of this latter magnitude may be formed. Van der Waals's coefficient b (p. 87) is intimately connected with the ratio of the volume of the particles of a gas to the whole volume occupied by the gas at normal temperature and pressure, being theoretically four times this ratio. For air b=0·0017, and therefore the volume of the air particles in 1 cc. will be 0.00042 cc. approximately. If there are n particles in 1 cc. of air at normal temperature and pressure, we have for the sum of the volumes of these particles Σ vol. = N 4 IL ¹³ 1= = 0·00042 if r is the radius of each particle. n For the sum of the cross-sections we have Zarea = 1. #1² = 18,400, whence 4 0.00042 3′‍ = 18,400* 1.7 x 10-8 cm. 27 = 0·34 × 10-6 mm. We thus arrive at a value for the molecular diameter which is of the same order of magnitude as that derived from a consideration of continuity and from direct observation with the ultramicroscope, namely, something less than 1 µµ. If it is thought that the value 1b for the volume of the particles is too uncertain, we may take the volume of the liquid derived from the gas as a maximum estimate of the volume of the particles, since they cannot be greater than the volume of the liquid. The specific gravity of liquid air is 0.9; the volume occupied by 1 gram is thus 1.1 cc. Now 1 gram of air at normal temperature and 374 CHAP. INTRODUCTION TO PHYSICAL CHEMISTRY pressure occupies 22,400 29 770 cc. The particles of air then cannot occupy more than 0.0014 of the volume occupied by the gas under the conditions considered. This value, it is true, is more than thrice the value deduced from van der Waals's b, but it must be remembered that it is a maximum value, the volume of liquid air under 1 atm. at its boiling point being necessarily much greater than the actual volume of the particles. A very direct method of estimating molecular magnitudes has been found in the study of the a-radiation emitted from radio-active sub- stances (cp. Chapter XXXIII.). It has been shown that the a-radiation consists of a rapidly moving stream of molecules (in this case atoms) of helium, each helium molecule carrying two unit charges of positive electricity; in other words, the charge on 4 grams of helium in the form of a-particles is 2 faradays. By selecting a suitable source of a-radiation it is possible not only to measure the total charge carried by the a-particles in a given time, by observing the positive charge assumed by a receiver which absorbs them, but also to count the number of a-particles by which this charge has been generated. Two methods have been devised for the purpose of counting these particles. One is electrical, the apparatus being so arranged that the passage of each separate a-particle through a small orifice into a detecting chamber causes a measurable movement of the needle of an electro- meter. The other depends on the scintillation produced when an a-particle strikes a screen of phosphorescent zinc sulphide. Under suitable conditions movements of the needle or the scintillations may be counted, and thus the number of a-particles entering the detecting chamber or striking the screen in a given time can be ascertained. Knowing now the total electrical charge carried by the stream of a-particles, the relationship between the charge and the mass of the a-particles, and finally the number of a-particles in the stream, we are in a position to state the electrical charge carried by each particle, and consequently the mass of each particle. It was found in this way that the double charge carried by a helium atom as a-particle is 3.14 × 10-19 coulombs or 3.25 × 10-24 faradays, the weight of the helium atom being in consequence 6.5 × 10-24 g. Exact investigations have been carried out by Millikan on the rates of motion of minute drops of oil hovering between the horizontal plates of a charged condenser, the movement of the brightly illuminated droplets being observed through a microscope provided with a micro- meter scale. If the droplet has no electric charge it falls at a rate depending only upon gravity and on the resistance it experiences from the gas in which it is suspended. If it is charged electrically the electrical attraction may act with or against the gravitational attrac- tion, and is superadded to the latter, the rate of fall being increased or diminished thereby, or even made negative, in which case the XXXII DIMENSIONS OF ATOMS AND MOLECULES 375 droplet rises against gravitation. The droplets are charged electrically by the action of X-rays either directly or indirectly through the medium of the gas. Under the influence of X-rays the gas becomes ionised (p. 377), and occasionally a gaseous ion attaches itself to the droplet, conferring on it a positive or negative charge. This process may be frequently repeated so that the charge on the droplet varies. From the rate and direction of motion of the droplet and other ascertainable data, the sign and magnitude of the charge at any period may be calculated. The results show that the charge on a droplet varies discontinuously and is always an integral multiple of a fundamental unit charge, which is denoted by the symbol e, and of which the most probable value, expressed in different electrical units, is e=1·648 × 10 faraday, =1·591 × 10−19 coulomb, =4·77 × 10-10 electrostatic unit. K This value is of fundamental importance and represents the ultimate indivisible quantity of electricity. Determined in this way, the number is in excellent accordance with half the value obtained by the wholly independent method indicated in the preceding paragraph, namely 1.62 × 10-24 faraday for each of the two charges on the a-particle. Knowing the value of e, we can calculate from the ratio of charge to mass of hydrogen as ion in solution the weight of the atom of hydrogen, and from this the weight of any other atom or molecule. Hydrogen bears the charge of 1 faraday for 1.0076 g. when it acts as ion in solution. The weight of hydrogen which carries 1648 × 10−24 faraday is therefore N= 1.0076 × 1.648 × 10-24 = 1·66 × 10-24 X g., and this must be regarded as the weight of the atom of hydrogen. The weight of any other atom or molecule is 1·648 × 10−24 W, sense. where is the relative atomic or molecular weight in the ordinary Avogadro's constant or Avogadro's number, N, .e. the number of molecules in a gram-molecule, or the number of atoms in a gram-atom, is consequently W 1.648 × 10-24 W 6'06 x 1023. This number is of importance in many theoretical calculations. When an electric discharge is passed through a highly evacuated tube, faint bluish rays may be seen to stream from the cathode. Where they strike the glass wall of the tube they excite a greenish luminescence. These rays are termed cathode rays and proceed 376 CHAP. INTRODUCTION TO PHYSICAL CHEMISTRY from the cathode in straight lines, as may be shown by the com- parison of an object placed in their path with its shadow cast on the luminescent wall. By the deflection which they suffer when subjected to the action of a magnetic field they can be proved to carry negative electricity, and by the joint action of known electric and magnetic fields, the ratio of their charge to their mass has been determined and found equal to 1.77 × 108 coulombs, or 1830 faradays, per gram. A gram of hydrogen in electrolytic solutions only carries about 1 faraday, so that if the cathode rays consist of singly charged negative particles, as hydrogen in the ionic condition consists of singly charged positive particles, then each negative particle will have only about 1/1800 of the mass of the hydrogen atom. The ratio of charge to mass has been found to be always the same, no matter what gas the discharge tube contains, or how the cathode rays are excited, and has been also proved to be identical with that for the B-rays from radio-active substances. We have learnt above that the value of the unit electrical charge has been definitely determined, and leads to magnitudes for the mass and dimensions of atoms in accordance with other and independent methods of investigation. The conclusion is thus inevitable that corpuscles charged with negative electricity exist which possess. a mass much smaller than that of the hydrogen atom, and that these particles may apparently originate in any kind of matter. That they are practically universal has been further established by the fact that they are emitted by any substance exposed to the action of X-rays. These particles are now regarded as the ultimate units of negative electricity and generally receive the name of electrons.¹ The small mass may be regarded as arising from the electric inertia of the electron caused by the magnetic field surrounding it, any change in the motion of the electron in the field being resisted. This 2 €2 mass m is practically constant at low speeds and equal to where 3 7* e is the unit charge and r the radius of the electron. Since both m and e are known (pp. 374, 375) we find for r the value 1·87 × 10-12 mm. or about 1/100,000 part of the radius of an atom or molecule (p. 373). When the speed of the electron closely approaches that of light, viz. グ ​X 3 × 1010 cm. per second, its electrical mass increases, becoming about double its ordinary value when the velocity is 0.9 of the velocity of light. Since in the electron we have a particle of universal occurrence, much smaller in mass and size than the smallest atom, it is natural to inquire what relationship may be conceived to exist between the 1 The term electron was originally applied to the unit charge of electricity whether positive or negative, and whether associated with matter or not. It is still often used in this sense, but generally speaking by electron is meant the unit charge of negative electricity regarded apart from any association with matter. XXXII DIMENSIONS OF ATOMS AND MOLECULES 377 electron and the atom, and to inquire in particular if electrons are essential constituents of atoms. If an atom, which is normally in a condition of electric neutrality, contains electrons, these constitute a negative charge, and the atom must also contain a balancing charge of positive electricity. No positive electron corresponding to the negative electron, i.e. with a similar value of the ratio e/m, has been discovered. The unit positive charge is always associated with a mass equal to or greater than that of an atom of hydrogen. The atom then, according to Rutherford's conception, probably consists of a positively charged nucleus, in which resides practically the whole mass of the atom, and round which electrons are arranged as planets round a sun. It is noteworthy that the size of the electron bears much the same ratio to the size of the atom as the diameter of the earth bears to that of its orbit. + An atom on this view then is not one "solid" indivisible unit, but a system of electrically charged particles, the volume of which is almost infinitesimal compared with the volume of the atom itself, ¿.e. the volume into which no other atom under ordinary conditions can penetrate. Should the second atom, however (of like nature to the first), be moving with a very high velocity it may overcome the electrical repulsion exercised by the first and pass clear through it. Such penetration of the atom occurs, for example, when the swiftly moving helium particles constituting the a-rays pass through continuous. thin sheets of aluminium, glass, etc. When gases are subjected to the action of X-rays, of cathode rays. and B-rays, or of a-particles, they are ionised and become conductors of electricity instead of being nearly perfect insulators as they are in their normal state. The ionising agent probably expels an electron from an atom of the gaseous molecule, leaving a positively charged. residue. The free electron is, however, speedily attracted by other gas-molecules, and serves to charge them negatively. Thus in the gas there are now positive and negative ions as in an electrolytic solution, but it should be noted that in the gas the ions are much less persistent than those in solution, and mutually discharge each other in a comparatively short time. Even in very strongly ionised gases the degree of ionisation is almost inconceivably small, notwithstand- ing the marked phenomena they exhibit, the number of atoms ionised. per second being only about 10-12 of the total number of atoms exposed to the radiation. When any charged particle such as an electron is accelerated or retarded, an electro-magnetic wave is set up in the surrounding ether, which, if the 'disturbance is of sufficient magnitude, may reveal itself as light or X-rays. Conversely X-rays and light of short wave-length are capable of liberating electrons from bodies on which they fall. In this way phenomena of ionisation in gases are produced which are susceptible of exact measurement, and have thrown much light both 378 CHAP. INTRODUCTION TO PHYSICAL CHEMISTRY on the character of the vibrations themselves and on the constitution of matter. If we conceive an electron to be in vibration with a definite period, a periodic disturbance will be transmitted to the ether, and light or X-rays of definite frequency or wave-length will be produced. Such electronic vibrations give rise to line-spectra, and attempts have been made by reference to them to account for the regularities observed in the frequencies of the lines of a given spectrum. For example, Balmer's formula for a certain series of lines emitted by hydrogen is n = N 1 22 m² 11712) where n is the frequency, N a constant, and m the series of whole numbers greater than 2. By substituting 3, 4, 5, etc., in succession for m the frequencies of the observed lines are given with the utmost accuracy. The conclusion is almost irresistible that the lines of this spectral series are due to the same system in different modes of vibration. The X-ray spectra of the elements have in recent years received much attention, and have led to many results of great theoretical importance. When excited under certain conditions by cathode rays. each element emits not only diffuse or scattered X-rays, but also a comparatively simple spectrum consisting of lines of a few definite wave-lengths which are characteristic of each element. These wave- lengths are much shorter than those of visible light or even of ultra- violet light. The wave-length of the sodium D line is 589 µµ; in the extreme ultra-violet region the wave-lengths are of the order 100 μu, whilst the characteristic X-ray wave-lengths are less than 1 μμη i.e. of the same order of magnitude as atomic diameters (p. 373). Ruled diffraction gratings, with lines about 1200 μμ apart, may be used to disperse a beam of ordinary light into a spectrum, because the distance between the rulings is of the same order as the wave- length of the light falling on them. Such gratings are useless for the analysis of X-rays, being much too coarse for the purpose. It seemed to Laue, however, that the supposed regular arrangement of particles in crystals might act as a grating, and experiment at once. showed his supposition to be justified. If we consider a cubic crystal of sodium chloride, we might attribute its cubical form to each crystalline particle being a cube, but this assumption has long been discarded in favour of another, namely that the symmetry of the crystal depends not on the symmetry of the crystal units but on the symmetry of their arrangement in space. Each crystal particle of sodium chloride may be of any shape itself, but if each be supposed to reside in the centre of a cube, surrounded by other similar cubes in contact with it, the desired cubic symmetry is attained. XXXII DIMENSIONS OF ATOMS AND MOLECULES 379 If we imagine the crystal particle to be the sodium chloride mole- cule, we can calculate the distance d between the centres of the unit cubes, or the length of the edge of each cube, as follows. The volume of the gram molecule of crystalline sodium chloride is its molecular weight divided by its specific gravity, i.e. 58.5/2.17270 cc. But there are N=6·06 × 1023 molecules in the gram molecule (p. 375), so that the volume occupied by each molecular cube is 27/(6·06 × 102³) cc. But this volume is equal to d³, so we have d³ 27/(6.06 x 1023) cc. d = 3·54 × 10-8 cm. or d= 0.354 μμ. The crystal of sodium chloride may then, on the above assumption, be regarded as a series of planes parallel to the cube face, each plane packed with molecules and 0.354 ми apart from neighbouring planes. Here we have a regular arrangement corre- sponding to the rulings of a grating. If a monochromatic X-ray falls on such a series of planes, reflection takes place from them, but the reflected rays from the various planes inter- fere and annul each other except when the following equation is satisfied, 1 1 nλ= 2d sin Ons - FIG. 59. where is the wave-length, d the distance between the planes, & the glancing angle, i.e. the complement of the angle of incidence, and n the order of the spectrum. At certain glancing angles then there will be reflection of the X-ray, which may be detected either photo- graphically or by means of an ionisation chamber. These reflections correspond to the spectra of the first, second, third order, etc. all other angles no reflection occurs. At If we know then the distance d, the wave-length of any kind of X-ray may be determined. On the other hand, if the wave-length 380 INTRODUCTION TO PHYSICAL CHEMISTRY CH. XXXII for a definite source of X-rays is known, the value of d for any aspect of any crystal may be ascertained. An investigation of sodium chloride crystals not only parallel to the cubic faces, but parallel to the octahedral and dodecahedral faces led to the conclusion that the above assumption of the crystal particles being molecules of sodium chloride is not justified. The reflection from octahedral faces showed that there are two kinds of reflecting planes, arranged alternately, one kind showing more marked reflec- tion than the other. This result is most easily interpreted by assuming that the particles causing the reflection are not molecules of sodium chloride, but are atoms of sodium and atoms of chlorine arranged as in Fig. 59. Each atom of sodium has for nearest neighbours six equidistant atoms of chlorine, and each atom of chlorine six equidistant atoms of sodium. The conception of the molecule of sodium chloride is thus relinquished in favour of a kind of com- plete dissociation, with immobile instead of mobile ions. In each plane parallel to the cube face there are equal numbers of sodium and chlorine atoms; therefore all these planes are alike. On the other hand, in the planes parallel to the octahedron faces, indicated in Fig. 59 by dotted triangles, there occur in alternate planes only sodium atoms (white) and only chlorine atoms (black), which are unequally active in reflection. If d is the distance between the cube planes, the distance between the octahedron planes is 2d/ 3. The ratio of the distance between the cube planes to that between the octahedron planes was found by observation with X-rays to be 1 to 2/3 in accordance with the interpretation just given. The distance between the planes parallel to the cube faces is not, however, d=0.354 µµ, the value calculated on the assumption that the reflection is caused by complete molecules of sodium chloride. Each molecule now contributes two particles spaced as shown in Fig. 59, so that d³ = 27/(2 × 6·06 × 1023) cc. d=2·81 × 10-8 cm. or 0.281 μp. The intimate structure of many crystals has been investigated by this method and results of the utmost importance have been obtained. By appropriate modifications the X-ray method has been made to throw light on the nature of colloidal particles (p. 228), and even on the situation of electrons in crystalline structures. R. A. MILLIKAN, The Electron (Chicago, 1917). J. PERRIN, Atoms (trans. by D. Ll. Hammick, London, 1916). G. W. C. KAYE, X-Rays (London, 1919). W. H. BRAGG and W. L. BRAGG, X-Rays and Crystal Structure (London, 1919). J. A. CROWTHER, Molecular Physics (London, 1919). CHAPTER XXXIII RADIO-ACTIVE TRANSFORMATIONS OWING to the discovery of the phenomena of radio-activity in some of the well-known elements, in particular uranium and thorium, and still more to the discovery of intensely radio-active elements, such as radium and polonium, our whole conception not only of the nature of the chemical atom, but of the energy associated with it, has in the last few years undergone a radical change. Until the advent of radio- activity the atom as utilised by the chemist, even for purely theoretical purposes, was in the main Dalton's atom, indivisible and of definite weight, and the quantity of energy inherent in it was presumed to be of the same order as the amount liberated when the element entered into chemical action with other elements. Now we must conceive the atom in its essence as a complex structure, capable, in some instances at least, of spontaneous disintegration, and as a storehouse of energy millions of times greater in quantity than had been formerly suspected. Notwithstanding this revolution in our conception of the atom, the chemist in his ordinary work is entirely unaffected by the change. The practical conceptions of atomic weight, asymmetric carbon atom, and the like persist unaltered in the deepening of our knowledge of the atom itself. The radio-active elements differ from the ordinary inactive elements by possessing certain properties which the latter do not share. When a photographic plate is exposed in the dark to the action of a radio- active substance, it is affected as if the substance were a source of light, and it was owing to this property of affecting a photographic plate that radio-activity was first discovered, namely in the element uranium and its compounds. An investigation of uranium ores then led to the discovery of the much more active elements radium and polonium. Minute quantities of these substances or their compounds have a great and rapid effect on the photographic plate. They produce in addition luminescence when placed near a screen of phosphorescent zinc sulphide, and rapidly neutralise the charge of an electroscope brought into their vicinity. Investigation by means of the electroscope 381 382 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. has been the most productive in quantitative radio-active research, since under proper conditions the instrument is so delicate as to lead to the recognition of a quantity of radio-active material less in amount than a millionth of a millionth of a milligram. on a As the name radio-activity implies, the peculiarity of the active substances is that they appear to emit radiations, and it is these radiations which produce directly or indirectly the action on photographic plate, a phosphorescent screen, or a charged electroscope. Three clearly marked types of radiation have been recognised, named respectively a-, B-, and y-radiation. The intensity of these radiations depends only on the nature and amount of the radio-active elements present, being quite independent of external conditions and of any combination with other atoms. The a-particles constituting the a-radiation are produced by the disintegration of the atom of the radio-active element, and are identical with atoms of helium, each helium atom, however, carrying two unit charges of positive electricity, in other words, being deprived of two electrons. These charged helium atoms are shot out from the disin- tegrating atom of the radio-active element with enormous velocity, which varies according to the nature of the disintegrating atom, but which for any one radio-active element is probably constant. The actual rate at which an a-particle travels is of the order of 15 × 109 cm. (about 10,000 miles) per second. A minute particle moving at this high speed has considerable powers of penetration. Not only can it penetrate a layer of gas, but it is even able to pass through a thin layer of a continuous solid-for example, aluminium or mica. When a-particles, moving at this rate, traverse a gas, they necessarily encounter from time to time the comparatively sluggish molecules of the gas, and one effect of the encounter is that the gaseous molecule may become charged with electricity. This occurs when an a-particle detaches an electron from an atom of the gaseous molecule, with resulting ionisation of the gas (p. 377). In order that the swiftly-moving helium atom may ionise a gas it must have a velocity greater than a certain limit. If the speed at which it moves is less than this limiting velocity, its encounters with the gaseous molecules produce no ions. We find, therefore, that after a-particles have passed through a certain thickness of a given material, whether solid, liquid, or gas, they lose their power of ionising gases, owing to the loss of velocity which they experience in their encounters with other molecules. The greater the original velocity of the a-particle, the greater is the thickness of material which the particle can penetrate without losing its power of ionising gases. As has been said above, each type of atomic disintegration appears to produce a-particles all moving with the same velocity. For each type of atomic disintegration, then, we have a certain thickness of material through which the a-particles produced in the transformation XXXIII RADIO-ACTIVE TRANSFORMATIONS 383 can penetrate without losing their ionising power. This affords us a valuable means of analysing a complex radio-active change into the successive atomic disintegrations which may constitute it. The material generally chosen for reducing the velocity of the a-particles below the limiting value at which they can ionise gases, is atmospheric air at one atmosphere pressure and 15°, and the thickness of air which will just deprive the a-particles of their ionising power is spoken of as the range of the a-particle. This range varies in the case of radium and its decomposition products from about 3.5 cm. for the slowest a-particle to 8.7 cm. for the fastest a-particle. It will be seen from what has been said above that a-particles are ordinary gaseous molecules which have acquired special properties owing to the great speed at which they move, and also to some extent to the positive charge of electricity which they carry. The power of the a-particle to excite fluorescence, and to act upon a photographic plate, disappears at about the same limiting velocity as the power of producing ionisation. Though the B-radiation likewise is due to a stream of electrically charged particles, it is entirely different in character from the a-radia- tion. It consists of negative electrons each of a mass equal to 1/1800 of that of an atom of hydrogen (cp. p. 376). Owing to their very great speed, these ẞ-particles have much greater penetrative power than the a-particles, as can be judged from the much greater thickness of material which they can traverse without losing their power of ionising gases. Whilst the a-rays move at velocities of about 1.5 x 109 cm. per second, the B-particles move with a velocity more than ten times this value, and may, in extreme cases, move at such a rate that their speed approximates to that of light, namely 3 x 1010 cm. per second. The value 0·98 × 3 × 1010 cm. per second has been recorded for the fastest B-rays. The B-rays, like the a-rays, are not always emitted in radio-active transformations. Thus, for example, in the nine successive trans- formations of radium and its disintegration products which are at present known, only five emit B-rays. The effect of the B-rays may be studied apart from that of a-rays, with which they may be associated by passing the radiation through such a thickness of material as will stop the activity of the a-particles without greatly diminishing that of the B-particles. The B-particles produced in any given radio-active transformation do not all move with the same velocity, but in some transformations they are produced in homo- geneous groups with approximately equal velocities. The y-radiation is very penetrative and shows all the ordinary phenomena of exciting fluorescence, affecting the photographic plate, and ionising gases. It is probable, however, that these effects are not primary, but are due to ẞ-rays which are known to be produced when y-rays encounter ordinary molecules. This radiation is of the 384 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. same nature as X-rays. It has no electrical charge, and does not con- sist of a stream of particles, but is a wave-motion of the luminiferous ether of very short wave-length. When y-rays are observed in radio- active transformations they are always associated with B-rays, from which, however, they may be separated by their still higher penetrative power. A sheet of lead one cm. thick will stop all a- and ẞ-particles, but will still allow y-radiation to pass. As an example of radio-active transformation, we may take the successive stages produced by the disintegration of the element radium itself. Radium is an element belonging to the group of metals of the alkaline earths. In their ordinary chemical properties the radium compounds closely resemble the corresponding compounds of barium. The sulphate of radium, RaSO,, for example, like sulphate of barium is insoluble in water, and the chloride and bromide of radium are soluble and can be purified by crystallisation like the corresponding chloride and bromide of barium, with which they are isomorphous. Metallic radium decomposes water briskly with evolution of hydrogen and formation of radium hydroxide Ra(OH)2. The spectrum exhibited by radium compounds is of the same type as that of the other metals of the alkaline earths. No radio-activity has been detected in either. barium or any of the other elements of this group, radium being the only element of the family of the alkaline earths which is radio-active in any measurable degree. When a radium salt is confined in a vacuous space it is found to give off, besides helium, minute quantities. of a gas which is itself intensely radio-active-much more so than the radium compound from which it was derived. This gas is called the radium emanation. From experiment on its rate of diffusion. it appears to be a gas of high molecular weight, and if it is brought, say by means of liquid air, to a sufficiently low temperature, it can not only be condensed to a liquid, but solidified. With regard to its chemical properties the radium emanation is entirely inert, being a member of the helium and argon family of inert gases. Its atomic weight has been determined from its density to be 222, and it has received a place in the list of well-defined elements under the name of niton. Although it is chemically inactive in the ordinary sense, it yet displays great activity in the sense that it decomposes in a com- paratively short time into helium and another element, radium A, which is of quite a different character. Radium A, like all the elements in the stages of the disintegration of radium which follow it, is a solid at the ordinary temperature, and changes rapidly into the next member of the series, radium B, with evolution of helium. The transformation of radium B into radium is not accompanied by the production of a-particles, only ẞ and y radiations being emitted in this transformation. Radium C is short- lived and changes with production of B-rays into radium C', which at once disintegrates with great violence into helium and radium D, XXXIII RADIO-ACTIVE TRANSFORMATIONS 385 the a-particles in this instance having the greatest velocity of any yet measured. The next transformation of radium D into radium E is slow, and is unaccompanied by the production of a-particles. It is followed by a similar but more rapid transformation into radium F. Radium F is identical with the radio-active element polonium which was discovered in the early days of radio-active investigation in the same material, namely pitchblende, as that in which radium was found. This element, radium F or polonium, is the last of the radio- active series of elements derived from radium. The element into which it is transformed with evolution of helium is, as has been said, one of the ordinary stable, that is inactive, elements, and apparently lead is the stable form into which the unstable elements derived from radium ultimately settle down. Besides this main line of disintegration there is a short branch line, beginning at radium C. The great proportion of radium C undergoes as stated a ß-ray transformation into radium C', but a minute proportion undergoes simultaneously an independent a-ray disintegration into radium C2, which speedily by a ẞ-ray transforma- tion yields an inactive element, resembling radium D in every respect except its activity. When we investigate the rate at which radio-active transformations take place it is found that the formula of unimolecular reactions is strictly followed (p. 287). This might be expected, for the transforma- tion is undergone by the individual atoms of the radio-active elements, unaffected by the presence of other atoms. Each atom transforms itself independently of all others, and therefore the type of re- action must be unimolecular. In order to indicate the speed of the radio-active transformation it is customary to give, instead of the velocity constant, the "period" or time required for the transforma- tion to proceed half-way to the next stage. The relation between these two magnitudes is easily ascertained from the unimolecular equation (p. 287) or log a or a 2 — Ο α- X a = ekt If a=1, then, when the substance is half transformed at the time t。.¿ we have a − x=0·5 and the equation becomes '0'5 log 2 =0·434pk h 0.434th. 0.69 P 0.5 where pto is the "period" of the substance undergoing dis- integration. The periods of radium and the main series of radio-active 2 C 386 CHAP. INTRODUCTION TO PHYSICAL CHEMISTRY elements derived from it are given in the following table, together with an indication of the radiation emitted at each stage, and of the range of the a-particles in air at one atmosphere pressure and 15°. Element. Radium ↓ Niton ↓ Ra A ↓ Ra B ↓ Ra C ↓ Ra C' ↓ Ra D ↓ Ra E ↓ Ra F (Polonium) ↓ (Lead) Atomic Weight. 226 222 218 214 214 214 Period. 210 1800 years 3.85 days 3.1 min. 27 min. 20 min. 16 years 210 5 days 210 136 days ca. 10-6 sec. = ∞ Minutes. 9.4 × 108 5.5 × 103 •31 × 10 2.7 × 10 2.0 × 10 ca. 10-8 8.4 × 106 7 × 103 1.9 × 105 ∞ Radiation emitted. a a α β, γ B, Y = α slow B β, γ α Range of a-particle. 3.1 cm. 4.2 "" 4.8 | 7·0, | "" 3.8 ") 206 The period of radium emanation is 5.5 × 103 min. The velocity constant for its decay is therefore k 1·25 × 10-4, i.e. 0.69 5.5 × 103 this fraction of a given portion of the emanation would be transformed in one minute, the original quantity being assumed to be constant. Now, if we consider an intermediate product of the decay of radium, say the radium emanation, we find that, at any time, it is being formed by the disintegration of radium, and at the same time it is disappear- At first the ing by its own disintegration into helium and radium A. radium emanation will accumulate, but after a time its quantity will have increased to such an extent that in a given interval as much of At this it will pass into radium A as is formed from radium itself. point then what is known as radio-active equilibrium has set in, so far as the production of the emanation from radium and its disintegra- tion into radium A is concerned. In the same way, after a certain interval has elapsed, radium A will enter a state of radio-active equilibrium with radium emanation on the one hand, and radium B on the other hand, and so it will be If we suppose for the other disintegration products of radium. complete radio-active equilibrium to be reached with radium and all its XXXIII RADIO-ACTIVE TRANSFORMATIONS 387 disintegration products, we find from the unimolecular equation that there must be fixed ratios between all these substances, and that these ratios may be easily determined. Thus if r is an amount of radium and e the amount of emanation when there is equilibrium, we have for the production of the emanation and for its disintegration and dx dt dx dt 210 = k₁₂¹², for equilibrium these rates are equal, so that k₁r = k₂e r_k2_P1 k₁ P₂ = - k₂e; or the equilibrium amounts of the various transition products between the original radium and the final inactive product are in ratios inversely proportional to their disintegration constants, or directly proportional to their periods. Thus the periods expressed in the table of p. 386 in minutes give the relative amounts of radium and its products when the whole system is in radio-active equilibrium. We may look upon the ordinary elements as radio-active elements of excessively long period, so that the amounts in which they are found in the earth are necessarily large, when compared with such elements as radium, or still more polonium. Radium itself is a product of the slow disintegration of uranium, several intermediate radio-active substances occurring between these well-defined elements. The series is Uranium I → Uranium X1 → Uranium X2 → Uranium II → Ionium → Radium. Ordinary uranium is an inseparable mixture of Ur and UII, which are very feebly active, the period of the former being 5 × 109 years and of the latter about 2 × 100 years. The actinium series of radio-active elements forms a branch of the uranium series. We have seen that RaC disintegrates in two directions, about 0.03 per cent suffering an a-ray change and the remainder a ẞ-ray change, giving rise to RaC2 and RaC' respectively. Similarly it is supposed that UII disintegrates in two directions, although in this instance both changes are of the a-ray type, 92 per cent becoming ionium, the immediate parent of radium, and 8 per cent becoming uranium Y, which gives rise to ekatantalum, and his to actinium, which then disintegrates on lines roughly parallel to he disintegration of radium, yielding lead as the final stable product. The thorium series is apparently unconnected with the uranium eries. The period of thorium itself is about 18 x 1010 years. Again A 388 INTRODUCTION TO PHYSICAL CHEMISTRY there is a distinct parallelism with the other series, and once more lead would seem to be the ultimate product of the disintegration. Amongst the common elements potassium and rubidium have been found to exhibit a very slight ẞ-radio-activity. There is plainly a close connection between the valency of an atom and its electrical charges, and we should therefore expect that when an atom loses positive or negative charges its valency will change. The following generalisation concerning the variation of valency in a radio- active series was independently arrived at by Soddy and by Fajans. When an atom loses an a-particle it loses at the same time two positive unit charges, and its positive valency diminishes by 2. Thus the dyad radium by losing one a-particle becomes the non-valent radium-emanation. On the other hand, in a ß-ray change the atom loses an electron, that is one charge of negative electricity, and its positive valency increases by 1. So the tetrad UX1 becomes the pentad UX2 and the hexad UII by two successive B-ray changes. UX1 is undoubtedly tetrad as it is chemically indistinguishable from the tetrad thorium, and UII is chemically indistinguishable from the hexad UI. CHAP. Fig. 60 (p. 389) gives a diagrammatic representation of the disin- tegration of the main uranium series. Atomic weights are marked off on a vertical scale, and positive valency on a horizontal scale. Thus the elements in any vertical column have the same valency. Each a-ray change diminishes the atomic weight by 4 and the valency by 2; each B-ray change leaves the atomic weight unaltered and increases the valency by 1. The a-ray changes are consequently represented by diagonal lines directed upwards and towards the left; the B-ray changes are represented by horizontal lines directed to the right. The diagram should properly be bent round into a cylinder so that the right and left vertical axes coincide. As the weight of each atom has its seat in the nucleus (p. 377), each a-ray change must arise in the nucleus itself, the valency change being a concomitant alteration in the arrangement of the external ring of electrons. Similarly each ẞ-ray change must have its origin in the inner and not in the external part of the atom, since no change in external conditions has the slightest influence on the mode or rate of the disintegration. As far as can be ascertained, elements in the same vertical column of the table have not only the same valency, but are indistinguishable from one another by any chemical means, and therefore cannot be separated chemically. Such elements are said to be isotopic, as they occupy the same place in the periodic table. This statement, however only applies to elements on the same side, vertically, of the dotted line in the figure. The a-ray change which occurs when radium passes into niton brings about different results from the other a-ray changes expressed in the diagram (cp. p. 399). Thus uranium X1 and ionium XXXIII RADIO-ACTIVE TRANSFORMATIONS 389 are isotopic with each other, but not with radium B, radium D, and lead, which, however, form an isotopic group with one another and belong to the same natural family as the first isotopic group. It may be noted that elements of the actinium and thorium series are isotopic with elements of the main uranium series. Elements of the actinium series have necessarily on the theory under consideration atomic weights identical with those of the main uranium series (cp. Table, p. 392). Elements of the thorium series on the other hand 3 0 205 210 215 220 225 230 2.35 240 1 2 Ra RaC2 4 RaB RaD RaE Pb Io Th UXI 1 + 5 FIG. 60. RaC 1 UX2 6 RaA Po RaC' UII UI 7 Nt have atomic weights, throughout different from those of the other series, the starting-point being thorium with atomic weight 232. Its position is indicated in Fig. 60, which shows it to be isotopic with uranium X1 and with ionium. A very striking confirmation of this disintegration theory has been found in its application to the atomic weight of lead. Lead is the ultimate product of the disintegration of uranium, atomic weight 238, after 8 a-ray changes. Leaving aside fractional numbers, the atomic weight of lead should thus be 238 − (8 × 4) = 206. But lead is also the ultimate product of the disintegration of thoriumi, atomic weight 390 INTRODUCTION TO PHYSICAL CHEMISTRY CH. XXXIII 232, after 6 a-ray changes. Its atomic weight should therefore be 232(6 × 4) = 208. The atomic weight of ordinary lead is 207.2. Thus theoretically the atomic weight of lead from uranium should be below that of ordinary lead, whereas the atomic weight of lead from thorium should be above that of ordinary lead. Now lead found associated with uranium in uranium minerals, and presumably derived from uranium, has been found to have an atomic weight as low as 206′0 and 206·1, and lead from thorium minerals an atomic weight as high as 207.77. Here we are dealing then with isotopic elements of a non-radio-active character, chemically identical and inseparable. It is interesting to note that the specimens have densities varying in the proportion of their atomic weights so that their atomic volumes (p. 34) are identical. E. RUTHERFORD, Radio-active Substances and their Radiations (Cambridge Univ. Press, 1913). F. SODDY, The Chemistry of the Radio-Elements (London, 1917); The Interpretation of Radium (1920). CHAPTER XXXIV ATOMIC NUMBER AND ISOTOPY EXPERIMENTS on the scattering of X-rays had led to the conclusion that the number of free negative electrons in an atom was approxi- mately equal to half the atomic weight. Wholly independent experi- ments on the deflection suffered by a-particles in passing through thin metallic sheets pointed to the number of free positive charges in an atom being equal to about half the atomic weight. These two con- clusions are mutually confirmatory, since, the atom being neutral, the number of free positive charges on the nucleus must be equal to the number of free negative electrons circulating round it. The relationship thus established is only approximate, but even so it might suggest that as we pass from one element to that of next higher atomic weight, the nuclear charge would increase by one unit and also the number of free electrons by 1, since the increase in atomic weight at each step lies on the average between 2 and 3. The researches of Moseley on the X-ray spectra of the elements from aluminium (at. wt. 27) to gold (at. wt. 197) provide a firm basis for such a hypothesis. Moseley determined accurately by photo- graphy of the characteristic X-rays reflected from a crystal surface (p. 379) the frequency n of the principal lines given out by the elements when excited by cathode rays, and found that the frequencies could be expressed by a general formula n = A (N-b)², where A and b are constant for a given type of line and N is an integer, the atomic number characteristic of the element investi- gated. Knowing the values of A and b, the atomic number N of the element may be ascertained by measurement of n. As we proceed from one element to that of next higher atomic weight, experiment shows that the value of N increases by 1. Assuming the atomic number of aluminium, the thirteenth element in order of atomic weight, to be 13, known elements correspond with every whole number between 13 (Al) and 79 (Au) except 43, 72, and 75, and no element exists for which N has a fractional value. 391 392 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. At. No. TABLE I ELEMENTS ARRANGED IN ORDER OF ATOMIC NUMBERS At. Wt. At. No. 1 1. Hydrogen 2. Helium 3. Lithium 4. Glucinum 5. Boron 6. Carbon 7. Nitrogen 8. Oxygen 9. Fluorine 10. Neon 11. Sodium 12. Magnesium 13. Aluminium 14. Silicon 15. Phosphorus 16. Sulphur 17. Chlorine 18. Argon 19. Potassium 20. Calcium 21. Scandium 22. Titanium 23. Vanadium 24. Chromium 25. Manganese 26. Iron 27. Cobalt 28. Nickel 29. Copper 30. Zinc 31. Gallium 32. Germanium 33. Arsenic 34. Selenium 35. Bromine 36. Krypton 37. Rubidium 38. Strontium 39. Yttrium 40. Zirconium 41. Columbium 42. Molybdenum 43. 44. Ruthenium 45. Rhodium 46. Palladium 47. Silver 48. Cadmium 49. Indium 50. Tin 51. Antimony 52. Tellurium 53. Iodine 54. Xenon 55. Cæsium 56. Barium 57. Lanthanum 58. Cerium 59. 60. Neodymium 61. Praseodymium .... 62. Samarium 63. Europium H HEGAOZOFŽŽ He Li Gl B с Ne Na Mg Al ˜ˆÙÕÃ×Õ¿E>&££8Z68 Sc Ti Mn Fe Ni Ga Ge As Se Br Kr Rb HOFFOR LEE: FREE I Xe 별명​: 봄​부응​닭​볶음 ​Nd Sa 4 7 9 11 12 14 16 19 20 23 24 27 28 31 32 35 40 39 40 44 48 51 52 55 56 59 59 64 65 70 72 75 79 80 83 85 88 89 91 94 96 102 103 107 108 112 115 119 120 127 127 130 133 137 139 140 141 144 150 152 64. Gadolinium 65. Terbium 66. Dysprosium 67. Holmium 68. Erbium 69. Thulium 70. Ytterbium 71. Lutecium 72. 73. Tantalum 74. Tungsten 75. 81. 76. Osmium 77. Iridium 78. Platinum 79. Gold 80. Mercury Thallium Actinium D Thorium D Radium C2 84. Thorium-lead 82. Radium-lead 2 88. 12 ….. 89. 90. ...... 83. Actinium C Thorium C Radium C (Polonium Thorium C Radium C' Actinium A Thorium A Radium A 91. 92. Actinium-lead Radium-lead 1 Lead 85. Actinium-eman. 86. Thorium-eman. Niton 87. Radium D Actinium B Thorium B Radium B Bismuth Radium E ••• ………….. Actinium X Thorium X Radium Mesothorium 1 f Actinium Mesothorium 2 Radio-actinium Radio-thorium Ionium Uranium Y Thorium Uranium X1 Ekatantalum Uranium X2 Uranium 11 Uranium I ˜€±á€² £Þ 8פäA Gd Tb Ds Ho Er Tu Yb Lu Ta W • Os Ir Pt Au Hg TI AcD ThD RaC2 AcPb RaPbl Pb ThPb RaPb2 RaD AcB Th B RaB ... At. Wt. 157 159 162 163 168 169 173 175 AcX ThX Ra MsTh1 Ac MsTh2 Rd Ac RdTh Io UY Th UX1 181 184 Bi RaE AcC The RaC Po ThC' 212 RaC' 214 AcA 214 ThA 216 RaA 218 191 193 195 197 201 204 206 208 210 206 206 207 208 210 210 210 212 214 AcEm 218 ThEm 220 Nt 222 208 210 210 212 214 210 ... 222 224 226 228 226 228 226 228 230 230 232 234 EkTa 230 UX2 234 UII 234 UI 238 XXXIV 393 ATOMIC NUMBER AND ISOTOPY The atomic number is plainly the expression of some fundamental property or constituent of the atom, and it was identified by Moseley with the number of free positive charges existing in the nucleus, and consequently with the number of free negative electrons rotating about the nucleus. According to this identification hydrogen has one 0. 2. He 10. Ne 18. Ar 36. Kr 54. Xe TABLE II PERIODIC CLASSIFICATION ACCORDING TO ATOMIC NUMBERS 1. 1. H 3. Li 11. Na 19. K 29. Cu 37. Rb 47. Ag 55. Cs 79. Au AcEm 86. ThEm $7. Nt .. 2. 4. Gl 12. Mg 20. Ca 30. Zn 38. Sr 48. Cd 56. Ba SO. Hg SS. ľ AcX ThX Ra MsTh1 5. B 13. Al 21. Sc 3. 31. 39. Y $1. Ga 49. In 57. La 58. Ce 59. Pr 60. Nd 61. ... 62. Sm 63. Eu 64. Gd 65. Tb 66. Ds 67. Ho 68. Er 89. CX 69. Tu 70. Yb 71. Lu { ΤῚ AcD ThD RaC2 Ac MsTh2 6. C 14. Si 22. Ti 4. 32. Ge 40. Zr 50. Sn 72. 90. AcPb RaPbl ThB RaB Rd Ac RdTh Io UY Th UXI 5. 7. N 15. P 23. V 33. As 41. Cb 51. Sb Pb Bi ThPb RaE $2. RaPb2 | 83.√ AcC RaD The RaC AcB 73. Ta 91. EkaTa UX2 S. O 16. S 24. Cr 6. 34. Se 42. Mo 52. Te 74. W $4. 92. Po ThC' RaC' ACA ThA RaA GUIL UI 7. 9. F 17. Cl 25. Mn 35. Br 43. 53. I 75.. $5. ** ·· 1 8. 26. Fe 27. Co 28. Ni 44. Ru 45. Rh 46. Pd 76. Os 77. Ir 78. Pt free electron, helium two, lithium three, etc., and it will be seen that for elements of low atomic weight at least, the atomic number is approximately half the atomic weight, in accordance with the con- clusion previously mentioned. Table I. shows the arrangement of the elements in the order of their atomic numbers. Atomic weights are also given to the nearest integer. It will be noted that with few exceptions (e.g. argon and 394 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. potassium) the order of atomic number is also the order of atomic weights until an atomic weight of 200 is reached. Thereafter we have groups of isotopic elements, often of very different atomic weight, under each atomic number, and considerable overlapping of atomic weight occurs as we pass to successive atomic numbers. The majority of the atomic weights in the last column have, of course, not been determined directly, but are deduced from the theory of disintegration referred to in Chapter XXXIII.; nevertheless the ex- perimental confirmation in the case of lead is so striking (p. 389) that the general arrangement may be accepted with confidence. The elements of the actinium series are the least certain. Taking as basis the atomic number instead of the atomic weight, the periodic classification of the elements assumes the form shown in Table II. In the vertical columns are found elements of the same valency and general chemical character, the numbers at the head of the columns indicating the positive valency. Isotopic elements are bracketed together. It will be seen that there are still a few blanks left in the table. These blanks in all probability correspond to elements that have yet eluded discovery. Thus no element is known with an atomic weight. between those of molybdenum and ruthenium. Yet Moseley's work shows molybdenum to have the atomic number 42, and ruthenium the atomic number 44. It is therefore extremely probable that an element of atomic number 43 exists, but is hitherto undiscovered. A glance at the periodic table shows the probable general chemical character of the missing elements. They may be summarised as follows: 43. and 75. Analogues of manganese. 61. A rare earth metal. 72. An analogue of zirconium. 85. A halogen element. 87. An alkali metal. So far we have encountered isotopy only amongst the radio-active elements or their final disintegrated products, but the method of positive rays has greatly extended the concept and thrown much light on the nature of the elements. We have seen (p. 375) that in a highly evacuated discharge - tube a stream of negative electrons flows constantly outwards from the cathode; but this is only one side of the phenomenon. Positively charged particles are at the same time moving towards the cathode, and if the cathode is perforated they pass through the perforations and appear behind the cathode in the form of luminescent rays known originally as canal- rays, which have the power of exciting fluorescence, discharging an electroscope, and affecting a photographic plate. They are known to carry a positive charge from the effect of electric or magnetic fields XXXIV ATOMIC NUMBER AND ISOTOPY 395 upon them. The deflection which they suffer through such fields may be recorded by a photographic method, and from measurement of the deflection their velocity and the ratio of mass to charge may be calculated. If the magnetic and electric fields are coincident (J. J. Thomson) particles having the same ratio m/e but different velocities produce a parabolic trace on the photographic plate. If the fields are applied in succession at right angles to each other (Aston), then for small ranges of velocity particles of constant m m/e form a sharp line across a photographic strip. The position of the line, which resembles a line of definite wave-length in a light spectrum, may be accurately measured, and the corresponding value of m/e deduced. From this superficial analogy such photographic records have been termed mass-spectra, the scale being not one of wave- lengths but of m/e ratios. The ratios show that an element may under the conditions of experiment be charged with an amount of electricity altogether different from that corresponding to its electro-valency in compounds. Thus the atoms of non-valent elements appear with one, two, or more unit-charges of positive electricity. Mercury appears not only with one or two positive charges, but with three or four, and even some- times with negative charges. It should be borne in mind that the conditions are altogether abnormal; the gas is charged in an intense electric field, the proportion of charged to uncharged atoms or molecules is almost inconceivably small, and the gas is so tenuous that the effect of collisions is greatly reduced. When compounds are under observation charged groups of atoms may appear that under ordinary conditions can lead no independent existence; thus in hydrocarbon vapours there is evidence of the presence of the molecules CH, CH₂, CH, etc. An early observation of the positive rays from neon showed that, besides molecules having the normal ratio m/e=20 in grams per faraday, there was a faint parabola corresponding to the ratio 22. As a constituent with this ratio could neither be removed nor accounted for as an impurity, it was attributed to an isotope of neon. This observation has been repeatedly confirmed, but attempts to separate the two isotopes by diffusion owing to their difference in density have not hitherto met with much success. An extended investigation of the elements from this point of view has led to remarkable results. Since it is estimated that the error in the m/e ratio does not greatly exceed 0.1 per cent, it is possible to compare with each other to this degree of accuracy the atomic weights of elements giving mass-spectra. Some elements would appear to be simple; others would appear to be mixtures of two or more isotopes. The following list embodies the chief results hitherto obtained. Under the heading m/e are the relative masses of the atoms derived from the mass-spectra. Amongst the isotopic 396 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. elements the numbers are in the order of intensity of the observed lines, numbers in brackets being provisional only. Element. H He 4.00 9.1 12.00 14.00 16.00 JOZOFAS Simple or Non-isotopic. At. Wt. Gl с N P As I m/e. Element. At. Wt. 1.008 7.00 Li 4.00 B (9) 12.0 14.0 16.0 19.0 31.0 32.07 32.0 75.0 75.0 126.92 127.0 1.0077 19.0 31.0 11.0 20.2 28.3 35.46 39.9 79.92 Kr 82.9 Ne Si Mixed or Isotopic. m/e. Ar Br Xe 130.2 Hg 2004 (6, 7) 11.0, 10.0 20.0, 220, (210) 28·0, 290, (30·0) 35·0, 37·0, (39⋅0) 40·0, (360) 79.0, 81·0 84, 86, 82, 83, 80, 78 129, 132, 131, 134, 136 (197-200), 202, 204 It is immediately apparent from the table that the simple elements all have atomic weights closely approximating to whole numbers. All atomic weights, on the other hand, which depart greatly from integers are found amongst the isotopic elements. We may infer, provisionally at least, that elements in the strict sense have atomic weights expressible by whole numbers, those elements with fractional atomic weights being in reality not single elements but mixtures of isotopes. Since these isotopes are inseparable by ordinary chemical means, the apparent fractional atomic weight is constant, no matter to what method of purification the substance may have been subjected. Diffusion experiments on gaseous isotopic elements like chlorine, or compounds like hydrochloric acid, may lead to the actual separation of the isotopes. Indeed, a physical method of separa- tion has apparently met with some success in the case of mercury. When the carefully purified metal is vaporised in vacuo without boiling, the vapour being at once condensed on a cold surface near the surface of the evaporating liquid, a minute difference in the density of the mercury remaining and that distilled has been observed. This difference remains unaltered by subsequent ordinary distillation. The position of hydrogen in the system is peculiar. The divergence from unity might be due to the presence of a heavier isotope, say of atomic weight 2 or 3, but no trace of such an isotope has been found, and the position of the line in the mass-spectrum is accurately 1.008 and not 1.000. A plausible reason for the abnormality will be referred to in the next chapter. From what has been said, the atomic weight of our elements as ordinarily conceived is not a characteristic number, but rather in many cases an apparently fortuitous average of weights which have XXXIV ATOMIC NUMBER AND ISOTOPY 397 real significance. For classification and comparison atomic number and not atomic weight should therefore be taken as the fundamental magnitude. H. G. J. MOSELEY, "The High Frequency Spectra of the Elements," Part I. Phil. Mag. 6th series, 26, p. 1024 (1913); Part II. ibid. 27, p. 703 (1914). F. W. ASTON, "The Mass-Spectra of Chemical Elements," Phil. Mag. 6th series, 39, p. 611; 40, p. 628 (1920); 42, p. 140 (1921). CHAPTER XXXV THE STRUCTURE OF ATOMS In the foregoing chapters we have had occasion to discuss properties of the elements and their compounds which lead to definite speculations as to the nature and structure of atoms. We have seen that atoms in the present state of physical science must be regarded as structures of positive and negative electricity-the negative electricity being in the form of electrons of negligible mass, the positive electricity on the other hand being definitely associated with the massive part of the atom. This massive part is universally recognised as forming the nucleus of the atom; in it there is always found an excess of positive electricity, though in every instance save that of hydrogen the nucleus of the atom contains negative electrons as well. About the nucleus and far external to it are electrons, the sum of the negative charges on which exactly balances the positive excess of the nucleus. It was at first thought that the number of external electrons in an atom was comparatively large, but we have seen in the previous chapter that this idea is erroneous, Moseley's assumption that the atomic number represents the number of extranuclear electrons, or the number of positive charges of the nucleus, being in good agreement with all the known experimental data. With this quantitative basis the problem of constructing an atomic model therefore resolves itself for a given element into fixing an arrangement and a mode of motion of a definite number of electrons about a nucleus of known mass and charge. A satisfactory model should give some account not only of the physical properties of the elements, e.g. their spectra, but also of their chemical properties, e.g. their valency and their mode of union to form compounds. It may be admitted that no generally satisfactory model has yet been arrived at, but considerable success in special directions has been attained. If we take into consideration the periodic character of the system of the elements we naturally conclude that the arrangement of electrons in successive elements must present a periodic recurrence, 398 CH. XXXV THE STRUCTURE OF ATOMS 399 and that the atoms of elements belonging to the same family must resemble each other in structure. This conclusion may be most readily represented by the assumption that the electrons are arranged round the nucleus in successive rings or layers. Only the outer ring or layer is affected by other atoms, so the chemical character is due to this outer portion, which should therefore be similar for elements. of the same natural family. If the atom readily loses electrons from this outer layer, thereby assuming a positive charge, it is what we term electropositive; if it readily takes up electrons from other atoms. it is electronegative. Thus the metals of the alkalies may be conceived as being composed of a stable kernel (which consists of a nucleus and one or more layers of electrons) together with a single external electron which the atom readily loses, thereby assuming a unit positive charge. The metals of the alkaline earths similarly would have two such easily detachable external electrons. The halogens, on the other hand, may be regarded as having such an external layer as would rather take up an extra electron than part with those it already contains, and would thus tend to assume a negative charge. The non-valent elements are those of maximum stability, their external layers neither giving nor taking electrons. A systematic treatment on these lines was originated by G. N. Lewis and developed by Irving Langmuir. It may be illus- trated by the consideration of the elements between neon and potassium. Atomic number Outer layer. Positive valency Negative valency. • Ne Na Mg Al 10 12 13 8 2 3 4 0 2 3 4 0 4 11 1 1 Si P CI Ar K 14 15 16 17 18 19 S 1 0 1 0 त 10 10 00 5 5 3 Τ 6 6 2 7 7 1 • The atomic number gives the total number of extranuclear electrons. In passing from neon to sodium a new outer layer is begun; the extra electron is easily lost and we have a univalent positive element. In magnesium we have a bivalent, and in aluminium a trivalent positive element. The atom of silicon has four outer electrons, and may reach the stable arrangement of eight found in neon and argon either by acting as a quadrivalent positive element and losing four electrons, or as a quadrivalent negative element and gaining four electrons from other atoms. Silicon is actually capable of acting in both of these ways, forming both SiCl, in which it is electropositive, and SiH, in which it is electronegative. Similarly phosphorus can form both PC1, and PH,, and chlorine Cl₂O, and ClH, the sum of the positive and negative valencies when both are developed being always eight. After the new stable group of eight is formed in argon, the addition of a further electron in potassium leads to the beginning of still another outer layer, and we have a repetition of the arrangement. found in sodium with consequent similar properties of the element. 5 400 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. The conception of a stable octet of electrons is the fundamental idea of the Lewis atomic model. If an atom has already an external octet, it is stable and inert, and tends to enter into no combination with other atoms: such atoms are found in neon and argon. If, on the other hand, the atom has not an external octet it strives to form one by combination with other atoms; and this it may do in two ways, which may be exemplified on the one hand by the combination of sodium and chlorine atoms to form a molecule of sodium chloride, and on the other hand by the combination of two chlorine atoms to form a molecule of chlorine. These two modes of combination are illustrated in Fig. 61. The sodium atom has an outer layer of one + 11 17 (11) 17 g g áà g Na Na Cl 18 Ar 17 17 99 99 17 (17 C1 C12 FIG. 61. electron, the chlorine atom has an outer layer of seven electrons. By the passage of the single electron from sodium to chlorine two external octets are formed. The outer layer of the sodium atom now corresponds to that of neon; the outer layer of the chlorine atom now corresponds to that of argon. It will be observed, however, that the nuclei are different from the nuclei of these inert gases, argon for example having 18 positive charges and the completed chlorine octet only 17. In consequence of the resulting want of balance of positive and negative electricity the sodium atom in the compound has one positive charge, the chlorine atom one negative charge, and the compound is held together by the mutual attraction of the oppositely charged atoms. This type of combination is termed polar and is shown most markedly in ionisable compounds. The combination of THE STRUCTURE OF ATOMS 401 chlorine atoms on the other hand is of a non-polar type. Here the octets are completed (Fig. 61) by two electrons becoming common to two octets, just as in naphthalene two carbon atoms are common to two rings. Neither of the atoms is more positive or negative than the other, and the compound is in the ordinary sense non-ionisable. In this scheme of formulation one pair of electrons common to two octets corresponds to a single bond in the ordinary graphic formulation, two pairs of electrons to a double bond, and so on. XXXV The student should note that the lines (cube edges) in the diagrams have no physical significance; they are present merely to aid the eye in grouping the electrons. Nor should too literal an interpretation be placed on the cubical arrangement, the cubical form being liable to distortion by the mutual attractions and repulsions. of the charged particles. It is noteworthy in particular that if we arrange the four external electrons of the carbon atom at alternate corners of a cube, leaving the other four corners blank, we obtain a tetrahedral carbon atom in correspondence with the ordinary assump- tion of organic chemistry. Langmuir has laid down a series of rules for the arrangement of electrons in the atoms of all the elements, taking as his stable starting-points the inert gases, the numbers of electrons in which are as follows: He Ne 2 10 Ar 18 Kr Xe 36 54 i Nt 86 In these stable arrangements, as pointed out by Rydberg, the numbers are expressed by the following formula: 2(1² + 2² + 2² + 3² + 3² + 4º) which may be easily interpreted by making each term within the brackets represent a half-layer of electrons. The system so developed corresponds with the long and short periods found in the classifica- ion of the elements, and accounts in a simple manner for many of he observed properties of the elements and their compounds, both physical and chemical. │ So far we have assumed, for the purpose of explaining valency and hemical properties generally, that the electrons are fixed, but if we re to explain the spectra of the elements we must assume that the lectrons are in motion. An atomic model due to Bohr and modified y Sommerfeld postulates that the electrons revolve in circular or liptical orbits round the nucleus, and has met with striking success. 1 dealing with the spectra of the simpler atoms. The Balmer series br hydrogen, for example (p. 378), has been deduced not only in its rm but in the value of the constant N. This type of motion of the ectrons, however, is inadequate to account for the general phenomena valency. A compromise may conceivably be reached on the basis at the electrons of the Lewis-Langmuir atom move in paths about 2 D 402 CHAP. INTRODUCTION TO PHYSICAL CHEMISTRY centres which are the valency positions represented as stationary points in the formulae. It might also be that both types of model. correspond to actual but different states of the atom, the Lewis- Langmuir model corresponding to the atom in its ordinary state, and the Bohr-Sommerfeld model to the independent radiating atom for which questions of valence and combination are nugatory, and which may be regarded as a kind of isomer of the ordinary atom. - Turning now to the nucleus of the atom wherein mass in the ordinary sense resides, we recognise in the first place that it is remote from external interference, being protected by the outer electrons. It thus plays no part in chemical combination, and the properties due to it remain unaltered, irrespective of the mode of combination of the atom. This is obviously true as regards mass (p. 148) and radio- activity (p. 382). The nuclei of the radio-active elements are un- stable to a greater or less degree, and by the expulsion of electrons. or of helium nuclei they strive to attain a state of greater stability. For any given kind of radio-active atom the rate of disintegration is constant, as it is a property of the nucleus alone, aud is independent of temperature or mode of combination. Since the energy change involved in such disintegrations is very great, we must conclude in general that the energy content of the nucleus is enormous compared with the energy associated with the valency electrons. Isotopic elements are those with the same arrangement of extra- nuclear electrons but with different nuclei. These nuclei may be of different mass, in which case the isotopes are heterobaric, i.e. of different atomic weight, and physically separable; or the nuclei may be of the same mass in which case the isotopes are isobaric, i.e. of the same atomic weight, and are physically inseparable. The nuclei in the latter case are isomeric, but if radio-active they are still distinguish- able owing to their different degrees of stability and consequent different types of radiation, e.g. uranium-Y and ionium. As to the structure of the massive portion of the nuclei it would now seem that we must return to the view of Prout who suggested a century ago that all atomic weights are multiples of that of hydrogen. for the reason that all other atoms are built up of atoms of hydrogen First we have the fact that radio-active nuclei emit helium nuclei ir their a-ray transformations. Helium nuclei must, therefore, be con sidered proximate constituents of the larger nuclei. Then there i the fact that all the elements proper yet investigated by the positiv ray method (p. 396) have atomic weights which are exact multiple of one number, namely 1 on the basis of O = 16. This fundamenta number is interpreted as the characteristic weight of a massive co stituent of all atoms, a constituent which has received the nan proton. Proton is naturally identified with the nucleus of hydroge the only difficulty being that the atomic weight of hydrogen, estimated either from chemical data or by the positive ray method, ་ XXXV 403 THE STRUCTURE OF ATOMS not identical with the characteristic weight of proton, namely 1 exactly, but has the value 1.0076, nearly 1 per cent greater. It is, how- ever, not unlikely that by close packing of the hydrogen nuclei, together with electrons, a change of mass may have been brought about, the difference appearing as the mass of the electrical energy liberated during the condensation of the protons to form the nuclei of other elements. These protons may be conceived be conceived as being grouped with cementing electrons into subsidiary structures such as the helium nucleus, which may be written (Prt,e,) and appears as a constituent of larger nuclei. Groups of two and of three protons are also probable, cementing electrons being present in each case. Hydrogen remains exceptional in its intimate atomic structure as it is in its general properties, and is outside the classification which embraces all the other elements. The nucleus of the atom of hydrogen is the only nucleus which contains no negative electrons. The nuclei of the radio-active elements are essentially unstable: those of the other elements are essentially stable and require great concentration of energy to disrupt them. That the disruption, however, is possible has been shown by Rutherford, who, by exposing oxygen and nitrogen to the impact of swiftly moving a-particles, obtained particles which had a greater range and velocity than the activating a-particles themselves. These rapid light particles have been shown to have a mass approximately equal to 3, and a charge equal to that of the a-particle. They would therefore correspond to the formula (Prt,e). It would thus appear that a group of three protons has been detached from the nuclei of oxygen and of nitrogen atoms. In virtue of the double charge the group may be regarded as an isotopic helium nucleus. Single protons (hydrogen nuclei) were also produced simultaneously from nitrogen, but were absent from oxygen. Thus disintegration of atomic nuclei has been brought about by the application of energy external to them: the reverse phenomenon of the synthesis of nuclei from hydrogen may possibly be found in the observations of Collie and others on the appearance of helium and neon in vacuum tubes originally containing only hydrogen, after long passage of an electric discharge. ·· · G. N. LEWIS, "The Atom and the Molecule," Jour. Amer. Chem. Soc., $8, p. 762 (1916). 18, IRVING LANGMUIR, "Arrangement of Electrons in Atoms and Molecules," our. Amer. Chem Soc., 41, p. 868 (1919). W. D. HARKINS, "Stability of Atoms," Jour. Amer. Chem. Soc., 42, 1956 (1920); "The Constitution and Stability of Atom Nuclei," Phil. Tag., 42, p. 305 (1921). E. RUTHERFORD, "Nuclear Constitution of Atoms," Proc. Roy. Soc., eries A, 97, p. 374 (1920). 404 INTRODUCTION TO PHYSICAL CHEMISTRY CH. Xxxv COLLIE, PATTERSON, AND MASSON, "Production of Neon and Helium by the Electric Discharge," Proc. Roy. Soc., Series A, 91, p. 30 (1914). N. BOHR, "Structure of the Atom," Phil. Mag., 30, p. 394 (1915). J. J. THOMSON, "On the Origin of Spectra and Planck's Law," Phil. Mag., 37, p. 419 (1919); "On the Structure of the Molecule and Chemical Combination," ibid. 41, p. 570 (1921). L. VEGARD, “On the X-ray Spectra and the Constitution of the Atom," Phil. Mag. 35, p. 293 (1918); 37, p. 237 (1919). A. SOMMERFELD, Atombau und Spektrallinien (1921). CHAPTER XXXVI THERMODYNAMICAL PROOFS THE experience of practical and scientific men alike goes to show that it is impossible to construct a perpetual motion machine. In the general acceptance of the term, a perpetual motion machine is one from which more energy can be obtained than is put into it from the outside. This may be termed a perpetual motion of the first class. But another kind of perpetual motion machine might exist—one, namely, which could by a recurring series of processes continuously afford mechanical energy at the expense of the heat of surrounding bodies at the same temperature as itself. This kind of perpetual motion is also impossible, and has been termed perpetual motion of the second class. It must be clearly understood that not only have such machines never been constructed, but that no increase in our experimental skill at present conceivable could lead to their construction. If there- fore in argument we find that an imaginary series of processes would lead to a perpetual motion of either of the kinds mentioned above, we conclude that such a series of processes can have no real existence. The denial of the possibility of the existence of the two sorts of perpetual motion is contained in the two following positive and general statements, which are known as the First and Second Laws of Thermodynamics respectively. I. The energy of an isolated system remains constant. II. The entropy of an isolated system tends to increase. With the second law in its general and formal aspect we shall have here little to do, and shall use in its stead the negative proposi- tion given above. Entropy is a function which, while theoretically of great value as indicating the direction in which chemical or other processes take place, and in fixing the conditions of equilibrium, is somewhat difficult to grasp and not susceptible of direct measurement, ko that it is of less obvious and immediate practical importance. The First Law states the principle of the Conservation of Energy- ›y an isolated system being meant a system which can neither give ip energy to its environment nor absorb energy from it. The sum of 405 406 CHAP. INTRODUCTION TO PHYSICAL CHEMISTRY the different kinds of energy in such a system is always the same, no matter what forms the energy may assume. In order to ascertain practically that the sum of the energies is constant, we must obviously have one kind of unit in which all energies may be expressed. If we work with only one kind of energy, we express its amount in the appropriate unit. Thus we express amount of heat in calories; electrical energy in volt-coulombs, or the like; and mechanical energy in foot-pounds or gram-centimetres. In dealing with different kinds of energy, we may take any of these units as the standard in which we express all the different varieties, for we know the factors necessary for converting one unit into any of 'the others. One calorie, for example, is under all circumstances equivalent to 42,720 gram- centimetres, so that if we wish to add heat energy and mechanical energy together, we must either multiply each calorie of heat energy by 42,720 to convert it into gram-centimetres, or divide the number of gram-centimetres by the same number in order to find their equivalent in calories. A few simple deductions from the First Law, involving only the consideration of thermal and mechanical energy, will now be given. When a small quantity of heat dQ is supplied to a gram-molecular weight of a perfect gas at constant volume, the heat goes entirely to raise where the temperature of the gas, and we may therefore write dQ = C„dT; C, is the heat capacity of the gram molecule at constant volume (cp. p. 30), and dT the small rise of temperature which this amount of gas experiences. If the heat is supplied to the gas under constant pressure, it not only goes to raise the temperature, but is also partially converted into work done by the gas during expansion. If p is the constant pressure, and dv the change of volume, this work is repre- sented by the product pdv. Now according to the First Law dQ = C¿dT + pdv, (1) since no other kind of energy is involved, the internal energy of a perfect gas being independent of its volume (cp. p. 32). In the gas equation for the gram molecule (pv = RT), p, v, and T are variable, so that on differentiation it becomes pdv + vdp = RdT. (2) By eliminating dT from equations (1) and (2) we obtain Cv + R R dQ= = pdv + Cvdp; R dQ = or since C₂ = C₂+ R, as was shown on p. 30, Cv ppdv + vdp. R R Now, if the pressure, volume, and temperature of the gas b XXXVI THERMODYNAMICAL PROOFS 407 allowed to change adiabatically, i.e. in such a way that heat neither enters nor leaves the system, we have dQ = 0, and consequently Сppdv + С₂vdp = 0. Introducing into this equation the ratio of specific heats k = Cp/C, we have kpdv + vdp = 0 ; dv dp k + 0. V p This equation, when integrated between the values pv and p₁₁, gives or whence or k(logev₁ − logev) + logeP₁ − logep = 0 ; loger - logel1; k logev₁ – logev 1 P 1 2- (4) P1 since pv = RT. This result is of importance as it enables us to ascertain the ratio of the specific heats of gases by observations of pressures and volumes under circumstances in which heat can neither leave nor enter the gas considered. For example, during the rapid compressions and dilatations which constitute the passage of sound through a gas, there is no time for the temperature change at any portion of the gas to communicate itself by conduction to neighbouring portions, so that the process instead of being isothermal, is, so far as any given portion of the gas is concerned, an adiabatic process, the gas being cooled at each dilatation and warmed at each compression. The gas then does not in these circumstances obey Boyle's Law, which holds good only for isothermal change of pressure and volume, but the law given above, where the pressure is not inversely proportional to the volume, but to the k-th power of the volume. It is owing to this circumstance that we can deduce the ratio of specific heats of a gas from the speed at which sound travels in it (cp. p. 31). We have now to find the law connecting the volume of a gas with its absolute temperature when it is heated adiabatically, i.e. by com- pression, since no heat as such must enter the system. This can easily be done by eliminating pdv from equations (1) and (2), the result being dQ = (C₂+ R)dT – vdp = СpdT – vdp = = C₂dT - k dp P. • RT, 408 CHAP. INTRODUCTION TO PHYSICAL CHEMISTRY For an adiabatic compression dQ = 0, so that C₂dT - dp¸RT = 0, p dT dp Cp Cp - Cv T p k dT ᏧᎢ - dp k-1'T P Integrating between T, p, and T₁, P₁, we obtain k or k. k S ·(logeT₁ - logeT) – (logep₁ – logep) = 0, 1 T T log. = (-1) log, h-1 T k ())*=())**! དང But since in an adiabatic process Tk (+1)* · T T Τ P1 p V V (+1) = k-1 =(%) k = 0; " 0. k) k - l P1, 'p' we obtain finally This result we shall find useful in calculating the maximum amount of work which can be obtained from a given quantity of heat under given conditions, a problem which we shall now proceed to solve. According to our statement of the second law, no work can be obtained by a cyclical process from the heat contained in a number of bodies all having the common temperature of their surroundings.. In order to convert heat into work, we must have temperature differences. A body in cooling through a certain range of temperature parts with a certain amount of heat, and a definite fraction of that heat may be transformed into work. In actual practice, different engines will effect the conversion of different amounts of the heat, but there is a theoretical limit which no engine, however perfect, can exceed, and it is our task to find what that limit is. The process of reasoning is essentially the same as that of Carnot, who introduced the conceptions necessary for the solution of the problem, and arrived at the desired result, although to him heat was a material substance, and not, as we now believe, a form of energy. The fundamental conception is that of a reversible cycle of opera- tions. By a cycle of operations is understood a series of processes which leaves the system considered in exactly the same state as it was initially, and the term reversible applies not merely to the direction XXXVI 409 THERMODYNAMICAL PROOFS of the mechanical operation, but to the complete physical reversibility of all the processes involved. A reversible heat-engine after converting a certain amount of heat into work will return to its original state in every respect if made to act backwards step by step, so that the work obtained is entirely converted into heat by its agency. Such an engine is of the maximum possible efficiency, for if any engine were more perfect, a perpetual motion could be obtained. This may be shown as follows. Let A be a reversible engine, and let B be an engine which under the same conditions can convert a larger proportion of heat into work than A. Let the two engines work between the same temperatures, and sup- pose that when a quantity of heat Q is given to A, the proportion q is converted into mechanical work. If the work corresponding to q is now done upon this engine so that the processes are reversed, the system will arrive exactly at its initial state, the quantity of heat Q – q being raised from the lower to the higher temperature. Now instead of letting the engine A act directly, take in its stead the more perfect engine B, and supply it at the higher temperature with the quantity of heat Q. A greater proportion of this than before is converted into work, say q', the quantity of heat Q-q' falling to the lower tempera- ture. By using the reversible engine to perform the reverse trans- formation of work into heat, we can regain the original quantity of heat Q at the original temperature, by expending mechanical energy equivalent to q, the quantity of heat Q-q being at the same time raised to the higher temperature. In the whole series of operations, then, the heat qʻ- q has been taken in at the lower temperature and an equivalent amount of work has been gained. This cycle of operations can be repeated as often as we choose, so that here we should have a system capable of giving an indefinitely large amount of work at the expense of heat at the lower temperature, which might be the uniform temperature of the surroundings, i.e. we should have realised a perpetual motion of the second class. We conclude, then, that an engine more perfect than the reversible engine A cannot exist. It will be noticed that nothing is said as to the nature of the working substances in the reversible or the other engine, so that the conclusion is perfectly general. We are at liberty therefore to use any kind of reversible engine in our calculations in order to ascertain the maximum quantity of heat which can be converted into work, and we shall find it convenient to take for our working substance a perfect gas, on account of the simplicity of the laws which it obeys. We must first of all investigate if the processes through which we put the working substance are really reversible. The condition of reversibility is that the state of the system at any time does not differ sensibly from equilibrium, for then the slightest variation. in the conditions will determine the occurrence of the process in the one direction or the other. If, therefore, we communicate heat 410 CHAP. INTRODUCTION TO PHYSICAL CHEMISTRY to the gas, we must so arrange that the gas and the heat source have temperatures differing from each other by an infinitely small amount. Similarly, if the gas is to part with heat, it must do so to a heat-sink with a temperature lower than its own only by an infinitesimal quantity. If the gas is to be compressed, the external pressure at any instant must be only greater by an infinitely small amount than the pressure of the gas itself; and if the gas is to be expanded, the external pressure must be less than the pressure of the gas by an infinitesimal difference. The machine of course must be absolutely frictionless, for otherwise some work would have to be expended in moving the machine, and thus converted into heat, inde- pendent of the gaseous working substance which is alone considered. From all this it is obvious that a reversible engine is an engine which can never be realised in practice. For an engine to be strictly reversible, there should be no departure from the conditions of equilibrium at any stage, in which case no process could occur at all, for the occurrence of any process naturally involves a departure from equilibrium. If the departure from the equilibrium conditions were infinitely small, the process would occupy an infinitely long time in its performance. We see then that a reversible process is an ideal, just as a perfect gas is an ideal. Neither can ever be met with in practice, but this in no way detracts from the value of the theoretical conclusions deduced by their aid. The series of operations which we shall perform on the gas will be best seen in the pressure-volume diagram, Fig. 62. We begin with the gas in the state represented in the diagram by the point 1. At the con- stant temperature T we let the gas slowly expand until its pressure and volume are indicated by the point 2. The form of the curve obtained during p RY V₁ 2 3 te δ β 7 FIG. 62. て ​the expansion is the rectangular hyper- bola of gases (cp. p. 73). We next isolate the gas from the heat source of constant temperature T, and let the expansion continue adiabatically until the point 3 is reached. Since the pressure varies more rapidly with the volume in an adiabatic than in an isothermal process (p. 407), the line 2, 3 will be more inclined to the volume axis than 1, 2, as is shown in the diagram. As no heat can enter the system during the adiabatic expansion, the temperature will fall, say, to T'. We now bring the gas into contact with a heat reservoir at the temperature T', and compress it isothermally until a point 4 is reached, such that when the compression is continued adiabatically, the adiabatic curve will pass through the initial point 1, where the process is stopped and the cycle thus completed. XXXVI THERMODYNAMICAL PROOFS 411 In this cycle a certain quantity Q of heat has been absorbed by the gas at the higher temperature T, and the quantity has been given out by the gas at the lower temperature T. At the same time a certain amount of work has on the whole been performed. The gas on expanding does work, and this work is measured by the product of each pressure into the corresponding change of volume. In the diagram, therefore, the work performed by the gas on expansion is measured by the area a, 1, 2, 3, y. When the gas was compressed, work was done upon it, and this work in the diagram appears as the area y, 3, 4, 1, a. The total work obtained then from the gas during the cycle is the difference of these areas, viz. the quadrilateral, 1, 2, 3, 4. To obtain actual numerical relations we may consider a gram molecule of the gas, for which the previous equations of this chapter are valid. During the isothermal expansion 1, 2, the gas absorbed Q units of heat at the temperature T, while it expanded from the volume v to the volume v. In equation (1), p. 406, dQ = C¿dT + pdv, we may put dT equal to zero, since we are considering an isothermal RT process, and for p we may substitute We thus obtain V dQ = RT Rpdv V which, when integrated between the limits v₁ and v„, gives V1 or Q = RT loge V2 V. as the amount of heat absorbed by the gas. The right-hand member of the equation, expressed in mechanical units, is of course the work done by the gas during isothermal expansion from v₁ to v2. Similarly for the isothermal compression 3, 4 we obtain – Q′ = RT' loge Q' = RT' loge Q 1 مانی Q T loge V* T' loge ? 23 លិZ.. The sign of Q' is in the first equation of this pair different from that of Qabove, since in one case the system gains heat, and in the other loses it. Division now gives } A V2 A1 V3 (la) กา (3) 412 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. For the adiabatic expansion 2, 3, we have (p. 408) T F = (2)*¹ T' and for the adiabatic compression 4, 1 k-1 F-()* T whence V3 V4 12 V1 21 or Equation (3) therefore reduces to Q T Q' ~ I" Vz UA i.e. the heat absorbed is to the heat given out as the absolute tempera- ture of the absorption is to the absolute temperature at which the heat is lost by the system. A slight alteration gives the equation in the form Q-Q T-T Q T i.e. the proportion of the absorbed heat which is converted into work is equal to the temperature difference between the two isothermal operations divided by the absolute temperature of absorption. A form which we shall find useful in subsequent calculations is Q-Q= T-T T •Q. (4) These results, although derived from a consideration of the be- haviour of gases, are valid for all reversible cycles, and can therefore be applied in every case for which we can show all the operations involved to be reversible. The first application of equation (4) will be to the process of vaporisation. Let us consider a quantity of liquid under a pressure P which is equal to the vapour pressure of the liquid at the constant temperature chosen. An infinitesimal diminution of the pressure on the liquid will cause it gradually to pass into vapour if this pressure and the constant temperature are maintained. Suppose one gram- molecular weight of vapour to be produced in this way, and let the isothermal process be represented in the indicator diagram (Fig. 63), by the line 1, 2, which is parallel to the axis of volumes, the pressure being constant. Now expand the vapour adiabatically from the original pressure P to a pressure which is dP lower, the temperature at the same time falling. The pressure and volume may then be represented by the point 3. At a temperature T- dT for which the XXXVI 413 THERMODYNAMICAL PROOFS vapour pressure of the substance is PdP, compress the vapour isothermally to liquid and finish the compression adiabatically until the original pressure, temperature, and volume are regained. The work done upon the system is equal to the area of the figure 1, 2, 3, 4, which is practically the product of the line 1, 2 into the vertical distance between 1, 2 and 3, 4. Now the line 1, 2 in the diagram represents 1 Pressure difference in the volume between the liquid and the vapour, and the distance be- tween the two hori- zontal lines lines repre- sents the difference of vapour pressure dP due to a difference of temperature dT. For the work done then we have the product dP(V-v), where Vis the molecular volume of the vapour, and v the corresponding volume of the liquid. The quantity of heat absorbed at the higher temperature 7 is Q, the molecular heat of vaporisation of the liquid at this temperature. The dT quantity which has been transformed into work is consequently Q T according to equation (4). If we express the heat in mechanical units, as we can do by multiplying the heat units by J= 42,720 (see p. 6) we obtain the equation 4 dP(V ~ v) = JQ - = Τ 2ᎫᎢ . dP P dT I' JQ dT T' Τ Volume FIG. 63. > a result of considerable practical importance and of wide applicability. As the volume of a liquid is only a fraction of a per cent of the volume of vapour derived from it at ordinary pressures, it is often per- missible to write simply V instead of V-v, which brings about a numerical simplification. For the gram-molecular volume of a gas we have PV = RT, where R in mechanical units is equal to 2J very 2JT We may therefore write equation nearly (see p. 24), so that V: (5) in the form P 2 (5) 414 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. dP Q PdT 272 i.e. (6) (7) We may calculate from this result in the form of equation (6) the latent heat of vaporisation of benzene from the change of its vapour pressure with the temperature. At 5° the vapour pressure of benzene is 34.93 mm. of mercury, or 47.50 g. per sq. cm.; at 5·58° the pres- sure is 36·06 mm. or 49'04 g. per sq. cm. For dP then we have 1.54, for dT we have 0.58, for P we have the mean value 48.27, and for T the mean value 278.3. The value of Q is therefore or 2TdP PdT d logeP dT Q 27'2* 2(278-3)2 x 1.54 48.27 × 0.58 dT = = 8520 cal. The value of the heat of vaporisation actually found is 8420 cal., so that the agreement between calculation and experiment is fairly close, the difference not being greater than the error of experiment. Equation (5) not only holds good for the liquid and gaseous phases, i.e. for vaporisation, but also for the equilibrium between any other pair of phases, for example, solid and liquid, or two solid phases such as the different crystalline modifications of sulphur. In the form T(V – v) dP Jq (8) it is useful for ascertaining the effect of pressure on the temperature of equilibrium. V, v, and q may all refer either to molecular quantities or to unit weight of the substance considered, as the molecular factor cuts out in the right-hand member. If we consider the transformation brought about by supplying heat to the substance, q is a positive quantity, while J and T also are necessarily positive. Consequently dP will have the same sign as dT, or the opposite sign, according as V-vis positive or negative, i.e. if the substance expands on being transformed by application of heat, dP and dT will have the same sign, whilst if it contracts dP and dT will have different signs. In the first case then the transition point will be raised by increase of pressure; in the second case it will be lowered. Rhombic sulphur on melting expands ; V – v is therefore positive and the melting point of rhombic sulphur is raised by pressure. Again rhombic sulphur expands on passing into monoclinic sulphur, and consequently the transition point is raised by application of pressure. Water, on the other hand, occupies a smaller volume than the ice from which it is produced; V-v is therefore negative, and increase of pressure lowers the melting point. As a numerical example of the application of formula (8), we may calculate the effect of one atmosphere increase of pressure on the XXXVI 415 THERMODYNAMICAL PROOFS melting point of ice. A cubic centimetre of water at 0° is obtained from 1·09 cc. of ice; the change of volume on liquefaction is therefore 0.09 cc. per gram of water. The latent heat of liquefaction per gram is 80 cal., and the temperature of liquefaction is T = 273. We have, therefore, if dP = 1 atm. = 1033 g. per sq. cm., T(V – v) - v) ¿P 'dP Jq dT 273 x (-0.09) × 1033 42720 × 80 – 0·0074; that is, the melting point of ice is lowered 0.0074° for each atmo- sphere increase of pressure. A corresponding diminution of pressure causes the same rise in the melting point. Thus ice which melts under atmospheric pressure at 0°, melts at 0·0074° under the pressure of its own vapour, so that the triple point (p. 98) lies at this tempera- ture and not at 0°. Dilute Solutions When we dissolve a substance in any liquid, the process is not, under ordinary conditions, a reversible one, for we have not in general during dissolution a state bordering on equilibrium. In certain cir- cumstances, however, it is possible to conduct the process reversibly. If we are dealing, for example, with the solution of a gas in a non- volatile liquid, we can proceed reversibly as follows. Let the gas and liquid be taken in such proportions that the gas will just dissolve in the liquid at the pressure p and the constant temperature of experi- ment t. Suppose the liquid and gas to be contained in a cylinder with a movable gas-tight piston. At first let there be a partition separating the gas and the liquid. Without removing this partition, expand the gas by gradually diminishing the pressure in such a way that at no instant during the expansion the condition of the gas differs sensibly from equilibrium. According to Henry's Law, the quantity of gas dissolved by the liquid is proportional to the pressure of the gas. Let the expansion be continued until the pressure of the gas is so small that practically none of it dissolves in the liquid when the separating partition is removed. After removal of the partition let the pressure on the gas be increased by insensible gradations. At no time does the state of the system deviate sensibly from equilibrium, and the pressure may be gradually raised until at the pressure p all the gas has dissolved. The solution of a gas in a liquid then may be made part of a reversible cycle if the process is carried out as here indicated. The concentration of a solution of a non-volatile substance in any liquid can be changed reversibly by bringing the solution into contact with the solvent under equilibrium conditions, and then by an in- finitely small alteration of the conditions determining a process of concentration or dilution. It is apparent at once that we cannot bring 416 CHAP. INTRODUCTION TO PHYSICAL CHEMISTRY the solution into direct contact with the liquid solvent, either by mixing directly or allowing the dissolved substance to diffuse slowly into a fresh quantity of solvent as in the experiment described on p. 163, for no slight change in the conditions can at any stage make the action proceed in the reverse direction, i.e. make a solution separate into a more concentrated solution and the solvent. If the solvent is in the form of vapour or solid, however, the dilution or concentration may be effected reversibly. Suppose the solution to be in presence of its own saturated vapour at the constant temperature of experiment. An increase of external pressure, however slight, will cause part of the vapour to condense, i.e. will dilute the solution; while a slight diminution of the external pressure will cause part of the solvent to evaporate, i.e. will concentrate the solution. Similarly, if the solution is in equilibrium with the solid solvent, a very slight rise of tempera- ture will bring about partial liquefaction of the solid solvent, and thus dilute the solution, and a correspondingly slight diminution of tempera- ture will produce partial separation of solid solvent and thus con- centrate the solution. There is still another way of bringing the solution into contact with the pure solvent under equilibrium conditions, namely, through a diaphragm which is permeable to the solvent and not to the dissolved substance. If the solution is enclosed in a cylinder with a semi- permeable end and a movable piston, it will be in equilibrium with the liquid solvent through the diaphragm when there is a certain pressure on the piston, the osmotic pressure. If the pressure on the piston is increased ever so slightly, solvent flows outward through the semipermeable diaphragm and the solution becomes more concentrated; if the pressure on the piston is diminished, solvent flows inwards through the diaphragm, and the solution is thereby diluted. All these methods of changing the concentration of a solution can therefore be adopted as parts of reversible cycles of operations, and we shall see that by removing a portion of solvent from a solution by one method, and by adding it to the solution again by another method, we obtain a series of results which are of great theoretical and practical importance. In the first place we shall consider a cycle in which a gas is dis- solved in a non-volatile liquid by the reversible process given on p. 415, and the system then brought back to its original condition by means of semipermeable diaphragms. We start with a volume v, of the gas under pressure po, and with a volume of liquid just sufficient to dissolve the gas under this pressure, and we propose to find what amount of work (positive or negative) must be done in order to bring the gas into solution reversibly at constant temperature. During the first stage contact between gas and liquid is prevented by a partition inserted at the surface of the liquid. If the cylinder in which the gas and liquid are contained have unit cross-section, and XXXVI THERMODYNAMICAL PROOFS 417 the initial distance of the piston from the liquid surface is zo, we have for this state x。 = v。• At any stage of the expansion (x) the pressure p is given by the equation p-Pove, and the work done by the gas during the expansion is represented by the expression X x dx Polo f Vo Povologe being a very large multiple of vo The partition is then removed, and the pressure on the gas increased. The pressure on the piston in a given position x is less than before, for the gas which was previously confined to the space 2 is now partly in solution. s denote the solubility (p. 50), the available volume is practically increased in the ratio x:x + sV, so that the pressure in position x is now given by If X or P X X = Povo x + s V and the work required to be done during the compression is dx Povo fa da 。 x + st = Po%。 loge x + sv SV Po% {log Povologe > On the whole, the work done on the system during the double opera- tion is + s V P(log-log."); + - log!) Vo X The quantity within the brackets of the second expression may be seen to be zero, since x is indefinitely great, so that x + 8 V s X = X = 1, and since by supposition the quantity of liquid is just capable of dissolving the gas, so that s = v。· The conclusion, then, is that there is no gain or loss of work in dissolving the gas reversibly in the liquid. The gas may now be removed from solution and restored to its original state reversibly by means of semipermeable membranes. arranged as in Fig. 63. One membrane gg, permeable to gas but not to liquid, is introduced at the surface of the liquid, on which the piston KK rests at the commencement of the operation. A second membranell, permeable to liquid but not to gas, is substituted as a piston for the bottom of the cylinder, and is backed upon its lower side by the pure solvent. By suitable proportional movements of the two pistons, KK being raised through the space v., while l is raised 2 E 418 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. through the space V, the gas may be expelled, the pressure of the gas retaining the constant value Po, and the solution which remains retaining a constant concentration, and therefore a constant osmotic pressure P. When the expulsion is complete, i.e. when the two semi- permeable membranes have come together, the work done upon the lower piston is PV, and the work done by the gas in raising the upper piston is povo = The system is now in its original state, and all the operations have been conducted reversibly. A reversible cycle has therefore been T-T performed, and the equation Q-Q': Q is applicable. The T temperature has remained constant throughout, so that T- T′ = 0, and therefore Q-Q' 0, as in all reversible isothermal cycles. Since no heat has been converted into work, or vice versa, the work done on the system must on the whole be equal to the work done by the system. In K the first stage, i.e. the process of solution, it has been shown that there is neither loss nor gain of work, so that for the second stage we must have K વ Gas 2 Povo = PV. If the gas which occupies the volume v at the pressure po, is made to occupy the volume , its pressure will assume a new value p, and according to Boyle's Law we shall have P。。=pV; combining this with the previous Povo equation, we obtain pV PV and p= P. The osmotic pressure then of the gas in solution is equal to the gaseous pressure which it would exert in absence of the solvent if it occupied the same space at the same temperature-a result in harmony with the calculation from experiment made on p. 178. The above result holds good only for ideal substances and for very dilute solutions, since in its deduction we have assumed that the gas is a perfect gas which obeys the laws of Boyle and Henry exactly, and that the volume of the solution is exactly the same as the volume of the solvent which it contains-an assumption which can only be justifiably made when the solution is extremely dilute. The solvent, too, was supposed to be non-volatile (although the proof may be extended to volatile solvents), and the reversible processes themselves are purely ideal. Notwithstanding all these assump- tions the conclusion arrived at is practically important, and holds good with close approximation for all dilute solutions under ordinary conditions. Solution Solvent مه FIG. 64. argling – XXXVI THERMODYNAMICAL PROOFS 419 We shall now consider an isothermal reversible cycle performed with a solution of a non-volatile solute in a volatile solvent. Let the solution contain n gram molecules of dissolved substance in W grams of solvent, and let the constant absolute temperature of all the processes be T. From the solution let there be removed by means of a piston and a diaphragm permeable only to the solvent, a quantity of the latter which in the solution contained one gram molecule of solute dissolved in it. This quantity is W/n grams, and the solution is supposed to be present in such large proportions that the removal of this amount of solvent does not sensibly affect its concentration, the osmotic pressure thus remaining constant during the operation. Since the change in volume of the solution is the gram-molecular volume, the work done on the system in removing the solvent is the product of this into the osmotic pressure, and is therefore equal to RT, if the gas laws apply to dissolved substances. This quantity of liquid solvent is now con- verted into vapour reversibly by expanding at the vapour pressure of the liquid, which we shall call f. The vapour pressure of the solution is smaller than this and equal to f'. Let the vaporous solvent there- fore expand reversibly till its pressure diminishes to this value. The gaseous solvent is then in equilibrium with the solution, and may be brought into contact with it and condensed reversibly at the pressure f', so that the whole system regains its initial state. We have now to consider the work involved in the expansion and contraction. The work done by the system on expanding from liquid to vapour under the pressure f, is equal to the work done on the system in condensing the gas to liquid under the constant pressure f', being equal in each case to RT for the gram molecule, if we neglect the volume of the liquid (cp. p. 24). There remains then the work done by the gas on expanding from f to f'. For isothermal expansion we have the work dv ᎡᎢ per gram molecule of gas (equation la, p. 411). Now V • dp dv p V - for a gas, since pv const., and therefore pdv + vdp = 0. For the work done during the expansion of the gas then we have - RTI-F Rp dp RT or p per gram molecule if the difference between ƒ and ƒ' is very f small. For the actual amount of solvent considered we have conse- ƒ -ƒ' ᎡᎢ . IV quently ゲ ​Mn where M is the molecular weight of the solvent in the gaseous state. Since in the whole cycle no heat is con- verted into work or vice versa, the work done by the system must be numerically equal to the work done on the system during the osmotic 420 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. expulsion of the solvent, i.e. or RT = W. RT-T ゲ ​ᎡᎢ . Mn f ƒ-ƒ' Mn f IV which is identical with the result arrived at for very dilute solutions. by the method of calculation given on p. 187. Thus by a direct thermodynamical proof we can arrive at the relation between the osmotic pressure and the lowering of the vapour pressure of liquids by substances which are dissolved in them. If the solution considered be not very dilute, we cannot write ƒ -ƒ' f for dp For greater concentrations we integrate this expression, and P thus obtain loge. By the same process of reasoning as before, we f Mn loge f = W then get This expression holds good for all concentrations of solutions for which the total volume of the liquid does not change when the quantity of solvent considered is added to or removed from the solution. If this condition is not observed the result is only approximate, for in the deduction of the relation we assumed that the work done by the expressed liquid in expanding at the constant gaseous pressure of the solvent was equal to the work done on the vapour when it was con- densed at the constant vapour pressure of the solution, an assumption which is only valid if the volume of the liquid is the same before and after the reversible mixing of the solvent with the solution. The relation between the logarithmic and the usual expression may be obtained by writing the former as follows:- loge(1 + ††). ナーザ ​f' If we expand the logarithm in this second form, the first term of the expansion is, practically identical with the usual expression. Having thus obtained the formula for the lowering of the vapour pressure of a solvent, we may now proceed to deduce the formula for the corresponding elevation of the boiling point. From equation (6) we obtain the expression dP Q dT P = 27dr Q TEAT " XXXVI 421 THERMODYNAMICAL PROOFS ing is equal to to represent the concomitant variations of temperature and vapour pressure of a solvent. Consider now a solution containing n gram molecules of dissolved substance in W grams of solvent. Let this solution have the vapour pressure P at the temperature 7+ dT, T being the temperature at which the solvent has the same pressure. At the temperature T + dT the solvent will have the pressure P + dP. Now d P is the lowering of the vapour pressure of the solvent, but since P+dP dP is very small compared with P we may write instead of this the But we found above that this lower- dP expression for the lowering. P Mn W' ΟΙ so that Mn Q W = 272dT. QT Now is the latent heat of a gram molecule of the solvent, and M is its molecular weight (both molecular quantities referring to the gaseous state), so that Q/Mq the latent heat of vaporisation per gram. We thus obtain = N w = " ; 2 TdT; W 272 2T2 а T ***** N W q T + dT − T = dT is the elevation of the boiling point of the solvent caused by the substance dissolved in it, and we have now obtained an expression for this in terms of the boiling point of the solvent itself, its latent heat of vaporisation and the concentration of the solution, to which the elevation is proportional. The elevation of the boiling point caused by the solution of one gram molecule of substance in 100 grams of solvent is sometimes referred to as the "molecular elevation" (p. 203). For this concentration n becomes 1 and I becomes 100, 0.0272 q so that the molecular elevation is For a solution containing one gram molecule per 1000 grams, i.e. very nearly one gram molecule 0.00272 per litre for aqueous solutions, the elevation is 9 Since both boiling point and heat of vaporisation vary with the pressure at which ebullition takes place, there is no definite "molecular elevation" for any one solvent unless the pressure is specified. For ordinary purposes the pressure is of course the atmo- spheric pressure, and the fluctuations to which this is subject have so little effect on the molecular elevation that it may be taken as a constant magnitude in the practical molecular weight determination 422 CHAP. INTRODUCTION TO PHYSICAL CHEMISTRY by the boiling-point method. In order to give an example of the agreement between the calculated molecular elevation and the same magnitude as determined experimentally, we may take the common solvent ether. The boiling point of ether is 35°, and therefore T = 308. The latent heat of vaporisation per gram at this temperature is 0.0272 has thus the value 21.1. The average 9 90. The expression molecular elevation observed in the case of nine different substances in moderately dilute ethereal solution was found to be 213, the extreme values being 2010 and 21.8. The expression for the molecular depression of the freezing point has a similar form, and may be deduced by means of a reversible cycle as follows. Let a solution containing n gram mole- cules of dissolved substance in W grams of solvent be contained in a cylinder provided with a semipermeable end, and a movable piston. At T-dT, the freezing point of the solution, let such a quantity of the solvent freeze out as originally contained one gram W molecule of substance dissolved in it, viz. N grams. The quantity of solution is supposed to be so great that the freezing out of this amount of solvent does not appreciably affect its concentration, so that the temperature of equilibrium between solution and solid solvent does not change during the process of freezing. The solid is now separated from the solution, and the whole system is raised to the temperature T, the melting point of the solvent, and at this temperature the solid solvent is allowed to melt. In doing so it absorbs q calories, if q W N represents the latent heat of fusion per gram. The fused solvent is now brought into contact with the solution through the semipermeable diaphragm under equilibrium conditions, viz. with the pressure on the solution equal to the osmotic pressure P. By raising the piston under this constant osmotic pressure, the solvent passes through the diaphragm and mixes reversibly with the solution, the concentration as before remaining unchanged. The work done on the piston is equal to the product of the constant osmotic pressure P into the volume v which contains one gram molecule of solute. But this amount of work, accord- ing to the osmotic pressure theory, is equal to RT; or if we express the gas constant in thermal units, approximately to 27. The system after mixing is finally cooled to the original temperature T-dT so as to complete the cycle. By selecting the solution sufficiently dilute we may make the depression of the freezing point dT as small as we choose, and consequently the heat absorbed and evolved in warming and cooling the system through this small range of temperature may be made negligible in comparison with the finite amount of heat -9 absorbed by the solvent on melting. In the reversible cycle, then, we IF N XXXVI 423 THERMODYNAMICAL PROOFS W have the amount 9 absorbed at the higher temperature T, and the n ΟΙ dT W 9. But the only work amount of this converted into work is Τη done by the system is the osmotic work, since the external work brought about by the volume-changes on freezing and melting are so small as to be negligible. We have consequently Solvent. AT W Τ 92 Water Formic acid Acetic acid Benzene Phenol dT = Nitrobenzene Ethylene dibromide • • q= 27; 272 N 9 The depression of the freezing point is thus seen to be proportional to the concentration of the solution, and if we make the concentration such that n = 1 and IV – 100,--that is, if we dissolve one gram molecule in 100 grams of solvent, we get the molecular depression equal to 0.0272 IV q The expression is exactly the same as that for the elevation of the boiling point, q referring here, however, to heat of fusion instead of to heat of vaporisation. The following table exhibits the nature of the agreement between the calculated and observed values of the molecular depression in various solvents :- ❤ 0.0272. q 18.5 28.4 38.8 51 76 69.5 119 Mol. Dep. 18.7 27.7 39 49 74 70.7 118 Electrical and Chemical Energy It is possible, as Helmholtz showed, to establish a relation between the energy of the chemical processes occurring in a galvanic cell and the electrical energy produced by the action. At one time it was supposed that all the energy which could be obtained under ordinary circumstances as the heat of the chemical reaction, could be converted into electrical energy and made available as an electrical current. Thus if Q were the thermal effect for one gram equivalent of the sub- stances entering into chemical action in the cell, it was assumed that an amount of electrical energy equivalent to this could be obtained from the cell for each gram equivalent of chemical transformation. In a Daniell cell, for example, where the reacting system is Cu | CuSO4 solution | ZnSO, solution | Zn, 424 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. the total chemical action is Zn + CuSO₁ = Cu + ZnSO4 4 zinc being dissolved up at one pole of the battery and copper being deposited at the other. Now the thermal effect of this reaction is obtained by subtracting the heat of formation of the zinc sulphate from that of the copper sulphate, both numbers referring to aqueous solutions of the salts (cp. p. 132). We thus get 106,090 – 55,960 50,130 cal. per gram molecule, or 25,065 cal. per gram equivalent. That is, when 32.5 grams of zinc displace an equivalent amount of copper from a solution of copper sulphate, 25,065 calories are evolved. Now, if the displacement takes place indirectly in a galvanic cell with production of electric current, 96,540 coulombs of electricity will be obtained, according to Faraday's Law, for every 32.5 grams of zinc. dissolved. To express the numerical equivalence of thermal and electrical energy we have the equation 1 volt-coulomb = 0.2387 cal. The electrical energy then equivalent to the heat of the chemical reaction is 25,065 0.2387 105,000 volt-coulombs. If we now divide this number by 96,540, the number of coulombs produced, we get 1.08 for the electromotive force of the cell expressed in volts, on the assumption that all the chemical energy of the reacting substances has been converted into electrical energy. Direct measurement of the E.M.F. of the Daniell cell gives 1.09 to 110 volt. The assumption then that the chemical energy is wholly converted into electrical energy is in this case very nearly true, and several other cells are known to which a similar simple calculation for their electromotive force is applicable. These cells, however, are all of a special type, and we shall therefore proceed by means of a reversible cycle of operations to deduce a formula of more general application. In the first place the cell considered must be, like Daniell's, of a completely reversible or non-polarisable type (cp. p. 357), so that if we pass a current through it in the reverse direction of the current which the cell would of itself generate, the chemical action in the cell will be exactly reversed. Thus when the Daniell cell is in action the positive current within the cell moves with the positive ions from the zinc electrode to the copper electrode. If we now by using an external electromotive force make the current pass from the copper pole to the zinc pole, the chemical action which occurs is Cu + ZnSO4 = Zn + CuSO4, which is the reverse of the primary action of the cell, copper being dissolved and zinc deposited. = Let the cell act at the constant temperature 7, and generate the quantity C=1 faraday of electricity. If the electromotive force of the cell is E, the electrical energy produced by the cell is EC, and this may or may not be equal to Q, the diminution in internal energy, XXXVI 425 THERMODYNAMICAL PROOFS which under ordinary circumstances would be the heat of the chemical action. Suppose the electrical energy produced to be less than the fall in internal energy, then the element on working will give out Q - EC as heat at the constant temperature T. Let now the system be heated to the slightly higher temperature T+ dT, and let the quantity C of electricity be sent through the cell in the reverse direction, the temperature being kept constant. If the electromotive force has diminished by the amount dE owing to the change of temperature, the work done on the cell will be C(E - dE), and the amount of heat absorbed will be Q-C(E-dE) on the supposition that the heat of the reaction does not change appreciably with the temperature. The system is finally cooled to the original temperature T, so that everything regains its initial state, a reversible cycle having been performed. On the whole, the system has done the external electrical work C(E – dE) - CE - CdE, which must be equal to the fraction dT/T of the heat given out at the lower temperature. Now the quantity of heat given out is Q- CE, so that the heat transformed into work is — d T or = -(Q – CE) ; and we have therefore the equation - CdE E ་ ¿T Τ Q Ꮯ (Q − CE) ; E +qdE d T From this relation we see that in order to ascertain the electromotive force of an element from the heat of the chemical action within it, we must know the rate of change of the electromotive force with the temperature. If the electromotive force does not vary with the temperature, i.e. if dE/dT be zero, then the simple formula Q C may be used. The electromotive force of a Daniell cell has a very small temperature coefficient, so in its case the simpler formula gives a result closely approximating to the truth. The more accurate formula, however, gives a still closer approximation to the observed electromotive force, and has been experimentally verified in many other instances for which the temperature coefficient is larger. If we write the expression in the form dE EC = Q + CT- 07d dT" we see at once that the electrical energy and the chemical energy of 426 CHAP. INTRODUCTION TO PHYSICAL CHEMISTRY the process are equal if dE/dT is zero; that the electrical energy is greater than the chemical energy if the temperature coefficient is positive, i.e. if the electromotive force increases with rise of tempera- ture; and that the electrical energy is less than the chemical energy if the temperature coefficient is negative, i.e. if the electro- motive force falls with rise of temperature. If a cell with a positive temperature coefficient of electromotive force is allowed to act, it will make up for the difference between the electrical and chemical energies by abstracting heat from neighbouring bodies; or, if no external heat is available, it will cool itself by working. The student is apt to imagine that this is a contradiction of the Second Law of Thermodynamics; but, like the self-cooling of a freezing mixture, the process here involved is not a cycle of operations, and the system cannot regain its original state without work being done upon it. What the Second Law contradicts is the existence of a system which, working in a cycle, by repeated self-cooling, can convert into work the heat of neighbouring bodies all of the same temperature. Concentration Cells.-From the discussion of such a case as that of two silver electrodes immersed in a solution of a silver salt through which a current is being passed, it is apparent that a change of concentration round the two electrodes may be brought about by the passage of electricity through the system. Now such a process may be performed reversibly if the changes in concentration round the electrodes are relatively very small, although the total amount of electrolyte transported may be large. But the concentration changes round the electrodes may also be executed reversibly in the other ways already referred to in the previous chapter, viz. by reversible evaporation or freezing of the solvent, or by the use of a semipermeable membrane. It is thus evident that we obtain, for example, a thermodynamical relation between osmotic work on the one hand and electrical energy on the other by changing the concentration first by means of a semipermeable membrane, and then bringing the system back to its original condition reversibly by the passage of an electric current. The osmotic work and the electric energy will here be equivalent to each other, since mechanical energy and electrical energy are completely interconvertible, so that knowing one, the other may be calculated. Assuming the gas laws for dilute solutions, the osmotic work may be very simply determined, and thus the electrical energy, and from that the electromotive force of any concentration cell, may be deduced. can For simplicity of calculation we shall as heretofore consider a binary electrolyte with univalent cation and univalent anion, say silver nitrate. Let there be two solutions of the salt of osmotic pressure P and P+dP respectively. The number of cubic centi- metres in which 1 gram molecule of the salt is dissolved may then be denoted by Vand -d respectively. If u and v are the rates XXXVI 427 THERMODYNAMICAL PROOFS U u + v of migration of the cation and anion, the passage of 1 faraday through the solution will necessitate the passage of gram mole- cules of cation through the surface of separation of the solutions in one direction, and of gram molecules of anion in the opposite ย U + v direction. Since the anode dissolves up to supply cation to replace that which leaves the anodic compartment and also to neutralise the anion which arrives, there will be on the whole (cp. p. 239) an increase V of concentration of gram molecules of salt in the anodic compart- ment and an equal diminution of concentration in the cathodic com- partment. The passage then of this amount of electricity is equivalent to the transference of gran molecules from the cathodic to the V U + v U + V anodic compartment. Let us now consider the osmotic work which would have to be done on the supposition that the solution in the anodic compartment is originally the more concentrated solution. Cut off from the more dilute solution of osmotic pressure P a portion containing V U + v gram molecules of the salt. This may be done by interposing a semiperme- able membrane at the proper place. Concentrate the portion cut off by forcing water through the semipermeable membrane into the rest of the more dilute solution until the osmotic pressure in the separated portion reaches the value PdP. It is assumed that the total bulk of the original solutions is so great that the addition or removal of the quantity of water containing gram molecules has no appreciable effect on V U + V their dilution or osmotic pressure. Since the degree of dilution of the solution of osmotic pressure P is V, and that of the solution of osmotic pressure P+dP is V - dV, the process of concentrating one gram molecule of salt from the first to the second dilution would involve the removal of dV cubic centimetres of water, and so the concentration of V V W + v U + V centimetres through the osmotic membrane. This volume has been moved against an osmotic resistance rising uniformly from zero at the beginning of the compression to dP at the end. On the average the pressure difference is dP. The osmotic work done during this part of the operation is therefore gram molecules necessitates the expulsion of 21 dP. U + V · dv. • d cubic The separated portion of solution being now of the same osmotic 428 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. pressure as the more concentrated anodic solution may be mixed with it. To restore to the more dilute solution the (V – dV) cubic ย - U + v centimetres of water necessary to bring it back to the original volume we may force this quantity through the semipermeable membrane separating the solutions. The difference of osmotic pressure on the two sides of this membrane is and remains dP, so that the osmotic work done during this operation is dP. or dP. dP. V On the whole U + v gram molecules of dissolved salt have now been transferred to the concentrated from the dilute solution at an expendi- ture of work equal to ย u + v • = ข U + v L = ________ dV + dP. ጎ U + Ÿ Since the difference in dilution d is very small in comparison with V, the total work may be written. dP = d • ย и то iRT ᎡᎢ d(iKT) L = RT. (V – ždV) gram centimetres. (V – dV). (9) without sensible error. In order now to evaluate this expression in quantities accessible to measurement, we may make use of the gas laws for dilute solutions. For one gram molecule of dissolved salt, the van 't Hoff coefficient of which is i (p. 258), we have PV V v и + и • id V di = RT ( − 2 = 2 + 7 ᎡᎢ V2 V :(. u + v dP.V • Substituting this value of dP in (9) we obtain id V (− dV); Ꮴ 2). + di). If the same transference of one gram molecule from the concentrated to the dilute solution were carried out electrically, the electric energy expended would be CaE, dE representing the electromotive force (due to the difference in dilution d) which must be overcome, i.e. the electromotive force of the concentration cell. The work and electric XXXVI THERMODYNAMICAL PROOFS 429 energy are equivalent, so that when R is expressed in electrical units, id V (10) V If we are dealing with very dilute solutions for which i may be assumed to be constant and equal to 2, we obtain dE * dE E E === RT V E = For finite differences of dilution we may imagine a great number of concentration cells of the kind just considered arranged in series one after the other. The finite electromotive force between the first and last of these will then be obtained by integration between the extreme dilutions V₁ and V2, and will be 1 C' u + v loge!! 2 If, as is usual, we substitute the inverse ratio of concentrations the ratio of dilutions we obtain Evaluating the constant natural logarithms, we have E Je 2RT C 2RT V dV じ ​u + v Ꮴ 2RT V с 2RT C V u + v =0·114. 2 × 8.31 x 288 x 2.30 X 96,540 U + 2 -di). V u + v + loge C.2 1. (11) for 15°, and substituting decadic for • V 2RT no 2 + v .log10 • V и про го C1. C2 The value of the gas constant R is of course given in electrical units (ep. p. 24). If the electrovalency of the ions of the binary electrolyte is n, then nC coulombs of electricity must be passed through the solutions for the transference of gram molecules from the cathodic to the ገ W+V anodic compartment, and equation (11) becomes loge • ໖໑, C1 C1 10$100₂ for those derived from it experiencing the same division of the right-hand member by n (cp. p. 360). It is an easy matter to calculate from the osmotic theory the 430 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. diffusion electromotive force existing at the junction of two differ- ently concentrated solutions of the same electrolyte, the speeds of migra- tion of whose cation and anion are u and v respectively. We will again assume for simplicity that the ions are univalent and that the electro- lyte is completely ionised. The osmotic pressure of each ion will be P₁ in one solution and P₂ in the other. Let C coulombs be passed through the solution so that the cation is transferred from the higher pressure P₁ to the lower pressure P2. The amount of cation which will pass 2 1 U gram molecules, and similarly the junction from P₁ to P₂ will be 1 2 V gram molecules. U + v Applying now the gas laws, we find that when a gas is isothermally expanded, the work it does per gram molecule (cp. p. 411) is the amount of anion passing from P₂ to P, will be 2 1 from P, to P, we have 1 2 or, since Boyle's Law here holds good, P₁ 1 L = RT loge p Consequently for the transference of Lc and for the transference of we have so that L La M = V = RT loge 77 1 E' u + v 2 u + v CE' V U + v L = Le + La = V 20 + 0 The total work during the passage of C coulombs when the cation moves from the higher to the lower concentration is therefore W + v P₁ RT loge p ; 21 and since this work is equivalent to the electrical energy, P₁ .RT loge P₂ 1 P. gram molecules of anion from P₂ to P₁ 2 U V P 2 RT loge Pi U + v U 27 W + v RT u-v C и + г gram molecules of cation 1 • P 1 RT loge p 2 1 Loge P Substituting the ratio of concentrations 1 for the ratio of osmotic C₂ XXXVI 431 THERMODYNAMICAL PROOFS pressures, we obtain E' RT u C E E' (12) For c₁c₂ or u= the expression vanishes, i.e. there is no electro- motive force between two solutions of the same concentration, and none at the junction of two solutions of different concentrations when the speeds of anion and kation are equal. Since the net electromotive force of the concentration cell is from equation (11) U + V V 2RT ca and the diffusion electromotive force at the junction of the liquids is RT C V Ꮯ U + V RT/ 2v Cu + v ᎡᎢ C - loge C2 U บ U + V it follows that the electromotive force due to the electrode potentials is the sum of these two terms, i.e. E+ E' 1 -loge Co 1 volt. • + . • C1 loge (13) The same result may be obtained from equation (11) by assuming the velocities of anion and cation to be equal, thus eliminating diffusion RT loge с C1 potential. Then 2 u + v and equation (11) becomes E= ໖ ໑, as in (13) above. If we evaluate 1 loge Ca ι u + 1) log? ca • volts. decadic for natural logarithms, we obtain for the difference of the electrode potentials at 15° the value 0·057 log10 C1 1 and the difference of electrode C Now if c₁ = 10c₂, we have log10 RT as before, and if at the same time we substitute C C potentials is equal to 0.057 volts. This is the value which we have used in Chap. XXX. for the increase of electrode potentials with a tenfold increase of dilution. With regard to the sign of the electromotive forces, i.e. to the direction of the positive current due to them, the following points should be noted. If in equation (12) u>v, the expression on the right hand is positive, since we assumed c> ; if u c₂, E should be taken with the negative sign, if we adopt the sign-convention of the preceding paragraph. The electrode and diffusion potentials thus act against each other. For a combination with reversible anion, they act so as to reinforce each other. The direction of the net external current is always the same as that due to the electrode potentials, for this is always numerically greater than the diffusion potential, as may be seen from the following consideration. Reference to equations (12) and (13) shows that the diffusion electromotive force is to the difference of electrode potentials U ·V to 1, i.e. the former is to the latter as u v to u + v. U + v Since u and v are essentially positive numbers expressing the rates of the ions, u v is always less than u+v, and the diffusion electromotive force thus appears as a correcting term of the difference of electrode potentials which for salts is sometimes negligible on account of the speeds of the anion and cation being nearly equal. In the case of sodium chloride, for example, u v is 24.7 and u + v is 125.7 (cp. table, p. 256). In this relatively unfavourable case for a salt the error involved by neglecting diffusion potential is 20 per cent. In the case of acids and bases, the error is much more serious. v is 272, and u + v is 422. The error Thus for hydrochloric acid u here involved by neglecting diffusion potential is over 60 per cent. In dealing with acids and bases then, it is essential to adopt the device alluded to on p. 357 of joining the solutions by means of a concen- trated solution of a salt with nearly equal speeds for anion and kation. as - E. BUCKINGHAM, Outline of the Theory of Thermodynamics, 1900. O. SACKUR, Textbook of Thermochemistry and Thermodynamics (trans. by G. E. Gibson, London, 1917). J. R. PARTINGTON, Textbook of Thermodynamics. F. H. MACDOUGALL, Thermodynamics and Chemistry. VAN 'T HOFF's original paper on Osmotic Pressure and the Thermo- dynamic Deductions for Dilute Solutions" will be found in the Philosophical Magazine, 5th series, 26, p. 81 (1888), and also in Zeitschrift für physikalische Chemie, 1, p. 481 (1887). W. NERNST, "The Electromotive Activity of the Ions," Zeitschrift für physikalische Chemie, 4, p. 129 (1889); The Applications of Thermodynamics to Chemistry (London, 1907). Absolute temperature, 6 Absorption-coefficient, 50 spectra, 160 Accelerating influence of acids, 288, 302, 330 of bases, 309 of water, 295 Accumulators, 366 Acids, dissociation constants, 158, 254, 271, 304, 312 electric conductivity, 254, 303 strength (avidity), 297-307, 319 Actinium, 387 Active mass, 267, 275 solvents, 219 Additive properties, 148 Affinity constants, 158, 254, 271, 304, 312 Alligation formula, 313 Alloys, 111-119 Alpha particles, 374, 382, 402 Ammonia, synthesis of, 273, 283 Ammonium cyanate, 293 salts, gaseous dissociation, 276, 281- 283, 296 Amorphous state, 57 Ampère, 7 Amphoteric electrolytes, 333 Angström unit, 2 Association, molecular, 213 Asymmetry, 152 Atomic heat, 26 INDEX hypothesis, 8, 371 nuclei, 377, 398-403 number, 391-397 refraction, 151 volume, 34, 149, 209 weights, 8-20, 396 standard of, 12 table of, 21 Atoms, 8, 10 dimensions of, 371-376 structure of, 398-403 Autolytic conduction, 231 Avidity, 297-307, 319 Avogadro's constant, 375 433 Avogadro's Law, 11, 23, 83, 161, 371 for solutions, 178, 418 Balanced actions, 265-284 Balmer's formula, 378, 401 Bases, conductivity, 254, 307 strength, 307-309, 329 Benzene formula, 156-158 Beta radiation, 383 Bibasic acids, dissociation constant, 312 Bimolecular reactions, 289 Blagden's Law, 58 Bohr's atomic model, 401 Boiling points in series, 142, 217 of solutions, 78, 189, 198, 420- Border curve, 74 Boyle's Law, 22, 72, 83, 407, 418 deviations from, 86, 88 for solutions, 177 Brine-bath, 77 Brownian motion, 224 Cadmium cell, 7, 355, 358 Calcium carbonate, dissociation of, 274 Calomel electrode, 356 Calorie, 6, 7 Calorimeter, 137 Calorimetric bomb, 136 Cane sugar, inversion of, 286, 302 Capillarity, 206 Capillary electrometer, 355 Carbon dioxide, isothermals, 73, 90 Catalysis of cane sugar, 286 of diazoacetic ester, 303 of esters, 288, 302 Cathode rays, 375 Cells, accumulator, 366 concentration, 360, 426 galvanic, 353 standard, 7, 355, 358 C.G.S. units, 3 Charles' (Gay-Lussac's) Law, 22 for solutions, 177 Circular polarisation, 151, 155, 164 Clark's cell, 7, 355, 358 2 F 434 INTRODUCTION TO PHYSICAL CHEMISTRY Coagulation, 224 Colligative properties, 161 Colloidal solutions, 221-229, 333 Colloids, 227 Combining capacity, 39 proportions, 9 Combustion, heat of, 133, 141, 157 Complex ions, 339, 345, 346, 348, 362 molecules, 212-220 Concentration cells, 360, 426 Condensation and vaporisation, 69-80, 85, 412 Conductivity, molecular, 241, 243, 250-256 specific, 7, 241, 250 Conservation of energy, 5, 130, 405 Constant boiling mixtures, 78-79 freezing mixtures, 61, 65, 68 proportions, 9 Constitution and physical properties, 148 Constitution, relation to properties, 148- 161 Constitutive properties, 149 Contact process for SOŽ, 273 Continuity of state, 74, 90, 99 Copper sulphate, hydrates, 123-125, 275 Corresponding conditions, 93 Coulomb, 7 Co-volume, 209 Critical constants, 69, 92 solution temperature, 72 temperature, 69, 73, 93 voluine, 69, 79 Cryohydrates, 61 Cryohydric points, 60-65, 122 Cryoscopy, 204-206 Crystal structure, 378-380 Crystalline liquids, 57, 104 nuclei, 56, 58 Crystallisation, 55-68, 294 rate of, 56, 109, 291 Crystalloids, 227 Dalton's Law of partial pressures, 51, 54, 76, 94, 220 Daniell cell, 129, 359, 423-425 Degree of ionisation, 252, 256, 263, 304 Dehydration of hydrates, 123, 275 of salts, 275 Deliquescence, 126 Density, 4 of gases, 13, 89, 191 of liquids, 139, 208 of solutions, 169, 210, 301 Depression of freezing point, 58, 189, 204, 422 Desmotropy, 284 Deviation from gas laws, 86 Diacetene-alcohol, catalysis of, 309 Dialysis, 227 Diazoacetic ester, catalysis of, 303 Dibasic acids, dissociation constant, 312 Dielectric constant, 231, 262, 263 Dieterici's equation, 93 Diffusion, 162, 179 e.m.f., 429 of gases, 85, 163 in solids, 163 in solution, 163, 179, 227 Dilatometer, 101, 309 Dilute solutions, 415 Dilution formulæ, 254, 257, 262 Dimensions of atonis and molecules, 371- 380 of colloid particles, 223, 228 Diminution of dissociation, 281, 310, 344 of solubility, 321, 342 Discharging potentials, 359, 367 Disintegration of elements, 8, 131, 384, 388 radio-active, 384 Dispersed phase, 221 Dispersion in sols, 221 Displacement of acids, 297-302, 338 Dissociation constants, 158, 254, 271, 304, 312 degree of, 252, 256, 263, 304 electrolytic, 215, 247-264, 271, 336 gaseous, 272-284 hydrolytic, 326-335 of molecules, 213 pressure, 275, 276, 283 Dissolved substances, properties of, 162- 171 Distillation of mixtures, 77, 106 with steam, 80 Double decomposition, 346 Dulong and Petit's Law, 17, 25 Dynamic isomerism, 160, 284, 295 Dyne, 5 Efflorescence, 126 Electrical conductivity of acids, 254, 303 conductivity of bases, 254, 307 dispersion, 222 energy, 7, 423 units, 7 Electro-analysis, 369 Electro-chemical lists, 360, 367 Electrodes, 232 normal, 356 potential, 353, 359, 367, 431 reversible, 357 Electrolysis, 230-246, 365-370 Electrolytes, 230, 333 Electrolytic analysis, 369 conductivity, 230-246 decomposition of water, 368 dissociation, 215, 247-264, 271, 336 solution pressure, 352 Electrometer, capillary, 355 INDEX 435 Electromotive force, 7, 352-366, 423-432 Electrons, 376-377, 383, 398 Electro-valency, 235 Elements, 8, 33 classification of, 33-43, 394-397 disintegration of, 8, 131, 384, 388 table of, 21, 38, 392 Elevation of boiling points, 189, 198, 420 Emanations, 384 Emulsoids, 222, 225 Enantiomorphism, 152 Endothermic compounds, 134 Energy, 6 conservation of, 5, 130, 405 intrinsic, 129, 131, 406 Entropy, 405 Equations, thermochemical, 131 Equilibrium, 96-127 chemical, 265-284 constant, 270 of electrolytes, 310-323 radio-active, 386 Equivalent weights, 9, 17 determination of, 17 Erg, 5 Esters, catalysis of, 288, 302 saponification of, 289, 290, 308 Ether, extraction with, 52 Eutectic mixtures, 112-116 Evaporation of solutions, 76 Exothermic compounds, 134 Explosions, 293 Explosion wave, 293 Extraction with ether, 52 Faraday, 7, 234 Faraday's Law, 233 Ferric chloride, hydrates, 63, 121 Fluidity of electrolytic solutions, 243, 251, 253 Formation of phases, 108 Free path, 372 Freezing machine, 74 mixtures, 60, 62 point, 56, 58 depression of, 58, 189, 204, 422 effect of pressure on, 97 of mixtures, 66, 112 of solutions, 58, 189, 258, 422 Fusion and solidification, 55-68 Galvanic cells, 353 Gamma radiation, 383 Gas cell, 370 constant, 24 for solutions, 178 Gas laws, 22, 86 deviations from, 86 for solutions, 176-178, 184-190, 418 Gas molecules, 81, 84, 212, 373 Gaseous diffusion, 85, 163 Gases, density, 13 ideal, 14, 24, 32, 70, 89, 406 inert, 17, 31, 39, 43, 50 kinetic theory, 81 liquefaction, 70 solubility, 50-51, 54 solvent action, 76 specific heat, 29-32, 407 Gay-Lussac's law of volumes, 11 of expansion, 22 for solutions, 177 Gel, 225 Ghosh's dilution formula, 257, 263 Glass, 155, 228 Gold number, 226 Gram-molecular volume, 14, 89 Guldberg and Waage's Law, 267, 280 Half period, 385 Heat, atomic, 26, 32 mechanical equivalent, 6 molecular, 28, 30 specific, 25-32, 36, 84, 407 units, 6, 7 of combustion, 133, 141, 157 * AN UNI SITY Patas » Ang Magi: T Par JY JO A