METHUEN'S TEXT-BOOKS OF SCIENCE 
 
 OUTLINES OF PHYSICAL CHEMISTRY 
 
OUTLINES OF 
 
 PHYSICAL CHEMISTRY 
 
 BY I 
 
 GEORGE SENTER 
 
 D.Sc. (LOND.), PH.D. (LEIPZIG), F.I.C. 
 
 PRINCIPAL OF BIRKBECK COLLEGE. LONDON 
 EXAMINER IN CHEMISTRY, UNIVERSITY OF LONDON 
 
 EXTERNAL EXAMINER IN CHEMISTRY, UNIVERSITY OF BIRMINGHAM 
 
 FORMERLY READER IN CHEMISTRY IN THE UNIVERSITY OF LONDON 
 
 LECTURER ON CHEMISTRY AT ST. MARY'S HOSPITAL MEDICAL SCHOOL AND 
 
 EXAMINER IN CHEMISTRY TO THE ROYAL COLLEGE OF PHYSICIANS OF 
 
 LONDON AND THE ROYAL COLLEGE OF SURGEONS OF ENGLAND 
 
 S2VENTH EDITION 
 
 METHUEN & CO. LTD. 
 
 36 ESSEX STREET W.C. 
 
 LONDON 
 
S4 
 
 First Piiblished 
 Second Edition 
 Third Edition 
 Fourth Edition 
 Fifth Edition 
 Sixth Edition 
 Seventh Edition 
 
 January 28 th rgog 
 February igir 
 September 1912 
 January I 9 I 4 
 
 November 1915 
 April 1918 
 
 A::/: vU : *H J A',-* 
 r**- ** 
 
PREFACE TO FIRST EDITION 
 
 THE present book is intended as an elementary in- 
 troduction to Physical Chemistry. It is assumed 
 that the student taking up the study of this subject has 
 already an elementary knowledge of chemistry and phy- 
 sics, and comparatively little space is devoted to those 
 parts of the subject with which the student is presumed 
 to be familiar from his earlier work. 
 
 Physical chemistry is now such an extensive subject 
 that it is impossible even to touch on all its important 
 applications within the limits of a small text-book. I 
 have therefore preferred to deal in considerable detail 
 with those branches of the subject which usually present 
 most difficulty to beginners, such as the modern theory 
 of solutions, the principles of chemical equilibrium, elec- 
 trical conductivity and electromotive force, and have 
 devoted relatively less space to the relationships between 
 physical properties and chemical composition. The prin- 
 ciples employed in the investigation of physical proper- 
 ties from the point of view of chemical composition are 
 illustrated by a few typical examples, so that the student 
 should have little difficulty in understanding the special 
 works on these subjects. Electrochemistry is dealt with 
 
 A 1 
 
vi OUTLINES OF PHYSICAL CHEMISTRY 
 
 rather more fully than has hitherto been usual in ele- 
 mentary works on Physical Chemistry, and the book is 
 therefore well suited for electrical engineers. 
 
 From my experience as a student and as a teacher, I 
 am convinced that one of the best methods of familiarising 
 the student with the principles of a subject is by means 
 of numerical examples. For this reason I have, as far as 
 possible, given numerical illustrations of those laws and 
 formulae which are likely to present difficulty to the be- 
 ginner. This is particularly important with regard to cer- 
 tain formulae more particularly those in the chapter on 
 Electromotive Force which cannot easily be proved by 
 simple methods, but which even the elementary student 
 must make use of. The really important thing in this 
 connection is not that the student should be able to prove 
 the formula, but that he should thoroughly understand 
 its meaning and applications. 
 
 I have throughout the book used only the most ele- 
 mentary mathematics. In order to make use of some of 
 the formulae, particularly those in the chapters on Velo- 
 city of Reaction and Electromotive Force, an elementary 
 knowledge of logarithms is required, but sufficient for the 
 purpose can be gained by the student, if necessary, from 
 a few hours' study of the chapter on " Logarithms " in 
 any elementary text-book on Algebra. 
 
 The experiments described in the sections headed 
 " Practical Illustrations " at the conclusion of the chap- 
 ters can in most cases be performed with very simple 
 apparatus, and as many as possible should be done by 
 the student. The majority of them are also well adapted 
 for lecture experiments. The more elaborate experi- 
 
PREFACE vli 
 
 ments which arc usually performed during a course of 
 practical Physical Chemistry are also mentioned for the 
 sake of completeness ; for full details a book on Practical 
 Physical Chemistry should be consulted. 
 
 In drawing up my lectures, which have developed into 
 the present book, I have been indebted most largely to 
 the text-books of my former teachers, Ostwald and 
 Nernst, more particularly to Ostwald's Allgemeine Chemie 
 (2nd Edition, Leipzig, Engelmann) and to Nernst's Theo- 
 retical Chemistry (4th Edition, London, Macmillan). 1 The 
 following works, among others, have also been consulted : 
 Van't Hoff, Lectures on Physical Chemistry ; Arrhenius, 
 Theories of Chemistry ; Le Blanc, Electrochemistry; Dan- 
 neel, Elektrochemie (Sammlung Goschen) ; Roozeboom, 
 Phasenlehre ; Findlay, The Phase Rule ; Mellor, Chemi- 
 cal Statics and Dynamics ; Abegg, Die elektrolytische 
 Dissociationstheorie. In these books the student will 
 find fuller treatment of the different branches of the 
 subject. References to other sources of information on 
 particular points are given throughout the book. 
 
 The importance of a study of original papers can 
 scarcely be overrated, and I have given references to a 
 number of easily accessible papers, both in English and 
 German, some of which should be read even by the 
 beginner. In the summarising chapter on " Theories of 
 Solution" references are given which will enable the 
 more advanced student to put himself abreast of the pre- 
 sent state of knowledge in this most interesting subject. 
 
 In conclusion, I wish to express my most sincere 
 thanks to Assistant-Professor A. W. Porter, of University 
 
 x The fifth German edition of Nernst's text-book has now appeared. 
 
viii OUTLINES OF PHYSICAL CHEMISTRY 
 
 College, London, for reading and criticising the sections 
 on osmotic pressure and allied phenomena, and for valu- 
 able advice and assistance on many occasions; also 
 to Dr. H. Sand, of University College, Nottingham, and 
 Dr. A. Slator, of Burton, for criticising the chapters on 
 Electromotive Force and on Velocity of Reaction respec- 
 tively. Lastly, I wish to acknowledge my indebtedness 
 to my assistant, Mr. T. J. Ward, in the preparation of the 
 diagrams and for reading the proofs. 
 
 G. S. 
 
 November, 1908 
 
PREFACE TO SECOND EDITION 
 
 AS less than two years have elapsed since the appear- 
 ance of the First Edition, only a few slight altera- 
 tions have been rendered necessary by the progress of 
 the subject in the interval. The opportunity has, how- 
 ever, been taken to revise the text thoroughly ; in one 
 or two places the wording has been slightly altered for 
 the sake of greater clearness, and some misprints have 
 been corrected. 
 
 A few additions of some importance have also been 
 made. In conformity with the elementary character of 
 the book, the mathematical proofs of the connection be- 
 tween osmotic pressure and the other properties of solu- 
 tions which can be made use of for molecular weight 
 determinations were omitted from the first edition. The 
 book has, however, been more largely used by advanced 
 students than was anticipated, and at the request of 
 several teachers the proofs in question have now been 
 inserted as an appendix to chapter V. The section 
 dealing with the relationship between physical properties 
 and chemical constitution has been rendered more com- 
 plete by the insertion of brief accounts of absorption 
 spectra and of viscosity. 
 
x OUTLINES OF PHYSICAL CHEMISTRY 
 
 I am again indebted to Professor Porter for much 
 kind advice and assistance, and take this opportunity of 
 expressing to him my grateful thanks. I wish also to 
 acknowledge my indebtedness to a number of friends and 
 correspondents, more particularly to Dr. A. Lapworth, 
 F.R.S., Dr. J. C. Philip, Dr. A. E. Dunstan, Dr. W. 
 Maitland and Mr. W. G. Pirie, M.A.. for valuable sug- 
 gestions. 
 
 G. S. 
 
 December, 1910. 
 
PREFACE TO FIFTH EDITION 
 
 THE fact that a fifth large edition is called for within 
 seven years of the first appearance of this book 
 shows that the object with which it was written to 
 assist in spreading a knowledge of the principles and 
 methods of Physical Chemistry is being satisfactorily 
 fulfilled. 
 
 The opportunities afforded by the calls for successive 
 editions have been taken advantage of to keep the book 
 up to date and further to increase its usefulness by in- 
 troducing certain new sections and amplifying others. 
 Thus for the Third Edition a new chapter on Colloidal 
 Solutions was written and a selection of questions and 
 numerical problems was added. In the Fourth Edition 
 the latter important section was amplified, and the last 
 chapter, on Electromotive Force, was considerably ex- 
 tended. For the Present Edition additions have been 
 made to the chapters on Thermochemistry and on 
 Heterogeneous Equilibrium, and a number of minor 
 alterations have been made. 
 
 In the Preface to the First Edition the great educa- 
 tional value of numerical problems and examples was 
 emphasized, and further experience has served to con- 
 firm and strengthen the opinion then stated. In addition 
 to working out the problems in the book, the student 
 would do well to make use of one or other of the excel- 
 lent books on Physico-chemical Calculations which have 
 recently appeared. 
 
 G. S. 
 
 LUMSDEN, ABERDEENSHIRE, 
 September, 1915. 
 
TABLE OF CONTENTS 
 
 (The numbers refer to pages) 
 
 CHAPTER I 
 
 PAGE 
 
 FUNDAMENTAL PRINCIPLES OF CHEMISTRY. 
 
 THE ATOMIC THEORY i 
 
 Elements and compounds, i Laws of chemical combination, 3 
 Atoms and molecules, 5 Fact. Generalisation or natural 
 law. Hypothesis. Theory, 6 Determination of atomic 
 weights. General, 8 Volumetric method. Gay-Lussac's 
 law of volumes. Avogadro's hypothesis, 9 Dulong and 
 Petit's law, n Isomorphism, 13 Determination of atomic 
 weights by chemical methods, 14 Relation between atomic 
 weights and chemical equivalents. Valency, 15 The values 
 of the atomic weights, 16 The periodic system, 20. 
 
 CHAPTER II 
 GASES ||| . ' . 25 
 
 The gas laws, 25 Deviations from the gas laws, 28 Kinetic 
 theory of gases. General, 29 Kinetic equation for gases, 
 30 Deduction of gas laws from the equation pv = %mnc z , 
 31 Van der Waals' equation, 33 Avogadro's hypothesis 
 and the molecular weight of gases. General, 36 Density 
 and molecular weight of gases and vapours, 36 Results 
 of vapour density determinations. Abnormal molecular 
 weights, 40 Association and dissociation in gases, 41 Ac- 
 curate determination of molecular and atomic weights from 
 gas densities, 42 Specific heat of gases. General, 43 
 
TABLE OF CONTENTS x iii 
 
 Specific heat at constant pressure, Cj,, and constant volume, 
 C v , 44 Specific heat of gases and the kinetic theory, 46 
 Experimental illustrations, 47. 
 
 CHAPTER III 
 LIQUIDS 49 
 
 General, 49 Transition from gaseous to liquid state. Critical 
 phenomena, 49 Behaviour of gases on compression, 51 
 Application of Van der Waals' equation to critical pheno- 
 mena, 53 Law of corresponding states, 56 Liquefaction of 
 gases, 58 Relation between physical properties and chemi- 
 cal composition of liquids. General, 59 Atomic and mole- 
 cular volumes, 60 Additive, constitutive, and colligative 
 properties, 62 Refractivity, 63 Rotation of plane of 
 polarization of light, 66 Absorption of light, 69 Viscosity, 
 73 Practical illustrations, 77 
 
 CHAPTER IV 
 SOLUTIONS 80 
 
 General 80 Solution of gases in gases, 81 Solubility of gases 
 in liquids, 82 Solubility of liquids in liquids, 84 Distilla- 
 tion of homogeneous mixtures, 87 Distillation of non- 
 miscible or partially miscible liquids ; steam distillation, go 
 Solution of solids in liquids, 91 Effect of change of tem- 
 perature on the solubility of solids in liquids, 92 Relation 
 between solubility and chemical constitution, 94 Solid 
 solutions, 94 Practical illustrations, 95. 
 
 CHAPTER V 
 
 DILUTE SOLUTIONS . . . . . .97 
 
 General, 97 Osmotic pressure. Semi-permeable membranes, 
 97 Measurement of osmotic pressure, 99 Van't Hoff's 
 theory of solution, 101 Recent direct measurements of 
 osmotic pressure, 104 Other methods of determining osmo- 
 tic pressure, 105 Mechanism of osmotic pressure, 106 
 Osmotic pressure and diffusion, 108 Molecular weight of 
 dissolved substances. General, IOQ Molecular weights 
 
xiv OUTLINES OF PHYSICAL CHEMISTRY 
 
 PACE 
 
 from osmotic pressure measurements, no Lowering of 
 vapour pressure, in Elevation of boiling-point, 114 Ex- 
 perimental determination of molecular weights by the 
 boiling-point method, 116 Depression of the freezing-point, 
 119 Experimental determination of molecular weights by 
 the freezing-point method, 120 Results of molecular weight 
 determinations in solution. General, 121 Abnormal mole- 
 cular weights, 123 Molecular weight of liquids, 125 The 
 results of measurements, 127 Nature of surface tension, 
 129 Practical illustrations, 129. Mathematical deduction 
 of formulae, 131. 
 
 CHAPTER VI 
 THERMOCHEMISTRY I39 
 
 General, 137 Hess's law, 139 Representation of thermo- 
 chemical measurements. Heat of formation. Heat of 
 solution, 141 Heat of combustion, 145 Thermochemical 
 methods, 145 Results of thermochemical measurements, 
 147 Relation of chemical affinity to heat of reaction, 148 
 Practical illustrations, 153. 
 
 CHAPTER VII 
 EQUILIBRIUM IN HOMOGENEOUS SYSTEMS. 
 
 LAW OF MASS ACTION . . . . 154 
 
 General, 154 Law of mass action, 155 Strict proof of the law 
 of mass action, 160 Decomposition of hydriodic acid, 161 
 Dissociation of phosphorus pentachloride, 163 Equili- 
 brium in solutions of non-electrolytes, 164 Influence of 
 temperature and pressure on chemical equilibrium. General, 
 166 Le Chatelier's theorem, 169 Relation between chemi- 
 cal equilibrium and temperature. Nernst's views, 169 
 Practical illustrations, 170. 
 
 CHAPTER VIII 
 HETEROGENEOUS EQUILIBRIUM. THE PHASE 
 
 RULE -V, . * . . . . , . 172 
 
 General, 172 Application of law of mass action to hetero- 
 geneous equilibrium, 172 Dissociation of salt hydrates, 174 
 
TABLE OF CONTENTS xv 
 
 PAGE 
 
 Dissociation of ammonium hydrosulphide, 176 Analogy 
 between solubility and dissociation, 177 Distribution of a 
 solute between two immiscible liquids, 177 The phase rule. 
 Equilibrium between water, ice and steam, 179 Equilibrium 
 between four phases of the same substance. Sulphur, 183 
 Systems of two components. Salt and water, 186 
 Freezing mixtures, 189 Systems of two components. 
 General, 190 Hydrates of ferric chloride, 194 Transition 
 points, 197 Practical illustrations, 197. 
 
 CHAPTER IX 
 
 VELOCITY OF REACTION. CATALYSIS* . . 200 
 
 General, 200 Unimolecular reaction, 202 Other unimolecular 
 reactions, 205 Bimolecular reactions, 207 Trimolecular 
 reactions, 209 Reactions of higher order. Molecular- 
 kinetic considerations, 211 Reactions in stages, 212 
 Determination of the order of a reaction, 213 Complicated 
 reaction velocities, 215 Catalysis. General ,217 Charac- 
 teristics of catalytic actions, 217 Examples of catalytic 
 action. Technical importance of catalysis, 219 Biological 
 importance of catalysis. Enzyme reactions, 221 Mechan- 
 ism of catalysis, 222 Nature of the medium, 224 Influence 
 of temperature on the rate of chemical reaction, 225 
 Formulae connecting reaction velocity and temperature, 228 
 Practical illustrations, 229. 
 
 CHAPTER X 
 ELECTRICAL CONDUCTIVITY .... 234 
 
 General, 234 Electrolysis of solutions. Faraday's laws, 236-=- 
 Mechanism of electrical conductivity, 238 Freedom of the 
 ions before electrolysis, 240 Dependence of conductivity on 
 the number and nature of the ions, 242 Migration velocity 
 of the ions, 243 Practical determination of the relative 
 migration velocities of the ions, 246 Specific, molecular 
 and equivalent conductivity, 249 Kohlrausch's law. Ionic 
 velocities, 251 Absolute velocity of the ions, Internal 
 
xvi OUTLINES OF PHYSICAL CHEMISTRY 
 
 PAGE 
 
 friction, 253 Experimental determination of conductivity of 
 electrolytes, 254 Experimental determination of molecular 
 conductivity, 257 Results of conductivity measurements, 
 258 Electrolytic dissociation, 260 Degree of ionisation 
 from conductivity and osmotic pressure measurements, 261 
 Effect of temperature on conductivity, 263 Basicity of 
 acids from conductivity measurements, 264 Grotthus' hypo- 
 thesis of electrical conductivity, 264 Practical illustrations, 
 264. 
 
 CHAPTER XI 
 
 EQUILIBRIUM IN ELECTROLYTES. STRENGTH 
 
 OF ACIDS AND BASES. HYDROLYSIS . 266 
 
 The dilution law, 266 Strength of acids, 269 Strength of 
 bases, 274 Mixture of two electrolytes with a common ion, 
 276 Isohydric solutions, 277 Mixture of electrolytes with 
 no common ion, 278 Dissociation of strong electrolytes, 
 279 Electrolytic dissociation of water. Heat of neutrali- 
 sation, 283 Hydrolysis, 285 Hydrolysis of the salt of a 
 strong base and a weak acid, 287 Hydrolysis of the salt 
 of a weak base and a strong acid, 290 Hydrolysis of the 
 salt of a weak base and a weak acid, 292 Determination of 
 the dissociation constant for water, 293 Theory of indica- 
 tors, 296 The solubility product, 298 Applications to 
 analytical chemistry, 300 Experimental determination of 
 the solubility of difficultly soluble salts, 301 Complex ions, 
 303 Influence of substitution on degree of ionisation, 
 304 Reactivity of the ions, 306 Amphoteric electrolytes, 
 307 Practical illustrations, 308. 
 
 CHAPTER XII 
 COLLOIDAL SOLUTIONS. ADSORPTION . :; 313 
 
 Colloidal solutions. General, 313 Preparation of colloidal solu- 
 tions, 315 Osmotic pressure and molecular weight of 
 colloids, 316 Optical properties of colloidal solutions, 317 
 Brownian movement, 318 Electrical properties of col- 
 loids, 319 Precipitation of colloids by electrolytes, 320 
 
TABLE OF CONTENTS xvii 
 
 Suspensions, suspensoids and emulsoids, 322 Filtration of 
 colloidal solutions, 323 Adsorption, general, 324 Adsorp- 
 tion of gases. Adsorption formulae, 328 The cause of 
 adsorption, 329 Further illustrations of adsorption, 330. 
 
 CHAPTER XIII 
 THEORIES OF SOLUTION - 333 
 
 General, 333 Evidence in favour of the electrolytic dissociation 
 theory, 335 lonisation in solvents other than water, 337 
 The old hydrate theory of solution, 339 Mechanism of 
 electrolytic dissociation. Function of the solvent, 342 
 Hydration in solution, 345. 
 
 CHAPTER XIV 
 ELECTROMOTIVE FORCE . . . . .348 
 
 The Daniel cell, 348 Relation between chemical and electrical 
 energy, 351 Measurement of electromotive force, 354 
 Standard of electromotive force. The cadmium element, 
 356 Solution pressure, 358 Calculation of electromotive 
 force at a junction metal/salt solution, 360 Differences of 
 potential in a voltaic cell, 362 Influence of change of con- 
 centration of salt solution on the E.M.F. of a cell, 365 
 Concentration cells, 367 Cells with different concentrations 
 of the electrode materials (substances producing ions), 371 
 Electrodes of the first and second kind. The calomel elec- 
 trode, 373 Single potential differences. The capillary 
 electrometer, 378 Gas cells, 383 Potential series of the 
 elements, 387 Cells with different gases, 391 Oxidation- 
 reduction cells, 393 Electromotive force and chemical 
 equilibrium, 396 Electrolysis and polarization, 398 Sep- 
 aration of ions (particularly metals) by electrolysis, 400 
 The electrolysis of water, overvoltage at electrodes, 401 
 Electrolysis and polarization (continued) 404 Accumulators, 
 405. The electron theory, 407 Practical illustrations, 410. 
 
 b 
 
DEFINITIONS AND UNITS 1 
 
 In this section the centimetre-gram-second (C.G.S.) system of units 
 is used throughout, length being measured in centimetres (cms.), mass in 
 grams, and time in seconds. 
 
 Density is mass per unit volume : unit, gram per c.c. (cubic centi- 
 metre). 
 
 Specific Volume (i/density) is volume per unit mass : unit, c.c. per 
 gram. 
 
 Velocity is rate of change of position : unit, cm. per sec. or cm. /sec. 
 
 Acceleration is rate of change of velocity : unit, cm. per sec. per sec. 
 or cm./sec. 2 . 
 
 Momentum is mass x velocity : unit, gram-cm, per sec. 
 
 Force is mass x acceleration (rate of change of momentum). Unit, 
 the dyne, is that force which is required to produce an acceleration of i cm. 
 per sec. per sec. in a mass of i gram. As a gram-weight, falling freely, 
 obtains an acceleration of 980-6 cm. per sec. (owing to the attraction of 
 the earth) the force represented by the gram-weight = g8o'6 dynes at 
 a latitude of 45 and at sea-level. 
 
 Energy may be defined as that property of a body which diminishes 
 when work is done by the body ; and its diminution is measured by the 
 amount of work done. 
 
 Work Done is force x distance (the work done by a force is measured 
 by the product of the force and the distance through which the point of 
 application moves in the direction of the force). The unit of work 
 
 1 The more important constants made use of in physical chemistry 
 are collected here for convenience of reference, 
 
 xviii 
 
DEFINITIONS AND UNITS xix 
 
 (which is also the unit of energy) is the dyne-centimetre or erg. The 
 gram-centimetre unit is sometimes used; i gram-centimetre = 980-6 
 ergs; also the joule ( = io 7 ergs) is frequently used, especially in electrical 
 work (see below). 
 
 Power is rate of doing work, unit, erg per second. 
 
 There are six chief forms of energy: (i) mechanical energy, (2) 
 volume energy, (3) electrical energy, (4) heat, (5) chemical energy, (6) 
 radiant energy. These forms of energy are mutually convertible, and 
 according to the law of conservation of energy, there is a definite and 
 invariable relationship between the quantity of one kind of energy which 
 disappears and that which results. 
 
 The unit of energy, the erg, has already been defined. It is some- 
 times convenient to express certain forms of energy in special units, heat* 
 for example, in calories ; in the following paragraphs the equivalents in 
 ergs of these special units are given. 
 
 Volume Energy is often measured in litre-atmospheres. When a 
 volume, v lt of a gas expands to the volume v% against a constant pressure 
 p, say that of the atmosphere, the external work done by the gas (gained) 
 is p (v z - vj. The (average) pressure of the atmosphere on unit area 
 (i sq. cm.) supports a column of mercury 76 cm. high and i sq. cm. in 
 cross-section. Hence the pressure on i sq. cm. = 76 x 13*596 = 1033-3 
 grams weight (as the density of mercury is 13*596), or 1033-3 x 980-6 = 
 1,013,200 dynes. As the work done is the product of the constant pres- 
 sure and the increase of volume, i litre-atmosphere (the work done when 
 the increase in the volume of a certain quantity of a gas is i litre or 
 1000 c.c.) = 1,013,200 x 1000 c= 1,013,200,000 ergs. 
 
 Electrical Energy is the product of electromotive force and quantity 
 of electricity, and is usually measured in volt-coulombs or joules. The 
 practical unit of quantity of electricity is the coulomb ; it is that quantity 
 of electricity which under certain conditions liberates o'ooniS grams of 
 silver from a solution of silver nitrate. If a coulomb passes through a 
 conductor in i second, the strength of current is i ampere ; the latter is 
 therefore the practical unit of strength of current. The practical unit of 
 resistance is the ohm, which is the resistance at o offered by a column of 
 mercury 106*3 cm * l n g an< ^ weighing 14-4521 grams. The practical unit 
 of electromotive force is the volt; when a current of i ampere passes in 
 i second through a conductor of resistance i ohm, the electromotive 
 force is i volt. 
 
xx OUTLINES OF PHYSICAL CHEMISTRY 
 
 The definitions of the C.G.S. units of electromotive force, current 
 strength and resistance are to be found in text-books of physics, and 
 cannot be given here. It can be shown that i ohm = io 9 C.G.S. units 
 and i ampere = i/io C.G.S. unit; hence, by Ohm's law, i volt = io 8 
 C.G.S. units. Further, i volt-coulomb or i joule = io 8 x io- 1 = io 7 
 C.G.S. units or io 7 ergs. 
 
 Heat Energy is measured in calories. The mean calorie is i/ioo of 
 the amount of heat required to raise i gram of water from o to 100 and 
 does not differ much from the amount of heat required to raise i gram of 
 water from 15 to 16. i calorie = 42,650 gram-centimetres = 41,830,000 
 ergs (the mechanical equivalent of heat) = 4*183 joules. One joule = 
 0-2391 calories. There is no special unit for chemical energy ; it is 
 usually measured in volt-coulombs or calories. 
 
 The value of R, in the general gas equation (p. 27) for a mol of gas = 
 34,760 gram-centimetres = 83,150,000 ergs = 8-315 joules = 1*985 calories 
 = 0*08205 litre-atmospheres. 
 
 USE OF SIGNS IN ELECTRO-CHEMISTRY. 
 There has always been much confusion in Electro-chemistry as to the 
 proper use of positive and negative signs, and even now no general 
 agreement has been reached on the subject. Recently, however, a simple 
 convention has been suggested by the German Electro-chemical Society 
 (Bunsen-Gesellschaft) which promises to find general acceptance. The 
 potential difference has the positive sign if the metal is charged positively 
 with respect to the solution, and negative if the metal is negatively 
 charged, when metal and solution are combined with a comparison 
 electrode to form a cell (cf. p. 358). In the present book, while this con- 
 vention is adopted for the potential series of the elements, etc., the 
 potential differences are often given in absolute value and the E.M.F. of 
 combinations illustrated by the graphic method described on pp. 365, 377, 
 386 and elsewhere. As a result of considerable experience, it has been 
 found that the graphic method is much more useful in avoiding errors of 
 sign than any convention with regard to the use of signs. 
 
OUTLINES OF 
 PHYSICAL CHEMISTRY 
 
 CHAPTER I 
 
 FUNDAMENTAL PRINCIPLES OF CHEMISTRY. 
 THE ATOMIC THEORY 
 
 Elements and Compounds Definite chemical substances 
 are divided into the two classes of elements and chemical com- 
 pounds. Boyle, and later Lavoisier, denned an element as a 
 substance which had not so far been split up into anything 
 simpler. The substances formed by chemical combination of 
 two or more elements were termed chemical compounds 
 This definition proved to be a very suitable one, and retained 
 its value even when many of the substances classed as ele- 
 ments by Lavoisier proved to be complex. In course of 
 time it came to be recognised that the substances which resisted 
 further decomposition possessed certain other properties in com- 
 mon, for example, the so-called atomic heat of solid elements 
 proved to be approximately 6*4 (p. 12), and it was found possible 
 to assign even newly-discovered elements with more or less 
 certainty to their appropriate positions in the periodic table of 
 the elements p. 21). There are, therefore, conclusive reasons, 
 apart from the fact that they have so far resisted decomposi- 
 tion, for regarding elements as of a different order from chemical 
 compounds, and these reasons remain equally valid when full 
 allowance is made for the remarkable discoveries of the last 
 few years in this branch of knowledge. 
 
CHEMISTRY 
 
 Until lately no case of the transformation'of one element into 
 another was known, but recent work on radium, by Ramsay and 
 Soddy and others, has shown that this element is continuously 
 undergoing a series of transformations, one of the final products 
 of which is the inactive gas helium. It might at first sight be 
 supposed that the old view of the impossibility of transforming 
 the elements could be maintained, radium being looked upon 
 as a chemical compound of helium with another element, but 
 further consideration shows that this suggestion is not tenable, 
 as radium fits into the periodic table, and, so far as is known, 
 possesses all those other properties which have so far been con- 
 sidered characteristic of elements as distinguished from chemical 
 compounds. 
 
 Evidence is gradually accumulating which indicates that the 
 slow disintegration, with final production of other elements, is 
 not confined to radium alone, but is shown more particularly 
 by certain elements of high atomic weight such as uranium and 
 thorium. It is true that the change is spontaneous, as so far 
 there is no known means of initiating it or even of influencing 
 its rate, but further progress in this direction is doubtless only 
 a matter of time. As the phenomenon in question is probably 
 a general one, it seems desirable to retain the term " element " 
 to indicate a substance which has a definite position in the 
 periodic table, and has the other properties usually regarded as 
 characteristic of elements. 
 
 From what has been said, it will be evident that it is difficult 
 to define an element in a few words, but in practice there will 
 probably not be much difficulty in drawing the distinction 
 between elements and compounds. Ostwald l (i 907) defines an 
 element as a substance which only increases in weight as the 
 result of a chemical change, and which is stable under any 
 attainable conditions of temperature and pressure, but in this 
 definition the question of radio-active substances is left out of 
 account. 
 
 1 Prinzipien dw Qhentie, Leipzig, 1907, p, 266,. 
 
FUNDAMENTAL PRINCIPLES OF CHEMISTRY 3 
 
 Laws of Chemical Combination Towards the end of 
 the eighteenth century, Lavoisier established experimentally 
 the law of the conservation of mass, which may be expressed 
 as follows : When a chemical change occurs, the total iveight 
 (or mass) of the reacting substances is equal to the total weight 
 (or mass) of the products. As the weight is proportional to 
 the mass or quantity of matter, the above law may also be 
 stated in the form that the total quantity of matter in" the uni- 
 verse is not altered in consequence of chemical (or any other) 
 changes. It is, of course, impossible to prove the law with 
 absolute certainty, but the fact that in accurate atomic weight 
 determinations no results in contradiction with it have been 
 obtained shows that it is valid at least within the limits of the 
 unavoidable experimental error. 
 
 The enunciation of the law of the conservation of mass by 
 Lavoisier, and the extended use of the balance, facilitated the 
 investigation of the proportions in which elements combine, and 
 soon afterwards the first law of chemical combination was estab- 
 lished by the careful experimental investigations of Richter and 
 Proust. This law is usually expressed as follows : 
 
 A definite compound always contains the same elements in the 
 same proportions. 
 
 The truth of this law was called in question by the famous 
 French chemist Berthollet. Having observed that chemical 
 processes are greatly influenced by the relative amounts of 
 the reacting substances (p. 155), he contended that when, for 
 example, a chemical compound is formed by the combination 
 of two elements, the proportion of one of the elements in the 
 compound will be the greater the more of that element there 
 is available. This suggestion led to the famous controversy 
 between Berthollet and Proust (1799-1807), which ended in 
 the firm establishment of the law of constant proportions. All 
 subsequent work has shown that the law in question is valid 
 within the limits of experimental error. 
 
 In certain cases, elements unite in more than one proportion 
 
4 OUTLINES OF PHYSICAL CHEMISTRY 
 
 to form definite chemical compounds. Thus Dalton found by 
 analysis that two compounds of carbon and hydrogen methane 
 and ethylene contain the elements in the ratios 6 : 2 and 6 : i 
 by weight respectively ; in other words, for the same amount of 
 carbon, the amounts of hydrogen are in the ratio 2:1. Similar 
 simple relations were observed for other compounds, and on 
 this experimental basis Dalton (1808) formulated the Law of 
 Multiple Proportions, as follows : 
 
 When two elements unite in more than one proportion, for a 
 fixed amount of one element there is a simple ratio between the 
 amounts of the other element. 
 
 Dalton's experimental results were not of a high order of 
 accuracy, but the validity of the law was proved by the 
 subsequent investigations of Berzelius, Marignac and others. 
 
 Finally, there is a third comprehensive law of combination, 
 which includes the other two as special cases. It has been 
 found possible to ascribe to each element a definite relative 
 weight, with which it enters into chemical combination. The 
 Law of Combining Proportions, which expresses this conception, 
 is as follows : 
 
 Elements combine in the ratio of their combining weight 's, or in 
 simple multiples of this ratio. 
 
 The combining weights are found by analysis of definite com- 
 pounds containing the elements in question. When the com- 
 bining weight of hydrogen is taken as unity, the approxi- 
 mate values for chlorine, oxygen and sulphur are 35-5, 8 and 
 1 6 respectively. These numbers also represent the ratios in 
 which the elements displace each other in chemical com- 
 pounds. Water, for example, contains 8 parts by weight of 
 oxygen to i of hydrogen, and when the former element is dis- 
 placed by sulphur (forming hydrogen sulphide) the new com- 
 pound is found to contain 16 parts by weight of the latter 
 element. 16 parts of sulphur are therefore equivalent to 8 parts 
 of oxygen, and the combining weights are therefore often termed 
 chemical equivalents. The chemical equivalent of an element is 
 
FUNDAMENTAL PRINCIPLES OF CHEMISTRY 5 
 
 that quantity of it which combines with, or displaces, one part 
 (strictly i'oo8 parts) by weight of hydrogen (cf. p. 18). 
 
 It must be clearly understood that the above generalisations 
 or laws are purely experimental ; they express in a simple form 
 the results of the investigations of many chemists on the com- 
 bining powers of the elements, and are quite independent of 
 any hypothesis as to the constitution of matter. As they have 
 been established by experiment, we are certain of their validity 
 only within the limits of the unavoidable experimental error, 
 and cannot say whether they are absolutely true. It is possible 
 that when the methods of analysis are greatly improved, it will 
 be possible to detect small variations in the composition of 
 definite compounds, but up to the present the most careful 
 investigations, in the course of atomic weight determinations, 
 have failed to show any deviation from the results to be expected 
 according to the laws. 
 
 Atoms and Molecules The question now arises as to 
 whether a theory can be suggested which allows of a convenient 
 and consistent representation of the laws enunciated above. 
 The atomic theory, first brought forward in its modern form by 
 Dalton (1808), answers these requirements. Following out an 
 idea of the old Greek philosophers, Dalton suggested that 
 matter is not infinitely divisible by any means at our disposal, 
 but is made up of extremely small particles termed atoms ; the 
 atoms of any one element are identical in all respects and 
 differ, at least in weight, from those of other elements. By the 
 association of atoms of different kinds, chemical compounds 
 are formed. The laws of chemical combination find a simple 
 explanation on the atomic theory. Since a chemical compound 
 is formed by the association of atoms, each of which has a 
 definite weight, it must be of invariable composition. Further 
 when atoms combine in more than one proportion, for a fixed 
 amount of atoms of one kind the amount of the other must in- 
 crease in steps, depending on the relative atomic weight which 
 is the law of multiple proportions. It is here assumed that 
 
6 OUTLINES OF PHYSICAL CHEMISTRY 
 
 the ultimate particles of a compound are formed by the associa- 
 tion of comparatively few atoms, and this holds in general for 
 inorganic compounds. Finally, the law of combining weights 
 is also seen to be a logical consequence of the atomic theory, 
 the empirically found combining weights, or chemical equiva- 
 lents, bearing a simple relation to the (relative) weights of the 
 atoms (p. 15). 
 
 When Dalton brought forward the atomic theory, the number 
 of facts which it had to account for was comparatively small. 
 As knowledge has progressed, the atomic theory has proved 
 capable of extension to represent the new facts, and its applica- 
 tion has led to many important discoveries. At the present 
 day, the great majority of chemists consider that the atomic 
 theory has by no means outgrown its usefulness. 
 
 Fact. Generalisation or Natural Law. Hypothesis. 
 Theory 1 Chemistry, like most other sciences, is based on 
 facts, established by experiment. A few such facts have already 
 been mentioned, for example, that certain chemical compounds, 
 which have been investigated with the greatest care, always 
 contain the same elements in the same proportions. A mere 
 collection of facts, however, does not constitute a science. 
 When a certain number of facts have been established, the 
 chemist proceeds to reason from analogy as to the behaviour of 
 systems under conditions which have not yet been investigated. 
 For example, Proust showed by careful analyses that there are 
 two well-defined oxides of tin, and that the composition of 
 each is invariable. From the results of these and a few other 
 investigations, he concluded from analogy that the composition 
 of all pure chemical compounds is invariable, although of course 
 very few of them had then been investigated from that point of 
 view. To proceed in this way is to generalise^ and the short 
 statement of the conclusion arrived at is termed a generalisation 
 
 1 H. Poincare", La Science et I'Hypothese, Paris, Flammarion ; Ostwald, 
 Vorlesungen uber Naturphilosophie, Leipzig, 1902; Alexander Smith, 
 General Inorganic Chemistry, London, 1906. 
 
FUNDAMENTAL PRINCIPLES OF CHEMISTRY 7 
 
 or law. It will be evident that a law is not in the nature of an 
 absolute certainty ; it comprises the facts experimentally estab- 
 lished, but also enables us (and herein lies its value) to foretell a 
 great many things which have not been, but which if necessary 
 could be, investigated experimentally. The greater the number 
 of cases in which a law has been found to hold, the greater is the 
 confidence in its validity, until finally a law may attain practically 
 the same standing as a statement of fact. We may confidently 
 expect that however greatly our views regarding natural pheno* 
 mena may change, such generalisations as the law of constant 
 proportions will remain eternally true. 
 
 Natural laws can be discovered in two ways : (i) by corre- 
 lating a number of experimental facts, as just indicated ; (2) by 
 a speculative method, on the basis of certain hypotheses as to 
 the nature of the phenomena in question. The meaning to be at- 
 tached to the term " hypothesis " is best illustrated by an example. 
 In the previous section we have seen that the laws of chemical 
 combination are accounted for satisfactorily on the view that 
 matter is made up of extremely small, discrete particles, the 
 atoms. Such a mechanical representation, which is more or less 
 inaccessible to experimental proof, is termed a hypothesis A 
 hypothesis may then be defined as a mental picture of an 
 unknown, or largely unknown, state of affairs, in terms of some- 
 thing which is better known. Thus, the state of affairs in 
 gases, which is and Avill remain unknown to us, is represented, 
 according to the kinetic theory, in terms of an enormous number 
 of rapidly moving perfectly elastic particles, and on this basis 
 it is possible, with the help of certain assumptions, to deduce 
 certain of the laws which are actually followed by gases (p. 25). 
 
 There does not appear to be any fundamental distinction in 
 the use of the terms hypothesis and theory. A theory may be 
 defined as a hypothesis, many of the deductions from which 
 have been confirmed by experiment, and which admits of the con- 
 venient representation of a large number of experimental facts. 
 
 There is some difference of opinion as to the value of 
 
8 OUTLINES OF PHYSICAL CHEMISTRY 
 
 hypotheses and theories for the advancement of science. 1 The 
 majority of scientists, however, appear to consider that the 
 advantages of hypotheses, regarded in the proper light and not 
 as representing the actual state of affairs, are much greater than 
 the disadvantages. Boltzmann, 2 indeed, maintains that " new 
 discoveries are made almost exclusively by means of special 
 mechanical conceptions ". 
 
 DETERMINATION OF ATOMIC WEIGHTS 
 
 General After the laws of chemical combination had been 
 established, the next problem with which chemists had to deal 
 was the determination of the relative atomic weights of the 
 elements. This might apparently be done by fixing on one 
 element, say hydrogen, as the standard ; a compound containing 
 hydrogen and another element may then be analysed, and the 
 amount of the other element combined with one part of hydrogen 
 will be its atomic weight. It is clear, however, that this will 
 be the case only when the binary compound contains one atom 
 of each element, and it was just this difficulty of deciding the 
 relative number of atoms of the two elements present that 
 rendered the decision between a number and one of its multi- 
 ples or sub-multiples so difficult. 
 
 It has already been pointed out that the amount of an ele- 
 ment which combines with, or displaces, part by weight of 
 hydrogen (strictly speaking, 8 parts by weight of oxygen) is 
 termed the combining weight or chemical equivalent of an ele- 
 ment. The first step in determining the atomic weight of an 
 element is to find the chemical equivalent as accurately as pos- 
 sible by analysis and then to find the relation between the 
 atomic weight and chemical equivalent by one of the methods 
 described below. The atomic weight may be equal to, or a 
 simple multiple of, the chemical equivalent. 
 
 1 In one or two recent books, Ostwald has treated certain branches of 
 chemistry on a system free from hypotheses. 
 
 2 Gas Theorie, Leipzig, 1896, p. 4. 
 
FUNDAMENTAL PRINCIPLES OF CHEMISTRY 9 
 
 Dalton, working on the assumption that when two elements 
 unite in only one proportion one atom of each is present, drew 
 up the first table of atomic weights. Water was found by 
 analysis to contain i part of hydrogen to 8 parts of oxygen by 
 weight ; the atomic weight of oxygen was therefore taken as 8. 
 In the same way, since ammonia contained i part of hydrogen 
 to 4-6 parts of nitrogen, the atomic weight of the latter element 
 was taken as 4*6. Great advances in this subject were then 
 made by the Swedish chemist Berzelius. For fixing the pro- 
 portional numbers, he depended to some extent, like Dalton, 
 on the assumption of simplicity of composition, but was able 
 to check the numbers thus obtained by the application of Gay- 
 Lussac's law of volumes and Dulong and Petit's law. Later 
 still, the discovery of isomorphism by Mitscherlich afforded yet 
 another means of checking the atomic weights. Besides these 
 physical methods, chemical methods may also be used for fixing 
 the atomic weights of the elements. Each of these methods 
 ivill now be shortly referred to. 
 
 (a) Volumetric Method. Gay-Lussac's Law of Volumes. 
 Avogadro's Hypothesis Gay-Lussac, on the basis of an 
 extensive series of experiments on the combining volumes of 
 gases, established the law of gaseous volumes, which may be 
 expressed as follows : 
 
 Gases combine in simple ratios by volume, and the volume of 
 the gaseous product bears a simple ratio to the volumes of the re- 
 acting gases, when measured under the same conditions. 
 
 A few years before, the same chemist had discovered that 
 all gases behave similarly with regard to changes of pressure 
 and temperature, and this fact, taken in conjunction with the 
 law of volumes and the atomic theory, seemed to point to some 
 simple relation between the number of particles in equal volumes 
 of different gases. Berzelius suggested that equal volumes of 
 different gases, under corresponding conditions of temperature 
 and pressure, contain the same number of atoms. It was soon 
 found, however, that this assumption was untenable, and the 
 
io OUTLINES OF PHYSICAL CHEMISTRY 
 
 view held at the present day was first enunciated by the Italian 
 physicist Avogadro. He drew a distinction between atoms, the 
 smallest particles which can take part in chemical changes, and 
 molecules, the smallest particles which can exist in a free con- 
 dition, and expressed his hypothesis as follows : 
 
 Equal volumes of all gases, under the same conditions of 
 temperature and pressure, contain the same number of molecules. 
 
 In expressing the results of determinations of the densities 
 of different gases, hydrogen, as the lightest gas, is taken as 
 standard, and the number expressing the ratio of the weights 
 of equal volumes of another gas (or vapour) and hydrogen, 
 measured under the same conditions, is the density of the gas 
 (or vapour density in the case of a vapour). From Avogadro's 
 hypothesis it follows at once that the ratio of the vapour densities 
 of another gas and hydrogen, being a comparison of the relative 
 weights of an equal number of molecules, is also the ratio of the 
 molecular weights. It is usual to refer both atomic and mole- 
 cular weights to the atom of hydrogen as unity, 1 and therefore the 
 molecular weight, being referred to a standard half that to which 
 the vapour density is referred, is double the vapour density. 
 
 When the molecular weight is known, it is a comparatively 
 simple matter to establish the atomic weight. As an example, 
 we may employ the volumetric method to fix the atomic weight 
 of beryllium, a matter of great historical interest. It was found 
 by analysis that beryllium chloride contains 4*55 parts of beryllium 
 to 3 5 '5 parts of chlorine by weight; in other words, the chemi- 
 cal equivalent of beryllium is 4*55. If beryllium be regarded as 
 a bivalent metal (p. 16), the formula for the chloride will be BeCl 2 , 
 and its atomic weight 2. x 4-55 = 9-1. If, however, it is trivalent, 
 the formula for the chloride must be BeCl 3 , and, to obtain the 
 ratio for Be : Cl found experimentally, its atomic weight must be 
 4'55 x 3 ** I 3'^5- The vapour density of the chloride was de- 
 termined by Nilson and Petterson, and from the result the mole- 
 cular weight calculated as 8o'i. The molecule of beryllium 
 
 1 Strictly speaking, to the atom of oxygen as 16 (p. 18). 
 
FUNDAMENTAL PRINCIPLES OF CHEMISTRY n 
 
 chloride cannot therefore contain more than 35-5 x 2 = 71 parts 
 of chlorine, the formula for the chloride is BeCl 2 , and the atomic 
 weight of beryllium 9*1. 
 
 The determination of atomic weights by the volumetric 
 method thus reduces to finding the smallest quantity of an 
 element present in a molecule, referred to the atom of hydrogen 
 as unity. If the molecular weights of a large number of vola- 
 tile compounds containing a particular element are determined, 
 it is practically certain that at least some of the compounds will 
 contain only one atom of the element in question, and the pro- 
 portion in which the element is present in these compounds is 
 its atomic weight. In the above example, for instance, it has 
 been assumed that only one atom of beryllium is present in 
 the molecule of beryllium chloride of weight 8o'i, and the 
 justification for this assumption is that no compound is 
 known the molecule of which contains less than 9-1 parts of 
 beryllium. It is clear that the numbers thus obtained are 
 maximum values, and the possibility is not excluded that the 
 true values may be fractions of those thus arrived at. The 
 values generally accepted are, however, confirmed by so many 
 independent methods that every confidence can be placed in 
 their trustworthiness. 
 
 (b) Dulong and Petit's Law In 1818, the French chemists 
 Dulong and Petit enunciated the important kw that for solid 
 elements the product of the specific heat and atomic weight is 
 constant, amounting to about 6-4. This law is a very striking 
 one when the great differences in the magnitude of the atomic 
 weights are taken into account. Thus, the specific heat of lead 
 the ratio of the quantity of heat required to raise i gram 
 of the metal i in temperature to that required to raise the 
 temperature of the same weight of water i is 0-031, and its 
 atomic weight 207, the product being 6*4; whilst for lithium, 
 with a specific heat of 0*9 and an atomic weight of 7, the 
 product is 6-3. Since quantities of the different elements in 
 the proportion of their atomic weights require the same amount 
 
12 OUTLINES OF PHYSICAL CHEMISTRY 
 
 of heat to raise the temperature by a definite number of degrees, 
 the law may also be expressed as follows : The atoms of all ele- 
 ments have the same capacity for heat. 
 
 It is clear that this law can be used to determine the atomic 
 weight of an element when the specific heat is known, the 
 quotient of the constant by the specific heat giving the re- 
 quired value. Dulong and Petit's law was largely used by 
 Berzelius in fixing the values of the atomic weights. 
 
 Like many other empirical laws, that of Dulong and Petit is 
 only approximately true, the " constant " varying from about 
 6-0 to 6-7. This degree of concordance is, of course, quite 
 sufficient for fixing the values of the atomic weights, as it is 
 only necessary for this purpose to choose between a number 
 and a simple multiple or submultiple. Moreover, the specific 
 heat varies with the allotropic form of the element and with the 
 temperature, and there is much uncertainty as regards the proper 
 conditions for comparison. Regnault, who made a series of 
 very careful determinations of specific heats, showed that most 
 elements of small atomic weight, more particularly carbon, silicon 
 and boron, have exceptionally small atomic heats. Later, how- 
 ever, it was found that the specific heats of these elements 
 increase rapidly with rise of temperature, and at high tempera- 
 tures their behaviour is in approximate accordance with Dulong 
 and Petit's law. This is clear from the accompanying table, 
 showing the behaviour of carbon (diamond) and boron. 
 
 CARBON (DIAMOND). 
 
 BORON. 
 
 Temp. 
 
 Sp. Heat. 
 
 Atomic Heat. 
 
 Temp. 
 
 Sp. Heat. 
 
 Atomic Heat. 
 
 10 
 
 206 
 600 
 
 1000 
 
 0-II28 
 
 0-2733 
 0-4408 
 
 0-4589 
 
 i'33 
 3-25 
 5-28 
 
 5-5i 
 
 27 
 
 126 
 
 177 
 233 
 
 0-2382 
 0-3069 
 0-3378 
 0-3663 
 
 2'6l 
 3'4<> 
 370 
 4-02 
 
 The few elements which show this abnormal behaviour are of 
 
FUNDAMENTAL PRINCIPLES OF CHEMISTRY 13 
 
 low atomic weight, but the converse does not hold, as the 
 atomic heat of lithium is normal. 
 
 Some years after the introduction of Dulong and Petit's law, 
 a similar law for compounds was enunciated by Neumann. He 
 showed that, for compounds of similar chemical character, the 
 product of specific heat and molecular weight is constant in 
 other words, the molecular heats of similarly constituted com- 
 pounds in the solid state are equal. In 1864, Kopp extended 
 Neumann's law by showing that the molecular heat of solid 
 compounds is an additive property, being made up of the 
 sum of the atomic heats of the component atoms. It follows 
 that in certain cases atoms have the same capacity for heat 
 before and after entering into chemical combination. For ex- 
 ample, the specific heat of calcium chloride is 0-174, the mole- 
 cular heat is therefore 0-174 x in = 19-3, and the atomic heat 
 of each atom 6-4. 
 
 This law may be used to estimate the atomic heats of sub 
 stances which cannot be readily investigated in the solid form, 
 The atomic heat of solid oxygen in combination is about 4-0 
 and of solid hydrogen 2*3. 
 
 (c) Isomorphism Mitscherlich observed that the corre- 
 sponding salts of arsenic acid, H 3 AsO 4 , and phosphoric acid, 
 H 3 PO 4 , crystallize with the same number of molecules of water, 
 are identical, or nearly so, in crystalline form, and can be obtained 
 from mixed solutions in crystals containing both salts, so-called 
 mixed crystals. On the basis of these and similar observations, 
 Mitscherlich established the Law of Isomorphism, according to 
 which compounds of the same crystalline form are of analogous 
 constitution. Thus, when one element replaces another in a 
 compound without altering the crystalline form, it is assumed 
 that one element has displaced the other atom for atom. The 
 replacing quantities of the different elements are therefore in 
 the ratio of their atomic weights, and if the atomic weight of 
 one of them is known, that of the other can be calculated. 
 This principle was largely used by Berzelius for fixing atomic 
 
i 4 OUTLINES OF PHYSICAL CHEMISTRY 
 
 weights before the establishment of Dulong and Petit's law, and 
 afforded a welcome corroboration of those obtained by the use 
 of the law of volumes. The converse to the law of isomorphism 
 does not hold, as elements may displace one another atom for 
 atom with complete alteration of crystalline form. 
 
 The principle of isomorphism is, however, somewhat in- 
 definite, inasmuch as even the most closely related compounds 
 are not completely identical in crystalline form, and it is 
 difficult to decide where the line between similarity and want 
 of similarity is to be drawn. Thus the corresponding angles 
 for the naturally-occurring crystals of the carbonates of cal- 
 cium, strontium and barium are: Aragonite, 116 10'; Stron- 
 tianite, 117 19'; Witherite, 118 30'. Tutton, 1 from a careful 
 comparative study of the sulphates and selenates of potassium, 
 rubidium and caesium, has shown that each salt has its own 
 specific interfacial angle, but the differences produced by dis* 
 placing one metal of the alkali series by another does not exceed 
 i of arc, and is usually much less. The three most important 
 characteristics for the establishment of isomorphism are : 
 
 (1) The capacity of forming mixed crystals. The miscibility 
 must be complete, or within fairly wide limits of concentration. 
 
 (2) Similarity of crystalline form, which must include at least 
 approximate agreement in the values of the geometrical con- 
 stants. 
 
 (3) The capacity of crystals of one substance to increase in 
 size in a saturated solution of the other. 
 
 (d) Determination of Atomic Weights by Chemical 
 Methods It is evident from the considerations advanced in 
 the section on the determination of atomic weights by volu- 
 metric methods (p. 10) that if the composition of a binary 
 compound containing one or more atoms of an element such as 
 hydrogen or chlorine for each atom of an element of unknown 
 atomic weight has been determined by analysis, and if further 
 the relative number of atoms of hydrogen or chlorine present is 
 
 1 Science Progress, 1906, i, p. 91. 
 
FUNDAMENTAL PRINCIPLES OF CHEMISTRY 15 
 
 known, the atomic weight of the other element can at once be 
 calculated. In the case of beryllium chloride, it has been 
 shown that the number of chlorine atoms present can be deter- 
 mined by a physical method (p. 10), but such determinations 
 can sometimes be made by purely chemical methods. As an 
 illustration, we will consider the determination of the atomic 
 weight of oxygen. Analysis shows that water contains approxi- 
 mately i part of hydrogen to 8 parts of oxygen by weight. If 
 the molecule of water contains one atom each of hydrogen and 
 oxygen, its formula must be HO and the atomic weight of 
 oxygen will be 8 ; if, on the other hand, two atoms of hydrogen 
 are present, the formula must be H 2 O and the atomic weight of 
 oxygen must be 1 6 in order to obtain the ratio between the 
 weights of the elements found experimentally. It has been 
 found that by the action of metallic sodium half the hydrogen 
 in water can be displaced, and as by definition atoms are indi- 
 visible, this indicates that the molecule of water contains two (01 
 a multiple of two) atoms of hydrogen. The (probable) formula 
 for water is therefore H 2 O and the atomic weight of oxygen 16. 
 
 It will be evident that, as in the case of the volumetric 
 method, the value thus obtained for the atomic weight is a 
 maximum and further experiments are necessary to fix the 
 value definitely. 
 
 Relation between Atomic Weights and Chemioal 
 Equivalents. Yalency The exact proportions in which 
 the elements enter into chemical combination are determined 
 by analysis, and the numbers thus obtained, referred to a 
 definite standard, represent, according to the atomic theory, the 
 atomic weights, or simple multiples or sub-multiples of the 
 atomic weights, of the respective elements. The choice between 
 several possible numbers is based on the methods discussed in 
 the foregoing paragraphs, more particularly on Avogadro's hypo- 
 thesis, and the fact that these independent methods give the 
 same values affords strong evidence that the numbers thus ob- 
 tained are the true ones a view which obtains still further 
 
1 6 OUTLINES OF PHYSICAL CHEMISTRY 
 
 support from the periodic classification of the elements due to 
 Mendeleeff (1869). 
 
 The ordinary chemical formulae with which the student is 
 familiar are based on the atomic weights thus obtained. For 
 example, the formulae of a number of compounds containing 
 only hydrogen and one other element are as follows : HC1, 
 H 2 O, NH 3 , CH 4 . It is evident from these formulae that the 
 power of different elements to combine with hydrogen is very 
 different ; whilst one atom of chlorine combines with only one 
 atom of hydrogen, one atom of carbon can become associated 
 with no less than four atoms of hydrogen. The combining 
 capacity of an element for hydrogen or other univalent element 
 is termed the valency of an element, chlorine being a univalent 
 and carbon a quadrivalent element. A little consideration of 
 the above formulae will make clear the relationship between the 
 atomic weight and the chemical equivalent of an element. It 
 is clear that an amount of one element equivalent to its atomic 
 weight may combine with (or displace) i, 2, 3 or more parts 
 of hydrogen by weight, depending on its valency. Since the 
 chemical equivalent of an element is that amount of it which 
 can combine with or displace one part by weight of hydrogen it 
 follows that 
 
 Atomic weight ~, , , 
 
 s_ = Chemical equivalent. 
 
 Valency 
 
 As has been indicated in the foregoing paragraphs, the 
 atomic weight and the chemical equivalent are often deter- 
 mined more or less independently and the quotient of the two 
 values is the valency. In other cases, however (for example, 
 the volumetric method and the chemical method), the chemical 
 equivalent and the valency are determined, and the product of 
 the two is the atomic weight. 
 
 The Values of the Atomic Weights After the establish- 
 ment of the law of multiple proportions and the formulation of 
 the atomic theory by Dalton, it became a matter of the utmost 
 importance for chemists to determine the atomic weights of 
 
FUNDAMENTAL PRINCIPLES OF CHEMISTRY 17 
 
 the elements with the greatest possible accuracy. This task 
 was undertaken by Berzelius, who, in the course of about six 
 years (1810-1816), fixed the combining weights of most of 
 the known elements. Since then, the determination of atomic 
 weights has proceeded regularly, but on two occasions a 
 special impulse was given to these investigations. The first 
 occasion was a suggestion by Prout that the atomic weights 
 are exact multiples of that of hydrogen. The idea under- 
 lying this assertion was that hydrogen is the primary element, 
 the other elements being formed from it by condensation. 
 The results of Berzelius were incompatible with Prout's 
 hypothesis, but as the atomic weights of certain elements un- 
 doubtedly approximated to whole numbers, Stas made a number 
 of atomic weight determinations with a degree of accuracy which 
 has only been improved upon in quite recent times. The results 
 obtained by Stas completely disposed of Prout's hypothesis in 
 its original form. 
 
 The second event which stimulated atomic weight investiga- 
 tions was the development of the periodic classification of the 
 elements. In certain cases, the order of the elements, arranged 
 according to their atomic weights, did not correspond with their 
 chemical behaviour, and Mendeleeff asserted that in these 
 cases the commonly accepted atomic weights were inaccurate. 
 The investigations undertaken to test these suggestions afforded 
 striking confirmation of Mendeleeff's views in some cases, but 
 not in others. As regards recent progress in this branch of 
 investigation special mention should be made of the determina- 
 tion of the combining ratios of hydrogen and oxygen by Morley l 
 and of the comprehensive and masterly investigations of T. W. 
 Richards and his co-workers. 2 
 
 As the combining weights are relative, it is necessary to fix 
 on a standard to which they may be referred. Dalton took the 
 atomic weight of hydrogen, the lightest element, as unit, but 
 
 1 Smithsonian Contributions to Knowledge, 1895. 
 
 2 See, for example, J. Amer. Chem. Soc., 1907, 29, 808-826. 
 
 2 
 
1 8 OUTLINES OF PHYSICAL CHEMISTRY 
 
 Berzelius, from practical considerations, proposed oxygen as 
 standard, putting its atomic weight = 100. The justification 
 for this procedure is that very few elements form compounds 
 with hydrogen suitable for analysis ; the majority of determina- 
 tions have been made with compounds containing oxygen, and 
 until comparatively recently the ratio of the atomic weights of 
 hydrogen and oxygen was not accurately known. Although 
 the hydrogen standard again came into use after the time of 
 Berzelius, mainly because hydrogen was taken as a standard for 
 other properties, yet in more recent times the oxygen standard 
 has again come most largely into use, the atomic weight of that 
 element being taken as 16*00. The unit to which atomic 
 weights are referred is therefore ^ of the atomic weight of 
 oxygen, and is rather less than the atomic weight of hydrogen. 1 
 Besides the advantage already mentioned, the atomic weights of 
 more of the elements approximate to whole numbers when the 
 oxygen standard is used, which is a distinct advantage, since 
 round numbers are generally used in calculations. 
 
 Chemical equivalents, like atomic weights, should be referred 
 to the oxygen standard, and the chemical equivalent of an 
 element is a number representing that quantity of it which 
 combines with, or displaces, 8 parts by weight of oxygen. 
 
 The arguments in favour of the oxygen standard seem con- 
 clusive, and as it is very confusing to have two standards in 
 general use, it is very satisfactory that the International Com- 
 mittee on Atomic Weights now use the oxygen standard only. 
 
 1 According to the recent determinations of Morley, Rayleigh and others 
 the atomic weight of oxygen is about 15*88 when hydrogen is taken as 
 unit. It follows that on the oxygen standard, O = 16-00, the atomic weighf. 
 of hydrogen is about i'oo8. 
 
FUNDAMENTAL PRINCIPLES OF CHEMISTRY 19 
 INTERNATIONAL ATOMIC WEIGHTS (1912) 
 
 Elements. 
 
 Sym- 
 bols. 
 
 At. Wt. 
 = i6. 
 
 Elements. 
 
 Sym- 
 bols. 
 
 At. Wt. 
 O = 16. 
 
 Aluminium 
 
 Al 
 
 2 7 -I 
 
 Neodymium . 
 
 Nd 
 
 I 44'3 
 
 Antimony . 
 
 Sb 
 
 I20'2 
 
 Neon 
 
 Ne 
 
 20'2 
 
 Argon 
 
 A 
 
 39-88 
 
 Nickel 
 
 Ni 
 
 58-68 
 
 Arsenic 
 
 As 
 
 74-96 
 
 Niton 
 
 
 
 Barium 
 
 Ba 
 
 I3737 
 
 (Radium Emana- 
 
 
 
 Bismuth 
 
 Bi 
 
 208-0 
 
 tion) 
 
 Nt 
 
 222-4 
 
 Boron . 
 
 B 
 
 ii-o 
 
 Nitrogen . 
 
 N 
 
 14-01 
 
 Bromine . 
 
 Br 
 
 79-92 
 
 Osmium 
 
 Os 
 
 190-9 
 
 Cadmium . . 
 
 Cd 
 
 112-40 
 
 Oxygen 
 
 
 
 16*00 
 
 Caesium . 
 
 Cs 
 
 132-81 
 
 Palladium . 
 
 Pd 
 
 106-7 
 
 Calcium 
 
 Ca 
 
 40-07 
 
 Phosphorus 
 
 P 
 
 31-04 
 
 Carbon 
 
 C 
 
 I2'OO 
 
 Platinum . 
 
 Pt 
 
 195-2 
 
 Cerium 
 
 Ce 
 
 140-25 
 
 Potassium . 
 
 K 
 
 39-10 
 
 Chlorine . 
 
 Cl 
 
 35-46 
 
 Praseodymium . 
 
 Pr 
 
 140-6 
 
 Chromium . 
 
 Cr 
 
 52-O 
 
 Radium 
 
 Rd 
 
 226-4 
 
 Cobalt 
 
 Co 
 
 58-97 
 
 Rhodium . 
 
 Rh 
 
 1 02 -9 
 
 Columbium 
 
 Cb 
 
 93-5 
 
 Rubidium . 
 
 Rb 
 
 85-45 
 
 Copper 
 
 Cu 
 
 63-57 
 
 Ruthenium 
 
 Ru 
 
 101-7 
 
 Dysprosium 
 
 Dy 
 
 162-5 
 
 Samarium . 
 
 Sa 
 
 150-4 
 
 Erbium 
 
 Er 
 
 167-7 
 
 Scandium . 
 
 Sc 
 
 44-1 
 
 Fluorine . 
 
 F 
 
 19-0 
 
 Selenium . 
 
 Se 
 
 79-2 
 
 Gadolinium 
 
 Gd 
 
 I 57'3 
 
 Silicon 
 
 Si 
 
 28-3 
 
 Gallium 
 
 Ga 
 
 69-9 
 
 Silver . 
 
 Ag 
 
 107-88 
 
 Germanium 
 
 Ge 
 
 72-5 
 
 Sodium 
 
 Na 
 
 23-00 
 
 Glucinum . 
 
 
 
 Strontium . 
 
 Sr 
 
 87-63 
 
 (beryllium) . 
 
 Gl 
 
 9-1 
 
 Sulphur 
 
 S 
 
 32-07 
 
 Gold ... 
 
 Au 
 
 197-2 
 
 Tantalum . 
 
 Ta 
 
 181-5 
 
 Helium 
 
 He 
 
 3*99 
 
 Tellurium . 9 . 
 
 Te 
 
 127-5 
 
 Hydrogen . . 
 
 H 
 
 1-008 
 
 Terbium 
 
 Tb 
 
 159-2 
 
 Indium 
 
 In 
 
 114-8 
 
 Thallium . 
 
 Tl 
 
 204-0 
 
 Iodine . 
 
 I 
 
 126-92 
 
 Thorium . 
 
 Th 
 
 232-4 
 
 Iridium . 
 
 Ir 
 
 193-1 
 
 Thulium . 
 
 Tm 
 
 168-5 
 
 Iron . 
 
 Fe 
 
 55-84 
 
 Tin . 
 
 Sn 
 
 119-0 
 
 Krypton 
 
 Kr 
 
 82-9 
 
 Titanium . 
 
 Ti 
 
 48-1 
 
 Lanthanum 
 
 La 
 
 139-0 
 
 Tungsten . 
 
 W 
 
 184-0 
 
 Lead . 
 
 Pb 
 
 207-10 
 
 Uranium . 
 
 U 
 
 238-5 
 
 Lithium 
 
 Li 
 
 6-94 
 
 Vanadium . 
 
 V 
 
 51-0 
 
 Lutecium . 
 
 Lu 
 
 174-0 
 
 Xenon 
 
 Xe 
 
 130-2 
 
 Magnesium 
 
 Mg 
 
 24-32 
 
 Ytterbium 
 
 
 
 Manganese 
 Mercury 
 
 Mn 
 H g 
 
 54-93 
 200-6 
 
 (Neoytterbium). 
 Yttrium 
 
 Yb 
 Yt 
 
 172*0 
 89-0 
 
 Molybdenum 
 
 Mo 
 
 96-0 
 
 Zinc . 
 
 Zn 
 
 65-37 
 
 
 
 
 Zirconium . . 
 
 Zr 
 
 90-6 
 
 Only significant figures are given in the table. Where no 
 figure follows the decimal point, the value of the first decimal 
 is uncertain. 
 
20 OUTLINES OF PHYSICAL CHEMISTRY 
 
 The Periodic System It was early observed that there 
 are some remarkable relationships between the magnitude of 
 the atomic weights of the elements and their chemical behaviour. 
 The most important observation in this connection is that the 
 differences in the atomic weights of successive members of the 
 same group of elements are approximately 16 or a multiple of 
 that number. Thus for the halogen group, F = 19, Cl = 35*5, 
 Br = 80, I = 127, the differences between each element and 
 its immediate predecessor are 16-5, 44-5 and 47 respectively, 
 the latter two numbers being approximately 3x16. Further, 
 for the members of the alkali group, Li = 7, Na = 23, K = 39, 
 Rb = 85*5, Cs = 132*9, the differences are 16, 16, 46*5 and 
 47*2 respectively. 
 
 In 1864, a considerable advance was made by the English 
 chemist Newlands, which is summarised in the law of octaves. 
 He pointed out that when the elements, beginning with lithium, 
 are arranged in the order of ascending atomic weights, there is 
 a gradual variation in properties till the eighth element is 
 reached ; this element (sodium) shows a strong resemblance to 
 the first element, lithium, the ninth element, magnesium, is 
 similar in chemical behaviour to beryllium, and so on. The 
 first fourteen elements may therefore be arranged as follows : 
 
 Li=*7 Be = 9 B = n C=i2 N = i4 O=i6 F=i9 
 Na=2 3 Mg=2 4 Al=2 7 Si = 28 P = 3 i 8 = 32 = 35-5 
 
 and the elements which show similar chemical behaviour are 
 thus brought into the same vertical row. On these lines, but 
 without any knowledge of the views of Newlands, a complete 
 system for classifying the elements, termed the periodic system 
 was later developed by Mendeleeff. The system is based on 
 the observation that when the elements are arranged in the 
 order of ascending atomic weights, elements with similar chemical 
 properties recur at regular intervals. 
 
O 
 
 Ii 
 
 w 
 
 
 3 
 
 I 
 
 
 
 ro 
 
 Jl 
 
 u 
 
 CQ 
 
 B >. 
 
 so 
 
 fc fc 
 
 s 
 
 ro 
 II 
 
 CO 
 
 
 N 
 
 8 
 
 II 
 
 ii 
 
 r 
 
 O vo j^ 
 
 ^T " w i 
 
 ! ii 4 
 
 & 
 
 w w 
 
 
22 OUTLINES OF PHYSICAL CHEMISTRY 
 
 The accompanying table (p. 2 i), in which the atomic weights 
 are only approximate, is practically the same as that proposed 
 originally by Mendeleeff. Starting with helium = 4 (which 
 was unknown in Newlands' time) we have the arrangement 
 shown in the first line of the table, in which the properties 
 vary regularly from the first member to fluorine. The next 
 element, neon = 20, is an inactive gas, and is therefore placed 
 below helium, sodium falls into its proper place below lithium, 
 and so on. The first and second periods possess 8 elements 
 each. A third period is started with potassium, but in this case 
 it is necessary to pass over 18 elements before another metal 
 (rubidium), bearing a close resemblance to potassium, is reached. 
 Such a period of 18 elements is termed a long periodic contrast 
 to the two short periods of 8 elements each. The whole table 
 is made up of two short and five long periods, but four of the 
 long periods are . incomplete and the last one contains only 
 three elements. The positions of the elements in these periods 
 are fixed by their chemical relationships with those above them 
 (in the vertical rows), and it is assumed that the blanks indicate 
 the positions of elements which have not yet been discovered. 
 The arrangement of the three intermediate elements in each of 
 the first three long periods presented a certain difficulty, and 
 Mendeleeff put them in a group by themselves, the so-called 
 eighth group (group vm. in the table). 
 
 A study of the elements arranged as above reveals many 
 striking regularities. Thus the valency with regard to hydrogen 
 increases regularly up to the middle of a short period and then 
 falls to unity, whilst the valency for oxygen increases regularly 
 from the beginning to the end of a period. Helium and the 
 elements in the same vertical row do not enter into chemical 
 combination, and may therefore be regarded as having zero 
 valency. The valency relations in the long periods are not 
 quite so regular, being complicated by the fact that most of 
 the elements have several valencies. 
 
 Many of the physical properties of the elements, such as the 
 
FUNDAMENTAL PRINCIPLES OF CHEMISTRY 23 
 
 melting-point, the atomic volume and the density, also vary 
 regularly within each period. For example, the melting-points 
 in the first series gradually rise from helium to carbon and fall 
 again to fluorine, and similarly the elements of highest melting- 
 point (iron, cobalt, nickel, etc.) occur in the middle of the 
 long periods. Besides this variation of physical properties in 
 the horizontal series, there is a similar, but much less marked, 
 variation in the vertical series ; in the case of the alkali metals, 
 for example, there is a gradual fall in the melting-point from 
 lithium to caesium. 
 
 Not only the physical properties, but also the chemical 
 properties of the elements vary regularly within the periods. 
 Thus the elements on the extreme left hand of the table are 
 inactive gases, those in the second group decompose water and 
 are strongly electropositive, at the middle of the period they 
 appear to be electrically indifferent (carbon, silicon) and towards 
 the right hand strongly electronegative. 
 
 The statements in the last three paragraphs are summarised 
 in the periodic law, due to Mendeleeff, which may be expressed 
 as follows : The properties of the elements^ as well as the proper- 
 ties of their compounds, are periodic functions of the atomic 
 weights. 
 
 The arrangement of the elements according to the periodic 
 system is not in all respects satisfactory. Copper, silver and 
 gold do not fit very well into their positions beside the alkali 
 metals, and it has been suggested that they belong more 
 properly to the eighth group, coming after nickel, palladium 
 and platinum respectively. Further, the atomic weight of argon 
 is greater than that of potassium, but the former element must 
 undoubtedly precede the latter in the periodic table. The 
 question which has raised most discussion in this connection, 
 however, is the relative position of tellurium and iodine. 
 Although from its chemical relationships the latter element 
 must follow tellurium, yet experiment shows that the atomic 
 weight of tellurium is greater than that of iodine. It is at first 
 
24 OUTLINES OF PHYSICAL CHEMISTRY 
 
 natural to suppose that there must have been some mistake in 
 determining the atomic weights, but the recent work of Laden - 
 burg, Puccini, Kothner, Brereton Baker 1 and others leave 
 practically no room for doubt that the facts are as stated. It is 
 unquestionable that the periodic system is of great value, but 
 the above considerations indicate that it is only a first approxi- 
 mation to a satisfactory system. 
 
 The periodic system is of use mainly in three ways : 
 
 (a) As a system of classification which indicates in a fairly 
 satisfactory way the chemical and physical relationships of the 
 elements. 
 
 () For predicting the existence and properties of elements 
 hitherto undiscovered. 
 
 (c) For enabling us to fix the correct values of the atomic 
 weights of elements which do not form volatile compounds. 
 
 When the periodic system was first brought forward, there 
 were more blanks in the table than there are at the present day, 
 and MendeleefT not only suggested that the positions of these 
 blanks corresponded with hitherto undiscovered elements, but 
 even foretold the properties of the missing members of the 
 series from the known properties of the elements near them in 
 the periodic table. It is an interesting historical fact that 
 within a few years three of the blanks had been filled by ele- 
 ments gallium (1875), scandium (1879), germanium (1886) 
 having in all respects the properties foretold by Mendeleeff. 
 
 The use of the periodic system, for fixing atomic weights will 
 be readily understood from the foregoing. When the equivalent 
 of the element has been determined, it is usually possible to 
 decide which multiple of it is to be taken, as there will in 
 general be only one position in the table into which the element 
 can be satisfactorily fitted. 
 
 1 Trans. Chetn. Soc., 1907, 91, 1849. 
 
CHAPTER II 
 GASES 
 
 The Gas Laws The gaseous form of matter is charac- 
 terised by its tendency to fill completely and to a uniform 
 density any available space. In general, gases are less dense 
 than other forms of matter, and their internal friction is 
 much less. In consequence of this, the laws expressing the 
 behaviour of gases under varying conditions are much simpler 
 than those holding for liquids and solids. The most striking 
 fact about these laws is that they are to a great extent inde- 
 pendent of the nature of the gas : the volume of all gases is 
 affected by changes of temperature and pressure to much the 
 same extent. 
 
 The well-known laws which represent more or less accurately 
 the behaviour of all gases under varying conditions may be 
 enunciated as follows : - 
 
 1. At constant temperature, the volume, v, of a given mass of 
 any gas is inversely proportional to the pressure, p ; otherwise 
 expressed, pv = constant (Boyle, 1662). 
 
 2. At constant pressure, the volume, v, of a given mass of 
 any gas is proportional to its absolute temperature, T (273 + 
 temp. Centigrade] (Gay-Lussac, 1802). 
 
 3. At constant volume, the pressure of a given mass of any 
 gas is proportional to its absolute temperature. 
 
 These laws are not independent ; when any two of them are 
 known, the third can readily be deduced. 
 
 The three laws just given may be summarised in a single 
 
 25 
 
26 OUTLINES OF PHYSICAL CHEMISTRY 
 
 equation, which represents the behaviour of a gas when any two 
 of the determining factors are varied. Let / , v and T repre- 
 sent the original pressure, volume and temperature of a definite 
 quantity of a gas, and p lt v l and 1\ the final values. Suppose 
 that at first the pressure is altered from / to its final value/! 
 at constant temperature, then, by Boyle's law, the gas will have 
 a new volume, V, given by the equation / # = p-^T. Then, 
 keeping the pressure constant at p lt alter the temperature from 
 T to the final value T lf the final volume, v lt will, by Gay-Lussac's 
 law, be given by the equation V/T = s^/Tj. Substituting in the 
 last equation the value of V obtained from the former equation 
 (V = /oV/i) we obtain 
 
 P Q V Q p^v, 
 
 ^ = S*H = constant. 
 L o M 
 
 This may be written in the form pv rT where r is a constant ; 
 in other words, the product of the pressure and volume of a gas 
 is proportional to the absolute temperature. 
 
 At a definite temperature and pressure, the volume of the 
 gas', and consequently the value of r, will be proportional to 
 the quantity of gas taken. Further, according to Avogadro's 
 hypothesis, the molecular weight in grams of all gases occupies 
 the same volume under the same conditions. It follows that for 
 these quantities the constant r will have the same value for all 
 gases, quite independent of the conditions under which the 
 gases are measured. This special value of the constant may 
 conveniently be represented by R, and we then obtain the 
 equation 
 
 PV = RT, 
 
 where V is the volume occupied by the molecular weight of a 
 gas in grams, at the absolute temperature T and under the 
 pressure P an equation which is of fundamental importance 
 for the behaviour of gases and also for dilute solutions. The 
 molecular weight of a gas in grams, which, according to the 
 atomic theory, represents the weight of the same number of 
 molecules in each case, is conveniently termed a mol (Ostwald). 
 
GASES 27 
 
 The numerical value of R, in C.G.S. units, may readily be 
 calculated from the accurate observations of the densities of 
 gases made by Regnault, Rayleigh and others. There is, how- 
 ever, a little uncertainty in the calculation owing to the fact that 
 the volumes occupied by a mol of different gases under equiva- 
 lent conditions are not quite the same, although very nearly so. 
 Thus 2 -oi 6 grams of hydrogen, 32-00 grams of oxygen and 
 28*02 grams of nitrogen occupy 22*43, 22 '39 ar >d 22*40 litres 
 respectively at o and 76 cms. Taking 22*40 litres as a mean 
 value, and substituting the values for T (273) and P (76 x 
 13*59 = 1033*3 grams per sq. cm.), we obtain 
 
 PV low* x 22,400 
 
 ergs 
 
 As the pressure is measured in gram/cm. 2 , the volume is of the 
 dimensions cm. 8 , and T is merely a number, the above value 
 for R is of the form gram x cms. or gram-centimetres. The 
 calorie, the ordinary heat unit, is equal to 42,640 gram-centi- 
 metres = 41,830,000 ergs, so that the value of R is almost ex- 
 actly double (accurately 1*99 times) that of a calorie. We may 
 therefore write the gas equation in the simplified form PV = 
 i*99T, but the approximate form PV = 2T is sufficiently accu- 
 rate for many purposes. In this form the gas equation is repre- 
 sented in thermal units. 
 
 The product P V in the gas equation is of the nature of energy, 
 as is clear from the fact that when a volume, v t of a gas is gener- 
 ated under constant external pressure (say that of the atmosphere) 
 the work done is proportional to the volume and to the pressure 
 overcome, and therefore to their product. The work done by 
 or upon a gas when it changes its volume under a constant ex- 
 ternal pressure can readily be obtained in thermal units by using 
 the second form of the gas equation given above. Thus if a 
 mol of a gas is generated at o under the pressure of the atmos- 
 phere, the amount of heat absorbed in performing the external 
 work of expansion, PV, is 2! = 2 x 273 = 546 cal. 
 
 Since PV is constant, the pressure under which a definite 
 
a8 OUTLINES OF PHYSICAL CHEMISTRY 
 
 mass of a gas is generated at a definite temperature has no 
 influence on the external work of expansion. Thus, to take 
 the above illustration, if a mol of a gas is generated under a 
 
 pressure of atmosphere, the volume will be 22-4 x 10 = 224 
 
 litres, and the work done will again be 546 cal. at o. 
 
 Deviations from the Gas Laws Careful experiment 
 shows that although the gas laws, which are summarised in 
 the general formula PV = RT, give a general idea of the 
 behaviour of gases, yet they do not represent accurately the 
 behaviour of any single gas, the deviations depending both on 
 the conditions of observation and on the nature of the gas. 
 It may be said, in general, that the laws are the more nearly 
 obeyed the higher the temperature and the smaller the pressure, 
 and, as regards the nature of the gas, the further it is removed 
 from the temperature of liquefaction. A gas which would folloM 
 the gas laws accurately is called a perfect or ideal gas, and 
 ordinary gases approach more or less nearly to this ideal 
 behaviour. 
 
 The accompanying figure (Fig. i) gives a graphic representa- 
 tion of the behaviour of the three typical gases, hydrogen, 
 nitrogen and carbon dioxide, according to Amagat. The pro- 
 duct PV, in arbitrary units, is represented on the vertical axis 
 and the pressure P, in atmospheres, along the horizontal axis. 
 If PV were constant (Boyle's law), the curves would be straight 
 lines parallel to the horizontal axis. Actually, PV increases 
 continuously with the pressure in the case of hydrogen, and for 
 nitrogen and carbon dioxide it first decreases, reaches a mini- 
 mum, and beyond that point increases with increase of pres- 
 sure. All gases except hydrogen show a minimum in the 
 curve, which indicates that the compressibility is at first greater 
 than corresponds with Boyle's law, reaches a point (which differs 
 for different gases) at which for a short interval Boyle's law is 
 followed, and beyond that point is less compressible than the 
 law indicates. Hydrogen, on the other hand, is always less 
 
GASES 
 
 29 
 
 compressible than the law requires at ordinary temperatures, 
 but at very low temperatures it would also probably show a 
 minimum in the curve. That the deviations from the simple 
 law become less the higher the temperature, is very well illus- 
 trated by the curves for carbon dioxide at 35*1 and 100. 
 
 Mercury) 
 
 2O 40 6O 80 IOO 120 140 160 1 80 20O 22O 240 260 28O 3<X) 
 FlG. I. 
 
 The general behaviour of gases, and the deviations from the 
 simple laws, find a very satisfactory interpretation on the basis 
 of the kinetic theory of gases, which we are now to consider. 
 
 KINETIC THEORY OF GASES 
 
 General The fact that the laws followed by gases are so 
 simple in character makes it readily intelligible that attempts 
 to account for their properties on a mechanical basis were 
 made very early in the history of science (Bernoulli, 1738). 
 Our present views on the subject, known as the kinetic theory 
 
30 OUTLINES OF PHYSICAL CHEMISTRY 
 
 of gases, are due more particularly to the labours of Clerk 
 /Maxwell, Clausius and Boltzmann. 
 
 According to the theory, gases are regarded as being made 
 up of small, perfectly elastic particles (the chemical molecules) 
 which are in continual rapid motion, colliding with each other 
 and with the walls of the containing vessel. ' The space actually 
 filled by the gas particles is supposed to be small compared with 
 that which they inhabit under ordinary conditions, so that the 
 particles are practically free from each other's influence except 
 during a collision. Owing to the comparatively large free space 
 in which the particles move, an individual particle will move over 
 a certain distance before colliding with another particle; the 
 average value of this distance is termed the " mean free path " 
 of the molecule. . On this theory the pressure exerted by the 
 gas on the walls of the containing vessel (which is equal to the 
 pressure under which it is confined) is due to the bombardment 
 of the walls of the vessel by the moving particles. It is clear that 
 the magnitude of the pressure must depend on the frequency 
 of the collisions, as well as on the mass and velocity of the 
 particles. 
 
 Kinetio Equation for Gases The pressure exerted by a 
 gas can be calculated quantitatively in terms of the number 
 and velocity of the molecules as follows : Imagine a definite 
 mass of a gas contained in a cube, the sides of which are of 
 length /; let n be the number of particles, each of mass m and 
 velocity c. The particles will impinge on the walls in all direc- 
 tions, but the velocity of each may be resolved into three com- 
 ponents, x, y and z, parallel to the edges of the cube, the 
 components being related to the original velocity c by the 
 equation y? + y* + z 2 = r 2 . This means that the action on 
 the wall which the molecule would exert if it reached it in the 
 original path with the velocity c is the same as the sum of the effects 
 if the collisions took place successively in directions perpendicular 
 to the three walls at right angles to each other with the velocities 
 x % y and z respectively. If we consider at first one of the com- 
 
GASES 31 
 
 ponents, the particle will strike the wall at right angles with 
 velocity x, and will fly off with the same velocity in the opposite 
 direction : as the wall and the particle are considered perfectly 
 elastic, there will be no loss of energy in this process. The 
 momentum before the impact is mx, after the impact mx, so 
 that the total change of momentum due to the impact is 2mx. 
 As the distance between the two parallel walls is /, the particle 
 
 will perform impacts on the walls in unit time, and as in each of 
 these the change of momentum is 2 MX, the total effect of a single 
 particle in one direction in unit time will be 2mx - = 2mx z /t. 
 
 The same applies to the other components of the velocity, so 
 that the total action of a molecule on all six sides of the cube 
 
 2ttl , o o\ 21HC 
 
 is (x 2 + y* + 2r) = - ror the total number of mole- 
 cules, , the effect is therefore 2mnc i \L In order to obtain 
 the pressure on unit area, the above value must be divided by 
 the total interior surface of the cube, 6^ ; we then obtain / = 
 
 , p > which, since the volume, v, of the cube is /*, can be put 
 
 in the more convenient form, p = ^mnc^jv or pv = ^mn<?. 
 
 Although the above expression has been deduced only for 
 a cube, it can readily be extended to a vessel of any shape, as 
 follows. The total volume of the latter can be partitioned up 
 into small cubes, and as the pressures on the two sides of the 
 interior walls of these cubes neutralize each other, they do not 
 affect the pressure on the outer walls of the exterior cubes, 
 which in the limit constitute the wall of the containing vessel. 
 
 Deduction of Gas Laws from the Equation /z> = \rnnc* 
 From the above equation, which has been derived on certain 
 more or less plausible assumptions regarding the constitution of 
 gases, some of the laws which have been obtained experimentally 
 may readily be deduced. 
 
 Since on the assumptions made in deducing the general 
 
32 OUTLINES OF PHYSICAL. CHEMISTRY 
 
 formula the right-hand side of the equation is made up of 
 factors which are constant at constant temperature, the product 
 of pressure and volume must also be constant, which is Boyle's 
 law. 
 
 Moreover, the above equation may be written in the form 
 pv = . \rnnc 1 . As shown in mechanics, the expression \m<? 
 represents the kinetic energy of a particle of mass m and velo- 
 city f, and therefore n . \mc 2 represents the kinetic energy of n 
 particles, so that the product of the pressure and volume of a 
 gas is equal to f of the kinetic energy of the molecules. From 
 this it follows that for equal volumes of different gases under the 
 same pressure ', the total kinetic energy of their molecules is the 
 same. 
 
 It has been shown (p. 25) that at constant volume the pres- 
 sure of a gas varies proportionately with the absolute tempera- 
 ture. In the previous paragraph, it has further been shown 
 that at constant volume the pressure of a gas is proportional 
 to the mean kinetic energy of progressive motion of its particles. 
 Hence the mean kinetic energy of the molecules of a gas is pro- 
 portional to its absolute temperature. 1 
 
 Avogadro's hypothesis may now be deduced from the kinetic 
 theory. For equal volumes of two gases at the same pressure 
 we have, from the above equation, 
 
 pv^^mrf-^mj* . (i) 
 
 where n^ and n 2 , m l and m% and ^ and c 2 represent the number, 
 mass and velocity of the molecules of the respective gases- 
 Further, if the gases are at the same temperature, the mean 
 kinetic energy of a single molecule is the same for each gas, 2 
 hence 
 
 Dividing equation (i) by equation (2) we obtain n = n. p or, 
 otherwise expressed, equal volumes of two gases, under the same 
 
 1 In this case, instead of deducing the experimental law from the kinetic 
 theory, we have made use of the experimental result to obtain a definition 
 of temperature on the kinetic theory. The temperature of a gas is deter- 
 mined by the mean kinetic energy of progressive movement of its molecules. 
 2 This was proved by Clerk Maxwell ; it dees not foHow directly from 
 the above statement. 
 
GASES 
 
 33 
 
 conditions of temperature and pressure, contain the same num- 
 ber of molecules, which is Avogadro's hypothesis. 
 
 Finally, the mean velocity, c, of the molecules 1 may be cal- 
 culated by substituting the appropriate values for the other 
 magnitudes in the general equation pv = tyrinc 1 . Thus for i 
 mol 2 'oi 6 grams of hydrogen under standard conditions, 
 P IO 33'3 gram/cm. 2 or 1033*3 x 981 dynes, mn = 2-016 
 grams, v = 22,400 c.c., we have 
 
 , = JIE = /3 x .033-3 x 980-6 x 22,400 _ 
 y mn \ 2*016 
 
 cm. per sec. Therefore the molecule of hydrogen moves at o at 
 the enormous speed of 1-839 kilometres or rather more than i 
 mile per second. The magnitude of this velocity may seem 
 surprising, in view of the fact that the diffusion of gases, which 
 on the kinetic theory is conditioned by the speed of the mole- 
 cules, is comparatively slow, but it must be borne in mind that 
 this speed is only attained in the " free path," and owing to 
 collisions and consequent change of path, the actual progress of 
 a particle is very much less. The average speed of the mole- 
 cules of any other gas may be calculated by substitution in a 
 similar way. It is clear from the form of the equation (mnfv = d] 
 that at constant temperature, the rates are inversely proportional 
 to the square root of the densities of the different gases a result 
 which is connected with the laws of gaseous diffusion (Graham's 
 law) and of gaseous effusion. 
 
 Yan der Waals' Equation The kinetic theory of gases 
 not only admits of a simple deduction of the gas laws, but also 
 gives a plausible explanation of the deviation of gases from the 
 simple laws, discussed in the previous sections. In deducing 
 Boyle's law in the previous section two simplifying assumptions 
 have been made which are not strictly justifiable. In the first 
 place, it has been assumed that in its motion backwards and for- 
 
 1 For a full discussion of the kinetic theory of gases, more particularly 
 with reference to the different velocities of the individual particles, see 
 O. E. Meyer, Kinetic Theory of Gases (Longmarjs, 1899). 
 3 
 
34 OU1 LINES OF PHYSICAL CHEMISTRY 
 
 wards between the opposite walls of the cube, the molecule has the 
 whole length available for its motion, but a little consideration 
 shows that, if the size of the molecule is not negligible in com- 
 parison with this distance, the number of impacts on the walls, 
 and therefore the pressure of the gas, is greater than it would be 
 on the original assumption. To take an extreme case, let us 
 assume that the distance between the walls is 20 times that of 
 the diameter of the molecule, the space available for movement 
 is only 19 molecular diameters, and therefore the pressure will 
 be greater, in the ratio of 20 : 19, than if the size of the molecule 
 could be neglected. This error will clearly be negligible at low 
 pressures (when the total volume is large) but will become im- 
 portant when the gas is strongly compressed. If b is the 
 volume occupied by the molecules, it seems plausible to correct 
 the general gas equation, PV = RT, by substituting for V, the 
 total volume occupied by the gas, V - b, the " free " volume avail- 
 able for the movement of the molecules, so that the general gas 
 equation becomes P(V - b) =* RT. In reality, for reasons 
 which cannot be entered into here, b must be taken, not as 
 the volume filled by the material of the molecules, but as a 
 multiple of it, the actual magnitude of which is uncertain. 1 
 
 Another tacit assumption we have made in deducing Boyle's 
 law is that the gas particles exert no mutual attraction, an 
 assumption which is not justified when the gas is strongly com- 
 pressed. A little consideration shows that when increased 
 pressure is applied to a gas, the resulting volume will be less 
 than the calculated value owing to molecular attraction ; in 
 other words, the gas behaves as if the pressure applied is greater 
 than it actually is. It was shown by van der Waals that this 
 correction is inversely proportional to the square of the volume, 
 and as, from the above considerations, it affects the gas as an 
 increase of pressure, we must substitute for P, in the gas equation, 
 the expression P + a/V 2 where a is a constant. The corrected 
 gas equation then becomes 
 
 1 According to van der Waals b is 4 times, according to O. E. Meyer 
 4/X/2 times, the actual volume of the molecules.. 
 
GASES 
 
 an expression, first proposed by van der Waals (1879), which 
 not only represents with fair accuracy the behaviour of strongly 
 compressed gases, but throws considerable light on the con- 
 stitution of liquids. The detailed consideration of van der 
 Waals' equation is postponed to the next chapter (p. 53) to 
 which it more properly belongs ; at present it will be sufficient 
 to show that it affords a satisfactory qualitative explanation of 
 the deviations from the gas laws, discussed on page 28. 
 
 When V is very large, b will be negligible in comparison and 
 0/V 2 will be very small, so that van der Waals' equation then 
 approximates to the simple gas equation PV = RT. It may 
 therefore be expected that any influence which tends to increase 
 the volume, such as raising the temperature at constant pressure, 
 or diminishing the pressure at constant temperature, will affect 
 the gas in such a way that it follows the simple gas laws more 
 closely, and this expectation is quite borne out by the results of 
 experiment (p. 29). Further, we can employ the equation to 
 explain the fact already referred to, that PV for all gases except 
 hydrogen at first diminishes with increasing pressure, reaches a 
 minimum value, and beyond that point increases steadily as far 
 as the observations extend. From the form of van der Waals' 
 equation it is clear that the volume correction acts in the 
 opposite direction to that for the attraction, the value of PV 
 being increased by the former and diminished by the latter. 
 At low pressures the effect of the attraction preponderates, 
 whilst at high pressures the latter is negligible in comparison 
 with the volume correction. At some intermediate pressure 
 (about 50 metres for nitrogen at 30 ; see Fig. i) the two cor- 
 rections just balance, and in this short region the gas exactly 
 follows Boyle's law. 
 
 To explain the behaviour of hydrogen, it is assumed that the 
 attraction correction at ordinary temperatures is from the first 
 counterbalanced by the volume correction. 
 
36 OUTLINES OF PHYSICAL CHEMISTRY 
 
 AVOGADRO'S HYPOTHESIS AND THE MOLECULAR WEIGHT 
 OF GASES 
 
 General As we have already learnt, the application of 
 Avogadro's hypothesis permits of the determination of the 
 molecular weight of any substance which can be obtained in 
 the gaseous form; it is only necessary to find the weight in 
 grams of the substance in question which occupies in the 
 gaseous form a volume of 22-40 litres at o and 76 cm. pressure. 
 In determining molecular weights by this method, it is neither 
 necessary nor practicable to work at the temperature or under 
 the pressure referred to ; in practice the volume occupied by 
 a known weight of the gas or vapour under suitable and known 
 conditions of temperature and pressure is determined, then with 
 the help of the gas laws the weight in grams which will occupy 
 22-40 litres under standard conditions is determined. Suppose 
 it is found that g grams of a gas or vapour occupy v c.c. at 
 T Abs. and / millimetres pressure, what weight in grams will 
 occupy 22,400 c.c. at 273 Abs. and 760 mm. pressure? 
 
 It has already been shown (p. 26) that the expression pvjl 
 is proportional to the mass of the gas, independent of the 
 conditions under which it is measured. It follows that 
 
 pv ^, 760 x 22,400 
 ^ : T : -^r~ 
 
 where M is the molecular weight of the gas in grams. Hence 
 M ^x 760 x 22,400 x (273 + /C) ,, 
 
 273x^x2; 
 
 A shorter method is to use the alternative expression for the 
 gas equation l pv = n RT, where n represents the number of 
 gram-molecules of gas in the volume v. Since n =/M we have 
 
 . 
 
 pv 
 
 Density and Molecular Weight of Gases and Vapours 
 
 The determination of the volume occupied by a known 
 
 1 Care must be taken to express R in the proper units, corresponding 
 with those in which the pressure and volume are given. (Cf. p. in.) 
 
GASES 37 
 
 weight of a gas or vapour under definite conditions is equivalent 
 to determining its density, which is the mass per unit volume. 
 The actual determinations may be made in various ways. It 
 will be sufficient for our present purpose to describe Regnault's 
 method, which is particularly suitable for the permanent gases, 
 and the method first suggested by Victor Meyer, which is 
 now used almost exclusively for determining the density, and 
 therefore the molecular weight, of vapours. 
 
 (i) Regnaulfs Method Two glass bulbs of approximately 
 equal capacity and provided with well-ground stop-cocks are 
 used. One is exhausted as completely as possible by means 
 of a pump, weighed; filled at a known temperature and pressure 
 with the gas the density of which has to be determined, and 
 again weighed, the other bulb being used as counterpoise. The 
 volume of the bulb may be obtained by weighing it empty and 
 then filled with distilled water at a known temperature. One 
 of the advantages of using a second bulb is that any risk of 
 error arising from a change in the temperature, pressure or 
 humidity of the air in the balance-case during the weighing 
 is avoided. From the results, the density referred to hydrogen 
 as unity can readily be calculated, and hence the molecular weight, 
 which is double the density (p. 10). The molecular weight can 
 also be obtained by substituting in the formula given above. 
 This method, with slight modifications, has been employed in 
 recent years by Lord Rayleigh, Morley, Ramsay and others for 
 determining the densities of the permanent gases, and is cap- 
 able of giving results of the highest accuracy. 
 
 (2) Victor Meyer s Method This method differs from all the 
 others inasmuch as it is not the volume of the vapour itself 
 which is measured, but that of an equal volume of air which 
 has been displaced by the vapour. 
 
 The apparatus consists of a cylindrical vessel, A, of about 
 200 c.c. capacity, ending in a long neck provided with two 
 side tubes, as shown in Fig. 2. One of these side tubes, /, 
 from which the displaced air issues during an experiment, is 
 
OUTLINES OF PHYSICAL CHEMISTRY 
 
 bent in such a way that its free end can conveniently be brought 
 under the surface of water in a suitable vessel. Into the other 
 tube, / 2 , fits a rubber tube enclosing a glass rod which can be 
 moved outwards and inwards, and, at the commencement of the 
 experiment, serves to retain in place the small glass bulb shown 
 in the figure, containing a weighed quantity of the liquid, the 
 vapour density of which is to be determined. The top of the 
 main tube is closed by a cork which is kept in place throughout 
 an experiment, and a little asbestos or 
 mercury is placed in the bottom to guard 
 against fracture of the glass when the bulb 
 drops. The apparatus is heated at a con- 
 stant temperature throughout the greater 
 part of its length by means of the vapour 
 of a liquid boiling in the outer bulb-tube, 
 B ; the temperature should be at least 20 
 above the boiling point of the liquid to 
 be vaporized. 
 
 At the commencement of an experi- 
 ment, the bulb and rod are placed in 
 position and the cork inserted, the jacket- 
 ing liquid is then boiled till air ceases to 
 issue from the end of the tube, /, and 
 bubble through the water, showing that 
 the temperature inside the bulb, A, is 
 constant. A graduated measuring tube, 
 C, full of water, is then inverted over the end of the delivery 
 tube, and the small bulb allowed to drop by drawing back 
 the glass rod. When air ceases to issue from the end of the 
 delivery tube, the graduated tube is closed by the thumb, 
 removed to a deep vessel containing water, allowed to stand 
 till the temperature is constant, and the volume of air read off 
 when the water outside and inside are at the same level. 
 
 The temperature inside the tube, A, is the same before and 
 after the experiment, the only difference in the conditions is 
 
 B 
 FIG. 2. 
 
GASES 39 
 
 that a certain volume of air is displaced by an equal volume 
 of vapour. The observed volume of air is therefore that which 
 the vapour would occupy after reduction to the temperature 
 and pressure at which the air is measured (provided that the 
 vapour and air are equally affected by changes of temperature 
 and pressure, which is approximately the case under suitable 
 conditions). The temperature is that of the water, and the 
 pressure that of the atmosphere less the vapour pressure of 
 water at the temperature of observation. It is clear that it is 
 not necessary to know the temperature at which the substance 
 is vaporized, and this is one of the advantages of the method. 
 
 The mode of calculating molecular weights from the observed 
 data may be illustrated by the following example: 0-220 grams 
 of chloroform when vaporized displaced 45-0 c.c. of air, 
 measured at 20 and 755 mm. pressure. As the vapour 
 pressure of water at 20 is 17-4 mm., the actual pressure 
 exerted by the gas is 755 - i7'4 = 737'6 mm. Therefore, as 
 0-220 grams of vapour, at 20 and 737*6 mm. pressure, occupy 
 45 'o c.c., we have to find what weight in grams will occupy 
 22,400 c.c., at 273 abs. and 760 mm. pressure, and this will 
 be the required molecular weight. Substituting in the general 
 formula (p. 36) we have 
 
 o'22o x 760 x 22,400 x (273 + 20) 
 
 JV1 = =r I I 7. 
 
 273X737-6X45 
 
 This result is in fair agreement with the molecular weight 
 of chloroform (119-5) calculated from its formula. 
 
 The results obtained by this method are only accurate if the 
 vapour follows the gas laws as closely as air (since only under 
 these circumstances will the vapour replace an exactly equivalent 
 quantity of air), and as the vapour is often not very remote 
 from its temperature of condensation, this condition is not in 
 general fulfilled. In the great majority of cases, however, the 
 results are sufficiently near for the purpose, as the composition 
 of the substance can be determined with great accuracy by 
 chemical analysis, and the vapour density method is only used 
 
40 OUTLINES OF PHYSICAL CHEMISTRY 
 
 to distinguish between simply related numbers. In the example 
 given above, for instance, analysis shows that the molecular 
 weight of chloroform must be 119*5 or a simple multiple of 
 that number, and the density determination proves that the 
 former alternative is correct. 
 
 Results of Yapour Density Determinations. Ab- 
 normal Molecular Weights The most important result of 
 the numerous molecular weight determinations which have been 
 made by this method is that in general the values obtained are 
 in complete agreement with those based on chemical considera- 
 tions. There are certain exceptions to this rule, but for all 
 these plausible explanations have been suggested. In the case 
 of elements, the values found are simple multiples of the atomic 
 weights (very often twice the atomic weight), whilst in the case 
 of compounds they are simple multiples of the sum of the 
 atomic weights. 
 
 All the metals which have been obtained in gaseous form, 
 including mercury, zinc, cadmium, potassium, sodium, antimony 
 and bismuth, 1 are monatomic, as are the rare elements argon, 
 helium, krypton, etc., discovered in the atmosphere by Ramsay 
 and his co-workers. Many of the non-metals, such as oxygen, 
 nitrogen and chlorine, are diatomic under ordinary conditions. 
 Whilst arsenic and phosphorus are tetratomic, sulphur at low 
 temperatures gives results corresponding with the formula S 8 . 
 Several elements, such as carbon and silicon, and many metals 
 have not yet been obtained in the gaseous form. 
 
 The determination of molecular weights at high temperatures 
 has been greatly developed in recent years, more particularly 
 by Victor Meyer and his co-workers, and by Nernst. The air- 
 displacement method has proved most suitable for this purpose, 
 but the chief difficulty has been to obtain vessels which stand 
 high temperatures, and are not porous for the contained 
 vapours. Victor Meyer at first replaced glass by porcelain, 
 
 1 Compare H. von Wartenberg, Zeitsch. anorg. Chem., 1907, 56, 320; 
 Abstracts Chem. Society, 1908, ii., 86. 
 
GASES 41 
 
 and the latter by an alloy of platinum and iridium, and at the 
 time of his death was experimenting with vessels of magnesium 
 oxide. Satisfactory results were obtained up to 1700-1800. 
 
 One of the most striking results obtained by Victor Meyer is 
 that the molecular weight of iodine, which at 600 corresponds 
 with the formula I 2 , becomes smaller as the temperature is 
 further raised, until at 1500 it reaches half the initial value> 
 indicating that at the latter temperature it is completely split 
 up into iodine atoms. Bromine is also partially decomposed 
 at 1500, and chlorine commences to split up about the same 
 temperature. It had previously been shown by Deville and 
 Troost that the molecular weight of sulphur also diminishes 
 with increasing temperature, and above 800 gives results which 
 indicate that only diatomic molecules are present. 
 
 Nernst 1 has quite recently succeeded in extending this method 
 up to 2000 by using a vessel of iridium coated outside and 
 inside with a paste of magnesia and magnesium chloride, 
 and heated in an electric furnace. At this temperature, the 
 molecular weight of mercury is 201, indicating that the atoms of 
 this element have undergone no further simplification, whilst 
 sulphur, between 1800 and 2000, has a density of about 24, 
 indicating that the diatomic molecules are split up, to the ex- 
 tent of about 33 per cent., into single atoms. 
 
 Association and Dissociation in Gases We have al- 
 ready seen that such substances as sulphur and arsenic have 
 abnormally high molecular weights at low temperatures ; such 
 substances are said to be associated. This peculiarity is not 
 confined to elements, as the molecular weight of acetic acid, 
 which, as determined by chemical methods, is 60, exceeds 100 
 when determined by the vapour density method at comparatively 
 low temperatures. The conclusion that acetic acid in the form 
 of vapour at comparatively low temperatures consists largely of 
 double molecules (CH 3 COOH) 2 , is in satisfactory agreement 
 with other considerations. 
 
 1 Compare Wartenberg, loc. cit. 
 
42 OUTLINES OF PHYSICAL CHEMISTRY 
 
 An apparent deviation from Avogadro's hypothesis of a 
 different nature is met with, for example, in the case of gaseous 
 ammonium chloride. On chemical grounds, the molecular 
 formula, NH 4 C1, is given to this substance, corresponding with 
 a molecular weight of 53-5, whereas the observed value, obtained 
 from its vapour density, is only half as great. This behaviour 
 could be accounted for on the assumption that, at the tem- 
 perature of the experiment, the molecule is to a great extent 
 split up, or dissociated, into NH 3 and HC1 molecules, and 
 the experimental justification for this assumption has been 
 obtained by Pebal (1862), who effected a partial separation 
 of the decomposition products by taking advantage of their 
 different rates of diffusion. 
 
 It may be added that in the complete absence of moisture, 
 ammonium chloride can be vaporized without dissociation, and 
 then has the normal molecular weight deduced by means of 
 Avogadro's hypothesis. 1 
 
 Accurate Determination of Molecular and Atomic 
 Weights from Gas Densities We have seen that Avogadro's 
 hypothesis does not hold strictly for actual gases, and that the 
 reason for this is probably to be found in the mutual attractions 
 and finite volumes of the gas particles. We may, therefore, 
 assume that it would be strictly true for an ideal gas, and, on 
 the basis of van der Waals' equation, apply a correction to 
 actual gases to find their true molecular weights, that is, the 
 relative masses which would occupy equal volumes at great 
 rarefaction, when the gas laws would be strictly followed (p. 28). 
 The method followed is therefore to determine the volume of 
 a definite mass of a gas under two or more pressures (the 
 compressibility of the gas), and find by extrapolation the relative 
 densities of different gases as the pressure approaches zero. 
 This method has been used more particularly by Daniel 
 Berthelot and by Lor d Rayleigh. From the results, the follow- 
 
 *H. Brereton Baker, Trans. Chem. Society, 1894, 65, 6n ; 1898,73, 
 422. Compare Johnston, Zeitsch. physikal Chem., 1908, 61, 457. 
 
GASES 43 
 
 ing molecular weights (vapour density x 2) were calculated by 
 Berthelot (oxygen = 32 being taken as the standard) : 
 
 H 2 N 2 CO O 2 CO 2 N 2 O HC1 
 
 2*0145 28*013 28*007 32*000 44*000 44*000 36-486 
 
 From these observations, the following atomic weights have 
 been obtained, the values derived by chemical methods being 
 placed below for comparison : 
 
 O H C N Cl 
 
 Gas density 16*000 1-0075 12-000 14-005 35*479 
 Chemical 16*000 1*008 12-00 14*01 35*45 
 
 The agreement, except in the case of chlorine, is excellent. 
 As a matter of fact, the chemical value for chlorine is probably 
 too low ; the recent investigations of Richards and Wells l appear 
 to show that the true value is 35*473, almost identical with that 
 obtained by the density method. 
 
 The above striking results lend strong support to the assump- 
 tion that in the limit Avogadro's hypothesis is strictly valid for 
 all gases. 
 
 SPECIFIC HEAT OF GASES 
 
 General The specific heat of any substance may be defined 
 as the ratio of the amounts of heat required to raise i gram of 
 the substance in question and i gram of water through a given 
 range of temperature. The amount of heat required to raise i 
 gram of water i in temperature is termed a calorie, and hence 
 the specific heat may also be defined as the quantity of heat in 
 calories required to raise i gram of the substance i in tem- 
 perature. This statement has only a definite meaning, however, 
 when the conditions under which the heating is carried out 
 are stated, and this is particularly true of gases. If a gas is 
 suddenly compressed it becomes warmer, although no heat has 
 been supplied to it, and, conversely, if a gas is allowed to expand 
 against pressure it becomes cooled, although no heat has been 
 abstracted from it. According to the above definition, 
 i y. A mer. Chem. Soc., 1905, 27, 459. 
 
44 OUTLINES OF PHYSICAL CHEMISTRY 
 
 amount of heat supplied 
 
 Specific heat = -. *-*- 
 
 nse in temperature 
 
 so that if a gas is warmed by compression its specific heat is zero. 
 Moreover, if, while a gas is expanding against pressure, sufficient 
 heat is supplied to keep its temperature constant, the heat 
 supplied has a certain finite value whilst the change of tempera- 
 ture is zero, so that the specific heat, according to definition, is 
 infinite. It is clear that the specific heat of a gas may have any 
 value whatever, unless the conditions under which it is measured 
 are stated. 
 
 Specific Heat at Constant Pressure, G p , and Constant 
 Volume, C, There are two important cases in which the term 
 " specific heat of a gas " is clearly defined : (a) the specific heat 
 at constant volume, C,, (b) the specific heat at constant pressure, 
 C p . In the former case, the volume is kept constant whilst 
 the gas is being heated, and no external work is done. In the 
 latter case, the volume is allowed to increase whilst heat is being 
 supplied, work is therefore done against the pressure of the 
 atmosphere, which tends to cool the gas. Sufficient heat must 
 therefore be supplied not only to raise the temperature, but to 
 make up for the cooling due to the external work performed. 
 It is clear that the specific heat at constant pressure is greater 
 than that at constant volume, and the difference is the heat equi- 
 valent of the amount of work done against the external pressure. 
 
 The difference between the two specific heats may readily be 
 obtained in thermal units by using the general gas equation. 
 For this purpose, it is convenient to deal with a mol of a gas. 
 It has already been shown (p. 27) that when a gas expands at 
 constant pressure, the work done is measured by the product of 
 the pressure and the change of volume. If at first the absolute 
 temperature is T lf we have the equation PV l = RTj where V 1 is 
 the molecular volume. If the temperature is raised to T 2 , and 
 the new molecular volume is V 2 , the work done during the 
 expansion is 
 
 P(V 2 -V 1 ) = R(T 2 -T 1 ). 
 
GASES 
 
 45 
 
 In the present case, T 2 - Tj is i, therefore P(V 2 - V t ) = R. 
 Further, the difference in the molecular heats of a gas at 
 constant pressure and constant volume is the external work 
 done when a mol of gas is raised i in temperature, and there- 
 fore M (C P C v ) (where M is the molecular weight of the gas) 
 is also = P (V 2 - Vj). Hence M (C p - C,) - R. In thermal 
 units, R is approximately 2 calories, so that the difference of 
 the specific heats of a mol of any gas in other words, the 
 difference of the molecular heats of any gas at constant pressure 
 and at constant volume is 2 calories. 
 
 The specific heat of a gas at constant pressure can readily be 
 determined by passing a known quantity of it, heated to a 
 definite temperature, through a metallic worm in a calorimeter, 
 at such a rate that there is a constant difference of temperature 
 between the entering and issuing gas. It is more difficult to 
 determine directly the specific heat at constant volume, and this 
 has only been accomplished satisfactorily in comparatively recent 
 times. 1 
 
 The molecular heats MC, and MC P (molecular weight x 
 specific heat) of a few of the commoner gases are given in the 
 accompanying table, the values of C being obtained from those 
 of C p by subtracting 2 calories : 
 
 
 Specific 
 
 
 
 
 Gas. 
 
 Heat, 
 
 MCj,, 
 
 MC tf 
 
 CJC V 
 
 
 C P 
 
 
 
 
 Argon . 
 
 _ 
 
 4-98 
 
 2-98 
 
 1-66 
 
 Helium 
 
 
 
 
 . 
 
 66 
 
 Mercury 
 
 
 
 
 
 2-965 
 
 66 
 
 Hydrogen 
 
 3-409 
 
 6-880 
 
 4-880 
 
 412 
 
 Oxygen 
 
 0-2175 
 
 6-960 
 
 4-960 
 
 40 
 
 Hydrogen chloride 
 Chlorine 
 
 0-1876 
 0-1241 
 
 6-84 
 8-820 
 
 4-84 
 6-820 
 
 409 
 29 
 
 Nitrous oxide 
 
 0-2262 
 
 9'99 
 
 7-99 
 
 247 
 
 Ether . 
 
 0-4797 
 
 35-5I 
 
 33'5i 
 
 060 
 
 1 Joly, Proc. Roy. Soc., 1889 47, 218. Compare Preston, Theory of 
 Heat, p. 239. 
 
46 OUTLINES OF PHYSICAL CHEMISTRY 
 
 For diatomic molecules, the average value of the molecular 
 heat at constant volume is about 4-8 calories in the neighbour- 
 hood of 100; but chlorine and bromine are exceptions. For 
 triatomic molecules the average value of MC t is 6-5 cal. and 
 the value increases with the complexity of the compound, as is 
 illustrated in the table. 
 
 Specific Heat of Gases and the Kinetic Theory- 
 Much light is thrown on the question of the specific heat of 
 gases by the kinetic theory. According to this theory the 
 energy-content of a gas is made up of three parts: (i) the 
 energy of rectilineal (progressive motion of the molecules), the 
 so-called kinetic energy (p. 32) ; (2) the energy of intramolecular 
 motion ; (3) the potential energy due to the mutual action of 
 the atoms ; 1 and when heat is supplied to a gas at constant 
 volume all three factors of the energy may be affected. For 
 monatomic gases, however, such as mercury vapour, the factors 
 (2) and (3) are presumably absent, and the heat supplied must 
 simply be employed in increasing the kinetic energy of the 
 molecules. We have already learnt (p. 32) that the kinetic energy 
 of i mol of any gas = f PV = 3T if expressed in thermal units. 
 When a gas is raised at constant volume from the absolute 
 temperature Tj to T 2 we have for the kinetic energies at the 
 two temperatures the equations f PjV = 3^ and f P 2 V = 3T 2 , 
 where Pj and P 2 are the pressures at T 1 and T 2 respectively. 
 Subtracting the first equation from the second, we obtain 
 f (Pj-PjJV = 3(T 2 -T!) and for a rise of temperature of i 
 f (P 2 - Pj)V = 3 (calories). Therefore the molecular kinetic 
 energy of a monatomic gas is increased by 3 calories for a rise of 
 i in temperature, or, in other words, the molecular heat MC, 
 of a monatomic gas at constant volume is 3 calories. As the 
 specific heat at constant pressure is 3 + 2 = 5 calories, the ratio, 
 MC P /MC V , for a monatomic gas must be r66, if the assumptions 
 we have made on the basis of the kinetic theory are justified. 
 
 As has already been mentioned, C e is somewhat difficult to 
 
 1 Boltzmann, he. cit., ?, 54.. 
 
GASES 47 
 
 determine directly, and to test the above deduction from the 
 kinetic theory it is simpler to determine the ratio C P /C, in- 
 directly, which can be done in various ways, for example, by 
 measuring the velocity of sound in a gas. Kundt and Warburg 
 (1876) therefore determined the velocity of sound in mercury 
 vapour and obtained for the above ratio the value r66, in exact 
 accord with the theoretical value, undoubtedly one of the most 
 striking triumphs of the kinetic theory. 
 
 Conversely, a gas for which the ratio C P /C, is r66 must be 
 monatomic, and by this method Ramsay showed that the rare 
 gases argon and helium are monatomic. 
 
 For gases containing two or more atoms in the molecule, the 
 heat supplied is employed not only in accelerating the rectilinear 
 motion of the particles, but also in performing internal work in 
 the molecule. As the former effect alone requires 3 calories, 
 the total molecular heat of a polyatomic gas will be 3 + 
 calories, where a is a positive quantity, constant for any one gas. 
 The value of MCp will be 5 + a calories, and the ratio of the 
 specific heats will be 
 
 MC P 5 + a 
 
 less than 1-67 but greater than i. 
 
 A comparison of the numbers given in the table shows that 
 this deduction is in complete accord with the experimental 
 facts. It may be expected that the more complex the molecule 
 the greater will be the amount of heat expended in performing 
 internal work and therefore the greater will be the specific heat. 
 In accordance with this, MC, for i mol of ether vapour is 33*5 
 calories, and for turpentine (C 10 H 16 ) 66'8 calories. 
 
 The specific heat of monatomic gases is independent of 
 temperature, that of polyatomic gases usually increases slowly 
 with temperature. 
 
 Experimental Illustrations Experiments with gases are 
 usually somewhat difficult to perform, and require special ap- 
 paratus. Experimental illustrations of the simple gas laws are 
 
48 OUTLINES OF PHYSICAL CHEMISTRY 
 
 described in all text-books on physics, and need not be con- 
 sidered here. 
 
 The determination of the density l and hence the molecular 
 weight of such a gas as carbon dioxide by Regnault's method 
 may be performed as follows : One of the bulbs is first exhausted 
 as completely as possible by means of a pump, the stop-cock 
 closed, and the bulb weighed. It is then filled with water by 
 opening the stop-cock while the end of the tube dips under the 
 surface of water and again weighed. The volume of the bulb 
 is obtained by dividing the weight of the water by its density 
 at the temperature of the experiment. The water is removed 
 from the bulb by means of a filter-pump, the interior of the bulb 
 is dried (by washing out with alcohol and ether and warming), 
 pkced nearly to the stop-cock in a bath at constant temperature, 
 the end is then connected to a T piece bv means of rubber 
 tubing ; one of the free ends of the T piece is connected, through 
 a stop-cock or rubber tube and clip, to a pump, the other, also 
 through a stop-cock or rubber tube and clip, to an apparatus 
 generating carbon dioxide. The bulb is evacuated by means 
 of the pump, the stop-cock connecting it with the latter is then 
 closed, that connecting it with the carbon dioxide apparatus 
 opened, the bulb filled with carbon dioxide, disconnected and 
 weighed. As the apparatus fills it with carbon dioxide at rather 
 more than atmospheric pressure, the stop-cock is opened for a 
 moment to adjust it to atmospheric pressure before weighing. 
 The weight of a known volume of the gas at known temperature 
 and pressure having thus been determined, its density and 
 molecular weight can readily be calculated. 
 
 The determination of vapour densities by Victor Meyer's 
 method is fully described on page 38, and may readily be 
 performed by the student with ether or chloroform, steam being 
 used as jacketing vapour. 
 
 i For full details as to the manipulation of gases, consult Travers 1 
 Experimental Study of Gases (Macmillan, 1901). 
 
CHAPTER III 
 LIQUIDS 
 
 General Liquids, like gases, have no definite form, but, 
 unlike the latter, they have a definite volume, which is only 
 altered to a comparatively small extent by changes of tempera- 
 ture and pressure. 
 
 In contrast to the simple gas laws, the formulae connecting 
 temperature, pressure and volume of liquids are very compli- 
 cated and empirical in character, and depend also on the nature 
 of the liquid. This is, of course, connected with the fact that 
 liquids represent a much more condensed form of matter than 
 gases. i c.c. of liquid water at 100, when converted into 
 vapour at the same temperature, occupies a volume of over 
 1600 c.c. It seems plausible to suggest that the main reason 
 why the formulae representing the behaviour of liquids are so 
 much more complicated than the gas laws is that the mutual 
 attraction of the particles, which is almost negligible in the case 
 of gases (p. 34), is of predominant importance for liquids. 
 
 As is well known, gases can be liquefied by increasing the 
 pressure and lowering the temperature ; and, conversely, by 
 raising the temperature and diminishing the pressure a liquid 
 can be changed to a gas. It is shown in the next section that 
 there is no difference in kind, but only a difference in degree, 
 between liquids and gases. 
 
 Transition from Gaseous to Liquid State. Critioal 
 Phenomena If gaseous carbon dioxide, below 31, is con- 
 fined in a tube and the pressure on it gradually increased, a 
 point will be reached at which liquid makes its appearance in 
 4 40 
 
OUTLINES OF PHYSICAL CHEMISTRY 
 
 the tube, and the whole of the gas can be liquefied without 
 appreciable increase of pressure. If, however, carbon dioxide 
 above 31 is continuously compressed, no separation into two 
 layers (liquid and gas) occurs, no matter how high the pressure 
 applied. Similarly, if carbon dioxide is contained in a sealed 
 tube, under such conditions that both liquid and gas are 
 present, and the temperature is gradually raised, it will be 
 noticed that when the temperature reaches 31 the boundary 
 between liquid and vapour disappears, and the contents of the 
 tube become homogeneous. Other liquids show 
 the same remarkable phenomena, but at tem- 
 peratures which are characteristic for each sub- 
 stance. This temperature is known as the critical 
 temperature; above its critical temperature no 
 pressure, however great, will serve to liquefy a 
 gas, below its critical temperature any gas can be 
 liquefied by pressure. That pressure which is 
 just sufficient to liquefy a gas at the critical tem- 
 perature is termed the critical pressure, and the 
 specific volume under these conditions is called 
 the critical volume. 
 
 The critical phenomena may be observed, and 
 rough measurements of the constants obtained, 
 with an apparatus (Fig. 3) used by Cagniard de 
 la Tour, who discovered these phenomena in 
 1822. The upper part of the branch A contains 
 a suitable volume of the liquid to be examined, 
 the branch B, the upper pait of which is gradu- 
 ated, contains a little air to act as a manometer, 
 the remainder of the apparatus (the shaded part 
 in the figure) is filled with mercury. The tube 
 at first contains both vapour and liquid, but on gradually raising 
 the temperature, a point is ultimately reached at which the 
 boundary between liquid and vapour becomes faint, and finally 
 disappears ; the tube is momentarily filled with peculiar flicker- 
 
 FIG. 3. 
 
LIQUIDS 
 
 ing striae, and then the contents become quite homogeneous. 
 On allowing to cool, a mist suddenly appears in the tube at a 
 certain temperature, and separation into liquid and vapour 
 again occurs. The temperature at which the boundary dis- 
 appears on heating or reappears on cooling approximates to 
 the critical temperature, and the critical pressure can be cal- 
 culated from the volume of air in B. Accurate measurements 
 of the constants may be made by methods described by Young l 
 and others. As the temperature rises, the density of the liquid 
 in the sealed tube naturally decreases, whilst that of the vapour 
 increases, and it has been shown that at the critical temperature 
 the densities of liquid and vapour are equal. The critical 
 temperatures and pressures of a few substances are given in 
 the accompanying table : 
 
 
 Critical 
 Temperature, C. 
 
 Critical Pressure 
 (Atmospheres). 
 
 Helium 
 
 - 267-268 
 
 2-3 
 
 Hydrogen 
 Nitrogen .* 
 
 -2 3 8 
 -149 
 
 *5 
 27 
 
 Oxygen . 
 
 - 119 
 
 58 
 
 Carbon dioxide 
 
 3i 
 
 72 
 
 Ethyl ether 
 
 195 
 
 35 
 
 Ethyl alcohol . 
 
 243 
 
 63 
 
 Behaviour of Gases on Compression We have already 
 learnt that if gaseous carbon dioxide is compressed at a tem- 
 perature below 31 it can be liquefied, but if the compression 
 is carried out above 31 no separation into two layers occurs. 
 These relations are best shown diagrammatically, as in Fig. 4, in 
 which the ordinates represent the pressures and the abscissae 
 the corresponding volumes at constant temperature. If the 
 gas in question obeys Boyle's law, the curves obtained by 
 plotting the pressures against the corresponding volumes at 
 constant temperature (the so-called isothermals) are hyperbolas, 
 corresponding with the equation/^ constant, and this condi- 
 tion is approximately fulfilled by air, as shown in the upper right- 
 
 1 Phil. Mag., 1892, [v.], 33* J 53. 
 
52 OUTLINES OF PHYSICAL CHEMISTRY 
 
 hand corner of the diagram. An examination of the isothermals 
 for carbon dioxide shows that the same is nearly true of this 
 gas at 48-1, but at 35-5, and still more at 32-5, the isothermals 
 deviate from those of an ideal gas. At the latter temperature 
 it is very interesting to observe that the compressibility at 75 
 Y 
 
 t 
 
 Air. 
 
 Carbon 
 
 FIG. 4. 
 
 atmospheres is very great for a short part of the curve, and 
 beyond that point extremely small ; in the latter respect the 
 highly compressed gas resembles a liquid. At the critical 
 point, 31 -i, the curve is for a short distance practically hori- 
 zontal, thus representing a great decrease of volume for a small 
 change of pressure in other words, a high compressibility. 
 Finally, at 21*1 and 13*1, separation of liquid takes place, the 
 
LIQUIDS 53 
 
 curves run horizontal whilst the gas is changing to liquid at 
 constant pressure, and then the curves run almost vertical, 
 indicating a small decrease of volume with increase of pres- 
 sure (f.e. t a small compressibility), characteristic of liquids. It 
 is evident from the foregoing that at any point within the 
 dotted line ABC both vapour and liquid are present ; at any 
 point outside only one form of matter, either vapour or liquid. 
 
 The above considerations serve to show that there is no 
 fundamental distinction between gases and liquids: a highly- 
 compressed gas above its critical point cannot be definitely 
 classified either as liquid or gas. It is evident from the figure 
 that as regards compressibility, highly compressed carbon dioxide 
 behaves more like a liquid than a gas. // is, in fact, possible 
 to pass from the typically liquid to the gaseous form, and vice 
 versa, without a separation into two layers. Thus liquid carbon 
 dioxide below 31, under the conditions represented by the 
 point x l in Fig. 5, may be compressed above its critical pressure 
 along x- l y l and then warmed above its critical temperature 
 whilst the pressure is kept constant at y^ During this process 
 no separation into two layers will be noticed, and by now re- 
 ducing the pressure the fluid can be obtained in as dilute a 
 form as desired (what is ordinarily termed a gas), say the con- 
 dition represented by x, whilst remaining quite homogeneous. 
 Similarly a gas can be completely converted to a liquid without 
 discontinuity, along xyy^x Y 
 
 Application of van der Waals' Equation to Critical 
 PhenomenaWe have already learnt that no actual gas 
 follows the gas laws quite strictly, and, further, that the 
 behaviour of actual gases can be represented with fair ac- 
 curacy, even up to high pressures, by van der Waals' equation, 
 
 + ^ J (V - b) = RT. If this equation is arranged in 
 descending powers of V it becomes 
 
54 
 
 OUTLINES OF PHYSICAL CHEMISTRY 
 
 This is a cubic equation, V being treated as the variable 
 and P and T, as well as a, b, and R, as constants. Ac- 
 cording to the conditions the equation has either three 
 real roots or one real and two imaginary roots. Otherwise 
 expressed, the magnitudes of a and b may be such that at 
 one temperature and pressure, the volume V has three real 
 values, whilst at another temperature and pressure it may 
 
 have only one real 
 value. We will now 
 compare these theo- 
 retical deductions with 
 the actual case of car- 
 bon dioxide. It is clear 
 from Fig. 4 that at 
 13*1 and 48 atmo- 
 spheres, carbon dioxide 
 has two volumes, as a 
 gas (represented by the 
 point D) and as a liquid 
 (the point E), but the 
 third volume demanded 
 by the equation is not 
 shown. At 48- 1, on 
 the other hand, there is 
 only one volume for 
 each pressure, corres- 
 ~~~ ponding with one real 
 
 Volume > 
 
 root of the equation 
 under these conditions. 
 
 FIG. 5. 
 
 Some light is thrown on the question of the missing third 
 real volume when the isothermal curves for carbon dioxide are 
 plotted by substituting the values of a and b> found experi- 
 mentally (p. 34) in equation (i). The wavy curve ABCDE 
 shown in Fig. 5 was obtained in this way ; on comparison with 
 the experimental curve for carbon dioxide (Fig. 4), it will be 
 
LIQUIDS 55 
 
 seen that whereas the points D and E in the latter isothermal 
 for 13 are joined by a straight line DE, in the former the 
 corresponding points E and A are joined by the dotted line 
 ABCDE, which represents a change from the gaseous to the liquid 
 form without discontinuity. The point C, at which the line of 
 constant pressure cuts the isothermal, represents the third of 
 the volumes required by the above cubic equation, but it 
 probably cannot be realised in practice, as the part DCB of 
 the curve on which it occurs represents decrease of volume 
 with diminishing pressure, quite contrary to our usual experi- 
 ence. On the other hand, the sections AB and ED have 
 a real meaning. When a vapour is compressed till saturated, 
 it does not necessarily liquefy; in the complete absence of 
 liquid it may be compressed considerably beyond the point 
 at which liquefaction occurs in the presence of traces of liquid; 
 in other words, a part of the curve ED may be experimen- 
 tally realised. Similarly, water may be heated in a carefully 
 cleaned vessel several degrees above its boiling point, that is, 
 it does not necessarily pass into vapour when the superincum- 
 bent pressure is less than its vapour pressure, and a part of 
 the curve AB may thus be experimentally realised. Similar 
 phenomena will be met with later ; it often happens that when 
 a system is under such conditions that the separation of another 
 phase (form of matter) is possible, the change does not occur 
 in the absence of the new phase. 
 
 Van der Waals' equation can also be employed to obtain 
 important relations between the critical constants and the other 
 characteristic constants representing the behaviour of gases. It 
 has already been pointed out that the densities and consequently 
 the volumes of liquid and gas become equal at the critical 
 temperature, and as this must also be true for the intermediate 
 third volume, it follows that the three roots of the equation 
 , T , /, RT\ _ TO a^ r ab . ^ 
 
 v \ + T) v + P v " T = (l) 
 
 become equal under these conditions. If, in this general 
 
56 OUTLINES OF PHYSICAL CHEMISTRY 
 
 equation, we call the three roots V lf V 2 and V 3 , then the equation 
 ( v - v i)( v ~ V 2)( V - V 3 ) = must hold > which, when the roots 
 are equal, becomes 
 
 (V - V*) 3 - V - 3 V*V 2 + 3 V* 2 V - V* 3 = o (2), 
 
 where V^ is the critical volume. Equating the coefficients of 
 the identical equations (i) and (2), we have 
 
 sV* (i) ; - 3 vi (ii) ; g - v (iii). 
 
 From the last three equations, the values of the critical con- 
 stants can readily be obtained in terms of R, a and b. We 
 have 
 
 Critical volume V* = 3^ (from ii and iii), 
 
 Critical pressure P* = -=* (from ii), 
 Critical temperature T^ = (from i). 
 
 We thus reach the interesting result that the critical constants 
 may be calculated from the deviations from the gas laws, when 
 the latter are expressed in terms of the constants a and b of 
 Van der Waals' equation. As an illustration of the satisfactory 
 agreement between the observed and calculated values, we will 
 take the data for ethylene, for which a = 0*00786, b = 0*0024, 
 R = 0-0037. 
 Vj, = 0*0072 (observed value 0*006), 
 
 P t = -=r = ^o't; (observed value si atmospheres) 
 
 27 x (0*0024)2 
 
 8 x 0*00786 , . 
 
 T fc = * = 262 Abs. (observed value 282 ). 
 
 27 x 0*0024 x 0*0037 
 
 Law of Corresponding States Van der Waals has further 
 pointed out that if the pressure, volume and temperature of a 
 substance are expressed as multiples of the critical values, that 
 is, if we put P = aP*, V = /3V A , T = yT*, and then substitute 
 in the equation (P + a/V 2 )(V - b) = RT, P, V* and T* being 
 
LIQUIDS 57 
 
 replaced by their values in terms of a, b and R, the equation 
 simplifies to 
 
 This equation does not contain the constants characteristic 
 of any particular substance, and ought therefore to hold for 
 all substances in the gaseous and liquid state. Experiment 
 shows, however, that it is only to be regarded as a first ap- 
 proximation, the deviation in many cases being much greater 
 than the experimental error. 
 
 For our present purpose, these considerations are chiefly of 
 importance as affording information regarding the proper con- 
 ditions for comparison of the physical properties of liquids. If 
 we wish, for example, to compare the molecular, volumes of 
 ether and alcohol, it would probably not be satisfactory to 
 compare them at the ordinary temperature of a room, as this 
 would be near the boiling-point of ether, 35, but much below 
 that of alcohol, 78. According to van der Waals, the proper 
 temperatures for comparison, the so-called " corresponding tem- 
 peratures," are those which are equal fractions of the respective 
 critical temperatures. Thus, if we choose 20, or 293 Abs., 
 as the temperature of experiment for ether, the critical tem- 
 perature of which is 195, or 468 Abs., the proper temperature, 
 /, for comparison with alcohol (critical temperature, 243 C.) 
 
 will be given by 27320 = 273 / , whence /= 5 i. The 
 
 same considerations apply to the pressures. 
 
 The theoretical basis for this method of comparison is that, 
 as mentioned above, the choosing of pressures, volumes or tem- 
 peratures which for different substances bear the same proportion 
 to their respective critical constants leads, when substituted in 
 van der Waals' equation, to an equation which is the same for 
 all substances, and the practical justification for choosing these 
 as corresponding conditions is that more regularities are actually 
 
58 OUTLINES OF PHYSICAL CHEMISTRY 
 
 observed by this method than when "the comparison is made 
 under other circumstances. 
 
 Liquefaction of Gases As already indicated, all gases 
 can be liquefied by cooling them below their respective critical 
 temperatures and applying pressure. The methods employed 
 for this purpose by Cailletet, Pictet, Wroblewski and others 
 are fully described in text-books of physics. In recent years 
 the older methods have been almost completely displaced, in 
 the case of the less condensible gases, such as air and hydro- 
 gen, by a method introduced almost simultaneously by Linde 
 and by Hampson. The principle of the method is that when a 
 gas is allowed to pass from a high to a low pressure through a 
 porous plug without performing external work it becomes cooled 
 (Joule-Thomson effect). The cooling effect is due to the 
 performance of internal work in overcoming the mutual attrac- 
 tion of the particles, and is therefore only observed for " im- 
 perfect " gases (p. 34). The effect is the greater the lower 
 the temperature at which the expansion takes place, and the 
 greater the difference of pressure on the two sides of the plug. 
 The cooling effects thus obtained are summed up in a very 
 ingenious way by the principle of " contrary currents," the same 
 quantity of gas being made to circulate through the apparatus 
 several times, and after passing through the plug being caused 
 to flow over and cool the tube through which a further quantity 
 of gas is passing on its way to the plug (or small orifice). 
 
 The apparatus employed is represented diagrammatically in 
 Fig. 6. By means of the pump A, the gas is compressed in 
 B to (say) 100 atmospheres, the heat given out in this process 
 being absorbed by surrounding B with a vessel through which 
 a continuous current of cold water is passed. The cooled, com- 
 pressed gas then passes down the central tube G, towards the 
 plug E, being further cooled on the way by the gas passing up 
 the wide tube D, which has just expanded through the plug. 
 After passing through E and thus falling to its original pressure, 
 the gas passes upwards over the central tube G and again 
 
LIQUIDS % 
 
 59 
 
 reaches A by the tube C and the left-hand valve at the bottom 
 of A. The direction of the circulating stream of gas is indicated 
 by the arrows. In course of time, the temperature becomes so 
 low that part of the gas is liquefied 
 and collects in the vessel F. 
 More air is drawn into the appar- 
 atus as required, and the process 
 is continuous. 
 
 By means of an apparatus con- 
 structed on this principle Dewar, 
 and, somewhat later, Travers, 
 succeeded in obtaining liquid 
 hydrogen in quantity. All known 
 gases have now been liquefied. 
 The liquefaction of helium was 
 effected quite recently by Kam- 
 merlingh Onnes. 
 
 RELATION BETWEEN PHYSI- 
 CAL PROPERTIES AND 
 CHEMICAL CONSTITUTION 
 
 General The foregoing para- 
 graphs of this chapter represent 
 an introduction to the relation- 
 ship between the physical pro- 
 perties of liquids and their chemi- 
 cal composition, inasmuch as in- 
 formation has been gained as 
 to the conditions under which 
 measurements should be made 
 with different liquids in order to 
 obtain comparable results (theory 
 
 Lit 
 
 FlG - 
 
 of corresponding states). Although in this chapter we are 
 mainly concerned with the physical properties of pure liquids, 
 it is convenient to include also some observations with solu- 
 
6o OUTLINES OF PHYSICAL CHEMISTRY 
 
 tions. We will deal shortly with the following physical pro- 
 perties: (i) Atomic and molecular volumes; (2) refractivity ; 
 (3) rotation of plane of polarization of light ; (4) absorption 
 of light; (5) viscosity. 
 
 Atomic and Molecular Volumes In the case of gases, 
 we have seen that simple relations are obtained when the 
 volumes occupied by different substances in the ratio of their 
 molecular weights are compared ; at the same temperature and 
 pressure, the volumes are equal. The justification for taking 
 the molecular weights (in grams, for instance) as comparable 
 quantities is that, according to the molecular theory, equal 
 numbers of molecules of different substances are thus compared. 
 Similarly, in dealing with liquids, it is usual to determine the 
 molecular volume of the liquid, i.e., the volume occupied by the 
 molecular weight of the liquid in grams, which is, of course, ob- 
 tained by dividing the molecular weight in grams by the density 
 of the liquid at the temperature of experiment. As the specific 
 volume, v, of a liquid is inversely as the density, the molecular 
 volume may also be defined as molecular weight in grams x 
 sp. volume. Similarly, the atomic volume = (atomic weight in 
 grams) 4- density, or, (atomic weight in grams) x sp. volume. 
 
 Kopp was the first chemist to carry out an extended series of 
 observations on this subject, and he found that the most regular 
 results were obtained when molecular volumes were determined, 
 not at the same temperature, but at the boiling-points of the 
 respective liquids under atmospheric pressure. It is interesting 
 to observe that this purely empirical method of procedure was 
 found much later to be theoretically justifiable, as the boiling- 
 points of most liquids are approximately two-thirds of their 
 respective critical temperatures (both measured on the absolute 
 scale). The boiling-points are therefore corresponding tempera- 
 tures, (p. 57). 
 
 Kopp found that as a first approximation the molecular 
 volume could be regarded as the sum of numbers representing 
 
LIQUIDS 6 1 
 
 the volumes of the component atoms. The atomic volumes of 
 the commoner elements occurring in organic compounds are as 
 follows : 
 
 C H Cl Br I S 0(0 -H) 0(0 = ) 
 
 ii 5*5 22'8 27*8 37*5 22*6 7*8 12*2 
 
 In some cases the atomic volume depends on the way in which 
 the element is bound, thus oxygen joined to hydrogen (hydroxyl 
 oxygen) has the atomic volume 7*8, whilst for oxygen doubly 
 linked to carbon (carbonyl oxygen) the volume is 12*2. As an 
 illustration the calculated and observed volumes of acetic acid 
 may be compared as follows : 
 
 2C = 22 
 
 4 H - 22 
 
 (Carbonyl) O = 12-2 
 (Hydroxyl) Q = 7-8 
 
 64*0 
 
 As the molecular weight of acetic acid is 60 and its density at 
 the boiling-point is 0-942 the observed molecular volume M/d 
 
 = 637. 
 
 It should be mentioned that the atomic volumes given 
 above are not obtained directly, but by comparison of chemical 
 compounds with definite differences of composition (e.g., the 
 difference in the molecular volumes of the compounds C 4 H 10 
 and C 4 H 8 gives the volume of two atoms of hydrogen), and are 
 therefore not necessarily the same as those for the free elements. 
 In some cases, however, the two values coincide, thus the atomic 
 volumes of the free halogens, chlorine and bromine, at their boil- 
 ing-points are 23*5 and 27' i respectively, whilst their values in 
 combination are 22-8 and 27*8, so that the halogens have ap- 
 proximately the same volume in the free and combined condi- 
 tion. The same is approximately true for certain other elements 
 for which comparison is possible. 
 
 The extended investigations of Thorpe, Lossen and Schiff 
 
62 OUTLINES OF PHYSICAL CHEMISTRY 
 
 afford a general -confirmation of Kopp's conclusions; the cal- 
 culated and observed values generally agree within about 4 per 
 cent. 
 
 Although, strictly speaking, the consideration of the mole- 
 cular volume of a substance in solution does not belong to this 
 section, it is convenient to refer to it here. The molecular solu- 
 tion volume ', M#, of a substance may readily be calculated from the 
 
 formula Mv = j =-, where M is the molecular weight 
 
 a a 
 
 of the solute in grams, n the weight of the solvent containing M 
 grams of solute, d the density of the solution, and d' that of the 
 solvent. This formula is derived on the assumption, which is 
 certainly not justifiable, that the density of the solvent itself is 
 not affected by dissolving a substance in it, and therefore Mz/is 
 only the " apparent " solution volume. The molecular volume 
 is sometimes nearly the same in the free state and in solution 1 
 (e.g.) bromine in carbon tetrachloride), but is often much less 
 (e.g., many salts in water). 
 
 Additive, Constitutive and Colligative Properties 
 The molecular volume is a good example of what is termed 
 an additive property, since it can be represented as the sum of 
 volumes pertaining to the component atoms. It is not, how- 
 ever, strictly additive, since it is influenced somewhat by the 
 arrangement of the atoms in the molecule, and therefore the 
 atoms to some extent influence each other. Properties which 
 depend largely on the constitution of the molecule, in other 
 words, on the arrangement of the atoms in the molecule, are 
 termed constitutive ; a typical constitutive property is the rota- 
 tion of the plane of polarization of light (p. 66). The only 
 strictly additive property is weight. Other properties, such as 
 the refractivity, the molecular volume, heat of combustion, etc*, 
 are more or less additive, but are to some extent complicated 
 by constitutive influences, probably due largely to the mutual 
 influence of the atoms. 
 
 There is a third class of properties, which always retain the 
 
 1 Lumsden, Trans. Chem. Soc., 1907, 91, 24 ; Dawson, ibid., 1910, 97, 
 1041. 
 
LIQUIDS 63 
 
 same value independent of the number and nature of the atoms 
 in a molecule or of their arrangement, and depend only on the 
 number of molecules. A good illustration of these properties 
 has already been met with in connection with gases. When these 
 are taken in quantities which, according to the atomic theory, are 
 proportional to their molecular weights, they all exert the same 
 pressure when occupying equal volumes at the same tempera- 
 ture. Such properties have been termed colligative by Ostwald. 
 
 Many illustrations of these three classes of properties will be 
 met with in the course of our work. 
 
 Refractivity The velocity with which light is propagated 
 through different substances is very ' different. The relative 
 velocities in two media can be 
 deduced when the change in 
 direction of a ray of light in 
 passing from one medium to 
 another is known. When the 
 ray passes from one medium to 
 the other, the incident ray, the 
 refracted ray (in the second 
 medium) and the normal to 
 the boundary between the two 
 media (a line drawn perpen. FIG. 7. 
 
 dicular to the boundary where 
 
 the incident ray meets it) are in one plane. If / is the angle of 
 incidence (the angle between the incident ray and the normal) 
 (Fig. 7) and r is the angle of refraction (the angle between the 
 normal and the refracted ray) and i\ and v 2 the respective velo- 
 cities of light in the two media, it can be shown that the ratio of 
 the sine of the angle of incidence to the sine of the angle of 
 refraction is constant, and is equal to the ratio of the velocity 
 of light in the two media. The ratio in question is termed the 
 index of refraction, and is usually represented by the symbol n. 
 
 We have therefore the relation n = 
 
 sin r 
 
 '2 
 
64 OUTLINES OF PHYSICAL CHEMISTRY 
 
 If the first medium is a vacuum, n is always greater than i 
 in other words, light attains its greatest velocity in a vacuum, 
 and is retarded on passing through matter. The refractive 
 index is, however, often referred to air as unity, and to convert 
 the values thus found to a vacuum they must be multiplied by 
 1-00029, giving what is termed the "absolute refractive index". 
 
 An instrument employed for the determination of refractive 
 indices is termed a re fracto meter. Among the more convenient 
 forms of refractometer, those due to Abbe* and to Pulfrich may 
 be specially mentioned. 
 
 Ordinary white light cannot be employed for refractivity 
 measurements, as the component rays are refracted or retarded 
 to a different extent on passing through matter, the rays thus 
 scattered or dispersed giving rise to a spectrum. This difficulty 
 is "avoided by using light of the same wave-length (so-called 
 monochromatic light), and for this purpose .sodium light, the 
 wave-length of which is represented by the letter D, is con- 
 venient. Measurements of the refractive index referred to 
 sodium light are represented by the symbol D . 
 
 The refractive index of a given substance, like any other 
 property, depends on the conditions of temperature, pressure, 
 etc., under which the measurements are made. It has been 
 found convenient to express the results of measurements not 
 
 simply in terms of D , but in terms of the function - 
 
 where d is the density of the liquid or gas (Gladstone and 
 Dale, 1858). This purely empirical function, known as the 
 refraction constant, 1 is, for any given substance, practically inde- 
 pendent of the temperature. In 1880, Lorenz and Lorentz 
 arrived simultaneously, from theoretical considerations, at the 
 
 n 2 - i i 
 somewhat more complicated expression, 2 . j, and showed 
 
 that, for the same substance, it remained fairly constant, not 
 only for widely differing temperatures, but even for the change 
 from liquid to gaseous form. Thus Eykman found that the 
 
 1 The expression here called the " refraction constant " is sometimes 
 called the " specific refractivity". 
 
LIQUIDS 65 
 
 value of this function for isosafrol, C 10 H 10 O 2 , amounted to 
 0-2925 and 0-3962 at 17-6 and 141 respectively, and Lorenz 
 obtained for water at 10 and water vapour at 100 the values 
 0*2068 and 0*2061 respectively. 
 
 For comparative purposes, it is usual to employ the atomic 
 refraction (atomic weight x refraction constant) and the mole- 
 cular refraction (molecular weight x refraction constant) ; the 
 latter, if we employ the second form of the refraction constant, 
 
 is given by -5 * -7, where M is the molecular weight. 
 
 In this case also it has been found, from measurements on 
 many organic liquids, that the molecular refraction may be 
 represented to a first approximation as the sum of the refrac- 
 tions of the component atoms, so that the refractive power is 
 largely an additive property. 
 
 Just as in the case of molecular volumes, however, there are 
 certain deviations from this additive behaviour (constitutive 
 influences) which may be connected with the arrangement of 
 the atoms in the molecule. Briihl, 1 who has been particularly 
 prominent in investigating this question, points out that the 
 molecular refraction of compounds containing double and triple 
 bonds is greater than the calculated value, and he takes account 
 of this constitutive influence by ascribing definite refractivities 
 to these bonds. The most recent values for a few of the 
 elements are as follows : 
 
 Oxygen Double Triple 
 
 C H (in CO group) (in ethers) (in OH group) Cl I bond bond 
 2-418 i-ioo 2-2ii 1-643 1-525 5-967 13-90 1733 2-398 
 
 Later investigations show that neighbouring double or triple 
 bonds exert a mutual influence, so that the matter becomes 
 somewhat complicated. 
 
 Conversely, it is sometimes possible, from measurements of 
 the refractive index (or other physical property), to draw con- 
 clusions as to the constitution of chemical compounds, but as 
 
 1 For a short summary of Bruhl's work, by himself, see Proc. Royal 
 Institution, 1906, 18, 122. 
 
 2 The numbers are referred to the D line (sodium light) and are calcu- 
 lated according to the Lorentz formula. 
 5 
 
66 OUTLINES OF PHYSICAL CHEMISTRY 
 
 our knowledge of the relations between physical properties and 
 chemical constitution is very imperfect, this method should only 
 be employed with great caution. It is evident that no conclu- 
 sions as to chemical constitution can be drawn from additive 
 properties, but only from constitutive properties. 
 
 Rotation of Plane of Polarization of Light The pro- 
 perties of liquids so far dealt with have been mainly additive, 
 the magnitude of properties such as volume, effect on the 
 speed of light, etc., being much the same in the free state and 
 in combination. We have now to deal with the property pos- 
 sessed by a few liquids (and dissolved substances) of rotating 
 the plane of polarized light a property which depends entirely 
 upon the arrangement of the atoms in the molecule. Isomeric 
 substances have in general nearly equal molecular refractivity 
 and molecular volume, but it often happens that of two isomeric 
 substances, such as the two lactic acids, one rotates the plane 
 of polarized light and the other does not. 
 
 Plane polarized light (light in which the vibrations are all in 
 one plane) is obtained by passing monochromatic light through 
 a polarizing prism (Nicol prism or tourmaline plate) which cuts 
 off all the rays except those vibrating in one plane. A prism 
 of this type is mounted at some distance from another similar 
 prism in such a way that light which has been polarized in the 
 first prism may be examined after it has passed through the 
 second prism, which is termed the analyser. If now the 
 analyser is rotated until it is perpendicular to the polarizer, all ; 
 the light which passes through the former will be cut off by the 
 analyser, and on looking through the eyepiece the field will 
 appear dark. If now, while the prisms are in this relative posi- 
 tion, a tube filled with turpentine is placed between them, the 
 field again appears clear, but becomes dark on rotating the 
 analyser through a certain angle. This observation is readily 
 accounted for on the view that the plane of polarization is 
 twisted through a certain angle whilst the light is traversing the 
 turpentine, and the analyser must therefore be rotated in ordej 
 
LIQUIDS 67 
 
 to bring it into the former relative position with regard to the 
 polarized ray. The angle through which the analyser has been 
 turned is read off on a graduated scale. The instrument used 
 for measuring the rotation of liquids, which consists essentially 
 of the two prisms and graduated scale, as described above, is 
 termed a polarimeter. The observed angle of rotation depends 
 on the nature of the liquid, on the wave-length of the light 
 employed in the measurements, and on the temperature, and 
 is proportional to the length of liquid traversed. For purposes 
 of comparison, the results are usually expressed in terms of the 
 specific rotation [a] for a fixed temperature / and a particular 
 wave-length of light (for example, sodium light) by means of 
 the formula 
 
 where a is the observed angle, / is the length of the column 
 of liquid in decimetres, and d is the density of the liquid at the 
 temperature /. The molecular rotation m[a] is obtained by 
 multiplying the specific rotation by the molecular weight of the 
 liquid. 
 
 The specific rotation of substances in solution is represented 
 by the analogous formula 
 
 r ,* iooa 
 
 [0] D = W 
 
 where g is the number of grams of solute in 100 grams of the 
 solution, d is the density of the solution at the temperature /, 
 and a and / have the same significance as before. 
 
 The liquids and dissolved substances which possess this re- 
 markable property are almost exclusively compounds containing 
 carbon. Further, only compounds containing an asymmetric 
 carbon atom, that is, a carbon atom joined to four different 
 groups, have the power of rotating polarized light ( van't Hoff and 
 Le Bel, 1874). For example, the graphic formula of ordinary 
 lactic acid, which is optically active, may be written as 
 follows : 
 
68 OUTLINES OF PHYSICAL CHEMISTRY 
 
 CH 8 
 
 : OH 
 
 H-i 
 
 COOH 
 
 showing that no two of the groups attached to the central 
 carbon atom are identical. A further 'remarkable fact is that 
 when one form of a substance, such as lactic acid, can rotate 
 the plane of polarized light to the right, a second modifica- 
 tion can always be obtained which, although identical with 
 the first in all other physical properties, rotates polarized light 
 to the left to the same extent as the first modification rotates 
 it to the right. The first is termed the dextro or d modi- 
 fication, the second the laevo or / modification. Van't Hoff 
 and Le Bel account for this on the hypothesis that the four 
 different groups are not in the same plane as the carbon 
 atom, but are arranged in the form of a tetrahedron with 
 the carbon atom in the centre. It can be shown, most 
 readily by means of a model, 1 that when all 4 four groups 
 are different there are two arrangements which cannot be 
 made to coincide by rotating one of the models. These 
 two arrangements behave to each other as object and mirror 
 image, or a right- and left-hand glove, which cannot be 
 brought into the same relative position, and, according to the 
 theory, correspond with the dextro and laevo modifications 
 respectively. If any two of the groups become identical, 
 however, the two arrangements can always be brought ;:o 
 coincidence, and there is no possibility of optical isomerism. 
 In calculating the specific rotation of a dissolved substance, 
 it is implicitly assumed that it is not affected by an indifferent 
 solvent. As a matter of fact, however, the molecular rotation 
 of a dissolved substance often differs considerably from its value 
 
 1 Models for this purpose can be bought for a few pence, or can be made 
 by the student with a piece of cork and four pins provided with differently 
 coloured balls at the ends. 
 
LIQUIDS 69 
 
 in the pure state. This is well shown by the following results 
 obtained by Patterson 1 for /-menthyl-^-tartrate in the pure state, 
 and in 1-2 per cent, solution. 
 
 Molecular Rotation of /-menthyW-tartrate. 
 
 Solvent. Molecular Rotation at 20. 
 None - 284 
 
 Ethyl alcohol -306-2 
 
 Benzene - 296-1 
 
 Nitrobenzene - 245-3 
 
 So far, no definite connection has been found between any 
 other property of the solvent and its effect on the rotation of a 
 solute. Almost the only regularity which has yet been dis- 
 covered in this branch of the subject is that the rotatory power 
 of the salt of an optically active acid (or base) in dilute solution 
 is independent of the nature of the base (or acid) with which it 
 is combined. This important result is further referred to at a 
 later stage. , 
 
 All transparent substances, when placed in a magnetic field, 
 rotate the plane of polarized light, and the late Sir William 
 Perkin, who devoted many years to the systematic investigation 
 of this subject, showed that this magnetic rotation is, like refrac- 
 tivity, largely an additive property. 
 
 Absorption of Light 2 When a ray of white light passes 
 through matter a greater or less amount of absorption invari- 
 ably occurs, and the spectrum of the issuing ray shows 
 dark bands corresponding with the rays which have been 
 absorbed. This spectrum is termed an absorption spectrum. 
 Absorption may be general or selective. In the former case 
 there is a general weakening throughout whole regions of the 
 spectrum ; in the latter case, there are independent relatively 
 narrow bands in different regions of the spectrum. 
 
 ^ee Trans. Chem. Soc., 1905, 87, 128. 
 
 2 Smiles, Chemical Constitution and some Physical Properties (Long- 
 mans, 1910), pp. 324-423. 
 
70 OUTLINES OF PHYSICAL CHEMISTRY 
 
 The absorption spectrum is one of the most characteristic 
 properties of a given substance or class of substances under 
 definite conditions. Thus, for instance, the absorption spectra 
 of permanganates in dilute solution are characterised by the 
 presence of five dark bands in the green. 
 
 In the case of liquids or solutions, absorption is most 
 conveniently measured in the transmitted light, as already 
 indicated. In the case of solids, however, observations may be 
 made with reflected light, as part at least of the reflection takes 
 place from layers a little below the surface, and the short distance 
 traversed is usually sufficient to produce some absorption. 
 
 The method which is most largely employed in the case of 
 liquids is to use as a source of illumination the electric arc 
 or spark passing between metallic (preferably iron) poles, the 
 spectrum of the light being photographed after the latter has 
 passed through the liquid. Substances which have high ab- 
 sorbing power are usually examined in solution, and the solvent 
 chosen should not itself show absorption in the part of the 
 spectrum studied. Ethyl alcohol is most largely used as solvent. 
 
 A very important question in such measurements is the 
 effect of dilution on the persistence and intensity of a band 
 it very often happens that a band appears only within certain 
 limits of dilution. For this reason, it is usual to carry out 
 measurements with varying thicknesses of the absorbing layer 
 and with varying concentrations. The solutions are made 
 more and more dilute till absorption is no longer observed. 
 For purposes of comparison, the solutions to be examined are 
 usually made up to contain simple fractions of a mol per litre. 
 
 The method usually employed in representing the results 
 graphically requires some explanation. It is illustrated in Fig. 
 8, which shows the absorption in the ultra-violet region of a 
 solution of the amino derivative of dimethyldihydroresorcin 
 in ethyl alcohol. The diagram shows the presence of one 
 band. The principle of the method is that the oscillation 
 frequencies of the limits of the absorption band are plotted 
 
LIQUIDS 71 
 
 against the logarithms of the relative thicknesses referred to 
 the thinnest layer of the most dilute solution examined. It is 
 
 more convenient to use 
 
 Oscillation Frequencies. 
 
 
 
 FIG. 8. 
 
 2000 
 1,500 
 
 1000 . 
 
 c 
 
 500 -| 
 
 X 
 
 250 g 
 200 o 
 
 the logarithms of the 
 relative thicknesses than 
 the thicknesses them- 
 selves as ordinates. The 
 latter are shown on the 
 right-hand side of the 
 figure, and it is evident 
 that if equal increments 
 of thickness were used, 
 the curve would become 
 inconveniently long. 
 The space above the 
 curve shows the wave 
 lengths of the light, 
 absorbed at different 
 dilutions, and the curve 
 itself represents the 
 limits of the absorption. 
 The lowest part of the 
 curve, usually termed the 
 " head," is that point at 
 which the strongest ab- 
 sorption takes place ; in 
 the diagram it occurs at 
 an oscillation frequency 
 of 3600 units. The so- 
 called " persistence " of 
 the band is the difference 
 of the logarithms of the 
 thicknesses at which the band makes its appearance and dis- 
 appears respectively, and is indicated on the diagram by its 
 depth (in the present case extending from i to 23 units). 
 
 34 
 32 
 
 I 30 
 I '28 
 I 26 
 24 
 *22 
 
 <J-4 
 
 20 
 
 100 
 50 
 
 30 
 
 20 
 15 
 10 
 
 5 
 
 0-6 
 
72 OUTLINES OF PHYSICAL CHEMISTRY 
 
 It may be mentioned that most of the measurements have 
 been made in the ultra-violet region of the spectrum, but a 
 certain number of observations have also been made in the 
 visible and ultra-red regions. 
 
 A great deal of discussion has taken place as to the 
 mechanism of light absorption, and a dynamical explanation 
 has found most favour. According to Drude, absorption in the 
 ultra-violet and visible regions is connected with the presence 
 of negative electrons (valency electrons), and there is doubtless 
 some intimate connection between the rate of vibration of the 
 electrons and the frequency of the light waves absorbed. It 
 appears, however, that absorption in the ultra-red region is due, 
 not to electrons, but to particles of much greater magnitude, 
 probably atoms or even groups of atoms. 
 
 As already indicated, the absorption spectra of chemical 
 compounds are closely related to their constitution, and in 
 recent years this method of elucidating chemical constitution 
 has come increasingly into use. The progress made in this 
 branch of investigation is mainly due to Hartley, and within 
 the last few years to Baly and his co-workers. 1 It is impossible 
 here to do more than to indicate in the briefest way the broad 
 differences between the absorption spectra of the main groups 
 of organic compounds. 
 
 According to Hartley, 2 all compounds which exert selective 
 absorption are of aromatic character, for example, benzene, 
 pyridine, and their derivatives. Those showing strong general 
 absorption have also a cyclic structure, but are not typical aro- 
 matic compounds ; examples, thiophene, piperidine, etc. Finally, 
 compounds which exert only a weak general absorption are 
 open chain compounds; examples, fatty alcohols, esters, etc. 
 Although recent investigation has led to the discovery of one 
 or two exceptions to these generalisations, they remain sub- 
 stantially accurate. 
 
 Apart from these broad differences in the absorption curves 
 
 1 Trans. Chem. Soc. from 1904 onwards. 
 a British Association Report, 1903. 
 
LIQUIDS 73 
 
 of different groups of organic compounds, it has been shown 
 by Hartley that substances of closely allied constitution have 
 absorption curves of similar type. The recent progress in the 
 establishment of the chemical constitution of organic compounds 
 from measurements of their absorption spectra is entirely due 
 to the application of this important rule. Thus if the con- 
 stitution of one compound is known with certainty from its 
 chemical behaviour, and a second compound can be represented 
 by alternative formulae, of which one is analogous to that of 
 the compound of known constitution, a comparison of absorp- 
 tion spectra often enables a decision to be made. 
 
 Like the other physical methods of determining chemical 
 constitution, the one under discussion should be used with 
 considerable caution in the present very imperfect state of our 
 knowledge of absorption phenomena. As a supplement to 
 purely chemical methods of determining constitution, however, 
 especially when such methods lead to contradictory results, it 
 is rendering valuable service. 
 
 As regards the additive character of absorption spectra, 
 comparatively little is known with certainty. The determina- 
 tion of the effect of different groups of atoms on the character 
 of the absorption spectrum is attended with considerable diffi- 
 culty, but some progress has been made in this direction. 
 Ultra-red absorption spectra have a pronounced additive char- 
 acter ; this is probably connected with the generally accepted 
 view that absorption in this case is connected with the vibrations 
 of the atoms themselves. 
 
 Viscosity The measurement of internal friction, or vis- 
 cosity, gives information as to the work done in the relative 
 displacement of the particles of a solid, liquid, or gas. In the 
 case of liquids, for which the property has been most fully 
 investigated, the viscosity is most conveniently studied by 
 observing the rate of flow through capillary tubes. We may 
 assume that the layer of liquid in contact with the wall of the 
 tube is at rest, that the layer next to it is moving slowly parallel 
 to the axis of the tube, and that the rate of movement gradually 
 
74 OUTLINES OF PHYSICAL CHEMISTRY 
 
 increases towards the interior, attaining its maximum at the 
 centre of the tube. It is evident that the rate of displacement 
 of the layers with regard to each other must be mainly deter- 
 mined by the amount of friction between them, and hence 
 measurements of the rate of flow afford information as to the 
 viscosity of liquids. The converse of viscosity is termedy?wz#//>> ; 
 a liquid of small viscosity, such as ether, is said to have a high 
 degree of fluidity. 
 
 The magnitude of the viscosity depends greatly on the 
 nature of the liquid. Thus the viscosity of warm ether is 
 very small, whereas that of treacle and of pitch is so great that 
 they approximate to the behaviour of solids, the internal friction 
 of which is extremely high. The internal friction of gases is 
 very small (p. 25). 
 
 The coefficient of viscosity, rj, is usually defined as the force 
 required to move a layer of unit area in unit time through a 
 distance of unit length past an adjacent layer unit distance 
 away. For water at 15, rj = o - oi34 in absolute units; for 
 glycerine at the same temperature, rj = 2-34. 
 
 The coefficient of viscosity of a liquid can be calculated 
 from the rate of outflow from a cylindrical tube by means of 
 the equation 
 
 where v is the volume of liquid discharged in the time /, p the 
 pressure under which the outflow takes place, r the radius and 
 / the length of the tube. In practice, however, the rate of 
 flow of the liquid is compared with that of a standard liquid, 
 usually water, under the same experimental conditions, and its 
 absolute viscosity, 17, calculated by means of the formula 
 
 "nho = */** 
 
 where r}^ is the absolute viscosity of water at the temperature 
 
 of the experiment, and t and / w are the times required for the 
 
 discharge of equal volumes of the liquid and water respectively. 
 
 The viscosity of liquids diminishes rapidly as the temperature 
 
LIQUIDS 
 
 75 
 
 O 
 
 rises, but so far no simple relationship of general applicability 
 connecting change of viscosity and temperature has been dis- 
 covered. 
 
 Measurement of Yiscosity The determination of vis- 
 cosity by the comparative method may be conveniently carried 
 out with the apparatus described by Ostwald 
 and illustrated in Fig. 9. It consists essenti- 
 ally of a capillary tube, db, connected at its 
 upper part with a bulb /, and at its lower 
 end with a wider tube, bent into U- shape 
 and provided with a bulb <?. Marks are etched 
 on the capillary tube at c and d above and 
 below the bulb k. The apparatus is filled at/ 
 with a definite volume of the liquid, which 
 is then sucked into the other limb till the 
 surface rises above the mark c. The time 
 required for the level of the liquid to fall 
 from c to d is then noted. Observations 
 have previously been made with water under 
 similar conditions. As the pressures under 
 which the discharge takes place are in the 
 ratio of the density, d, of the liquid to that of 
 water, the relative viscosity, x = rj/rj^ is given 
 by the equation 
 
 FIG. 9. 
 
 On account of the great influence of 
 temperature on the viscosity, the measure- 
 ments must be made at constant temperature, and for this 
 purpose the apparatus is so constructed that it can conveniently 
 be immersed in a thermostat. 
 
 Results of Yiscosity Measurements 1 Reference has 
 already been made to the enormous differences in the viscosity 
 of liquids, and also to the fact that temperature has a great 
 influence on the magnitude of the viscosity. In comparing 
 
 1 Smiles, loc. cit. pp. 51-105. 
 
7 6 
 
 OUTLINES OF PHYSICAL CHEMISTRY 
 
 different liquids with regard to viscosity, we are again confronted 
 with the difficulty of deciding at what temperatures the measure- 
 clients should be made. The choice of the boiling-points as 
 corresponding temperatures (p. 57) did not in this case lead 
 to very satisfactory results, and most regularities were observed 
 by using the data for points at which the rate of change of 
 viscosity with temperature is the same (Thorpe and Rodger). 
 
 A comparison of substances of allied chemical constitution 
 shows that viscosity is to some extent an additive property, but 
 is greatly affected by constitutive influences. 
 
 The absolute viscosity of a number of pure liquids at o and 
 at 25 is given in the accompanying table (Walden) : 
 
 Liquid. 
 
 rj at 0. 
 
 i> at 25. 
 
 Acetone 
 
 0-00397 
 
 0-00316 
 
 Methyl alcohol 
 
 0*00846 
 
 0-00580 
 
 Acetic anhydride 
 
 O'OI3O 
 
 O'oo86o 
 
 Water 
 
 0-0178 
 
 0-00891 
 
 Ethyl alcohol 
 
 O-OI79 
 
 o-o 108 
 
 Benzonitrile 
 
 0*0194 
 
 0-0125 
 
 Nitrobenzene 
 
 0-0307 
 
 0-0182 
 
 In recent years the viscosity of mixtures of liquids has been 
 the subject of a good deal of investigation. If for simplicity 
 we confine our attention to mixtures of two components only, 
 three classes may be distinguished : 
 
 (1) The viscosity of the mixture lies between those of the 
 pure components. Example, ethyl alcohol and carbon disul- 
 phide. 
 
 (2) The viscosity of the mixture in certain proportions is 
 greater than that of either component. Example, pyridine 
 and water. 
 
 (3) The viscosity of the mixture in certain proportions is 
 less than that of either constituent. Example, benzene and 
 acetic acid. 
 
 A few numbers illustrating the behaviour of the mixtures 
 
LIQUIDS 
 
 77 
 
 cited as examples of classes (2) and (3) are given in the 
 accompanying table. 
 
 Pyridine and Water.l 
 
 Benzene and Acetic Acid.* 
 
 Per cent. 
 Pyridine. 
 
 T? at 25. 
 
 Per cent. 
 Benzene. 
 
 , at 25. 
 
 0'00 
 
 0-00890 
 
 o-oo 
 
 0-01174 
 
 IQ'28 
 
 0-01336 
 
 34'93 
 
 0*00734 
 
 39-84 
 
 0-01833 
 
 48-29 
 
 0-00666 
 
 59'7 
 
 0*02187 
 
 77-26 
 
 0-00597 
 
 66-6 5 
 
 O-O2225 
 
 8973 
 
 0-00591 
 
 80-15 
 
 0-01894 
 
 97-25 
 
 0-00594 
 
 lOO'OO 
 
 0-00885 
 
 IOO-00 
 
 0-00598 
 
 The results are very similar to those for the vapour pressures 
 of binary mixtures, described later (p. 87). 
 
 No general agreement has so far been reached as to the 
 explanation of these remarkable differences in the behaviour of 
 binary mixtures (compare p. 322). 
 
 Measurements of the viscosity of salt solutions (solutions 
 which conduct the electric current) have also led to interesting 
 results, but they cannot be usefully considered at the present 
 stage. 
 
 Practical Illustrations, (a) Critical Phenomena The 
 critical phenomena can be observed in an apparatus, con- 
 structed like that of Cagniard la Tour, but more simply as fol- 
 lows : A tube 3-4 mm. internal diameter and 3-4 cm. long is 
 constructed out of a piece of glass tubing, the walls of which 
 are 0*7-0-8 mm. thick, by closing one end in the blow-pipe and 
 drawing out the other at a distance 3-4 cm. from the closed end 
 into a fairly long (5-6 cm.) thick-walled capillary tube. The 
 capillary is then bent, at a point about i cm. from the commence- 
 ment of the wide part of the tube, at right angles to the latter and 
 then partly filled with ether as follows. The tube is warmed and 
 the capillary end dipped into ether, which is drawn into the tube 
 
 1 Hartley, Thomas and Appleby, Trans. Chem. Soc., 1908, 94, 538. 
 2 Dunstan, ibid., 1905, 87, 16. 
 
78 OUTLINES OF PHYSICAL CHEMISTRY 
 
 as the latter cools. The ether in the tube is then boiled gently 
 to expel all the air, the end of the capillary dipping all the time 
 in ether, and on again allowing to cool, ether is drawn in so as 
 practically to fill the tube. The excess of ether is then boiled 
 off till the tube is about three-quarters full, the end of the 
 capillary being in ether throughout, and then allowed to cool 
 till the liquid just begins to rise up the capillary tube, showing 
 that the pressure inside is somewhat less than atmospheric ; the 
 capillary is then rapidly sealed off near the bend on the side 
 remote from the tube. 
 
 In order to observe the capillary phenomena in the tube 
 thus prepared, the latter is suspended by a wire and heated by 
 means of a Bunsen burner held in one hand, the face being 
 protected, in the event of an explosion, by a large plate of 
 glass held in the other hand. In this way, the complete dis- 
 appearance of the liquid above a certain temperature, and its 
 reappearance on cooling, may be observed without the least 
 danger. 
 
 When practicable, the tube may be heated in an iron or 
 copper vessel provided with mica windows, and the critical 
 temperature may be read off on a thermometer placed side by 
 side with the tube in the air bath. 
 
 () Determination of Molecular Volume The determination 
 of the molecular volume of a liquid reduces to a determination 
 of its density at a definite temperature compared with that of 
 water as unity, and any of the well-known methods for deter- 
 mining the density of liquids may be employed for this purpose. 
 A direct method for determining the density, and hence the mole- 
 cular volume, of a liquid at its boiling-point has been described 
 by Ramsay. A tube is drawn out to a long neck and the latter 
 bent in the form of a hook. The vessel is then filled with 
 liquid, heated in the vapour of the same liquid till equilibrium 
 is reached and then weighed. If the volume of the vessel and 
 its coefficient of expansion are known, the molecular volume 
 of the liquid can at once be calculated. 
 
LIQUIDS 79 
 
 Measurements of the refractive index of liquids (water 
 alcohol, benzene) and of the rotation of the plane of polarization 
 of light by liquids or solutions (cane sugar in water) should be 
 made by the student ; the methods are fully described in text- 
 books of physics. The experiments on rotation of the plane 
 of polarization may conveniently be made in connection with 
 the hydrolysis of cane sugar in the presence of acids (p. 230). 
 
CHAPTER IV 
 SOLUTIONS 
 
 General Up to the present, we have dealt only with the 
 properties of pure substances which may exist in the gaseous, 
 liquid or solid state, or simultaneously in two or all of these 
 states. We now proceed to deal with the properties of mixtures 
 of two or more pure substances. When these mixtures are 
 homogeneous, they are termed solutions. 1 There are various 
 classes of solutions, depending on the state of the components, 
 The more important are : 
 
 (i) Solutions of gases in gases ; 
 
 (ii) Solutions of (a) gases, (b) liquids, (c) solids in liquids ; 
 
 (iii) Solutions of solids in solids, so-called solid solutions ; 
 and each of these classes will be briefly considered. 
 
 A distinction is often drawn between solvent and dissolved 
 substance, but, as will appear particularly from the sections 
 dealing with the mutual solubility of liquids, there is no sharp 
 distinction between the two terms. The component which is 
 present in greater proportion is usually termed the solvent. 
 The dissolved substance is sometimes called the solute. 
 
 When one of the compounds is present in very small pro- 
 portion, the system is termed a dilute solution, and as the laws 
 
 a The term solution is also applied to mixtures which appear homo- 
 geneous to the naked eye but heterogeneous when examined with a micro- 
 scope or ultramicroscope, e.g., colloidal solution of arsenic sulphide. 
 
 The usual definition of a solution is "a homogeneous mixture which 
 cannot be separated into its components by mechanical means," but the 
 last part of this definition is open to objection. 
 
 80 
 
SOLUTIONS 
 
 81 
 
 representing the behaviour of dilute solutions are comparatively 
 simple, they will be dealt with separately in the next chapter. 
 
 Solution of Gases in Gases This class of solution differs 
 from the others in that the components may be present in any 
 proportion, since gases are completely miscible. If no chemical 
 change takes place on mixing two gases, they behave quite in- 
 dependently and the properties of the mixture are therefore the 
 sum of the properties of the constituents. In particular, the 
 total pressure of a mixture of gases is the sum of the pressures 
 which would be exerted by each of the components if it alone 
 occupied the total volume a law 
 which was discovered by Dalton, and 
 is known as Dalton s law of partial 
 pressures. Dalton's kw is of the same 
 order of validity as Avogadro's hy- 
 pothesis ; it is nearly true under 
 ordinary conditions, and would in v 
 all probability become strictly true 
 at great dilution. 
 
 Dalton's law can of course be 
 tested by comparing the sum of the 
 pressures exerted separately by two 
 gases with that after admixture, but 
 it is of interest to inquire into the 
 possibility of measuring the partial 
 pressure of one of the components 
 in the mixture itself. It was pointed 
 
 out by van't Hoff that this is always possible if a membrane 
 can be obtained which allows only one of the gases to pass 
 through. This suggestion was experimentally realised by Ram- 
 say * in the case of a mixture of nitrogen and hydrogen as 
 follows: P (Fig. 10.) is a palladium vessel containing nitrogen, 
 the pressure of which can be determined from the difference 
 of level between A and B in the manometer, which contains 
 mercury. P is enclosed in another vessel, which can be filled 
 
 H 
 
 FIG. 10. 
 
 Phil. Mag., 1894, [v.], 38, 206. 
 
82 OUTLINES OF PHYSICAL CHEMISTRY 
 
 with hydrogen at any desired pressure. The vessel P is heated 
 and a stream of hydrogen at known pressure passed through the 
 outer vessel. As palladium at high temperatures is permeable 
 for hydrogen, but not for nitrogen, the former gas enters P till 
 its pressure outside and inside are equal. The total pressure in 
 P, as measured on the manometer, is greater than the pressure 
 in the outer vessel, and it is an experimental fact that the 
 excess pressure inside is approximately equal to the partial 
 pressure of the nitrogen. 
 
 If, on the other hand, we start with a mixture of hydrogen and 
 nitrogen, and wish to find the partial pressure of the latter, all 
 that is necessary is to put the mixture inside a palladium bulb, 
 keep the latter at a constant high temperature and pass a current 
 of hydrogen at known pressure through the outer vessel till equi- 
 librium is attained, as shown on the manometer. The difference 
 between the external and internal pressure is then the partial 
 pressure of the nitrogen. This very instructive experiment will 
 be referred to later in connection with the modern theory of 
 solutions (p. 101). 
 
 Solubility of Gases in Liquids In contrast to the com- 
 plete miscibility of gases, liquids are only capable of dissolving 
 gases to a limited extent. When a liquid will not take up any 
 more of a gas at constant temperature it is said to be saturated 
 with the gas, and the resulting solution is termed a saturated 
 solution. The amount of a gas taken up by a definite volume 
 of liquid depends on (a) the pressure of the gas, (b) the tem- 
 perature, (c) the nature of the gas, (d) the nature of the liquid. 
 
 The greater the pressure of a gas, the greater is the quantity 
 of it taken up by the solvent. For gases which are not very 
 soluble, and do not enter into chemical combination with the 
 solvent, the relation between pressure and solubility is expressed 
 by Henry's law as follows : The quantity of gas taken up by a 
 given volume of solvent is proportional to the pressure of the gas. 
 Another way of stating Henry's law is that the volume of a gas 
 taken up by a given volume of solvent is independent of the 
 
SOLUTIONS 83 
 
 pressure. This is clearly equivalent to the first statement, be- 
 cause when the pressure is doubled the quantity of gas absorbed 
 is doubled, but since its volume, by Boyle's law, is halved, the 
 original and final volumes dissolved are equal. 
 
 The question may be regarded from a slightly different point 
 of view, which is instructive in connection with later work. 
 When a definite volume of liquid is saturated with a gas at a 
 certain pressure, there is an equilibrium between the dissolved 
 gas and that over the liquid, and Henry's law may be expressed 
 in the alternative form : The concentration of the dissolved gas is 
 proportional to that in the free space above the liquid. We may 
 consider that the gas distributes itself between the solvent and 
 the free space in a ratio which is independent of the pressure. 
 The solubility of gases in liquids diminishes fairly rapidly with 
 rise of temperature. 
 
 For purposes of comparison, the solvent power of a liquid for 
 a gas is best expressed in terms of the " coefficient of solubility," 
 which is the volume of the gas taken up by unit volume of the 
 liquid at a definite temperature. 1 The so-called "absorption 
 coefficient " of Bunsen, in which solubility measurements are 
 still often expressed, is the volume of a gas, reduced to o and 
 76 cm. pressure, which is taken up by unit volume of a liquid 
 at a definite temperature under a gas pressure equal to 76 cm. 
 of mercury. 
 
 With regard to the influence of the nature of the gas on the 
 solubility, it may be said in general that gases which have 
 distinct basic or acidic properties, for example, ammonia and 
 hydrogen chloride, are very soluble, whilst neutral gases, such as 
 hydrogen, oxygen and nitrogen, are comparatively insoluble. 
 Further, gases which are easily liquefied, for example, sulphur 
 dioxide and hydrogen sulphide, are fairly soluble. 
 
 As regards the relation between solvent power and the nature 
 of the liquid, very little is known. In general, the order of 
 the solubility of gases in different liquids is the same, and 
 the solvent power of a liquid therefore appears to be to some 
 extent a specific property. 
 
 1 Alternatively : the ratio of the concentration in liquid and in gas space. 
 
OUTLINES OF PHYSICAL CHEMISTRY 
 
 The above remarks are illustrated by the following table, in 
 which the coefficients of solubility of some typical gases in water 
 and in alcohol are given : 
 
 Gas. 
 
 Water. 
 
 Alcohol. 
 
 Ammonia 
 
 1050 
 
 _ 
 
 Hydrogen sulphide 
 Carbon dioxide 
 
 80 
 1-8 
 
 18 
 4'3 
 
 Oxygen . 
 
 0-04 
 
 0-28 
 
 Hydrogen 
 
 O'O2 
 
 0-07 
 
 It may be mentioned that the solubility of gases in water is 
 greatly diminished by the addition of salts, and to a much 
 smaller extent by non-electrolytes. The interpretation of these 
 results has given rise to considerable difference of opinion. 1 
 
 Solubility of Liquids in Liquids As regards the mutual 
 solubility of liquids, three cases may be distinguished : (i) The 
 liquids mix in all proportions, e.g., alcohol and water ; (2) the 
 liquids are practically immiscible, e.g., benzene and water ; (3) 
 the liquids are partially miscible, e.g., ether and water. 
 
 (i) and (2) Complete miscibility and non-miscibility Very 
 little is known as to the factors which determine the miscibility 
 or non-miscibility of liquids. The separation of the compon- 
 ents by fractional distillation is discussed in succeeding sections. 
 
 (3) Partial miscibility If ether, in gradually increasing 
 amounts, is added to water in a separating-funnel, and the mix- 
 ture well shaken after each addition, it will be noticed that at 
 first a homogeneous solution is formed, but when sufficient 
 ether has been added, a separation into two layers takes place 
 on standing. The upper layer is a saturated solution of water 
 in ether, the lower layer a saturated solution of ether in water. 
 As long as the relative quantities of ether and water are such 
 that a separation into two layers takes place on standing, the 
 
 Compare Philip, Trans. Chem. Sec., 1907, 91, 711; Usher, ibid., 
 1910, 97, 66. 
 
SOLUTIONS 85 
 
 composition of these layers is independent of the relative amounts 
 of the components present^ since the composition is determined 
 by the solubility of ether in water and of water in ether at the 
 temperature of experiment. Further, the saturated vapours sent 
 out by the two layers have the same pressure and the same com- 
 position this follows from the fact that they are in equilibrium 
 with the two layers which are in equilibrium with each other. 
 
 In the majority of cases, the solubility of two partially 
 miscible liquids increases with the temperature, and it may 
 therefore be anticipated that liquids which in certain propor- 
 tions form two layers at the ordinary temperature may be com- 
 pletely miscible at higher temperatures. Several such cases 
 are known, for example, phenol and water, which has been 
 investigated by Alexieeff. At room temperature a saturated 
 solution of phenol in water contains about 8 per cent, of the 
 former component. When more phenol is added two layers 
 are formed and the temperature has to be raised in order to 
 secure complete miscibility. For further additions of phenol 
 up to 36 per cent, the temperature at which complete miscibility 
 occurs rises progressively to 68 '4. In this way the solubility- 
 temperature curve of phenol in water is obtained. Similarly, 
 a saturated solution of water in phenol at 20 contains 28 per 
 cent, of the former component and on further additions of 
 water the temperature of complete miscibility rises progres- 
 sively to 68-4, at which point the two solubility-temperature 
 curves meet. These results are represented graphically in 
 Fig. 1 1, the composition of the mixture being measured on the 
 horizontal axis and temperatures along the vertical axis. The 
 point D represents o per cent, phenol (100 per cent, water), 
 E represents 100 per cent, phenol. At all points outside the 
 curve ABC there is complete miscibility, at points inside the 
 curve two layers exist. The maximum represents the tem- 
 perature, 68*4, above which phenol and water are miscible 
 in all proportions. If, therefore, we start with a homogeneous 
 solution of phenol in water of the composition represented 
 
86 
 
 OUTLINES OF PHYSICAL CHEMISTRY 
 
 by the point x, and gradually add phenol at constant tem- 
 perature, the composition of the. .solution will alter along the 
 dotted line xx until the curve AB is reached at z. This 
 point represents a saturated solution of phenol in water, and 
 on further addition of phenol a separation into two layers 
 takes place, the compositions of which are represented by the 
 points s and z' respectively. As more phenol is added, the 
 composition of the layers remains unaltered, but the relative 
 amount of the second layer increases until at the point z' only 
 this layer is present, and its composition then alters along zx'. 
 
 60" 
 
 A 
 
 o/ 
 Phenol 
 
 Miscibility of Phenol and Water. 
 FlG. II. 
 
 100 / 
 
 Phenol 
 
 o% 100% 
 
 NICOTINE NICOTINE 
 
 Miscibility of Nicotine and Water. 
 FlG. 12. 
 
 If, however, phenol is added to the same solution at the tem- 
 perature corresponding with the point y the composition alters 
 along yy but no separation into two layers takes place. It is 
 evident that there is a striking analogy between the miscibility 
 of two liquids and the critical phenomena represented in Fig. 
 5. In both cases there is only one phase outside the curves 
 AOE and ABC respectively, and two phases at points inside 
 the curves. Further, above a certain temperature only one 
 phase can exist in each case, and the temperature of com- 
 plete miscibility for binary mixtures may therefore be termed 
 the critical solution temperature. Moreover, just as we can 
 
SOLUTIONS 87 
 
 pass without discontinuity from a gas to a liquid (p. 53), we can 
 pass from a solution containing excess of water to one contain- 
 ing excess of phenol without discontinuity. Starting with a 
 mixture represented by the point x, the temperature is raised 
 above the critical solution temperature along xy, phenol is then 
 added till the point y' is reached and the homogeneous mixture 
 then cooled along yx. 
 
 In some cases, however, the solubility of one liquid in another 
 diminishes with rise of temperature, thus if a saturated solution 
 of ether in water, prepared at the ordinary temperature, is gently 
 warmed, it becomes turbid, indicating partial separation of the 
 ether. An interesting example of this behaviour is seen in 
 nicotine and water, which are miscible in all proportions at the 
 ordinary temperature, but separate into two layers when the 
 temperature reaches 60. If a temperature can be reached 
 beyond which the mutual solubility again begins to increase 
 with rise of temperature, the components may again become 
 miscible in all proportions. This has been experimentally 
 realised so far only for nicotine and water, which again be- 
 come completely miscible when the temperature exceeds 210. 
 The remarkable solubility relations of these two liquids are 
 therefore represented by a closed curve (Fig. 12), which will 
 be readily understood by comparison with Fig. n. 
 
 Distillation of Homogeneous Mixtures A very impor- 
 tant matter with reference to binary homogeneous mixtures is 
 the possibility of separating them more or less completely into 
 their components by distillation. Much light is thrown on this 
 question by the investigation of the vapour-pressure of the mix- 
 ture as a function of its composition at constant temperature. 
 Experimental investigation shows that the curve representing the 
 relation between vapour pressure and composition at constant 
 temperature usually belongs to one of the three main types a, 
 b and c represented in Fig. 13, in which the abscissae represent 
 the composition of the mixture and the ordinates the corres- 
 ponding vapour pressures. 
 
88 
 
 OUTLINES OF PHYSICAL CHEMISTRY 
 
 (a) The vapour-pressure curve of the mixture may have a 
 minimum, as represented by the point U in the curve RUS ; 
 example, hydrochloric acid and water. (In the diagram the 
 ordinate PR represents the vapour-pressure of B, and QS that of 
 the other pure substance A.) 
 
 () The vapour-pressure curve may show a maximum, repre- 
 sented by the point T on the curve RTS ; example, propyl 
 alcohol and water. 
 
 P(o/ A) 
 
 Composition 
 FIG. 13. 
 
 Q (TOO o/ A) 
 
 (f) The vapour-pressure of the binary mixture may lie 
 between those of the pure components A and B, as repre- 
 sented by the curve RWS ; example, methyl alcohol and 
 water. 
 
 In considering these three typical cases with regard to their 
 bearing on the separation of two liquids by fractional distilla- 
 tion, the important question is the relation between the com- 
 position of the boiling liquid and that of the escaping vapour. 
 For a pure liquid, the composition of the escaping vapour is 
 necessarily the came as that of the liquid, but this is not in 
 general the case for a mixture of liquids, and therefore the 
 
SOLUTIONS 89 
 
 composition of the mixture may alter continuously during dis- 
 tillation. 
 
 Case (a). Since a liquid boils when its vapour pressure is 
 equal to the external pressure, it is clear that if a mixture the 
 vapour-pressure curve of which has a minimum (as in the curve 
 RUS) be boiled, the composition of the liquid will alter in 
 such a way that it tends to approximate to that represented by 
 the point U, since all other mixtures have a higher vapour 
 pressure, and will consequently pass off first. When finally 
 only the mixture U remains, it will distil at constant tempera- 
 ture like a homogeneous liquid, since the composition of the 
 vapour is then the same as that of the liquid. The best- 
 known example of such a constant-boiling liquid is a mixture of 
 hydrochloric acid and water, which boils at no . If a mixture 
 containing the components in any other proportion be heated, 
 either hydrochloric acid or water will pass off, and the com- 
 position of the liquid will move along the curve to the point 
 of minimum vapour pressure, beyond which it distils as a whole, 
 without further change of composition. 
 
 Case (b). When, on the other hand, there is a maximum in 
 the vapour-pressure curve, the mixture which has the highest 
 vapour tension will pass over first, and the composition of the 
 residue in the retort will tend towards the component which 
 was present in excess in the initial mixture. In the case of 
 propyl alcohol and water, for example, the mixture which has 
 the highest vapour tension contains from 70 to 80 per cent, 
 alcohol (the maximum being very flat) ; a mixture of this com- 
 position would boil at constant temperature, whilst for one con- 
 taining more water, some of the latter would finally remain in 
 the retort. 
 
 Case (c). In this case the composition of the vapour, and 
 therefore the composition of the liquid remaining, alter continu- 
 ously on distillation. The vapour, and therefore the distillate, 
 will contain the more volatile liquid, A, in greater proportion, and 
 the residue excess of the less volatile liquid, a partial separation 
 
OUTLINES OF PHYSICAL CHEMISTRY 
 
 being thus effected. If the distillate rich in A is again distilled 
 a mixture still richer in A is at first given off, and the process 
 may be repeated again and again. The more or less complete 
 separation of liquids by this method is termed fractional dis- 
 tillation. 
 
 It was long thought that constant-boiling mixtures were 
 definite chemical compounds of the two components for 
 example, HC1, 8H 2 O in the case of hydrochloric acid and 
 water, but this view was shown by Roscoe to be untenable. 
 He found that the composition of the mixtures does not 
 correspond with simple molecular proportions of the com- 
 ponents, and, further, that the composition alters con- 
 tinuously with alteration of the pressure under which the 
 distillation is conducted, which is not likely to be the case if 
 definite chemical compounds are present. 
 
 Distillation of Ncn-Miscible or Partially Miscible 
 Liquids. Steam Distillation If two immiscible liquids are 
 
 distilled from the same 
 vessel, since one does 
 not affect the vapour 
 pressure of the other, 
 they will pass over in 
 the ratio of the vapour 
 pressures till one of them 
 is used up. The tem- 
 perature at which the 
 mixture boils is that at 
 which the sum of the 
 vapour pressures is equal 
 
 p(o/ B) Composition 
 FlO. 14. 
 
 Q zoo / B) to the superincumbent 
 pressure. The curve re- 
 presenting the relation 
 between vapour pressure and composition of the mixture is 
 therefore a straight line (UU', Fig. 14) parallel to the axis of 
 composition, PU representing the sum of the vapour pressures 
 
SOLUTIONS 91 
 
 RP and QT, of the two components at the temperature in 
 question. 
 
 These considerations are very important in connection with 
 steam- distillation. This process is usually employed for the 
 separation of substances with a high boiling-point, such as 
 aniline, and will be familiar to the student. The relative 
 volumes of steam and the vapour of the liquid which pass over 
 are in the ratio of the vapour pressures, p^ and / 2 , at the tem- 
 perature of the experiment, and the relative weights which pass 
 over are therefore in the ratio p^ : / 2 </ 2 , in which d l and d% are 
 the densities of steam and of the other vapour respectively. 
 Assuming that the system aniline-water boils at 98, at which 
 temperature the vapour pressure of water is 707 mm. (/j), the 
 partial pressure of the aniline (/ 2 ) will be 760- 707 = 53 mm., 
 and the weights of aniline and water which pass over are in 
 the ratio 53 x 93 : 707x18 or 4929 : 12,726. This shows 
 that under ideal conditions the ratio of water to aniline in the 
 distillate is approximately 2-5 : i, the comparatively small vapour 
 pressure of aniline being partially compensated for by its rela- 
 tively high molecular weight. 
 
 Finally, we consider the distillation from the same vessel 
 of partially miscible liquids. The relation between vapour 
 pressure and composition of the mixture in this case is repre- 
 sented by the curve RSST (Fig. 14), RP representing the vapour 
 pressure of the pure component A, and QT that of the other 
 component B. In general, the vapour pressure of one liquid 
 is lowered by the addition of another. It follows that the 
 vapour pressure of partially miscible liquids, as long as two 
 layers are present, will be represented by a straight line parallel 
 to the axis of composition, but lower than the line UU' repre- 
 senting the sum of the separate vapour pressures. The pressures 
 when only one phase is present are represented by RS and ST 
 respectively. 
 
 Solution of Solids in Liquids In this case the solubility 
 is always limited, and depends on the nature of the solvent 
 
92 OUTLINES OF PHYSICAL CHEMISTRY 
 
 and solute and on the temperature. As in the other cases 
 of solubility we are really dealing with an equilibrium, in this 
 case between the dissolved salt and the solid. 
 
 Two principal methods are employed for determining the 
 solubility of solids in liquids, (i) Excess of the finely-divided 
 solid is shaken continuously with a definite quantity of the 
 solvent at a definite temperature till equilibrium is attained; 
 (2) the solvent is heated with excess of the solute to a tem- 
 perature higher than that at which the solubility is to be 
 determined, and then cooled to the desired temperature in 
 contact with the solid. The second method is to be recom- 
 mended when sufficient time is allowed for the attainment of 
 equilibrium at the temperature of experiment, but with suitable 
 precautions the methods give identical results. 
 
 It is of interest to note that the solubility depends somewhat 
 upon the state of division of the solids ; the more finely divided 
 the solid the greater is the solubility. For fairly soluble salts 
 the difference is negligible, but for slightly soluble salts it may 
 be relatively very considerable. Thus Ostwald has shown that 
 the solubility in water of yellow oxide of mercury is about 14 
 per cent, greater than that of the coarser-grained red modification. 
 
 The results of solubility measurements may be expressed in 
 various ways, for example, as the number of grams of solute 
 in TOO grams of the solvent, in 100 grams of the solution, or 
 in 100 c.c. of the solution. 
 
 When a solution saturated at a high temperature is allowed 
 to cool in the complete absence of the solid solute, the excess 
 of dissolved substance may not separate (cf. p. 55), and the 
 solution is then said to be supersaturated. 
 
 Effect of Change of Temperature on the Solubility of 
 Solids in Liquids The variation of the solubility in water of 
 certain salts with change of temperature is shown in Fig. 15, 
 the solubilities being represented as ordinates and the tempera- 
 tures as abscissae. The curves are not drawn to scale, but 
 represent diagram matically the effect of change of temperature. 
 
SOLUTIONS 
 
 93 
 
 The solubility of most solids in water increases fairly rapidly 
 with the temperature, e.g., potassium nitrate; but sodium 
 chloride is only slightly more soluble in hot than in cold water. 
 Calcium hydroxide and certain other calcium salts are less 
 soluble in hot than in cold water; a solution of calcium 
 hydroxide (lime water) saturated at the ordinary temperature, 
 
 Temperature-^ 
 FIG. 15. 
 
 becomes turbid on warming. In the case of calcium sulphate, 
 the solubility increases at first with rise of temperature and then 
 diminishes, so that there is a maximum in the solubility curve, 
 as shown in the figure. 
 
 Solubility curves are usually continuous, but that representing 
 the solubility of sodium sulphate shows a distinct change of 
 direction at 33. This is owing to the fact that we are dealing 
 
94 OUTLINES OF PHYSICAL CHEMISTRY 
 
 with the solubility curves of two distinct substances. Below 
 33, the dissolved salt is in equilibrium with the solid deca- 
 hydrate Na 2 SO4, ioH 2 O, but the latter splits up into the 
 anhydrous salt and water at 33, so that the solubility curve 
 at higher temperatures is that of the anhydrous salt. That 
 this explanation of the phenomenon is correct is shown by the 
 fact that the solubility of the anhydrous salt can be determined 
 for a few degrees below 33 in the complete absence of crystals 
 of the decahydrate ; the part of the curve thus obtained, repre- 
 sented by the dotted line, is continuous with the right-hand 
 curve. 
 
 It is important to remember that the change to which the 
 break in the solubility curve of sodium sulphate is due takes 
 place in the solid phase and not in the solution. No well- 
 defined case is known of a sharp break in a curve representing 
 the variation of a physical property in a homogeneous system 
 with temperature or composition. 
 
 Relation between Solubility and Chemical Constitution 
 Very little is known as to causes influencing the mutual 
 solubility of substances and its variation with temperature. In 
 general, it may be said that substances of similar chemical com- 
 position are mutually soluble, thus the paraffins are miscible 
 with each other in all proportions, and organic compounds 
 containing the hydroxyl group are all fairly soluble in water. 
 In these cases it does not seem probable that the solubility 
 is connected with anything in the nature of ordinary chemical 
 combination. Phenomena of an apparently different nature are 
 met with in the solubility of gases in liquids, already referred 
 to (p. 83), -where it was pointed out that the solvent power is 
 in general a specific property of the solute, and to some 
 extent independent of the nature of the gas. The solubility 
 of indifferent gases appears in the first instance to depend upon 
 the relative ease with which they can be liquefied. 
 
 Solid Solutions x In general, when the temperature of a 
 dilute solution is lowered until partial solidification takes place, 
 
 1 Compare Bruni, Feste Losungen und Isomorphismus, Leipzig, 1908. 
 
SOLUTIONS 95 
 
 the solvent separates in the solid form, practically uncontami- 
 nated with the solute. When, however, a solution of iodine in 
 benzene is partially frozen, crystals containing both substances 
 separate, and when solutions of varying concentration are used, 
 there is a constant ratio between the iodine remaining in solution 
 and that in the solidified benzene. This is illustrated in the 
 accompanying table ; in the top line is given the concentration 
 C x of the iodine in the liquid benzene, in the second line the 
 iodine concentration C 2 in the solid benzene, and in the third 
 line the ratio of the concentrations in solid and liquid, which is 
 approximately constant : 
 
 Q 3-39 2-587 0-945 percent. 
 
 C 2 1-279 '9 2 5 o'S 1 ? percent. 
 
 0-377 0-358 0-336 
 
 This phenomenon exactly corresponds with Henry's law regard- 
 ing the solubility of gases in liquids (p. 82), and as the crystals 
 containing the two substances are quite homogeneous, they may 
 be regarded as a solid solution of iodine in benzene. Similar 
 phenomena have been observed for many other pairs of sub- 
 stances, more particularly for certain metals (p. 192). 
 
 Besides crystalline solid solutions, of which an example has 
 just been given, non-crystalline or amorphous solid solutions 
 are known. The hydrogen absorbed by palladium appears to 
 be in solid solution in the metal, but the phenomenon is com- 
 plicated, and is not yet thoroughly understood. Van't Hoff 
 suggests that two solid solutions are present, hydrogen dissolved 
 in palladium and palladium dissolved in solid hydrogen, corre- 
 sponding with the two liquid layers formed by phenol and 
 water. 
 
 There is evidence that in some cases the dissolved substances 
 can diffuse slowly through the solid solvent, which indicates that 
 the solute exerts osmotic pressure (p. 97). 
 
 Practical Illustrations, (a) Partial Miscibility The 
 fact that certain liquids, ,such as phenol and water, are only 
 
yb OUTLINES QF PHYSICAL CHEMISTRY 
 
 partially miscible at ordinary temperatures, but completely 
 miscible above 69, may be illustrated as follows: 5-6 grams 
 of crystals of phenol are placed in a small separating funnel, 
 and on adding a little water and shaking it will be found that 
 a clear solution is formed, mainly due to the effect of water 
 in lowering the freezing-point of phenol. On further addition 
 of water two layers will be formed, which only disappear when 
 a large excess of water has been added. If, on the other 
 hand, phenol is warmed to 75 in a test tube, and water at 
 the same temperature is gradually added, no separation into 
 two layers will occur, corresponding with the fact that the 
 critical solution temperature has been exceeded. 
 
 (b) Solubility and Temperature The solubility in water of a 
 salt such as sodium chloride may conveniently be determined 
 as follows. A large test tube is partially filled with distilled 
 water and excess of powdered sodium chloride, and partially 
 immersed in a bath kept at constant temperature. The tube is 
 closed by means of a rubber cork through which passes a glass 
 stirrer, driven by means of a motor. At intervals samples of 
 the solution are removed by means of a pipette, a weighed 
 portion of the solution evaporated to dryness, and the residue 
 weighed. When the concentration of the solution no longer 
 alters, it is saturated. Observations should be made at differ- 
 ent temperatures at intervals of 10 and the results plotted on 
 squared paper, the temperatures being plotted as abscissae, and 
 the solubility expressed as grams of the salt in 100 grams of 
 solvent, as ordinates (Fig. 15). 
 
 (c) Supersaturated Solutions A supersaturated solution of 
 sodium sulphate may be prepared by heating the salt with half 
 its weight of water in a flask till a perfectly clear solution is 
 obtained; the flask is then plugged with cotton wool and set 
 aside to cool. If a small crystal of the sulphate is added to 
 the cold solution, crystallization at once starts, but crystals not 
 isomorphous with sodium sulphate (e.g., sodium chloride 
 crystals) are not efficient in starting crystallization. 
 
CHAPTER V 
 DILUTE SOLUTIONS 
 
 General So far, we have dealt in a general way with the 
 properties of solutions, mainly on the lines of the older methods 
 of investigation, as they were practised up to about twenty 
 years ago. In 1885, however, a new turn was given to the 
 subject by the enunciation of the modern theory of solution 
 by van't Hoff. Many of the experimental facts on which this 
 theory is based had previously been established by the work ot 
 Ostwald, Raoult and others, and these results were correlated, 
 and many fresh avenues opened for investigation, by van't Hoff's 
 theory, the most important point in which is the conception of 
 osmotic pressure, which will now be considered. 
 
 It has already been shown that the laws applicable to gases 
 are very simple and that these laws are most strictly followed in 
 the rarefied condition ; on the molecular theory this is accounted 
 for by assuming that the particles are then so far apart as to 
 exert little or no mutual influence, and that the space filled by 
 the material of the particles is negligible in comparison with 
 the space occupied. Similarly, we may reasonably anticipate 
 that the laws expressing the behaviour of dissolved substances 
 will be most simple in dilute solution, in other words, when 
 one of the components (the solvent) is present in large propor- 
 tion compared with the other (the solute), and this is quite 
 borne out by the facts. 
 
 Osmotic Pressure. Semi-permeable Membranes 
 When a few drops of bromine are carefully placed by means 
 7 97 
 
9 8 OUTLINES OF PHYSICAL CHEMISTRY 
 
 of a pipette at the bottom of a jar full of hydrogen or air, and 
 the jar is covered and allowed to stand, it will be found after 
 some time that the heavy bromine vapour has distributed itself 
 uniformly throughout the confined space against the force of 
 gravity. We may say that the bromine vapour exerts a pressure 
 in virtue of which it diffuses into those parts of the confined 
 space where the pressure is less, and equilibrium is only attained 
 when the pressure is equal throughout. In an exactly similar 
 way, if a sugar solution is carefully covered with a layer of 
 water, the dissolved sugar exerts a pressure with the result that 
 it ultimately becomes uniformly distributed in the solution. 
 This pressure cannot be determined directly, since the external 
 pressure is the sum of this and the pressure of the liquid, but 
 the principle of the method used for measuring it will be under- 
 stood from its analogy with the method already described 
 for measuring the partial pressure of a gas in a mixture (p. 81). 
 In the latter case, it will always be possible to determine the 
 partial pressure of a gas, A, mixed with another gas, B, when a 
 membrane is known which allows B, but not A, to pass through. 
 Such a membrane is said to be semi-permeable, and for 
 mixture of nitrogen and hydrogen heated palladium answers 
 the purpose. It is clear that, if we could find a membrane 
 which allows water, but not dissolved sugar, to pass through, 
 the pressure exerted by the latter in solution, its so-called 
 osmotic pressure, could be measured. 
 
 Such membranes were discovered by Traube, in the course 
 of his experiments on so-called artificial vegetable cells. The 
 most suitable membrane for the purpose was found to be copper 
 ferrocyanide, and the formation of this membrane and its im- 
 permeability for certain dissolved salts may be illustrated as 
 follows. A glass tube, provided with a rubber tube and clip 
 at one end and open at the other, is partly filled by suction 
 with an aqueous solution of copper acetate (about 2 '8 per cent.) 
 and ammonium sulphate (0-5 per cent.), and the open end, 
 in which the surface of the liquid has been made parallel by 
 
DILUTE SOLUTIONS 
 
 99 
 
 B 
 
 adjustment, is then dipped carefully into a 2 -4 per cent, aqueous 
 solution of potassium ferrocyanide, containing a little barium 
 chloride, and the tube supported in that position. A thin 
 membrane of copper ferrocyanide forms across the lower end of 
 the tube, and if the experiment has been carefully performed, 
 it will be found that even after standing some hours there is 
 no white precipitate (of barium sulphate) in the lower solution, 
 showing that the membrane is impervious to ammonium sul- 
 phate. Traube tried many other membranes, but none proved 
 so efficient as copper ferrocyanide. 
 
 Measurement of Osmotic Pressure The semi-permeable 
 membranes prepared by Traube were much too weak to with 
 stand fairly large pressures, and their use 
 for actual measurements was only ren- 
 dered possible by Pfeffer's idea of deposit- 
 ing them in the walls of a porous pot A 
 (Fig. 1 6) such as are used for experiments 
 on gas diffusion. The pot is first carefully 
 washed, soaked in water for some time, 
 then nearly filled with a solution of copper 
 sulphate (2*5 grams per litre), dipped 
 nearly to the neck in a solution of potas- 
 sium ferrocyanide (2*1 grams per litre) and 
 allowed to stand for some hours. The 
 salts diffuse through the walls of the pot 
 and at their junction form a membrane of 
 copper ferrocyanide, which, since it is im- 
 permeable for the salts from which it is 
 formed, remains quite thin but is capable 
 of withstanding fairly large pressures, 
 owing to its being supported by the walls. 
 The cell is then taken out, washed, filled 
 
 i 11 rIG. ID. 
 
 with a strong solution of sugar, and closed 
 
 with a well-fitting rubber cork through which an open glass tube, 
 
 B, passes. To ensure the tightness of the apparatus, it is of 
 
 ~ Water 
 
ioo OUTLINES OF PHYSICAL CHEMISTRY 
 
 advantage, before forming the membrane, to dip the upper part 
 of the pot, to the depth of about two inches, into melted paraffin, 
 and after the pot is filled and the cork and tube placed in position, 
 the upper surface should be covered with melted paraffin. When 
 a cell thus prepared is placed in water, the water passes in, the 
 pressure inside slowly increases (as shown by the rise in level 
 of the solution in the tube), and finally reaches a point at which, 
 when the cell has been properly prepared, it remains constant 
 for days. The maximum pressure thus attained, in other words, 
 the excess of pressure which must prevail inside the cell in order 
 to prevent more water flowing in through the semi-permeable 
 membrane, is termed the osmotic pressure of the solution. 
 
 In accurate measurements, it is preferable to close the cell with 
 a cork carrying a closed manometer, containing a definite 
 volume of air confined over mercury. Dilution of the solution 
 by the entry of a large volume of water is thus avoided. This 
 arrangement was used by Pfeffer in his original experiments. 
 
 The osmotic pressures measured in this way are very con- 
 siderable thus a i per cent, solution of cane sugar exerts a 
 pressure of more than half an atmosphere, and a i per cent, 
 potassium nitrate solution over two atmospheres. 
 
 The question as to the relation between osmotic pressure and 
 concentration of the solution was investigated by Pfeffer, and 
 the results obtained for aqueous solutions of sugar and of 
 potassium nitrate at room temperature are given in the ac- 
 companying table, in which C represents the concentration of 
 the solution (in grams per ioo c.c. of solution) and P the 
 osmotic pressure (in cm. of mercury). 
 
 Cane Sugar. Potassium Nitrate. 
 
 C P P/C C P P/C 
 
 1 53*5 53*5 o <8 130*4 163 
 
 2 101-6 50-8 1-43 218-5 J 53 
 4 208-2 52-1 3-33 436-8 133 
 6 307'5 5i'3 
 
 It is clear from these results that the ratio P/C is approximately 
 
DILUTE 
 
 constant for any one solution, in other words, the osmotic pres- 
 sure of a solution is proportional to its concentration. It will 
 be observed that the ratio is somewhat less for potassium nitrate 
 at the higher pressures, a result due in part to the slight per- 
 meability of the membrane for the salt under these conditions. 
 
 It was also shown by Pfeffer that the osmotic pressure at 
 constant concentration increases with the temperature ; some 
 of the results obtained in this connection are quoted on the next 
 page. 
 
 The magnitude of the osmotic pressure observed with 
 different membranes was not quite the same, but this must 
 be ascribed to the imperfection of most of the membranes, 
 as it can be shown theoretically that the numerical value of 
 the osmotic pressure is independent of the nature of the 
 membrane, provided the latter is perfectly semi-permeable. 
 
 Yan't Hoff' s Theory of Solutions Pfeffer's investiga- 
 tions were undertaken for botanical purposes, and their great 
 general importance was only recognised in 1885 by van't Hoff, 
 who used them as the experimental basis of a new theory of 
 solution. We have already seen that osmotic pressure may be 
 regarded as in some respects analogous to gas pressure, and as 
 ihe former can be measured as described in the previous section 
 we are now in a position, following van't Hoff, to investigate 
 the relationship between osmotic pressure, volume and tem- 
 perature as has already been done for gases. 
 
 As regards the relation between osmotic pressure and volume 
 at constant temperature, we have seen in the previous section 
 that P/C is constant for any one substance, and as the concen- 
 tration is inversely as the volume in which a definite weight of 
 substance is dissolved, we obtain, by substituting i/V for C, 
 the equation PV = constant, the exact analogue for solutions 
 of Boyle's law for gases. 
 
 With reference to the relation between osmotic pressure and 
 temperature at constant concentration, van't Hoff showed from 
 Pfeffer's observations that the osmotic pressure, P, like the gas 
 
^PUT^ES pK PHYSICAL CHEMISTRY 
 
 pressure^ is proportional to the absolute temperature, T. Some 
 of the observations on which this conclusion is based are given 
 in the accompanying table : 
 
 Cane Sugar. Sodium Tartrate. 
 
 / T P t T P 
 
 14-2 287-2 51-0 13-3 285-3 9' 8 
 
 32-0 305-0 54-4 (54-6) 37-0 310-0 98-3 (98-2) 
 
 The observed values are given in the third column, with the 
 calculated values in brackets ; the agreement is within the limits 
 of experimental error. 
 
 From these two equations, PV = const, and P oc T, we can 
 derive an equation for dilute solutions corresponding to that 
 already obtained for gases (p. 26), 
 
 PV = /r, 
 
 in which P represents the osmotic pressure of a solution con- 
 taining a definite weight of a solute in the volume, V, of solution, 
 and r is a constant. 
 
 It remains to calculate the numerical value of r for a definite 
 amount, say a mol of a dissolved substance, as has already been 
 done for gases. This can readily be done from Pfeffer's observa- 
 tion that a i per cent, solution of cane sugar at o exerts an 
 osmotic pressure of 49*3 cm. of mercury. The molecular 
 weight of cane sugar is 342, the volume in which it is contained 
 34,200 c.c., the pressure is 49*3 x 13*59 gram/cm. 2 , and the 
 absolute temperature 273. Hence 
 
 r = PV = 49-3 x 13-59 x 3 . _ g 3)9OO fm gram . cm . units)> 
 
 a value which almost exactly corresponds with that obtained 
 for gases. As the same value for r is obtained for a mol of 
 other organic compounds, such as urea and glucose, we will 
 represent it by R, to indicate that it is a factor of general im- 
 portance. We have thus obtained two results of the greatest 
 importance : (i) the equation PV = RT is valid for dilute solu- 
 
DILUTE SOLUTIONS 103 
 
 tions ; (2) the numerical value of R is the same for dissolved 
 substances as for gases. The latter statement implies, as is 
 clear from the general equation, that the osmotic pressure of a 
 definite quantity of cane sugar or other substance in solution is 
 equal to the gas pressure which it would exert if it occupied the 
 same volume in the gaseous form. We may therefore say with 
 van't Hoff that " the osmotic pressure exerted by any substance 
 in solution is the same as it would exert if present as gas in the 
 same volume as that occupied by the solution, provided that the 
 solution is so dilute that the volume occupied by the solute is 
 negligible in comparison with that occupied by the solvent", 
 This statement holds for all temperatures, as is at once clear 
 from the fact that the solution obeys the gas laws. Certain 
 important exceptions to the above rule, more particularly in 
 the case of solutions which conduct the electric current, will be 
 discussed in a later chapter. 
 
 Some important consequences of the validity of the general 
 equation, PV = RT, for dissolved substances will be dealt with 
 in detail later. In particular, the molecular weight of a dis- 
 solved substance is the quantity in grams which, when dissolved 
 in 2 2 '4 litres at o, exerts an osmotic pressure of i atmosphere, 
 a definition almost exactly analogous to that for gases (p. 36). 
 The same fact may be expressed somewhat differently as fol- 
 lows : Quantities of different substances in the ratio of their 
 molecular weights, when dissolved in equal volumes of the 
 same solvent, exert the same osmotic pressure. 
 
 So far, we have considered only the experimental basis of the 
 theory of solution. It has, however, been shown theoretically 
 by van't Hoff, by thermodynamical reasoning, that the osmotic 
 pressure and gas pressure must have the same absolute value, 
 if the solution is sufficiently dilute, and this conclusion has 
 been confirmed by Lord Rayleigh and by Larmor, among 
 others. The latter writer puts the matter as follows : " The 
 change of available energy on further dilution, with which 
 alone we are concerned in the transformations of dilute solu- 
 
^YSICAL CHEMISTRY 
 
 tions [(/". p. 150], depends only on the further separation of 
 the particles ... and so is a function only of the number 
 of dissolved molecules per unit volume and of the temperature, 
 and is, per molecule, entirely independent of their constitution 
 and that of the medium," 1 the assumption being made that 
 the particles are so far apart that their mutual influence is 
 negligible. " The change of available energy " is thus brought 
 into exact correlation with that which occurs in the expansion 
 of a gas. 
 
 Recent direct Measurements of Osmotic Pressure 
 It is a remarkable fact that, although Pfeffer's osmotic pres- 
 sure measurements were made as early as 1877, the degree of 
 accuracy attained by him has not been improved upon until 
 quite recently. Accurate measurements are, however, very 
 desirable, because although the relation between osmotic pres- 
 sure and concentration can be calculated from the gas laws in 
 dilute solution, there is still much uncertainty as to how far the 
 gas laws are applicable, or what is the exact relationship between 
 osmotic pressure and concentration, in concentrated solutions. 
 In particular, it is, or was until quite recently, uncertain whether 
 V in the general equation, PV = RT, should represent the volume 
 of the solvent or that of the solution. This uncertainty has 
 been to some extent removed by the very careful measurements 
 carried out by Morse and Frazer 2 since 1903 by Pfeffer's method 
 with slight modifications. Their results show that, if V in the 
 general equation be taken as the volume of the solvent, aqueous 
 solutions of cane sugar approximately follow the gas laws up to 
 a concentration of 342 grams of the solute in 1000 grams of 
 water. A few of their results, illustrating the above statement, 
 are given in the accompanying table; the numbers under 
 "gaseous" are calculated on the assumption that the sub- 
 stance as gas occupies the same volume as the solvent in the 
 solution. 
 
 1 Larmor, Encyc. Britannica, loth ed., vol. xxviii., p. 170. 
 
 *Amer. Chem. J., 1905,34, i; 1906, 36, i, 39; 1007, 37, 324, 425, 
 558; 38, 175; 1908, 39, 667; 40, i, 194; 1909, 4*1 * 2571 *9"i 4S 554. 
 etc. 
 
DILUTE SOLUTIONS 
 
 105 
 
 Concentration of Solution. 
 
 Pressure at constant temp. Ratio of 
 
 A 
 
 
 (20) (atmospheres). osn 
 
 lotic pres- 
 re to gas 
 ressure. 
 
 Mols per 1000 grams 
 of water. 
 
 Mols per litreU! 
 solution. 
 
 Gaseous. 
 
 Osmotic. p 
 
 O'lO 
 
 0-09794 
 
 2'39 
 
 2-522 
 
 055 
 
 0*20 
 
 OT9I92 
 
 4-78 
 
 5^23 
 
 051 
 
 0-40 
 
 0-36886 
 
 9-56 
 
 9-96 
 
 038 
 
 o - 6o 
 
 0-532S2 
 
 J 4'34 
 
 15-20 
 
 060 
 
 0-80 
 
 0-68428 
 
 19-12 
 
 20"6o 
 
 077 
 
 I '00 
 
 0-82534 
 
 23-90 
 
 26-12 
 
 093 
 
 The table shows that only when concentrations are referred to 
 a definite weight (or volume) of solvent is there proportionality 
 between concentration and osmotic pressure ; if they are re- 
 ferred to a constant volume of solution ( column 2) the osmotic 
 pressure increases faster than the concentration. The numbers 
 in the fifth column show, however, that the osmotic pressure 
 at 20 is on the average about 6 per cent, greater than the 
 gas pressure. Curiously enough, the ratio is about the same 
 at all temperatures from o to 25. 
 
 Other Methods of Determining Osmotic Pressure The 
 difficulties inherent in the direct determination of osmotic 
 pressure can often be avoided by determining it indirectly 
 by comparison with a solution of known osmotic pressure. 
 Solutions which have the same osmotic pressure are said to 
 be iso tonic or isosmotic. 
 
 One such method, used by de Vries, depends on the use 
 of plant cells as semi-permeable membranes. The protoplasmic 
 layer which surrounds the cell-sap is permeable to water, but 
 impermeable to many substances dissolved in the cell-sap, such 
 as glucose and potassium malate. If such a cell is placed in 
 contact with a solution of higher osmotic pressure than the 
 cell-sap, water is withdrawn from the cell (just as a sugar 
 solution absorbs water through a semi-permeable membrane) 
 and the protoplasm shrinks away from the cell-wall ; a pheno- 
 menon which is termed plasmolysis. If, however, the solution 
 has a smaller osmotic pressure than that of the cell-sap, water 
 enters the cell, the protoplasm expands and lines the cell-wall. 
 The behaviour of the protoplasm, especially if coloured, can be 
 1 Zeitsch. physikal. Chem., 1888, 2, 415. 
 
106 OUTLINES OF PHYSICAL CHEMISTRY 
 
 followed under the microscope, and by trial a solution can be 
 found which has comparatively little effect upon the appearance 
 of the cell, and is therefore isotonic with the cell contents. 
 
 A method depending on the same principle, in which red 
 blood corpuscles are used instead of vegetable cells, has been 
 described by Hamburger, 1 and the cell-walls of bacteria may 
 also be used as semi-permeable membranes. 
 
 The following table contains some " isotonic coefficients " as 
 given by de Vries and by Hamburger ; the numbers represent 
 the ratio of the osmotic pressures of equimokcular or equimolar 
 solutions of the compounds mentioned. 
 
 Isotonic Coefficients. 
 
 Plasmolytic With Red Blood 
 
 Substance. Method. Corpuscles. 
 
 Cane sugar . . . 1-81 172 
 
 Potassium nitrate . . 3-0 3-0 
 
 Sodium chloride . .3*0 3-0 
 
 Calcium chloride . 4'33 4*05 
 
 It will be observed that, although the results obtained by the 
 two methods agree fairly well, the osmotic pressures for equi- 
 valent solutions are not equal, as would be expected according 
 to Avogadro's hypothesis. The deviation from the expected 
 result is such that potassium nitrate, for example, exerts an 
 osmotic pressure about 17 times greater than that due to an 
 equimolar solution of cane sugar. This observation is of 
 fundamental importance in connection with modern views as 
 to salt solutions. 
 
 The Mechanism of Osmotic Pressure The foregoing 
 considerations are quite independent of any hypothesis as to 
 the exact nature of osmotic pressure, and so far no general 
 agreement has been reached on this point. Van't Hoff inclines 
 to the view that the pressure is to be accounted for on kinetic 
 grounds, 2 perhaps as being due to the bombardment of the 
 
 1 Zeitsch. physikal. Chem., 1890, 6, 319. 
 
 2 For a discussion between van't Hoff and Lothar Meyer on this point, 
 see Zeitsch. physikal. Chem. t 1890, 5, 23, 174. 
 
DILUTE SOLUTIONS 
 
 107 
 
 walls of the vessel by solute particles, in the same way as the 
 pressure of a gas is produced according to the kinetic theory, 
 and the fact that the osmotic pressure is proportional to the 
 absolute temperature appears rather to support this suggestion. 
 Other views are that it is connected with attraction between 
 solvent and solute, or perhaps with surface tension effects. It 
 may be pointed out that the equivalence of osmotic pressure 
 and gas pressure in great dilution is no evidence that they arise 
 from the same cause. 
 
 As regards semi -permeable membranes, their efficiency does 
 not depend, as might at first sight be sup- 
 posed, on anything in the nature of a 
 sieve action, only the smaller molecules 
 being allowed to pass, but rather upon a 
 difference in their solvent power for the two 
 components of the mixture. The action 
 of the palladium in Ramsay's experiment 
 (p. 81) is very probably to be accounted 
 for in this way, and that the same is true 
 for solutions is well illustrated by an in- 
 structive experiment due to Nernst and 
 illustrated in Fig. 17. The wide cylin- 
 drical glass tube A is closed at the bottom 
 with an animal membrane (bladder) which 
 has previously been thoroughly soaked in 
 water; it is then filled with a, mixture 
 of ether and benzene and fitted with a 
 
 
 FIG. 17. 
 
 cork and narrow tube, as shown in the figure. The cell is 
 supported on a piece of wire gauze in a beaker partly filled with 
 moist ether and loosely closed by a cork, B. After a time it 
 will be observed that the liquid has risen to a considerable 
 height in the narrow tube. What occurs in this case is that 
 the ether dissolves in the water with which the membrane is 
 soaked, and in this way is transferred inside the cell, whilst 
 the benzene, being insoluble in water, is unable to pass out. 
 
loS OUTLINES OF PHYSICAL CHEMISTRY 
 
 Similarly, the efficiency of the copper ferrocyanide membrane 
 may depend on its solvent power for water but not for 
 sugar. 
 
 Osmotic Pressure and Diffusion It has already been 
 pointed out that there is a close connection between osmotic 
 pressure, as defined above, and diffusion; it is the difference 
 in the osmotic pressure of cane sugar in different parts of a 
 system which causes it in time to be uniformly distributed 
 through that system. The diffusion of dissolved substances 
 was very fully investigated by Graham, but the general law of 
 diffusion was first enunciated by Fick. Fick's law is comprised 
 in the equation 
 
 ds = - DA^<#, 
 ax 
 
 which states that the amount of solute, ds, which passes through 
 the cross-section of a diffusion cylinder is proportional to the 
 area, A, of the cross-section, to the difference of concentration, 
 dc t at two points at a distance dx from one another, to the time, 
 dt t and to a constant, D, characteristic for the substance, and 
 termed the diffusion-constant. As an illustration of the above 
 formula, it was found that when dc=- 1 gram per c.c., dx=i cm., 
 A = i sq. cm. and dt= one day, that 0-75 grams of sodium chloride 
 passed between the two surfaces. This is excessively slow, in 
 comparison with the high osmotic pressures set up even by 
 dilute solutions, and the explanation is to be found in the great 
 friction due to the smallness of the particles. As the driving 
 force the osmotic pressure and the rate of diffusion are 
 known, the resistance to the movement of the particles can be 
 obtained. It has been calculated that the enormous force of 
 four million tons weight is needed to force i gram mol of cane 
 sugar through water at a velocity of i cm. per sec. The more 
 rapid diffusion in gases may plausibly be ascribed to the much 
 smaller resistance to the movement of the particles. 
 
 The rate of diffusion is much influenced by temperature 
 and, curiously enough, to about the same extent for all solutes 
 
DILUTE SOLUTIONS 109 
 
 the average increase is about -fa of the value at 1 8 for every 
 degree C. 
 
 MOLECULAR WEIGHT OF DISSOLVED SUBSTANCES. 
 
 General It has already been pointed out (p. 103) that since 
 Avogadro's hypothesis is valid for solutions, the molecular weight 
 of a dissolved substance can readily be calculated when the 
 osmotic pressure exerted by a solution of known concentration 
 at known temperature and pressure is known. An illustration of 
 this is given on the next page. As, however, the direct measure- 
 ment of osmotic pressure is a matter of considerable difficulty, 
 it has been found more convenient for the purpose to measure 
 other properties of solutions, the relationship of which to the 
 osmotic pressure is known. The only three methods which can 
 be dealt with here are : 
 
 (1) The lowering of vapour pressure ; 
 
 (2) The elevation of boiling-point ; 
 
 (3) The lowering of freezing-point, 
 
 brought about by adding a known weight of solute to a known 
 weight or volume of solvent. 
 
 It can be shown by thermodynamical reasoning (p. 131) that un- 
 der certain conditions the lowering of vapour pressure, the eleva- 
 tion of the boiling-point and the lowering of the freezing-point due 
 to the addition of a definite quantity of solute to a definite volume 
 of solvent are each proportional to the osmotic pressure of the 
 solution. Further, the equations expressing the exact relation- 
 ships between these three factors and the osmotic pressure have 
 also been established, 1 and all these theoretical deductions have 
 been fully confirmed by experiment. // follows that just as 
 equimolecular quantities of different substances in equal volumes 
 of the same solvent exert the same osmotic pressure, so equimole- 
 cular quantities of different substances in equal volumes of the 
 same solvent raise the boiling-point) lower the freezing-point, and 
 lower the vapour tension to the same extent. These statements 
 find a very simple representation on the molecular theory. Since 
 1 Appendix, pp. 131-138. 
 
no OUTLINES OF PHYSICAL CHEMISTRY 
 
 equimolecular quantities of different substances contain the 
 same number of molecules, it follows that the magnitude of 
 the osmotic pressure, lowering of vapour pressure, etc., depends 
 only on the number of particles present and is independent of 
 their nature (colligative properties, p. 63). 
 
 The molecular weight of the solute could, of course, be 
 obtained by determining one of the factors (i), (2), (3) and 
 then calculating the value of the osmotic pressure, but it is 
 much simpler to obtain the molecular weight by comparison 
 with a substance of known molecular weight. 
 
 It may be mentioned that besides the methods just indicated, 
 there are other analogous methods for determining molecular 
 weights which, from considerations of space, cannot be referred 
 to here. Nernst has pointed out that any process involving the 
 separation of solvent and solute can be used for determining 
 molecular weights, and a little consideration will show that 
 the four methods just mentioned come under this heading. 
 Moreover, the osmotic effect of the solute is to diminish tha 
 readiness with which part of the solvent may be separated from 
 the solution, and the effect of the solute on the boiling- and 
 freezing-points of the solvent must therefore be in the direction 
 already indicated. The mathematical proof of the connection 
 between these four properties depends upon the equivalence of 
 the work done in removing part of the solvent from the solu- 
 tion (Appendix). 
 
 The four different methods for determining molecular weights 
 in solution and the general nature of the results obtained will 
 now be considered in detail. 
 
 Molecular Weights from Osmotic Pressure Measure- 
 ments, (a) from absolute values of the osmotic pressure The 
 principle of this method has already been discussed (p. 103). 
 If g grams of substance, dissolved in v c.c. of solvent, gives an 
 osmotic pressure of/ cm. at T abs., the molecular weight, 
 m, will be that quantity which, when present in 22,400 c.c. of 
 solvent, will give an osmotic pressure of 76 cm. Hence, since 
 is proportional to the amount of substance used (p. 26), 
 
DILUTE SOLUTIONS in 
 
 pv 22,400 x 76 
 
 g x 22,400 x 76 x (273 + f) 
 
 or /;/ = - 7 - 
 273/2? 
 
 As an example, we will take an experiment of Morse and Frazer (p. 105) 
 
 in which a solution containing 34-3 grams of sugar in 1000 c.c. (really 
 
 1000 grams) of water gave an osmotic pressure of 2*522 atmospheres = 
 
 191*6 cm. at 20. Hence 
 
 m = 34-2 x 22,400 x 76 x 293 = 6 . Q 
 273 x 191*6 x 1000 
 
 as compared with the theoretical value of 342. 
 
 Alternatively, by formula (2) p. 36 
 
 m = g RT -342 x 0-08205 x 293 = 6 
 
 pv 2*522 x i 
 
 (b] By comparison of osmotic pressures Since equimolecular 
 
 solutions in the same solvent have the same osmotic pressure, 
 
 it is only necessary to find the strengths of two solutions which 
 are in osmotic equilibrium (isotonic), and if the molecular weight 
 
 of one solute is known that of the other can be calculated; 
 
 De Vries found that a 3*42 per cent, solution of cane sugal 
 was isotonic with a 5*96 per cent, solution of raffinose, the 
 molecular weight of which was then unknown. If it be re 
 presented by x, then 3-42 : 5-96 : : 342 : : x, whence # = 596. 
 This result has since been confirmed by chemical methods. 
 
 Lowering of Yapour Pressure It has long been known 
 that the vapour pressure of a liquid is diminished when a non- 
 volatile substance is dissolved in it, and that the diminution is 
 proportional to the amount of solute added. In 1887 Raoult, 
 on the basis of a large amount of experimental work, established 
 the following important rule : Equimolecular quantities of differ- 
 ent substances, dissolved in equal volumes of the same solvent, 
 lower the vapour pressure to the same extent. On comparing 
 the relative lowering (i.e., the ratio of the observed lowering and 
 the original pressure) in different solvents, the same observer 
 discovered another important rule, which may be expressed as 
 follows : The relative lowering of vapour pressure is equal to the 
 ratio of the number of molecules of solute and the total number of 
 
ii2 OUTLINES OF PHYSICAL CHEMISTRY 
 
 molecules in the solution. Putting p^ and p 2 for the vapour 
 pressures of solvent and solution respectively, the rule may be 
 put in the form 
 
 A -A = 
 p l N + n 
 
 in which n and N represent the number of molecules of solute 
 and solvent respectively, In order to illustrate the validity of 
 this rule, some results given by Raoult are quoted in the ac- 
 companying table; the relative lowering is that due to the 
 addition of i mol of solute to 100 mols of the various sol- 
 vents : 
 
 Solvent. H 2 O PC1 3 CS 2 CC1 4 CH 3 I (C 2 H 5 ) 2 O CH 3 OH 
 
 Relative lowering croio2 0*0108 0*0105 0*0105 0-0105 0-0096 0*0103 
 
 The results agree excellently among themselves, and fairly well 
 with the calculated value, i/ioi = 0*0099. 
 
 About the same time van't Hoff introduced the conception 
 of osmotic pressure, and showed by a thermodynamical method 
 that the relation between the relative lowering of vapour pres- 
 sure and the osmotic pressure is given by the equation 
 
 _M. P , r v 
 
 jRT' 
 
 where M = molecular weight of solvent, in the form of vapour, s 
 is its density, and the other symbols have their usual significance. 
 The expression M/^RT is therefore constant, since it depends 
 only on the nature of the solvent, and consequently the relative 
 lowering of vapour-pressure is proportional to the osmotic pres- 
 sure P. By using the general equation, PV = RT (where n is 
 the number of mols of solute), P in equation (i) can be elimin- 
 ated, 1 and we finally obtain 
 
 i P -s -y , where V is the volume of the solvent. If N represents 
 
 the number of mols of solvent, M its molecular weight and s its 
 
 * *fT?npc 
 
 density, the volume V of the solvent = MN/s. Hence P = -j^f ' and 
 
 when this value is substituted in equation (i) we obtain (pi~Pz)IPi = n l^' 
 
DILUTE SOLUTIONS 113 
 
 A "A SL 
 
 A = N- 
 
 This equation differs from that of Kaoult in that the de- 
 nominator on the right-hand side is N instead of N + n % but 
 they become identical " at infinite dilution " when the volume 
 of the solute is negligible in comparison with that of the solvent. 
 By substituting for n and N gjm and W/M, where g and W 
 are the weights of solute and solvent respectively, m is the 
 (unknown) molecular weight of the solute and M that of the 
 solvent in the form of vapour, we obtain the equation 
 
 which enables us to calculate the molecular weight of a dissolved 
 substance when the relative lowering produced by a known 
 weight of solute in a known weight of solvent is known. As 
 an illustration, an experiment of Smits may be quoted. He 
 found that at o the lowering of vapour pressure produced by 
 adding 29-0358 grams of sugar to 1000 grams of water is 
 0-00705 mm., the vapour pressure of water at that temperature 
 being 4*62 mm. Hence 
 
 0-0070=1 20-0^8 x 18 
 
 --^ = y ^ , and m = 342, 
 
 4-02 looom 
 
 in exact agreement with the theoretical value. 
 
 As the lowering of vapour pressure is very small and not 
 very easy to determine accurately by a statical method, it has 
 not been very largely used for molecular weight determinations, 
 the closely allied method depending on the elevation of the 
 boiling-point being preferred. It has, however, one great 
 advantage, inasmuch as, unlike the boiling-point and freezing- 
 point methods, it can be used for the same solution at widely 
 different temperatures. For this purpose, a dynamical method 
 suggested by Ostwald and worked out by Walker l has certain 
 advantages. A current of air is drawn in succession through 
 (i) a set of Liebig's bulbs containing the solution of vapour 
 
 1 Zeitsch. physikal Chern., 1888, 2, 602. 
 8 
 
ti4 ' OUTLINES OF PHYSICAL CHEMISTRY 
 
 pressure / 2 (2) similar bulbs containing the pure solvent vapoui 
 pressure p l (3) a U-tube containing concentrated sulphuric 
 acid. In the first set of bulbs it becomes saturated up to 
 / 2 with the vapour of the solvent, in the second set up to/^ in 
 the U-tube the moisture is completely absorbed. The loss of 
 weight in the second set of bulbs is proportional to p^ -/.,, and 
 the gain in the U-tube to/ x (cf. p. Qi). 1 
 
 Elevation of Boiling-point A little consideration shows 
 -that there is a close connection between this method of deter- 
 mining molecular weights and that depending on the lowering 
 of vapour pressure. A liquid boils when its vapour pressure 
 is equal to that of the atmosphere. The presence of a solute 
 lowers the vapour pressure, and to reach the same pressure 
 as before we require to raise the temperature a little; it is 
 evident that, to a first approximation, this elevation must be 
 proportional to the lowering of vapour pressure. It follows 
 that, in this case also, equimolecular quantities of different 
 solutes, in equal volumes of the same solvent, raise the boiling- 
 point to the same extent. The molecular weight of any soluble 
 substance may therefore be found by comparing its effect on 
 the boiling-point of a solvent with that of a substance of known 
 molecular weight. 
 
 For this purpose, it is convenient to determine the molecular 
 elevation constant, K, for each solvent, that is, the elevation of 
 boiling-point which would be produced by dissolving a mol of 
 any substance in 100 grams or 100 c.c. of the solvent. Actually, 
 of course, the elevation is determined in fairly dilute solution, 
 and the value of the constant calculated on the assumption that 
 the rise of boiling-point is proportional to the concentration. 
 Then the weight in grams of any other compound which, when 
 dissolved in 100 grams or 100 c.c. of the solvent, produces a 
 rise of K degrees in the boiling-point' is the molecular weight. 
 
 If g grams of substance, of unknown molecular weight, m, 
 dissolved in L grams of solvent raises the boiling-point 8 
 
 1 A modification of this method has recently been used by Lord Berkeley 
 and Hartley for the indirect determination of the osmotic pressure of con- 
 centrated solutions of cane sugar (Proc. Roy. Soc. t 1906, 77A, 156. 
 
DILUTE SOLUTIONS 115 
 
 degrees, whilst m grams in 100 grams of solvent give a rise of 
 K degrees, it follows, since loog/L is the number of grams 
 of substance in 100 grams of solvent, that 
 
 loop- , ioo,e-K 
 
 T e : 8 : : m : : K, whence m = =r-f . 
 L Lo 
 
 In the course of the last few years, the constants for 100 grams 
 and 100 c.c. have been very carefully determined for a large 
 number of solvents, and some of the more important data are 
 given in the accompanying table : 
 
 Solvent. 
 
 muiecuiar .c-ievc 
 100 grams. 
 
 Ltion v^onsii 
 100 C.C. 
 
 Water 
 
 5*2 
 
 5-4 
 
 Alcohol 
 
 II'5 
 
 15-6 
 
 Ether 
 
 2I'O 
 
 30-3 
 
 Acetone 
 
 . . 167 
 
 22'2 
 
 Benzene . 
 
 267 
 
 32-8 
 
 Chloroform 
 
 39' 
 
 27-7 
 
 Van't Hoff has shown that these constants, some of which 
 had previously been obtained empirically by Raoult, can be 
 calculated from the latent heat of vaporization, H, per gram of 
 solvent, and its boiling-point, T, on the absolute scale, by 
 means of the formula 
 
 0-02T 2 
 
 -IT- 
 
 As an example, the calculated value for the molecular elevation 
 constant for water, the latent heat of vaporization of which at 
 its boiling-point is 537 calories, is 
 
 K = (0-02 x (373) 2 )/S37 = 5' 2 
 
 in satisfactory agreement with the experimental value. For all 
 solvents which have been carefully investigated, the experi- 
 mental and calculated values are in good agreement. 1 
 
 1 The observed and calculated values for a large number of solvents 
 are given in Landolt and Bornstein's tables. 
 
n6 OUTLINES OF PHYSICAL CHEMISTRY 
 
 Experimental Determination of Molecular Weights 
 by the Boiling-point Method The ease and certainty 
 with which such determinations can now be made is largely 
 due to the work of Beckmann. One of the methods sug- 
 gested by him will first be considered, and then a method 
 due to Landsberger, based on a different principle. 
 
 (a) Beckmann s Method The apparatus used is represented 
 in Fig. 1 8. The boiling-tube, A, is provided with two side 
 tubes, /j, by means of which the solute (solid or liquid) is 
 introduced, and / 2 , which is connected to a small condenser, 
 by the action of which the amount of solvent is kept fairly 
 constant. The solution is made to boil by the heat from a 
 small screened burner, B, which can be carefully regulated, 
 and the boiling liquid is insulated by means of an air jacket 
 between the outer cylindrical glass tube, G, and the boiling 
 tube. As the temperature of the vapour which escapes from 
 a boiling solution is little, if any, above the boiling-point of the 
 pure solvent, it is necessary to place the thermometer in the 
 boiling liquid so that the bulb is completely immersed. The 
 liquid tends to become superheated, and to eliminate this source 
 of error Beckmann recommends filling up the boiling-tube 
 nearly to the level of the liquid with glass beads or garnets, or, 
 still better, with platinum tetrahedra. The thermometer repre- 
 sented in the figure, which was specially designed by Beck- 
 mann for this work, has a large bulb and an open scale, 
 covering only 5-6, and graduated in T ^. To render the 
 thermometer available for widely different temperatures, there 
 is an arrangement by means of which the amount of mercury 
 in the bulb can be so adjusted that the top of the thread can 
 be brought on the scale at any desired temperature. The sol- 
 vent, of which 10 to 15 grams is usually sufficient, is measured 
 with a pipette, or weighed by difference in the boiling-tube 
 itself ; the solute, if solid, may be conveniently introduced in 
 the form of a compressed pastille or, if liquid, by means of a 
 bent pipette. The boiling-point of the solvent is determined 
 
DILUTE SOLUTIONS 
 
 117 
 
 by causing it to boil fairly vigorously, and the temperature should 
 remain constant within o-oi - 0-015 foi about twenty minutes 
 
 FIG. 18. 
 
 while readings are being taken. The temperature is then allowed 
 to fall several degrees by removing the source of heat, the solute 
 
n8 OUTLINES OF PHYSICAL CHEMISTRY 
 
 rapidly introduced, the boiling-point again determined, a fresh 
 quantity of solute introduced, the boiling-point re-determined, 
 and so on. The thermometer should be tapped before each 
 reading. The amount of solute added may conveniently be 
 such that the boiling-point is raised 0-15- 0-2 after each addi- 
 tion. It may be pointed out that more satisfactory results are 
 usually obtained when differences produced by the addition of 
 more solute are used in the calculation than when differences 
 in the boiling-point of solvent and solution are used. 
 
 As an illustration of the calculation of the results, an. experi- 
 ment with camphor in ethyl alcohol may be quoted. The 
 addition of 0*56 grams of camphor to 16 grams of the solvent 
 raised its boiling-point 0-278. Hence 
 
 loo^K 100 x 0^56 x 1 1*5 
 ~TS~ 16 x 0-278 
 
 the theoretical value for C 10 H 16 O being 142. 
 
 With proper precautions, the results obtained by this method 
 are accurate within 3-4 per cent. 
 
 (b) Landsberger s Method This method depends upon the 
 fact that a solution can be heated to its boiling-point by passing 
 into it a stream of the vapour of the boiling solvent. In this 
 case there is little or no risk of superheating, as the temperature 
 of the vapour is lower than the boiling-point of the solution. 
 The boiling-point of the solvent is first determined by passing in 
 vapour till the temperature ceases to rise, some of the solute 
 is then added, and more vapour passed in until the boiling- 
 point of the solution is reached. As, during the heating, the 
 amount of solvent increases by condensation of vapour, the 
 final amount of solution, upon which of course the observed 
 boiling-point depends, is obtained by weighing after the experi- 
 ment. If no great accuracy is required, the final volume may 
 be read off in the boiling-tube, graduated for the purpose. 
 Radiation may be minimised by jacketing the inner tube with 
 the vapour of the boiling solvent. 
 
DILUTE SOLUTIONS 119 
 
 Depression of the Freezing-point This is the most 
 accurate and most largely employed method for the deter- 
 mination of molecular weights in solution. The two necessary 
 conditions for its applicability are (i) the pure solvent, free from 
 any of the solute, must separate out when the freezing-point is 
 reached; (2) only a little of the solvent must have separated 
 when the measurement is taken, otherwise the concentration 
 of the solution will be appreciably altered. As in solubility 
 determinations, we are dealing with an equilibrium (p. 84) in 
 this case between ice and solution, and the experimental fact 
 is that the more concentrated the solution the lower is the tem- 
 perature at which equilibrium is reached. It is thus evident 
 that if a large amount of the solvent separates in the solid form, 
 the observed freezing-point is the temperature of equilibrium 
 with a more concentrated solution than that originally prepared. 
 
 In this case also, the osmotic pressure, and hence the mole- 
 cular weight, could be calculated from the formula connecting 
 osmotic pressure and depression of the freezing-point (p. 138), but 
 the comparison method is always used. Just as for the boiling- 
 point (p. 114) the molecular freezing-point depression , i.e., the 
 depression produced by dissolving i mol of solute in 100 grams 
 or 100 c.c. of the solvent, has been determined for a large 
 number of solvents, and some of the more important data are 
 given in the accompanying table. 
 
 Molecular Depression. 
 Solvent. 100 grams. 100 c.c. 
 
 Water . .18-5 18-5 
 
 Benzol . . 50 56 
 
 Acetic acid -39 41 
 
 Phenol . . 74 
 
 Naphthalene .69 
 
 The molecular depression, K, can be calculated from the 
 latent heat of fusion, H, of the solvent and its freezing point 
 on the absolute scale by means of the expression. 
 
120 OUTLINES OF PHYSICAL CHEMISTRY 
 
 0-02T 2 
 
 K 
 
 H 
 
 analogous to that which holds for the boiling-point elevation. 
 Thus for water we have K = (0-02 x (273) 2 )/8o = i8'6. 
 
 It may be mentioned, as a matter of historical interest, that 
 the experimental values for K obtained with solutions of cane 
 sugar by Raoult, Jones and others, were at first much greater 
 than 1 8-6, but the careful experiments of Abegg, Loomis, 
 Wildermann, and later of Raoult himself made it clear that 
 the high values previously obtained were due to experimental 
 error, and that, with proper precautions, the value of K deduced 
 on the basis of the theory of solution was fully confirmed by 
 experiment. 
 
 If g grams of solute, in L grams of solvent, caused a depres- 
 sion, A, of the freezing-point of the solvent, the molecular weight 
 of the solute can be calculated from the formula 
 
 loqg-K 
 
 which exactly corresponds with that already given for elevation 
 of boiling-point. 
 
 Experimental Determination of Molecular Weights 
 by the Freezing-point Method The apparatus which is 
 used almost exclusively for this purpose was also designed by 
 Beckmann, and is shown in Fig. 19. The inner tube, A, which 
 contains the solvent, has a side tube by which the solute may 
 be introduced, and is provided with a Beckmann thermometer, 
 D, and a stirrer, preferably of platinum. The remainder of the 
 apparatus consists of a tube, B, rather wider than A, and fitted 
 into the loose cover of the large beaker, C, which contains water 
 or a freezing-mixture (ice, or ice and salt), the temperature of 
 which is 2-3 below the freezing-point of the solvent. 
 
 In making an experiment, 15-20 grams of the solvent are 
 weighed or measured into the tube, A, the stirrer and ther- 
 mometer are put in place, and A is then placed in the wider 
 
DILUTE SOLUTIONS 
 
 121 
 
 tube B, which acts as an air mantle. The liquid is then stirred 
 
 continuously and the thermometer observed. Owing to super- 
 
 cooling, the temperature falls below the freezing-point of the 
 
 solvent, but as soon as solid begins to 
 
 separate, it rises rapidly, owing to the 
 
 latent heat set free, and the highest 
 
 temperature observed is taken as the 
 
 freezing-point of the solvent. The tube 
 
 is then removed from the bath, the solid 
 
 allowed to melt, a weighed amount of the 
 
 solute added, and the determination of the 
 
 freezing-point repeated. A further por- 
 
 tion of solute may then be added, and 
 
 another reading taken. With some sol- 
 
 vents there is considerable supercooling, 
 
 and as this would be a source of error 
 
 owing to separation of much solvent when 
 
 solidification finally occurs, a small par- 
 
 ticle of solid solvent is added to start 
 
 solidification when the temperature has 
 
 fallen 1-2 below the freezing-point. 
 
 As an illustration of the calculation of 
 the results, an experiment with napthalene 
 in benzene may be quoted. The addition 
 of 0*142 grams of the compound to 
 20*25 grams of the solvent lowered the 
 freezing-point 0*2 84. Hence 
 
 IPO x 0*142 x 51*2 
 
 126 
 
 AL 0*284 x 20*25 
 
 as compared with the theoretical value Fio. 19* 
 
 128. 
 
 Results of Molecular Weight Determinations in Solu- 
 tion. General The most important result of the numerous 
 molecular weight determinations of dissolved substances which 
 
122 OUTLINES OF PHYSICAL CHEMISTRY 
 
 have been made in recent years is that in general the molecular 
 weight in dilute solution is the same as that deduced from the simple 
 chemical formula of the solute, as based on vapour density deter- 
 minations or on its chemical behaviour. For example, the 
 empirical formula of naphthalene is C 5 H 4 , and since one-eighth 
 of the hydrogen can be replaced, the simplest chemical formula 
 must be C 10 H 8 , and the molecular weight 128. Cryoscopic 
 determinations in benzene gave a value 126, so that naphthalene 
 is present as simple molecules in solution. 
 
 The Van't Hoff-Raoult formulae (p. 112) on which the 
 determination of molecular weights in solution depend, have 
 been deduced on certain assumptions which hold only for dilute 
 solutions, and it is of the utmost importance to bear in mind 
 that there is no a priori reason why they should give trust- 
 worthy results for concentrated solutions. The question as to 
 how far the gas laws hold for concentrated solution, or what 
 modifications are necessary, has been much debated, but so far 
 no definite conclusions have been arrived at. It is mainly a 
 matter for further experiment. It has already been shown 
 (p. 105) that when V in the general formula is taken as the 
 volume of the solvent, the normal molecular weight is obtained 
 for cane sugar up to very high concentrations on the assumption 
 that the gas laws are valid for these solutions. The same is 
 true for other compounds, more particularly in organic solvents, 
 as may be illustrated by the values obtained by Beckmann for 
 camphor in benzene 1 (theoretical value 152) : 
 
 Concentration. Value of m. Concentration. Value of m. 
 
 0*411 144 I2"ll 149 
 
 1-253 143 23-12 152 
 
 2791 145 26-59 J 54 
 
 5'897 147 
 
 The observed molecular weights depend not only on the nature 
 of the solute and on the concentration, but also very largely on 
 the nature of the solvent. Examples will be given in the follow- 
 ing pages showing that in certain solvents the observed mole- 
 1 Concentration in grams per 100 grams of benzene. 
 
DILUTE SOLUTIONS 
 
 123 
 
 cular weights are often higher than those deduced from the 
 chemical formula of the solute. The solute is then said to form 
 complex molecules or to be associated, and the solvent is termed 
 an associating solvent. In other solvents, on the contrary, the 
 molecular weight may be equal to or less than that deduced 
 from its chemical formula. In the latter case the solute is said 
 to be dissociated, and the solvents in question are termed 
 dissociating solvents. 
 
 Abnormal Molecular Weights In order to illustrate the 
 results of molecular weight determinations from a slightly differ- 
 ent point of view, the following table contains the values for the 
 molecular freezing-point depression, K, for three typical sol- 
 vents, water, acetic acid and benzene. The data are mainly due 
 to Raoult, and in calculating K it is assumed that the molecular 
 weight corresponds with the ordinary chemical formula of the 
 solute : 
 
 Solvent Water. Solvent Acetic Acid. 
 
 Solvent Benzene. 
 
 Solute. K. 
 
 Solute. K. 
 
 Solute. K. 
 
 Cane sugar 18*6 
 
 Methyl iodide . 38-8 
 
 Methyl iodide. 50-4 
 
 Acetone . 17-1 
 
 Ether . . . 39-4 
 
 Ether . . .497 
 
 Glycerol . 17*1 
 
 Acetone . . 38*1 
 
 Acetone . . 49-3 
 
 Urea . . 187 
 
 Methyl alcohol 357 
 
 Aniline . . . 46*3 
 
 HC1 . .39-1 
 
 HC1. . . . 17-2 
 
 Methyl alcohol 25*3 
 
 HN0 3 . 35'8 
 
 H 2 SO 4 . . .18-6 
 
 Phenol . . . 3 2 '4 
 
 KN0 3 . . 35'8 
 
 (CH 3 COO) 2 Mg 18-2 
 
 Acetic acid . 25-3 
 
 NaCl . . 36-0 
 
 
 Benzoic acid . 25*4 
 
 This very instructive table shows that, for all three solvents, 
 there are two sets of values for K, one of which is approximately 
 double the other. The question now arises as to which of 
 these are the normal values, obtained when the solute exists as 
 single molecules in solution. This can at once be settled by 
 using van't HofFs formula, K = (o'o2T 2 )/H (p. 120), and we 
 find that the normal depressions are 18-6, 39-0 and 51-2 for 
 water, acetic acid and benzene respectively. This means that 
 acetic acid, phenol and some other compounds dissolved in 
 
i2 4 OUTLINES OF PHYSICAL CHEMISTRY 
 
 benzene produce only half the depression, in other words, exert 
 only about half the osmotic pressure that would be expected ac- 
 cording to their formulae, whilst in water certain acids and salts 
 have an abnormally high osmotic pressure. The osmotic pressure 
 of certain mineral acids in acetic acid is abnormally low. 
 
 On the molecular theory, an abnormally small osmotic 
 pressure shows that the number of particles is smaller than was 
 anticipated. The experimental results can be satisfactorily 
 accounted for on the view that acetic acid and benzoic acid 
 exist as double molecules in benzene solution, and that phenol 
 is polymerized to a somewhat smaller extent. This explanation 
 seems the more plausible inasmuch as acetic acid contains com- 
 plex molecules in the form of vapour (p. 41). 
 
 It is mainly compounds containing the hydroxyl and cyanogen 
 groups which are polymerized in non-dissociating solvents ; in 
 dissociating solvents, such as water and acetic acid, 1 these com- 
 pounds have normal molecular weights. 
 
 It may be anticipated that the molecular complexity of 
 solutes will be greater in concentrated solutions, and the avail- 
 able data appear to show that such is the case. The results 
 are, however, somewhat uncertain, inasmuch as in concentrated 
 solution the gas laws are no longer valid (p. 122). 
 
 Solvents such as benzene are sometimes termed associating 
 solvents, but this probably does not mean that they exert any 
 associating effect. There is a good deal of evidence to show 
 that the substances existing as complex molecules in benzene 
 and chloroform solution are complex in the free condition, and 
 that the complex molecules are only partly broken up in 
 so-called associating solvents. 
 
 The explanation of the behaviour of solutes in water is by 
 no means so simple, and can only be dealt with fully at 
 a later stage. The data in the table indicate that cane 
 sugar, urea, acetone, etc., are present as single molecules in 
 solution, but hydrochloric acid, potassium nitrate, etc., be- 
 have as if there were nearly double the number of molecules to 
 
 1 That acetic acid is in some cases at least a dissociating solvent is 
 evident from the fact that the molecular weight of methyl alcohol in it is 
 almost normal. 
 
DILUTE SOLUTIONS 125 
 
 be anticipated from the formulae. When van't Hoff put forward 
 his theory of solutions he was quite unable to account for this 
 behaviour, and contented himself with putting in the general 
 gas equation a factor, /', to represent the abnormally high os- 
 motic pressure, so that for salts and the so-called " strong " 
 acids and bases in aqueous solution the equation became 
 PV = iRT. 
 
 The factor / can of course be obtained for aqueous solutions by 
 dividing the experimental value of the molecular depression by 
 the normal constant, 18*6, so that for potassium nitrate, for 
 example, / = 3 5*8/1 8*6 = 1*92. 
 
 Van't Hoff recalled the fact that ammonium chloride, in the 
 form of vapour, exerts an abnormally high pressure, which is 
 simply accounted for by its dissociation according to the equa- 
 tion NH 4 C1 = NH 3 + HC1, but it did not appear that the 
 results with salts, etc., could be explained in an analogous way. 
 We shall see in detail later that the elucidation of the signifi- 
 cance of the factor / was of the highest importance for the 
 further development of the theory of solution. According to 
 our present views, the substances which show abnormally high 
 osmotic pressures are partially dissociated in solution, not into 
 ordinary atoms, but into atoms or groups of atoms associated 
 with electrical charges. The equation representing the partial 
 splitting up of potassium nitrate, for example, may be written 
 
 KNO 3 = K + NO 3 , which indicates that the solution contains 
 potassium atoms associated with positive electricity, and an 
 equal number of NO 3 groups, associated with negative elec- 
 tricity. These charged atoms, or groups of atoms, are termed 
 ions. 
 
 Molecular Weight of Liquids Our knowledge as to the 
 molecular weight of pure liquids is due mainly to the investi- 
 gations of Eotvos (1886) and of Ramsay and Shields (1893), 
 and is based on the remarkable rule, discovered by Eotvos, 
 that the rate of change of the "molecular surface energy " of many 
 
126 OUTLINES OF PHYSICAL CHEMISTRY 
 
 liquids with temperature is the same. If y represents the sur- 
 face tension and therefore the energy per sq. cm. of surface 
 and s the "molecular sur f ace," the rule in question may be 
 writ en 
 
 < ....(<> 
 
 where c is a constant. 1 The molecular surface, s, can be ex- 
 pressed in terms of readily measurable quantities as follows : 
 The molecular volume of any liquid is represented by M# 
 where M is the molecular weight and v the specific volume. 
 If the molecular volume is regarded as a cube, one edge of 
 the cube will measure (Mz;)*, and the area of one side of it 
 (Mv)L (M#)ir may therefore be called the molecular sur- 
 face, s, and just as the relative molecular volumes of different 
 liquids contain an equal number of molecules so the relative 
 molecular surfaces for different liquids are such that an equal 
 number of molecules lie on them. Equation (i) then becomes 
 
 dt 
 
 .. , _ , , 
 
 1 The student should make himself familiar with this method of repre- 
 senting rate of change, as it is largely used in physical chemistry. It is 
 perhaps most readily understood by considering the rate of change of 
 position of a body as discussed in mechanics. If a body is moving with 
 uniform velocity, the velocity can at once be found by dividing the distance, 
 
 5, traversed by the time, t, taken to traverse it, hence velocity = *li. The 
 
 velocity may, however, be continually altering, and it is often desirable to 
 express the velocity at any instant. It is not at first sight evident how 
 this can be done, as it requires some time for the particle to traverse any 
 measurable distance, and the velocity may be altering during that time. 
 The nearest approach to the real velocity at any instant will be obtained by 
 taking the time, and therefore the distance traversed, as small as possible. 
 We might then imagine an ideal case in which 5 and t are taken so small 
 that any error due to the variation of speed during the time t can be 
 neglected. If we represent these values of s and t by ds and dt respectively, 
 the speed of the particle at any instant will be given by ds{dt. 
 
 In the example given in the text d(ys)jdt represents the rate of change 
 of the product with temperature, and the equation shows that the rate ol 
 change is constant. 
 
DILUTE SOLUTIONS 127 
 
 where y l and y 2 are the surface tensions of a liquid at the 
 temperatures / x and / 2 respectively. From equation (2) we 
 obtain, for the molecular weight M, 
 
 M f_(fiL^\*. . . (3) 
 
 The surface tension of a large number of pure liquids at 
 different temperatures has been measured by Ramsay and 
 Shields by observing the height to which they rose in capillary 
 tubes. The results show that, if M is taken as the molecular 
 weight corresponding with the simplest formula of the liquid, 
 the value of c for the majority of substances is about 2*12. 
 
 The method may be illustrated l by a determination of the 
 molecular weight of liquid carbon disulphide. The experi- 
 mental data are that y 33-6 ergs per sq. cm. at i9'4, and 
 29*4 ergs at 46'!: the specific volume (i/density) of carbon 
 disulphide at 19-4 is 1/1-264; at 46*1 it is 1/1-223. Hence 
 
 M=/- - 2-12(19-4 - 46-1) i* = 
 ---- 
 
 as compared with the value 76 calculated from the formula. 
 
 As already indicated, the surface tension of a liquid in 
 contact with its vapour diminishes as the temperature rises and 
 becomes zero at the critical temperature, where the surface of 
 separation beween liquid and vapour disappears (p. 50). If 
 temperatures are measured downwards from the critical tem- 
 perature as zero, dt in equation (i) p. 126 has a positive value, 
 and therefore c is positive. In the next section for convenience 
 positive values of the constant will be used. It should be 
 added that the rule regarding the constancy of the expression 
 d (ys)jdt only holds for temperatures at some distance (say 50) 
 below the critical temperature. 
 
 Results of Measurements Among the liquids which give 
 
 values for c about 2-12 are the following : benzene 2'iy, carbon 
 
 tetrachloride 2*11, silicon tetrachloride 2-03, ethyl iodide 2-10, 
 
 ethyl ether 2-17, benzaldehyde 2-16, aniline 2-05. On the other 
 
 Ramsay and Shields, Trans. Chem. Soc., 1893, 63, 1096. 
 
128 OUTLINES OF PHYSICAL CHEMISTRY 
 
 hand, many substances give values for c which are much smaller 
 than 2-12 and which vary with the temperature. Thus for ethyl 
 alcohol the values of d [y (Mz>) 2/3 ]/<// in the neighbourhood of 
 the temperatures indicated are as follows : i'i8 at 30, 1*31 at 
 90, 1-46 at 130, i '7 7 at 185 and 1*94 at 225. Among sub- 
 stances which give low values for c are the alcohols, the organic 
 acids (acetic acid 0-90 at 16-46), acetone and water. 
 
 The most plausible explanation of these observations is that 
 liquids which give constant values for c approximating to 2*12 
 are non-associated, whilst those giving smaller values for this 
 factor are associated. We may assume that association would 
 tend to lower the molecular volume and thus give a smaller 
 value for c, as is actually found. The fact that for liquids with 
 abnormally small values for c the latter increases steadily with 
 the temperature is also in harmony with this explanation, since 
 it may be assumed that the molecular complexity diminishes 
 with rise of temperature. 
 
 Attempts have been made to deduce from the observed 
 values of d (ys)ldt the degree of complexity of associated liquids, 
 but the results are by no means conclusive. According to 
 Ramsay, the association factor of water at 5, 25, 45 and 85 
 is 3*81, 3'44, 3*13 and 279 respectively; van der Waals, how- 
 ever, deduces from the same data considerably smaller values 
 for this factor. 1 It is quite certain that water under ordinary 
 conditions is a complex mixture of molecules of the formulae 
 H 2 O, (H 2 O) 2 , (H 2 O) 3 and perhaps still more complicated 
 aggregates, but the average degree of association at any given 
 temperature is not definitely known. 
 
 Recently Walden 2 has shown that the value of d (ys}ldt for 
 palmitic and stearic acids is greater than 5 between 60 and 
 1 20. On the basis of the above interpretation of abnormally 
 small values of the temperature coefficient in question, this 
 would appear to indicate that the two acids are highly dissociated, 
 whilst direct determinations in solution show that the mole- 
 cular weights are normal. This affords further evidence in 
 
 l Zeitsch. Physikal Chem., 1894, 13, 713. See also General Dis- 
 cussion, Trans. Faraday Soc., 1910, 6, 71-123. 
 
 a jTfittrJt Pfivtikal tlhsm TniT. 7C CCC. 
 
DILUTE SOLUTIONS 129 
 
 favour of the conclusion indicated above, that the rule of 
 Eotvos is only approximately valid. 
 
 The Nature of Surface Tension A deeper insight into 
 surface tension is obtained on the basis of the molecular theory. 
 We assume that liquid particles attract each other with a force 
 which falls off very rapidly with the distance. A particle in 
 the interior of a liquid is equally attracted on all sides, but a 
 particle in the surface layer is attracted inwards by all the 
 particles of liquid within its sphere of influence, the corre- 
 sponding attraction by the few particles in the vapour space 
 being negligible in comparison. It follows that at the surface 
 of liquids there is a force the so-called surface tension act- 
 ing inwards, the liquid behaving as if it were covered by an 
 elastic skin. 
 
 It is evident that work must be done against molecular 
 attraction in bringing a particle to the surface layer, and there- 
 fore the formation of a larger surface involves an expenditure 
 of energy. The surface energy is proportional to the product 
 of the surface tension y and the area of the surface, and therefore 
 the molecular surface energy is represented by the expression 
 y (M.v) 2/3 , as already mentioned. A liquid tends to diminish the 
 area of its surface as much as possible in virtue of the force 
 tending to draw the particles on the surface towards the interior. 
 The tendency of liquid drops to assume a spherical shape and 
 of minute drops to aggregate to larger drops is thus readily 
 explained. 
 
 Practical Illustrations. Osmotic Pressure The nature 
 of semi-permeable membranes may readily be illustrated by 
 Traube's experiment, described on page 98, and also by 
 allowing drops of a fairly concentrated solution of potassium 
 ferrocyanide to fall into a moderately dilute solution of copper 
 sulphate in such a way that the drops, which are immediately 
 surrounded by a film of copper ferrocyanide, remain suspended 
 at the surface of the solution. It will be observed that the 
 cells grow fairly rapidly owin.g to passage inwards of water from 
 the copper sulphate solution, a$,d, further, that in consequence 
 
130 OUTLINES OF PHYSICAL CHEMISTRY 
 
 of the increased concentration of the copper sulphate solution 
 round the drop, the concentrated solution slowly flows down 
 through the less concentrated solution. The stream of con- 
 centrated solution can readily be recognised by the difference of 
 refractivity, especially if a bright light is placed behind the vessel 
 
 Selective Action of Semi-permeable Membrane This can be 
 illustrated by Nernst's experiment, which is fully described and 
 figured on page 107. 
 
 A simple experiment illustrating the same principle has been 
 described by Kahlenberg. 1 At the bottom of a cylindrical jar 
 is placed a layer of chloroform, above that a layer of water, and 
 at the top a layer of ether and the jar is then corked. After 
 some time it will be noticed that the chloroform layer has 
 increased in depth, the water layer having moved higher up 
 the tube. This phenomenon depends on the fact that ether is 
 much more soluble in water than chloroform. The water there- 
 fore acts like a semi-permeable membrane, absorbing the ether 
 and giving it up to the chloroform. At the same time the 
 chloroform is dissolving in the water and passing through tc 
 the ethereal layer, but owing to its much smaller solubility, the 
 current upwards is negligible in comparison with that downwards. 
 
 In the same paper, Kahlenberg describes a number of experi- 
 ments with rubber membranes, which are in many respects 
 instructive. 
 
 Separation of Solvent and Solute in Freezing-point Experi- 
 ments This point, which is of fundamental importance for the 
 applicability of the freezing-point method of determining mole- 
 cular weights, can be illustrated by partially freezing an aqueous 
 solution of a highly coloured substance such as potassium 
 permanganate (cri per cent, solution). When the solution is 
 poured off, it will be found that the ice which has separated 
 is practically colourless. 
 
 The determination of molecular weights by the boiling-point 
 method (p. 116) and by the freezing-point method (p. 120) are 
 fully described in the course of the chapter. 
 
 1 7, Physical Chew,, I96, 10, I4iv 
 
APPENDIX. 
 MATHEMATICAL DEDUCTION OF IMPORTANT FORMULA. 
 
 In the course of the present chapter, several formulas of fundamental 
 importance have been made use of, and their meaning has been fully 
 illustrated by numerical examples. For the sake of the more advanced 
 student, simple deductions of these formulae are given. It must bs under- 
 stood that the deductions are not mathematically strict, as certain of the 
 assumptions on which they are based are only approximately true. 
 
 The fundamental equation (p. 112), 
 
 .* ..... > 
 
 which gives the connection between the relative lowering of vapour 
 pressure and the osmotic pressure, and the Raoult-van't Hoff formula 
 (P- IX 3). 
 
 which is readily derived from equation (i) by means of the gas laws, can 
 be deduced by a statical-thermodynamical method due to Arrhenius and 
 also by a cyclical therrnodynamical method due to van't Hoff. These 
 deductions will now be given. 
 
 (i) The Statical Method A long tube R containing a solution of n 
 mols of a non-volatile solute in N mols of solvent, 1 is closed at its lower 
 end by a semi-permeable membrane and placed upright in a vessel C 
 which contains pure solvent (fig. 20). The arrangement is covered by a 
 bell-jar and all air is removed from the interior. When equilibrium be- 
 tween solvent and solution is established through the semi-permeable 
 membrane it is evident that the osmotic pressure is measured by the hydro- 
 static pressure of the column of liquid (height h) in the tube. Now the 
 pressure of vapour at the level, a, cf the surface of the solution must be 
 the same inside and outside the tube. If this were not the case, evapora- 
 tion or condensation of vapour would take place at the surface, a, and in 
 either case the concentration of the solution would be aliered and the 
 equilibrium between solution and solvent disturbed, which is contrary to 
 the original postulate that the system is in equilibrium. If p 1 is the 
 vapour pressure of the solvent and /> 2 that of the solution, the difference 
 Pi - Pz ' s tne difference of pressure at the surface of the solvent and at 
 
 1 In calculating the number of mols of solvent, its molecular weight, 
 M, is taken as that in the form of vapour (p. 112). 
 
132 OUTLINES OF PHYSICAL CHEMISTRY 
 
 the level a. This difference is due to the weight of a column of vapour 
 of height h on unit area, therefore 
 
 p^-p^hd. . (a) 
 where d is the density of the^yapour. 
 We have now to express d and 
 h in a different form. If v l is the 
 volume of i mol of the vapour, and 
 M the molecular weight of the 
 vapour in the gaseous form, we have 
 d = M/z^ or t>! = Mfd. When/this 
 value of z>! is substituted in/ the 
 general gas equation p^^ = RT, we 
 obtain , _ M/> t 
 
 - j^r. 
 
 Further, as the osmotic pressure, P, 
 is measured by the weight of the 
 column h, therefore P = hs' where 
 s' is the density of the solution. ' If 
 very dilute solutions are used, no 
 appreciable error will be committed 
 by substituting s, the density of the 
 solvent, for 5', the density of the 
 solution. Substituting these values 
 of d and h in equation (a) we obtain 
 
 RT 
 
 FIG. 20. 
 
 Pi ~ * RT ' 
 
 which is equation (i), p. 112. 
 
 From this equation, we obtain 
 the formula, 
 
 as already described (p. 112). 
 
 (2) The Cyclical Method This thermodynamical proof of the above 
 formula depends upon the performance of a cyclic process in which the 
 system is finally brought back to its initial condition reversibly at con- 
 stant temperature. The fundamental point to bear in mind in connection 
 with such processes is that they must be conducted throughout under 
 equilibrium conditions. It has already been pointed out (p. no) that the 
 thermodynamical proof of the connection between osmotic pressure and the 
 lowering of vapour pressure depends on the work done in removing solvent 
 reversibly from a solution. There are two principal methods by which 
 removal (or addition) of solvent can be accomplished; 
 
 (a) If the solution is brought into contact with its own saturated 
 vapour at constant temperature, the slightest diminution of the external 
 pressure will effect the removal of part of the solvent ; on the other hand, 
 the slightest increase of the external pressure will bring about condensation 
 of vapour. If the change of volume of the solution is in each case very 
 small compared with the total volume, the change in concentration can 
 be neglected. 
 
APPENDIX 133 
 
 (b) A solution is placed in a cylinder closed at the bottom with a 
 semi-permeable membrane, the cylinder is immersed in the pure solvent, 
 and a movable piston rests on the upper surface. The solution and 
 solvent will be in equilibrium through the semi-permeable membrane 
 when the pressure on the piston is equal to the osmotic pressure. If the 
 pressure is diminished ever so slightly by raising the piston, solvent will 
 enter ; if the pressure on the piston is slightly increased, solvent will pass 
 out through the membrane. We have, therefore, a second method by 
 which solvent can be separated from a solution in a reversible n.anner, 
 equilibrium being maintained throughout. The cyclic process, in which 
 both these methods are used, will now be described. 
 
 (1) From a solution containing n mols of solute to N mols of solvent, 
 a quantity of solvent which originally contained i mol of solute is 
 squeezed out reversibly by means of the piston and cylinder arrangement; 
 the quantity thus removed is N/w mols. As the original quantity of solu- 
 tion is supposed to be very great, its concentration, and therefore its 
 osmotic pressure, are not appreciably altered in the process.' As the 
 volume removed is that which contained i mol of solute, the work done 
 on the system, which is the product of the change of volume and the 
 pressure on the piston (the osmotic pressure), is equal to 
 
 - RT ..... (i) 
 
 (p. 27) if the gas laws apply. 
 
 (2) The quantity of solvent is now converted reversibly into vapour 
 by expansion at the pressure, p lt of the solvent ; the work gained is ap- 
 proximately p l v l for i mol of vapour (the volume of the liquid being 
 regarded as negligible in comparison), or 
 
 -Pi*! - . () 
 
 altogether. 
 
 (3) The vapour is now allowed further to expand till its pressure falls 
 to p 2 , the vapour pressure of the solution : in this process, an amount of 
 work is done by the system represented approximately by 
 
 per mol of vapour, where (p t + /> 2 )/ 2 ' s tne mean pressure during the small 
 expansion. The quantity of vapour actually used is N/n mols, hence the 
 total work done by the system is 
 
 (4) The vapour, at the pressure p z , is now brought into contact with 
 the solution, with which it is in equilibrium, and condensed reversibly, so 
 that the system regains its initial state. The work done on the system in 
 condensing the gas to liquid at the pressure / 2 i fi approximately 
 
 - ~M 2 ..... ( iv ) 
 
 As the entire cycle is carried through at constant temperature, there 
 has on the whole been no transformation of heat into work or vice versa ; 
 as the system is finally brought back to its initial condition, the work 
 done on the system must on the whole be equal to the work done by the 
 system ; in other words (i) + (ii) + (iii) + (iv) must be zero. 
 
134 OUTLINES OF PHYSICAL CHEMISTRY 
 
 Combining in the first place (ii), (iii), and (iv), we have 
 
 which reduces to 
 
 N /a, + zi\ , . , . N ., . 
 
 - (-~^) Mi ~ PJ = n V ^ 1 ~ ^ 
 where v is the mean volume of i mol of vapour. 
 
 Substituting for v its value from the general gas equation, v = RT//>, 
 we have finally l 
 
 n p l 
 
 as the work done by the system in the last three stages of the cy le. This 
 must be equal to the work done on the system during the osmotic removal 
 of solvent, hence 
 
 Pi 
 
 or &JL& = .? 
 
 A N 
 
 as before. 
 
 The same result may be obtained still more simply by integration. 
 The work done by the system in step (ii) is exactly balanced by that done 
 on the system in (iv), as is evident from the factors themselves, if the pas 
 laws hold. In (iii) bot'i the pressure and the volume change during the 
 exoansion, hence work done by the system for i mol of vapour is equal to 
 
 = I 
 J 
 
 = RT log 2 = RT log 1 (since p^ = / 
 "V v 
 
 = - RT log = - RT log, i - 
 
 Pi \ Pi / Pi 
 
 approximately, 2 or, for the total volume of vapour, 
 
 
 
 The remainder of the proof is as above. 
 
 Lowering Of Freezing-point The above formula has been deduced 
 by an isothermal cyclic process, but the cyclic process by which the 
 freezing-point formula is deduced cannot be carried through at constant 
 temperature. We are therefore concerned witfi a new question, that of 
 the relationship between heat and work. The law which applies in this 
 case is the second law of thermodynamics, the deduction of which is to be 
 
 1 When the solution is dilute, p in the denominator may be put equal 
 to p l without sensible error. 
 
 2 RT ,, >s the first term of the expansion of the logarithmic 
 
 ,, 
 . Th 
 
 function. The more accurate form of the van't Hoff-Raoult formula is 
 = CT, to which the usual form approximates in dilute solution. 
 
APPENDIX 13$ 
 
 found in any advanced book on Physics, and which states that the maxi- 
 mum work, <7A, obtainable from a given quantity of heat Q, in a reversible 
 cycle is given by 
 
 dT 
 dA = q^ 
 
 where the symbols have the usual significations (p. 151). 
 
 A solution containing n mols of solute in N mols (W grams) of solvent 
 is contained in the cylinder with semi-permeable membrane and movable 
 piston already described. The freezing-point of the solvent is taken as T 
 and that of the solution as T - dT. The stages in the cyclic process are 
 as follows : 
 
 1 i) At the temperature T - dT an amount of solvent which originally 
 contained i mol of solute is frozen out ; the amount in question is N/w 
 mols or MN/w grams. The separation can be carried out at constant 
 temperature provided that the amount of solution is so great that its con- 
 centration is not thereby appreciably affected. The solidified solvent is 
 then separated from the solution and the temperature of both raised to T. 
 
 (2) The solidified solvent is fused, in which process H . calories 
 
 are taken up, H being the heat effusion per gram. 
 
 (3) The fused solvent is then brought into contact with the solution 
 through the semi-permeable membrane under equilibrium conditions, that 
 is, when the pressure on the piston is equal to the osmotic pressure of the 
 solution (p. 133) and is allowed to mix reversibly with the solution. The 
 work done by the system in this process is represented by the product of 
 the osmotic pressure, P, and the volume, v, in which i mol of solute was 
 diss Ived and is, therefore, according to the gas laws, equal to RT. 
 
 (4) The system is finally cooled to the original temperature, T - dT, 
 in order to complete the cycle. 
 
 We have now to consider the work done in the different stages of the 
 cycle. The heat expended in warming solution and solvent in (i) is 
 practically compensated 1 by the heat given out in (4). Further, an 
 amount of heat HW/w is taken in at the higher temperature T and a 
 somewhat k ss amount given out at T - dT ; hence, by the second law of 
 thermodynamics, the work done on the system is 
 
 H W dT 
 H . - . T . 
 
 The only work done by the system is that expended in the osmotic 
 readmission of the solvent, hence 
 
 RT H W dT 
 
 RT = H 'VT 
 
 JT RT 2 , . 
 
 or dT = -jj- . ^r . . . , . (i) 
 
 1 The two amounts are not exactly equal, but the difference can be 
 made negligible in comparison with the heat taken up in the second stage 
 of the cycle. 
 
136 OUTLINES OF PHYSICAL CHEMISTRY 
 
 If instead of H we use the molecular heat of fusion, \, we have \ = 
 MH, and, further, N = W/M. Substituting these values in equation (i), 
 the latter reduces to 
 
 From this it is evident that the lowering of the freezing-point, like 
 the relative lowering of vapour pressure, is proportional to the ratio of the 
 number of mols of solute to the number of mols of solvent. 
 
 From the above formula, or more readily from equation (i), above, 
 an expression for K, the depression produced when i mol of solute is 
 dissolved in 100 grams of solvent, can readily be obtained. R is approxi- 
 mately = 2 whe.n expressed in calories, n = i and W = TOO. Hence we 
 obtain, for this particular value of dT, 
 
 aT - K - ^ -L - ' 02T2 (o) 
 
 - H 'loo" H ' 
 
 which is the formula given on p. 120. 
 
 Elevation of Boiling-point By means of a cyclic process exactly 
 corresponding with that already used in establishing the freezing-point 
 formula, the formula connecting the elevation of the boiling-point with 
 the latent heat of vaporisation of the solvent is obtained in the form 
 
 where H is the heat of vaporisation of i gram of solvent at the tempera- 
 ture of the experiment, and T is the boiling-point of the solvent en the 
 absolute scale. 
 
 Summary Of Formulae (a) Osmotic pressure and relative lowering 
 of vapour-pressure. From formula (r) (p. 112) we obtain by substitution 
 
 P = 82 ^ Pi-^P, atmospheres . 
 
 where P is the osmotic pressure, expressed in atmospheres, s is the density 
 of the solvent at the absolute temperature T, M is the molecular weight 
 of the solvent in the form of vapour, and/> x and p% are the vapour-pressures 
 of solvent and solution respectively. 
 
 (b) Osmotic pressure and lowering of freezing-point From formula 
 (i) (p. 137), by substitution 
 
 roooHs dT 
 
 P = - . -=r- atmospheres ; 
 24*22 T 
 
 where H is the latent heat of fusion of the solvent in calories per gram, 
 T is the freezing-point of the solvent on the absolute scale and dT is the 
 freezing-point depression. 
 
 Osmotic pressure and elevation of boiling-point The formula, which 
 corresponds exactly with that for the freezing-point depression, is 
 _ roooHs dT 
 24*22 ' T 
 
 where H is the latent heat of vaporisatipn for i gram of solvent at its 
 boiling-point, T is the boiling-point of the solvent on the absolute scale, 
 and dT is the boiling-point elevation. 
 
CHAPTER VI 
 THERMOCHEMISTRY 
 
 General It is a matter of every-day experience that 
 chemical changes are usually associated with the develop- 
 ment or absorption of heat. When substances enter into 
 chemical combination very readily, much heat is usually given 
 out (for example, the combination of hydrogen and chlorine 
 to form hydrogen chloride), but when combination is less 
 vigorous, the heat given out is usually much less, and, in 
 fact, heat may be absorbed in a chemical change. These 
 facts, which were noticed very early in the history of 
 chemistry, led to the suggestion that the amount of heat 
 given out in a chemical change might be regarded as a 
 measure of the chemical affinity of the reacting substances. 
 Although, as will be shown later, this is not strictly true, there 
 is, in many cases, a parallelism between chemical affinity and 
 heat liberation. In thermochemistry, we are concerned with 
 the heat equivalent of chemical changes. 
 
 Heat is a form of energy, and therefore the laws regarding 
 the transformations of energy are of importance for thermo- 
 chemistry. It is shown in text-books of physics that there 
 are different forms of energy, such as potential energy, kinetic 
 energy, electrical energy, radiant energy and heat, and that 
 these different forms of energy are mutually convertible. 
 Further, when one form of energy is converted completely 
 into another, there is always a definite relation between the 
 amount which has disappeared and that which results. The 
 
138 OUTLINES OF PHYSICAL CHEMISTRY 
 
 best-known example of this is the relation between kinetic 
 energy and heat, which has been very carefully investigated by 
 Joule, Rowland and others. Kinetic energy may be measured 
 in gram-centimetres or in ergs, and heat energy in calories 
 (see p. xvii). The investigators just referred to found that 
 i calorie = 42,650 gram-centimetres = 41,830,000 ergs, an 
 equation representing the mechanical equivalent of heat. 
 From the above considerations it follows that when a certain 
 amount of one form of energy disappears an equivalent amount 
 of another form of energy makes its appearance. These results 
 are summarised in a law termed the Law of the Conservation of 
 Energy, which may be expressed as follows : The energy of an 
 isolated system is constant, i.e., it cannot be altered in amount 
 by interactions between the parts of the system. The proof of 
 this law lies in the experimental impossibility of perpetual motion 
 it has been found impossible to construct a machine which 
 will perform work without the expenditure of energy of some 
 kind. 
 
 In dealing with chemical changes, it has been found con- 
 venient to employ the term chemical energy, and when two 
 substances combine with liberation of heat, we say that chemical 
 energy has been transformed to heat. To make this clear, we 
 will consider a concrete case, the burning of carbon in oxygen 
 with formation of carbon dioxide, a reaction which, as is well 
 known, is attended with the liberation of a considerable amount 
 of heat. The reaction can be carried out under such condi- 
 tions that the heat given out when a definite weight of carbon 
 combines with oxygen can be measured, and it has been found 
 that when 12 grams of carbon and 32 grams of oxygen unite, 
 94,300 calories are liberated. This result may conveniently be 
 represented by the equation 
 
 C + O 2 = CO 2 + 94,300 cal. 
 
 in which the symbols represent the atomic weights of the 
 reacting elements in grams. The above equation is an illus- 
 tration of the conversion of chemical energy into heat 12 
 
THERMOCHEMISTRY 1 39 
 
 grams of free carbon and 32 grams of free oxygen possess 
 94,300 cal. more energy than the 44 grams of carbon dioxide 
 formed by their union. From these and similar considerations 
 it follows that the free elements must have much intrinsic 
 energy, but the absolute amount of this energy in any par- 
 ticular case is quite unknown. Fortunately, this is a matter 
 of secondary importance, as chemical changes do not depend 
 on the absolute amounts of energy, but only on the differences 
 of energy of the reacting systems. 
 
 So far, we have implicitly assumed that the increase or de- 
 crease of internal energy when a system A changes to a system 
 B is measured by the heat absorbed or given out during the 
 reactions ; but this is not necessarily the case. In particular, 
 external work may be done during the change, by which part of 
 the energy is used up, or heat may be produced at the expense 
 of external work (cf. p. 27). If the total diminution of internal 
 energy in the change A -> B is represented by U, the heat given 
 out by - q, and the external work done by the reacting sub- 
 stances during the transformation by A, we have, by the prin- 
 ciple of the conservation of energy, 
 
 U = A - q. 
 
 When no external work is done the total diminution of energy, 
 U, is numerically equal to Q, the heat evolved in the reaction. 
 The factor A is only of importance when gases are involved in 
 the chemical change. 
 
 Hess's Law It is an experimental fact that when the same 
 chemical change takes place between definite amounts of two 
 substances under the same conditions the same amount of heat 
 is always given out provided that the final product or products 
 are the same in each case. Thus when 12 grams of carbon 
 combine with 32 grams of oxygen with formation of carbon 
 dioxide, 94,300 cal. are always liberated, quite independently of 
 the rate of combustion or of the nature of the intermediate pro- 
 ducts. This law was first established experimentally by Hess in 
 1840, and may be illustrated by the conversion, by two different 
 methods, of a system consisting of i mol of ammonia and of 
 
MO OUTLINES OF PHYSICAL CHEMISTRY 
 
 hydrochloric acid respectively and a large amount of water, each 
 taken separately, into a system consisting of i mol of ammonium 
 chloride in a large excess of water. By the first method we 
 measure (a) the heat change when i mol of gaseous ammonia 
 and i mol of gaseous HC1 combine, (I)) the heat change when 
 the solid ammonium chloride is dissolved in a large excess of 
 water ; by the second method we measure the heat changes 
 when (c) i mol of ammonia, (d) i mol of hydrochloric acid are 
 dissolved separately in excess of water, and (e) when the two 
 solutions are mixed. The results obtained were as follows : 
 
 First Way. 
 
 (a) NH 3 gas + HC1 gas = +42,100 cal. 
 
 (b) NH 4 C1 + aq = - 3,900 cal. 
 
 38,200 cal. 
 Second Way. 
 
 (c) NH 3 gas -f aq = + 8,400 cal. 
 
 (d) HClgas + aq + 17,300 cal. 
 
 (e) HC1 aq + NH 3 <*q = + 12,300 cal. 
 
 38,000 cal. 
 
 As will be seen, a+b=c+d+e within the limits of ex- 
 perimental error. 
 
 It can easily be shown that Hess's law follows at once from 
 the principle of conservation of energy. 
 
 This law is of the greatest importance for the indirect deter- 
 mination of the heat changes involved in certain reactions 
 which cannot be carried out directly. For example, we cannot 
 determine directly the heat given out when carbon combines 
 with oxygen to form carbon monoxide. The heat given out 
 when 12 grams of carbon burn to carbon dioxide is 94,300 
 cal., which is, by Hess's law, equal to that produced when the 
 same amount of carbon is burned to monoxide and the latter 
 then converted to dioxide. The latter change gives out 68,100 
 
THERMOCHEMISTRY 141 
 
 cal., and the reaction C + O = CO must therefore be associated 
 with the liberation of 94,300 - 68,100 = 26,200 cal. 
 
 Representation of Thermochemical Measurements. 
 Heat of Formation. Heat of Solution As has already 
 been pointed out, the results of thermochemical measure- 
 ments may be conveniently represented by making the or- 
 dinary chemical equation into an energy equation, for example, 
 
 C + O 2 = CO a + 94,300 cal. 
 
 Sometimes, if the final condition of the system is assumed to 
 be known, the shorter form 
 
 C, O 2 = 94,300 cal. 
 may be used. 
 
 When, as is frequently the case, the reacting substances are 
 used in aqueous solution, this is indicated by adding aq to the 
 formula in question. Thus the neutralisation of dilute hydro- 
 chloric acid by sodium hydroxide is represented as follows : 
 NaOH aq + HC1 aq = NaCl aq + 13,700 cal. 
 
 The heat of formation of a compound is the heat given out 
 when a mol of the compound is formed from its component ele- 
 ments. Thus the heat of formation of carbon dioxide (at 
 constant volume) is 94,300 cal. The above energy equations, 
 e.g., that representing the formation of carbon dioxide, are, 
 however, not complete, inasmuch as we do not know the in- 
 trinsic energy associated with free carbon and oxygen re- 
 spectively, nor do we know the differences of energy between 
 the various elements, as they are not mutually convertible by 
 any known means. We may therefore choose any arbitrary 
 values for the intrinsic energies of the elements, and it has 
 been found most convenient to put them all equal to zero. 
 On this basis the intrinsic energy of carbon dioxide, being 
 94,300 cal. less than the sum of the intrinsic energies of the 
 component elements, is - 94,300 cal., and, in general, the in- 
 trinsic energy of a compound is numerically equal to its heat 
 of formation, but with the sign reversed. 
 
i 4 2 OUTLINES OF PHYSICAL CHEMISTRY 
 
 When the heats of formation of all the substances taking 
 part in a reaction are known, the heat set free in the reaction 
 can be calculated. One method of doing so is to apply the 
 law that the heat of reaction is equal to the sum of the heats 
 of formation of the substances formed minus the sum of the 
 heats of formation of the substances used up. This law follows 
 at once if we imagine the reacting substances first decomposed 
 into their elements and these elements then combined to form 
 the final products. In the first stage there would be absorbed 
 an amount of heat equal to the sum of the heats of formation 
 of the reacting substances, and in the second stage an amount 
 of heat would be given out equal to the sum of the heats of 
 formation of the products. 
 
 An alternative method, the basis of which will be evident on 
 a little consideration, is to write an energy equation in which 
 the formulae of the various compounds are replaced by their 
 intrinsic energies (the respective heats of formation with the 
 signs reversed). As an example of the method, we may cal- 
 culate the heat of reaction, x, when copper is displaced from 
 copper sulphate in dilute solution by metallic zinc. The heat 
 of formation of copper sulphate (from its elements) in dilute 
 solution is 198,400 cal. and of zinc sulphate under the same 
 conditions 248,500 cal. The energy equation for the chemical 
 change is therefore 
 
 Zn + CuSO 4 aq = Cu + ZnSO 4 aq 
 o + ( 198,400) = o + (- 248,500) + x cal. 
 whence x, the total heat liberated in the reaction, is 248,500 - 
 198,400 = 50,100 cal. 
 
 In the same way an unknown heat of formation can be cal- 
 culated when all the other heats of formation and the heat of 
 reaction are known a method which, as shown in the last 
 section, is particularly useful for obtaining the heats of forma- 
 tion of substances such as carbon monoxide and methane, 
 which cannot be determined directly, As an example, the 
 
THERMOCHEMISTRY 143 
 
 heat of formation of methane will be calculated. The heat 
 given out when i mol of this compound is burned completely 
 in oxygen is 213,800 cal., and the heat of formation of the 
 products, carbon dioxide and water, are 94,300 and 68,300 
 cal. respectively. Representing the heat of formation of 
 methane by x, its intrinsic energy therefore by - x, we have 
 the equation 
 
 CH 4 + 2O 2 = CO 2 + 2H 2 O 
 -x + o = -94,300 + (-2 x 68,300) + 213,800 cal. 
 
 Whence x = 17,100 cal. 
 
 A compound such as methane, which is formed with libera- 
 tion of heat, is termed an exothermic compound, whilst one 
 which is formed with absorption of heat is termed an endo- 
 thermic compound. 
 
 The majority of stable compounds are exothermic. Among 
 the best-known endothermic compounds are carbon disulphide, 
 hydriodic acid, acetylene, cyanogen and ozone. It is not 
 always easy to determine directly whether a compound is 
 exothermic or endothermic, but this may be done indirectly 
 by carrying out a chemical change with the compound itself 
 and with the components separately and comparing the heat 
 changes in the two cases. The method may be illustrated by 
 reference to carbon disulphide. When burnt completely in 
 oxygen, the gaseous compound gives out 265,100 cal. accord- 
 ing to the equation 
 
 CS 2 + 3<D 2 = CO 2 + 2SO 2 + 265,100 cal. 
 
 Hence, representing the intrinsic energy of the compound by 
 - x, we have, for the energy equation, 
 
 - x + o = + (- 94,3) + (- 2 x 71,000) + 265,100, 
 and x + 28,800 cal. The intrinsic energy of carbon disul- 
 phide is therefore 28,800 cal. ; that is, the compound has 28,800 
 cal. more energy than the elements from which it is formed. 
 
 The heat of solution is the quantity of heat given out or 
 absorbed by the solution of a mol of the substance in so much 
 
144 OUTLINES OF PHYSICAL CHEMISTRY 
 
 of the solvent that no further heat change is observed when 
 more of the solvent is added. The heat of solution as thus 
 defined is usually different from the heat change observed 
 when a mol of substance is dissolved in sufficient solvent to 
 form a saturated solution, and the two quantities may even be 
 of opposite sign. Thus the heat of solution of cupric chloride 
 dihydrate, CuCl 2 , 2H 2 O, is positive in dilute and negative in 
 very concentrated solution. It is the heat of solution in nearly 
 saturated solution which is of importance in predicting the 
 effect of temperature on the solubility in accordance with Le 
 Chatelier's theorem (p. 169). 
 
 It has already been mentioned (p. 141) that the heat given 
 out in a chemical change differs according to whether changes 
 of volume occur with the consequent performance of internal 
 work by or on the system. When only soiids and liquids are 
 concerned, no appreciable changes of volume occur and the 
 internal work is negligible. When gases are concerned, how- 
 ever, the heat change at constant volume, when no internal 
 work is done, differs slightly from that at constant pressure. 
 As already pointed out (p. 27), the work associated with the 
 production or disappearance of a mol of gas is 2 T calories 
 = 2 x 290 = 580 calories at room temperature, 17, and the 
 heat change at constant pressure, Q p , differs from that at 
 constant volume, Q v , by 2 (n l n 2 ) T, where n l and 2 represent 
 the number of mols of gas in the initial and final products 
 respectively. If (n l - n 2 ) is positive, then Q p > Q v ; if, on the 
 other hand, (n-^ n 2 ) is negative, Q v > Q p . 
 
 The heats of formation at constant volume of some important 
 compounds are given in the accompanying table. The state- 
 ments in brackets refer either to the state of the reacting sub- 
 stances or of the product : 
 
THERMOCHEMISTRY 
 
 Substance. 
 
 Heat of Formation 
 (Calories). 
 
 H.p ( iquid) 
 
 + 67,520 
 
 CO 2 (diamond) 
 
 + 94.3 
 
 CO (diamond) 
 
 + 26,600 
 
 SO 2 (rhombic sulphur) 
 HF (gaseous fluorine) 
 
 + 71,080 
 + 38,600 
 
 HC1 (gaseous chlorine) 
 HBr (liquid bromine) 
 
 + 22,000 
 + 8,400 
 
 HI (solid iodine) 
 
 - 6,100 
 
 NH S 
 
 + 12,000 
 
 NO' 
 
 - 21,600 
 
 NO 9 
 
 - 7,700 
 
 KCl" 
 
 + 105,600 
 
 KBr 
 
 + 95,300 
 
 Heat of Combustion Whilst a great many inorganic re- 
 actions are suitable for thermochemical measurements, this is 
 .iot in general the case for organic reactions ; in fact, the only 
 reaction which is largely used for the purpose is combustion in 
 oxygen to carbon dioxide and water. The heat given out 
 when a mol of a substance is completely burned in excess 
 of oxygen is termed the heat of combustion^ and from this, 
 by application of Hess's law, the heats of formation can be cal- 
 culated, as has been done for methane and carbon disulphide, 
 in the preceding section. Further, the heat given out in a 
 chemical change can readily be calculated by Hess's law when 
 the heats of combustion of the reacting substances are known 
 it will clearly be equal to the sum of the heats of combustion 
 of the substances which disappear less the sum of the heats of 
 combustion of the substances formed. As an example, the heat 
 of formation of ethyl acetate from ethyl alcohol and acetic acid 
 may be calculated. The heat of combustion of ethyl alcohol 
 is 34,000 cal, of acetic acid 21,000 cal., and of ethyl acetate 
 55,400 cal., whence the heat of formation of ethyl acetate is 
 34,000 + 21,000 55,400 = - 400 cal. 
 
 Thermochemical Methods Two principal methods are 
 employed in measuring the heat changes associated with 
 chemical reactions. If the reaction takes place in solution, the 
 water calorimeter, so largely used for purely physical measure- 
 ments, may be employed. For the determination of heats of 
 
i 4 o OUTLINES OF PHYSICAL CHEMISTRY 
 
 combustion, on the other hand, in which solids or liquids are 
 burned completely in oxygen, special apparatus has been de- 
 signed by Thomsen, Berthelot, Favre and Silbermann and others. 
 
 (a) Reactions in Solution The change (chemical reaction, 
 dilution or dissolution), the thermal effect of which is to be 
 measured, is brought about in a test-tube deeply immersed in 
 a large quantity of water, and the rise of temperature of the 
 water is measured with a sensitive thermometer. When the 
 weight of the water and the heat capacity of the calorimeter 
 are known, the heat given out in the reaction can readily be 
 calculated. Allowance must, of course, be made for the heat 
 capacity of the solution in the test-tube. 
 
 A simple modification of Berthelot's calorimeter, used by 
 Nernst, is shown in Fig. 21. It consists of two glass beakers, 
 the inner one being supported on corks, as shown, and nearly 
 filled with water. Through the wooden cover, X, of the outer 
 beaker pass a thin-walled test-tube, A, in which the reaction 
 takes place, an accurate thermometer B, and a stirrer C of 
 brass, or, better, of platinum. The water in the calorimeter 
 is stirred during the reaction, which must be rapid, and the 
 heat of reaction can then be calculated in the usual way 
 when the weight of water in the calorimeter and the rise of 
 temperature are known. Experiments on neutralization and 
 on heat of solution are conveniently made in the inner beaker, 
 the solution itself serving as calorimetric liquid. For dilute 
 aqueous solutions, it is sufficiently accurate to assume that the 
 heat capacity of the solution is the same as that of water. 
 
 The chief source of error in the measurements is the loss 
 of heat by radiation, which is minimised (a) by choosing for 
 investigation reactions which are complete in a comparatively 
 short time ; (b) by making the heat capacity of the calorimeter 
 system large. It is of advantage so to arrange matters that 
 the temperature of the calorimeter liquid is 1-2 below the 
 atmospheric temperature before the reaction, and 1-2 above 
 it after the reaction. 
 
THERMOCHEMISTRY 
 
 147 
 
 d\\\\\\\\ 
 
 () Combustion in Oxygen This may conveniently be carried 
 out in Berthelot's calorimetric bomb, a vessel of steel, lined 
 with platinum and provided with an air-tight lid. The sub- 
 stance for combustion is placed in the bomb, which is filled 
 with oxygen at 20-25 atmo- 
 spheres' pressure. The whole 
 apparatus is then sunk in the 
 water of the calorimeter, and 
 the combustion initiated by 
 heating electrically a small 
 piece of iron wire placed in 
 contact with the solid. 
 
 Results of Thermochemi- 
 cal Measurements Some 
 of the more important results 
 of thermochemical measure- 
 ments have already been inci- 
 dentally referred to in the 
 preceding paragraphs. In 
 stating the results of thermo- 
 chemical measurements, the 
 condition of the substances 
 taking part in the reaction must 
 always be clearly stated. This 
 applies not only to the physical 
 state, in connection with which 
 allowance must be made for 
 heat of vaporization, heat of 
 fusion, etc., but also to the 
 
 different allotropic modifications of the solid. Thus mono- 
 clinic sulphur has 2300 cal. more internal energy than rhombic 
 sulphur, and yellow phosphorus 27,300 cal. more than the red 
 modification. 
 
 The correction for change of state is often very great. For 
 the transformation of water to steam at 100, it amounts to 
 
i 4 8 OUTLINES OF PHYSICAL CHEMISTRY 
 
 about 537 x 18 = 9566 calories per mol. If, instead of the 
 heat of formation of liquid water, which is 68,300 cal., the 
 heat of formation of water vapour is required, it is 68,300 - 
 9570 = 58,730 cal. in round numbers. 
 
 As regards the thermochemistry of salt solutions, one or two 
 experimental results may be mentioned which will find an 
 interpretation later. When dilute solutions of two salts, such 
 as po'a^sium nitrate and sodium chloride, are mixed, heat is 
 neither given out nor absorbed. This important result is 
 termed the Law of thermoneutrality of salt solutions (p. 279). 
 Further, when a mol of any strong monobasic acid is neutralized 
 by a strong base, the rame amount of heat, 13,700 cal., is 
 always liberated (p. 284). 
 
 The heat of formation of salts in dilute aqueous solution is 
 obtained by the addition of two factors, one pertaining to the 
 positive, the other to the negative part of the molecule ; in 
 other words, the heat of formation of salts in dilute solution 
 is a distinctly additive property. The same is true to some 
 extent for the heat of combustion of organic compounds. For 
 example, the difference in the heat of combustion of methane 
 and ethane is 158,500 cal., and in general, for every increase 
 of CH 2 , the heat of combustion increases by about 158,000 cal. 
 From these and similar results, we can deduce the general rule 
 that equal differences in composition correspond to approxi- 
 mately equal differences in the heat of combustion. We may go 
 further, and obtain definite values .for the heat of combustion 
 of a carbon atom and a hydrogen atom as has already been 
 done for atomic volumes ; the molecular heat of combustion 
 is then the sum of the heats of combustion of the individual 
 atoms. Experience shows that when allowance is made for 
 double and triple bindings, the observed and calculated values 
 for the heats of combustion of hydrocarbons agree fairly well. 
 
 Relation of Chemical Affinity to Heat of Reaction- 
 Very early in the study of chemistry, it becomes evident that 
 chemical actions may be divided into two classes : (i) those 
 
THERMOCHEMISTRY 
 
 149 
 
 which under the conditions of the experiment are spontaneous 
 or proceed of themselves, once they are started, e.g., the com- 
 bination of carbon and oxygen ; (2) those which only proceed 
 when forced by some external agency, e.g., the splitting up of 
 mercuric oxide into mercury and oxygen. In this section we 
 are concerned only with spontaneous changes. 
 
 The direction in which a chemical change takes place in a 
 system depends on the energy relations of the system. We 
 are accustomed to say that the direction of the change is 
 determined by the chemical affinity of the reacting substances, 
 and it is a matter of the utmost importance to obtain a 
 numerical expression for the chemical affinity or driving force 
 in a chemical system, the driving force being defined in such 
 a way that the chemical change proceeds in the direction in 
 which it acts, and comes to a standstill when the driving force 
 is zero. 
 
 Most reactions in which there is a considerable transformation 
 of chemical energy, and therefore a considerable development 
 of other forms of energy, such as heat or electrical energy, 
 proceed very rapidly (for example, the combination of hydrogen 
 and chlorine), whilst reactions in which less chemical energy is 
 transformed are usually much less vigorous (for example, the 
 combination of hydrogen and iodine). It seems, therefore, at 
 first sight plausible to measure the chemical affinity in a system 
 by the amount of heat liberated in the reaction (Thomsen, 
 Berthelot). As, however, chemical affinity has been defined as 
 acting in the direction in which spontaneous chemical change 
 takes place, it would follow that only reactions in which heat 
 is given out can take place spontaneously. This deduction is 
 contrary to experience. Water can spontaneously pass into 
 vapour, although in the process heat is absorbed, and many 
 salts, such* as ammonium chloride, dissolve in water with 
 absorption of heat. It is clear, therefore, that chemical affinity, 
 as above defined, cannot be measured by the total heat liberated 
 in the reaction. 
 
iSc OUTLINES OF PHYSICAL CHEMISTRY 
 
 The importance for technical purposes of such a reaction as 
 the burning of coal in oxygen is not so much the total heat 
 obtainable by the change as the amount of work which the 
 change may be made to perform. In a similar way, it has been 
 found convenient to measure the chemical affinity of a system by 
 the maximum amount of external work which, under suitable 
 conditions, the reaction may be made to perform* This is a 
 special case of a very comprehensive natural law, which may 
 be expressed as follows : All spontaneous reactions (in the 
 widest sense, including neutralization of electrical charges, 
 falling of liquids to a lower level, etc.) can be made to 
 perform work, and all reactions which can be made to per- 
 form work are spontaneous, /.<?., can proceed of themselves 
 without the application of external forces. The available 
 energy of a chemical reaction, that is, that part of the total 
 energy which at constant temperature and under suitable condi- 
 tions can be made to perform an equivalent of work, has been 
 termed " free energy " by Helmholtz. The chemical affinity 
 or driving force of a reaction is not proportional to the total 
 change of energy, but to the change in the available or free energy'. 
 
 The total energy, U, of a chemical change can be obtained 
 in the form of heat by carrying out the reaction under such 
 conditions that no external work is done (p. 141). We have 
 now to consider what is the connection between the total 
 decrease of energy, U, and the decrease of available or free 
 energy, which may be termed A. This question is closely 
 connected with the conditions under which heat can be con- 
 tinuously transformed into work. We have to find an expres- 
 sion for the maximum work performed in a cycle in which the 
 heat is taken in at the temperature T, and given out at the 
 slightly lower temperature T - dT. The principle to be used for 
 this purpose is that employed in the theory of the steam-engine, 
 but it has universal applicability. According to this, the maxi- 
 mum work, </A, obtainable from a given quantity of heat Q is 
 given by 
 
THERMOCHEMISTRY 151 
 
 Equation (i) is th3 mathematical expression of the second law 
 of thermodynamics. 
 
 The first law of thermodynamics may be expressed in the 
 form (p. 139) 
 
 U = A-? . . . . (2) 
 
 in which A represents the external work done when the total 
 diminution of energy is U and the heat given out is - q. Now 
 dA, in equation (i), is the difference of the maximum amounts 
 of work obtainable in isothermal processes at the temperatures 
 T and T - dT ; thus A has the same significance here as in 
 equation (2). We may therefore obtain an expression in which 
 q does not occur by substituting for q in equation (i) its value 
 A-U, from equation (2). We thus obtain 
 
 A-U = T^ ... (3) 
 
 in which U represents the total change of energy in the re- 
 action, A represents the free energy or chemical affinity, and 
 dA/dT the rate of change of the free energy with tempera- 
 ture. Equation (3) is the fundamental equation for isothermal 
 chemical changes, that is, for chemical changes which take 
 place at constant temperature. The equation shows that the 
 change in the free energy differs from the change in the total 
 energy by an amount q = T(dA/dT), and that the two 
 quantities can only be identical when the right-hand side of 
 the equation is zero, q = - T(^A/OT) is sometimes termed 
 the latent heat of the reaction. 
 
 As the above equation has been derived from the second 
 law of thermodynamics, it follows that the free energy can be 
 determined only for reactions which can be made completely 
 reversible. By complete reversibility we mean that if a system 
 in changing from the state A to B, performs an amount of 
 
152 OUTLINES OF PHYSICAL CHEMISTRY 
 
 work, X, it can be restored to the condition A by the expendi- 
 ture of the same amount of work. Reactions such as the dis- 
 sociation of calcium carbonate in a closed space by heat are 
 reversible (p. 173), but a reaction in which gases escape from 
 the system, as when zinc is dissolved in acid, are, of course, 
 not reversible. Many reactions which are not reversible under 
 ordinary conditions can be carried out reversibly in galvanic 
 elements (for example, the displacement of copper from solu- 
 tion by zinc in the Daniell element), and therefore measure- 
 ments of electromotive force are largely used for determinations 
 of the free energy in a system. As will be shown later (p. 332), 
 when a chemical reaction takes place reversibly in a galvanic 
 cell, the electromotive force of the cell is proportional to the 
 free energy of the reaction. 
 
 It is beyond the scope of this book to discuss fully the 
 many deductions which may be made from equation (3), and 
 only one important consequence will be mentioned. At the 
 absolute zero (T = o) the right-hand side of the equation 
 becomes zero, and therefore the total change of energy U 
 (which is equal to the heat of reaction Q when no external 
 work is done) is equal to the change in the free energy A. In 
 other words, at the absolute zero, all reactions would proceed 
 in the direction in which heat is given out, and the heat evolved 
 would then be a measure of the chemical affinity. As the 
 absolute zero is at present unattainable, this statement by itself 
 is of no practical importance, but the form of equation (3) 
 indicates that at temperatures not very far from the absolute 
 zero, U and A may often not be greatly different. As ordinary 
 temperatures are rektively not very far from the absolute zero, 
 we can now understand, from the approximation in the values 
 of U and A, why so many chemical reactions proceed of them- 
 selves in the direction in which heat is given out (cf. p. 149). 
 
 Although the dissolving of some salts in water is attended 
 with absorption of heat, it can be shown that the free energy has 
 diminished, and the same is true of the vaporization of water, 
 
THERMOCHEMISTRY 153 
 
 In these extreme cases, not only are A and U of very different 
 numerical value, but they even have opposite signs. 
 
 Practical Illustrations As already mentioned, the heat 
 evolved in certain chemical reactions can conveniently be 
 measured by causing the reaction to proceed in the glass tube, 
 A, Fig. 21, and obtaining the heat of the reaction from the rise 
 in temperature of the water. This form of apparatus is, how- 
 ever, more useful for measuring heats of dilution and of solution, 
 If, for example, we wish to determine the heat of solution of 
 potassium chloride, a known weight of water is placed in the 
 inner beaker, a known weight of salt in the tube, A, and when 
 the salt may be expected to be at the same temperature as 
 the water, the glass tube is broken with a glass rod, the salt 
 dissolved in the water by stirring, and the change of temperature 
 read off on the thermometer. 
 
 For reactions in dilute solution, the tube A may be dispensed 
 with, and the outer beaker supported on corks in a third beaker, 
 so as to minimise the loss of heat by radiation. In this apparatus 
 the heat of neutralization of a dilute acid (half normal hydro- 
 chloric acid) by an equal volume of dilute alkali (sodium 
 hydroxide) may be determined. litre of the hydrochloric 
 acid is placed in the inner beaker, at a temperature 2 to 3 
 below that of the atmosphere, litre of N/2 sodium hydroxide, 
 of known temperature, is rapidly poured into the acid with 
 constant stirring. The highest temperature attained is noted' 
 If the solutions are at the same temperature before mixing, the 
 rise of temperature will be about 3*4, corresponding with the 
 fact that the heat of neutralization of i mol of sodium hydroxide 
 by hydrochloric acid is 13,700 cal. (p. 148). Measurements of 
 heat of neutralization, heat of dilution, etc., may be made still 
 more conveniently with metal calorimeters, as used in physical 
 laboratories ; the vessels should be well polished so as to 
 minimise the loss of heat by radiation. 
 
 Measurements should also be made with some form of com- 
 bustion calorimeter, if available. 
 
CHAPTER VII 
 
 EQUILIBRIUM IN HOMOGENEOUS SYSTEMS. LAW 
 OF MASS ACTION 
 
 General In the last chapter we have been mainly concerned 
 with the heat equivalents of chemical charges. We have now 
 to deal with chemical transformations, with reference more 
 particularly to the dependence of the rate and extent of chemical 
 reactions on the conditions. 
 
 When a chemical reaction can take place between two sub- 
 stances, it is usual to say that they have a certain "'chemical 
 affinity " for each other. From very early times the question as 
 to the nature of this affinity has been discussed, but up to the 
 present with very little success. Newton was of opinion that the 
 small particles of different kinds attract each other much as the 
 heavenly bodies do (gravitational attraction), and that the attrac- 
 tion falls off very rapidly with the distance. According to this 
 view, if we have three substances, A, B and C, and the attraction 
 between A and B is greater than that between A and C, then B 
 will completely displace C from its combination with A; in 
 other words, the reaction AC + B - AB + C will be complete 
 in the direction indicated by the arrow. These views found 
 their expression in the so-called affinity tables drawn up by 
 Stahl, Bergmann and others, in which the elements were arranged 
 in the order in which they could displace each other from com- 
 bination. Somewhat later, Berzelius developed his electro- 
 chemical theory, according to which the attractions concerned 
 in chemical changes are electrical in character, but this theory 
 
 '54 
 
EQUILIBRIUM IN HOMOGENEOUS SYSTEMS 155 
 
 proved in many respects unsatisfactory. The importance of the 
 conditions, more particularly as regards the relative amounts of 
 the reacting substances and the temperature, on the direction 
 and amount of chemical change, only came to be recognised 
 very gradually. 
 
 In recent years, the question as to why certain chemical 
 changes take place has been relegated to the background and 
 attention has been directed to how they take place. As mentioned 
 in the last chapter, it has been found possible in many cases 
 to obtain numerical values for the chemical affinity, without 
 troubling about its exact nature. When for any reaction the 
 chemical affinities of the reacting substances are known, as well 
 as the dependence of the reaction on the conditions, the reaction 
 is completely described. 
 
 Law of Mass Action The importance of the relative 
 amounts of the reacting substances for the course of a chemical 
 change was first clearly established by Wenzel and by Berth- 
 ollet. The latter pointed out that though under ordinary 
 circumstances sodium carbonate and calcium chloride react 
 almost completely according to the equation Na 2 CO 3 -f 
 CaCl 2 -> 2NaCl + CaCO 3 , yet the sodium carbonate found 
 on the shores of certain lakes in Egypt is produced according 
 to the equation 2NaCl + CaCO 3 -> Na 2 CO 3 -f CaCl 2 , the 
 converse of the first equation. In the latter case, the sodium 
 chloride is present in solution in such large excess that the re- 
 action proceeds in the direction indicated by the arrow, so that, 
 according to Berthollet, an excess in quantity can compensate 
 for a weakness in specific affinity. 
 
 An important step forward was made in this subject by 
 Berthelot and Pean de St. Gilles in 1862, in the course of an 
 investigation on the formation of esters from acids and alcohol. 
 For acetic acid and ethyl alcohol, the reaction may be repre- 
 sented by the equation 
 
 C 2 H 5 OH + CH 3 COOH ^ CH 3 COOC 2 H 5 + H 2 O. 
 If one starts with equivalent amounts of acid and alcohol, 
 
156 OUTLINES OF PHYSICAL CHEMISTRY 
 
 the reaction proceeds till about 66 per cent, of the reacting 
 substances have been used up, and then comes to a standstill. 
 Similarly, if equivalent quantities of ethyl acetate and water are 
 heated, the reaction proceeds in the reverse direction (indicated 
 by the lower arrow) until 34 per cent, of the compounds have 
 been used up and the mixture finally obtained is of the same 
 composition as when acid and alcohol are the initial substances. 
 A reaction of this type is termed a reversible reaction, and the 
 facts are conveniently represented by the oppositely-directed 
 arrows. 
 
 When, however, for a fixed proportion of acid, varying 
 amounts of alcohol are taken, the equilibrium point is greatly 
 altered, as is shown in the accompanying table. The first and 
 third columns show the proportion of alcohol present for i 
 equivalent of acetic acid, and the second and fourth columns 
 the proportion of acid per cent, converted to ester. 
 
 Ester 
 Formed. 
 82-8 
 88-2 
 93*2 
 
 lOO'O 
 
 We here measure the amount of chemical action by the extent 
 to which the acid is converted into ester, and the table shows 
 very clearly the influence of the mass of the alcohol on the 
 equilibrium. 
 
 The influence of the relative proportions of the reacting 
 substances on chemical action was thus clearly recognised, but 
 was not accurately formulated till 1867. In that year, two Nor- 
 wegian investigators, Guldberg and Waage, enunciated the Law 
 of mass action, which may provisionally be expressed as follows : 
 The amount of chemical action is proportional to the active mass 
 of each of the substances reacting, active mass being defined as the 
 molecular concentration of the reacting substance. The im- 
 portant part of this statement is that the chemical activity of 
 
 Equivalents of 
 Alcohol. 
 
 Ester 
 Formed. 
 
 Equivalents of 
 Alcohol. 
 
 0'2 
 
 19*3 
 
 2'0 
 
 O'S 
 I'O 
 
 42*0 
 66-5 * 
 
 4-0 
 
 I2'0 
 
 1-5 
 
 77'9 
 
 50*0 
 
EQUILIBRIUM IN HOMOGENEOUS SYSTEMS 157 
 
 a substance is not proportional to the quantity present, but to 
 its concentration, or amount in unit volume of the reaction 
 mixture. The law applies in the first instance more particularly 
 to gases and substances in solution ; the active mass of solids 
 will be considered later. 
 
 The " amount of chemical action " exerted by a certain sub- 
 stance can be measured (a) from its influence on the equilibrium, 
 as in the formation of ethyl acetate, just referred to ; (ft) from 
 its influence on the rate of a chemical action, such as the inver- 
 sion of cane sugar. The law of mass action can therefore be 
 deduced from the results of kinetic or equilibrium experiments. 
 Conversely, once the law is established, it can be employed both 
 for the investigation of rates of reaction and of chemical 
 equilibria, and it is the fundamental law in both these branches 
 of physical chemistry. 
 
 In the above form, the law of mass action cannot readily be 
 applied, and it will therefore be formulated mathematically. 
 For purposes of illustration, we choose a reversible reaction 
 between two substances in which only one molecule of each 
 reacts ; a typical case is the formation of ethyl acetate and 
 water from ethyl alcohol and acetic acid, already referred to. 
 Calling the molecular concentrations of the reacting substances 
 a and b, the rate at which they combine is, according to the law 
 of mass action, proportional to a and to b separately, and there- 
 fore proportional to their product. We may therefore write for 
 the initial velocity of reaction at the time t Q 
 
 Rate^ oc ab or Rate <n = kab, 
 
 where k is a constant an affinity constant depending only on 
 the nature of the substances, the temperature, etc. As the re- 
 action proceeds, the active masses gradually diminish, since the 
 original substances are being used up in producing the new 
 substances. If, after an interval of time t, x equivalents of the 
 ester and water have been formed, the rate of the original 
 reaction will be 
 
158 OUTLINES OF PHYSICAL CHEMISTRY 
 
 Rate, = k(a-x)(b-x). 
 
 That this must be so is clear when one bears in mind that 
 a and b represent molecular concentrations, and that for every 
 molecule of ester and of water which are formed, an equal 
 number of molecules of acid and alcohol must be used up. 
 We have now to take into account the fact that the substances 
 formed react to produce the original substances. At the time 
 /, when the concentration of the ester and water is x, the rate 
 of the reverse reaction will be : Rate t = /fc^ 2 , where k is 
 another affinity constant. We then have two reactions pro- 
 ceeding in opposite directions, the velocity of the direct reaction 
 is continually diminishing owing to diminishing concentration, 
 that of the reverse reaction is continually increasing owing to 
 increasing concentration of the reacting substances. A point 
 must ultimately be reached when the velocity of the direct is 
 equal to that of the reverse reaction, and the system will no 
 longer change ; this is the condition of equilibrium. If the 
 particular value of x under these conditions is x we have the 
 equations 
 
 Rate d i re ct = k(a - x^)(b - xj and rate 
 and since these are equal 
 
 *(a - *J(* - *,) = 
 which may be written 
 
 (a - x^(b - arj _ i 
 
 rev erse 
 
 x 
 
 The facts are made still clearer if we represent the reaction 
 as follows, the initial concentrations being represented on the 
 upper, and the equilibrium concentrations on the lower line : 
 Commencement a b o o 
 
 C 2 H 6 OH + CH 3 COOH^CH 3 COOC 2 H 5 + H 2 O 
 Equilibrium a - x l b - x t Xj^ x 1 
 
 It is important to note that, since K, which is usually termed 
 the equilibrium constant, is the ratio of the two velocity con- 
 
EQUILIBRIUM IN HOMOGENEOUS SYSTEMS 159 
 
 slants, which are independent of the concentration, the above 
 equation holds for all concentrations. Hence, if the equilibrium 
 constant for any chemical change is obtained from one experi- 
 ment, the equilibrium conditions can be calculated for any value 
 of the original concentrations. Numerous applications of this 
 equation are given in the succeeding paragraphs. 
 
 When more than one molecular equivalent of a compound 
 takes part in a chemical change, each equivalent must be con- 
 sidered separately, as far as the law of mass action is concerned. 
 
 In order to illustrate this statement, we will consider the com- 
 bination of hydrogen and iodine to form hydriodic acid. The 
 reaction is reversible, and may therefore be represented by the 
 equation 
 
 H 2 + I 2 ^HI + HI. 
 
 The rate of the inverse reaction = ^iCfn, since it is propor- 
 tional to the concentration of each of the two mols of hydriodic 
 acid and therefore to their product. As the velocity of the 
 direct reaction = CH 2 Ci a , we obtain for the conditions at 
 equilibrium the equation 
 
 CH,C k== ^ L=Kj 
 
 The general equation for a reversible reaction may be written 
 in the form 
 
 where n molecules of the substance A 1 react with 2 molecules 
 of the substance A^ ... to form n^ molecules of the substance 
 A/ and 2 ' molecules of the substance A 2 '. The rates of the 
 direct and reverse actions are represented by the equations : 
 Rate direct ~*C*;C ... and rate invers e = ^C'jic^ .... 
 and in equilibrium 
 
 A ' ** ' 
 
 Aj A> 
 
 The above is the strict mathematical form of the law of mass 
 
160 OUTLINES OF PHYSICAL CHEMISTRY 
 
 action, which in words may be expressed as follows : At 
 equilibrium the product of the concentrations on one side, divided 
 by the product of the concentrations on the other side, is constant 
 at constant temperature. Thus for the reaction represented by 
 the equation :- 
 
 2FeCl 3 + SnCl 2 =SnCl 4 + 2FeCl 2 
 we have 
 
 k' CFeCIs CsnCl 3 
 
 r rs 
 
 ^SnC1 4 ^FeCl-j 
 
 Strict Proof of the Law of Mass Action The law of 
 
 mass action, the meaning of which has been illustrated in 
 the previous paragraphs, may be strictly proved by a thermo- 
 dynamical method (van't Hoff, 1885, cf. p. 416), or by a 
 molecular-kinetic method (van't Hoff, 1877). The latter proof 
 is comparatively simple, and depends on the assumption that 
 the rate of chemical change is proportional to the number of 
 collisions between the reacting molecules, which, in sufficiently 
 dilute solution, will be proportional to the respective concen- 
 trations. Taking again ester formation as an example, the 
 velocity of the direct change = C a i CO hoi Cadd an( * tnat ^ t ^ ie 
 reverse change = ^jCester C wa ter. At equilibrium, the rates will 
 just balance, and therefore 
 
 As before, this equation may be put in the form 
 
 ^alcohol ^acid i -r/- 
 
 r - r - = T "* K> 
 
 '-'ester Cwater ^ 
 
 where the respective concentrations are those under equilibrium 
 conditions, and K is the equilibrium constant. 
 
 It follows from the assumptions made both in the thermo- 
 dynamical and kinetic proofs that the law of mass action holds 
 strictly only for very dilute solutions, but the experimental 
 results show that it often holds with a fair degree of accuracy 
 even for moderately concentrated solutions.^ 
 
EQUILIBRIUM IN HOMOGENEOUS SYSTEMS 161 
 EQUILIBRIUM IN GASEOUS SYSTEMS 
 
 (a) Decomposition of Hydriodic Acid A typical example 
 of equilibrium in a gaseous system is that between hydrogen, 
 iodine and hydriodic acid, investigated by Bodenstein. 1 The 
 reaction, which is represented by the equation H 2 + I 2 ^ 2HI, 
 is a completely reversible one, the concentration at equilibrium 
 being the same whether one starts with hydrogen and iodine 
 or with hydriodic acid, if the conditions otherwise are the same. 
 
 Applying the law of mass action, we get at once 
 
 as shown in the previous paragraph. 
 
 It is clear from the equation that if from one observation the 
 respective molecular concentrations of iodine, hydrogen and 
 hydriodic acid are known, K, the equilibrium constant at the 
 temperature in question, can be calculated. 
 
 The question now arises as to how the progress of the 
 reaction can be followed, so that it may be known when 
 equilibrium is attained. It is further necessary to find a 
 method of measurement such that the equilibrium does not 
 alter while the observations are being made. In this case it 
 happens that both the direct and inverse reactions are extremely 
 slow at room temperature, but are fairly rapid at 445, the tem- 
 perature of boiling sulphur. If then the mixture is heated for 
 a definite time at a high temperature and then cooled rapidly, 
 the respective concentrations at high temperatures can be deter- 
 mined at leisure by analysis. The reacting substances, in varying 
 proportions, are heated at a definite temperature in sealed glass 
 tubes for definite periods, and the amount of hydrogen then 
 present measured after absorption of the iodine and hydriodic 
 acid by means of potassium hydroxide. 
 
 For the present, only results will be considered in which the 
 tubes were heated so long at 445 that equilibrium was attained. 
 Jn one experiment, 20-55 mols of hydrogen were heated with 
 
 1 Zeitsch. physikal. Cliem-, 1897, 2*, I. 
 II 
 
1 62 OUTLINES OF PHYSICAL CHEMISTRY 
 
 31*89 mols of iodine, and it was found that the mixture at 
 equilibrium contained 2*06 mols of hydrogen, 13*40 mols of 
 iodine and 36*98 mols of hydriodic acid in the same volume. 
 Hence _ [HJflJ _ (2-06 x 13-40) 
 
 *~" CTTTT9 7 7 c^\-> " O'O2OO. 
 
 [HI] 2 (36'98) 2 
 
 Equation (i) could, of course, be tested by finding if the 
 same value of K is obtained for different initial concentrations 
 of the reacting substances, but it is in some respects preferable 
 to calculate by means of the equation the proportion of hydriodic 
 acid formed at equilibrium when different initial concentrations 
 of the reacting substances are taken, and to compare the results 
 with those actually observed. In the calculation, K is taken as 
 0*0200 at 445. 
 
 If i mol of hydrogen is heated with a mols of iodine, and 
 2X mols of hydriodic acid are formed, i - x mols of hydrogen 
 and a x mols of iodine will remain behind. Substituting in 
 equation (i), 
 
 (i - x)(a - x) . N 
 
 - - ~ - - = K = 0*0200 . . (2) 
 4# 2 
 
 The first and second columns of the accompanying table 
 contain the initial concentrations of hydrogen and iodine 
 respectively, and the fourth and fifth columns the observed 
 and calculated concentrations of hydriodic acid at equilibrium, 
 the latter values being obtained from the expression 
 
 ' + a ~ ^* + a " as 
 
 s 
 
 where s = i - 4K = 0*92 
 
 obtained by solving the quadratic equation (2) above. 
 
 H 2 I a Lj/H 2 = a HI found 2* (calc.) 
 
 20*57 5*22 0-254 10-22 10-19 
 
 20*6 i4'45 0*702 25-72 25-54 
 
 20'55 31*89 1*552 36-98 37-13 
 
 20*41 52*8 2-538 38*68 39'oi 
 
 20*28 67*24 3*316 39-52 39-25, 
 
EQUILIBRIUM IN HOMOGENEOUS SYSTEMS 163 
 
 The close agreement between observed and calculated values 
 shows that the law of mass action applies in this case. 
 
 It can easily be shown from the fundamental equation that 
 in this case the position of equilibrium is independent of the 
 pressure or of the volume. Calling a, b and c the amounts of 
 hydrogen, iodine and hydriodic acid present at equilibrium, the 
 concentrations are a/V, </V and cjV respectively, where V is 
 the volume occupied by the mixture. Substituting in the 
 general equation, we obtain ab\c^ = K ; in other words, K is 
 independent of the volume. Bodenstein found that this re- 
 quirement of the theory was also satisfactorily fulfilled. 
 
 (b) Dissociation of Phosphorus Pentachloride Another 
 instructive example of equilibrium in a gaseous system is that 
 between phosphorous pentachloride and its products of de- 
 composition, represented by the equation PC1 5 ^ PC1 3 + Q 2 . 
 A decomposition of this type, in which a chemical compound 
 yields one or more products, is termed dissociation, and the 
 student will have met with many examples of dissociation in his 
 earlier work. As before, on applying the law of mass action, 
 we obtain 
 
 If we commence with a molecules of PC1 3> and x molecules 
 each of PC1 3 and C1 2 are formed, the concentrations of PQ 5> 
 PC1 3 and C1 2 at equilibrium are (a x)/V, x/V and x/V re- 
 spectively, and, substituting in the above equation, 
 
 (a - *)V 
 
 It will be observed that the equilibrium in this case depends 
 on the volume, and the larger the volume the smaller is (a - x) 
 in other words, the greater is the dissociation. 
 
 An important point in connection with chemical equilibrium 
 in general is the effect of the addition of excess of one of the 
 products of decomposition (dissociation) on the degree of 
 
164 OUTLINES OF PHYSICAL CHEMISTRY 
 
 decomposition. If, for example, a mols of PC1 5 are vaporized 
 in a volume V in which b mols of PC1 3 are already present, and 
 if #! is the degree of dissociation of the pentachloride under these 
 conditions, the relative concentrations of trichloride, penta- 
 ci/loride and chlorine will be b 4 x lt a - x lt and x l respec- 
 tively. The equilibrium equation is therefore 
 fa) (b + xj _ 
 (a - Xl )V 
 
 where K has the same numerical value as for the pentachloride 
 alone, provided that the volume V and the temperature are tfa 
 same. If it is assumed that the degree of dissociation when 
 PC1 6 is heated alone under the same conditions is 'not more 
 than say 25 per cent., it is clear that the proportion of undis- 
 sociated compound cannot bj very seriously increased by the 
 presence of excess of PC1 3 . Hence when &, the initial amount 
 of PC1 3 , is made very large, x lt the amount of chlorine present 
 .at equilibrium must become very small in order that the pro- 
 duct K (a - x-i) may retain approximately the same Value ; in 
 other words, the dissociation of PC1 5 must then be very small. 
 From these considerations we deduce the following important 
 general rule : The degree of dissociation of a compound is 
 diminished by addition of excess of one of the products oj 
 dissociation provided that the volume remains constant. 
 
 Equilibrium in Solutions of Non-Electrolytes As a* 
 illustration of an equilibrium in solution, that between acid, 
 alcohol, ester and water (p. 156) may be considered rather 
 more fully. For this equilibrium, according to the law of 
 mass action, we have 
 
 If at the commencement a, b and c mols of acid, alcohol and 
 water respectively are present in V litres, and under equi- 
 librium conditions x mols of water and ester respectively have 
 been formed, the respective concentrations are 
 
EQUILIBRIUM IN HOMOGENEOUS SYSTEMS 165 
 _ a-x r-_* . r - - . r _ + x 
 
 ^acid y I ^alc. - y > ^ester y I Crater y ' 
 
 whence, substituting in the above equation, 
 (a-x)(i>-x) _ 
 
 x(c + x) 
 
 In this case also, the position of equilibrium is independent of 
 the volume. 
 
 The value of K may be obtained from the observation already 
 mentioned, that when acid and alcohol are taken in equivalent 
 proportions, two-thirds is changed to ester and water under 
 equilibrium conditions. Hence 
 
 This equation may now be employed, as in the case of hydriodic 
 acid, to calculate the equilibrium conditions for varying initial 
 concentrations of the reacting substances. As an example, we 
 take the proportion of i mol of acetic acid converted to ester by 
 varying proportions of alcohol, when the initial mixture contains 
 neither ester nor water. The equation in this case simplifies to 
 
 (I -*)(*-*) 
 
 X* " ~ * 
 
 whence x = |(i + b- JW&_ + i). The observed and cal- 
 culated values of x are given in the table, and it will be seen 
 that the agreement is very satisfactory, although the solution is 
 sp concentrated that it is scarcely to be expected that the law of 
 mass action will apply strictly. 
 
 b 
 
 x (found) 
 
 x (calc.) 
 
 * 
 
 x (lound) 
 
 x (calc.) 
 
 '5 
 
 '5 
 
 0*049 
 
 0-67 
 
 0-519 
 
 0-528 
 
 0-08 
 
 0-078 
 
 0-078 
 
 I'O 
 
 0-665 
 
 0-667 
 
 0-18 
 
 0-171 
 
 0-17 1 
 
 '*5 
 
 0-819 
 
 o'75 
 
 0-28 
 
 0*226 
 
 0-232 
 
 2*0 
 
 0-858 
 
 0-845 
 
 o'SJ 
 
 0-293 
 
 0-311 
 
 2-24 
 
 0-876 
 
 0-864 
 
 0*50 
 
 0-414 
 
 0-42^ 
 
 8-0 
 
 0-966 
 
 '945 
 
166 OUTLINES OF PHYSICAL CHEMISTRY 
 
 As regards the practical investigation of this equilibrium, the 
 reacting substances are heated in sealed tubes at constant 
 temperature (say 100) till equilibrium is attained, cooled, and 
 the contents titrated with dilute alkali, using phenolphthalein as 
 indicator. As the concentrations of acid and alcohol before the 
 experiment are known and the acid concentration after the at- 
 tainment of equilibrium is obtained from the results of the 
 titration, the proportion of ester formed can readily be calcu- 
 lated. 
 
 The equilibrium in salt solutions will be more conveniently 
 dealt with at a later stage (Chapter XL). 
 
 Influence of Temperature and Pressure on Chemical 
 Equilibrium. General The equations for chemical equi- 
 librium deduced by means of the law of mass action hold for 
 all temperatures provided that all the components remain in the 
 system : the only effect of change of temperature is to alter the 
 value of the equilibrium constant. The displacement of equi- 
 librium is connected with the heat liberated in the chemical 
 change by the equation 
 
 which shows that the rate of change of the logarithm of the 
 equilibrium constant with temperature is equal to the heat 
 evolved in the complete reaction l divided by twice the square 
 of the absolute temperature at which the change takes place. 
 
 Strictly speaking, the above equation holds only for the dis- 
 placement of equilibrium due to an infinitely small change of 
 temperature r/T, and must be integrated before it can be applied 
 to a concrete case. This can readily be done on the assumption 
 that Q remains constant between the two temperatures, which 
 is in general only approximately true. Integration between the 
 absolute temperatures Tj and T 2 gives on this assumption 
 
 log.K, - log.K 2 = , - 
 
 \ 1 2' 
 
 1 The negative sign is taken in order that Q may denote the heat evolved 
 in the forward reaction (from left to right). 
 
EQUILIBRIUM IN HOMOGENEOUS SYSTEMS 167 
 
 in which K 1 and K 2 are the equilibrium constants at r l\ and T 2 
 respectively. When transformed to ordinary logarithms (by 
 dividing by 2-3026) and R is put = 1*99 (p. 27), the above 
 equation is obtained in the more convenient form 
 
 . (2) 
 
 In this equation, Q refers only to the heat used in doing 
 internal work, and does not include that used in external work 
 (p. 141). It applies, therefore, in the first instance, only to 
 systems in which there is no change of volume, and if there is 
 expansion or contraction, the corresponding correction must be 
 applied (p. 27). The equation shows that Q may be calculated 
 when the equilibrium constants for two near temperatures, T 1 
 and T 2 , are known. Conversely, when the heat change in a 
 chemical reaction and the equilibrium constant for any one 
 temperature are known, the condition of equilibrium at any other 
 temperature may be calculated. The equation is particularly 
 useful for the indirect determination of the heat of reaction at 
 high temperatures (in gas reactions, for example) when the 
 direct calorimetric determination is difficult or impossible. 
 
 As an example of the application of the general equation (2), 
 the heat of dissociation, Q, for hydrogen sulphide, represented 
 by the equation 2H.,S; 2H 2 + S 2 , will be calculated. Ac- 
 cording to Preuner, the equilibrium constant K of the equation 
 
 nf4fvf = K > has the value 2 '9 x I0 ~ 5 at I220 abs< and 
 
 L"-2^J 
 
 10-4 x io~ 5 at 1320 abs. Hence, substituting in equation 
 (2), we have 
 
 io'4 x io~ 5 Q (1320 - 1220) 
 
 log 
 
 2^90 x io~ 5 45&i * (1320 x 1220) 
 
 and Q = - 41,000 cal. approximately. 
 
 A specially interesting case is that in which there is no heat 
 change when the first system changes to the second. Since in 
 this case Q = o, the right-hand side of equation (i) becomes 
 
 zero, and therefore there should be no displacement of equili- 
 
 t 
 
1 68 OUTLINES OF PHYSICAL CHEMISTRY 
 
 brium with temperature. The condition of zero heat of reaction 
 is, as has already been pointed out (p. 145), approximately ful- 
 filled in ester formation, and in accordance with this, Berthelot 
 found that at 10 65*2 per cent, of the acid and alcohol change 
 to ester and at 220 66'5 per cent. ; the displacement of 
 equilibrium with temperature is therefore slight. 
 
 There are certain rules of great importance which show 
 qualitatively how the equilibrium is displaced with changes of 
 temperature and pressure. If Q is the heat developed when 
 the system A changes to the system B, and is positive, then 
 with rise of temperature A increases at the expense of B ; con- 
 versely, if Q is negative, B increases with rise of temperature 
 at the expense of A. These statements may be summarised as 
 follows : At constant volume increase of temperature favours 
 the system formed under heat absorption and conversely. 
 
 As an example, we may take nitrogen peroxide, N 2 O 4 ^ 2NO 2 , 
 for which the change represented by the lower arrow is attended 
 with the liberation of a large amount (12,600 cal.) of heat. 
 Increase of temperature favours the reaction for which heat is 
 absorbed, in this case the reaction represented by the upper 
 arrow, so that as the temperature rises N, 2 O 4 is split up more 
 completely into NO 2 molecules. 
 
 Another interesting example is the relationship between 
 oxygen and ozone, represented by the equation 2O 3 = 3O 2 + 
 2 x 29,600 cal. The equilibrium for the reaction 2O 3 ^3O,, 
 is very near the oxygen side at the ordinary temperature, but 
 increase of temperature must displace it in the direction repre- 
 sented by the lower arrow, since under these circumstances heat 
 is absorbed ; in other words, ozone becomes increasingly stable 
 as the temperature rises. The experimental results so far 
 obtained are in satisfactory agreement with the theory. 1 
 
 Compare Fischer and Marx, Berichte, 1907, 40, 443. At first sight 
 this appears to be in contradiction to the well-known fact that when a 
 mixture of oxygen and ozone is heated to 250 the ozone is pract cally 
 destroyed. It must be remembered, however, that the mixture contains 
 far too much ozone for equilibrium, but owing to the lo\v temperature it 
 attains its true equilibrium very slowly. At 250, however, the attainment 
 of equilibrium is fairly rapid. 
 
EQUILIBRIUM IN HOMOGENEOUS SYSTEMS 169 
 
 From the above considerations we conclude that endothermic 
 compounds, such as ozone, acetylene and carbon disulphide, 
 become increasingly stable as the temperature rises, whilst 
 exothermic compounds undergo further dissociation. 
 
 A similar law can be enunciated for the effect of pressure on 
 equilibrium as follows : On increasing th j . pressure at constant 
 temperature the equilibrium is displaced in the direction in which 
 the volume diminishes. Taking as an illustration the gaseous 
 equilibrium, PCl f ^PCl 3 + C1 2 in which the upper arrow 
 indicates the direction of increase of volume, the rule indicates 
 that increase of pressure will displace the equilibrium to the 
 left, whilst decrease of pressure will favour the reverse change. 
 As is well known, these deductions are in complete accord with 
 the experimental facts. 
 
 For reactions not attended by any appreciable change of 
 volume, such as the decomposition of hydriodic acid at high 
 temperatures, the equilibrium should not be altered by change 
 of volume, a conclusion borne out by experiment (p. 163). 
 
 Le Chatelier's Theorem Le Chatelier has pointed out 
 that the rules above referred to with regard to the effect of 
 changes of temperature and pressure on equilibria are special 
 cases of a much more general law which may be enunciated 
 as follows : IVhen one or more of the factors determining an equi- 
 librium are altered, the equilibrium becomes displaced in such a 
 way as to neutralize, as far as possible, the effect of the change. 
 A little consideration will show that this rule affords a satisfac- 
 tory interpretation of all the phenomena just mentioned. 
 
 Relation between Chemical Equilibrium and Tem- 
 perature. Nernst's Views Although the van't Hoff 
 equation connecting equilibrium and temperature enables us 
 to calculate the position of equilibrium at different tempera- 
 tures when the position of equilibrium at one temperature 
 and the heat of reaction are known, it has not until quite 
 recently been possible to calculate chemical equilibria from 
 thermal and thermochemical data alone. Within the last 
 
lyo OUTLINES OF PHYSICAL CHEMISTRY 
 
 two or three years, the latter problem appears to have been 
 to a great extent solved by Nernst. 1 The fundamental as- 
 sumption, on the basis of which it has been found possible 
 to deduce formulae connecting the equilibrium in a system 
 with the thermal data characteristic of the reacting substances, 
 is that the free energy, A, and the total heat of reaction, Q, 
 are not only equal at the absolute zero, as already pointed out 
 (p. 152), but their values coincide completely in the immediate 
 vicinity of that point. It is evident that this assumption cannot 
 be tested directly, but the fact that the formulae deduced on 
 this basis have been to a great extent confirmed by experiment 2 
 goes far to justify it. 
 
 There can be no doubt that the results just described con- 
 stitute one of the most important advances in physics and 
 chemistry of recent years. It is beyond the scope of the 
 present book to discuss the question more fully, but it may 
 be mentioned that the theory not only admits of the calcula- 
 tion of equilibria in homogeneous and heterogeneous systems 
 from thermal data, but also gives a formula representing the 
 variation of vapour pressure with temperature. 
 
 Practical Illustrations The law of mass action may be 
 illustrated most conveniently by the action of water on bismuth 
 chloride, represented by the equation 
 
 BiCl 3 + H 2 O^BiOCl + 2HCl. 
 
 When dilute hydrochloric acid is added to a mixture of the 
 salt and water, the equilibrium is displaced in the direction 
 represented by the lower arrow, and a homogeneous solution is 
 obtained. If excess of water is added to this solution, the 
 equilibrium is displaced in the direction represented by the 
 upper arrow, and a precipitate of bismuth oxychloride is formed. 
 
 The law may also be illustrated qualitatively by the inter- 
 action of ferric chloride and ammonium thiocyanate to form 
 
 1 Nernst, Applications of Thermodynamics to Chemistry. London: 
 Constable, 1907. Annual Reports, Chemical Society, 1906, pp. 2.0-22. 
 
 2 Nernst, loc. cit. 
 
EQUILIBRIUM IN HOMOGENEOUS SYSTEMS 171 
 
 blood-red ferric thiocyanate. 1 This reaction is of particular 
 interest, as it was one of the first reversible reactions to be 
 systematically investigated (J. H. Gladstone, i855). 2 The 
 equation representing the reaction is as follows : 
 
 FeCl 3 + 3NH 4 CNS ^ Fe(CNS) 3 + 3 NH 4 C1. 
 
 Solutions of the salts are first prepared ; the thiocyanate 
 solution contains 37 grams of the salt to 100 c.c. of water, 
 and the ferric chloride solution 3 grams of the commercial 
 salt and 12*5 c.c. of concentrated hydrochloric acid to 100 
 c.c. of water. 5 c.c. of each of the solutions are added to 
 2 litres of water and the solution divided between four beakers. 
 The solutions are pale-red in colour, as the equilibrium lies 
 considerably towards the left-hand side. To the contents of 
 two of the beakers are added 5 c.c. of the ferric chloride and 
 the thiocyanate solution respectively, and it will be observed 
 that the solutions become deep red, owing to the displace- 
 ment of the equilibrium in the direction of the upper arrow. 
 On the other hand, the addition of 50 c.c. of a concentrated 
 solution of ammonium chloride 3 to the solution in the third 
 beaker makes it practically colourless, the equilibrium being 
 displaced in the direction of the lower arrow, in accordance 
 with the law of mass action. 
 
 1 Lash Miller and Kenrick, J. Amtf. Chem. Soc., 22, 291. 
 
 2 Phil. Trans. Roy. Soc., 1855, 179. 
 
 :{ Strictly speaking, all the solutions should be made up to the same 
 volume in each case, but for qualitative purposes the method described is 
 sufficiently accurate. The deep-red colour is presumably due to the for- 
 mation ot non-ionised ferric thiocyanate (p. 260) ; it cannot be due to 
 Fe 1 " or to CNS 1 ions, which are practically colourless. 
 
CHAPTER VIII 
 
 HETEROGENEOUS EQUILIBRIUM. THE PHASE 
 RULE 
 
 General In contrast to homogeneous systems, in which 
 the composition is uniform throughout, heterogeneous systems 
 are made up of matter in different states of aggregation. The 
 separate portions of matter in equilibrium are usually termed 
 phases ; each phase is itself homogeneous, and is separated by 
 bounding surfaces from the other phases. Liquid water in 
 equilibrium with its vapour is a heterogeneous system made 
 up of two phases, the equilibrium in this case being of a 
 physical nature. Another heterogeneous equilibrium, formed 
 by calcium carbonate with its products of dissociation, consists 
 of three phases, two of which are solid, calcium carbonate and 
 calcium oxide, and one gaseous. A still more complicated case 
 is the equilibrium between a solid salt, its saturated solution, 
 and vapour, made up of a solid, a liquid and a gaseous phase. 
 It should be remembered that though each phase must be 
 homogeneous, both as regards chemical and physical pro- 
 perties, it may be chemically complex. For example, a mixture 
 of gases only forms a single phase, since gases are miscible in 
 all proportions. Further, a phase may be of variable composi- 
 tion, thus a solution only constitutes one phase, although it 
 may vary greatly in concentration. 
 
 Application of Law of Mass Action to Heterogeneoub 
 Equilibrium It has already been shown that equilibria in 
 homogeneous systems may be dealt with satisfactorily by 
 
 172 
 
HETEROGENEOUS EQUILIBRIUM 173 
 
 means of the law of mass action, provided that the molecular 
 condition of the reacting substances is known. The matter is, 
 however, somewhat more complicated for heterogeneous equi- 
 libria, more particularly when solid substances are present, as 
 in the equilibrium between calcium carbonate, calcium oxide 
 and carbon dioxide already referred to. Debray, who investi- 
 gated this system very carefully, showed that, just as water at a 
 definite temperature has a definite vapour pressure, independent 
 of the amount of liquid present, there is a definite pressure of 
 carbon dioxide over calcium carbonate and oxide at a definite 
 temperature, independent of the amount or the relative propor- 
 tions of the solids present. The question now arises as to 
 how the law of mass action is to be applied to systems in 
 which solids are present. This problem was solved by Guld- 
 berg and Waage, who found that the experimental results, such 
 as those for the dissociation of calcium carbonate, were satis- 
 factorily represented on the assumption that the active mass oj 
 a solid substance at a definite temperature is constant, i.e., inde- 
 pendent of the amount of solid present. 
 
 It was not at first clear what physical meaning is to be 
 attached to this statement, but Nernst pointed out that for 
 any such system it was sufficient to consider the equilibrium 
 in the gaseous phase, the active mass of a solid being repre- 
 sented as its concentration in the gaseous phase. In other 
 words, a solid, like a liquid, may be regarded as having a 
 definite vapour pressure at a definite temperature, independent 
 of its amount. At first sight it may seem surprising to ascribe 
 a definite vapour pressure to such a substance as calcium oxide, 
 but it is well known that solids like bismuth and cadmium have 
 definite vapour pressures at moderate temperatures, and there 
 is every reason for supposing that the diminution of vapour 
 pressure with fall of temperature is continuous. There is now 
 no difficulty in applying the law of mass action to equilibria 
 in which solid substances are concerned, for example, to the 
 dissociation of calcium carbonate. For convenience, we will 
 
174 OUTLINES OF PHYSICAL CHEMISTRY 
 
 use the partial pressures, />, of the components in the gaseous 
 phase as representing the active masses. 1 We then obtain 
 
 CO 
 
 whence pco = -7* - 3 = constant. 
 
 / 
 
 Otherwise expressed, since all the factors on the right-hand 
 side of the equation are constant at constant temperature, the 
 vapour pressure of carbon dioxide must be constant, which is 
 in accordance with the experimental facts. 
 
 It is clear from the form of the equation that the pressure 
 remains constant only within limits of temperature such that 
 both calcium carbonate and oxide are present. If the tem- 
 perature is so high that no calcium carbonate is present, the 
 pressure is no longer defined, but depends on the size of the 
 vessel, etc. 
 
 Dissociation of Salt Hydrates Other interesting examples 
 of heterogeneous equilibrium are those between water vapour and 
 salts with water of crystallisation. If, for example, crystallised 
 copper sulphate, CuSO 4 , 5H 2 O is placed in a desiccator over 
 concentrated sulphuric acid at 50, it gradually loses water and. 
 finally only the anhydrous sulphate remains. If arrangements. 
 are made for continuously observing the pressure during dehyd- 
 ration, it will be found to remain constant at 47 mm. until the: 
 salt has lost two molecules of water, it then drops to 30 mm. 
 and remains constant until other two molecules of water have- 
 been lost, when it suddenly drops to 4-4 mm. and v.mains 
 constant till dehydration is complete. The explanation of the 
 successive constant pressures observed during dehydration is 
 similar to that already given for the constant pressure of carbon 
 dioxide over calcium carbonate and oxide. At 50 the hydrates 
 CuSO 4 , 5H 2 O and CuSO 4 , 3H 2 O are in equilibrium with a 
 
 1 The partial pressure of a gas is proportional to the number of particles 
 present per unit volume and therefore to its molQQuJar Concentration or . 
 active mass (cf. p. 156). 
 
HETEROGENEOUS EQUILIBRIUM 175 
 
 pressure of aqueous vapour = 47 mm., and as long as any of the 
 pentahydrate is present, the pressure necessarily remains con- 
 stant. When, however, all the pentahydrate is used up, the 
 trihydrate begins to dehydrate, giving rise to a little of the 
 monohydrate, CuSO 4 , H 2 O. As a new substance is then taking 
 part in the equilibrium, the pressure of aqueous vapour neces- 
 sarily alters, and remains at the new value until the trihydrate is 
 used up. The successive equilibria are represented by the 
 following equations : 
 
 I. CuSO 4 , 5H 2 O;CuSO 4 , 3H 2 O + 2H 2 O. 
 II. CuSO 4 , 3H 2 O^CuSO 4 , H 2 O + 2H 2 O. 
 III. CuSO 4 , H 2 6^CuSO 4 + H 2 O. 
 
 By applying the law of mass action to any of the above 
 equations, it may easily be shown that the pressure of aqueous 
 vapour must be constant at constant temperature. Putting the 
 partial pressures of the pentahydrate and the trihydrate as Pj 
 and P 2 respectively, we have from equation I. : 
 
 whence / 2 H 2 o = 7-77- = constant. 
 *i l 2 
 
 It is important to realise clearly that the observed pressure is 
 not due to any one hydrate, it is only definite and fixed when 
 both hydrates are present. 
 
 The tension of aqueous vapour over hydrates, like the vapour 
 pressure of water, in creases rapidly with the temperature. This 
 is illustrated in the following table, in which the vapour pressures 
 (in mm.) over a mixture of Na 2 HPO 4 , 7H 2 O and Na.,HPO 4 , 
 and those of water at the same temperatures, are given : 
 Temperature . . 12-3 16-3 207 249 31-5 36-4 40-0 
 
 Na 2 HPO 4 + o-7H 2 O . 4-8 6-9 9-4 12-9 21-3 30-5 41-2 
 Water .... 10-6 13-8 18-1 23-4 34-3 45-1 54 - g 
 Ratio salt/water . . 0-46 0-50 0-52 0-55 0-62 0-68 075 
 
 The results throw light on the question of the efflorescence 
 (giving up of water) and deliquescence (absorption of water) of 
 
.76 OUTLINES OF PHYSICAL CHEMISTRY 
 
 hydrated salts in contact with the atmosphere. If a hydrate 
 (in the presence of the next lower hydrate) has a higher vapour 
 pressure than the ordinary pressure of aqueous vapour in the 
 atmosphere, it will lose water and form a lower hydrate. For 
 example, the salt Na 2 HPO 4 , i2H 2 O has a vapour pressure of 
 over 1 8 mm. at 25, which is greater than the average pressure 
 of aqueous vapour in the atmosphere at that temperature, though 
 less than the saturation pressure (see table), and therefore the 
 salt is efflorescent under ordinary conditions. On the other 
 hand, the vapour tension of the heptahydrate at 25 is only 
 13 mm. and it is therefore stable in air. The table shows that 
 the ratio of the vapour pressure of a hydrate to that of water 
 increases rapidly with the temperature and ultimately it becomes 
 greater than unity ; the vapour tension of the hydrate is then 
 greater than that of water. 
 
 Dissociation of Ammonium Hydrosulphide Solid 
 ammonium hydrosulphide partly dissociates on heating into 
 ammonia and hydrogen sulphide, according to the equation 
 NH 4 HS^NH 3 + H 2 S. This equilibrium is of a different type 
 to those already mentioned, as a solid dissociates into two 
 gaseous components. Representing molecular concentrations as 
 partial pressures, we obtain, on applying the law of mass action, 
 
 - K/ NH4 Hs = constant, 
 
 since the partial pressure of solid ammonium sulphide is 
 constant at constant temperature. 
 
 The equation indicates that the product of the partial 
 pressures of the two gases is constant at constant temperature. 
 
 When the gases are obtained by heating ammonium hydro- 
 sulphide, they are necessarily present in equivalent amount and 
 exert the same partial pressure. The above formula may, 
 however, be tested by adding excess of one of the products 
 of dissociation to the mixture. This was done by Isambert, 
 with the result indicated in the following table, which holds for 
 25*1; the volume being kept constant throughout: 
 
HETEROGENEOUS EQUILIBRIUM 177 
 
 250-5 250-5 62,750 
 
 208*0 294*0 60,700 
 
 453-0 143-0 64,800 
 
 In the first experiment the gases are present in equivalent 
 proportions, in the second experiment excess of hydrogen 
 sulphide, in the third excess of ammonia have been added. 
 The results indicate that the product of the pressures is con- 
 stant within the limits of experimental error, as the theory 
 indicates, and, further, that addition of excess of one of the 
 products of dissociation diminishes the amount of the other, as 
 already shown for phosphorus pentachloride (p. 163). 
 
 Analogy between Solubility and Dissooiation There 
 is a very close analogy between the solubility of solids in 
 liquids and the equilibrium phenomena just considered, more 
 particularly the dissociation of salt hydrates. In both cases 
 there is equilibrium between the solid as such and the same sub- 
 stance in the other (gaseous or liquid) phase. We have already 
 seen, in the case of the hydrates of copper sulphate, that the 
 vapour pressure (i.e., the concentration of vapour in the gas 
 space) depends on the composition of the solid phases, and 
 it is then easy to see that the solubility of sodium sulphate 
 (its concentration in the liquid phase) must also depend on 
 the composition of the solid phase. The solubility alters when 
 the solid decahydrate changes to the anhydrous salt, just as 
 does the vapour pressure when copper sulphate pentahydrate 
 disappears. A further analogy between the two phenomena is 
 that just as the addition of an indifferent gas to the gas phase 
 does not alter the equilibrium, except in so far as the volume is 
 changed, so the addition of an indifferent substance to a solu- 
 tion does not greatly affect the solubility of the original solute. 
 
 Distribution of a Solute between two Immiscible 
 
 Liquids The distribution of a solute such as succinic acid 
 
 bet-ween two immiscible liquids such as ether and water exactly 
 
 corresponds with the distribution of a substance between the 
 
 12 
 
178 OUTLINES OF PHYSICAL CHEMISTRY 
 
 liquid and gas phase (p. 83), and therefore the rules already 
 mentioned for the latter equilibrium apply unchanged to the 
 former. The most important results may be expressed as 
 follows (Nernst) : 
 
 (1) If the molecukr weight of the solute is the same in both 
 solvents, the distribution coefficient (the ratio of the concentra- 
 tions in the two solvents after equilibrium is attained) is con- 
 stant at constant temperature (Henry's law). 
 
 (2) In presence of several solutes, the distribution for each 
 solute separately is the same as if the others were not present 
 (Dalton's law of partial pressures). 
 
 The first rule may be illustrated by the results obtained by 
 Nernst for the distribution of succinic acid between ether and 
 water, which are given in the table : 
 
 C x (in water) C a (in ether) Cj/C, 
 
 0*024 0*0046 5*2 
 
 0*070 0*013 ' 5'4 
 
 o'i2r 0*022 5*4 
 
 The results were obtained by shaking up varying quantities 
 of succinic acid with 10 c.c. of water and 10 c.c. of ether in 
 a separating funnel, and determining the concentrations of acid 
 in the two layers after they had separated completely. The 
 fact that the ratio Q/Cg is approximately constant shows that 
 Henry's law applies. 
 
 When the molecular weight of the solute is not the same 
 in both solvents, the ratio of the concentrations is no longer 
 constant, and, conversely, if the ratio of the concentrations is 
 not constant at constant temperature, the molecular weight 
 cannot be the same in both solvents. This is illustrated by the 
 following results obtained by Nernst for the distribution of 
 benzoic acid between water and benzene : 
 
 Cj (in water) C 2 (in benzene) Cj/C 2 CJ */C^ 
 
 0*0150 0*242 0*062 0*0305 
 
 0*0195 0*412 0*048 0*0304 
 
 0*0289 0*970 0*030 0*0293 
 
HETEROGENEOUS EQUILIBRIUM 179 
 
 As the table shows, the ratio Cj/Cg is not even approximately 
 constant, but on the other hand, the ratio C,/ JC^ is constant 
 (fourth column). A little consideration shows that this is 
 connected with the fact already mentioned, that whilst benzoic 
 acid has the normal molecular weight in water, in benzene it is 
 present almost entirely as double molecules (p. 124). In this 
 case also we may assume that there is a constant ratio between 
 the concentrations of the simple molecules in the two phases, 
 and as it may easily be shown (p. 267) that the concentration 
 of the simple molecules in benzene is proportional to the square 
 root of the total concentration, the results are in accord with 
 the theory. 
 
 The Phase Rule. Equilibrium between Water, Ice 
 and Steam In the previous sections of this chapter, it has 
 been shown that many heterogeneous equilibria can be dealt 
 with satisfactorily by means of the law of mass action. This 
 holds not only for phases of constant composition, but within 
 limits also for phases of variable composition, such as solutions. 
 With reference to dilute solutions there is, of course, no difficulty, 
 as the active mass of the solute is proportional to its concentra- 
 tion. This is not the case, however, for strong solutions, and 
 the application of the law of mass action to these is attended 
 with considerable uncertainty. 
 
 As far back as 1874, a complete method for the representa- 
 tion of chemical equilibria was developed by the American 
 physicist, Willard Gibbs, which has come to be known as the 
 phase rule. The first point to notice with regard to this method 
 is that it is entirely independent of the molecular theory ; the 
 composition of a system is determined by the number of in- 
 dependently variable constituents, which Gibbs terms components. 
 He then goes on to determine the number of " degrees of free- 
 dom" of a system from the relation between the number of 
 components and the number of phases. It is for this reason- 
 that his method of classification is termed the phase rule. 
 
 In order to make clear the meaning of the terms employed' 
 it will be well, before enunciating the rule, to illustrate them by 
 
Solid 
 
 1 80 OUTLINES OF PHYSICAL CHEMISTRY 
 
 means of a very ^simple case, namely water. As regards the 
 equilibrium in this case, we may, according to the conditions 
 of the experiment, have one, two or more phases present. 
 Thus, under ordinary conditions of temperature and pressure, 
 there are two phases, water and water vapour, in equilibrium. 
 This equilibrium is represented in the accompanying diagram 
 by the line OA (Fig. 22), temperature being measured along 
 the horizontal and pressure along the vertical axes. It is only 
 at points on the curve that there is equilibrium. If, for example, 
 the pressure is kept below that represented by a point on the 
 
 curve OA (by continuously 
 increasing the volume) the 
 whole of the water will be 
 converted to vapour ; if, on 
 the other hand, it is kept at a 
 point a little above the curve 
 at a definite temperature, the 
 Vapour whole of the vapour will ulti- 
 
 mately liquefy. When the 
 temperature is a little below 
 o, only ice and vapour are 
 
 Temperature-* present, and the equilibrium 
 
 FIG. 22. between them is represented 
 
 on the diagram by the line 
 OC, which is not continuous with OA. 
 
 The two curves meet at O, and O is the point at which ice 
 and water are in equilibrium with water vapour. It is easy to 
 see that at this point ice and water have the same vapour 
 pressure. If this were not so, vapour would distil from the 
 phase with the higher vapour pressure to that with the lower 
 vapour pressure till the first phase was entirely used up, a 
 result in contradiction with the fact that the two phases remain 
 in equilibrium at this point. Since o is the temperature at 
 which ice and water are in equilibrium with their vapour under 
 atmospheric pressure, and as pressure lowers the melting-point 
 
HETEROGENEOUS EQUILIBRIUM 181 
 
 of ice, the point O, at which the two phases are in equilibrium 
 under the pressure of their own vapour (about 4'6 mm.), must 
 be a little above o ; the actual value is + 0-007 C. The 
 diagram is completed for stable phases by drawing the line OB, 
 which represents the effect of pressure on the melting-point of 
 ice ; the line is inclined towards the pressure axis because in- 
 creased pressure lowers the melting-point. 
 
 The point O is termed a triple point, because there, and 
 there only, three phases are in equilibrium. At points along 
 the curves two phases are in equilibrium, and under the con- 
 ditions in the intermediate spaces only one phase is present, as 
 the diagram shows. 
 
 So far, only stable conditions have been considered, but 
 unstable conditions may also occur. Thus water does not 
 necessarily freeze at o ; if dust is carefully excluded, it is pos- 
 sible to follow the vapour pressure curve for some degrees 
 below zero. The part of the curve thus obtained is represented 
 by the dotted line OA' which is continuous with OA and lies 
 above OC, the vapour pressure curve for ice. These results 
 illustrate two important rules : (i) there is no abrupt change 
 in the properties of a liquid at its freezing-point when the 
 solid phase does not separate ; (2) the vapour pressure of an 
 unstable phase is greater than that of the stable phase at the 
 same temperature. The last result may be anticipated, since it 
 is then evident how an unstable phase may change to a stable 
 phase by distillation. 
 
 The phase rule may now be enunciated as follows : If P 
 represents the number of phases in a system, C the number oj 
 components and F the number of degrees of freedom, the rela- 
 tion between the number of phases, components and degrees oj 
 freedom is represented by the equation C P 4- 2 = F. 
 
 The meaivng of the terms " component " and " degree of 
 freedom "will become clear as we proceed. The former has 
 already been defined as the smallest number of independent 
 variables of which the system under consideration can be built 
 
i82 OUTLINES OF PHYSICAL CHEMISTRY 
 
 up. Thus in the case of water, considered above, there is only 
 one component, and the system calcium carbonate-calcium 
 oxide-carbon dioxide can be built up from two components, 
 say calcium oxide and carbon dioxide. Particular instances of 
 the application of the phase rule will now be given. 
 
 If the number of phases exceeds the number of components by 
 2, the system has no degrees of freedom (F = o), and is said to 
 be non-variant. An illustration of this is the triple point O 
 in the diagram for water (p. 180), where there are three phases 
 (liquid water, ice and water vapour) and one component (water). 
 If one of j:he variables, the temperature or the pressure, is 
 altered and kept at the new value, one of the phases disappears ; 
 in other words, the system has no degrees of freedom. 
 
 If the number of phases exceeds the number of components by 
 one F = i , and the system is said to be univariant. As an illus- 
 tration, we take the case of water vapour, where there are two 
 phases and one component, say any point on the line OA. In 
 this case the temperature may be altered within limits without 
 altering the number of phases. If the temperature is raised, the 
 pressure will increase correspondingly, and the system will thus 
 adjust itself to another point on the curve OA. Similarly, the 
 pressure may be altered within limits, the system will re-attain to 
 equilibrium by a change of temperature at the new pressure. 
 If, however, the temperature be kept at an arbitrary value and 
 the pressure is then changed, one of the phases will disappear ; 
 the system has therefore one, and only one, degree of freedom. 
 
 If the number of phases is equal to the number of components, 
 the system lias two degrees of freedom, and is said to be divariant. 
 The areas in the diagram (Fig. 22) are examples of this case 
 there is one phase (vapour, liquid or solid) and one component. 
 If, for instance, we consider the vapour phase, the temperature 
 may be fixed at any desired point within the triangle AOC, and 
 the pressure may still be altered within limits along a line 
 parallel to the pressure axis without alteration in the number 
 of phases, as long as the curves OA and OC are not reached. 
 
HETEROGENEOUS EQUILIBRIUM 183 
 
 If the number of phases is less than the number of com- 
 ponents by one, the system is trivariant, and so on. This 
 particular instance cannot occur in the case of water, but does 
 so in a four-phase system, as described in the next section. 
 
 Equilibrium between Four Phases of the same Sub- 
 stance. Sulphur The diagram for water represents the equi- 
 librium between three phases of the same substance. We are 
 now concerned with sulphur, which is somewhat more compli- 
 cated inasmuch as there are two solid phases, monoclinic and 
 rhombic sulphur, in addition to the usual liquid and vapour 
 phases. 
 
 Rhombic sulphur is stable at the ordinary temperature, and 
 on heating rapidly melts at 1 1 5. On being kept for some time 
 in the neighbourhood of 100, however, it changes completely 
 to monoclinic sulphur, which melts at 120. Monoclinic sulphur 
 can be kept for an indefinite time at 100 without changing to 
 rhombic ; it is therefore the stable phase under these conditions. 
 Thus, just as water is the stable phase above o and ice is stable 
 below o, there is a temperature above which monoclinic sulphur 
 is stable, below which rhombic sulphur is stable, and at which 
 the two forms are in equilibrium with their vapour. This tem- 
 perature is termed the transition point, and occurs at 95 '6. 
 The change of one form into another is under ordinary condi- 
 tions comparatively slow, and it is therefore possible to determine 
 the vapour pressure of rhombic sulphur up to its melting-point, 
 and that of monoclinic sulphur below its transition point. 
 Although the vapour pressure of solid sulphur is comparatively 
 small, it has been measured directly down to 50. 
 
 The complete equilibrium diagram, which includes the fixed 
 points just mentioned, is represented in Fig. 23. O is the point 
 at which rhombic and monoclinic sulphur are in equilibrium 
 with sulphur vapour, and is consequently a triple point, analogous 
 to that /or water ; OB is the vapour-pressure curve of rhombic 
 sulphur, and OA that of monoclinic sulphur. OA', which is 
 continuous with OA, is the vapour-pressure curve of monoclinic 
 
184 OUTLINES OF PHYSICAL CHEMISTRY 
 
 sulphur in the unstable or metastable condition ; OB' similarly 
 represents the vapour-pressure curve of rhombic sulphur in the 
 unstable condition, and B' its melting-point. It will be observed 
 that in both cases the metastable phase has the higher vapour 
 pressure (p. 181). The line OC represents the effect of pressure 
 on the transition point O and is therefore termed a transition 
 curve ; since, contrary to the behaviour of water, pressure raises 
 the transition point, the line is inclined away from the pressure 
 
 Liquid 
 
 (131, 400atmos.) 
 
 Vapoui 
 
 Temperature > 
 FIG. 23. 
 
 axis. Similarly, the curve AC represents the effect of pressure 
 on the melting-point of monoclinic sulphur, and as it is less 
 inclined away from the temperature axis than OC, the two lines 
 meet at C at 131 under a pressure of 400 atmospheres. The 
 curve AD is the vapour-pressure curve of liquid sulphur above 
 120, where the liquid is stable, and AB', continuous with DA, 
 is the vapour-pressure curve of metastable liquid sulphur. As 
 already indicated, B' represents the melting-point of metastabl^ 
 
HETEROGENEOUS EQUILIBRIUM 185 
 
 rhombic sulphur; in other words, it is a uiclastable triple point 
 at which rhombic and liquid sulphur, both in the metastable 
 condition, are in equilibrium with sulphur vapour. OB', as 
 already indicated, represents the vapour-pressure curve of meta- 
 stable rhombic sulphur, and the diagram is completed, both for 
 stable and metastable phases, by B'C, which represents the effect 
 of pressure on the melting-point of rhombic sulphur. Mono- 
 clinic sulphur does not exist above the point C ; when fused 
 sulphur solidifies at a pressure greater than 400 atmospheres, 
 the rhombic form separates, whilst, as is well known, the 
 monoclinic form first appears on solidification under ordinary 
 pressure. The areas, as before, represent each a single phase, 
 as shown in the diagram. Monoclinic sulphur is of particular 
 interest, because it can only exist in the stable form within 
 certain narrow limits of temperature and pressure, represented 
 in the diagram by OAC. 
 
 The phase rule is chiefly of importance in indicating what are 
 the possible equilibrium conditions in a heterogeneous system, 
 and in checking the experimental results. To illustrate this, we 
 will use it to find out what are the possible non-variant systems 
 in the case of sulphur, just considered. From the formula 
 C P + 2 = F, since C = i, P must be three in order that 
 F may be zero ; in other words, the system will be non-variant 
 when three phases are present. As any three of the four phases 
 may theoretically be in equilibrium, there must be four triple 
 points, with the following phases : 
 
 (a) Rhombic and monoclinic sulphur and vapour (the point O). 
 
 (1)) Rhombic and monoclinic sulphur and liquid (the point C). 
 
 (f) Rhombic sulphur, liquid and vapour (the point B'). 
 
 (d) Monoclinic sulphur, liquid and vapour (the point A). 
 The phase rule gives no information, however, as to whether 
 the triple points indicated can actually be observed. In this 
 particular case they are all attainable, as the diagram shows, 
 but only because the change from rhombic to monoclinic 
 sulphur above the triple point is comparatively slow. If it 
 
1 86 OUTLINES OF PHYSICAL CHEMISTRY 
 
 happened to be rapid, the point B' could not be actually 
 observed. 
 Systems of Two Components. Salt and Water The 
 
 equilibrium conditions are somewhat more corAplicated on 
 passing from systems of one component to those of two com- 
 ponents, such as a salt and water. For simplicity, the equi- 
 librium between potassium iodide and water will be considered, 
 as the salt does not form hydrates with water under the con- 
 ditions of the experiment. There will therefore be only four 
 
 phases, solid salt, solu- 
 tion, ice and vapour. 
 There are three degrees 
 
 30 h :x / of freedom, as, in ad- 
 
 dition to the tempera- 
 ture and pressure, the 
 concentration of the 
 solution may now be 
 varied. 
 
 The equilibrium in 
 this system is repre- 
 sented in Fig. 24, the 
 10 20 30 .jo 50 GO 70 80 90 ordinates representing 
 FlG - 24. temperatures and the 
 
 abscissae concentra- 
 tions. At o (A in the diagram) ice is in equilibrium with 
 water and vapour, as has already been shown. If, now, a 
 little potassium iodide is added to the water, the freezing-point 
 is lowered, in other words, the temperature at which ice and 
 water are in equilibrium is lowered by the addition of a salt, 
 and the greater the proportion of salt present, the lower is 
 the temperature of equilibrium. This is represented on the 
 curve AO, which is the curve along which ice, solution and 
 vapour are in equilibrium. On continued addition of potassium 
 iodide, however, a point must be reached at which the solution 
 is saturated with the salt, and on further addition of potassium 
 
 _3o 
 
 Grams KI in 100 grams solution 
 
HETEROGENEOUS EQUILIBRIUM 187 
 
 iodide, the latter must remain in the solid form in contact with 
 ice and the solution. It is clear that, since the progressive 
 lowering of the freezing-point depends upon the continuous 
 increase in the concentration of the solution, the temperature 
 corresponding with the point O must represent the lowest 
 temperature attainable in this way under stable conditions. 
 
 To complete the diagram, it is further necessary to determine 
 the equilibrium curve for the solid salt, the solution and 
 vapour. Looked at in another way, this will be the solubility 
 curve of potassium iodide, which is represented by the curve 
 OB. This curve is only slightly inclined away from the tempera- 
 ture axis, corresponding with the fact that the solubility of 
 potassium iodide only increases slowly with rise of temperature. 
 The point O, which is the lowest temperature attainable with 
 two components, is known as a eutectic point ; in the special 
 case when the two components are a salt and water, it is termed 
 a cryohydric point. 
 
 The meaning of the diagram will be clearer if we consider 
 what occurs when solutions of varying concentration are pro- 
 gressively cooled. If, for example, we commence with a weak 
 salt solution above its freezing-point (x in the diagram) and 
 continuously withdraw heat, the temperature will fall (along xy) 
 till the line OA is reached ; ice will then separate, and as the 
 cooling is continued the solution will become more con- 
 centrated and the composition will alter along the line AO 
 until O is reached ; salt then separates as well as ice, and the 
 solution will solidify completely at constant temperature, that 
 of the point O. Similarly, if we start with a concentrated 
 salt solution and lower the temperature until it reaches the 
 curve OB, salt will separate and the composition will alter 
 along the curve BO until it reaches the point O, when the 
 mixture solidifies as a whole. Finally, if a mixture corre- 
 sponding with the composition of the cryohydric mixture is 
 cooled, the line parallel to xy representing the fall of tem- 
 perature will meet the curve first at the point O, and the 
 
i88 OUTLINES OF PHYSICAL CHEMISTRY 
 
 mixture will solidify at constant temperature. At the cryohydric 
 temperature, the composition of the solid salt which separates is 
 necessarily the same as that of the solution. Guthrie/who was 
 the first to investigate these phenomena systematically, was of 
 opinion that these mixtures of constant composition were definite 
 hydrates, which were therefore termed cryohydrates. At first 
 sight there seems much to be said for this view, as the separa- 
 tion takes place at constant temperature, independent of the 
 initial concentration, and the mixtures are crystalline. For the 
 following reasons, however, it is now accepted that the cryo- 
 hydrates are not chemical compounds : (a) the properties of 
 the mixture (heat of solution, etc.) are the mean of the pro- 
 perties of the constituents, which is seldom the case for a chemical 
 compound ; (b) the components are not usually present in 
 simple molar proportions ; and (c) the heterogeneous character 
 of the mixture can be recognised by microscopic examination. 
 The magnitude of the cryohydric temperature is 'of course con- 
 ditioned by the effect of the salt in lowering the freezing-point 
 of the solvent, and by its solubility at low temperatures. The 
 eutectic temperatures of solutions of sodium and ammonium 
 chlorides are 22 and 17 respectively, that of a solution of 
 calcium chloride - 37. 
 
 The application of the phase rule to this system will be 
 readily understood from what has been said above. When there 
 are four phases, ice, salt, solution and vapour, and two com- 
 ponents, we find, by substituting in the formula C-P+2 = F, 
 that F = o, that is, the four phases can only be in equilibrium 
 at a single point, the point O in the diagram. When there are 
 three phases, there is one degree of freedom, and the equi- 
 librium is represented by a line (OA and OB in the diagram). 
 If, for example, the condition of affairs is that represented by a 
 point on the line OA, and the concentration of the solution is 
 increased and kept at the new value (by further addition of 
 salt when necessary) ice will dissolve, and the temperature will 
 fall till it corresponds with that at which ice is in equilibrium 
 ^ Phil. Ma*., 1884, [5], 17,462. 
 
HETEROGENEOUS EQUILIBRIUM r 8 9 
 
 with the more concentrated solution and vapour. If, then, while 
 the concentration is still kept at the above value, the tempera- 
 ture is altered and kept at the new value, one of the phases will 
 disappear. The system is therefore univariant. If, instead of 
 the concentration, one of the other variables is changed, corre- 
 sponding changes in the remaining two variables take place, 
 and the system adjusts itself till the three phases are again in 
 equilibrium. 
 
 If there are only two phases, for example, solution and vapour, 
 the phase rule indicates that the system is bivariant, and it can 
 readily be shown, by reasoning analogous to the above, that 
 such is the case. 
 
 Freezing Mixtures The use of mixtures of ice and salt 
 as " freezing mixtures " for obtaining constant low temperatures, 
 depends upon the principles just discussed. Suppose, for 
 example, we begin with a fairly intimate mixture of ice and 
 salt and a little water. When a little of the salt dissolves, the 
 solution is no longer in equilibrium with ice. It will strive 
 towards equilibrium by some more ice dissolving to dilute the 
 solution, the latter, being now more unsaturated with regard 
 to the salt, will dissolve more of it, more ice will go into 
 solution and so on. As a consequence of these changes, heat 
 must be absorbed in changing ice to water (latent heat of fusion 
 of ice), and in connection with the heat of solution of the salt 
 if, as is usually the case, the heat of solution is negative. The 
 temperature, therefore, falls till the cryohydric point is reached, 
 and then remains constant, since it is under these conditions 
 that ice, salt, solution and vapour are in equilibrium. As 
 the temperature of a cryohydric mixture is so much below 
 atmospheric temperature, heat will continually be absorbed from 
 the surroundings, but as long as both ice and solid salt are 
 present, the heat will be used up in bringing about the change 
 of state, and the temperature will remain constant. When, 
 however, either ice or salt is used up, the temperature must 
 necessarily begin to rise. 
 
1 9 o OUTLINES OF PHYSICAL CHEMISTRY 
 
 Systems of Two Components. General The particular 
 case of a two-component or binary system already considered 
 potassium iodide and water is very simple, for two reasons f . 
 (i) the solid phases separate pure from the fused mass, in other 
 words, the phases are not miscible in the solid state ; (2) the 
 components do not enter into chemical combination. Com- 
 plications occur when chemical compounds are formed, and 
 when the solid phases separate as mixed crystals (p. 95) con- 
 taining the two components in varying proportions. We will 
 consider three comparatively simple cases of equilibrium in binary 
 systems, the components being in all cases completely miscible 
 in the fused state : 
 
 (a) The components do not enter into chemical combination, 
 and are not miscible in the solid state. 
 
 (b) The components do not enter into chemical combination, 
 but are completely miscible in the solid state. 
 
 (c) The components form one chemical compound, but are 
 not miscible with each other or with the compound in the 
 solid state. 
 
 Case (a). One example of this case is potassium iodide and 
 water which has just been discussed. Another, which will be 
 briefly considered, is the equilibrium between the metals zinc 
 and cadmium. 1 To determine the equilibrium curves, mixtures 
 of these metals in varying proportions are heated above the 
 melting-point, and then allowed to cool slowly, the rate of 
 cooling being observed with a thermocouple, one junction of 
 which is placed in the mixture and the other kept at constant 
 temperature. If the thermocouple is connected to a mirror 
 galvanometer, the rate of cooling can be followed by the move- 
 ment of a spot of light. Curves in which the times are plotted 
 against the corresponding temperatures of the mixtures are 
 represented in Fig. 256. The curves a, Z and b represent the 
 cooling of mixtures corresponding with the points a, Z, b on 
 the composition axis of the upper figure ; the curves o and 100 
 
 1 Hinrichs, Zeitsch. at<org. Chcm., 1907, 55, 415. 
 
HETEROGENEOUS EQUILIBRIUM 
 
 191 
 
 represent the cooling of pure zinc and cadmium respectively. 
 For the mixtures there are halts in the rate of cooling so- 
 called "breaks" at two points; the first point varies with the 
 composition of the mixture, whilst the second is practically con- 
 stant at 270. The curve obtained by plotting the temperatures 
 of the first break against the composition of the mixture is 
 
 419 
 
 EUTECTIC 
 
 I Cd 
 ' eUTECTIC 
 
 t 
 
 321" 
 
 270" 
 
 COMPOSITION * b 100?, Cd 
 
 4-19 
 
 Equilibrium Diagram for Zinc and Cadmium 
 FIG. 25A (top), 2515 (bottom). 
 
 represented in Fig. 25 A. The analogy of the diagram with 
 that for potassium iodide and water will be evident, as is its 
 interpretation. The point A represents the freezing-point of 
 zinc, B that of cadmium, the curve AC represents the effect of 
 the gradual addition of cadmium upon the freezing-point of 
 zinc, and BC the effect of zinc in lowering the freezing-point of 
 
iQ 2 OUTLINES OF PHYSICAL CHEMISTRY 
 
 cadmium, C is the eutectic point at which solid zinc and cad- 
 mium are in equilibrium with the fused mass. Consider first a 
 fused mixture rich in zinc. As the temperature falls, a point is 
 ultimately reached at which pure zinc begins to separate, and 
 as the change from liquid to solid is, as usual, attended with 
 liberation of heat, the rate at which the temperature falls will 
 diminish this represents the first break on cooling. As zinc 
 continues to separate, the composition of the mixture moves 
 along the curve AC, at C the solution is also saturated with 
 regard to cadmium, and therefore at the temperature repre- 
 sented by C, both zinc and cadmium separate in a mixture of 
 the same composition as the fused mass. This, as already 
 mentioned, is the eutectic point, and as the mixture at that 
 point behaves like a single substance, the temperature remains 
 constant till the whole mass has solidified. This is the second 
 break observed when the mixture cools. 
 
 If, on the other hand, we commence with a mixture rich in 
 cadmium, the latter separates along the curve BC till the point 
 C is reached, and then the mixture solidifies as a whole. Finally, 
 if a mixture is taken, the composition of which is that of the 
 eutectic mixture, there is only one break in the cooling curve, 
 at the eutectic point (curve Z). The composition of the eutectic 
 mixture is represented by the point Z on the axis of composi- 
 tion XY, and corresponds with 17*4 per cent, of zinc and 82*6 
 per cent, of cadmium by weight. 
 
 Case (). No cJiemical compound. Separation of mixed 
 crystals. A good . example of this case is the system 
 palladium-gold, 1 the equilibrium diagram for which is repre- 
 sented in F'ig. 26. In this case, when the fused mass is allowed 
 to cool, crystals containing both metals in varying proportions 
 separate, and therefore the ordinary rule with regard to the 
 lowering of melting-point of one substance by the addition of 
 another does not apply, and there is no eutectic point (p. 187). 
 As the composition of the crystals and the solution does not 
 
 1 Ruer, Zeitsch. anoyg. Ghent., 1906, tji, 391. 
 
HETEROGENEOUS EQUILIBRIUM 
 
 193 
 
 remain the same during solidification, there is in general an 
 interval of temperature, termed the crystallisation interval, be- 
 tween the beginning and end of crystallisation, and the cooling 
 curve shows two breaks or changes of direction representing 
 the beginning (curve ACB) and end (curve ADB) of solidi- 
 fication. 
 
 Gold and palladium are miscible in all proportions in the 
 solid state, but in many cases the miscibility is limited. 
 
 Case (^r). One 
 chemical compound. 
 No miscibility in solid 
 form. A typical 
 freezing-point curve 
 for a system of this 
 type is given in Fig. 
 27, representing thai- 
 Hum -mercury amal- 
 gams. It will be ob- 
 served that there is 
 a maximum on the 
 curve, corresponding 
 with the composition 
 
 t 
 
 LIQUID 
 
 SOLID 
 
 0% Au COMPOSITION*- IOO% Au 
 
 Equilibrium Diagram for Palladium and Gold 
 
 FIG. 26. 
 
 of a compound 
 HgjTl, containing 
 two equivalents of 
 mercury to one of 
 
 thallium. This compound melts at a definite temperature, 
 and it is evident from the curve that the melting-point is 
 lowered both by the addition of mercury (along CB) and of 
 thallium (along CD). From this it is clear that the components 
 of a chemical compound lower the freezing-point of the compound 
 just as foreign substances do. The remainder of the curve will 
 readily be understood from the previous paragraphs. A repre- 
 sents the freezing-point of mercury, AB the lowering of the 
 freezing-point of mercury by the progressive addition of the 
 13 
 
194 OUTLINES OF PHYSICAL CHEMISTRY 
 
 compound Hg2Tl, E the melting-point of thallium, and ED 
 the effect on the melting-point of thallium produced by 
 gradually increasing amounts of the same compound. The 
 points B and D are eutectic points, the eutectic mixtures 
 containing mercury and Hg 2 Tl, and Hg 2 Tl and thallium re- 
 spectively. 
 
 As in Fig. 25, above the curve ABCDE only liquid, and 
 below the dotted line only solid is present. The intermediate 
 regions represent heterogeneous systems in which both liquid 
 
 and solid are present. 
 
 The occurrence of a 
 maximum on the freez- 
 ing-point curve is 
 usually an indication of 
 chemical combination, 
 but, on the other hand, 
 
 LIQUID 
 
 Hg + EUTECTICB 
 
 0%Tl 
 
 COMPOSITION 
 
 Equilibrium Diagram for Mercury and Thallium 
 
 FIG. 27. 
 
 chemical compounds 
 are sometimes present, 
 although there are no 
 maxima on the curve. 
 Y This occurs more par- 
 Z(Hg 2 T/) IOO%TI ticularly when the 
 chemical compound de- 
 composes before its 
 melting-point is reached. 
 The systematic investigation of the freezing-point curve of 
 a binary mixture is one of the best methods for detecting 
 chemical compounds and establishing their formulae. This 
 can be illustrated by the equilibrium curve of ferric chloride 
 and water, which is of great historical interest, inasmuch as, 
 in the course of the investigation of this system by Rooze- 
 boom, many of the points we have been discussing were 
 elucidated. 
 
 The Hydrates of Ferric Chloride The freezing-point 
 curve of this system is represented in Fig. 28, and will be 
 
HETEROGENEOUS EQUILIBRIUM 195 
 
 readily understood by comparison with the previous diagrams, 
 more particularly Fig. 27. For convenience, the concentrations 
 in this case are expressed as the ratio of the number of mols 
 of ferric chloride present to 100 mols of water. 
 
 In interpreting the curve, we commence at the left-hand side 
 of the diagram. A represents the freezing-point of water, and 
 
 O 5 10 15 to 25 3 o 35 x 
 
 Moll. Fe 2 Cl 6 to loo mols. H 2 O 
 FIG. 28. 
 
 AB the lowering of freezing-point produced by the progressive 
 addition of ferric chloride. At the point B, - 55, the solution 
 is saturated with regard to ferric chloride, and B is therefore the 
 eutectic point, at which ice, solution, a hydrate of ferric chloride 
 (Fe 2 Cl 6 , i2H 2 O) and vapour are in equilibrium. On adding 
 more ferric chloride, the ice phase will disappear, and equilibrium 
 will be attained at a point on the curve BC, at which the 
 
196 OUTLINES OF PHYSICAL CHEMISTRY 
 
 dodecahydrate, Fe 2 Cl 6 , i2H 2 O, is in equilibrium with solution 
 and vapour. The curve BC may therefore be regarded as the 
 solubility curve of the dodecahydrate, and corresponds exactly 
 with the curve OB (Fig. 24). On continued addition of ferric 
 chloride, the equilibrium temperature continues to rise up to the 
 point C at 37, at which point the composition of the mixture 
 corresponds with that of the dodecahydrate. Further addition of 
 ferric chloride lowers the temperature along the line CD, so that 
 C is a maximum on the curve. At this point, the solid dodeca- 
 hydrate is in equilibrium with a liquid of the same composition ; 
 in other words, C is the melting-point of the compound 
 Fe 2 Cl 6 , i2H 2 O, just as the point C (Fig. 27) represents the 
 melting-point of Hg 2 Tl. On addition of more ferric chloride, 
 the melting-point is lowered, as represented by the section CD 
 of the curve. At the point D, the curve reaches another 
 minimum, or eutectic point, which corresponds with the point 
 B, except that a new hydrate, Fe 2 Cl 6 , 7H 2 O, takes the place of 
 ice. On further addition of ferric chloride, another maximum 
 is reached, which represents the melting-point of the hepta- 
 hydrate. In an exactly similar way, the other two maxima at 
 higher concentrations of ferric chloride indicate the existence 
 of two other hydrates, Fe 2 Cl 6 , 5H 2 O and Fe 2 Cl 6 , 4 H 2 O, in the 
 solid state. At the last eutectic point, K, the four phases in 
 equilibrium are the tetrahydrate, anhydrous ferric chloride, solu- 
 tion and vapour, and KL represents the effect of temperature 
 on the solubility of the anhydrous chloride. 
 
 The application of the phase rule to this system should be 
 made by the student. As at the point D (and the other corre- 
 sponding points) the solubility curves of the dodecahydrate 
 and heptahydrate intersect, D may be termed the transition 
 point for the two hydrates. Since at these points there are four 
 phases in equilibrium, they are also termed quadruple points. 
 
 In recent years, many other systems have been fully investi- 
 gated on analogous lines to that indicated above. The chief 
 experimental difficulty is to avoid supercooling, a source of 
 
HETEROGENEOUS EQUILIBRIUM 197 
 
 error which is much more troublesome in some systems than 
 in others. 
 
 Transition Points In the preceding sections, mention has 
 been made on several occasions of transition points. When a 
 substance is polymorphic (i.e., exists in more than one form) it 
 is often possible, as in the case of sulphur, to find a temperature 
 the so-called transition point at which two forms are in 
 equilibrium; at higher temperatures one of the forms is stable, 
 at lower temperatures the other. 
 
 It has been shown that at the transition point the two 
 modifications have the same vapour pressure, but at other 
 temperatures the metastable form has the higher vapour 
 pressure. By similar reasoning it can be shown that the 
 metastable phase has the higher solubility and the lower melt- 
 ing-point. 
 
 It may happen, however, as in the case of benzophenone, 
 that the modification stable at low temperatures melts before 
 the transition point is reached. It is evident that in this case 
 the transition can only take place in one direction, from the 
 modification the existence of which is favoured by high tem- 
 perature to that stable at low temperatures. Substances such 
 as sulphur, for which there is a reversible transition, are termed 
 enantiotropic, substances for which the transition is only in one 
 direction are termed monotropic. 
 
 It may be asked how it is possible to obtain the metastable 
 form of benzophenone, as it is necessarily unstable under all 
 conditions. The explanation is that when a fused polymorphic 
 substance is allowed to solidify, it is generally the most unstable 
 form which separates first (e.g., sulphur). 
 
 Similarly, the transition point of two hydrates is that tempera- 
 ture at which they are in equilibrium, and at which they have 
 the same solubility. 
 
 Practical Illustrations. Distribution of a solute between 
 two immiscible solvents To a nearly saturated aqueous solu- 
 tion of succinic acid (say 100 c.c.) in a stoppered bottle an 
 
198 OUTLINES OF PHYSICAL CHEMISTRY 
 
 approximately equal volume of ether is added, and the bottle 
 kept at room temperature for half an hour, shaking it at intervals. 
 When the separation into two layers is complete, part of the 
 ethereal layer is pipetted off and titrated with barium hydroxide, 
 and the same is done with the aqueous layer. Less concentrated 
 solutions of succinic acid are then used, and the concentrations 
 in the ethereal and aqueous layers determined as before. The 
 value of the distribution constant Cj/C 2 is then calculated from 
 the results, and should be constant, as the molecular weight is 
 the same in the two solvents (p. 178). 
 
 In a similar way the distribution of benzoic acid between 
 water and benzene may be investigated. In this case, as ben- 
 zoic acid is unimolecular in water and bimolecular in benzene 
 (p. 179), the ratio C x / \/C 2 , where C x represents the concentration 
 in the aqueous layer, C 2 the concentration in the benzene layer, 
 should be approximately constant. 
 
 Determination of transition points Transition temperatures, 
 such as that above which rhombic sulphur is unstable and 
 monoclinic sulphur stable, can be determined in various ways, 
 which depend in principle on the difference in properties of 
 the two phases. The dilatometer method is largely used ; it 
 depends on the fact that there is a more or less sudden change 
 of volume when the transition takes place. Into a fairly wide 
 glass tube, provided with a capillary tube and scale at the 
 upper end and open at the lower end, is placed a mixture of 
 the two modifications, e.g., monoclinic and rhombic sulphur, 
 the lower end is then sealed up, and the vessel rilled to the 
 lower end of the scale with an indifferent liquid such as oil. 
 The dilatometer is then placed in a bath the temperature of 
 which is gradually raised, and readings made of the level of the 
 liquid in the dilatometer. The rate of change of level will be 
 more or less steady till the neighbourhood of the transition 
 point is reached, when a relatively rapid change of level will be 
 observed. The chief drawback to the method is that the 
 change seldom takes place rapidly when the transition point is 
 
HETEROGENEOUS EQUILIBRIUM 199 
 
 reached. To facilitate the change as far as possible, some of 
 the second phase should always be present, (p. 55). 
 
 By solubility or vapour-pressure measurements The fact 
 that the two phases have the same solubility at the transition 
 point may be employed to determine the latter. To determine 
 the transition point in the system Na 2 SO 4 ,io H 2 O ^ Na 2 SO 4 + 
 10 H 2 O, it is only necessary to measure the solubility of the 
 anhydrous salt and of the decahydrate in the neighbourhood 
 of the transition point, and the point at which the solubility 
 curves intersect will be the required point. 
 
 Measurements of vapour pressure may be employed in a 
 similar way. Thus Ramsay and Young determined the vapour 
 pressures of solid and liquid bromine respectively and found 
 that the curves intersected at - 7, which is therefore the 
 transition temperature for solid and liquid bromine ; in other 
 -words, the melting-point of bromine. 
 
CHAPTER IX 
 VELOCITY OF REACTION. CATALYSIS 
 
 General In inorganic chemistry, the question as to the 
 rate of a chemical change does not often arise, because in 
 general the reactions are so rapid that it is impossible to measure 
 the speed. The neutralization of an acid by a base, for instance, 
 as shown by the change of colour of the indicator, is practically 
 instantaneous. However, some instances of slow inorganic re- 
 actions in homogeneous systems are known. The rate of com- 
 bination of sulphur dioxide and oxygen to form sulphur trioxide 
 is very slow under ordinary conditions, and the mixture of gases 
 has to be heated in contact with platinum as an accelerating 
 agent in order to obtain a good yield of trioxide. In organic 
 chemistry, on the other hand, slow chemical reactions are very 
 frequently met with. Thus the combination of an organic acid 
 and an alcohol to form an ester is very slow under ordinary 
 conditions, and the mixture of acid and alcohol, saturated 
 with hydrogen chloride, has to be boiled for a considerable 
 time in order to obtain a good yield of ester. 
 
 In this chapter, as in the preceding one on chemical equili- 
 brium, the law of mass-action is the guiding principle. It has 
 already been pointed out (p. 158) that the rate of a chemical 
 reaction at any instant may be regarded as the difference in 
 the speeds of the direct and the reverse reaction at that instant. 
 If we consider a simple reversible reaction, such as ester forma- 
 tion, in which a, b, c and d are the initial equivalent concentra- 
 tions of the reacting substances, and if x is the amount of 
 
 300 
 
VELOCITY OF REACTION. CATALYSIS 201 
 
 ester formed in the time /, the equation for the reaction velo- 
 city at that instant may be written 
 
 ^ -*(-*)(&-*)- ^fc + *)(<*+*) . (i) 
 
 in which dx represents the small increase in the amount of x 
 during the small interval of time, dt. This equation is a direct 
 consequence of the application of the- law of mass action to the 
 reaction in question. 
 
 It very often happens, however, that the equilibrium ^lies very 
 near one side, which can only mean that the rate of the reverse 
 reaction is small in comparison with that of the direct reaction. 
 This is clear from the following considerations. The splitting 
 up of hydrogen peroxide into water and oxygen is represented 
 by the equation H 2 O 2 -H 2 O + -JO 2 , and the equilibrium be- 
 tween this compound and its decomposition products by the 
 equation H 2 O 2 ^H 2 O + -JO 2 . Applying the law of mass action,* 
 we have 
 
 *[H 2 OJ^, [H 2 0][OJ 
 and 
 
 [H,OJ J!_ K 
 [H 2 0][0 2 ] k " 
 
 Now it is an experimental fact that the concentration of hydrogen 
 peroxide in equilibrium with water and oxygen at atmospheric 
 pressure is so small that it cannot be detected by analytical 
 means. It follows, from the above equation, that ^ is small in 
 comparison with k ; otherwise expressed, the rate of the con- 
 verse reaction, represented by the lower arrow, is negligible in 
 comparison with that of the direct reaction, represented by the 
 upper arrow. 
 
 The same considerations apply to the more general case re- 
 presented by. equation (i). If ^ is negligible in comparison 
 with k, the expression k^(c + x)(d + x) in equation (i) is 
 negligible in comparison with the remainder of the right-hand 
 side of the equation, and the latter therefore simplifies to 
 
202 OUTLINES OF PHYSICAL CHEMISTRY 
 
 -*(- *)(* - *). 
 
 Reactions of this type, in which the rate of the inverse reaction 
 is negligible, are by far the simplest from a kinetic standpoint, 
 and will therefore be considered first. 
 
 Unimolecular Reaction The simplest type of chemical 
 reaction is that in which only one substance is undergoing 
 change, and there is practically no back reaction. Such a 
 reaction, which can be readily followed, is the splitting up of 
 hydrogen peroxide to water and oxygen in the presence of 
 colloidal platinum or of certain enzymes. The reaction, which 
 is usually carried out in dilute aqueous solution, may be repre- 
 sented by the equation H 2 O 2 = H 2 O + O, and there is the 
 advantage that the solvent does not appreciably alter during 
 the reaction. As the colloidal platinum or the' enzyme remains 
 of constant activity during the reaction, the course of reaction 
 is determined solely by the peroxide concentration (cf. p. 205). 
 
 The reaction in the presence of haemase (blood-catalase) can 
 conveniently be followed by removing a portion of the solution 
 from time to time, adding to excess of sulphuric acid, which 
 immediately stops the reaction, and titrating with permanganate 
 solution. Some of the results obtained in this way are repre- 
 sented in the accompanying table : 
 
 a-x x 
 
 t (mins.). (c.c. KMnO 4 ). (c.c. KMnO 4 ). k 
 
 o 46*1 o 
 
 5 37'i 9'o 0*0435 
 
 10 29*8 16*3 0*0438 
 
 20 19*6 26*5 0*0429 
 
 30 12-3 33*8 0*0440 
 
 50 5*0 41*1 0*0444 
 
 The numbers in the second column represent the number of 
 c.c. of dilute permanganate solution equivalent to 25 c.c. of 
 the reaction mixture when the times represented in the first 
 
VELOCITY OF REACTION. CATALYSIS 203 
 
 column have elapsed after mixing the peroxide and enzyme, 
 and therefore represent the concentrations of peroxide in the 
 mixture at the times in question. By subtracting these numbers 
 from that representing the initial concentration of the peroxide, 
 46*1 c.c., the amounts of peroxide split up at the times / are 
 obtained ; these numbers are given in the third column. 
 
 The numbers illustrate very clearly the falling off in the rate 
 of the reaction as the concentration of the peroxide diminishes. 
 Thus in the first ten minutes an amount of peroxide equivalent 
 to 1 6 '3 c.c. of permanganate is split up, whilst in the second 
 interval of ten minutes only 26*5 - 16-3 10-2 c.c. are de- 
 composed. 
 
 According to the law of mass action, the rate of the reaction 
 at the time / should be proportional to the concentration of the 
 peroxide, a x, at that time, hence 
 
 =k(a-x) ... (2) 
 
 In this form, however, the equation cannot be applied directly 
 to /the experimental results, since dx t the amount of change of 
 x in the time dt t would have to be taken fairly large in order to 
 obtain accurate results, and during the interval a x would 
 naturally have diminished. Better results would be obtained 
 if a x were taken as the average concentration during the 
 interval dt within which dx of peroxide is being decomposed, 
 but even this method does not give accurate values for k. The 
 difficulty is got over by integrating the equation on principles 
 described in books on higher mathematics. In this way, and 
 bearing in mind that, when /= o, x = o, we obtain from equation 
 (2) above 
 
 It is, however, much more convenient to work with ordinary 
 logarithms (to the base 10) than with logarithms to the base e. 
 Making the transformation in the usual way, we obtain 
 
204 OUTLINES OF PHYSICAL CHEMISTRY 
 
 "7 log i^b = '4343* (4) 
 
 By substituting in the above equation corresponding values of 
 a, a-x and / from the table (p. 202), we obtain the values of k 
 given in the fourth column of the table, the average value being 
 0*0437. Since k is constant within the limits of experimental 
 error, it follows that the assumption on which the equation is 
 based, that the rate of the reaction is proportional to the 
 concentration of the peroxide, is justified. 
 
 It is instructive to compare the accurate value of k, obtained 
 by means of the integrated equation (4), with that calculated 
 from the differential equation (2), taking as the value of a x 
 the average value during an interval. If we choose the interval 
 between 5 and 10 minutes, dx = 7*3 c.c., dt 5 minutes, and 
 the mean value of a - x is 33*45 c - c ' Hence 
 
 y - * * 33-45, 
 
 and k = 0-0436, which is very close to the accurate average 
 value 0-0437. 
 
 A reaction of the above type, in which only one mol of a 
 single substance is undergoing change, is termed a unimolecular 
 reaction or a reaction of the first order, k is termed the velocity 
 constant, and fs t for unimolecular reactions, independent of the 
 units in which the concentration is expressed, but increases con- 
 siderably with rise of temperature. The meaning of k becomes 
 clear when we consider the initial velocity of the reaction. 
 Under these circumstances, the general equation 
 
 dxjdt = k (a - x) 
 
 becomes dxjdt = ka 0-043 7^ f r ^ ne example given above. 
 Otherwise expressed, the initial velocity of the reaction = velo- 
 city constant x initial concentration. If, by addition of fresh 
 peroxide, the concentration is kept throughout at its initial 
 value, then in unit time (i minute) 0-0437 f a r over 4 P er 
 cent, of a will be decomposed. 
 
VELOCITY OF REACTION. CATALYSIS 205 
 
 If the integrated equation (3) is written in the form 
 log a/(a - x) = kt, 
 
 we see that the left-hand side of the equation only becomes in- 
 finitely great (a = x) when / is infinitely great. Hence, for 
 finite values of /, x is always less than a, in other words, a 
 chemical reaction, even if irreversible, is never quite complete. 
 
 Other Unimolecular Reactions Many unimolecular re- 
 actions have been investigated, among the more interesting 
 being the hydrolysis of cane sugar and of esters under the 
 influence of acids. The hydrolysis of cane sugar to dextrose 
 and laevulose is represented by the equation 
 
 C 12 H 22 O n + H 2 - C 6 H 12 6 + C 6 H 12 6 , 
 
 and as the acid remains unaltered at the end of the reaction, it 
 does not occur in the equation. The reaction can be con- 
 veniently followed by measuring the change in rotation with a 
 polarimeter (p. 66) ; as cane sugar is dextrorotatory and the 
 mixture of dextrose and laevulose laevorotatory, the rotation 
 diminishes steadily as the reaction progresses, and finally 
 changes sign. 
 
 As both sugar and water take part in the reaction, the 
 velocity equation, according to the law of mass action, is 
 
 j- =5 A^sugar v^water- 
 
 As, however, water is present in great excess, its concentra- 
 tion, and therefore its active mass, remain practically constant 
 throughout the reaction, and the equation therefore reduces to 
 one of the first order, in satisfactory agreement with the ex- 
 perimental results. 
 
 Similar considerations enable us to understand why the 
 
 acid does not appear in the velocity equation. It is true that 
 
 he rate of the reaction depends upon the concentration of acid 
 
 present, and this, if necessary, could be expressed by putting in 
 
 a term C ac id on the right-hand side of the velocity equation. 
 
206 OUTLINES OF PHYSICAL CHEMISTRY 
 
 But as Cacid does not change in the course of the reaction, 
 the only effect is to multiply the right-hand side of the equa- 
 tion by a constant amount which, under ordinary circumstances, 
 we may consider as included in the velocity constant k. In the 
 same way, the enzyme concentration in the example quoted on 
 the previous page is included in the velocity constant, and the 
 course of the reaction is determined by the peroxide concentra- 
 tion. 
 
 The hydrolysis of an ester for example, ethyl acetate in 
 the presence of excess of a strong acid, such as hydrochloric 
 acid, is represented by the equation 
 
 CH 3 COOC 2 H 5 + H 2 O - CH 3 COOH + C 2 H 5 OH, 
 
 and is also of the first order, as the active mass of the water 
 is constant. The reaction may readily be followed by removing 
 a portion of the reaction mixture from time to time and titrating 
 with dilute alkali. The measurements may conveniently be 
 made as follows : To 40 c.c. of 1/2 normal hydrochloric acid, 
 previously kept in a 100 c.c. flask for some time at 25, 2 c.c. of 
 methyl acetate, warmed to the same temperature, is added, the 
 mixture shaken, and the flask well corked and allowed to remain 
 in the thermostat at constant temperature. At first every 20-30 
 minutes, and then at longer intervals, 2 c.c. of the mixture is 
 removed with a pipette, diluted, and titrated rapidly with N/io 
 barium hydroxide, using phenolphthalein as indicator. The 
 results are calculated in the usual way by substitution in equa- 
 tion (i). At 25 the value of k for this reaction is 0-0032. 
 
 It should be mentioned that it is not necessary to take for 
 the value of a the initial concentration of the reacting sub- 
 stance ; the calculation of the velocity constant may be com- 
 menced at any stage in the reaction, or may 6e made from 
 titration to titration. In the latter case the integrated equation 
 for a reaction of the first order is of the form 
 
VELOCITY OF REACTION. CATALYSIS 207 
 
 where a - x^ and a - x 2 are the respective concentrations at 
 the times ^ and t 2 . These methods of calculation gives satis- 
 factory results when, as is not infrequently the case, there are 
 irregularities at the beginning or in the course of a reaction. 
 
 Bimolecular Reactions When two substances react and 
 both alter in concentration, the reaction is said to be bimole- 
 cular, or of the second order. If the initial molar concentration 
 of one substance is a, that of the other b, and x the amount 
 transformed in the time /, the velocity equation is 
 
 g = k(a - x)(b - X ) . . . (i) 
 
 The simplest case is that in which the substances are present in 
 equivalent quantities. The velocity equation then becomes 
 
 dx ,. N9 
 = k(a- x}*, 
 
 which, on integration, gives the formula 
 
 -'* . w 
 
 As an illustration of a bimolecular reaction, the hydrolysis of an 
 ester by alkali may be adduced, a reaction which has been 
 thoroughly investigated by Warder, Ostwald, Arrhenius and 
 others. For ethyl acetate and sodium hydroxide, the equation 
 is as follows : 
 
 CH 3 COOC 2 H 5 + NaOH = CH 3 COONa + C 2 H 6 OH. 
 
 In carrying out an experiment, 1/20 molar solutions of ethyl 
 acetate and of sodium hydroxide are warmed separately in a 
 thermostat at constant temperature (25) for some time, equal 
 volumes of the solutions are then mixed, and from time to time 
 a portion of the reaction mixture is removed and titrated rapidly 
 with dilute hydrochloric acid. In the following table are given 
 some results obtained by Arrhenius : 
 
i (min.) 
 
 a x 
 
 k 
 
 t (min.) 
 
 a x 
 
 
 
 8-04 
 
 
 
 
 
 2*31 
 
 4 
 
 S-SO 
 
 o'oi6o 
 
 6 
 
 I*8 7 
 
 6 
 
 4*58 
 
 0*0156 
 
 12 
 
 r S7 
 
 8 
 
 3'9l 
 
 0*0164 
 
 18 
 
 i*35 
 
 10 
 
 3'S 1 
 
 0*0160 
 
 24 
 
 1*20 
 
 12 
 
 3*12 
 
 0*0162 
 
 30 
 
 I*IO 
 
 208 OUTLINES OF PHYSICAL CHEMISTRY 
 
 1/50 molar solution at 24*7. 1/170 molar solution at 247. 
 
 o'oiyo 
 0^0170 
 0-0171 
 0-0167 
 0-0163 
 
 The numbers for a - x in the second and fifth columns re- 
 present the concentrations of sodium hydroxide (and of ethyl 
 acetate) expressed as the number of c.c. of hydrochloric acid 
 required to neutralize 10 c.c. of the reaction mixture. The 
 reaction is very rapid, and therefore the experimental error 
 is somewhat large but the values obtained for k in the third 
 and sixth columns show that the assumptions on which formula 
 (2) is based are justified. 
 
 Another bimolecular reaction, which differs from the former 
 inasmuch as two molecules of the same substance react, is the 
 transformation of benzaldehyde to benzoin under the influence 
 of potassium cyanide. The reaction is represented by the 
 equation 
 
 2 C 6 H 5 CHO = C 6 H 5 CHOHCOC 6 H 5 , 
 
 the potassium cyanide remaining unaltered at the end of the 
 reaction. 
 
 When the reacting substances are not present in equivalent 
 proportions, the calculation is somewhat more complicated. On 
 integrating the equation dxfdt = k(a - x)(b - x\ we obtain 
 for this case 
 
 In order to illustrate the application of this equation, some 
 results obtained by Reicher for the saponification of ethyl 
 acetate by excess of alkali may be adduced. The reaction was 
 followed by titrating portions of the reaction-mixture from time 
 to time with standard acid, as already described, and the excess 
 
VELOCITY OF REACTION. CATALYSIS 209 
 
 of alkali was determined by titration of a portion of the solution 
 after the ethyl acetate was completely saponified (at the end of 
 twenty-four hours). 
 
 t (min.). a x b - x k 
 
 (alkali concentration). (ester concentration). 
 
 o 61-95 47'3 
 
 4-89 50-59 35-67 0-00093 
 
 11-36 42-40 27-48 0-00094 
 
 29*18 29-35 14*43 0-00092 
 
 oo r 4'9 2 o 
 
 It is important to note that the value of k for a bimolecular 
 reaction is not, as the case of a unimolecular reaction, indepen- 
 dent of the units in which the concentration is expressed. If, 
 for example, a unit i/th of the first is chosen, the value of 
 
 ix, i nx i x i 
 
 k = - f r becomes 7 r = - - . -, 
 
 / a(a - x\ t na.n(a-x) t a(a - x) n 
 
 so that the value of k diminishes proportionally to the increase 
 of the numbers expressing the concentrations. The truth of 
 this statement can be tested by means of the data for ester 
 saponification due to Arrhenius. If the titrations are made 
 with acid of half the strength actually used, (n = 2), the 
 numbers expressing the concentrations will be doubled, and it 
 will be found by trial that the value of k becomes half that 
 given in the table. ^ 
 
 Trimolecular Reactions When three equivalents take 
 part in a chemical change, the reaction is termed trimolecular, 
 and several such reactions have been carefully investigated. If, 
 as before, we represent the initial molar concentrations of the 
 reacting substances by a, b and c respectively, and if x is the 
 proportion of each transformed in the time /, the rate of reaction 
 at that time will, according to the law of mass action, be 
 represented by the differential equation 
 
2io OUTLINES OF PHYSICAL CHEMISTRY 
 
 g = k(a - *)(* - *)(, - *). 
 
 Such an equation is somewhat difficult to integrate, and we will 
 therefore confine ourselves to the simple case in which the 
 initial concentrations are the same. The equation then becomes 
 dxjdt = k (a - x) z , which on integration gives for k 
 
 , __ i x(2a x) 
 = 7 2a?(a - otf 
 
 Different cases arise according as the reacting molecules are the 
 same or different. The simplest case, in which the three re- 
 acting molecules are the same, is illustrated by the condensation 
 of cyanic acid to cyamelide, represented by the equation 
 
 3 HCNO = H 3 C 3 N 3 3 . 
 
 A case where two only of the reacting molecules are the same 
 is the reaction between ferric and stannous chlorides, represented 
 by the equation 
 
 2FeCl 3 + SnCl 2 = SnCl 4 + 2FeCl 2 . 
 
 finally, the reaction between ferrous chloride, potassium chlorate 
 and hydrochloric acid, represented by the equation 
 
 6FeQ 2 + KC1O 3 + 6HC1 = 6FeCl 3 + KC1 + 3H 2 O, 
 
 has been shown by Noyes and Wason to be proportional to the 
 respective concentrations of the three reacting substances, and 
 is therefore of the third order. 
 
 As an illustration of a trimolecular change, some of Noyes' 
 results for the reaction between ferric and stannous chlorides 
 are given in the table. The reacting substances were mixed 
 together at constant temperature, and from time to time a 
 measured quantity of the solution was removed with a pipette, 
 the stannous chloride decomposed with mercuric chloride, and 
 the ferrous salt still remaining titrated with potassium per- 
 manganate in the usual way. 
 
VELOCITY OF REACTION. CATALYSIS 211 
 
 SnCl 2 = FeCl 3 = 0*0625 normal = a. 
 t (min.) a - x x k 
 
 0*0625 o 
 
 1 0*04816 0*01434 88 
 3 0*03664 0*02586 81 
 7 0*02638 0*03612 84 
 
 17 0*01784 0*04502 89 
 
 25 0*01458 0*04792 89 
 
 Reactions of Higher Order. Molecular Kinetic Con- 
 siderations Whilst reactions of the first and second order 
 are very numerous, reactions of the third order are compara- 
 tively seldom met with, and reactions of a still higher order are 
 practically unknown. This is at first sight surprising, as the 
 equations representing many chemical reactions indicate that a 
 considerable number of molecules take part in the change; and 
 a correspondingly high order of reaction is to be expected. The 
 oxidation of ferrous chloride by potassium chlorate in acid solu- 
 tion, for example, is usually represented by the equation 
 
 6FeCl 2 + KC1O 3 + 6HC1 - 6FeCl s + KC1 + 3 H 2 O. 
 Applying the law of mass action (p. 143), we have therefore 
 
 that is, the reaction should be of the thirteenth order, whilst it 
 is actually of the third order (p. 210). . To account for this 
 result, it has been suggested by 'van't Hoff that complicated 
 chemical reactions take place in stages, and that the reaction 
 whose speed is actually measured is one in which only two or 
 three molecules take part, the velocity of the other reactions 
 being very great in comparison. This view is further considered 
 in the next section. 
 
 The molecular theory throws a good deal of light on this 
 question. On the assumption that the rate of chemical reaction 
 is proportional to the number of collisions between the reacting 
 molecules (p. 160), it follows that in a trimolecular reaction the 
 
212 OUTLINES OF PHYSICAL CHEMISTRY 
 
 three reacting molecules must collide simultaneously to produce 
 a chemical change. The probability of such a collision is ex- 
 tremely small compared with that between two molecules ; there- 
 fore, if at all possible, the reaction will take place between two 
 molecules or by the change of a single molecule. The proba- 
 bility of the simultaneous collision of four molecules is so small 
 as to be almost negligible. 
 
 Reactions in Stages It has just been pointed out that 
 many reactions which are represented by rather complicated 
 equations prove on investigation to be of the second or third 
 order, which seems to show that the reaction, the speed of which 
 is being measured, is in reality a comparatively simple one. When 
 a chemical change takes place in stages, a little consideration 
 shows that it is the slowest of a series of reactions which is the 
 determining factor for the observed velocity. This process has 
 been fittingly compared by Walker to the sending of a telegram ; 
 the time which elapses between dispatch and receipt is con- 
 ditioned almost entirely by the time taken by the messenger 
 between receiving-office and destination, as that is by far the 
 slowest in the successive stages of transmission. 
 
 An instructive example of a reaction which takes place in 
 stages is the burning of phosphorus hydride in oxygen, investi- 
 gated in van't HofFs laboratory by van der Stadt. The change 
 is usually represented by the equation 
 
 2 PH 3 + 4 2 = P 2 5 + 3 H 2 O, 
 
 according to which it would be a reaction of the sixth order 
 whilst the rate was actually found to be proportional to the 
 respective concentrations of the two gases, the reaction being 
 therefore of the second order. On allowing the gases to mix 
 gradually by diffusion, it was then found that the first stage of 
 the reaction is represented by the equation 
 
 PH 3 + O 2 = HPO 2 + H 2 
 that of a bimolecular reaction the subsequent changes by 
 
VELOCITY OF REACTION. CATALYSIS 213 
 
 which water and phosphorus pentoxide are produced being very 
 rapid in comparison. 
 
 Many other reactions proceed in stages, and in some cases 
 evidence as to the nature of the intermediate compounds has 
 been obtained. Thus the reaction between hydrobromic and 
 bromic acid, usually represented by the equation 
 
 HBr0 3 + 5HBr = 3 H 2 O + 3 Br 2 , 
 
 is bimolecular in the presence of excess of acid, and it is probable 
 that the first stage (the slow reaction) is as follows : 
 
 HBrO 3 + HBr = HBrO + HBrO 2 , 
 
 the subsequent changes, by which bromine and water are finally 
 produced, being comparatively rapid. 
 
 It is very probable that the equations which we ordinarily 
 use represent only the initial and final stages in a series of 
 changes, and the determination of the " order " of the reaction 
 is one of the most important methods for elucidating the nature 
 of the relatively unstable intermediate compounds. 
 
 Determination of the Order of a Reaction Three im- 
 portant methods which are largely used in determining the order 
 of reactions may be mentioned here. 
 
 (a) The Method of Integration According to this method, 
 the values of k given by the integrated equations for reactions 
 of the first, second and third order are calculated from the ex- 
 perimental results, and the order of the reaction is that in which 
 constant values are obtained for k. The method can be applied 
 to the numbers obtained for the reaction between equivalent 
 amounts of ethyl acetate and sodium hydroxide, when it will be 
 found that the values for k t calculated from the equation k = 
 i//loga/(a - x), continually decrease throughout the reaction; 
 the values obtained with an equation of the third order continu- 
 ally increase, and only for the equation k = ijt.xja(a x) is k 
 actually constant. The disadvantage of this method is that 
 disturbing causes, such as secondary reactions or the influence 
 of the reaction oroducts on the velocity, may so complicate the 
 
2i 4 OUTLINES OF PHYSICAL CHEMISTRY 
 
 results that a decision as to the order of the reaction is impossible 
 or an erroneous conclusion may be drawn. 
 
 (b) Ostwald's "Isolation" Method It has already been 
 pointed out that if one or more of the reacting substances is 
 taken in great excess, so that their concentration does not alter 
 appreciably during the reaction, the velocity, as far as these 
 substances is concerned, may be regarded as constant. It is 
 on this principle that the "isolation" method is based. Each 
 of the reacting substances in turn is taken in small concentra- 
 tion and all the others in excess, and the relation between the 
 reaction velocity and the concentration of the substance present 
 in small amount determined experimentally. As an example, 
 we will consider the reaction between potassium iodide and 
 iodate in acid solution, investigated by Dushman. 1 Regarding 
 the action of the acid as due to H- ions (p. 125), the reaction 
 velocity, according to the law of mass action, must be repre- 
 sented by the equation 
 
 When iodide and acid are used in large excess, the velocity is 
 proportional to the iodate concentration (n-^ = i) ; with a large 
 excess of iodate and acid it is proportional to the square of the 
 iodide concentration (n 2 = 2) ; and finally, when iodide and 
 iodate are in large excess, it is proportional to the square of 
 the acid concentration ( 3 = 2). Hence the velocity equation 
 becomes 
 
 = *[I0 3 IIT[H-?, 
 
 and when neither of the reagents is present in great excess, is 
 of the fifth order. 
 
 (c) Time taken to Complete the same Fraction of the Reaction 
 Measurements are made with definite concentrations of the 
 reacting substances, and with double and treble those con- 
 
 i y. Physical Chem., 1904, 8, 453. As the rate of reaction does not 
 depend upon the particular iodate or iodide used, we employ the formulas 
 for the iodate ion IO 3 ' and the iodide ion I'. 
 
VELOCITY OF REACTION. CATALYSIS 215 
 
 centrations, and the times taken to complete a certain stage (say 
 one-third) of the reaction noted. The order of the reaction 
 can then be determined from the following considerations 
 (Ostwald) : 
 
 (1) For a reaction of the first order the time taken to com- 
 plete a certain fraction of the reaction is independent of the 
 initial concentration. 
 
 (2) For a reaction of the second order, the time taken to 
 complete a certain fraction of the reaction is inversely propor- 
 tional to the initial concentration, e.g., if the concentration is 
 doubled, the time taken to complete a certain fraction of the 
 reaction is halved. 
 
 (3) In general, for a reaction of the nth order, the times 
 taken to complete a certain fraction of the reaction are inversely 
 proportional to the (n - i) power of the initial concentra- 
 tion. 
 
 In the experiments by Arrhenius, quoted on page 208, it will 
 be noticed that in the second experiment, in which the con- 
 centrations are less than J of those in the first experiment, the 
 time taken to complete half the reaction is about three times 
 as long in the former case as in the latter. 
 
 Complicated Reaction Velocities In the present chapter, 
 it has so far been assumed that chemical reactions proceed only 
 in one direction, and are ultimately complete, or practically 
 so, but in many cases these conditions are not fulfilled and 
 the course of the reaction is complicated. The more im- 
 portant disturbing causes are (a) side reactions; (b) counter 
 reactions ; (c) consecutive reactions. Each of these will be 
 briefly considered. 
 
 (a) Side Reactions In this case the same substances react 
 in two (or more) ways with formation of different products ; in 
 general the reactions proceed side by side without influencing 
 each other. An example is the action of chlorine on benzene, 1 
 which may substitute or form an additive product according to 
 the equations 
 
 1 Slator, Trans. Chem. Soc., 1903, 83, 729. 
 
*i6 OUTLINES OF PHYSICAL CHEMISTRY 
 
 (1) C 6 H 6 + C1 2 = C 6 H 5 C1 + HC1, 
 
 (2) C 6 H 6 + 3 C1 2 = C 6 H 6 C1 6 . 
 
 The relative amounts of the products formed in side reactions 
 depend on the conditions of the experiment, and it is usually 
 possible so to choose the conditions that one of the reactions 
 greatly predominates and can be investigated independently. 
 
 (b) Counter Reactions This term is applied when the pro- 
 ducts of a reaction interact to reproduce the original substances, 
 so that a state of equilibrium is finally reached, all the re- 
 acting substances being present (p. 201). Using the same 
 terminology as before, the rate of formation of an ester, for 
 example, will be represented by the differential equation 
 
 ~ = k(a - x)(b -x)- k& + x)(d + *). 
 
 The conditions can, however, usually be chosen in such a way 
 that either the direct or the reverse reaction predominates, and 
 the values of k and k^ can thus be determined separately. In 
 this way it has been shown experimentally by Knoblauch 1 that 
 the ratio k^k = K, the equilibrium constant, as the theory 
 requires (p. 158). 
 
 (c) Consecutive Reactions Consecutive reactions are those 
 in which the products of a chemical change react with each 
 other or with the original substances to form a new substance 
 or substances. They appear to be of very frequent occurrence 
 (p. 212). A good example is the saponification of ethyl succi- 
 nate, investigated by Reicher. It proceeds in the following two 
 stages : 
 
 (i) C 2 H 4 (COOC 2 H 5 ) 2 + NaOH = W CHOH 
 
 25 
 
 /COOC 2 H 5 
 
 H +NaOH = C 2 H 5 (COONa) 
 
 the product of the first reaction, ethyl sodium succinate, re- 
 acting further with sodium hydroxide to form the normal sodium 
 salt. 
 
 1 ?eitsch. physikal. Chem., 1897, 22, 268, 
 
VELOCITY OF REACTION. CATALYSIS 217 
 
 It is -important to remember that when one of the reactions 
 is very slow compared with the others, good constants cor- 
 responding with the slow reaction are obtained, but when the 
 rates are not very different, the observed velocity does not 
 correspond with any simple order of reaction. 
 
 CATALYSIS 
 
 General We have already met with instances in which the 
 rate of reaction is greatly increased by the presence of a third 
 substance, which itself is unaltered at the end of the reaction. 
 Thus cane sugar is hydrolysed very slowly by water alone, but 
 the change is greatly accelerated by the addition of acids. Such 
 phenomena are termed catalytic, and the substance which exerts 
 the catalytic or accelerating action is termed a catalysor or 
 catalyst. Ostwald, to whom much of our knowledge of catalytic 
 actions is due, defines a catalyst as " a substance which alters 
 the velocity of a reaction, but does not appear in the end 
 products ". From the point of view of the quantitative treat- 
 ment of reaction velocities, it is important to note that the equa- 
 tions already established remain valid in the presence of catalysts, 
 the only effect of the latter being to alter the value of the velocity 
 constant, k. The acceleration produced by a catalyst is in the 
 majority of cases proportional to the amount of the latter added. 1 
 
 Characteristics of Catalytic Actions From the above it 
 will be seen that the term catalysis does not include any explana- 
 tion of the observed phenomena ; it is merely a classification of 
 reactions which have certain features in common. In order to 
 make clear the exact bearing of the term, some characteristics 
 of catalytic actions will now be adduced. 
 
 (a) The catalyst is usually present in relatively small concen- 
 tration. This is, of course, connected with the fact that it is 
 
 1 One or two exceptions to this rule are known ; for example, the ac- 
 celerating influence of iodine monochloride on the rate of reaction between 
 chlorine and benzene is proportional to the square of the concentration of 
 the catalyst (Slator, Zeitsch. physikal. Chem., 1903, 45, 513). 
 
218 OUTLINES OF PHYSICAL CHEMISTRY 
 
 not used up during the reaction, so that a relatively small pro- 
 portion of catalyst can effect the transformation of large amounts 
 of the substance acted on. An apparent contradiction to this 
 rule (as regards the small concentration of the catalyst) is the 
 influence of the medium on the rate of reaction (p. 22/1), but 
 this can scarcely be termed a true catalytic action. 
 
 () The catalyst does not start a reaction, but only accelerates 
 a change which can proceed of itself, though perhaps extremely 
 slowly Although there is some difference of opinion with 
 regard to the general validity of this statement, it is now fairly 
 widely accepted, and seems to derive support from thermodyna- 
 mical considerations. As an illustration, we may consider the 
 combination of hydrogen and oxygen to form water. It is well 
 known that the mixed gases can be kept at the ordinary tem- 
 perature for an almost indefinite time without any apparent 
 combination, but when brought in contact with platinum they 
 combine fairly rapidly. It might at first sight be supposed that 
 the platinum actually initiates the combination. However, when 
 the gases are heated alone at 440, they combine with a measur- 
 able velocity, and at lower temperatures still combination can be 
 observed on long heating. Since the rate of reaction diminishes 
 greatly with fall of temperature (p. 225), it can readily be under- 
 stood that the rate of combination may be so slow at the ordinary 
 temperature as not to be measurable. 
 
 The alternative view is that the hydrogen and oxygen are 
 not entering into combination at all at the ordinary temperature, 
 that they are in a condition of unstable or false equilibrium and 
 that the catalyst actually initiates the combination. A direct 
 decision between these alternative hypotheses is difficult, but 
 there is some evidence of the existence of false equilibria. 
 For example, oxygen at fairly low pressures acts readily on 
 phosphorus, but when the pressure reaches a certain value, 
 which depends on the temperature, the reaction stops com- 
 pletely, and there is an apparent equilibrium. The conditions 
 determining this curious phenomenon are not well understood. 
 
VELOCITY OF REACTION. CATALYSIS 219 
 
 The limit of pressure above which oxidation does not occur 
 depends on the amount of moisture present, and doubtless also 
 on the presence of traces of impurities. 
 
 (<:) The presence of a catalyst does not affect the equilibrium ; 
 it alters the speed of the direct and inverse actions to the same 
 extent The truth of the first part of this statement follows at 
 once from the principle of the conservation of energy ; provided 
 that the catalyst is not combined with any of the products when 
 the reaction is complete, it does not produce any change in the 
 energy content of the system. It has also been proved experi- 
 mentally in many cases. Thus Kiister found that in the reaction 
 represented by the equation 2SO 2 + O 2 = 2SO 3 , the same equi- 
 librium point was reached in the presence of such different 
 catalysts as platinum, ferric oxide and vanadium pentoxide. 
 
 The second part of the above statement, that the catalyst 
 alters the speed of the direct and inverse actions to the same 
 extent, is a direct consequence of the first part. It has already 
 been shown that the equilibrium constant K = k^k, so that, if 
 k is increased, k^ must increase in the same ratio in order that 
 K may remain constant. In accordance with this rule, Baker 
 found that in the complete absence of water vapour, neither of 
 the actions represented by the oppositely-directed arrows in the 
 equation NH 4 C1 ^ NH 3 + HC1 took place, but the presence of 
 moisture accelerated the dissociation of ammonium chloride, as 
 well as its formation from its components. 
 
 Examples of Catalytic Action. Technical Importance 
 of Catalysis There appear to be very few chemical changes 
 which cannot be accelerated by the addition of certain sub- 
 stances, and many catalysts are known. Acids as a class 
 accelerate many chemical changes, more particularly those of 
 hydrolysis, such as the hydrolytic decomposition of cane sugar, 
 of amides, esters, etc., but not of chloroacetic acid. 1 As the 
 catalytic power is in general proportional to the extent to which 
 the acid is split up into its ions, we ascribe the catalytic pro- 
 
 1 Senter, Trans. Chem. Society, 1907, 91, 460. 
 
220 OUTLINES OF PHYSICAL CHEMISTRY 
 
 perty to that which is common to all acids, namely, hydrogen 
 ions. Only in dilute solution is there exact proportionality 
 between hydrogen ion concentration and catalytic power. 
 In stronger solutions, for a reason as yet unexplained, the 
 catalytic activity increases more rapidly than the hydrogen 
 ion concentration. 
 
 Bases in some cases also exert a catalytic effect, as in the 
 condensation of acetone to diacetonylalcohol, represented by 
 the equation 
 
 2 CH 3 COCH 3 = CH 3 COCH 2 C(CH 3 ) 2 OH ; 
 
 in this case the acceleration is proportional to the OH" ion 
 concentration. Bases have also a powerfully accelerating action 
 in certain isomeric changes of organic compounds. 1 
 
 The rare metals, more particularly finely-divided platinum, 
 also accelerate many chemical reactions, especially oxidation 
 reactions. Thus platinum is used commercially in the manu- 
 facture of sulphuric acid as a catalyst for the reaction 2SO 2 + 
 O 2 = 2SO 3 , as well as in the oxidation of methyl alcohol to 
 formaldehyde and of ammonia to nitric acid. Hydrogen and 
 oxygen also combine to form water at the ordinary temperature 
 in the presence of finely-divided platinum, and the same catalyst 
 also accelerates the splitting up of hydrogen peroxide into water 
 and oxygen. The latter reaction has been investigated fully 
 by Bredig and his pupils, who used the platinum in so-called 
 " colloidal " solution. The solution was obtained by passing 
 the electric arc between platinum poles immersed in cold 
 water ; under these circumstances the metal was torn from the 
 poles and remained suspended in the water in a very finely- 
 divided condition (p. 232). 
 
 Another interesting catalyst is water vapour. It has been 
 found, for example, that thoroughly dried carbon monoxide does 
 not burn in thoroughly dried air, but when a trace of moisture 
 is present, combination takes place at once. Reference has 
 
 1 Lowry and Magson, Trans. Chem. Society, 1908, 93, 107. 
 
VELOCITY OF REACTION. CATALYSIS 221 
 
 already been made to the fact that in the complete absence 
 of moisture ammonium chloride can be volatilised without 
 dissociation, and under the same circumstances ammonia and 
 hydrogen chloride do not combine. As the water does not 
 occur in the equation, it acts as a catalytic agent, but its mode 
 of action is quite unknown. 
 
 The importance of catalysis for technical processes will be 
 clear from the above. Many important technical reactions are 
 very slow, and the mixture has to be kept for a long period at 
 a high temperature to complete them, which adds much to the 
 cost of production. By using a suitable catalyst, the reaction 
 may be completed at a much lower temperature, and in a much 
 shorter time. As an illustration, the oxidation of naphthalene 
 to phthalic acid by means of sulphuric acid an important step 
 in the manufacture of indigo may be referred to. Under 
 ordinary circumstances, the reaction is slow even at high tem- 
 peratures. Owing to the accidental breaking of a thermometer 
 on one occasion when the reagents were being heated together 
 a little mercury fell into the mixture, and the observation that 
 the reaction then proceeded much more rapidly led to the dis- 
 covery that mercury was a catalyst for the reaction, and the 
 whole process was thus rendered commercially successful. 1 The 
 use of copper salts in the Deacon process production of 
 chlorine by oxidation of hydrogen chloride with free oxygen 
 and of oxides of nitrogen in the manufacture of sulphuric acid, 
 are other examples of technical catalysis. 
 
 Biological Importance of Catalysis. Enzyme Re- 
 actions Catalysis is also of great importance in physiology 
 and allied subjects, as the majority of the changes taking place 
 in the living organism are accelerated by those organic catalysts 
 the enzymes. When Berzelius brought forward the concep- 
 tion of catalysis, he adduced among other illustrations the 
 hydrolysis of cane sugar by invertase, and the hydrolysis of 
 starch in the presence of an extract of malt. It is now 
 generally recognised that these and allied changes, such as 
 
 1 Berichte, 1900, 33, Appendix. 
 
222 OUTLINES OF PHYSICAL CHEMISTRY 
 
 alcoholic fermentation of certain sugars, are due to the action 
 of catalysts of animal or vegetable origin which can be separated 
 from the living cells without losing their activity, an d which are 
 termed enzymes. 
 
 In recent years much progress has been made with the 
 investigation of enzyme reactions, and although little or nothing 
 is known as to the nature of the catalysts themselves, no 
 enzyme having so far been isolated in a state of purity, the laws 
 followed by many enzymes have been satisfactorily elucidated. 
 In general it may be said that enzymes behave like inorganic 
 catalysts, but there are certain characteristic differences. Just 
 as in the case of an inorganic catalyst, the acceleration pro- 
 duced by an enzyme is in the first instance proportional to its 
 concentration. The dependence of the speed of the reaction 
 on the concentration of the substance acted on is, however, 
 not so simple. To take a typical illustration, the rate of 
 hydrolysis of cane sugar in the presence of a constant concen- 
 tration of invertase increases with the concentration of the sugar 
 in dilute solution, but beyond a certain concentration of sugar, 
 further addition of the latter has no effect on the rate of the 
 reaction. In contrast to this behaviour, the rate of inversion 
 of cane sugar in the presence of a constant concentration of 
 acid increases with the sugar concentration as far as the reaction 
 has been followed. 
 
 As in the case of other catalysts, we may expect that the 
 enzyme will accelerate both the direct and inverse actions 
 when the reaction is reversible. The experiments of Croft Hill, 
 Emmerling, Kastle l and others, have shown that this expectation 
 is justified. 
 
 Mechanism of Catalysis The nature of catalysis becomes 
 
 1 Kastle and Loevenhart have shown that the reactions represented by 
 the upper and lower arrows in the equation 
 
 C a H B OH + C 3 H 7 COOH ^ C 3 H 7 COOC 2 H 5 + H a O 
 
 (formation and hydrolysis of ethyl butyrate) are both accelerated by lipase, 
 the enzyme which effects the hydrolysis of fats (Amer. Chem. y., 1900, 
 24, 491). 
 
VELOCITY OF REACTION. CATALYSIS 223 
 
 somewhat clearer when we represent reaction velocity, in a 
 manner analogous to Ohm's law, by means of the equation 
 
 TT . . Driving force 
 Velocity = -^ r 5 - 
 Resistance 
 
 The driving force of a chemical reaction is the same thing as 
 the free energy of the system (p. 149); of the resistance little 
 or nothing is known. It is clear from the above equation that 
 the velocity can be altered in two ways, by increasing the 
 driving force and by lessening the resistance. A catalyst can- 
 not to any extent affect the amount of energy in the system, 
 and we must therefore assume that in some way it increases 
 the velocity by diminishing the resistance. Ostwald compares 
 the action of a catalyst to that of oil on a machine, and it is 
 evident that the analogy is far-reaching. 
 
 It may be asked whether all catalytic accelerations are due 
 to a common cause. This is very unlikely ; it is much more 
 probable that the mechanism of the acceleration varies with 
 the nature of the catalyst and with that of the reacting sub- 
 stances. The suggestion of Liebig, that the catalyst sets up 
 certain molecular vibrations which lead to chemical changes, 
 has proved quite unfruitful. So-called explanations of this 
 nature, though often employed even at the present day, are 
 bad in principle, as we know very little of the nature of molecular 
 vibrations. It is, however, quite justifiable to inquire into the 
 mechanism of catalytic actions, in so far as it can be elucidated 
 by experimental investigation, and in recent years some light 
 has been thrown on this subject. 
 
 An explanation of catalytic acceleration which is much 
 favoured is that it depends on the formation of intermediate 
 compounds of the catalyst with the reacting substances. A 
 reaction represented by the equation A + B = AB may pro- 
 ceed very slowly directly, but in the presence of a catalyst C it 
 may proceed in the following two stages : 
 
 (a) A + C = AC, (b) AC + B = AB + C, 
 
224 OUTLINES OF PHYSICAL CHEMISTRY 
 
 much more rapidly to the same final products. For example, 
 it is known that the reaction SO 2 + O = SO 3 is a slow one, 
 and the accelerating effect of nitric oxide on the combination 
 may be represented in the following stages : 
 
 (a) NO + O = N0 2 ; (b) SO 2 + NO 2 = SO 3 + NO. 
 This explanation of the action of the oxides of nitrogen was 
 suggested more than a century ago by Clement and Desormes, 
 and remains the most plausible one at the present day. It is 
 possible that the finely-divided metals act as catalytic agents 
 mainly in a physical manner. In virtue of their large surface, 
 these metals condense gases and to some extent dissolved sub- 
 stances, and the local increase of concentration thus produced 
 must greatly increase the rate of chemical change. Similar 
 considerations may explain the catalytic effect of certain 
 enzymes. A chemical explanation of the action of finely-divided 
 metals and of enzymes based on the formation of intermediate 
 compounds is, however, to be preferred. 
 
 Nature of the Medium The rate of a chemical change 
 depends greatly on the nature of the medium in which it takes 
 place, but the way in which the influence is exerted is quite 
 unknown. The most complete data on this subject are due 
 to Menschutkin, who determined the rate of combination of 
 triethylamine and ethyl iodide, represented by the equation 
 
 (C 2 H 5 ) 3 N + C 2 H 6 I = (C 2 H 6 ) 4 NI 
 
 in more than twenty solvents at 100. A few typical results 
 are given in the accompanying table, in which , as usual, 
 represents the velocity constant : 
 
 Solvent. k. Dielectric Constant. 
 
 Hexane 0*00018 r86 (12-3) 
 
 Ethyl ether 0*000757 4-36(18) 
 
 Benzene 0-00584 2-26 (19) 
 
 Ethyl alcohol 0-0366 217 (15) 
 
 Methyl alcohol 0*0516 32-5 (16) 
 
 Acetone 0*0608 21*8 (15) 
 
VELOCITY OF REACTION. CATALYSIS 225 
 
 It will be seen that the rate varies enormously with change of 
 medium ; thus the ratio of the velocities in hexane and acetone 
 is approximately i : 340. 
 
 It is interesting to inquire whether there is any other pro- 
 perty of the different media which is parallel to the effect on 
 the reaction velocity. It was suggested on theoretical grounds 
 by J. J. Thomson, and somewhat later by Nernst, that the 
 " dissociating power of the medium " must be greater the 
 higher its specific inductive capacity or, to use the more 
 modern term, its dielectric constant. The dielectric constants 
 of the media are given in the third column of the table, and 
 it will be seen that there is a distinct parallelism, though not 
 direct proportionality, between dielectric constant and reaction 
 velocity. This question will be again referred to at a later stage. 
 
 Even a small change in the medium has sometimes a re- 
 markable effect on the rate of reaction. Thus it has recently 
 been shown that the rate of reaction between pure sulphuric 
 acid and oxalic acid is reduced to 1/17 of its original value 
 by the addition of o'i per cent, of water to the reaction 
 mixture. 
 
 On the other hand, a considerable alteration in the medium 
 may have very little effect on the velocity. Thus the rate of 
 reaction between chloracetic acid and silver nitrate, represented 
 by the equation 
 
 CH 2 C1C0 2 H + AgN0 3 + H 2 = CH 2 OHCO 2 H + AgCl + HNO 8 
 is practically the same in water and in 45 per cent, alcohol. 
 
 Influence of Temperature on the Rate of Chemical 
 Reaction It is a matter of common experience that the rate 
 of chemical reactions is greatly increased by rise of temperature. 
 Thus the combination of hydrogen and oxygen is so slow at 
 the ordinary temperature that it cannot be detected (p. 218), 
 but at high temperatures it proceeds with explosive rapidity. 
 It has already been pointed out that rise of temperature does 
 not usually affect the form of the velocity equation ; it can be 
 15 
 
226 OUTLINES OF PHYSICAL CHEMISTRY 
 
 represented simply as altering the magnitude of the velocity 
 constant. The more important facts in this connection are well 
 illustrated in the following table, which represents the effect of 
 temperature on the magnitude of the velocity constant for the 
 unimolecular reaction between dibromosuccinic acid and water, 
 represented by the equation 
 
 C 2 H 2 Br 2 (COOH) 2 = C 2 HBr(COOH) 2 + HBr. 
 
 Temperature. k (time in minutes). Temperature, k (time in minutes). 
 15 0*00000967 70*1 0*00169 
 
 40 0*0000863 80*0 0*0046 
 
 50 0*000249 89*4 0*0156 
 
 60*2 0^000654 101*0 0*0318 
 
 The table shows (i) that the velocity increases enormously 
 with rise of temperature; at 15 and 101 the relative rates are 
 in the ratio 0*00000967: 0*0318 or i : 3300; (2) the ratio 
 for a rise of 10 is approximately the same at different tem- 
 peratures, thus ^oAkro = 2 '7 2 > ^50/^40 = 2 '88. It is import- 
 ant to remember that the rate of most chemical reactions, as 
 in the above example, is doubled or trebled for a rise of 10. 
 This is shown in the accompanying table, which gives the quo- 
 tient for 10 (kt + tflkt) for a few typical chemical reactions. 1 
 
 1 The third column contains the average value of the quotient for 10 be- 
 tween the temperatures of observation. As data are not always available 
 at intervals of 10, the average value of kt+iolkt may be calculated approxi- 
 mately from the equation 
 
 Iog 10 * 2 - Iog 10 *! = A (T 2 - T,) 
 
 where k^ and h* are the velocity constants at the temperatures Tj and T 2 
 respectively. (See next section.) This equation gives us the value of A, 
 and the quotient for 10 is given by 
 
 log lo (**+ io)/*f = IOA or kt \\ 
 
 As an example, we will work out the quotient for 10 for the inversion of 
 cane sugar from the data given in the table. Log 10 35*5 - Iog 10 0765 is 
 1*66657 and as T - T : = 30 we obtain A = 0-05555. Hence log lo 
 (kt + w)lkt = 0-5555 and (k t + io)/kt = 3-60 approximately. 
 
VELOCITY OF REACTION. CATALYSIS 227 
 
 Reaction. 
 
 CH 3 COOC 2 H 5 
 CH 2 ClCOONa + NaOH 
 CH 2 ClCOONa+H 2 O . 
 C 2 H 5 ONa + CH 3 I 
 Inversion of cane sugar . 
 H 2 2 = H 2 + . 
 Fermentation by yeast . 
 
 Velocity Constants. 
 
 70 
 
 0*00089 
 0^000042 
 
 10-20 
 30-40 
 
 1347 = 
 44-90 = 
 
 ^102 ' 
 
 '648 
 
 0-01 5 
 
 / 10 = 0-00170 
 
 30=2-125 
 
 >& lft = o-oi8o 
 
 Quotient 
 for 10. 
 
 1-23 
 1*17 
 1-89 
 
 3*2 
 3'34 
 
 1*6 
 
 The quotient for 10 for reactions in solvents other than 
 water is also between 2 and 3 in the majority of cases. 
 
 According to the molecular theory, rise of temperature ought to 
 increase the rate of chemical change, owing to the accelerating 
 effect on molecular movements. This effect, however, would only 
 increase proportionally to the square root of the absolute tempera- 
 ture (p. 32), and it can easily be calculated on this basis that 
 the quotient for 10 for a bimolecular reaction at the ordinary 
 temperature would be about 1-04, much too small to account 
 for the large temperature coefficient actually observed. Up to 
 the present, no plausible explanation of the great magnitude of 
 the temperature coefficient of chemical reactions has been given. 
 The only other property which appears to increase as rapidly 
 with temperature is the vapour pressure, and it is not improbable 
 that there is a close connection between vapour pressure and 
 chemical reactivity. 
 
 At moderate temperatures the temperature coefficients of 
 enzyme reactions are approximately the same as those of 
 chemical reactions in general, but at temperatures in the neigh- 
 bourhood of o, the rate of change of k with the temperature 
 is often abnormally high, as the table shows. 
 
 It is interesting to note that the rate of development of 
 organisms, for example, the rate of growth of yeast cells, the 
 rate of germination of certain seeds, and the rate of develop- 
 
228 OUTLINES OF PHYSICAL CHEMISTRY 
 
 merit of the eggs of fish, is also doubled or trebled for a rise 
 of temperature of 10, and it has therefore been suggested that 
 these processes are mainly chemical. 
 
 Formulae connecting Reaction Yelocity and Tempera- 
 ture As has already been pointed out (p. 167), the law con- 
 necting the displacement of equilibrium with temperature is 
 known. So far, however, no thoroughly satisfactory formula 
 showing the relationship of rate of reaction and temperature 
 has been established, although many more or less satisfactory 
 empirical formulae have been suggested. If the relationship 
 which has been shown to hold approximately for the rate of de- 
 composition of dibromosuccinic acid that the quotient for 10 
 is the same at high as at low temperatures holds in general, 
 the equation connecting k and T must be of the form 
 
 dQQgjydT = A 
 where A is constant. On integration, this becomes 
 
 log* - AT + B, 
 
 B being a second constant. This formula holds for the decom- 
 position of nitric oxide and for certain other reactions, but is 
 not generally valid. As a matter of fact, the quotient k t +Jk t 
 generally diminishes with rise of temperature. An equation 
 which takes account of this has been proposed by Arrhenius ; 
 in its integrated form the equation is as follows : 
 
 log* - - A/T + B. 
 
 For two temperatures, T 1 and T 2 , for which the values of the 
 velocity constant are * x and * 2 respectively, the above equation 
 becomes 
 
 log,* 2 ~ 
 
 x M 
 
 and when A is known (from two observations) the velocity 
 constant for any other temperature can be calculated. 
 
 Arrhenius showed that the above empirical equation repre- 
 sents satisfactorily the influence of temperature on the rate of 
 hydrolysis of cane sugar and on certain other reactions, and it 
 
VELOCITY OF REACTION. CATALYSIS 2*9 
 
 has since been employed by many other observers with fairly 
 satisfactory results. A is a constant for any one reaction, but 
 differs for different reactions. 
 
 Practical Illustrations As the rate of chemical reactions 
 alters so greatly with change of temperature, it is necessary in 
 accurate experiments to work in a thermostat provided with a 
 regulator to keep the temperature constant. For purposes of 
 illustration, however, sufficiently accurate results can be obtained 
 by using so large a volume of solution that the temperature 
 does not alter appreciably during the reaction. 
 
 Unimolecitlar Reaction. (a) Decomposition of Hydrogen 
 Peroxide * To 200 c.c. of a mixture of one part of defibrinated 
 ox-blood and 10,000 parts of water, an equal volume of about 
 i/ 1 oo molar hydrogen peroxide (0-34 grams per litre) is added, 
 and the mixture shaken. At first every ten minutes, and then 
 at longer intervals, 25 c.c. of the reaction mixture is removed 
 with a pipette, added to a little sulphuric acid, which at once 
 stops the action, and titrated with i/ioo normal potassium 
 permanganate. If pure hydrogen peroxide is not available, 
 the commercial product should be neutralized with sodium 
 hydroxide before use. 
 
 Another portion of the original hydrogen peroxide solution 
 should be titrated with permanganate, and if the reacting 
 solutions have been measured carefully, the initial concentration 
 in the reaction mixture may be taken as half that in the original 
 solution. 
 
 The observations should be calculated by substitution in the 
 formula ijt log a/a-x *= 0-4343 k, valid for a unimolecular 
 reaction. 
 
 A corresponding experiment may be made with double the 
 peroxide concentration, in order to illustrate the fact that the 
 time taken to complete a certain fraction of the reaction is 
 independent of the initial concentration. 
 
 (b) Hydrolytic Decomposition of Cane Sugar in the Presence 
 of Acids Equal volumes of a 20 per cent, solution of cane 
 1 Senter, Zeitsch. physikal. Chem., 1903, 44, 257. The amount of or- 
 ganic matter is so small that the error in titration due to its reducing 
 action on permanganate is negligible. 
 
2 3 o OUTLINES OF PHYSICAL CHEMISTRY 
 
 sugar and of normal hydrochloric acid, previously warmed to 
 25, are mixed, the observation tube of a polarimeter is filled 
 with the mixture, and an observation of the rotation taken as 
 quickly as possible. The polarimeter tube is then immersed 
 in a thermostat at 25, and kept in the latter except when 
 readings of the rotation are being made. In order that it may 
 be conveniently immersed in a thermostat, the polarimeter tube 
 is provided with a side tube, through which it is filled, and the 
 end of which is not immersed. To prevent alterations while 
 readings are being made, the tube is provided with a jacket 
 filled with water at the temperature of the thermostat. 
 
 Several chemists have described arrangements according to 
 which the tube remains in the polarimeter throughout an ex- 
 periment, the temperature being kept constant by passing a 
 stream of water at constant temperature through the jacket of 
 the polarimeter tube. 1 With rapid working, however, the simpler 
 method described above gives excellent results. 
 
 If it is convenient to make an observation after the reaction 
 is complete (say 24 hours), the total change of reading is taken 
 as a in the formula for a unimolecular reaction, and the differ- 
 ence of the initial reading and that at the time t is proportional 
 to x, the amount of sugar split up. If the reaction is not 
 complete in a reasonable time, the final reading can be calculated 
 from the fact that for every degree of rotation to the right in 
 the original mixture, the wholly inverted mixture will rotate 
 0-315 to the left at 25. 
 
 (c) Hydrolysis of Ethyl Acetate in the Presence of Hydro- 
 chloric Acid The method of experiment in this case has 
 already been described (p. 206). 
 
 Bimolecular Reaction, (a) The rate of reaction is proportional 
 to the concentration of each of the reacting substances This state- 
 ment can be illustrated very satisfactorily by a method described 
 by Noyes and Blanchard, and depending upon the fact that 
 the time taken to reach a certain stage of a reaction is inversely 
 proportional to its rate. 
 
 1 C/. Lowry, Trans. Faraday Society, 1907, 3, 119. 
 
VELOCITY OF REACTION. CATALYSIS 231 
 
 In a two-litre flask a mixture of 1600 c.c. of water, 50 c.c. 
 half-normal hydrochloric acid and 20-30 c.c. of dilute mucilage 
 of starch is prepared, and 400 c.c. of this mixture is placed in 
 each of 4 half-litre flasks of clear glass, standing on white 
 paper. For comparison, a fifth halt-litre flask contains 400 
 c.c. of water, 10 c.c. of starch solution, and sufficient of a n/ioo 
 solution of iodine to give a blue colour of moderate depth. 
 
 Half-normal solutions of potassium bromate (7 grams per 
 half-litre) and of potassium iodide are also prepared. To the 
 flasks I. and II. 5 c.c. of the bromate solution is added, and to 
 III. and IV. 10 c.c. of the same solution. Then at a definite 
 time, 5 c.c. of the iodide solution is added to flask I., the 
 mixture rapidly shaken, and the time which elapses until the 
 solution has the same depth as the test solution carefully noted. 
 Then to flasks II., III. and IV. are added successively 10 c.c., 
 5 c.c. and ro c.c. of the iodide solution, and the times required 
 to attain the same depth of colour as the test solution carefully 
 noted in each case. If x is the time required when 10 c.c. of 
 each reagent is used, 2x will be the time required when 10 c.c. 
 of one solution and 5 c.c. of the other is used, and 4% when 
 5 c.c. of each solution is u^ed. 
 
 If for some reason the reactions are too rapid, the strengths 
 of the bromate and iodide solutions should be altered till 
 intervals convenient for measurement are observed. 
 
 (b) Quantitative measurement of a bimolecular reaction The 
 rate of reaction between ethyl acetate and sodium hydroxide may 
 be measured as described on a previous page. From a practical 
 point of view, however, the measurement presents certain draw- 
 backs because it is difficult to prepare a solution of sodium hy- 
 droxide free from carbonate and also to prevent absorption of 
 that gas from the air during the experiment. A reaction 1 free 
 from these disadvantages, which is very rapid in dilute solution, 
 is that between ethyl bromoacetate and sodium thiosulphate, 
 represented by the equation 
 CH 2 BrCOOC 2 H 5 + Na 2 S 2 O 3 = CH 2 (NaS 2 O 3 )COOC 2 H 6 + NaBr. 
 
 1 Slator, Trans. Chem. Society, 1905, 87, 484. 
 
232 OUTLINES OF PHYSICAL CHEMISTRY 
 
 The rate of the reaction can readily be followed by removing a 
 portion of the reaction mixture from time to time and titrating 
 with n/ioo iodine, which reacts only with the unaltered thio- 
 sulphate. 
 
 300 c.c. of an approximately 1/60 normal solution of sodium 
 thiosulphate is added to an equal volume of a dilute aqueous 
 solution of ethyl bromoacetate (2-2-1 grams per litre) in a 
 litre flask, the mixture shaken and the flask closed by a 
 cork. At first every 5 minutes, and then every 10 or 15 
 minutes, 50 c.c. of the reaction mixture is removed with a 
 pipette and titrated rapidly with n/ioo iodine, using starch as 
 indicator. Seven or eight such titrations are made, and then, 
 after an interval of 5-6 hours, when the reaction is presumably 
 complete, a final titration is made in order to determine the 
 excess of thiosulphate remaining. The initial concentration of 
 thiosulphate, expressed in c.c. of the iodine used in titrating 
 the mixture, can be obtained by titrating part of the original 
 thiosulphate solution, and the initial concentration of bromo- 
 acetate in the reaction-mixture is clearly equivalent to the 
 amount of thiosulphate used up. The case is exactly analogous 
 to that quoted on page 209, in which the sodium hydroxide is 
 in excess, and the value of k may be obtained by substitution 
 in equation (3), p. 208. 
 
 Catalytic Actions The decomposition of hydrogen per- 
 oxide by blood and of cane sugar in the presence of acids are 
 examples of catalytic actions. Hydrogen peroxide can also be 
 decomposed by a colloidal solution of platinum, which may be 
 prepared as follows (Bredig): Two thick platinum wires dip 
 into ice-cold water, and a current of about 10 amperes and 
 40 volts is employed. When the ends of the wires are kept 
 1-2 mm. apart, an electric arc passes between them, particles 
 of platinum are torn off and remain suspended in the water. 
 The solution is allowed to stand for some time, and filtered 
 through a close filter. It represents a dark-coloured solution 
 in which the particles cannot be detected with the ordinary 
 
VELOCITY OF REACTION. CATALYSIS 233 
 
 microscope. A very dilute solution may be used to decompose 
 hydrogen peroxide, and the reaction may be measured as 
 described above when blood is employed. 
 
 Catalytic Action of Water This may be illustrated by its 
 effect on the combustion of carbon monoxide in air, which 
 does not take place in the entire absence of water vapour. 
 Carbon monoxide is prepared by the action of strong sulphuric 
 acid on sodium formate and is carefully dried by passing 
 through two wash-bottles containing strong sulphuric acid, and 
 finally through a U tube containing phosphorus pentoxide. 
 Some time before the experiment is to be tried, a little strong 
 sulphuric acid is put in the bottom of a wide-mouthed bottle 
 with a close-fitting glass stopper, and the bottle allowed to 
 stand, tightly stoppered, for some time. 
 
 The carbon monoxide issuing from the apparatus burns 
 readily in air, but is immediately extinguished if the wide- 
 mouthed bottle is placed over it. If, however, a small drop of 
 water is placed in the tube whence the gas is issuing, the gas 
 will continue to burn when the bottle is placed over it. 
 
CHAPTER X 
 ELECTRICAL CONDUCTIVITY 
 
 Electrical Conductivity. General From very early times 
 it was noticed that electricity can be conveyed in two ways : 
 (i) In conductors of the first class, more particularly metals, 
 without transfer of matter ; (2) in conductors of the second 
 class salt solutions or fused salts with simultaneous decom- 
 position of the conductor. We are here concerned only with 
 conductors of the second class, but the use of the terms em- 
 ployed in electrochemistry may be illustrated by reference to 
 conduction in metals. 
 
 For conductivity in general, Ohm's law holds, which may be 
 enunciated as follows : The strength of the electric current 
 passing through a conductor is proportional to the difference 
 of potential between the two ends of the conductor, and in- 
 versely proportional to the resistance of the latter. Strength 
 of current is usually represented by C, difference of potential 
 or electromotive force by E, and resistance by R ; Ohm's law 
 may therefore be written symbolically as follows : 
 
 "I 
 
 The practical unit of electrical resistance is the ohm, that of 
 electromotive force the volt, and that of current the ampere. 
 The strength of an electric current can be measured in various 
 ways, perhaps most conveniently by finding the weight of silver 
 liberated from a solution of silver nitrate in a de finite interval of 
 time. An ampere is that strength of current which in one 
 
 234 
 
ELECTRICAL CONDUCTIVITY 
 
 second will deposit 0*001118 grams of silver from a solution of 
 silver nitrate under certain definite conditions. Quantity of elec- 
 tricity is current strength x time; the amount of electricity 
 which passes in one second with a current strength of i ampere 
 is a coulomb. When there is a difference of potential of one 
 volt between two ends of a conductor, and a current of i 
 ampere is passing through it, the resistance of the conductor 
 is i ohm. 
 
 The resistance of a metallic conductor is proportional to its 
 length and inversely proportional to its cross-section. Hence 
 if / is the length and s the cross section, the resistance R is 
 
 given by R = p -, where p is a constant depending only on the 
 
 material of the conductor, the temperature, etc., and is termed 
 the specific resistance. If both / and s are equal to unity 
 (i cm.) the resistance is equal to p. The specific resistance, p, 
 of a conductor is therefore the resistance in ohms which a cm. 
 cube of it offers to the passage of electricity. If there is a 
 difference of potential of i volt between two sides of the cube, 
 and the current which passes is i ampere, the specific resistance 
 of the cube is, by Ohm's law, = i. A conductor of low resist- 
 ance is said to have a high electrical conductivity, that is, it 
 readily allows electricity to pass. Conductivity is therefore the 
 converse of resistance, and specific conductivity ', K = i//o, where 
 p is specific resistance. Specific conductivity is measured in 
 reciprocal ohms, sometimes termed mhos. In order to illustrate 
 the magnitude of these factors, the specific resistance and the 
 specific conductivity of a few typical substances at 18 are given 
 in the table : 
 
 30 per cent 
 
 Substance. Silver. Copper. Mercury. Gas carbon, sulphuric 
 
 acid. 
 
 Sp. resistance, p 0-0000016 o'oooooiy 0*00009 0-0050 0-74 
 Sp. conductivity, K 624,000 587,000 10,240 200 1-35 
 
 Silver has the highest conductivity of all known substances , 
 
236 OUTLINES OF PHYSICAL CHEMISTRY 
 
 gas carbon is a comparatively poor conductor; and 30, per 
 cent, sulphuric acid, one of the best conducting solutions, is 
 enormously inferior to the metals in this respect. 
 
 Electrolysis of Solutions. Faraday's Laws We now 
 consider the phenomena accompanying the conduction of 
 electricity in aqueous solutions of salts. If, for example, two 
 platinum plates, one connected to the positive, the other to the 
 negative pole of a battery, are dipped into a solution of sodium 
 sulphate, it will be observed that hydrogen is immediately given 
 off at the plate connected to the negative pole of the battery, 
 and oxygen at the plate connected to the positive pole. 
 Further, if a few drops of litmus have previously been added 
 to the solution, it will be noticed that the solution round the 
 positive plate or pole becomes red, indicating the production 
 of acid, and that round the negative pole becomes blue, showing 
 the formation of alkali. An ammeter placed in the circuit will 
 show that a current is passing through the solution, so that the 
 chemical changes in question accompany the passage of the 
 current. Even if the poles are far apart, the gases are liberated, 
 and the acid and alkali appear immediately connection is made 
 through the solution, and if the current is continued, the acid 
 and alkali accumulate round the respective poles without any 
 apparent change in the main bulk of liquid between the poles. 
 These phenomena can scarcely be accounted for otherwise than 
 by supposing that matter travels with the current, and that part 
 travels towards the positive pole and part towards the negative 
 pole. To these travelling parts of the solution Faraday gave 
 the name of ions. 
 
 It will be well to mention here the nomenclature used in 
 this part of the subject. A solution or fused salt which con- 
 ducts the electric current is termed an electrolyte. The plate 
 in the solution connected to the positive pole of the battery 
 is termed the positive pole, positive electrode or anode, that 
 connected to the negative pole of the battery the negative 
 pole, negative electrode or cathode. The ions which move 
 
ELECTRICAL CONDUCTIVITY 237 
 
 towards the anode are often termed ant'ons, those travelling 
 towards the cathode cations. 
 
 We will now consider the relationship between the amount 
 of chemical action and the quantity of electricity passed through 
 a solution. The amount of chemical action might be estimated 
 by measuring the volume of gas liberated at one of the poles, 
 or by the amount of metal deposited on an electrode. This 
 question was investigated by Faraday, and as a result he 
 established a law which bears his name, and which may be 
 enunciated as follows : For the same electrolyte, the amount of 
 chemical action is proportional to the quantity of electricity which 
 passes. 1 Further, Faraday measured the relative quantities of 
 substances liberated from different solutions by the same quantity 
 of electricity, and was thus led to the discovery of his so-called 
 second law : The quantities of substances liberated at the elec- 
 trodes when the same quantity of electricity is passed through 
 different solutions are proportional to their chemical equivalents. 
 The chemical equivalent of any element (or group of elements) 
 is equal to the atomic weight (or sum of the atomic weights) 
 divided by the valency. The second law, therefore, states that 
 if the same quantity of electricity is passed through solutions of 
 sodium sulphate, cuprous chloride, cupric sulphate, silver 
 nitrate and auric chloride, the relative amounts of hydrogen 
 and the metals liberated are as follows : 
 Electrolyte Na,,SO 4 CuCl CuSO 4 AgNO 8 AuCl 3 
 
 Electrochem. A * * o 
 
 equivalent H= i ; O= f ; Cu = ^- 4 ; Cu = ^; Ag = ^?; Au=^ 
 
 The above result may also be expresstd rather differently as 
 follows : The electrochemical equivalents (the proportions of 
 different elements set free by the same current) are proportional 
 to the chemical equivalents. 
 
 That quantity of electricity which passes through an electro- 
 lyte when the chemical equivalent of an element (or group of 
 
 1 Faraday measured the amount of electricity by its action on a mag- 
 netic needle. 
 
238 OUTLINES OF PHYSICAL CHEMISTRY 
 
 elements) in grams is being liberated will obviously be a 
 quantity of very considerable importance in electrochemistry. 
 Since i ampere in i second (a coulomb) liberates 0*00001036 
 grams of hydrogen, it follows that when the chemical equivalent 
 of hydrogen or any other element is liberated, 1/0*00001036 
 = 96540 coulombs must pass through the electrolyte. It 
 is often designated by the symbol F (faraday). One cou- 
 lomb will liberate 35*45 x o'ooooio36 =0*000368 grams of 
 chlorine, 127 x o*ooooio36 = o - ooi3i6 grams of iodine, and 
 108 x 0*00001036 = 0*001 1 1 8 grams of silver. 
 
 Mechanism of Electrical Conductivity It has already 
 been pointed out (p. 236) that during the electrolysis of sodium 
 sulphate the products of electrolysis appear only at the poles, 
 the main bulk of solution between the poles being apparently 
 unaffected. This is most readily accounted for on the view 
 that part of the solute is moving towards the positive and part 
 towards the negative pole, these moving parts being termed 
 anions and cations respectively. We now assume further that 
 the cations are charged with positive electricity, and move 
 towards the negatively charged cathode owing to electrical 
 attraction ; similarly, the negatively charged anions are attracted 
 to the cathode. When the ions reach the poles, they give up 
 their charges, which neutralize a corresponding amount of the 
 opposite kinds of electricity on the anode and cathode respec- 
 tively, and then appear as the elements or compounds we are 
 familiar with. The process of electrolysis is illustrated in Fig. 
 29. Into the vessel containing sodium sulphate solution dip 
 two electrodes (on opposite sides of the vessel) connected with 
 the positive and negative poles of the battery respectively. The 
 direction of motion of the ions to the oppositely-charged poles 
 is illustrated by the arrows. 
 
 It is not always an easy matter to say what the moving 
 ions are. It is only rarely that they are set free as such, since 
 secondary reactions often take place at the electrodes. When 
 a strong solution of cupric chloride is electrolysed, copper 
 
ELECTRICAL CONDUCTIVITY 
 
 239 
 
 and chlorine are liberated at the cathode and anode respectively, 
 and it is probable that these substances are the ions. In the 
 case of sodium sulphate, however, for which hydrogen and 
 oxygen are the products of electrolysis, secondary reactions 
 must take place. The current is in all probability conveyed 
 through the solution by Na and SO 4 ions. When the former 
 reach the cathode, they give up their charges and form metallic 
 sodium, which immediately reacts with the water, forming sodium 
 hydroxide and hydrogen. In the same way the SO 4 ions, on 
 reaching the anode, give up their charges, and the free SO 4 
 group then reacts with the water according to the equation 
 SO 4 + H 2 O = H 2 SO 4 + O, oxygen being liberated and sul- 
 phuric acid regener- 
 ated. In this way 
 the phenomena al- 
 ready described are 
 readily accounted 
 for. 
 
 So far, we have 
 assumed that the 
 material of the elec- 
 trodes is not acted 
 on by the products 
 of electrolysis. This 
 is generally true 
 when the electrodes 
 are made of platinum 
 or other resistant FIG. 29. 
 
 metal, but in other 
 
 cases secondary reactions take place between the discharged 
 ions and the poles. Thus when a solution of copper sulphate 
 is electrolysed between copper poles, the SO 4 ions, after losing 
 their charges, react with the anode according to the equation 
 Cu + SO 4 = CuSO 4 , so that the net result of the electrolysis 
 of copper sulphate between copper poles is the transfer of copper 
 from the anode to the cathode. 
 
240 OUTLINES OF PHYSICAL CHEMISTRY 
 
 We have assumed that*m a solution of sodium sulphate the 
 moving ions are Na and SO 4 . As the Na ion moves towards 
 
 the negative electrode, it must already be positively charged ; 
 
 + 
 this may be indicated thus : Na (Fig. 29), or, more concisely, 
 
 by a dot, thus : Na\ As neither positive nor negative elec- 
 tricity accumulates in the solution during electrolysis, the 
 amount of positive electricity neutralized on the anode must 
 be equivalent to that neutralized on the cathode. Hence, since 
 two sodium ions are discharged for every SO 4 ion, the latter 
 must carry double the amount of electricity that a sodium 
 ion carries, and this is indicated by the symbols SO 4 or SO 4 " 
 (Fig. 29). 
 
 According to our present views, the metallic components of 
 salts in solution are positively charged, the number of charges 
 corresponding with the ordinary valencies of the metals. Some 
 important cations are K*, Na', Ag', NH 4 ', Ca", Hg 2 ", Hg", 
 Fe", Fe , etc. The remainder of the salt molecule constitutes 
 the negative ion, which, like the positive ion, may have one, two 
 or more (negative) electric charges. Among the more important 
 anions are Cl', Br', I', NO 3 ', SO/', CO 3 ", PO 4 '", etc. Acids and 
 bases deserve special consideration from this point of view. 
 Since salts are derived from acids by replacing the hydrogen by 
 metals, it is natural to suppose that the positive ion in aqueous 
 solutions of acids is H", and that the remainder of the molecule 
 constitutes the negative ion. On the other hand, aqueous solu- 
 tions of all bases contain the OH' group. These points are 
 dealt with fully at a later stage. 
 
 Freedom of the Ions before Electrolysis The fact that 
 the ions begin to move towards the respective electrodes im- 
 mediately the current is made appears to indicate that they are 
 electrically charged in the solution before electrolysis is com- 
 menced. The questions therefore arise as to the state of such 
 a salt as sodium chloride in dilute solution, and as to what 
 occurs when the circuit is completed. The view long held was 
 that the atoms are united to form a molecule, NaCl, at least 
 
ELECTRICAL CONDUCTIVITY 241 
 
 partly owing to the electrical attraction of their contrary charges, 
 and that the current pulls them apart during electrolysis. Careful 
 measurements show, however, that Ohm's law holds for electro- 
 lytes, from which it follows that the electrical energy expended in 
 electrolysis is entirely used up in overcoming the resistance of the 
 electrolyte, so that no work is done in pulling apart the components 
 of the molecule. On the basis of this observation, and in agree- 
 ment with certain views previously enunciated by Williamson as to 
 the kinetic nature of equilibrium in general (cf. p. 160), Clausius 
 showed that the equilibrium condition in electrolytes cannot be 
 such that the ions of contrary charge are firmly bound together ; 
 on the contrary, the equilibrium must be of a kinetic nature, so 
 that the ions are continuously exchanging partners, and must, 
 at least momentarily, be present in solution as free ions. The 
 average fraction of the ions free under definite conditions of 
 temperature and dilution was not estimated by Clausius, but he 
 considered that the fraction was probably very small. Clausius's 
 theory accounts for the qualitative phenomena of electrolysis, 
 as during their free intervals the ions would be progressing 
 towards the oppositely charged poles, and would finally reach 
 them and be discharged. 
 
 The views of Clausius were further developed in 1887 by 
 Arrhenius^who first showed how the fraction of the molecules 
 split up into ions could be deduced from electrical conduc- 
 tivity measurements, and independently from osmotic pressure 
 measurements. This constitutes the main feature of the theory 
 of electrolytic dissociation, which is dealt with in detail later 
 (p. 260), and the fact that the two methods for determining 
 the fraction of the molecules present as free ions gave results 
 in very satisfactory agreement contributed much to the general 
 acceptance of the theory. In a normal solution of sodium chlo- 
 ride, then, there is an equilibrium between free ions and non- 
 ionised molecules, represented by the equation NaCl ^ Na* + Cl', 
 in which, according to Arrhenius, about 70 per cent, of the 
 salt is ionised and the remaining 30 per cent, is present as 
 
 1 Zeitsch. physikal. Chetn., 1887, I, 631. 
 16 
 
242 OUTLINES OF PHYSICAL CHEMISTRY 
 
 NaCl molecules. According to this theory, the electrical con- 
 ductivity is determi?ied exclusively by the free ions, and not at 
 all by the non-ionised molecules or by the solvent. 
 
 Dependence of the Conductivity on the Number and 
 Nature of the Ions We are now in a position to form a 
 picture of the mechanism of electrical conductivity in a solu- 
 tion. Suppose there are two parallel electrodes i cm. apart 
 (Fig. 29) with the electrolyte between them, and that the differ- 
 ence of potential between the electrodes is kept constant, say 
 at i volt. Before the electrodes are connected with the battery, 
 the ions are moving about in all directions through the solution. 
 When connection is made in other words, when the electrodes 
 are charged they exert a directive force on the charged ions, 
 which move towards the poles with the contrary charges. 
 Those nearest the poles arrive first, give up their charges to 
 the poles, thus neutralising an equivalent amount of electricity 
 on the latter, and then either appear in the ordinary uncharged 
 form (e.g.) copper), react with the solvent (e.g., SO 4 when platinum 
 electrodes are used), with the electrodes (e.g., SO 4 with copper 
 electrodes), or with each other. It will be seen that the process 
 does not consist in the direct neutralisation of the electricity on 
 the positive electrode by that on the negative electrode, but 
 part of the charge on the anode is neutralised by the anions, 
 whilst an equivalent amount of charge on the cathode is 
 neutralised by the cations a process which has the same 
 ultimate effect as direct neutralisation. 
 
 On this basis it is clear that with a constant E.M.F. the rate 
 at which the charges on our two plates are neutralised, in other 
 words, the conductivity of the solution between them, depends 
 on three things : (i) the number of carriers or ions per unit 
 volume ; (2) the load or charge which they carry ; (3) the rate 
 at which they move to the electrodes. Each of these factors 
 will now be briefly considered. 
 
 (i) The Number of Ions Other things being equal, the 
 conductivity of a solution will clearly be proportional to the 
 
ELECTRICAL CONDUCTIVITY 243 
 
 number of ions per unit volume. For the same electrolyte, 
 the number of ions can, of course, be varied by varying the 
 concentration of the solution. In general, it may be said that 
 on increasing of concentration the ionic concentration also in- 
 creases, but the exact relationship will be dealt with later. For 
 different electrolytes of the same equivalent concentration, the 
 conductivity will depend on the extent to which the solute is 
 split up into its ions and on their speed. 
 
 (2) The Charge Carried by the Ions As has already been 
 pointed out, there is a simple relationship between the capacity 
 of different ions for transporting electricity, since the gram- 
 equivalent of any ion (positive or negative) conveys 96,540 
 coulombs. Thus if in an hour ( = 3600 seconds) a gram-equiva- 
 lent of sodium (23 grams) and of chlorine (35*47 grams) are 
 discharged at the respective electrodes, the current which has 
 passed through the cell is 
 
 ^l5!? = 26-8 amperes. 
 3600 
 
 (3) Migration Velocity of the Ions In this section we 
 will for simplicity consider only univalent ions, but the same 
 considerations apply to all electrolytes. Since positive and 
 negative ions are necessarily discharged in equivalent amount 
 (p. 240), and the number of positive and negative univalent ions 
 discharged in a given time is therefore equal, it might be 
 supposed that the ions must travel at the same rate. This, 
 however, is by no means the case. Our knowledge of this 
 subject is mainly due to Hittorf, who showed that the relative 
 speeds of the ions could be deduced from the changes in concen- 
 tration round the electrodes after electrolysis. 
 
 The effect of the unequal speeds of the ions on the concen- 
 trations round the poles is made clear by the accompanying 
 scheme (Fig. 30), a modified form of one given by Ostwald. 
 The vertical dark lines represent the anode and cathode respec- 
 tively, and the dotted lines divide the cell into three sections, 
 those in contact with the electrodes being termed the. anode 
 
244 OUTLINES OF PHYSICAL CHEMISTRY 
 
 and cathode compartments respectively. The positive ions are 
 represented by the usual + sign, and the negative ions by the 
 sign. I. represents the state of affairs in the solution before 
 connection is made ; the number of anions is the same as that 
 of the cations, and the concentration is uniform throughout. 
 The remaining lines represent the state of affairs in the solution 
 after electrolysis on different assumptions as to the relative 
 
 I. 
 
 Iff. 
 
 * '-K +> 
 
 + + + t + + 
 
 + f + -!- 
 
 4- -1- 4- f 
 
 4 + 4- + + + 
 
 + 4- 4 + 
 
 4- 4- 
 
 ' 
 
 + -1- -h + 4- -r 
 
 f + f -f -f + 
 
 4- 
 
 4- 4- 4- f + + 
 
 -f f + -f + + + 
 
 
 
 
 FIG. 30. 
 
 speeds of the ions. Suppose at first that only the negative 
 ions move. The condition of affairs in the solution when all 
 the negative ions have moved two steps to the left is shown in 
 II. Each ion left without a partner is supposed to be dis- 
 charged, and the figure shows that although the positive ions 
 have not moved an equal number of positive ions is discharged. 
 Further, whilst the concentration in the anode compartment 
 has not altered during the electrolysis, the concentration in the 
 cathode compartment has been reduced by half. 
 
 Suppose now that the positive and negative ions move at the 
 same rate. The state of affairs when each ion, positive or 
 
ELECTRICAL CONDUCTIVITY 245 
 
 negative, has moved two steps towards the oppositely-charged 
 pole is represented in III. It is evident that four positive and 
 four negative ions have been discharged, and that the con- 
 centration of undecomposed salt has diminished in both com- 
 partments, and to the same extent, namely by two molecules. 
 
 Finally, let us assume that both ions move, but at unequal 
 rates, so that the positive ions move faster than the negative ions 
 in the ratio 3:2. The state of affairs when the positive ions 
 have moved three steps to the right, and the negative ions two 
 steps to the left, is shown in IV. It is clear that five positive 
 and five negative ions have been discharged, and that whilst 
 there is a fall of concentration of two molecules round the 
 cathode, there is a fall of three round the anode. 
 
 These results show that the fall of concentration round any 
 one of the electrodes is proportional to the speed of the ion 
 leaving it. In II., for example, there is a fall of concentration 
 round the cathode, but not round the anode, corresponding 
 with the fact that the anion moves, but not the cation. 
 Similarly, in III., the fall of concentration round anode and 
 cathode is equal, corresponding with the fact that the anion 
 and cation move at the same. Finally, in IV., fall round 
 anode : fall round cathode 1:3:2, corresponding with the fact 
 that speed of cation : speed of anion = 3:2. From these 
 examples we obtain the important rule that 
 
 Fall of concentration round anode _ speed of cation 
 Fall of concentration round cathode ~~ speed of anion ' 
 
 The student often finds a difficulty in understanding how, as in 
 IV., five ions can be discharged at the anode when only two 
 anions have crossed the partitions. To account for this, it 
 must be assumed that " there is always an excess of ions in 
 contact with the electrodes, so that more are discharged than 
 actually arrive by diffusion. 
 
 The speed of the cations is often represented by u, and that 
 of the anions by v. The total quantity of electricity (say, unit 
 
246 OUTLINES OF PHYSICAL CHEMISTRY 
 
 quantity) carried is proportional to (u + v), and, of this total 
 n vj(u + v) is carried by the anions and i - n = u/(u + v) 
 by the cations, n, the fraction of the current carried by the 
 anion, is termed the transport number of the anion ; similarly, 
 i - n is the transport number of the cation. 
 
 It is evident from the figure that there is a central section of 
 the cell between the dotted lines in which no change of con- 
 centration* takes place when elec- 
 *f f I "I trolysis is not carried too far. 
 
 Therefore, in order to investigate 
 the changes in concentration, it is 
 simply necessary to remove the 
 solutions round the electrodes 
 after electrolysis and analyse 
 them, but the experiment will only 
 be successful if the intermediate 
 layer has not altered in strength, 
 
 Practical Determination of 
 the Relative Migration Velo- 
 cities of the Ions The experi- 
 ment may conveniently be made 
 in the modified form of Hittorf s 
 apparatus used in Ostwald's labo- 
 ratory (Fig. 31). It consists of 
 two glass tubes communicating 
 towards the upper ends ; one of 
 them is closed at the lower end, 
 and the other provided with a 
 stopcock, as shown. The elec- 
 trodes, A and K, are sealed into 
 FIG. 31. glass tubes which pass up through 
 
 the liquid, and communication 
 
 with a battery is made in the usual way by means of wires which 
 pass down the interior of the glass tubes. 
 
 As an illustration, the determination of the transport numbers 
 
 B 
 
ELECTRICAL CONDUCTIVITY 247 
 
 of the Ag' and NO 3 ' ions in a ^plution of silver nitrate will be 
 described. The anode A is of silver, and should be covered with 
 finely-divided silver by electrolysis just before the experiment ; 
 the cathode is of copper. The electrodes are placed in position, 
 the anode compartment filled up to the connecting tube with 1/20 
 normal silver nitrate, the cathode compartment up to B with a 
 concentrated solution of copper nitrate, and finally the apparatus 
 is carefully filled up with the silver nitrate solution in such a way 
 that the boundary between the two solutions at B remains fairly 
 sharp. The cell is then connected in series with a high adjust- 
 able resistance, an ammeter, and a silver voltameter, and then 
 joined to the terminals of a continuous current lighting circuit 
 (no volts) in such a way that the silver pole becomes the 
 anode. By means of the variable resistance, the current is so 
 adjusted that a current of about o'oi ampere is obtained (to be 
 read off on the ammeter), and the electrolysis continued for 
 about two hours. Finally, a measured amount (about 3/4) of 
 the anode solution is run off and titrated with thiocyanate in 
 the usual way. The strength of the current can be read off 
 on the ammeter, and from this and the time during which 
 the current has passed, the total quantity of electricity passed 
 through the solution can be calculated. It is, however, prefer- 
 able to employ for this purpose the silver voltameter above 
 referred to. It consists of a tube with stopcock similar to the 
 left-hand part of the transport apparatus (Fig. 31), and is 
 provided with a silver electrode (to serve as anode) similar to 
 that in the other apparatus, and placed in a corresponding 
 position (in the lower part of the tube). The tube is filled to 
 3/4 of its length with a 15-20 per cent, solution of sodium or 
 potassium nitrate, and carefully filled up with dilute nitric acid 
 so that the two solutions do not mix. The cathode, of platinum 
 foil, dips in the nitric acid. During electrolysis, the NO 3 ' ions 
 dissolve silver from the anode, and by titrating the whole of the 
 contents with ammonium thiocyanate after the experiment, the 
 amount of silver in solution can be determined, and from this 
 
248 OUTLINES OF PHYSICAL CHEMISTRY 
 
 the quantity of electricity which has passed through the solution 
 can readily be calculated (p. 238). 
 
 We now return to the transport apparatus. For our purpose 
 it will be sufficient to deal only with the change of concentra- 
 tion in the anode compartment. From this the transport 
 number of the cation is obtained, and the transport number 
 of the anion is then at once obtained by difference. During 
 electrolysis, the silver concentration round the anode diminishes 
 owing to migration of silver ions towards the cathode. The 
 process may conveniently be illustrated by III. of Fig. 30 where 
 the fall owing to migration is from 4 to 2. At the same time, 
 however, NO 3 ions reach the anode, and after being discharged 
 dissolve silver from it, the silver concentration in the anode 
 compartment therefore increasing. The latter effect is the 
 same as that taking place simultaneously in the silver volta- 
 meter, as described above, and therefore, if no silver migrated 
 from the anode, the total increase of concentration in this com- 
 partment would be equal to that in the silver voltameter, 
 which, as explained above, is a measure of the total quantity 
 of electricity which passes ; we will term this a. If b is the 
 (unknown) change in concentration due to the migration of 
 the silver ions, the observed change in concentration at the 
 anode will be a b. As a is known, and a b is found by 
 titrating the anode solution after the experiment, b can readily 
 be obtained. 
 
 In practice, the greater part of the anode solution after 
 electrolysis is run into a beaker, it is then weighed or an aliquot 
 part measured, and titrated. 
 
 The calculation of the results will be rendered clear from the 
 details of an experiment made in Ostwald's laboratory. Before 
 the experiment 12-31 grams of the silver nitrate solution re- 
 quired 26-56 c.c. of a 1/50 n potassium thiocyanate solution, 
 so that i gram of solution contained 0*00739 grams of silver 
 nitrate. After the experiment, 23-38 grams of the anode 
 solution required 69-47 c.c. of the thiocyanate solution, cor- 
 
ELECTRICAL CONDUCTIVITY 249 
 
 responding to 0*2361 grams of silver nitrate. The solution, 
 therefore, contained 23*14 grams of water, which before the 
 experiment contained 23*14 x 0*00739 = o'lyio grams of 
 silver nitrate, hence the increase of concentration at the anode 
 is 0*065 l grams = a b. The contents of the silver volta- 
 meter required 36*16 c.c. of thiocyanate = 0*1229 grams of 
 silver nitrate = a ; the same amount is dissolved at the anode 
 in the transport apparatus. As the actual increase of con- 
 centration was only 0*0651 grams, 0*1229 - 0*0651 = 0*0578 
 grams of silver must have left the anode compartment by migra- 
 tion. Hence the transport number for silver is 
 u 0*0578 
 
 
 
 u + v 0*1229 
 and for the NO' ion 
 
 0*470, 
 
 n = 
 
 u + v 0*1229 
 
 Hence, of the total current 47 per cent, is carried by the silver 
 ions, and 53 per cant, by the NO 3 ' ions. 
 
 It was shown by Hittorf that the transport numbers are 
 practically independent of the E.M.F. between the electrodes, 
 but depend to some extent on the concentration and on 'the 
 temperature. It is remarkable that at higher temperatures 
 they tend to become equal. Some of the numbers are given 
 in the next section. 
 
 Specific, Molecular and Equivalent Conductivity Just 
 as in the case of metallic conduction, the resistance of an electro- 
 lyte is proportional to the length, and inversely proportional to 
 the cross-section of the column between the electrodes. Hence 
 we may define the specific resistance of an electrolyte as the 
 resistance in ohms of a cm. cube, and its specific conductivity 
 as i /specific resistance, expressed in reciprocal ohms. Sines, 
 however, the conductivity does not depend on the solvent but 
 on the solute, it is much more convenient to deal with solutions 
 containing quantities of solute proportional to the respective 
 molecular weights. The so-called molecular conductivity, ^ is 
 
250 OUTLINES OF PHYSICAL CHEMISTRY 
 
 most largely used in this connection ; it is the conductivity, in 
 reciprocal ohms, of a solution containing i mol of the solute 
 when placed between electrodes exactly i cm. apart. It may 
 also be defined as the specific conductivity, /c, of a solution, 
 multiplied by v, the volume in c.c. which contain a mol of the 
 solute. Hence we have 
 
 /x- = KV. 
 
 As an example, the following values for the specific and mole- 
 cular conductivities of solutions of sodium chloride at 18, as 
 given by Kohlrausch, may be quoted. 
 
 Sp. Con- Molecular 
 
 Concentration of Solution. ductivity, Conductivity, 
 
 K. KV, 
 
 I'o molar (v= 1,000) 0*0744 74-4 
 
 0*1 molar (v= 10,000) 0*00925 92-5 
 
 o'oi molar (v= 100,000) 0*001028 102*8 
 
 o'ooi molar (v= 1,000,000) 0-0001078 107*8 
 
 0*0001 molar (v 10,000,000) 0*00001097 I0 9'7 
 
 It will be noticed that the molecular conductivity as defined 
 above increases at first with dilution, but beyond a certain 
 point remains practically constant on further dilution. 
 
 These numbers enable us to illustrate more fully the physical 
 meaning of the molecular conductivity. Imagine a cell of i cm. 
 cross-section and of unlimited height, two opposite walls through- 
 out the whole height acting as electrodes. If a litre of a 
 molar solution of sodium chloride is placed in the cell, it will 
 stand at a height of 1000 cms. We may regard the solution 
 as made up of cm. cubes, 1000 in number, and if the con- 
 ductivity of one of these cubes the specific conductivity is 
 K r the total conductivity (in other words the molecular con- 
 ductivity) is loooKj. If now another litre of water is added, the 
 height of the solution will be 2000 cms. and its molecular con- 
 ductivity is now 2oooK 2 , where * 2 is the specific conductivity of 
 the half-molar solution. In exactly the same way, the mole- 
 cular conductivity may be determined at still greater dilutions. 
 From the above it is clear that a measure of the conducting 
 
ELECTRICAL CONDUCTIVITY 251 
 
 power of a mol of the electrolyte in different dilutions is ob- 
 tained by multiplying the volume in c.cs. in which the electrolyte 
 is dissolved by its specific conductivity at that dilution, or in 
 symbols 
 
 JU= KV 
 
 as given above. 
 
 A glance at the last two lines in the table helps us to under- 
 stand the approximately constant value of /x in very dilute 
 solutions. When the solution is diluted from i/iooo to 
 1/10,000 molar, the specific conductivity is reduced to about 
 i/ 10, but as the volume is ten times as great, the molecular 
 conductivity is only slightly altered. 
 
 Besides the molecular conductivity, the term equivalent con- 
 ductivity, A, is sometimes used. As the name implies, it is the 
 specific conductivity of a solution multiplied by the volume 
 in c.c. which contains a gram-equivalent of the solute. 
 
 Kohlrausch's Law. Ionic Yelocities The numbers in 
 the third column of the above table show that the molecular 
 rt conductivity of sodium chloride increases, at first rapidly and 
 \|then very slowly, with dilution. This subject was investigated 
 for a number of solutions by Kohlrausch, who found that for 
 solutions of electrolytes of high conductivity (salts, so-called 
 " strong " acids and bases) the molecular conductivity increases 
 with dilution up to about i/ioooo molar solution, and beyond 
 that point remains practically constant on further dilution. 
 Kohlrausch showed further that this limiting value of the mole- 
 cular conductivity, which may be represented by p^, is different 
 for different salts, and may be regarded as the sum of two inde- 
 pendent factors one pertaining to the cation or positive part 
 of the molecule, the other to the anion, or negative part of the 
 molecule. This experimental result is termed Kohlrausch's law, 
 and is readily intelligible on the basis of the theory of electrical 
 conductivity developed above. The limiting value of the mole- 
 cular conductivity is reached when the molecule is completely 
 split up into its ions ; under these circumstances the whole of 
 the salt takes part in conveying the current. For simplicity we 
 will consider solutions of binary electrolytes. In very dilute 
 
252 OUTLINES OF PHYSICAL CHEMISTRY 
 
 equimolar solutions of different electrolytes, the number of the 
 ions and their charges are the same, and the observed differences 
 of /AOQ can only be due to the different speeds of the ions. The 
 limiting molecular conductivities of binary electrolytes are 
 therefore proportional to the sum of the speeds of the ions, 
 and when the units are properly chosen we have 
 
 Moc = + 
 
 where u is the speed of the cation, v that of the anion. This is 
 the mathematical form of Kohlrausch's law, and expresses the 
 very important result that in sufficiently dilute solution the speed 
 of an ion is independent of the other ion present in solution. 
 
 From the results of conductivity measurements, only the 
 sum of the speeds of the ions can be deduced, but, as has 
 already been shown, the relative values of u and v can be 
 obtained from the results of migration experiments (p. 246). 
 It was found by Kohlrausch that the value of /A^ = u + v 
 for silver nitrate at 18 is 115*5. The accurate value for the 
 transport number of the anion, NO 3 ', is n = vj(u- + v) = 0-518. 
 Hence v 0^518 x 115-5 = 60*8 and u = 0-482 x 115*5 = 
 557. The values of u and v, expressed in these units, are 
 termed the ionic velocities, under the conditions of the experi- 
 ment. The accompanying table gives the ionic velocities, 
 calculated from the results of conductivity and transport measure- 
 ments, for some of the more common ions in infinite dilution * at 
 1 8, expressed in the same units as the molecular conductivity 
 of sodium chloride (p. 250): 
 
 H- = 318 Li- = 36 OH' = 174 
 
 K* 65 NH 4 - = 64 Cr = 66 
 
 Na- - 45 Ag- =56 T = 67 
 
 NCV= 6 1 
 
 It is interesting to observe that the velocity of the H* ion is 
 relatively very high, about five times as great as that of any of the 
 metallic ions. The ion which comes next to it is the OH' ion, 
 
 1 The more concentrated the solution, the smaller are the ionic velocities, 
 owing to the increased resistance to their motion. 
 
ELECTRICAL CONDUCTIVITY 253 
 
 the speed of which is more than half that of the H* ion, and 
 much greater than that of any of the other ions. Since the 
 conductivity of a solution is, as we have already seen, propor- 
 tional to the speed of the ions, it follows that the solutions of 
 highly ionised acids and bases will have a relatively high con- 
 ductivity. Thus, under conditions otherwise equal as regards 
 concentration, ionisation, temperature, etc., the conductivities 
 of dilute solutions of hydrochloric acid and of sodium chloride 
 will be in the ratio (318 + 66) = 384 to (45 + 66) = in, or 
 about 3-5:1. 
 
 Absolute Velocity of the Ions. Internal Friction 
 The absolute velocity of the ions is proportional to the E.M.F. 
 between the electrodes, and inversely proportional to the re- 
 sistance offered to their passage by the solvent. When the 
 fall of potential is one volt per c.m. (i.e., when the difference of 
 potential between the electrodes is x volts and the distance be- 
 tween them is x cm.) it can be shown (p. 414) that the absolute 
 velocities, in cm. per second, are obtainable from the values 
 for the ionic velocities given above by dividing by 96,540 
 or, what is the same thing, by multiplying by 1*036 x io~ 5 . 
 Hence the absolute velocity of the hydrogen ion is, under the 
 conditions described, 318 x 1-036 x io~ 5 = 0*00332 cm. per 
 second, and of the potassium ion 0-00067 cm - P er second at 
 infinite dilution. The speed of the ions is therefore extremely 
 low ; even the hydrogen ions, under a driving force of i volt 
 per cm., only move about twice as fast as the extremity of the 
 minute hand of an ordinary watch. This very slow motion 
 of the moving particles indicates that the resistance to their 
 passage through the solvent is very great. Kohlrausch has 
 calculated that the force required to drive a gram of sodium 
 ions through a solution at the rate of i cm. per second is 
 153 x io 6 kilograms weight, or about 150,000 tons weight. 
 
 The absolute velocity of the ions can also be measured 
 directly by a method the principle of which is due to Lodge, 
 and which may be illustrated by an experiment described by 
 
254 OUTLINES OF PHYSICAL CHEMISTRY 
 
 Danneel. A U-tube is partly filled with dilute nitric acid, 
 and in the lowest part of the tube a solution of potassium 
 permanganate, the specific gravity of which has been increased 
 as much as possible by the addition of urea, is carefully placed, 
 by means of a pipette, in such a way that the boundary between 
 the acid and the permanganate remains sharp. When platinum 
 electrodes are dipped in the nitric acid in the two limbs, and 
 a current passed through the solution, the violet boundary (due 
 to the coloured MnO 4 ' ion) moves towards the anode. From 
 the observed speed of the boundary and the difference of 
 potential between the poles, the speed of the ion for a fall 
 of potential of i volt/cm, is obtained, and has been found to 
 agree exactly with the value obtained by conductivity and 
 transport measurements. 
 
 This method is not confined to salts with coloured ions, but 
 the moving boundary can also be observed with colourless 
 solutions when, as is usually the case, the refractive index of 
 the two solutions is different. There are, of course, two 
 boundaries, one due to the positive ions moving towards the 
 cathode, and the other due to the negative ions moving towards 
 the anode. When the conditions are such that both can be 
 observed, the relative speeds give the ratio u/v directly. 
 
 Measurements of ionic velocities on this principle have been 
 made by Masson, Steele and others, and the results are in 
 entire agreement with those obtained indirectly. 
 
 Experimental Determination of Conductivity of Elec- 
 trolytes The measurement of the conductivity of conductors 
 of the first class is a very simple operation. Until compara- 
 tively recently, however, no very satisfactory results for the 
 conductivity of electrolytes could be obtained, because when 
 a steady current is passed through a solution between platinum 
 electrodes the products of electrolysis accumulate at the poles 
 and set up a back E.M.F. of uncertain value, a phenomenon 
 known as polarization (p. 373). This difficulty is, however, com- 
 pletely got over by using an alternating instead of a direct 
 
ELECTRICAL CONDUCTIVITY 255 
 
 current (Kohlrausch, 1880) ; by the rapid reversal of the current 
 the two electrodes are kept in exactly the same condition, and 
 there is no polarization. 
 
 The arrangement of the apparatus, which in principle amounts 
 to the measurement of resistance by the Wheatstone bridge 
 method, is shown in Fig. 32. R is a resistance box, S a cell 
 with platinum electrodes, between which is the solution the 
 resistance of which is to be measured, ab is a platinum wire 
 of uniform thickness, which may conveniently be a metre long, 
 and is stretched along a board graduated in millimetres, c is 
 a sliding contact. By means of a battery (not shown in the 
 figure) a direct 
 current is sent 
 through a Ruhm- 
 korff coil, K, the 
 latter then gives 
 rise to an alter- 
 nating current, 
 which divides at 
 a into two 
 branches, reach- 
 ing b by the paths 
 adb and acb re- 
 spectively. As a galvanometer is not affected by an alter- 
 nating current, it is in this case replaced by a telephone T, 
 which is silent when the points c and d are at the same potential. 
 The contact-maker, c, is shifted along the wire till the telephone 
 no longer sounds. Under these circumstances, the following 
 relationship holds 
 
 Length of ac Length of cb 
 
 FIG. 32. 
 
 and since ac, cb and R are known, S, the resistance of the part 
 of the electrolyte between the electrodes, can ?t once be 
 calculated. 
 
 As the resistance of electrolytes varies within wide limits, 
 
256 OUTLINES OF PHYSICAL CHEMISTRY 
 
 different forms of cell are employed according to circum- 
 stances. For solutions of small conductivity, the Arrhenius 
 form represented in Fig. 33. is very suitable. The electrodes, 
 which are stout platinum discs 2-4 cm. in diameter, are fixed 
 (by welding or otherwise) to platinum wires, which are sealed 
 into glass tubes A and B, as shown in the figure. These glass 
 tubes are fixed firmly into the ebonite cover of the cell, so that 
 the distance between the electrodes remains constant, and electri- 
 cal connection is made in the usual way by wires passing down the 
 interior of the glass tubes. In order to expose a larger surface, 
 
 
 FIG. 33. 
 
 FIG. 34. 
 
 and thus minimise polarization effects, which would interfere 
 with the sharpness of the minimum in the telephone, the elec- 
 trodes are coated with finely-divided platinum by electrolysis 
 of a solution of chlorplatinic acid. For electrolytes of high 
 conductivity, a modified form of conductivity vessel, with 
 smaller electrodes placed further apart, has been found con- 
 venient (Fig. 34). 
 
 Experimental Determination of Molecular Conduc- 
 tivity It is clear that the observed resistance of the electro- 
 lyte must depend on what is usually termed the capacity of the 
 
ELECTRICAL CONDUCTIVITY 257 
 
 cell, that is, on the cross-section of the electrodes and the 
 distance between them. The specific conductivity, and hence 
 the specific resistance, of the electrolyte could be calculated if 
 these two magnitudes were known (p. 235); but it is much 
 simpler to determine the " constant " of the vessel, which is 
 proportional to its capacity, by using an electrolyte of known 
 conductivity. For this purpose, a 1/50 molar solution of potas- 
 sium chloride may conveniently be used for cells of the first 
 type. The method of procedure will be clear from an example, 
 Referring to the figure, we have, for the resistance, S, of the 
 electrolytic cell 
 
 R'fo , i ac 
 
 b = - and conductivity (^ = - = =-. 
 ac S R'&r 
 
 Further, since the specific conductivity, *, must be proportional 
 to the observed conductivity, we have 
 
 A/^ ac 
 
 = 
 
 where A is a constant. Since all the other factors, including *, 
 the specific conductivity of potassium chloride, are known, A, 
 the constant of the cell, can be calculated. If, now, with the 
 same distance between the electrodes, a solution of unknown 
 specific conductivity, K lt is put in the cell, and for the resistance 
 R' the new position of the contact is c, the specific conductiv- 
 ity in question is given by the formula 
 
 ac 
 
 *' = A RW' 
 
 By multiplying /^ by the number of c.c. containing i mol of the 
 solute, the molecular conductivity is obtained. 
 
 An alternative method of calculating KJ without reference to 
 the cell constant is as follows. If S and S x represent the resist- 
 ances of the cell containing N/5oKCl and the solution of specific 
 conductivity K-^ respectively, then 
 
 KS 
 
258 OUTLINES OF PHYSICAL CHEMISTRY 
 
 From the results of conductivity measurements in different 
 dilutions, ya^ can readily be obtained directly for salts, strong 
 acids and bases ; it is the value to which /x, approximates on 
 progressive dilution. /x^ cannot, however, be obtained directly 
 for weak electrolytes, such as acetic acid and ammonia ; before 
 the limiting value of the conductivity is reached with these 
 electrolytes, the solutions would be so dilute as to render accu- 
 rate measurement of the specific conductivity impossible. This 
 difficulty is got over by making use of Kohlrausch's law. The 
 value of /X-OQ for acetic acid must be the sum of the velocities 
 of the H* and CH 3 COO' ions. The former is obtained from the 
 results of conductivity and transport measurements with any 
 strong acid, and has the value 318 at 18. In a similar way 
 the velocity of the CH 3 COO' ion can be obtained from ob- 
 servations with an acetate for which the value of /^ can con- 
 veniently be found, e.g., sodium acetate, /x^ for the latter salt at 
 18 is 78-1, and as the velocity of the Na* ion is 44*4 at infinite 
 dilution, that of the CH 3 COO' ion must be 78-1 -44*4 = 33*7. 
 Hence for acetic acid 
 
 /AGO = + v = 337 + 318 = 351-7 at 1 8. 
 
 Results of Conductivity Measurements In general, it 
 may be said that the conductivity of pure liquids is small. 
 Thus the specific conductivity of fairly pure distilled water is 
 about io~ 6 reciprocal ohms at 18, and even this small con- 
 ductivity is largely due to traces of impurities. It is a remark- 
 able fact that the specific conductivity of a number of other 
 liquids, which have been purified very carefully by Walden, 1 is 
 of the same order as that given above for water. 
 
 Mixtures of two liquids have in many cases a very small 
 conductivity, not appreciably greater than that of the pure 
 liquids themselves ; this is true of mixtures of glycerine and 
 water, and of alcohol and water. On the other hand, a 
 mixture of two liquids which are practically non-conductors 
 may have a very high conductivity for example, mixtures of 
 sulphuric acid and water. The results obtained for this 
 1 Zeitsch. Physikal Chern., 1903, 46, 103. 
 
ELECTRICAL CONDUCTIVITY 
 
 259 
 
 ture are represented in Fig. 35, the acid concentration being 
 measured along the horizontal axis, and the specific conductivity 
 along the vertical axis. The figure shows that, on gradually 
 adding sulphuric acid to water, the specific conductivity of the 
 mixture increases till 30 per cent, of acid is present, reaches a 
 maximum value at that point, and on further addition of acid 
 diminishes. When pure sulphuric acid is present (100 per' cent, 
 on curve), the conductivity is practically zero, and is increased 
 
 07 
 
 0-6 
 
 0-5 
 
 0-4 r-e/ 
 
 0-3 
 
 H 2 SO 
 
 \ 
 
 Concentration 
 
 40 50 60 
 FIG. 35. 
 
 70 80 90 TOO per cent. 
 
 both by the addition of water (left-hand part of curve) and of 
 sulphur trioxide (right-hand part of curve). Further, the curve 
 has a minimum between 84 and 85 per cent, of acid, which, it 
 is interesting to note, exactly corresponds with the composition 
 of the monohydrate H 2 SO 4 , H 2 O. According to the electro- 
 lytic dissociation theory, the conductivity depends on the pres- 
 ence of free ions, and the curve for sulphuric acid and water 
 shows in a very striking way that the condition most favourable 
 for ionisation is the presence of two substances. Why ions are 
 formed in a mixture of sulphuric acid and water, and not 
 appreciably, if at all, in a mixture of alcohol and water, is not 
 well understood (cf. p. 317). 
 Analogous phenomena are met with for solutions of solids 
 
260 OUTLINES OF PHYSICAL CHEMISTRY 
 
 and of other liquids in liquids. An aqueous solution of 
 sugar has no appreciable conducting power. The so-called 
 " strong " acids and bases form well-conducting liquids with 
 water. The conductivity of most organic acids and bases is 
 small, and in corresponding dilution ammonium hydroxide is 
 a much poorer conductor than potassium hydroxide. On the 
 other hand, all salts, even the salts of organic acids which are 
 themselves weak, have a very high conductivity. 
 
 The conductivity of substances in solvents other than water 
 is usually small, but solutions in methyl and ethyl alcohols and 
 in liquid ammonia are exceptions. The dependence of electrical 
 conductivity on the nature of the solvent will be discussed later 
 
 It is interesting to note that many fused salts, such as silver 
 nitrate and lithium chloride, are good conductors, and thus form 
 an exception to the rule that pure substances belonging to the 
 second class of conductors have a very small conductivity. 
 
 Electrolytic Dissociation It has already been pointed 
 out (p. 124) that solutions of salts, strong acids and bases, have 
 a much higher osmotic pressure in aqueous solution than would 
 be the case if Avogadro's hypothesis was valid for these solu- 
 tions. According to the molecular theory, the solutions behave 
 as if there were more particles of solute present than would be 
 anticipated from the simple molecular formula, and van't Hoff 
 expressed this by a factor /, which represented the ratio between 
 the observed and calculated osmotic pressures. This was the 
 position of the theory of solution in 1885. 
 
 About that time, Arrhenius pointed out that there is a close 
 connection between electrical conductivity and abnormally high 
 osmotic pressures ; only those solutions which, according to van't 
 Hoff 's theory, have abnormally high osmotic pressures, conduct 
 the electric current. Kohlrausch had previously shown that the 
 molecular conductivity increases at first with dilution, and for 
 many electrolytes attains a limiting value in a dilution of 
 10,000 litres (p. 251). Arrhenius accounted for this increase 
 on the assumption that the solute consists of "active" and 
 
ELECTRICAL CONDUCTIVITY 261 
 
 " inactive " parts, and that only the active parts, the ions, convey 
 the current. The extent to which the solute is split up into 
 ions increases with the dilution until finally (when the molecular 
 conductivity has attained its maximum value) it is completely 
 ionised or completely " active " as far as the conduction of 
 electricity is concerned. 
 
 The theory of Arrhenius is based upon the views of Clausius 
 on conductivity, as has already been pointed out. Arrhenius, 
 however, went much further, inasmuch as he showed how, from 
 the results of conductivity and of osmotic pressure measure- 
 ments, the degree of dissociation can be calculated, as shown in 
 the following section. 
 
 Degree of lonisation from Conductivity and Osmotic 
 Pressure Measurements According to the theory of electro- 
 lytic dissociation, the conductivity of a solution depends only on 
 the number of the ions per unit volume, on their charges (which 
 are the same for equivalent amounts of different electrolytes) 
 and on their speed. For the same electrolyte we may assume 
 that the velocities remain practically unaltered on dilution (the 
 friction in a dilute solution being practically the same as that 
 in pure water), therefore the increase of molecular conductivity 
 with dilution must depend almost entirely on an increase in 
 the number of the ions. The molecular conductivity at infinite 
 dilution is given by the formula 
 
 where u and v are the speeds of cation an d anion respectively 
 and the molecular conductivity at any dilution, v , must therefore 
 be represented by the formula 
 
 p v = a(u + v), 
 
 where a represents the fraction of the molecules split up into 
 ions. Hence, dividing the second equation by the first, we 
 have 
 
262 OUTLINES OF PHYSICAL CHEMISTRY 
 
 that is, the degree of dissociation, a, at any dilution, is the 
 ratio of the molecular conductivity at that dilution to the mole- 
 cular conductivity at infinite dilution. For example, p v for molar 
 sodium chloride is 74-3 and /x,^ = 110-3, hence a = /A X ,//* oc = 
 74'3/iio'3 = 0*673. Hence sodium chloride in molar solution 
 is about two-thirds split up into its ions. 
 
 We have now to consider the deduction of the degree of 
 dissociation from osmotic measurements. The assumption made 
 in this case is that the osmotic pressure is proportional to the 
 number of particles present, the ions acting as separate entities. 
 If a molecule is partially dissociated into n ions and the degree 
 of dissociation the ionised fraction is a, then the number of 
 molecules will be i a, and the number of ions na. Hence 
 the ratio of the number of particles actually present to that 
 deduced according to Avogadro's hypothesis (van't Hoffs factor 
 i) will be 
 
 I = I a + a = I + o.(n - i), 
 i I 
 
 or a 
 
 n i 
 
 As an illustration, de Vries obtained for a 0^14 molar solution 
 of potassium chloride /= 1*81, hence, since n = 2, a = o'8i, 
 or the salt is dissociated to the extent of 81 per cent, into its 
 ions, /for a 0-18 molar solution of calcium nitrate is 2-48, 
 
 therefore, since = 3, a = - - = 074 in this case. 
 
 The agreement in the values of / obtained from conductivity 
 and osmotic measurements is strikingly shown in the accom- 
 panying table (van't Hoff and Reicher, 1889). The values of 
 i (osmotic) are from the results of de Vries, those of / (freezing 
 point) mainly from the observations of Arrhenius, and those 
 of i (conductivity) are calculated by means of the formulae 
 a = /AV//XQC and / = i + a(n - i) as explained above. It is 
 not certain that the results obtained by the different methods 
 can be expected to agree absolutely (cf. p. 281). 
 
ELECTRICAL CONDUCTIVITY 263 
 
 Concentration . (freezi . f (conduc- 
 
 Substance. (gram equiv. ^ .. * (osmotic). t i v i ty ). 
 per litre). 
 
 KC1 
 
 0*14 
 
 1-82 
 
 1*81 
 
 r86 
 
 LiCl 
 
 0-13 
 
 1-94 
 
 1-92 
 
 1-84 
 
 Ca(N0 3 ) 2 
 
 0-18 
 
 2-47 
 
 2*48 
 
 2*46 
 
 MgCl 2 
 
 0*19 
 
 2-68 
 
 279 
 
 2-48 
 
 CaCL 
 
 0-184 
 
 2-67 
 
 2-78 
 
 2-42 
 
 As regards the mode of ionisation, it is clear that univalent 
 compounds, such as potassium chloride, can ionise only in one 
 way, thus, KC1;K* + Cl'. For more complex molecules, how- 
 ever, there are other possibilities, thus calcium chloride may 
 ionise as follows : CaCl 2 ^ CaCl* + Cl' as well as in the normal 
 way CaClg^tCa" + 2C1'. If ionisation were complete according 
 to the last equation, / would be = 3, as compared with the 
 observed value, 2-67 for 0*184 normal solution, given in the 
 table. Similarly, sulphuric acid may dissociate according to 
 the equation H 2 SO 4 ^H* + HSO/, the latter ion then under- 
 going further ionisation as follows : HSO 4 'JH* + SO 4 ". 
 
 Effect of Temperature on Conductivity The conduc- 
 tivity of electrolytes increases considerably with rise of tempera- 
 ture. The temperature coefficient for salts is 0*020 to 0*023, 
 for acids and some acid salts 0*009 to 0*016, for caustic alkalis 
 about 0*020, and does not vary much with dilution. Conduc- 
 tivity data are usually given for 18, and the specific conductivity, 
 K, at any other temperature, is given by the formula 
 
 K t = K lg [l + C(t - 1 8)] 
 
 where c is the temperature coefficient. 
 
 As the conductivity of an electrolyte depends both on the 
 number and velocity of the ions, the question arises as to 
 whether the change of conductivity with temperature is due 
 to the alteration of only one or of both these factors. The 
 matter can be at once decided by calculating the degree of 
 dissociation at the higher temperature from conductivity 
 measurements in the ordinary way, and comparing with that 
 at the lower temperature. For normal sodium chloride at 
 50, the value of a = /*//*< = 132/203*5 = 0*65, which is 
 
264 OUTLINES OF PHYSICAL CHEMISTRY 
 
 only slightly less than the value at 18, 0*678. Hence, as the 
 considerable increase of conductivity with temperature cannot 
 be due to an increase in the number of ions, it must be due 
 to an increase in their speed. This increased velocity is doubt- 
 less connected with the diminution in the internal friction of 
 the medium with rise of temperature, and the consequent 
 diminished resistance to the passage of the ions (p. 253). 
 
 Basicity of Acids from Conductivity Measurements 
 (Empirical) The conductivity of N/32 and N/I024 solutions 
 (equivalent normal) of the sodium salt of the acid is deter- 
 mined. For monobasic acids the difference A 102 4 - A 32 is 
 about 10, for dibasic acids about 20, and so on (Ostwald). 
 
 Grotthus' Hypothesis of Electrolytic Conductivity- 
 Long before the establishment of the electrolytic dissociation 
 theory, Grotthus put forward a hypothesis to account for the 
 conductivity of electrolytes which is of considerable historical 
 interest. He assumed that under the influence of the charged 
 electrodes the molecules of the salt, e.g., potassium chloride, 
 arrange themselves in lines between the electrodes so that the 
 potassium atoms are all turned to the negative electrode, and 
 the chlorine atoms to the positive electrode. Electrolysis 
 takes place in such a way that the external potassium atom is 
 liberated at the cathode and the chlorine atom at the anode. 
 The potassium atom which is left free at the anode unites with 
 the chlorine atom of the molecule next to it, the chlorine atom 
 of the latter with a potassium atom of the molecule next in the 
 chain, and so on. A similar process takes place starting at the 
 anode, in other words, an exchange of partners takes place 
 right along the chain, from one electrode to the other. Under 
 the influence of the charged electrodes, the new molecules twist 
 round till they are in the former relative position, when the 
 end atoms are again discharged, and so electrolysis proceeds. 
 
 The fatal objection to this ingenious theory is that a con- 
 siderable E.M.F. would have to be employed before any 
 decomposition whatever takes place, hence Ohm's law would 
 not hold (cf. p. 241). 
 
 Practical Illustrations The following experiments, which 
 
ELECTRICAL CONDUCTIVITY 265 
 
 are fully described in the course of the chapter, may readily be 
 performed by the student : 
 
 (1) Experiment on the migration velocity of the ions (p. 246). 
 
 (2) Rough determination of the absolute velocity of the 
 MnO 4 ' ion (p. 254). 
 
 (3) Determination of the constant of conductivity vessel with 
 n /5 potassium chloride. (The specific conductivity, K, of this 
 solution at different temperatures is as follows : 0-001522 at o, 
 001996 at 10, 0-002399 at 18, and 0-002768 at 25.) 
 
 (4) Determination of the specific and molecular conductivities 
 of solutions of sodium chloride and of succinic acid. 
 
 As the conductivity of solutions varies greatly with the tem- 
 perature, the conductivity vessel must be partially immersed in 
 a thermostat while measurements are being made. 
 
 In the case of sodium chloride, measurements may be made 
 with n/i t n/io and n/ioo solutions, and the values obtained 
 for the molecular conductivity compared with those given in 
 Kohlrausch's tables. 1 The results in very dilute solutions are 
 not trustworthy unless great attention is paid to the purification 
 of the water used in making up the solutions. * 
 
 In the case of succinic acid, it is usual to start with a 1/16 
 Mwlar solution; 20 c.c. of this solution is placed in the con- 
 ductivity cell in the thermostat, and when the temperature is 
 constant the resistance is determined. 10 c.c. of the solution 
 is then removed with a pipette, 10 c.c. of water at the same 
 temperature added, the resistance again determined after 
 thoroughly mixing the solution, and so*on. Measurements are 
 thus made in dilutions of 16, 32, 64, 128, 256, 512 and 1024 
 litres. From the values of /* thus obtained, the degree of 
 dissociation can be calculated by the usual formula a = pJn^ . 
 /XOQ in this case can only be determined indirectly ; its value at 
 25 is about 381. From the values of a in different dilutions, 
 K, the dissociation constant of the acid may then be calculated ; 
 according to Ostwald, K = 0*000066 at 25. 
 
 1 Full details of electrical conductivity measurements and a large amount 
 of conductivity data are given by Kohlrausch and Holborn, Leilvermogen 
 der Elektrolyte, Leipzig, 1898. 
 
CHAPTER XI 
 
 EQUILIBRIUM IN ELECTROLYTES. STRENGTH OF 
 ACIDS AND BASES. HYDROLYSIS 
 
 The Dilution Law In a previous chapter it has been 
 shown that chemical equilibria, both in gaseous and liquid 
 systems, can be represented satisfactorily by means of the law 
 of mass action. We have now to apply this law to binary 
 electrolytes, on the assumption that the ions are to be re- 
 garded as independent entities. 
 
 According to the electrolytic dissociation theory, an aqueous 
 solution ot acetic acid contains molecules of non-ionised acid 
 in equilibrium with its ions, represented by the equation 
 
 CH 3 COO' + H-^CH 3 COOH. 
 
 Suppose in the volume v of solution the total concentration of 
 the acid is i, and that a fraction of it, represented by a, is 
 split up into ions. The concentration of the undissociated 
 
 acid is , that of each of the ions (since they are neces- 
 sarily present in equivalent amount) -. Hence, from the law 
 of mass action, 
 
 2 /i - a\ a 2 , \ 
 
 = K( - ) or 7 r- = K . . (i) 
 \ V ) (i - a)v 
 
 where K, as before, is the equilibrium constant. For conduc- 
 tivity measurements, the above formula may be put in a rather 
 different form by substituting pvlpvz for a. It then becomes 
 
 266 
 
EQUILIBRIUM IN ELECTROLYTES 267 
 
 It is preferable, however, to remember the dilution formula in 
 the first form, or in the form a?cl(i - a) = K. 
 
 This relationship, which is known as Ostwald's dilution law, 
 may be tested by substituting a value for a (from conductivity 
 or osmotic observations) at any dilution v, and calculating K, 
 the equilibrium constant ; the value of a at any other- dilution 
 may then be obtained from the formula and compared with 
 that determined directly. This was done (from conductivity 
 measurements) by van't Hoff and Reicher, and the results are 
 given in the accompanying table : 
 
 Acetic acid: K = 0-0000178 at 14*1; /A^, = 316. 
 
 v (in litres) , . 0-994 2-02 15*9 i8'i 1500 3010 7480 15000 
 
 A*. 1*27 i'94 5'26 5*63 46-6 64-8 95-1 129 
 
 rooa (observed) . 0*40 0-614 I66 1 '7 8 *4*7 20-5 30-1 40-8 
 
 looo (calculated) . 0-42 0-6 1-67 178 15-0 20-2 30-5 40-1 
 
 The agreement between observed and calculated values is 
 excellent ; it is, in fact, much closer than for any case of 
 ordinary dissociation so far investigated. The table also 
 shows how small is the dissociation of acetic acid solutions 
 under ordinary conditions ; a molar solution is ionised only to 
 the extent of 0*4 per cent, and even a 1/1500 molar solution 
 rather less than 15 per cent. 
 
 The dilution law holds for nearly all organic acids and bases, 
 but does not hold for salts, or for certain mineral acids and 
 bases. The latter point is discussed in a later section. 
 
 When the deg ree of dissociation is small, as in the case of 
 acetic acid for fairly concentrated solution, a can be neglected 
 in comparison with i, and the dilution law then becomes 
 
 2 
 
 = K or a = xKz . . . ( 2 ) 
 
 that is, for weak electrolytes the degree of dissociation is approxi- 
 mately proportional to the square root of the dilution. When 
 
268 OUTLINES OF PHYSICAL CHEMISTRY 
 
 a cannot be neglected in comparison with i, a is given by the 
 equation 
 
 (3) 
 
 obtained by solving equation (i) for a. In order to familiarise 
 himself with the use of the dilution formula, the student should 
 calculate a for acetic acid in different dilutions from the value 
 of K given above both by the approximate and accurate 
 formula. 
 
 The physical meaning of the constant K will be clear if 
 a in the dilution formula (i) is put = -J. Then 2K = i/z>, 
 that is, 2K is the reciprocal of the volume at which the 
 electrolyte is dissociated to the extent of 50 per cent. Acetic 
 acid, for instance, will be half dissociated at a dilution of 
 
 - o* = 2 7>777 ntres (ff- table). 
 2 x 0-000018 
 
 As the method of deriving it indicates, the dilution law 
 applies only to binary electrolytes, i.e., electrolytes which split 
 up into two ions only, and it is not therefore a priori probable 
 that it will hold for dibasic acids, such as succinic acid, which 
 presumably dissociate according to the equation 
 
 C 2 H 4 (COOH) 2 ^C 2 H 4 (COO) 2 " + 2 H-. 
 
 It is, however, an experimental fact that when the concentration 
 of succinic acid is expressed in mols (not in equivalents) per 
 litre, the values of K obtained by substitution in the dilution 
 formula remain constant through a wide range of dilution. 
 This indicates that the acid at first splits up into two ions only, 
 doubtless according to the equation 
 
 /COO' . 
 
 4 H 
 
 and that the second possible stage, represented by the equation 
 
 /COO' - CH / COO "+H- 
 <-^ 2 4 \COO 1 
 
EQUILIBRIUM IN ELECTROLYTES 269 
 
 is hot appreciable under the conditions of the experiment. In 
 other cases, however, e.g., fumaric acid, the value of K increases 
 with dilution before the dilution has progressed very far, which 
 indicates that the second stage of the dissociation early becomes 
 of importance. 
 
 Strength of Acids We are accustomed to estimate the 
 strength of acids in a roughly qualitative way by their relative 
 displacing power. Sulphuric acid, for example, is usually re- 
 garded as a strong acid, because it can displace such acids as 
 acetic and hydrocyanic from combination. This principle can 
 be developed to a quantitative method for estimating the rela- 
 tive strengths of acids (and bases) if care is taken to make the 
 comparison under proper conditions. This is sufficiently secured 
 by making the experiments in a homogeneous system under such 
 conditions that all the reacting substances and products of reaction 
 remain in the system. We learn in studying inorganic chemistry 
 that many reactions proceed wholly or partially in a particular 
 direction for two main reasons : (a) because an insoluble (or 
 practically insoluble) product is formed which is thus removed 
 from the reacting system, 
 
 e.g., Na 2 SO 4 + BaCl 2 -> 2NaCl + BaSO 4 (insoluble) ; 
 (b] because a volatile product is formed which under the con- 
 ditions of experiment leaves the reacting system, 
 
 e.g., 2NaCl + H 2 SO 4 - Na 2 SO 4 + 2 HC1 (volatile). 
 
 Such reactions are obviously unsuitable for determining the 
 relative strengths of the acids concerned. 
 
 Bearing these considerations in mind, we now proceed to 
 investigate the relative strengths of, say, nitric and dichloracetic 
 acids by bringing them in contact with an amount of base insuffi- 
 cient to saturate both of them, and find how the base distri- 
 butes itself between the two acids. If, for example, the acids 
 are taken in equivalent amount, and sufficient base is taken to 
 saturate one of them, we have to determine the position of 
 equilibrium represented by the equation 
 
270 OUTLINES OF PHYSICAL CHEMISTRY 
 
 CHCl a COOK + HN0 3 ^CHC1 2 COOH + KNO 3 . 
 
 It is evident that no chemical method would answer the purpose, 
 because it would disturb the equilibrium. When, however, a 
 physical property of one of the components, which alters with 
 the concentration, can be measured, the position of equilibrium 
 can be determined. A method used for this purpose by Ostwald, 
 depending on the changes of volume on neutralization, will be 
 readily understood from an example. 
 
 When a mol of potassium hydroxide is neutralized by nitric 
 acid in dilute solution, there is an increase of volume of about 
 20 c.c. When, on the other hand, the same quantity of alkali 
 is neutralized by dichloracetic acid, the increase of volume is 
 about 13 c.c. It is therefore clear that the complete displace- 
 ment of dichloracetic acid by nitric acid, according to the 
 equation 
 
 CHC1 2 COOK + HNO 3 -CHC1 2 COOH + KNO 3 / 
 would give an increase of volume of (20 - 13) = 7 c.c. ; if no 
 displacement took place, there would, of course, be no change 
 of volume. The change actually observed was 5*67 c.c., which 
 means that the reaction represented by the equation has gone 
 
 from left to right to the extent of - - = 80 per cent, approxi- 
 
 7-0 
 
 mately; in other words, the nitric acid has taken 80 per cent, of 
 the base, and 20 per cent, has remained combined with the 
 dichloracetic acid. The relative strength, or relative activity, 
 of the acids under these conditions is therefore 80 : 20 or 4 : i. 
 
 Any other physical property, which is capable of quantitative 
 measurement and differs from the two systems, can be equally 
 well employed for the determination of equilibrium. The heat 
 of neutralization has been used for this purpose by Thomsen, 
 and the measurement of the refractive index by Ostwald ; the 
 principle of the methods is exactly the same as in the example 
 just given. 
 
 Thomsen's therm ochemical measurements were the first to 
 
EQUILIBRIUM IN ELECTROLYTES 271 
 
 be made on this subject, and he arranged the different acids 
 in the order of their " avidities " or activities. Ostwald then 
 showed that the same order of the avidities was obtained by 
 the volume and refractivity methods, and, further, that the results 
 were independent of the nature of the base competed for, so 
 that the avidities are specific properties of the acids. 
 
 The relative strength of acids can also be determined on an 
 entirely different principle, depending on kinetic measurements. 
 It has already been pointed out that acids accelerate, in a 
 catalytic manner, the hydrolysis of cane sugar, of methyl acetate, 
 acetamide, etc. Ostwald made many experiments on this sub- 
 ject and reached the very important conclusion that the order 
 of the activity of acids is the same, whether measured by the 
 distribution method (which is, of course, a static method), or 
 by a kinetic method. This affords further evidence in favour 
 of the conclusion just mentioned, that the activity or affinity is 
 a specific property of the particular acid, independent of the method 
 by which it is measured. 
 
 Thus far had our knowledge of the subject progressed when 
 in 1884 the first paper of Arrhenius appeared. He showed 
 that the order of the "strengths" of the acids as determined 
 by the methods just described is also that of their electrical 
 conductivities in equivalent solution. This fundamentally im- 
 portant fact is illustrated in the accompanying table, in which 
 the conductivities of the acids in normal solution are quoted, 
 that of hydrochloric acid being taken as unity. 
 
 Relative Activity. 
 
 Acid. Thermochemical. Cane Sugar. Conductivity. 
 
 Hydrochloric 100 100 100 
 
 Nitric 100 100 99-6 
 
 Sulphuric 49 53-6 65-1 
 
 Monochloracetic 9 4-8 4-9 
 
 Acetic 0-4 1-4 
 
 We have seen that, according to the electric dissociation 
 theory, the electrical conductivity of an acid is mainly deter- 
 
272 OUTLINES OF PHYSICAL CHEMISTRY 
 
 mined by its degree of dissociation; for example, the con- 
 ductivity of a normal solution of acetic acid is small because 
 it is ionised only to a very small extent. Further, owing to the 
 predominant share taken by the hydrogen ions in conveying 
 the current (p. 252), the relative conductivities of acids will be 
 approximately proportional to their H' ion concentrations. It 
 is therefore natural to suppose that the activity of acids, as 
 illustrated by distribution and catalytic effects, is also due to 
 that which all acids have in common, namely, hydrogen ions, 
 This assumption is in complete accord with the experimental 
 results, as the following illustration shows. The velocity con- 
 stant for the hydrolysis of cane sugar in the presence of 1/80 
 normal hydrochloric acid is 0-00469 at 54-3 (time in minutes) ; 
 as the acid may be regarded as completely dissociated CH- = 
 0-0125. CH- for 1/4 normal acetic acid (# = 4, cf. p. 267) may 
 be calculated from the dilution formula or directly from the 
 equilibrium equation as follows : 
 
 [H-][CH 3 COO'] [H-] 2 [H-] 2 
 
 [CH 3 COOH] [CH 3 COOH] [0-25 - H-]" 
 
 whence CH- = 0-002. 
 
 On the assumption that the catalytic effect of acids is propor- 
 tional to the H* ion concentration, the value of the velocity 
 constant, x, for the hydrolysis of cane sugar by 0-25 normal 
 acetic acid at 54*3 should be 
 
 0-0125 : 0*00469 : : 0-002 : x, 
 
 whence x = 0-00075. This is identical with the result obtained 
 experimentally by Arrhenius, and we have here a very striking 
 confirmation of the electrolytic dissociation theory. 
 
 From the above considerations, we conclude that the charac- 
 teristic properties which acids have in common, such as sour 
 taste, action on litmus, catalytic activity, property of neutralizing 
 bases, etc., are due to the presence of H- ions. It must be 
 remembered that direct proportionality between H- ion .concen- 
 tration and conductivity is neither observed nor to be expected 
 
EQUILIBRIUM IN ELECTROLYTES 273 
 
 from the theory ; the approximate proportionality is due to the 
 great velocity of the hydrogen ions, and would be altogether 
 absent if the anions were the more rapid. 
 
 As there is a simple relationship between the H> ion con- 
 centration of weak acids and their dissociation constants (p. 267), 
 it is clear that the behaviour of an acid can be to a great extent 
 foretold when its dissociation constant has been measured. Such 
 determinations have been made for a great number of weak 
 acids by Ostwald and others, and some of the results are given 
 in the accompanying table,. which shows very clearly how greatly 
 the value of K differs for different acids, and the influence of 
 substitution : 
 
 Acid. 
 
 Acetic H-CH 3 COO 
 Monochloracetic H CH 2 C1COO 
 Trichloracetic H C1 3 COO 
 Cyanacetic H CH 2 CNCOO 
 Formic H HCOO 
 Carbonic H HCO 3 
 Hydrogen sulphide H SH 
 Hydro yanic H CN 
 Phenol H OC 8 H 5 
 
 Value of K at 25 (v in litres). 
 0-000018 = 180,000 x io~ 10 
 
 3040 x io- 10 
 
 570 x io- 10 
 
 13 x io~ 10 
 
 i '3 x io~ 16 
 
 1*21 
 
 0-0037 
 0*000214 
 
 For a weak acid, a= >jKv, where a is the degree of dis- 
 sociation at the volume v. Hence, for two acids at the same 
 dilution a/a' = \/K/K', or the ratio of the degrees of dissocia- 
 tion is equal to the square root of the ratio of the dissociation 
 constants. From the data given in the table it can readily be 
 calculated that in solutions of monoch loracetic and acetic acid 
 of the same concentration the ratio of the H- ion concentrations 
 is approximately 9-3 : i. The effect of replacing one of the 
 hydrogens in acetic acid by chlorine is thus to form a much 
 stronger acid and the CN group has a still greater effect, as the 
 table shows. 
 
 A further important point is the effect of dilution on the 
 "strength" of an acid. As the degree of dissociation in- 
 creases regularly with dilution, it is evident that the activity of 
 a weak acid will approach nearer and nearer to that of a strong 
 18 
 
274 OUTLINES OF PHYSICAL CHEMISTRY 
 
 acid (which is completely active in moderate dilution) until 
 finally, when the weak acid is completely ionised, it will have 
 the same strength as the strong acid in equivalent dilution. It 
 follows that the strength of acids is the more nearly equal the 
 more dilute the solution, and that at " infinite dilution " all 
 acids are equally strong. It can readily be calculated from the 
 conductivity tables that the relative strengths of hydrochloric 
 and acetic acids in different dilutions are as follows : 
 
 Concentration n/i n/io H/IOO w/iooo n/io,ooo 
 
 a for HC1 0*81 0-91 0-97 0-99 i-o 
 
 a for HC 2 H 3 O 2 0-004 0-013 0-04 0*13 0-4 
 
 Ratio HC1/HC 2 H 3 O 2 200 70 24 7-5 2-5 
 
 Strength of Bases Just as the strength of acids depends 
 on their concentration in hydrogen ions, so the strength of 
 bases depends on the concentration in hydroxyl ions. On this 
 view, potassium hydroxide is a strong base, because in moderate 
 dilution it is almost completely ionised according to the equa- 
 tion KOH 1^ K* + OH' ; ammonium hydroxide, on the other 
 hand, is a weak base, because its aqueous solution contains only 
 a relatively small concentration of OH' ions. Since certain 
 organic compounds, including amines and alkaloids, have basic 
 properties, their aqueous solutions must also contain OH' ions. 
 Thus, solutions of pyridine, C 5 H 5 N, contain not only the free 
 base, but a certain concentration of C 5 H 6 N' and OH' ions, in 
 equilibrium with the undissociated hydrate, as represented by 
 the equation 
 
 C 5 H 6 NOH;C 5 H 6 N- + OH'. 
 
 The strength of bases may be determined by distribution or 
 catalytic methods, corresponding with those already described 
 for determining the strength of acids, as well as by conductivity 
 methods. A fairly satisfactory catalytic method is the effect 
 on the rate of condensation of acetone to diacetonyl alcohol, 1 
 represented by the equation 
 
 2CH 3 COCH 3 -> CH 3 COCH 2 C(CH 3 ) 2 OH. 
 
 1 Koelichen, Zeitsch. Physikal.Chem., 1900, 33, 129. 
 
EQUILIBRIUM IN ELECTROLYTES 275 
 
 The order of the strength of bases as determined by this method 
 agrees with the results of conductivity measurements. Another 
 method, which is not purely catalytic, since the base is used up 
 in the process, is the effect on the hydrolysis or saponification 
 of esters (p. 207). This process is usually represented by the 
 typical equation 
 
 CH 3 COOC 2 H 5 + KOH = CH 3 COOK + C 2 H 5 OH. 
 Experience shows, however, that for the so-called strong bases, 
 which are almost completely ionized in moderate dilution, the 
 rate of hydrolysis is practically independent of the nature of 
 the cation (whether K, Na, Li, etc.), a fact which is readily 
 accounted for on the view that the ions exist free in solution, 
 as the ionic theory postulates, and that the OH' ions are alone 
 active in saponification. The general equation for the hy- 
 drolysis of ethyl acetate by bases may, therefore, be written as 
 follows 
 
 CH 3 COOC 2 H 5 -i- OH' = CH 3 COO' + C 2 H 5 OH. 
 The relative strength of bases, as obtained from their efficiency 
 in saponifying esters, is in excellent agreement with their 
 strength as deduced from conductivity measurements. The 
 ionization view of the saponification of esters is further sup- 
 ported by the fact that the reaction between ethyl acetate and 
 barium hydroxide is bimolecular and not trimolecular, as would 
 be anticipated if it proceeded according to the equation 
 
 2CH 3 COOC 2 H 5 + Ba(OH) 2 = (CH 3 COO) 2 Ba + 2 C 2 H 5 OH. 
 
 The alkali and alkaline earth hydroxides are very strong 
 bases, being ionized to about the same extent as hydrochloric 
 acid in equivalent dilution. Bases differ as greatly in strength 
 as do acids; the dissociation constants for a few of the more 
 important are given in the table. 
 
 Base. 
 
 Ammonia NH 4 'OH 
 Methylamine CH 3 NH 3 'OH 
 Trimethylamme (CH 3 ) 3 NH-OH 
 Pyridine C 5 H 6 N-OH 
 Aniline C 6 H 5 NH 3 -OH 
 
 Value of K (25) (v in litres) 
 
 0*000023 = 230,000 x 10 
 
 0*00050 
 
 0*000074 
 
 23 x io~ 10 
 4*6 x io~ 10 
 
 -10 
 
276 OUTLINES OF PHYSICAL CHEMISTRY 
 
 Interesting results have been obtained as to the effect of sub- 
 stitution on the strength of bases. Thus the table shows that 
 the basic character is increased by replacing one of the hydrogen 
 atoms in ammonia by the CH 3 group, but is greatly diminished 
 by the C 6 H 5 group. 
 
 Mixture of two Electrolytes with a Common Ion The 
 dissociation of weak acids and bases is greatly diminished by 
 the addition of a salt with an ion common to the acid or base. 
 For example, the equilibrium in a solution of acetic acid is re* 
 presented by. the equation [H-] [CH 3 COO'] = K [CH 3 COOH], 
 and if by adding sodium acetate the CH 3 COO' ion con- 
 centration is greatly increased, the H* ion concentration must 
 correspondingly diminish, since the concentration of the undis- 
 sociated acid cannot be greatly altered (nearly the whole of the 
 acid being present in that form in the original solution) and 
 therefore the right-hand side of the equation is practically 
 constant. 
 
 The exact equations representing the mutual influence of 
 electrolytes with a common ion are somewhat complicated, but 
 an approximate formula which is often useful can be obtained 
 as follows : If the total concentration of a binary electrolyte is 
 c and its degree of dissociation is a, we have, from the law of 
 mass action, 
 
 ^W K. ':. (i) 
 
 (l -a)c 
 
 If now an electrolyte with a common ion is added, and the 
 concentration of the latter is c ot the above equation becomes 
 
 (a!c}(a'c + c ) _ K ,. 
 
 (l - a> 
 
 This equation is quite accurate. As K, c and c are known, a', 
 where a is the new degree of dissociation, can be calculated, 
 perhaps most readily by successive approximations. 
 
 If the degree of dissociation of the first electrolyte is small, 
 i - of can be taken as unity without appreciable error ; further, 
 
EQUILIBRIUM IN ELECTROLYTES 277 
 
 if the second electrolyte is highly ionised and is added in con- 
 siderable proportion, aV can be neglected in comparison with 
 c ot an d equation (2) simplifies to 
 
 (aV) C = K, . . . . (3) 
 
 Otherwise expressed, the concentration of one of the ions of a 
 weak electrolyte is inversely proportional to the ionic concentra- 
 tion of a highly-dissociated salt having an ion in common with 
 the other ion of the weak electrolyte. 
 
 As an illustration of the application of the last equation, we 
 will consider the effect of the addition of an equivalent amount 
 of sodium acetate on the strength of 0*25 molar acetic acid. In 
 this dilution a for the acid is 0-0085 and ac = 0*0085 x 0*25 = 
 0-0021 = C H \ In 0*25. molar solution, sodium acetate is 
 dissociated to the extent of 69-2 per cent., hence c = 
 0*25 x 0*692 = o'i73. We have, therefore, 
 
 a' x 0*173 0*000018, whence a' = 0*0001. 
 
 aV is therefore 0*25 x 0*0001 =0*000025= C n . in presence of 
 0*25 molar sodium acetate, so that the strength of the acid is 
 diminished in the ratio 85 : i. In an exactly similar way, the 
 strength of ammonia as a base is greatly reduced by the addi- 
 tion of ammonium salts. 
 
 This action of neutral salts on weak bases and acids is largely 
 taken advantage of in analytical chemistry. For example, the 
 concentration of OH' ions in ammonia solution is sufficient to 
 precipitate magnesium hydroxide from solutions of magnesium 
 salts, but in the presence of ammonium chloride the OH' ion 
 concentration is so greatly reduced that precipitation no longer 
 occurs. Similarly, the addition of hydrochloric acid diminishes 
 the concentration of S" ions in hydrogen sulphide to such an 
 extent that zinc salts are no longer precipitated (cf. p. 300). 
 
 Isohydric Solutions It is of particular interest to inquire 
 what must be the relation between two solutions with a common 
 ion two ^acids, for example in order that, when mixed, they 
 may exert no mutual influence. This problem was investigated 
 
278 OUTLINES OF PHYSICAL CHEMISTRY 
 
 both theoretically and practically by Arrhenius, who showed 
 (cf. p. 414) that no alteration in the degree of dissociation of 
 either of the salts (acids or bases) takes place when the con- 
 centration of the common ion in the two solutions before 
 mixing is the same. Such solutions are termed isohydric. 
 
 The relative dilutions in which two acids or other electrolytes 
 with a common ion are isohydric can readily be calculated from 
 their dissociation constants. The value of K for acetic acid is 
 o-ooooj.8, and for cyanacetic acid 0-0037, both at 25. Since 
 a = <s/K approximately, it is clear that the degree of dissocia- 
 tion, a, will be the same for the two acids when the dilutions 
 are inversely as the dissociation constants. The dilution of the 
 cyanacetic acid must therefore be 3700 : 1 8, or 205 times that 
 of acetic acid for isohydric solutions. 
 
 Arrhenius has shown that by the principle of isohydric solu- 
 tions the mutual influence of electrolytes with a common ion 
 even of strong acids and their neutral salts, can be calculated 
 with a considerable degree of accuracy, but the methods are 
 somewhat complicated and cannot be given here. Not only is 
 the degree of dissociation of a weak acid greatly influenced by 
 the addition of a strong acid, or other electrolyte with a common 
 ion, but the latter is affected, though to a much smaller extent, 
 by the presence of the former. It can, for instance, be calcu- 
 lated that when a mol of acetic acid and a mol of cyanacetic 
 acid are present in a litre of water, the dissociation of the 
 former is only about 1/14 of its value in aqueous solution, 
 whilst the presence of the acetic acid only diminishes the 
 dissociation of the cyanacetic acid by 0^25 per cent. 
 
 Mixture of Electrolytes with no Common Ion The 
 equilibrium in a mixture of two electrolytes without a common 
 ion can be calculated when the concentrations and dissociation 
 constants are known, but the calculation is somewhat compli- 
 cated. If solutions of two highly-dissociated salts, such as 
 potassium chloride and sodium bromide, are mixed, small 
 amounts of undissociated sodium chloride and potassium 
 
EQUILIBRIUM IN ELECTROLYTES 279 
 
 bromide will be formed ; but as the salts are all highly dis- 
 sociated, the mutual effect is very small. If dissociation is 
 complete, the process will be represented by the equation 
 
 K- + Cl' + Na- + Br' = K- + Cl' + Na- + Br', 
 otherwise expressed, the salts will exert no mutual influence. 
 These considerations account for the observation of Hess 
 (p. 148) that the thermal effect of mixing dilute solutions of 
 two binary salts is very slight. At the time Hess made his 
 observation, it was extremely puzzling, because it was known 
 that the heats of formation of different salts were very different, 
 and therefore heat should either be absorbed or given out as an 
 accompaniment of the double decomposition. On the basis of 
 the electrolytic dissociation theory, however, Hess's results at 
 once become intelligible, since both before and after admixture 
 the solution contains mainly the same free ions. 
 
 To the question which is very often asked as to what com- 
 pounds are present in a mixed salt solution, it must therefore 
 be answered that all the undissociated salts and ions are present 
 which can be formed by interaction of the components, and 
 that the proportions in which the various molecules and ions 
 are present depend on the concentrations and dissociation 
 constants of the various salts. 
 
 Dissociation of Strong Electrolytes l It has been pointed 
 out (p. 267) that the law of mass action holds for weak (i.e., 
 slightly ionised) electrolytes, the ions being regarded as inde- 
 pendent units. The proof of the applicability of the law has 
 been brought more particularly by Ostwald from the results 
 of measurements with organic acids. It is a remarkable fact, 
 however, that the dilution formula, a 2 /(i - a)^ = K, which is 
 a direct consequence of the law of mass action, does not appear 
 to be valid for the so-called " strong " or highly dissociated 
 electrolytes ; when the values of a, obtained from osmotic or 
 conductivity measurements, are substituted in the above formula, 
 
 l Cf. Drucker, "Die Anomalie der starken Elektrolyte," Ahrens' 
 Sammlung, Stuttgart, 1905. 
 
2 8o OUTLINES OF PHYSICAL CHEMISTRY 
 
 K diminishes greatly with dilution. This is well shown in the 
 following table for silver nitrate. The first column contains 
 v, the volume in litres in which i mol of the salt is dissolved, 
 in the second column is given the value of a calculated from con- 
 ductivity measurements at 25, and in the third column are given 
 the values of K calculated by means of the dilution formula. 
 
 V. o = /*r//ioc- K - K i- 
 
 16 0-8283 ' 2 53 'ii 
 
 32 0*8748 0-191 -16 
 
 64 0-8993 0-127 -06 
 
 128 0-9262 0-122 -07 
 
 256 0-9467 0-124 -08 
 
 512 0-9619 0*125 *9 
 
 The deviations from the simple law appear to be fairly 
 
 regular in character, and van't Hoff has proposed an empirical 
 
 formula which represents with a fair degree of accuracy the 
 
 behaviour of the great majority of strong electrolytes. The 
 
 formula in question is of the form 
 
 a*/(i - a.yv = Kj 
 which may also be written 
 
 CX = K, 
 
 where Ca represents the concentration of the dissociated part 
 (the ions), C that of the undissociated part. The application 
 of this formula to solutions of silver nitrate is illustrated in 
 the table, and it will be observed that the values of K x in 
 the fourth column are fairly constant. The slightly different 
 formula suggested by Rudolphi 
 
 is scarcely as satisfactory as that of van't Hoff. Some observers 
 have suggested a generalised form of van't HofFs formula, 1 as 
 follows : 
 
 C^/C = K 3 
 
 "^C/ Bancroft, Zeiisch, Physikal Chem., 1899, 31, 188. 
 
EQUILIBRIUM IN ELECTROLYTES 281 
 
 and experiment shows that in many cases n does not differ 
 much from 1-5. When n is 1-5 the general formula reduces to 
 that of van't Hoff. 
 
 The reason why the law of mass action does not apply to 
 strong electrolytes, although it holds so accurately for weak 
 electrolytes, has not been satisfactorily elucidated. It was long 
 thought that the values of a obtained from the results of con- 
 ductivity measurements were not the true values of the degree 
 of dissociation, but recent very careful comparison shows that 
 the results obtained by conductivity and freezing-point deter- 
 minations in dilute solution for the best-investigated substances 
 do not, as a rule, differ by more than 2 per cent, on the 
 average, and even these differences may be due largely to 
 experimental error. 1 In a few cases only does there appear to 
 be a real difference between the results obtained by the two 
 methods. As neither of the values for a gives a constant value 
 for K when substituted in the ordinary dilution formula, there 
 can be no doubt that strong electrolytes do behave in an 
 anomalous way. 
 
 Several disturbing causes might be suggested to account for 
 this behaviour, including (a) the formation of complex ions by 
 combination of the ions with non-ionised molecules; (b) 
 mutual influence of the ions ; (c) interaction of the ions and 
 the solvent, including more particularly hydration of the ions, 
 and one or more of these effects may be operative in any 
 one solution. The existence of complex ions in solutions has 
 been definitely proved by Hittorf and others. In solutions of 
 cadmium iodide, for instance, there is evidence that I' ions unite 
 with CdI 2 molecules to form complex ions of the formula CdI 4 ". 
 The formation of complex ions would diminish the number of 
 non-ionised molecules but not the total number of ions ; it would 
 thus affect the osmotic pressure, but not to any extent the con- 
 ductivity. This disturbing effect will be greatest in fairly concen- 
 trated solutions. In dilute solutions of salts of the alkalis and 
 1 Drucker, loc. cit. ; Noyes, Technology Quarterly, 1904, 17, 293. 
 
282 OUTLINES OF PHYSICAL CHEMISTRY 
 
 alkaline earths, the values of a obtained by conductivity and os- 
 motic pressure methods are very nearly equal, and this result 
 appears to show that there is little or no complex ion formation 
 in these solutions. 
 
 As to the possible effect of the ions on each other, practically 
 nothing can be said with certainty. The fact that weak electro- 
 lytes follow the law of mass action through a very wide range 
 of dilution may be connected with the fact that under all 
 circumstances the ionic concentration is small, a condition 
 which no longer holds in solutions of strong electrolytes. As 
 to how and to what extent the mutual influence is exerted, 
 little or nothing is known. 
 
 The possibility of the interaction of the ions with the solvent 
 has been much discussed, but as no general agreement has been 
 reached on the matter, a short reference to the subject here will 
 suffice. One way in which this effect might influence the results 
 would be if the ions became associated with a large proportion 
 of water which no longer acted as solvent. The effective con- 
 centrations of an ion would then be the ratio of the amount 
 present to that of the " free " solvent, instead of to the total 
 solvent, as usually calculated. As, however, no satisfactory 
 method of estimating the relative proportions of free and com- 
 bined solvent in the solution of an electrolyte has yet been 
 suggested the question is at present mainly of theoretical interest 
 
 (P- 325> 
 
 The most logical method so far suggested for dealing with the 
 
 problem of strong electrolytes is due to Nernst. He considers 
 that, owing to their mutual influence, the activity of the various 
 substances (ions and non-ionised substances) present is not pro- 
 portional to their respective concentrations, but certain correct- 
 ing factors have to be applied depending on the extent of the 
 mutual influence. Among these effects, that of the ions on each 
 other and on the non-ionised part of the molecules, as well as the 
 mutual influence between ions and solvent, seem to be of special 
 importance, but so far comparatively little progress has been 
 
EQUILIBRIUM IN ELECTROLYTES 283 
 
 made with the determination of the relative magnitudes of the 
 correcting factors. 
 
 Electrolytic Dissociation of Water, Heat of Neutral- 
 ization So far we have regarded the usual solvent water 
 simply as a medium for dissociation, but there is evidence to 
 show that it is itself split up to a very small extent into ions, 
 according to the equation 
 
 Applying to this equation the law of mass action, we have, as 
 usual, 
 
 [H-][OH'] = K[H 2 0], 
 
 where K is the dissociation constant for water. As the ionic 
 concentrations are extremely small the concentration of the 
 water is practically constant, and therefore the product of the 
 concentration of the ions 
 
 [H-] [OH'] = K w , a constant. 
 
 It has been found by different methods, which will be re- 
 ferred to later (p. 293), that the value of the above constant at 25 
 is about 1*2 x io 14 . In pure water the concentrations of the 
 ions are necessarily equal, hence CH = COH' = N/i'2 x io~ 14 = 
 i -i x 10 ~ 7 at 25. Otherwise expressed, this means that pure 
 water contains rather more than i mol of H* and OH' ions, that 
 is, i gram of H' ions and 17 grams of OH' ions, in io 7 or 
 10,000,000 litres. The ionic product is independent of whether 
 the solution is acid or alkaline, and therefore in a normal solu- 
 tion of a (completely dissociated) acid, since CH- = i, COH' is 
 only io ~ 14 , and in a solution of a normal alkali Cu- is corre- 
 spondingly small. 
 
 These considerations are of great importance in connection 
 with the process of neutralization. Assuming that the solutes 
 are completely ionised, the neutralization of i mol of sodium 
 hydroxide by hydrochloric acid in dilute solution may be repre- 
 sented as follows : 
 
 Na + OH' + H- + Cl' = Na- + Cl' + H 2 O. 
 
284 OUTLINES OF PHYSICAL CHEMISTRY 
 
 Since Na* and Cl' ions occur in equivalent amount on each 
 side, they may be neglected, and the equation reduces to 
 
 or, otherwise expressed, the combination of hydrogen and 
 hydroxyl ions to form water. The same equation applies to 
 the neutralization of any other strong base by a strong acid ; 
 provided that the solutions are so dilute that dissociation is prac- 
 tically complete, the process in all cases consists in the combination 
 of H' and OH' ions to ?ion-ionised water. It may, therefore, be 
 anticipated that for equivalent amounts of different strong bases 
 and acids the heat of neutralization will be the same, and that 
 this is actually the case is shown in the first part of the table. 
 The magnitudes of the heats of neutralization apply for molar 
 quantities. 
 
 Heats of Neutralization. 
 Acid and Base. Heat of Neutralization. 
 
 HC1 and NaOH 13,700 cal. 
 
 HBr and NaOH 13,700 cal. 
 
 HNO 3 and NaOH 13,700 cal. 
 
 HC1 and JBa(OH) 2 13,800 cal. 
 
 NaOH and CH 3 COOH 13,400 cal. 
 
 NaOH and HE 16,300 cal. 
 
 HC1 and ammonia 12,200 cal. 
 
 HC1 and dimethylamine n, 800 cal. 
 
 The fact that the heat of neutralization of strong acids and 
 bases is independent of the nature of the acid and base was 
 long a puzzle to chemists, and the simple explanation given 
 above is one of the conspicuous triumphs of the electrolytic 
 dissociation theory. 
 
 Below the dotted line in the above table are given the heats 
 of neutralization of two weak acids by a strong base and of two 
 weak bases by a strong acid. As the table shows, the heat 
 development in these cases may be more or less than 13,700 
 cal. for molar quantities, and a little consideration affords a 
 plausible explanation. The neutralization of acetic acid, which 
 
EQUILIBRIUM IN ELECTROLYTES 285 
 
 is very slightly ionised, by sodium hydroxide, may be repre- 
 sented by the equation 
 
 which may be regarded as taking place in two stages 
 (i) CH 3 COOH = CH 3 COO' + H-; (2) H- + OH' = H 2 O. 
 
 The heat of neutralization is, therefore, the sum of two effects 
 (i) the heat of dissociation of the acid ; (2) the reaction 
 H' + OH' = H 2 O, which gives out 13,700 cal. Hence, since 
 the observed thermal effect is 13,400 cal. the dissociation of 
 the acid must absorb 300 calories. For hydrofluoric acid, on 
 the other hand, the reaction HF= H' + F' is attended by a heat 
 development of 16,300 - 13,700 = 2,600 cal. We have thus, 
 an approximate method of determining the heat of ionisation 
 of electrolytes, which may be positive or negative. 
 
 In the above paragraphs the total heat change has been 
 regarded as the algebraic sum of the heats of neutralization 
 and of ionisation, but it is probable that other phenomena, for 
 example, changes of hydration, also play a part. 
 
 Hydrolysis It is a well-known fact that salts formed by a 
 weak acid and a strong base, such as potassium cyanide, show an 
 alkaline reaction in aqueous solution, whilst salts formed by the 
 combination of a weak base and a strong acid, for example, 
 ferric chloride, have an acid reaction. In the previous section 
 it has been mentioned that water is slightly ionised, according 
 to the equation H 2 O = H- + OH', and may therefore be re- 
 garded as at the same time a weak acid (since H- ions are 
 present) and a weak base (owing to the presence of OH' ions). 
 It will now be shown that the behaviour of aqueous solutions 
 of such salts as potassium cyanide and ferric chloride are quanti- 
 tatively accounted for on the assumption that water is electro- 
 lytically dissociated. 
 
 In a previous section (p. 271) it has been pointed out that 
 when two acids are allowed to compete for the same base, the 
 latter distributes itself between the acids in proportion to their 
 
286 OUTLINES OF PHYSICAL CHEMISTRY 
 
 avidities, and it has also been shown that the ratio of the 
 avidities of two acids is the ratio of the extent to which 
 they are electrolytically dissociated. The same applies to a salt 
 in aqueous solution, water, in virtue of its hydrogen ion concen- 
 tration, being regarded as one of the competing acids. In the 
 case of a salt of a strong acid, such as sodium chloride, it 
 would not be anticipated that such a weak acid as water would 
 take an appreciable amount of the base, and the available 
 experimental evidence quite bears out this expectation. In 
 other words, an aqueous solution of sodium chloride contains 
 only Na* and Cl' ions and undissociated sodium chloride in 
 appreciable amount, and is therefore neutral. 
 
 The case is quite different for a salt formed by a strong base 
 and a weak acid, such as potassium cyanide. Here water as 
 an acid is comparable in strength to hydrocyanic acid, and 
 therefore there is a distribution of the base between the acid 
 and the water according to the equation 
 
 KCN + H 2 O ^ KOH + HCN, 
 
 the proportions of potassium cyanide and potassium hydroxide 
 depending upon the relative strengths of water and hydrocyanic 
 acid. 
 
 From the equation it is evident that potassium hydroxide and 
 hydrocyanic acid must be present in equivalent amount ; and 
 since the hydroxide is much more highly ionised than hydro- 
 cyanic acid, the solution contains an excess of OH' ions, and 
 must therefore be alkaline, as is actually the case. 
 
 This process is termed hydrolysis, i.e., decomposition by 
 means of water. Similar considerations apply to the salts 
 formed by combination of weak bases and strong acids, such 
 as aniline hydrochloride. As water is comparable in strength 
 to aniline as a base, an equilibrium is established according to 
 the equation 
 
 C 6 H 5 NH 3 C1 + HOH^C C H 6 NH 3 OH + HC1. 
 In this case there is an excess of H* ions, as hydrochloric acid 
 
EQUILIBRIUM IN ELECTROLYTES 287 
 
 is much more highly ionised than anilinium hydroxide, and 
 therefore the solution has an acid reaction. 
 
 A salt formed by the combination of a weak acid and a 
 weak base, e.g., aniline acetate, is naturally hydro lysed to a 
 still greater extent. These three types of hydrolytic action 
 will now be considered quantitatively. 
 
 (a) Hydrolysis of the Salt of a Strong Base and a 
 Weak Acid A typical salt of this type is potassium cyanide, 
 the hydrolytic decomposition of which is represented by the 
 equation 
 
 KCN + HOHJKOH + HCN, 
 
 or, according to the electrolytic dissociation theory, 
 K- + CN' + HOH ^ K- + OH' + HCN, 
 
 on the assumption, which is only approximately true, that 
 potassium cyanide is completely ionised and hydrocyanic acid 
 non-ionised. 
 
 The equilibrium can now be investigated, and the extent of 
 the hydrolysis determined, if a means can be found of deter- 
 mining the equilibrium concentration of one of the reacting 
 substances, for example, the OH' ions. This could not, of 
 course, be done by titrating the free alkali, as the equilibrium 
 would thus be disturbed, but one of the methods given on 
 p.ayi may conveniently be used. The method which has been 
 most largely used is to determine the effect of the mixture on 
 the rate of saponification of methyl acetate, which, as has 
 already been pointed out, is proportional to the OH' ion con- 
 centration. The amount of hydrolysis per cent., loox, for 
 different concentrations, c, of potassium cyanide (mols per litre) 
 at 25, determined by the above method, is as follows: 
 
 c . 0*947 0*235 '95 0*024 
 IQOX . 0*31 0*72 i'i2 2*34 
 
 K* . 0'9 I'2 I'2 I^XIO 5 
 
 The table shows that, as is to be expected, the degree of 
 hydrolysis increases with dilution. 
 
288 OUTLINES OF PHYSICAL CHEMISTRY 
 
 A general equation, by means of which the equilibrium con- 
 dition can be calculated when the acid and base are not neces- 
 sarily present in equivalent proportions, can readily be obtained 
 by applying the law of mass action to the general equation 
 
 B* + A' + H 2 ; B* + OH' + HA, 
 
 where B' and A' represent the positive and negative ions respec- 
 tively. As B' occurs on both sides of the above equation, the 
 latter can be simplified to 
 
 A' + H 2 O ^ OH' + HA. 
 
 As the salt and the base are practically completely ionised, and 
 the acid is not appreciably ionised, A' and OH' are proportional 
 to the concentrations of salt and base respectively, and HA to 
 that of the acid. Hence, from the law of mass action, 
 
 [OH'][HA] _ [base][acid] 
 
 A' ~ [unhydrol. salt] " ( ' 
 
 a constant, as the concentration of the water may be regarded 
 as constant. KA is termed the hydrolysis constant, and, like 
 the ordinary equilibrium constant, is independent of the relative 
 concentrations of the substances present at equilibrium, but de- 
 pends on the temperature. 
 
 In order to illustrate the use of the above formula, the values 
 of K;, may be calculated from the data for potassium cyanide 
 already quoted. In 0*095 molar solution, potassium cyanide is 
 hydrolysed to the extent of 1-12 per cent., hence 
 
 0-095 x I<12 
 
 V-base = Cacid = = O*OOIOO4, 
 
 IOO 
 
 and C sa it = 0*095 0*001064 = 0*094. 
 
 (o*ooio64)(o*ooio64) 
 
 Hence K A = * 22 Z> = r2 x io~ 5 . 
 
 0*094 
 
 The values of the hydrolysis constant, calculated from the 
 other observations, are given in the table, and are approxi- 
 mately constant, thus confirming the above formula. Con- 
 versely, when from one set of observations the value of 
 
EQUILIBRIUM IN ELECTROLYTES 289 
 
 K A has been obtained, the degree of hydrolysis at any other 
 dilution can be obtained by substitution in the general formula. 
 For convenience of calculation, the simple formula in which 
 the acid and base are present in equivalent proportions, may 
 be written in the form 
 
 in which x represents the proportion of acid and base formed 
 by hydrolysis from i mol of the salt and v is the dilution. 
 This form of the equation shows at a glance that the degree of 
 hydrolysis, that is, the value of x, increases with dilution. More- 
 over, from the great similarity of the formula (10) to the dilution 
 formula, it is evident that when the hydrolysis is small it is 
 greatly diminished by the addition of a strong base, just as the 
 degree of dissociation of acetic acid is greatly diminished by the 
 addition of an acetate. 
 
 The quantitative relation between the hydrolysis constant, 
 KA, and the dissociation constants for the weak acid and water 
 respectively may be obtained as follows : The electrolytic dis- 
 soc'ation of the acid, HA, is represented by the equation 
 
 [H-][A'] - K, [HA] . . . . (2) 
 
 where K fl is the dissociation constant of the acid. In the 
 solution there is the other equilibrium [H - ] [OH'] = K w (3) 
 where K w is the ionic product for water. Dividing equation 
 (3) by equation (2) we obtain 
 
 [OH'] [HA] K w 
 
 A' B K~ w 
 
 The left-hand side of the above equation is simply equation (i) 
 for the hydrolytic equilibrium (p. 288), hence 
 
 [base] [acid] K, , 
 
 [unhydrol. salt] " ' K fl 
 
 that is, tJie hydrolysis constant Kh is the ratio of the ionic 
 product fig, for water to thf. dissociation constant of the acid. 
 
2 9 o OUTLINES OF PHYSICAL CHEMISTRY 
 
 It has already been deduced from general principles (p. 286) 
 that the hydrolysis is the greater the more nearly the strength 
 of water as an acid approaches that of the competing acid, and 
 the above important result is the mathematical formulation of 
 that statement. 
 
 In order to illustrate this point more fully, the degree of 
 hydrolysis of a few salts in i/io molar solution at 25 is given 
 in the accompanying table. 
 
 Salt. Degree of hydrolysis. 
 Sodium carbonate . . . 3-17 per cent. 
 Sodium phenolate . . . 3-05 
 Potassium cyanide . , . 1*12 
 Borax ..... 0*05 
 Sodium acetate . . ... 0*008 
 
 The numbers illustrate in a very striking way the fact that only 
 the salts of very weak acids are appreciably hydrolysed. Thus 
 although acetic acid is a fairly weak acid (K = i'8 x 10 ~ 5 ), 
 sodium acetate is only hydrolysed to the extent of 0*008 per 
 cent, at 25, and even potassium cyanide is only hydrolysed to 
 the amount of about i per cent, in i/io normal solution, although 
 the dissociation constant of the acid is only 1-3 x 10 ~ 9 . A com- 
 parison of the above table with the dissociation constants of the 
 acids (p. 273) is very instructive. From the known values of K a 
 and K w for hydrocyanic acid and water respectively at 25, we 
 have 
 
 X lO 
 
 " 14 
 
 .-3 x xe- 
 
 
 m very satisfactory agreement with the observed value of n x 
 io~ 6 (p. 288). As a matter of fact, however, it is easier to deter- 
 mine the hydrolysis constant than the dissociation constant for 
 a very weak acid, and therefore the latter is often calculated 
 from the observed value of the hydrolysis constant by means 
 of the above formula* 
 
EQUILIBRIUM IN ELECTROLYTES 291 
 
 (<*) Hydrolysis of the Salt of a Weak Base and a Strong 
 Acid The same considerations apply in this case as for the 
 salt of a strong base and a weak acid. The general equation 
 for the equilibrium is of the form 
 
 B- + A' + H 2 O ^ BOH + H- + A' 
 which simplifies to 
 
 B- + H 2 O^BOH + H\ 
 Applying the law of mass action, we obtain 
 [BOH][H-] [base] [acid] 
 
 B- [unhydrol. salt] V } 
 
 exactly the same equation as is applicable to the hydrolysis of 
 he salt of a strong base and a weak acid. Further, it may be 
 shown, by a method exactly analogous to that employed in the 
 previous section, that in this case 
 
 *-. . . . w 
 
 that is, the hydrolysis constant Khfvr the salt of a weak base and 
 a strong acid is the ratio of the ionic product for water, K^, and 
 the dissociation constant of the base, K&. 
 
 A typical case is the hydrolysis of urea hydrochloride, 1 which 
 may be represented thus 
 
 CON 2 Hg + Cl' + H 2 O^CON 2 H 5 OH + H- + CY. 
 It is clear that the degree of hydrolysis can at once be obtained 
 when the H- ion concentration in the solution has been deter- 
 mined, and for this purpose any of the methods previously 
 described can be employed (p. 275), such as the effect on the 
 rate of hydrolytic decomposition of cane sugar or of methyl 
 acetate or by electrical conductivity measurements. In the 
 experiments quoted in the table, gradually increasing amounts 
 of urea were added to normal hydrochloric acid, and the H' 
 ion concentration deduced from a comparison of the velocity 
 constant for the hydrolysis of cane sugar in the presence of the 
 free acid (k ), and with the addition of urea (k). 
 
 Walker, Proc. Roy. Soc. (Edin.), 1894, 18, 255. 
 
292 OUTLINES OF PHYSICAL CHEMISTRY 
 Normal hydrochloric acid + c normal urea. 
 
 k/k l-k/ko C-I + kjko 
 
 c. k. =free HC1. = salt formed. = free urea. Kh. 
 
 o 0*00315=^0 i 
 
 0-5 0-00237 0753 0-247 '253 '77 
 
 ro 0-00184 o'5 8 5 o'^S '5 8 5 ' 8 2 
 
 2-0 0*00114 0-36 0-64 1-36 0-77 
 
 4-0 0*0006 0-19 0*81 3*19 0*75 
 
 On the assumption that the rate of hydrolytic decomposition 
 is proportional to the H* ion concentration, kjk represents the 
 concentration of the --free" hydrochloric acid, and (i - kjk ) 
 that of the bound acid, which is, ofcourse, that of the unhydro- 
 lysed salt. The concentration of the free urea, that is, of the 
 hydrolysed salt, is therefore c - (i + k\k^. Substituting in the 
 general formula 
 
 [base] [acid] fr-(i +*/*,)] [*/*] 
 [unhydrol. salt] i - k\k 
 
 The values of the hydrolysis constant are given in the sixth 
 column of the table, and are approximately constant, as the 
 theory requires. 
 
 The degree of hydrolysis of a few salts of weak bases and 
 strong acids in i/io molar solution at 25 is given in the table, 
 and the results should be compared with the values for the 
 dissociation constants of weak bases (p. 275) in order to illustrate 
 equation (2). 
 
 Salt. Degree of hydrolysis. 
 
 Ammonium chloride . . . 0-005 P er cent - 
 
 Aniline hydrochloride . . . 1*5 
 
 Thiazol hydrochloride . . .19 
 
 Glycocoll hydrochloride . 20 
 
 Hydrolysis of the Salt of a Weak Base and a Weak 
 Acid The hydrolysis of aniline acetate, a typical salt of this 
 classes represented by the equation 
 C 6 H 5 NH 3 CH 3 COO + HOH ^ C 6 H 5 NH 8 OH + CH 3 COOH, 
 
EQUILIBRIUM IN ELECTROLYTES 293 
 
 which, on the assumption that the salt is completely, the base 
 and acid not at all, ionised, may be written as follows 
 C 6 H 5 NH 3 - + CH 3 COO' + HOH^C 6 H 5 NH 3 -OH + CH 3 COOH. 
 Applying the law of mass action, and using the former sym ols, 
 [BOH][HA] [basejacid] K w 
 
 [B-][A-J ~ [unhydrol. salt]' " h ~ K a K 6 * 
 If we express the amounts (in mols) of the base, acid and 
 salt respectively in volume v of the solution by b, a and s 
 respectively, the above equation becomes 
 
 K;,, 
 
 that is, the degree of hydrolysis of the salt of a weak base and a 
 weak acid is independent of the dilution. Experiment shows 
 that in dilutions of 12*5 and 800 litres, aniline acetate is 
 hydrolysed to the extent of 45-4 and 43-1 per cent, respectively ; 
 the slight deviation from the requirements of the theory is 
 doubtless due to the fact that the assumptions made in deducing 
 the above formula are only approximately true. 
 
 Determination of the Dissociation Constant for Water 
 It has been shown above that the process of hydrolysis in 
 the case of a salt of a strong base and a weak acid may be 
 looked upon as a distribution of the base between the weak 
 acid and water acting as an acid, and the degree of hydrolysis 
 therefore depends on the relative strengths of the weak acid 
 and water. The relationship between these three factors is 
 expressed by the equation 
 
 [acid][base] K. 
 
 [unhydrol. salt] * = K a 
 
 where K/, is the hydrolysis constant, K a the dissociation con- 
 stant for the acid, and K w the ionic product for water. It is 
 clear that if the degree of hydrolysis of a salt and the dissocia- 
 tion constant of the acid are known, K w can be calculated, and 
 
294 OUTLINES OF PHYSICAL CHEMISTRY 
 
 this is one of the most accurate methods for determining the 
 degree of dissociation of water. As an illustration, we may 
 calculate K w from Shields's value for the hydrolysis of sodium 
 acetate crooS percent, in o'i molar solution at 25 (Arrhenius, 
 Feb., 1893). We have 
 
 Cacid = Cbase = O'OOOoS X O'l, 
 
 Csait = 0*1 (the amount hydrolysed being negligible 
 in comparison). 
 
 (o'ooooS x o'i) 2 
 Hence v - '- = 0-64 x io~ 9 = K h . 
 
 ' 
 
 O'I 
 
 Now K w = K h K a = (0-64 x io- 9 ) x (1-8 x icr 5 ) = ri6 x lo" 14 . 
 
 Since [H-][OH'] is thus found to be approximately 1-2 x 
 io~ 14 , the concentration of H* or OH' ions (mols per litre) in 
 water at 25 is i'i x io~ 7 . 
 
 K w can also be calculated from measurements of the hydro- 
 lysis of salts of strong acids and weak bases, and, perhaps with 
 still greater accuracy, from measurements with salts of weak 
 ac.ds and weak bases l by means of the formula 
 Kfc = Ku,/K a Kft. 
 
 The degree of dissociation of water has been determined by 
 three other methods at 25 with the following results : 
 
 E.M.F. of hydrogen-oxygen cell (Ostwald, January, 
 1893) (corrected value), i-o x io - 7 at 25. 
 
 Velocity of hydrolysis of methyl acetate (Van't Hoff- 
 Wijs, March, 1893), 1-2 x io ~ 7 at 25. 
 
 Conductivity of purest water (Kohlrausch, 1894), 
 1*05 x io " 7 at 25. 
 
 When it is borne in mind how small the dissociation is, the 
 close agreement in the values obtained by these four inde- 
 pendent methods is very striking, and forms a strong justifica- 
 tion for the original assumption that water is split up to an 
 extremely small extent into ions. 
 
 The question can be still further tested by applying van't 
 
 1 Lunden, jf. Chim. Phys., 1907, 5, 574 ; Kanolt, J. Amer. Chem. Soc., 
 1907, 29, 1402. 
 
EQUILIBRIUM IN ELECTROLYTES 295 
 
 HofFs equation connecting heat development and displacement 
 of equilibrium to the equilibrium between water and its ions. 
 For the degree of dissociation at different temperatures, the 
 following values were obtained by Kohlrausch from conduc- 
 tivity measurements : 
 
 Temperature . . o 2 10 15 26 34 42 50 
 Degree of dissociation 035 0-39 0-56 0*8 1-09 1-47 1-93 2-48 xio~ 7 
 
 From any two of these measurements the heat development, Q, 
 of the reaction, H 2 O ^ H* + OH', can be calculated by sub- 
 stitution in the general formula (p. 167). 
 
 log loKl - log 10 K 2 = - 
 
 From the values l of K at o and 50, Q = - 13,740 cal., and 
 from that at 2 and 42, Q = 13,780 cal. 
 
 In a previous section (p. 284) it has been pointed out that, 
 according to the electrolyte dissociation theory, the neutraliza- 
 tion of a strong base by a strong acid consists essentially in the 
 combination of H* and OH' ions to form water. The heat 
 given out in the reaction is about 13,700 cal. for molar quanti- 
 ties, in excellent agreement with the above value. This sup- 
 ports the assumption that the variation of the conductivity of 
 pure water with temperature is due to the displacement of the 
 equilibrium H' + OH' ^H 2 O, in the direction indicated by 
 the lower arrow. The value of Q, obtained directly as above, 
 may be termed the heat of ionisation of water ; it is the heat 
 given out when i mol of H* and OH' ions combine to form 
 water. 
 
 The heat of ionisation of any electrolyte can naturally be 
 calculated in the same way from the displacement of the equili- 
 brium with temperature. The effect of increased temperature 
 on the degree of ionisation is almost always slight, and in the 
 majority of cases the ionisation is slightly diminished. As an 
 illustration, the degree of electrolytic dissociation for i/io molar 
 sodium chloride over a wide range of temperature, as deter- 
 
 1 K, = (2-48 x io- 7 ) 2 at 50; K! = (0-35 x io- 7 ) 2 at o (p. 283). 
 
296 OUTLINES OF PHYSICAL CHEMISTRY 
 
 mined by Noyes and Coolidge, 1 may be quoted. The values 
 obtained were 84 per cent, at 18, 79 per cent, at 140, 74 per 
 cent, at 218, 67 per cent, at 281, and 60 per cent, at 306. 
 Corresponding with the small variation in the degree of ionisa- 
 tion with temperature, the heat of ionisation is small, and may 
 be positive (as in the present case) or negative. 
 
 Theory of Indicators The indicators used in acidimetry 
 and alkalimetry have the property of giving different colours 
 depending on whether the solution is acid or alkaline. Ac- 
 cording to Ostwald's theory, which has met with fairly general 
 acceptance, such indicators, including methyl-orange, phenol- 
 phthalein and /-nitrophenol, are weak electrolytes, and their 
 use depends on the fact that the ions and the non-ionised com- 
 pounds have different colours. Since salts are almost always 
 highly ionised, it is clear that only weak acids and bases can 
 be employed as indicators. 
 
 Phenolphthalein is a very weak acid, the non-ionised acid is 
 colourless, and the negative ion red. In aqueous solution it is 
 ionised according to the equation 
 
 HP^H- + P'(red) 
 
 (where P' is the negative ion), but so slightly that the solution is 
 practically colourless. If now sodium hydroxide is added, the 
 highly-dissociated sodium salt is formed, and the solution is 
 deeply coloured owing to the presence of the red anion, P'. 
 If, on the other hand, the solution contains a slight excess of 
 acid, the increased H- ion concentration drives back the ioni- 
 sation of the phenolphthalein in the direction indicated by the 
 lower arrow, and the solution becomes colourless (cf. p. 277). 
 Finally, if a weak base, such as ammonium hydroxide, is added, 
 the ammonium salt will be partly hydrolysed, according to the 
 equation 
 
 NH 4 P + HOH^NH 4 OH + HP, 
 and excess of the base will be required in order to drive back 
 
 1 Zeitsch. Physikal Chem., 1904, 46, 323. 
 
EQUILIBRIUM IN ELECTROLYTES 
 
 297 
 
 the hydrolysis (p. 289) ; in other words, there will not be a sharp 
 change of colour when ammonium hydroxide is added. 
 
 Methyl-orange is an acid of medium strength, the non-ionised 
 acid is red, the negative ion yellow. As the acid is ionised in 
 aqueous solution to some extent, according to the equation 
 
 HM(red)^H- + M' (yellow), 
 
 the solution shows a mixed colour. The effect of the addition 
 of acids and alkalis is similar to that on phenolphthalein, and 
 only differs in degree. Since the aqueous solution already con- 
 tains a considerable proportion of H* ions, it is evident, from 
 the considerations advanced on p. 276, that a considerable 
 excess will be required to drive back the dissociation in the 
 direction indicated by the lower arrow, so as to turn the solu- 
 tion red. The proportion of H* ions in such a weak acid as 
 acetic acid is not sufficient for this purpose, and therefore 
 methyl-orange should not be employed as an indicator for 
 weak acids. It is, on the other hand, a suitable indicator for 
 weak bases, such as ammonia, as the salt formed is much less 
 hydrolysed than the corresponding phenolphthalein salt, and 
 therefore the change of colour is sharper. 
 
 Basic indicators are not in use. 
 
 The considerations to be borne in mind in selecting an 
 indicator, or in choosing a suitable alkali for titrating an acid, 
 or vice versa, may be put concisely as follows (Abegg) : 
 
 Solutions used. 
 
 
 
 Acid. 
 
 Base. 
 
 
 
 Strong 
 
 Strong 
 
 Any 
 
 Any 
 
 Strong 
 Weak 
 
 Weak 
 Strong 
 
 Strong acid 
 Weak acid 
 
 Methyl-orange, /-nitrophenol 
 Phenolphthalein, litmus 
 
 Weak 
 
 Weak 
 
 None satisfactory 
 
 Should be avoided 
 
 Some investigators maintain that the ionisation theory does 
 not give a satisfactory representation of the behaviour of indi- 
 
298 OUTLINES OF PHYSICAL CHEMISTRY 
 
 % 
 
 cators, but that the changes of colour are due to changes of 
 constitution, usually from the benzenoid to the quinonoid type 
 and vice vetsd. 1 In an aqueous solution of phenolphthalein, for 
 instance, there are only traces of the quinonoid (coloured) modifi- 
 cation, and the solution is colourless, but on the addition of 
 alkali the phenolphthalein salt is formed, the negative ions of 
 which, being of the quinonoid type, are strongly coloured. 
 
 The Solubility Product We have now to consider an 
 equilibrium of rather a different type in which the solution 
 is saturated with regard to the electrolyte. In such a case 
 there is equilibrium between the solid salt and the non-ionised 
 salt in the solution, so that the concentration of the non-ionised 
 salt remains constant at constant temperature. Further, there 
 is equilibrium in the solution between the non-ionised salt and 
 its ions, which may be represented, in the case of silver chloride, 
 for example, by the equation Ag* + Cl'^AgCl. Applying 
 the law of mass action, we have, for the latter equilibrium, 
 
 [Ag-][Cl'] = KfAgCl] = S, 
 
 where S is the product of the concentrations of the two ions 
 the so-called solubility product and is constant, since the 
 right-hand side of the above equation is constant. The equi- 
 libria in the heterogeneous system may be represented as 
 follows : 
 
 Ag- + Cl' ^ AgCl (in solution) 
 
 I t 
 AgCl (solid). 
 
 As will be shown later, the solubility product for silver chloride 
 at 25 is 1-56 x io- 10 , when the ionic concentrations are ex- 
 pressed in mols per litre. If the solution has been prepared 
 by dissolving the salt in water, the ions are necessarily present 
 in equivalent proportions, so that a solution of silver chloride, 
 saturated at 25, contains x/i'56 x io- 10 = 1-25 x io~ 6 mols 
 of Ag' and of Cl' ions. 
 
 The ions need not, however, be present in equivalent propor- 
 1 C/. Hewitt, Analyst, 1908, 33, 85. 
 
EQUILIBRIUM IN ELECTROLYTES 
 
 299 
 
 tions ; if by any means the solubility product is exceeded, for 
 example, by adding a salt with an ion in common with the 
 electrolyte, the ions unite to form undissociated salt, which falls 
 out of solution, and this goes on till the normal value of the 
 solubility product is reached. Perhaps the best-known illustra- 
 tion of this is the precipitation of sodium chloride from its 
 saturated solution by passing in gaseous hydrogen chloride. 
 In this case the original equilibrium between equivalent amounts 
 of Na* and Cl' ions is disturbed by the addition of a large 
 excess of Cl' ions, and sodium chloride is precipitated till the 
 original solubility product is regained, when the solution con- 
 tains an excess of Cl' ions and relatively few Na' ions. 
 
 As already mentioned, the difference between the present 
 form of equilibrium and those previously considered is that the 
 concentration of the non-ionised salt in the solution is constant 
 at constant temperature. If it is diminished in any way, salt 
 is dissolved till the original value is reached ; if it is exceeded, 
 as in the case just mentioned, salt falls out of solution till the 
 original value is reached. 
 
 Since the equilibrium equation for a binary salt is symmetrical 
 with regard to the two ions, it follows that the solubility of such a 
 salt should be depressed to the same extent by the addition of 
 equivalent amounts of its common ions, whether positive or 
 negative. This consequence of the theory was tested by Noyes, 
 who determined the influence of the addition of equivalent 
 amounts of hydrochloric acid and of thallous nitrate on the 
 solubility of thallous chloride, with the following results : 
 
 Solubility of thallous chloride at 25 : 
 
 Concentration of 
 Substance added 
 (mols per litre). 
 
 T1NO S added. 
 
 HC1 added. 
 
 O 
 0*0283 
 0-147 
 
 0*0161 
 
 0*0084 
 0*0032 
 
 0*0161 
 
 0*0083 
 0*0033 
 
300 OUTLINES OF PHYSICAL CHEMISTRY 
 
 The figures in the first column show the amounts of thallous 
 nitrate and of hydrochloric acid added, those in the second and 
 third columns represent the solubility of thallous chloride in 
 mols per litre. The results show that the requirements of 
 the theory are satisfactorily fulfilled. 
 
 The above results hold independently of the relative amounts 
 of ions and non-ionised salt in the solution. Since in dilute 
 solution all salts are highly ionised, it may, however, be as- 
 sumed that difficultly soluble salts, such as silver chloride, are 
 almost completely ionised in solution ; in other words, the con- 
 centration of non-ionised salt in solution may be regarded as 
 negligible in comparison with that of the ions. This deduction 
 is of great importance in estimating the solubility of difficultly 
 soluble salts (see next page). 
 
 Applications to Analytical Chemistry The above con- 
 siderations with regard to the solubility product are of the 
 greatest importance for analytical chemistry. A precipitate can 
 only be formed when the product of the ionic concentrations 
 attains the value of the solubility product, which for every salt 
 has a definite value depending only on the temperature. For 
 example, magnesium hydroxide is precipitated from solutions of 
 magnesium salts by ammonia because the solubility product 
 [Mg"] [OH'] 2 is exceeded. When, however, ammonium chloride 
 is previously added in excess to the hydroxide, the OH' ion 
 concentration is diminished to such an extent (p. 277) that the 
 solubility product is not reached, and precipitation no longer 
 occurs. Similarly, zinc sulphide is precipitated when the 
 product [Zn"][S"J exceeds a certain value. In alkaline solu- 
 tion, an extremely small concentration of hydrogen sulphide 
 suffices for this purpose, as the sulphide is considerably ionised, 
 but in acid solution the depression of the ionisation of the 
 hydrogen sulphide, in other words, the diminution in the con- 
 centration of S" ions, is so great that the solubility product is 
 not reached. On the other hand, the solubility product for 
 certain heavy metals, such as lead, copper, and bismuth, is so 
 
EQUILIBRIUM IN ELECTROLYTES 301 
 
 small that it is reached even in acid solution. It is, however, 
 possible to increase the acid concentration (and therefore to 
 diminish the S" ion concentration) to such an extent that the 
 ionic product is not reached even for some of the above metals, 
 for example, lead sulphide in concentrated hydrochloric acid is 
 not precipitated by a current of hydrogen sulphide. 
 
 On the same basis, a fact which has long been familiar in 
 quantitative analysis, that precipitation is more complete when 
 excess of the precipitant is added, can readily be accounted for. 
 A saturated aqueous solution of silver chloride contains about 
 1*25 x io~ 5 gram equivalents of the salt per litre, and the ad- 
 dition of ten times that concentration of Cl' ions (added in the 
 form of sodium chloride) will diminish the amount of silver in 
 solution to about i/io of its original value (on the assumption 
 that the concentration of the non-ionised salt is negligible in 
 comparison with that of the ions) (cf. p. 277). It is thus evident 
 that in the gravimetric estimation of combined chlorine as silver 
 chloride there might be considerable error owing to the solubility 
 of silver chloride in water, but if a fair excess of the precipitant 
 is used, the error is quite negligible. 
 
 The concentration in saturated solution at 25, and the 
 solubility product of a few difficultly soluble salts are given in 
 the accompanying table. 
 
 Salt. Saturation Concentration Solubility Product 
 
 (mols per litre). (mols per litre). 
 
 Silver chloride Ag* = 1*25 x io~ 5 1*56 x io~ 10 
 
 bromide ,, = 6-6 x io~ 7 4-35 x io~ 13 
 
 iodide = i'o x io~ 8 i'o x io~ 16 
 
 Thallous chloride Tl- = 1-6 x io~ 2 2-6 x IQ- 4 
 
 Cuprous chloride Or = i'i x io~ 3 1-2 x io~ 6 
 
 Lead sulphide Pb" = 5'i x io~ 8 2*6 x io- 15 
 
 Copper sulphide Cu" = IT x io~ 21 1*2 x'io~ 42 
 
 Experimental Determination of the Solubility of Diffi- 
 cultly Soluble Salts When a saturated solution of a rt-la- 
 
302 OUTLINES OF PHYSICAL CHEMISTRY 
 
 lively insoluble salt is so dilute that complete ionisation may be 
 assumed (//, = /x^ ), the solubility of the salt may readily be 
 obtained from electrical conductivity measurements. The 
 molecular conductivity at infinite dilution, p,^ , can be obtained 
 indirectly (p. 258), the specific conductivity of the saturated 
 solution is determined in the usual way, and, by substitution in 
 the formula /-i^-, = KV, we obtain the value of v, that is, the 
 volume in c.c., in which a mol of the substance is dissolved. If 
 the solubility is required in mols per litre, then 
 
 v = 1000 V, and 
 /A oo = i ooo *V, 
 
 where V is the volume in litres in which a mol of the sub- 
 stance is dissolved. As the specific conductivity of such a 
 solution is small, the conductivity of the water becomes of 
 importance, and it is necessary to subtract from the observed 
 specific conductivity of the solution the conductivity of the 
 water, determined directly. For such measurements, "con- 
 ductivity " water, of a specific resistance not much less than 
 io 6 ohms, should be used. 
 
 As an example of the determination of solubilities by this 
 method, Bottger found that a solution of silver chloride saturated 
 at 20 had K = 1-33 x io ~ 6 after subtracting the specific con- 
 ductivity of the water. Hence, as /XQQ for silver chloride at 20, 
 determined indirectly (p. 258), is i25'5, we obtain, by substitu- 
 tion in the above formula, 
 
 125*5 = 1000 x i '33 x io~ 6 V, 
 
 that is, 94,400 litres of a solution of silver chloride, saturated 
 at 20, contain i mol of the salt. In one litre of solution there 
 is, therefore, 1/94,400 = 1*06 x io ~ 5 mol, or 0-00152 grams 
 of silver chloride. The values for the solubility of a number of 
 difficultly soluble salts obtained by this method are given in 
 the previous section (p. 301), 
 
EQUILIBRIUM IN ELECTROLYTES 303 
 
 Complex Ions Complex ions have already been defined as 
 being formed by association of ions with non-ionised molecules 
 It is well known that though silver halogen salts are only 
 slightly soluble in water they are readily soluble in the presence 
 of ammonia. This phenomenon is due to the formation of 
 complex ions (p. 281), in which the Ag' ions are associated 
 with ammonia molecules, fprming univalent ions of the type 
 Ag(NH 3 )^. As nearly all the silver is present in this form 
 and very little in the form of Ag' ions, it is evident that a 
 solution may contain a very considerable amount of a silver salt 
 before the solubility product [Ag-] [X'] is reached. 
 
 The determination of the exact composition of the complex 
 ions in a solution is sometimes a matter of difficulty, but the 
 law of mass action is often of great assistance in this respect. 
 If, for example, the solution of a silver salt in ammonia 
 contains mainly complex ions of the formula Ag(NH 3 )-, 
 there must be an equilibrium represented by the equation 
 Ag(NH 3 )j ^ Ag- + 2NH 3 , and by the law of mass action the 
 
 expression , ^ * must be constant. To test this point 
 Ag(NH 3 )- 
 
 the concentrations of the ammonia and of the silver ions were 
 systematically varied, and it was found that the above expres- 
 sion remained constant, thus showing that the assumption as 
 to the composition of the complex ions is correct. The azure 
 blue solutions obtained by adding ammonia in excess to solu- 
 tions of cupric salts appear from the results of distribution (p. 
 178) and optical measurements to contain the copper exclusively 
 as Cu(NH 3 ) 4 - ions. The solutions obtained by the action of 
 ammonia on cobalt and chromium salts have been thoroughly 
 investigated by Werner : l they contain complex ions of the 
 formulae Co(NH 3 ) 6 - and Cr(NH 3 ) 6 -. 
 
 Other compounds besides ammonia can become associated 
 with positive ions to form complex cations. For example, 
 Werner has shown that the ammonia groups in the com- 
 plex cobalt ions mentioned above can be successively dis- 
 1 Summary, Anorganischf Chemie* 2nd edition, Brunswick, 1909. 
 
304 OUTLINES OF PHYSICAL CHEMISTRY 
 
 placed by water molecules, forming compounds of the type 
 [Co(NH 3 ) 5 H 2 0]-, [Co(NH 3 ) 4 (H 2 0) 2 ]- . . . 
 
 and finally 
 
 [Co(H 2 0) 6 ]- 
 
 It is probable that in the aqueous solutions of many salts com- 
 plex ions containing water are present. 
 
 Complex anions, formed by th^ association of negative ions 
 with neutral molecules, are also known. Solutions of cadmium 
 iodide contain CdI 4 " ions (p. 281), and the solution obtained by 
 the addition of excess of potassium cyanide to solutions of silver 
 salts contains potassium silver cyanide, KAg(CN) 2 , which is 
 largely ionised according to the equation 
 
 Influence of Substitution on Degree of lonisation 
 
 Reference has already been made to the influence of substitu- 
 
 tion on the strength of acids. As the effect of substitution on 
 
 the degree of ionisation has been most extensively investigated 
 
 for this class of compound, a few further examples may be 
 
 given. In the accompanying table, the affinity or dissociation 
 
 constants for some mono-substituted acetic acids are given, the 
 
 value of K holding for 25 (concentrations in mols per litre). 
 
 Acetic acid CH 3 COO'H . . . . t , ^ j . 0-000018 
 
 Propionic acid CH 3 CH 2 COO'H * <K> . 0*000013 
 
 Chloroacetic acid CH 2 C1COO'H . . 0-00155 
 
 Bromoacetic acid CH 2 BrCOO'H . *. 0-00138 
 
 Cyanacetic acid CH 2 CNCOO'H ;;- . 0*00370 
 
 Glycollic acid CH 2 OHCOO-H r; . 0-000152 
 
 Phenylacetic acid C 6 H 5 CH 2 COO'H . 0-000056 
 
 Amidoacetic acid CH 2 NH 2 COO'H -"..-'.- 3-4 x lo" 10 
 
 As the carboxyl group only is concerned directly in ionisation, 
 the above table affords an excellent illustration of the influence 
 of a group on a neighbouring one. The table shows that when 
 one of the alkyl hydrogens in acetic acid is displaced by 
 Cl, Br, CN or OH or C 6 H 5 an increase in the activity of the 
 
EQUILIBRIUM IN ELECTROLYTES 305 
 
 acid is brought about ; the effect is least for the phenyl group 
 and greatest for the cyanogen group. On the other hand, the 
 methyl group (in propionic acid) diminishes the activity slightly, 
 and the amido group diminishes it enormously. 
 
 These observations can readily be accounted for 1 on the 
 assumption that the atoms or groups take their ion-forming 
 character into combination. Thus the Cl, Br, CN and OH 
 groups, which tend to form negative ions, increase the tendency 
 of the groups into which they enter to form negative ions. 
 The " negative favouring " character of the phenyl group is 
 slight but distinct. On the other hand, the so-called basic 
 groups, such as NH 2 , lessen the tendency of the group into 
 which they enter to form negative ions, as is very strikingly 
 shown in the case of amidoacetic acid. The methyl group has 
 also a slight diminishing efiect on the tendency of a group to 
 form negative ions. 
 
 The magnitude of the influence of a substituent on a particu- 
 lar group depends on its distance from that group. This is 
 very well shown by the influence of the hydroxyl group on the 
 affinity constant of propionic acid. 
 
 Propionic acid CH 3 CH 2 COO'H . . . 0*0000134 
 Lactic acid CH 3 CHOHCOO'H . . 0-000138 
 
 /?-oxypropionic acid CH 2 OHCH 2 COO'H . 0-0000311 
 
 When the OH group is in the a (neighbouring) position its 
 effect on the dissociation constant is more than four times as 
 great as when it is in the @ position. 
 
 It seems plausible to suppose that a comparison of the 
 influence of groups in the ortho, meta and para positions on the 
 carboxyl group of benzoic acid might throw some light on the 
 question of the relative distances between the groups in the 
 benzene nucleus. The dissociation constants of benzoic acid 
 and the three chlor-substituted acids are as follows: 
 
 1 A complete theory of the phenomena in question has recently been 
 worked out by Flurscheim (Trans. Chem. Soc. t 1909, 95, 718 ; Proc., 1909, 
 J 93)- 
 
 2O 
 
3 o6 OUTLINES OF PHYSICAL CHEMISTRY 
 
 Benzoic acid C 6 H 5 COOH . . . 0-000060 
 o- Chlorobenzoic acid C 6 H 4 C1COOH . . 0*00132 
 m- 0-000155 
 
 p- .1 0-000093 
 
 It will be observed that the presence of the halogen in the ortho 
 position greatly increases the strength of the acid, and it is a 
 general rule that the influence of substituents is always greatest 
 in this position. The effect of substituting groups in the meta 
 and .para positions is much smaller, and the order of the two is 
 not always the same. As the table shows, ;;*-chlorobenzoic 
 acid is rather stronger than the para acid, but on the other hand 
 />-nitrobenzoic acid is somewhat stronger than the ?neta acid. 
 
 Similar considerations apply to the influence of substituents 
 on the strength of bases, but, as is to be expected, the effect of 
 the various groups is exerted in the opposite direction to that on 
 acids. Thus the displacement of a hydrogen atom in ammonium 
 hydroxide by the methyl group gives a stronger base (methyl 
 amine) but the entrance of a phenyl group gives a much weaker 
 base (aniline) (cf. p. 275). 
 
 Reactivity of the Ions It is a well-known fact in qualita- 
 tive analysis that in the great majority of cases the positive 
 component of a salt (e.g., the metal) answers certain tests, quite 
 independently of the nature of the acid with which it is combined, 
 and in the same way acids have certain characteristic reactions, 
 independent of the nature of the base present. These facts are 
 plausibly accounted for on the electrolytic dissociation theory 
 by assuming that the positive and negative parts of the salts 
 (the ions) exist to a great extent independently in solution, and 
 that the well-known tests for acids and bases are really tests for 
 the free ions. Thus silver nitrate is not a general test for 
 chlorine in combination, but only for chlorine ions. It is well 
 known that potassium chlorate gives no precipitate with silver 
 nitrate, although it contains chlorine ; this is readily accounted 
 for on the electrolytic dissociation theory because the solution 
 of the salt contains no Cl' ions, but only C1O 3 ' ions, which give 
 
EQUILIBRIUM IN ELECTROLYTES 307 
 
 their own characteristic reactions. These views appear still 
 more plausible when cases are considered in which the usual 
 tests fail, for example, mercuric cyanide does not give all the 
 ordinary reactions for mercury. This could be accounted foi 
 by supposing that the compound is not appreciably ionised in 
 solution, so that practically no Hg" ions are present, and as a 
 matter of fact the aqueous solution of mercuric cyanide is 
 practically a non-conductor. 
 
 The chief characteristic of ionic reactions is their great rapidity ; 
 they are for all practical purposes instantaneous, and it is doubt- 
 ful if the speed of a purely ionic reaction has so far been 
 measured. It is well known that silver nitrate reacts with the 
 chlorine in organic compounds such as ethyl chloride and chlor- 
 acetic acid, but very slowly as compared with its action on 
 sodium chloride. There is good reason for supposing that the 
 reactions last mentioned are not ionic actions, but that the 
 changes take place between the silver salt and combined chlorine. 
 
 The great reactivity of the ions in cases where it is known 
 that they are actually present has led Euler and others to 
 postulate that all reactions are ionic, and that in very slow 
 reactions we are dealing with excessively small ionic concen- 
 trations. 1 This question cannot be adequately considered here, 
 but it may be mentioned that the available experimental evi- 
 dence does not seem to lend any support to Euler's theory. 
 There is good reason to suppose that chemical reactions may 
 take place between non-ionised molecules as well as between 
 ions. 
 
 Amphoteric Electrolytes. It is a familiar fact that the 
 hydroxides of certain polyvalent metals show both basic and 
 acidic properties, since they form salts both with acids and 
 bases, e.g., lead hydroxide, Pb(OH) 2 ; aluminium hydroxide 
 A1(OH) 3 . In terms of the ionisation theory, these compounds 
 must give both hydrogen and hydroxyl ions on dissociation. 
 Substances of this type, which can ionise in more than one 
 way, are termed amphoteric electrolytes. 
 
 Lead hydroxide, Pb(OH) 2 , can dissociate according to the 
 following equations : 
 
 1 Compare Arrhenius, Electrochemistry (English Edition), p. 180. 
 
308 OUTLINES OF PHYSICAL CHEMISTRY 
 
 (i.a)Pb(OH) 2 ^Pb(OH)- + OH'; (i.b)Pb(OH)-^Pb- + OH' 
 (2.a)Pb(OH) 2 ^PbO(OH)'+H- ; (2.b)PbO(OH)'^PbO a "+H- 
 and doubtless all these compounds are present in greater or 
 less concentration in an aqueous solution of lead hydroxide. 
 It will of course be understood that the concentrations of H' 
 and of OH' ions cannot both be considerable in the same solu- 
 tion since the equilibrium [H*] [OH 7 ] = k always holds (p. 283). 
 
 When an acid is added to the hydroxide, the OH' ions com- 
 bine with the H' ions of the acid to form water, more hydr- 
 oxide dissociates according to equations la and ib, the fresh 
 OH' ions combine with the H- ions to form water, and so on, 
 till ultimately, if sufficient acid is added, the solution contains 
 chiefly Pb" ions and anions derived from the acid. If, on the 
 other hand, alkali is added to the hydroxide, the OH' ions 
 combine with the H' ions derived from the hydroxide and dis- 
 sociation proceeds progressively according to the upper arrows 
 in equations 2a and 2b till ultimately the solution contains chiefly 
 PbO 2 " ions and cations derived from the alkali added. The proof 
 of the above statements is that on electrolysis the lead in acid 
 solution travels to the cathode, in alkaline solution to the anode. 
 
 Some more complicated compounds can split off both H- and 
 OH' from a single molecule, leaving an uncharged ion or rather 
 an ion which is both positively and negatively charged. For 
 glycine (amidoacetic acid) we have the following equilibrium : 
 
 + 
 OH-H 3 NCH 2 COOH^H 3 NCH 2 COO + H- + OH'. 
 
 Ions of this type are termed zwitter ions or hermaphrodite 
 ions. Since they are electrically neutral, they are not affected 
 by an electric current. 
 
 Practical Illustrations. Dilution Law. Conductivity of 
 Acids and Salts. Many of the results discussed in this chapter 
 can be conveniently illustrated by means of the apparatus 1 
 shown in Fig. 36. The glass vessels each contain two circular 
 electrodes of platinized platinum, the lower one is connected 
 with a wire which passes through the bottom of the vessel 
 and is connected through a lamp to the wire E. The upper 
 electrode, which is movable, is connected to a wire which 
 
 1 Noyes and Blanchard, J. Amer. Chem. Soc.. 1900, 22, 726. 
 
EQUILIBRIUM IN ELECTROLYTES 
 
 39 
 
 passes through the cork loosely closing the vessel, and is 
 connected to the upper wire F. The electrodes in each 
 vessel should be of approximately the same cross-section, and 
 the four lamps of equal resistance. The wires E and F are 
 connected to the terminals of a source of alternating current, 
 and as they are at constant potential throughout, the fall of 
 potential through each of the vessels from E to F must be 
 the same when a current is passing. 
 
 FIQ. 36. 
 
 The use of the arrangement may be illustrated by employing 
 it to prove the dilution law in the form a = >Jkv. Solutions of 
 monochloracetic acid containing i mol of the salt in i, 4 and 
 1 6 litres respectively are prepared, and the vessels A, B and C 
 nearly filled with them. When connection is made with the 
 alternating current, it will be found that the brightness of the 
 lamps is very different, but the positions of the upper electrodes 
 can be so adjusted that the lamps are equally bright. Under 
 these circumstances it is evident that the resistance of each 
 
310 OUTLINES OF PHYSICAL CHEMISTRY 
 
 solution is the same. On measuring the distances between the 
 electrodes, it will be found that for the solutions v = i, v = 4, 
 v = 1 6, the distances are in the ratio 4:2:1. Hence, as 
 the conductivities are inversely proportional to the distances 
 between the electrodes, aoc *Jv. 
 
 Further, it may be shown that, although acids differ very 
 greatly in conductivity, neutral salts, even of weak acids, have 
 a conductivity nearly as great as that of strong acids. The 
 vessels are filled with 1/4 normal solutions of hydrochloric acid, 
 sulphuric acid, monochloracetic acid and acetic acid respectively, 
 and when the distances are altered till the lamps are equally 
 bright, it will be found that ths electrodes are very near in the 
 acetic acid solution, far apart in the hydrochloric acid solution, 
 and at intermediate distances for the other two acids. Sufficient 
 sodium hydroxide to neutralize the acid is now added to each 
 vessel, and after stirring and again adjusting to equal brightness 
 of the lamps, it will be found that the distances for all four 
 solutions are approximately equal. 
 
 Eqtdlibrium Relations as shoivn by Indicators The equi- 
 librium relations in the case of weak acids and bases may 
 be shown very well by means of indicators. Each of two 
 beakers contains 100 c.c. of water, i c.c. of n/i sodium 
 hydroxide, and a few drops of methyl-orange. To the con- 
 tents of one beaker n hydrochloric acid is added drop by 
 drop by means of a pipette, and to the other n acetic acid 
 is added in the same way till both solutions just become red. 
 It will be observed that whereas about i c.c. of hydrochloric 
 acid brings about the change of colour (owing to its relatively 
 high concentration in H' ions), several c.c. of acetic acid are 
 required to produce the same effect, owing to the much smaller 
 H' ion concentration of the latter solution. If now a strong 
 solution of sodium acetate is added to the last solution, the 
 yellow colour of the methyl-orange will be restored; the 
 acetate reduces the strength of the acid to such an extent 
 (p. 277) that the H' ion concentration is no longer sufficient 
 to drive back the ionisation of the methyl-orange. 
 
EQUILIBRIUM IN ELECTROLYTES 311 
 
 The effect of hydrolysis may be illustrated with indicators as 
 follows : Each of two beakers contains 50 c.c. of 1/2 n hydro- 
 chloric acid and a few drops of methyl-orange and phenol- 
 phthalein respectively. If n ammonium hydroxide is slowly 
 added to the solution containing methyl-orange, the colour 
 will change when about 25 c.c. of ammonia has been added, 
 but a much greater quantity of the same solution will be re- 
 quired to redden the phenolphthalein solution. The explanation 
 of this behaviour has already been given. Owing to the fact 
 that the ammonium salt of phenolphthalein is considerably 
 hydrolysed, it is necessary to add a fair excess of the base 
 before the coloured phenolphthalein ions are produced in 
 considerable amount. 
 
 The Solubility Product The conception of the solubilit) 
 product may be illustrated by the method employed by Nernst 
 in proving the formula experimentally. A saturated solution 
 of silver acetate is prepared by shaking the finely-powdered 
 salt with water for some time. To a few c.c. of the solution 
 in a test-tube a few c.c. of a fairly concentrated solution of 
 silver nitrate are added, and to another portion of the acetate 
 solution a solution of sodium acetate equivalent in strength 
 to the silver nitrate solution, and the mixtures are well shaken. 
 In each tube a precipitate of silver acetate will be formed. 
 
 Complex Ions The evidence in favour of the view that in 
 solutions of silver salts in potassium cyanide the silver is 
 mainly present as a constituent of a complex anion, Ag(CN)' 2 , 
 is that the silver migrates towards the anode during electrolysis. 
 The copper in Fehling's solution is also mainly present as a 
 component of a complex anion, as may readily be shown 
 qualitatively by a simple experiment described by Kiister. A 
 U-tube is about half-filled with a dilute solution of copper 
 sulphate, and the two limbs are then nearly filled up with a 
 dilute solution of sodium sulphate in such a way that the 
 boundaries remain sharp. A second U-tube is filled in an 
 exactly similar way with Fehling's solution in the lower part 
 and an alkaline solution of sodium tartrate in the upper part, 
 
3 i2 OUTLINES OF PHYSICAI CHEMISTRY 
 
 the arrangement being such that the Fehling's solution in the 
 one tube stands at the same level as the copper sulphate solu- 
 tion in the other tube. The U-tubes are then connected by a 
 bent glass tube, filled with sodium sulphate solution, and with the 
 ends dipping in the sulphate and tartrate solutions respectively. 
 A small current is then sent through the U-tubes in series, by 
 means of poles dipping in the outer limbs of the tubes, 
 and after a time it will be observed that the copper sulphate 
 boundary has moved with the positive current, whilst the 
 coloured boundary in the other tube has moved against the 
 positive current towards the anode. It is therefore evident 
 that in the latter case the copper is present in the anion, as 
 stated above. 
 
 Chemical Activity and lonisation The great difference in 
 chemical and electrical activity produced by ionisation is well 
 shown by comparing the properties of solutions of hydrochloric 
 acid gas in water and in an organic solvent such as toluene. 
 Whilst the former solution conducts the electric current and 
 dissolves calcium carbonate rapidly, the latter solution is a non- 
 conductor, and has little or no effect on calcium carbonate. 
 
 The same fact is illustrated by the interaction of silver nitrate 
 with potassium bromide, ethyl bromide and phenyl bromide 
 respectively in alcoholic solution. Approximately 5 per cent, 
 solutions of the bromides in ethyl alcohol are prepared, and 
 to each solution is added a few c.c. of a saturated solution of 
 silver nitrate in alcohol. With the potassium bromide there is 
 an immediate precipitate, the action being ionic. With ethyl 
 bromide the reaction is very slow, and there is no apparent 
 reaction with phenyl bromide. The reaction between silver 
 nitrate and ethyl bromide is a good example of a chemical change 
 which is not ionic, as far as one of the reacting substances (the 
 ethyl bromide) is concerned. 
 
 Non-ionic chemical changes may, however, be very rapid. 
 A solution of copper oleate in perfectly dry benzene reacts 
 immediately with a solution of hydrochloric acid gas in dry 
 benzene, with precipitation of cupric chloride (Kahlenberg), 
 
CHAPTER XII 
 COLLOIDAL SOLUTIONS. 1 ADSORPTION 
 
 Colloidal Solutions. General Up to the present we 
 have dealt with substances which on the basis of their osmotic 
 and electrical behaviour may be classed either as electrolytes 
 or non -electrolytes. In the present chapter we are concerned 
 with a new type of substance which differs in many respects 
 both from typical electrolytes and non-electrolytes. The first 
 discoveries in this field we owe to Thomas Graham (1861) who 
 found that whilst certain substances diffuse rapidly in solution 
 and readily pass through animal and vegetable membranes, 
 other substances diffuse very slowly in solution and are unable 
 to pass through membranes. To the first class of substances, 
 which can readily be obtained in crystalline form, Graham gave 
 the name crystalloids, whilst the members of the other class, 
 which cannot as a rule be obtained in crystalline form, were 
 termed colloids. Most inorganic acids, bases and salts and 
 many organic compounds, such as acetic acid, cane sugar and 
 urea are crystalloids ; starch, gum, gelatine, caramel and pro- 
 teins in general belong to the group of colloids. The dif- 
 ferences in the rates of diffusion in aqueous solution of typical 
 crystalloids and colloids are illustrated in the following numbers, 
 valid for 10, which represent the relative times required for the 
 same amount of diffusion of different substance's. 
 
 1 For fuller details of the subjects treated of in this chapter see Philip, 
 Physical Chemistry : Its Bearing on Biology and Medicine (Arnold, 1910) ; 
 Freundlich, Kapillarchemie (Leipzig, 1909) ; Wolfgang Ostwald, Kolloid* 
 chemie (Dresden, 1911). 
 
314 OUTLINES OF PHYSICAL CHEMISTRY 
 
 
 Crystalloids. 
 
 Colloids. 
 
 Substance 
 Relative times of 
 equal diffusion . 
 
 HC1 
 
 i 
 
 NaCl 
 2-3 
 
 Cane Sugar 
 7 
 
 Albumen 
 49 
 
 Caramel 
 98 
 
 As under equivalent conditions the rate of diffusion is pro- 
 portional to the osmotic pressure of the solute, it follows that the 
 osmotic pressure of dissolved colloids is very small and therefore 
 that their molecular weights are very high. This view as to the 
 high molecular weight of colloids was held by Graham, who 
 suggested that the differences in behaviour of the two classes 
 of substance might be connected with the much greater size 
 of colloidal particles as compared with dissolved particles of 
 crystalloids. It may be said at once that later investigation has 
 fully confirmed the view as to the high molecular weight of 
 colloids in solution. 
 
 As was to be anticipated, the later developments of the 
 subject have led to modifications of Graham's views in some 
 essential respects. In the first place it has been shown that 
 colloids are not a special class of substances ; the colloidal 
 state is a condition into which practically all chemical substances 
 can be brought by suitable methods. For example, metals such 
 as silver and platinum, and even salts such as silver chloride 
 and sodium chloride, all of which are ordinarily met with in 
 crystalline form, can be obtained in colloidal solution. There 
 are, however, great differences in the readiness with which dif- 
 ferent substances can be brought into the colloidal state, and 
 some substances, such as starch and gelatine, are only met 
 with in solution in the colloidal form. 
 
 A further point, which has been established within the last 
 few years, is that colloidal solutions are not solutions in the 
 ordinary sense of the term. A true solution has been defined 
 as a homogeneous mixture, and therefore consists of a single 
 
COLLOIDAL SOLUTIONS. ADSORPTION 315 
 
 phase. A colloidal solution, on the other hand, such as col- 
 loidal platinum, can be shown to be heterogeneous ; that is, it 
 consists of two phases at least. As we shall see later, however, 
 all intermediate stages exist between colloidal solutions and 
 true solutions on the one hand, and between colloidal solutions 
 and ordinary suspensions on the other. Within the last few 
 years it has become usual to speak of the phase present in 
 separate particles as the disperse phase and the liquid in which 
 it is distributed as the dispersion medium. 
 
 The preparation of some typical colloidal solutions, and 
 the properties characteristic of the colloidal state, will now be 
 considered. 
 
 Preparation of Colloidal Solutions A suitable colloidal 
 solution for demonstration purposes is that of arsenious sul- 
 phide. It is prepared by passing hydrogen sulphide through a 
 cold aqueous solution of arsenious oxide, free from electrolytes, 
 sufficiently long to ensure conversion to arsenious sulphide. 
 Excess of hydrogen sulphide is then removed as far as possible 
 by a stream of hydrogen. The resulting solution, after filtra- 
 tion, is yellowish in colour and clear by transmitted light, but 
 appears turbid by reflected light. The degree of dispersion of 
 the sulphide (that is, the size of the particles) varies greatly 
 with the mode of preparing the solution. 
 
 Silicic acid is obtained in colloidal solution by slowly add- 
 ing a solution of sodium silicate to excess of hydrochloric acid 
 and then removing the sodium chloride and free hydrochloric 
 acid by dialysis. The simplest form of dialyser is a tube of 
 parchment paper into which the mixture is poured. The tube 
 is then suspended by its ends in water which is continually re- 
 newed, and in course of time the crystalloids are completely 
 removed by diffusion through the membrane, leaving a pure 
 colloidal solution of silicic acid. 
 
 Ferric hydroxide is obtained in colloidal solution (so called 
 " dialysed iron ") by dissolving the freshly precipitated hydrox- 
 ide in a dilute solution of ferric chloride, and removing the 
 
316 OUTLINES OF PHYSICAL CHEMISTRY 
 
 ferric chloride by dialysis. Other colloidal hydroxides may be 
 obtained by an analogous method. 
 
 The preparation of colloidal platinum according to Bredig 
 has already been described (p. 232). Other colloidal metals 
 (e.g. gold, silver, palladium) have been prepared by the same 
 method. Colloidal gold and other metals can also be pre- 
 pared by reducing the corresponding salts in aqueous solution. 
 
 Gelatine, gum and certain other substances form colloidal 
 systems on simple solution in water. 
 
 Osmotic Pressure and Molecular Weight of Colloids 
 It has already been mentioned that, corresponding with their 
 slow rate of diffusion, the osmotic pressure of colloidal solutions 
 is very small. This is fully confirmed by recent investigations, 
 but direct quantitative measurements by different observers 
 have not led to very concordant results. One of the principal 
 sources of error has been the difficulty of freeing colloids com- 
 pletely from electrolytes, which even in very small concentra- 
 tion have considerable osmotic pressure. This difficulty is to 
 some extent overcome by using another colloid, such as parch- 
 ment paper, as semi-permeable membrane ; one colloid, whilst 
 usually permeable for crystalloids, is impermeable to other col- 
 loids. Hence, as parchment paper and other membranes are 
 permeable for dissolved salts, the latter cannot set up a lasting 
 osmotic pressure, and a pressure which persists for a consider- 
 able time may be regarded as due to the colloid only. 
 
 As illustrating the nature of the results obtained, Lillie, 1 
 using a collodion membrane, found that a solution of egg albu- 
 men containing 12*5 grams per litre gave an osmotic pressure 
 of 20 mm. of mercury at room temperature. According to 
 Waymouth Reid the osmotic pressure of a i per cent, solution 
 of haemoglobin is about 4 mm. of mercury, but much higher 
 values, indicating a molecular weight of about 16,000, were ob- 
 tained by Roaf (1910). 
 
 1 Amer. Journal of Physiology, 1907, 30, 127. 
 
COLLOIDAL SOLUTIONS. ADSORPTION 317 
 
 Moore and Roaf l observed a pressure of about 70 mm. of 
 mercury for a 10 per cent, solution of gelatine, which remained 
 fairly steady fur two months. 
 
 The effect of electrolytes on the magnitude of the osmotic 
 pressure depends on the nature of the colloid. Neutral salts 
 in many cases lower the osmotic pressure of colloids, a result 
 probably due to partial coagulation of the colloidal particles. 
 Acids and bases often raise the osmotic pressure of colloids, 
 probably in consequence of chemical combination. 
 
 As the osmotic pressure of colloids is so small when measured 
 by the direct method it will readily be understood that the 
 freezing points and boiling points of colloidal solutions scarcely 
 differ from those of pure water. This is evident when we 
 consider that a solution of osmotic pressure 70 mms. (as 
 observed in the experiments just described) would have a 
 freezing-point less than yj^ below that of water. 
 
 Optical Properties of Colloidal Solutions The majority 
 of colloidal solutions appear homogeneous even under the 
 highest power of the microscope, but their heterogeneous char- 
 acter is established by means of the so-called "Tyndall phe- 
 nomenon ". When a ray of light enters a darkened room its 
 path is recognised by the scattering of the light at the surface 
 of dust particles. Similarly, the path of a beam passed through 
 a colloidal solution can be detected by the scattering of the 
 light at the surface of the ultramicroscopic particles, whereas 
 no indication is afforded of the path of a beam passed through 
 a solution which contains no particles exceeding a certain 
 magnitude. Light which has passed through a colloidal solu- 
 tion is partially or completely polarised. 
 
 The Tyndall phenomenon has recently been utilised in the 
 construction of the ultra microscope, by means of which our 
 knowledge of colloidal solutions has been greatly extended. 
 An intense beam of light (the arc light or, better, sunlight) is 
 
 1 Moore and Roaf, Biochem. Jonrn. 1906, 2, 34. 
 
3 i8 OUTLINES OF PHYSICAL CHEMISTRY 
 
 directed on a very thin layer of the colloid and the latter 
 examined by a microscope at right angles to the direction of 
 the beam, the entrance of light from other sources being pre- 
 vented. When a homogeneous liquid is used the field remains 
 quite dark, but when the liquid contains discrete particles their 
 presence is indicated by the appearance of colourless or (for 
 smaller particles) characteristically coloured luminous moving 
 points on a dark background. It must be emphasised that 
 the ultramicroscope does not render the particles themselves 
 visible, but only shows the light reflected from them, so that 
 such observations afford no information as to the shape, colour, 
 etc., of the particles. 
 
 The average size of the particles in a colloidal solution can 
 be estimated indirectly by counting the number in a given 
 volume and determining the total amount of substance by 
 analysis. In this way it has been shown that the particles vary 
 greatly in magnitude, depending on the nature and mode of 
 preparation of the colloidal solution, from such as are visible in 
 the ordinary microscope to those not resolvable even by the 
 ultramicroscope. Particles visible in the ordinary microscope 
 (diameter exceeding 250 /A/A, where /x = 0*001 mm. and /A/A = 
 o-oooooi mm.) are termed by Zsigmondy microns, those detected 
 only by the ultramicroscope (diameter 6-250 /A/A) are termed 
 submicronS) and those of diameter less than 6 /A/A amicrons. 
 
 For comparative purposes it may be mentioned that the wave- 
 length of sodium light is 589 /A/A, It has been calculated l that 
 the diameter of an ether molecule is about o'6 x io~ 6 mm. 
 = 0-6 /A/A, so that the smallest particle which can be detected 
 by the ultramicroscope has a diameter only ten times greater 
 than that of an average chemical molecule. 
 
 Brownian Movement When a colloidal solution contain- 
 ing microns (e.g. mercuric sulphide, suspension of gum mastic) 
 is examined under the microscope, the particles are seen to be 
 
 1 Perrin, loc. /., p. 50. 
 
COLLOIDAL SOLUTIONS. ADSORPTION 319 
 
 performing continuous irregular movements (R. Brown, 1827) 
 "They go and come, stop, start again, mount, descend, remount 
 again, without in the least tending towards immobility " (Perrin). 
 Observations with the" ultra-microscope show that the move- 
 ments are the more brisk the smaller the particles and the less 
 the viscosity of the liquid, and they become more rapid with 
 rise of temperature. The phenomenon persists for years; it is 
 not due to any external cause, such as alterations of temperature 
 or of illumination, and it is now generally agreed that it is a 
 consequence of " the incessant movements of the molecules of 
 the liquid which, striking unceasingly the observed particles, 
 drive them about irregularly through the fluid, except in the 
 case where these impacts exactly counterbalance one another " 
 (Perrin, loc. "/.). It has been shown within the last few years, 
 . more particularly by Perrin, that the rates of movement of the 
 particles are in entire accord with those deducted on the basis 
 of the molecular -kinetic theory, which amounts to an experi- 
 mental proof of the atomic constitution of matter and of the 
 kinetic nature of heat J (cf. p. 32). 
 
 Electrical Properties of Colloids When two plates are 
 placed at some distance apart in a colloidal solution and con- 
 nected with a source of E.M.F. it will be found as a rule that the 
 particles move slowly towards the anode or cathode ; in other 
 words, they behave as if they are electrically charged. The 
 simplest method of making the experiment is to place the 
 colloidal solution in the lower part of a U-tube, which is filled 
 up on both sides with distilled water in which the electrodes 
 are placed. The latter are then connected with the terminals 
 of the lighting circuit (100-200 volts) and the speed of the 
 moving boundary observed directly. The results show that 
 particles of all kinds move at the rate of 10-40 x io~ 5 . cm 
 per second for a potential gradient of i volt per cm. As we have 
 seen, this is also the order of the migration velocity of the ions 
 
 'Ostwald, Giundriss der Allg. Chemie., p. iv, 545. 
 
320 OUTLINES OF PHYSICAL CHEMISTRY 
 
 (p. 253) and we have therefore the remarkable fact that particles 
 of all sizes microns, submicrons, amicrons, ions move with 
 approximately the same speed in the electric field. 
 
 In the case of the noble metals (gold, platinum, silver, etc.) and 
 the sulphides (arsenic and antimony trisulphides) the particles 
 are negatively charged and move towards the anode, whilst 
 hydroxides (ferric and aluminium hydroxides, etc.) and haemo- 
 globin are positively charged. The charge on some colloids 
 can, however, be altered in sign by certain additions to the 
 medium. Thus Hardy has shown that when acid is added to 
 egg albumen it migrates to the cathode, whilst in alkaline solu- 
 tion it moves towards the anode. The condition in which the 
 colloid is uncharged is known as the isoelectric point> which in 
 the case of egg albumen occurs in approximately neutral solu- 
 tion. 
 
 Precipitation of Colloids by Electrolytes It is a re- 
 markable fact that many colloidal solutions are readily coagu- 
 lated by the addition of electrolytes. When, for example, a 
 few drops of barium chloride solution are added to a colloidal 
 solution of arsenic sulphide the solution becomes turbid, and in 
 a few minutes the sulphide has completely separated in flocks. 
 The process can be followed under the ultramicroscope, and 
 is seen to consist in a gradual aggregation of the particles 
 (amicrons to submicrons, then to microns and finally to large 
 flocks) the Brownian movement becoming slower and slower 
 and finally ceasing. 
 
 The efficiency of different electrolytes in the coagulation of 
 arsenic sulphide depends mainly on the valency of the anion 
 and is largely independent of its nature. The molar concen- 
 trations of A1C1 3 , BaCl 2 , and KC1 required to produce same 
 degree of coagulation under conditions otherwise equivalent 
 areas follows: i: 7*4: 532 (Freundlich). With solutions of 
 ferric hydroxide, on the other hand, the coagulating power of 
 electrolytes is practically independent of the valency of the 
 cation, and is determined chiefly by the valency of the anion. 
 
COLLOIDAL SOLUTIONS. ADSORPTION 321 
 
 Thus the molar concentrations of K 2 SO 4 and KC1 which pro- 
 duced the same effect are in the ratio i : 45. 
 
 When it is remembered that the particles of arsenious sul- 
 phide are negatively charged and those of ferric hydroxide 
 positively charged the bearing of these results at once becomes 
 evident. The ion which brings about the coagulation of a 
 colloidal solution is the one carrying a charge of opposite sign to 
 that on the colloidal particles (Hardy). Further investigation 
 has shown that this rule can be extended to the reciprocal 
 action of colloidal particles, inasmuch as two colloidal solu- 
 tions containing particles of contrary sign coagulate on mixing 
 (e.g. colloidal platinum and ferric hydroxide) whilst colloids 
 of the same sign are practically without influence on each 
 other. 
 
 As regards the nature of the coagulation, it has been shown 
 that in certain cases at least the electrolyte is partially decom- 
 posed, the precipitating ion being carried down along with the 
 precipitate and the inactive ion left in solution in combination 
 with another ion. In order to understand this phenomenon it 
 is necessary to consider rather more fully the question of the 
 stability of a colloidal solution. It has been shown by Hardy, 
 Burton and others that certain colloids reach their point of maxi- 
 mum instability (that is, coagulate most readily) when the 
 charge on the particles (as indicated by their behaviour 
 under the influence of a potential gradient) reaches a minimum. 
 Taking as illustration the coagulation of arsenious sulphide by 
 potassium chloride solution we may assume that some of the 
 salt is taken up by the colloidal particles, the negative charges 
 on the latter are neutralized by the K- ions, with the result that 
 the particles become unstable, aggregate and fall out of solution 
 carrying the K- ions along with them (presumably as a salt). 
 The Cl* ions are left in the solution along with an equivalent 
 of H- ions derived from the hydrogen sulphide always associated 
 with the colloidal sulphide. This is the so-called "adsorp- 
 tion " theory of coagulation. Other theories of the pheno- 
 
322 OUTLINES OF PHYSICAL CHEMISTRY 
 
 menon, notably Billiters "condensation" theory, have also 
 been proposed, but cannot be dealt with here. 1 
 
 Suspensions, Suspensoids and Emulsoids All the pro- 
 perties discussed in the previous sections (with the possible 
 exception of the action of electrolytes on the stability) are 
 characteristic of colloidal solutions in general, as well as of 
 suspensions of particles easily visible under the microscope. 
 As typical ' suspensions " may be mentioned clay, finely divided 
 charcoal or gum mastic stirred up with water. They consist 
 of a practically insoluble solid phase, distributed in a liquid, 
 usually water. The particles settle to the bottom of the vessel 
 more or less rapidly, depending on their magnitude, but the 
 system " clears " much more rapidly when electrolytes are added 
 (cf. previous section). From the suspensions we pass through 
 a series of intermediate stages to the suspensoids or suspension 
 colloids, the heterogeneous character of which is only recognised 
 by Tyndall's phenomenon or by the ultramicroscope. Like 
 the suspensions, they consist of a solid phase distributed in a 
 liquid, generally water. " Colloidal solutions " are divided into 
 two fairly well-defined classes, the suspension colloids or sus- 
 pensoids just mentioned and the emulsion colloids or emulsoids. 
 The suspensoids are scarcely more viscous than water, do not 
 gelatinise and are readily precipitated by electrolytes. The 
 emulsoids are viscous, become gelatinous under certain condi- 
 tions and are not readily precipitated by electrolytes. The 
 colloidal metals, sulphides and hydroxides are suspensoids ; 
 silicic acid, gelatine, gum, mucilage of starch and proteins in 
 general are emulsoids. Emulsoids, like suspensoids, are two- 
 phase systems, but consist of two liquid phases, one a honey- 
 comb-like structure, rich in coJloid, in the meshes of which 
 the other phase, composed of a dilute solution of the colloid, 
 is distributed. Thus an aqueous solution of gelatine is 
 made up of two phases, one rich in water and containing a 
 
 1 Wo. Ostwald, Kolloidchemie, p. 499. 
 
COLLOIDAL SOLUTIONS. ADSORPTION 323 
 
 little gelatine in true solution, the other rich in gelatine, but 
 containing a little water (Hardy). 
 
 It is a familiar fact that an emulsoid such as silicic acid can 
 be obtained as a clear, apparently homogeneous solution 
 (p. 315) which on long standing, more rapidly on boiling or on 
 treatment with electrolytes, changes to a semi-solid amorphous 
 mass. The clear solution is termed a so/, the gelatinous mass 
 a gel. The term sol is also applied to suspensoids. When the 
 electrolyte is removed by washing and the gel is again treated 
 with water certain emulsoids, such as the proteins, return to the 
 sol modification (more readily on warming) and are therefore 
 termed reversible colloids. Suspensoids in general and certain 
 emulsoids, such as silicic acid, do not return to the soluble 
 form under these conditions and are therefore known as irre- 
 versible colloids. 
 
 The coagulation of emulsoids by electrolytes seems to be 
 entirely different to the action on suspensoids, but is by no 
 means well understood. Whether the electrical character of 
 the particles and of the electrolyte plays any part in the process 
 is doubtful ; in fact silicic acid sol seems to be most stable in 
 the electrically neutral condition. The addition of neutral 
 salts in considerable concentration causes the separation of the 
 solid phase, but the ratio of the activities of different electrolytes 
 is quite different from that observed for suspension colloids 
 and resembles the "salting out " observed, for instance, in the 
 effect on the solubility of gases in water (p. 84). 
 
 Filtration of Colloidal Solutions It has already been 
 pointed out that systems of all degrees of dispersion are met 
 with, from those containing large particles easily visible under 
 the microscope to molecular dispersed systems, which we term 
 true solutions. It is evident, however, that true solutions are 
 only apparently homogenous ; the solute particles are so minute 
 as to escape our present methods of detecting heterogeneity. 
 As already explained, the size of colloidal particles can be 
 roughly estimated by counting the number in a given volume 
 
324 OUTLINES OF PHYSICAL CHEMISTRY 
 
 of solution containing a known weight of the disperse phase. 
 Another method which has recently come into use for this pur- 
 pose is to use filters with pores of different sizes. Bechhold, 1 
 who has done much work on this subject, uses filter- papers 
 impregnated with gelatine solutions of different concentrations, 
 and finds that a filter with 2 per cent of gelatine retains all 
 particles of diameter less than 44/^, one containing 4-4*5 P er 
 cent, is required to retain the much smaller particles of serum- 
 albumen, the average molecular weight of which is about 10,000 
 (3,000-15,000). The permeability of such filters is of course 
 influenced by the pressure under which filtration is carried out. 
 A very early form of the "ultra-filter," introduced by Martin, 
 consists of an ordinary porcelain filter impregnated with gela- 
 tine. 
 
 Adsorption. General It is a familiar fact that when 
 water containing a colouring matter such as caramel or litmus 
 is shaken up with finely divided charcoal the latter on settling 
 carries down the colouring matter with it, leaving the water 
 practically colourless. Further investigation shows that other 
 substances, including electrolytes and non-electrolytes as well 
 as colloids, are largely taken up by charcoal from aqueous solu- 
 tion, and that other finely divided substances have the same 
 property. Charcoal has also the power of taking up gases, es- 
 pecially those which are easily liquefied, such as ammonia and 
 sulphur dioxide. 
 
 The nature of this phenomenon will be more readily under- 
 stood in the light of some quantitative observations, and for 
 this purpose the results of a series of experiments carried out 
 by Schmidt 2 on the taking up of acetic acid from aqueous 
 solution by charcoal are quoted. Animal charcoal (in quan- 
 tities of 5 grams) was shaken up with aqueous solutions of acetic 
 acid (100 ccs. in each case) of different concentrations and the 
 
 1 Zeitsch. Chem. Ind. Kolloide, 1907, 2, 3. 
 
 2 Zeitsch. physikal Chem., 1910, 74, 689. 
 
COLLOIDAL SOLUTIONS. ADSORPTION 325 
 
 amount of acid remaining in the water phase determined by 
 titration. In the accompanying table A c represents the amount 
 of acetic acid taken up by the charcoal and A w the amount left 
 in solution at equilibrium. 
 
 DISTRIBUTION OF ACETIC ACID BETWEEN WATER AND 
 CHARCOAL. 
 
 A c 0*93 1*15 1*248 i '43 i '62 
 
 A w 0*0365 0*084 0*13 0*206 '35o 
 
 Cc/Cw 205 208 180 203 197 
 
 As the volume of the solution and the amount of charcoal 
 are kept constant, the amounts given in the table are propor- 
 tional to the respective concentrations, C c and C w , in the two 
 phases. The figures show (i) that in very dilute solution the acid 
 is almost completely taken up by charcoal ; (2) that the con- 
 centration in the charcoal increases much less rapidly than the 
 concentration in the aqueous phase. That we are dealing with 
 true equilibria is shown by the fact that the same results are 
 obtained from either side (starting from concentrated or from 
 dilute solutions of the acid). 
 
 The question now arises as to how these observations are to 
 be interpreted. In the first instance we will consider whether 
 the process is a physical or a chemical one, and if the former, 
 whether it is mainly a surface condensation or whether solid 
 solutions are formed. 
 
 It appears highly improbable for several reasons that the 
 phenomena are chemical in nature. In the first place the 
 most various substances, including argon and the other in- 
 active gases, which do not, as far as is known, enter into 
 chemical combination, are taken up by charcoal. Further, a 
 definite chemical compound is constant in composition and, 
 if undissociated, its composition is independent of the concentra- 
 tion in the other phase, whereas, as the table shows, the composi- 
 tion of the carbon-acetic acid system varies continuously 
 
326 OUTLINES OF PHYSICAL CHEMISTRY 
 
 within wide limits. At first sight it would appear possible to 
 explain the results as being due to the formation of a partially 
 dissociated solid compound in equilibrium with its products of 
 dissociation, but it can easily be shown that this assumption also 
 is incompatible with the facts. Applying the law of mass action 
 to such an equilibrium (in the liquid phase) we have (cf. p. 173) 
 
 [Absorbent] n i [Substance taken up] ^/[Compound] w 8= Const. 
 
 where the square brackets represent concentrations, and i, ;zaand 
 ^3 represent the number of molecules of the absorbent (charcoal), 
 the substance taken up (acetic acid) and the compound re- 
 spectively taking part in the equilibrium. Further, since the 
 active masses of the charcoal and the compound are constant 
 
 [Substance taken up] = Constant (in liquid phase) 
 
 that is, the concentration of the acetic acid in the solution 
 must be constant as long as both solid phases are present. 
 As a matter of fact, the concentration of acetic acid in the 
 solution increases continuously with the total concentration 
 (compare table), so that no second solid phase (no chemical 
 compound) can be present. 
 
 The formation of a solid dissociating compound from a solid 
 phase and a substance in solution has been investigated by 
 Walker and Appleyard in the case of diphenylamine and picric 
 acid, which combine to form the slightly soluble brown com- 
 pound diphenylamine picrate. 1 Until the concentration of 
 the acid in the aqueous layer reached 0-06 mols per litre the 
 solid diphenylamine (which is practically insoluble in water) 
 remained colourless, on further addition of picric acid the brown 
 diphenylamine picrate began to form, and finally practically all 
 the diphenylamine was converted into picrate, the concentration 
 of the picric acid in the solution remaining all the time 
 practically constant at 0*06 mols per litre. It is evident that 
 
 1 Walker and Appleyard, 1896, 69, 1334. 
 
COLLOIDAL SOLUTIONS. ADSORPTION 327 
 
 the system exactly corresponds with the calcium carbonate 
 calcium oxide carbon dioxide equilibrium already considered 
 (p. 174), except that in the latter case the substance of variable 
 concentration (the carbon dioxide) is in a gaseous and not in 
 a liquid phase. 
 
 It remains to consider whether the phenomena in 
 question, such as the taking up of acetic acid by charcoal, are 
 due to surface condensation or whether solid solutions are 
 formed. It would seem possible to decide this question at 
 once by observing the rate of establishment of equilibrium, 
 since surface condensation must be a very rapid process, and 
 the formation of a solid solution, whereby (in the case under 
 consideration) one substance has to diffuse into the interior of 
 the other, must be very slow. As a matter of fact the estab- 
 lishment of equilibrium in many cases (but not in all cases, see 
 below) is practically instantaneous, which lends strong support 
 to the surface condensation theory. The strongest evidence in 
 favour of the latter theory, however, is based on a considera- 
 tion of the ratio of the distribution of the substance between 
 the two phases. It has been shown (p. 178) that when a 
 substance distributes itself between two phases the ratio of the 
 distribution is independent of the concentration provided the 
 molecular weight of the solute is the same in both solvents, but 
 if the molecular weight in the solvent A is n times that in the 
 solvent B then /s/C A /C B is constant, which may be written more 
 conveniently thus : C^*/C B = Constant. Now the table on 
 p. 325 shows that for the distribution of acetic acid between 
 water and charcoal the formula holds approximately 
 
 C C 4 /C W = Constant 
 
 where C c and C w represent the concentrations in charcoal 
 and in water respectively. Comparing this with the distri- 
 bution formula, C*fQ^ = Constant, we find that 
 
 i/x = 4 or x = 1/4; 
 that is, if charcoal and water may be regarded as two solvents 
 
328 OUTLINES OF PHYSICAL CHEMISTRY 
 
 between which the acetic acid is distributed then the molecular 
 weight of the acid in charcoal is 1/4 that in water. Now it was 
 shown by Raoult that acetic acid exists as single molecules 
 in aqueous solution, so that its molecular weight in charcoal, 
 deduced on the assumption that it is present in solid 
 solution, is an impossible one. Analogous results are obtained 
 with other solutes and other absorbing agents and it follows at 
 once that the " solid solution " explanation of the phenomena 
 under consideration is definitely disproved. There is evidence, 
 however, that in some cases solid solution may play a subsidi- 
 ary part in the phenomena. Thus Davis 1 found that when 
 iodine is shaken up with charcoal a very rapid action is 
 followed by a slow action, the latter being presumably due 
 to the slow diffusion of the iodine into the interior of the 
 charcoal. Similarly McBain 2 has shown that when hydrogen 
 which has been in contact with charcoal for a long time is 
 pumped out the greater part of it (that condensed on the 
 surface) can be drawn off immediately, but a small residue 
 (presumably present in solid solution) can only be removed 
 very slowly. 
 
 It has now been established that the phenomenon under 
 consideration is physical in nature and mainly at least due to 
 surface condensation. In order to distinguish it from such 
 a process as the absorption of gases in liquids, an example of 
 true solution, the process is termed Adsorption, and the 
 substance which is condensed on the surface of the solid phase 
 is said to be adsorbed. 
 
 Adsorption of Gases. Adsorption Formulae. So far 
 we have been concerned mainly with the adsorption of sub- 
 stances from solution. It is now necessary to deal a little more 
 in detail with the fact already mentioned, that porous substances 
 have a considerable adsorptive power for gases, and that those 
 gases which are most easily liquefied are most largely adsorbed. 
 The nature of the results is well shown by the recent accurate 
 
 1 Trans. Chem. Soc., 1907, 91, 1666. 2 Phil. Mag., 1909, 18, 816. 
 
COLLOIDAL SOLUTIONS. ADSORPTION 329 
 
 work of Homfray l and of Titoff 2 on the adsorption of gases by 
 charcoal. The amount of gas adsorbed is proportional to the 
 adsorbing surface and is the greatei the lower the temperature 
 and the higher the pressure. Titoff found that the adsorption 
 of hydrogen follows Henry's law, so that the formula 
 
 C A /C B = Constant 
 
 applies, where C A represents the concentration in the solid 
 phase, C B that in the gas phase. The other gases at low tem- 
 peratures do not follow Henry's law, but the results are repre- 
 sented fairly satisfactorily by a formula of the type C^/*/C B = 
 Constant. The adsorptive power of charcoal for traces of gas, 
 especially at low temperatures, has been used by Dewar to 
 obtain the highest vacua yet reached; the pressures were 
 too low to be capable of measurement. 
 
 It has been shown above that a formula of the type 
 C A '*/C B = Constant an exponential formula affords a 
 fairly satisfactory representation of the adsorption both of 
 gases and dissolved substances. In the literature it is met 
 with in a slightly different form, which will now be given. 
 Instead of writing C A ^/C B we may put CA/ C* = Constant. 
 When for C A we put x/m, where n represents the amount of sub- 
 stance adsorbed by m grams of adsorbent, we obtain, putting 
 p for C B and i/n for x, the formula 
 
 x/m = ftp l/n 
 
 where /3 and n are constant at constant temperature. When 
 i/n = i the adsorption follows Henry's law, but in almost 
 every instance i/n is considerably less than i. This expresses 
 the important fact that adsorption is relatively greatest from 
 dilute solution and falls off rapidly with the concentration (p-325). 
 
 The Cause of Adsorption Adsorption of gases and liquids 
 occurs more or less at all solid surfaces, a well-known case in 
 point being the adsorption of moisture by glass surfaces, but 
 
 1 Proc. Roy Soc., 1910, 84, A, 99. 
 
 2 Zeitsch. physikal Chem., 1910, 74, 641. 
 
330 OUTLINES OF PHYSICAL CHEMISTRY 
 
 it is only when the surface is very large in comparison with the 
 weight of the solid as in the case of porous and finely divided 
 substances that it can readily be measured. We have now to 
 consider why the concentration in the surface layers differs in 
 many cases so greatly from that in the main bulk of the 
 liquid or gas phase. It seems probable at the outset that this 
 must be connected with molecular attraction at the boundary 
 of the phases, in other words with the surface tension (p. 129) 
 and the connection between surface tension and adsorption has 
 been deduced theoretically by Willard Gibbs and by J. J. 
 Thomson. From the general standpoint we must assume that 
 not only increased concentration, but in certain systems a 
 lowering of concentration at the surface, as compared with that 
 in the main bulk of liquid, may occur. Calling an increase of 
 concentration positive adsorption and a diminution negative 
 adsorption, the rule may be expressed as follows : l A dissolved 
 substance is positively adsorbed when it lowers the surface tension^ 
 negatively adsorbed when it raises the surface tension. The first 
 case is met with in most solutions of organic compounds ; the 
 second in solutions of highly ionised inorganic salts. 
 
 Further Illustrations of Adsorption One very impor- 
 tant process in which adsorption plays a prominent part is the 
 dyeing of fibres such as wool and silk. Whether dyeing is 
 purely an adsorption phenomenon or whether chemical action 
 also plays a part has given rise to a great deal of discussion, and 
 is by no means finally settled. It has recently been shown 
 that the distribution of crystal violet, new magenta and patent 
 blue between wool, silk and cotton on the one hand and water 
 on the other is satisfactorily represented by the adsorption 
 formula, and the value of the exponent */ is approximately 
 the same as when charcoal is used as absorbent, a result which 
 supports the adsorption theory. On the other hand, Knecht 
 showed some years ago that when the basic dye crystal violet 
 
 1 Cf. Freundlich, Kapillarchemie., p. 52. 
 
COLLOIDAL SOLUTIONS. ADSORPTION 331 
 
 (the hydrochloride of an organic base) is shaken up with wool 
 or silk the dye is decomposed, the cation combining with the 
 fibre and the anion (in this case Cl') remaining in the solution. 
 This result was first described as a case of double decomposi- 
 tion between the dye and the fibre, the dye combining with an 
 organic acid in the fibre to form a salt, and ammonia originally 
 associated with the fibre combining with the chlorine to form 
 ammonium chloride. Freundlich and Neumann l have shown, 
 however, that in certain cases at least the chlorine is not left 
 in the solution as a salt, but in the form of hydrochloric acid. 
 The exact form in which the adsorbed dye occurs on the adsorb- 
 ent does not seem to have been properly established the 
 colour appears to indicate that it is present as a salt and not 
 as the free base. 
 
 The process just described would at first sight appear to be 
 an ordinary chemical change, but further investigation shows 
 that charcoal and even glass pellets split up dyes in an exactly 
 analogous way, the cation being adsorbed and the anion re- 
 maining in solution. It can scarcely be supposed that the 
 charcoal or the glass enter into chemical action with the dyes. 
 Phenomena of an exactly similar nature have already been met 
 with in connection with the precipitation of colloids by electro- 
 lytes (p. 319), and it has been shown that they are connected 
 with the electrical character of the colloidal particles, that ion 
 being most largely adsorbed which carries a charge of opposite 
 sign to that on the colloid. The splitting of basic dyes de- 
 scribed in the present section might be accounted for on similar 
 lines, as also the well-known fact that an "acid" fibre adsorbs 
 more particularly basic dyes and a "basic'' fibre "acid" dyes. 
 The above is a brief outline of the adsorption theory of dyeing, 
 but the process in any particular case is doubtless complicated 
 by other factors, and at present is far from being understood. 
 
 It has been suggested by Bayliss and others that adsorption 
 
 1 Zeitsch. physikal Chem., igog, 67, 538. 
 
332 OUTLINES OF PHYSICAL CHEMISTRY 
 
 plays an important part in enzyme reactions; the substance 
 acted on is first adsorbed by the colloidal enzyme particles and 
 chemical change follows. The interesting fact that colloids 
 such as gelatine increased the stability of suspension colloids 
 such as silver bromicje or colloidal gold towards electrolytes, 
 may also be accounted for on the basis of adsorption. In the 
 case under consideration it is assumed that the gelatine is ad- 
 sorbed as a thin film on the surface of the particles, so that the 
 latter do not come directly in contact with the electrolyte. It 
 has quite recently been shown that certain dyes, more parti- 
 cularly erythrosine, also exert a protective action on colloidal 
 silver bromide. Substances acting in this way are termed 
 " protective " colloids. 
 
CHAPTER XIII 
 THEORIES OF SOLUTION 
 
 General The nature of solutions, 1 more particularly as 
 regards the connection between their properties and those of 
 the components, has long been one of the most important 
 problems of chemistry. It was early ** recognized that the 
 properties of a solution are very seldom indeed the mean of 
 the properties of the components, as must necessarily be the 
 case if solvent and solute exert no mutual influence. Thus 
 we know that when two liquids are mixed either expansion or 
 contraction may occur, the boiling-point of a mixture may be 
 higher or lower than those of either of its components (p. 88), 
 and a mixture of two liquids may have a high conductivity, 
 although the components in the pure condition are practically 
 non-conductors (p. 259). 
 
 The most obvious way of accounting for observations of this 
 nature is to assume that they are connected with the formation 
 of chemical compounds between the two components of the 
 solution. As a matter of fact, explanations of the observed 
 phenomena on these lines were formerly in great favour. As 
 water was the substance most largely used as a component of 
 solutions (as solvent), the explanation of the properties of 
 aqueous solutions on the basis of formation of chemical com- 
 pounds between water and the solute was termed the hydrate 
 theory of solution. This theory appeared the more plausible 
 
 1 For simplicity, only mixtures of two components will be considered in 
 this chapter. 
 
 333 
 
334 OUTLINES OF PHYSICAL CHEMISTRY 
 
 as a very large number of hydrates compounds of substances, 
 more particularly salts, with water are known in the solid 
 state, thus showing that there is undoubtedly considerable 
 chemical affinity between certain solutes and water. 
 
 In spite of the plausible nature of the hydrate theory, how- 
 ever, it did not prove very successful in representing the 
 properties of aqueous solutions, and some . facts were soon 
 discovered in apparent contradiction with it. Thus, as already 
 mentioned, Roscoe showed that the composition of the mixture 
 of hydrochloric acid and water with minimum vapour pressure, 
 alters with the pressure, and therefore could not be connected 
 with the formation of a definite chemical compound of acid 
 and water, as had previously been assumed (p. 90). 
 
 The development of the electrolytic dissociation theory, 
 which has been discussed in the previous chapters, led to a 
 considerable change of view with regard to the influence of 
 the solvent on the properties of aqueous solutions. The pro- 
 perties of the solvent were to some extent relegated to the 
 background, 1 and it was looked upon simply as the medium 
 in which the molecules and the ions of the solvent the really 
 active things moved about freely. The fact that such great 
 advances in knowledge have been made by working along those 
 lines naturally goes far to justify the method of procedure. 
 
 Within the last few years, however, mainly as a result of the 
 investigation of solutions in solvents other than water, it has 
 come to be recognized that the solvent may play a more direct 
 part in determining the properties of dilute solutions than some 
 chemists were formerly inclined to suppose. Although the main 
 properties of aqueous solutions can be accounted for without 
 express consideration of affinity between solvent and solute, it 
 appears probable that the latter effect must be taken into con- 
 
 x The properties of the solvent are, of course, all-important in deter- 
 mining whether a substance becomes ionised or not. But it was pot found 
 necessary to take the question of affinity into account directly, and the 
 equations representing ionic equilibria did not contain any term referring 
 directly to the solvent. 
 
THEORIES OF SOLUTION 335 
 
 sideration in order to account for certain secondary phenomena 
 (and possibly also in connection with ionisation) (p. 344). 
 
 In the previous chapters the evidence in favour of the 
 electrolytic dissociation theory could not be dealt with as a 
 whole, owing to the fact that it belongs to different branches 
 of the subject. In the present chapter a short summary of the 
 more important lines of evidence bearing on the theory will be 
 given, and then a brief account of the investigation of solutions 
 in solvents other than water. Finally, after dealing with the 
 older hydrate theory of solution, the possible mechanism of 
 electrolytic dissociation will be considered. 
 
 Evidence in Favour of the Electrolytic Dissociation 
 Theory The evidence in favour of the electrolytic dissociation 
 theory is partly electrical and partly non-electrical. The non- 
 electrical evidence goes to show that there are more particles 
 in dilute solutions of salts, strong acids and bases, than can be 
 accounted for on the basis of their ordinary chemical formulae, 
 and that in dilute solution the positive and negative parts of the 
 molecule behave more or less independently. The electrical 
 evidence goes to show that the particles which result from the 
 splitting up of simple salt molecules are associated with electric 
 charges, either positive or negative. 
 
 The main points are as follows : 
 
 (a) If Avogadro's hypothesis applies to dilute solutions, 
 gram-molecular (molar) quantities of different substances, dis- 
 solved in equal volumes of the same solvent, must exert the 
 same osmotic pressure. As a matter of experiment, salts, strong 
 acids and bases exert an osmotic pressure greater than that due 
 to equivalent quantities of organic substances (p. 124). The 
 electrolytic dissociation theory accounts for this on the same 
 lines as the accepted explanation for the abnormally high 
 pressure exerted by ammonium chloride ; it postulates that 
 there are actually more particles present than that calculated 
 according to the ordinary molecular formula. 
 
 (b] Many of the properties of dilute salt solutions are additive, 
 
336 OUTLINES OF PHYSICAL CHEMISTRY 
 
 that is, they can be represented as the sum of two independent 
 factors, one due to the positive, the other to the negative part 
 of the molecule. This is true of the density, the heat of 
 formation of salts (p. 148), the velocity of the ions (p. 251), 
 the viscosity, and more particularly of the ordinary chemical 
 reactions for the "base" and "acid" as used in analysis 
 (p. 306). 
 
 A very striking illustration of the independence of the pro- 
 perties of one of the ions in dilute solution on the nature of 
 the other is the colour of certain salt solutions, investigated 
 by Ostwald. He examined the solutions of a large number oi 
 metallic permanganates, and found that all had exactly the same 
 absorption spectra. This is exactly what is to be expected 
 according to the electrolytic dissociation theory, the effect being 
 exerted by the permanganate ion. Similarly, salts of rosaniline 
 with a large number of acids in very dilute solution gave 
 identical absorption spectra (due to the rosaniline cation) but 
 rosaniline itself, which is very slightly ionised, gave a quite 
 different spectrum. 
 
 Too much stress should not be laid on this criterion, how- 
 ever, as certain properties, e.g., molecular volumes of organic 
 compounds (p. 61) and the heat of combustion of hydro- 
 carbons (p. 148), are more or less additive, although nothing 
 in the nature of ionisation is here assumed. 
 
 (c) The magnitudes of the degree of dissociation, calculated 
 on two entirely independent assumptions (i) that the con- 
 ductivity of solutions is due to the ions alone, and not to the 
 non-ionised molecules or to the solvent; (2) that the abnormal 
 osmotic pressures shown by aqueous solutions of electrolytes 
 are due to the presence of more than the calculated number 
 of particles owing to ionisation show excellent agreement 
 (p. 263). 
 
 (d) The heat of neutralization of molar solutions of all 
 strong monoacidic bases by strong monobasic acids is 13,700 
 calories, in excellent agreement with the value for the reaction 
 H- + OH' = H 2 O, calculated by van't Hoffs formula from 
 
THEORIES OF SOLUTION 337 
 
 Kohlrausch's measurements of the change of conductivity of 
 pure water with the temperature (p. 295). 
 
 (e) The results obtained by four entirely independent 
 methods for the degree of ionisation of water are in striking 
 agreement, in spite of the fact that the assumed ionisation is 
 very minute (p. 294). 
 
 (/) The formula for the variation of electrical conductivity 
 with dilution, obtained by application of the law of mass action 
 to the assumed equilibrium between ions and non-ionised mole- 
 cules in solution, represents the experimental results in the case 
 of weak electrolytes with the highest accuracy (p. 267). 
 
 (g) As shown in the next chapter, our present views as to 
 the origin of differences of potential at the junction of two 
 solutions, or at the junction of a metal and a solution of one 
 of its salts, are based on the osmotic and electrolytic dissocia- 
 tion theories, and the good agreement between observed and 
 calculated values goes far to justify the assumptions on which 
 the formulae are based. 
 
 Many other illustrations of the utility of the electrolytic 
 dissociation theory are mentioned throughout the book. 
 
 Ionisation in Solvents other than Water 1 In ac- 
 cordance with the mode in which the subject has developed, 
 we have up to the present been mainly concerned with 
 aqueous solutions, and the justification for this order of 
 treatment is that the relationships in aqueous solution are 
 often very simple in character, as shown in detail in the last 
 chapter. The importance of a theory would, however, be 
 much less if it only applied to aqueous solutions, and it is 
 therefore satisfactory that in recent years a very large number 
 of liquids, both organic and inorganic, have been employed as 
 solvents. 
 
 Although the progress so far made in this branch of know- 
 ledge is not great, the available data appear to ' show that 
 
 1 Carrara, Ahrens' Sammlung, 1908, 12, 404. 
 22 
 
OUTLINES OF PHYSICAL CHEMISTRY 
 
 the rules which have been found to hold for aqueous solutions 
 also apply to non-aqueous solutions 1 
 
 Some solvents, such as ethyl and methyl alcohol, acetic acid, 
 formic acid, hydrocyanic acid and liquefied ammonia, form solu- 
 tions of fairly high conductivity with salts and other substances ; 
 these are termed dissociating solvents (p. 123). Solutions in 
 certain other solvents, such as benzene, chloroform and ether, 
 are practically non-conductors, and the solutes are often present 
 in such solutions in the form of complex molecules. These 
 solvents are therefore often termed associating solvents, but it 
 is not certain whether they actually favour association or 
 polymerization of the solute, or have only a slight effect in 
 simplifying the naturally polymerized solute. 
 
 It is natural to inquire whether there is any connection 
 between the ionising power of a solvent and any of its other 
 properties. It has been found that as a general rule those 
 solvents with the greatest dissociating power have high 
 dielectric constants (p. 225) (J. J. Thomson, Nernst, 1893). 
 This observation is easily understood when it is remembered 
 that the attraction between contrary electric charges is inversely 
 proportional to the dielectric constant of the medium ; it is 
 evident that the existence of the ions in a free condition must 
 be favoured by diminishing the attraction between the contrary 
 charges. The dielectric constants of a few important solvents 
 (liquids and liquefied gases) at room temperature are given in 
 the table. 
 
 Solvent. 
 
 D.C. 
 
 Solvent. 
 
 D.C. 
 
 Hydrocyanic acid 
 
 95 
 
 Acetone 
 
 21 
 
 Water . 
 
 . 81 
 
 Pyridine 
 
 2O 
 
 Formic acid 
 
 57 
 
 Ammonia 
 
 16-2 
 
 Nitro benzene . 
 
 36-5 
 
 Sulphur dioxide 
 
 137 
 
 Methyl alcohol . 
 
 . 32-5 
 
 Chloroform 
 
 5' 2 
 
 Ethyl alcohol . 
 
 . 21-5 
 
 Benzene 
 
 2 '3 
 
 1 C/. Walden, Zeitsch. physikal. Chem., 1907, 58, 479. 
 
THEORIES OF SOLUTION 339 
 
 The data are not usually available for an accurate comparison 
 of the dissociating power of a solvent with its dielectric constant, 
 as the values of //,oo for electrolytes in solvents other than water 
 have been determined in only a few cases. It is important to 
 remember that a comparison of the conductivities of solutions 
 of the same concentration in different solvents is in no sense a 
 measure of the respective ionising powers of the solvents, as 
 the conductivity also depends on the ionic velocity (p. 252). 
 The available data are, however, sufficient to show that 
 although there is parallelism, there is not direct proportionality 
 between dielectric constant and ionising power. There appears 
 also to be some connection between the degree of association 
 of the solvent itself and its ionising power. The examples 
 already given show that water, the alcohols and fatty acids, 
 which are themselves complex, are the best ionising solvents. 
 There are, however, exceptions to this as to all other rules 
 in this section ; liquefied ammonia, though apparently not 
 polymerized, is a good ionising solvent. 
 
 Briihl has suggested that the ionising power of a solvent 
 depends on what he calls subsidiary valencies (the "free 
 affinity " of Armstrong) ; in other words, the best ionising 
 solvents are those which are unsaturated. It is by no means 
 improbable that the dielectric constant, the degree of poly- 
 merization, and the degree of unsaturation of a solvent are 
 in some way connected. 
 
 The ionising power of a solvent may be partly of a physical 
 and partly of a chemical nature. The effect of a high dielectric 
 constant would appear to be mainly physical, on the other 
 hand, if the effect of a solvent depends on its unsaturated 
 character, it would most likely be chemical in character. 
 
 The Old Hydrate Theory of Solution 1 As already men- 
 tioned, attempts have been made to account for the properties 
 of aqueous solutions of electrolytes on the basis of chemical 
 combination between solvent and solute. Among those who 
 
 1 Pickering, Watts's Dictionary of Chemistry, Article 
 Arrhenius, Theories of Chemistry (Longmans, 1907), chap. iii. 
 
340 OUTLINES OF PHYSICAL CHEMISTRY 
 
 have supported this view of solution, the names of Mendeleeff, 
 Pickering, Kahlenberg l and Armstrong 2 may be mentioned. 
 
 Mendeleeff made a number of measurements of the densities 
 of mixtures of sulphuric acid and water, and drew the con- 
 clusion that the curve representing the relation between density 
 and composition is made up of a number of straight lines 
 meeting each other at sharp angles, the points of discontinuity 
 corresponding with definite hydrates, for example, H 2 SO 4 , H 2 O ; 
 H 2 SO 4 , 2H 2 O ; H 2 SO 4 , 6H 2 O and H 2 SO 4 , i5oH 2 O. Pickering 
 repeated Mendeleeff's experiments, and found no sudden breaks 
 in the density curve, but only changes in direction at certain 
 points. He also drew the conclusion that these points cor- 
 respond with the composition of definite compounds of the 
 acid and water. 
 
 In this connection it may be recalled that the curve obtained 
 by plotting the electrical conductivity of mixtures of sulphuric 
 acid and water against the composition (p. 259) shows two dis- 
 tinct minima, at 100 per cent, and 84 per cent, of sulphuric 
 acid respectively, corresponding with the compounds H 2 SO 4 
 (SO 3 , H 2 O) and H 2 SO 4 , H 2 O respectively. As it is a general 
 rule that the electrical conductivity of pure substances is small, 
 there is little reason to doubt that the 84 per cent, solution 
 consists mainly of the monohydrate H 2 SO 4 , H 2 O. The con- 
 tention of Mendeleeff and Pickering, that aqueous solutions of 
 sulphuric acid contain compounds of the components, is thus 
 partially confirmed by the electrical evidence. 
 
 There is not much reason to doubt the truth of the first 
 postulate of the hydrate theory, that in many cases hydrates 
 are present in aqueous solution. The hydrates are, however, 
 in all probability more or less dissociated in solution, and 
 it will not usually be possible to determine the presence of 
 definite hydrates from the measurement of physical properties. 
 
 1 For a summary of Kahlenberg's views on Solution, see Trans. Faraday 
 Soc., 1905, I, 42. 
 
 2 Proc. Roy. Soc., 1908, 81 A, 80-95. 
 
THEORIES OF SOLUTION 341 
 
 It is probable that in general the equilibria are somewhat com- 
 plicated, and are displaced gradually by dilution in accordance 
 with the law of mass action, which accounts for the experimental 
 fact that in general the properties of aqueous solutions alter 
 continuously with composition. 
 
 Having proved the existence of hydrates in salt solutions in 
 certain cases, Pickering l attempted to account for the properties 
 of aqueous solutions (osmotic pressure, electrical conductivity, 
 etc.) on the basis of association alone, but as his views have not 
 met with much acceptance, a reference to them will be sufficient 
 for our present purpose. Kahlenberg, 2 who has carried out 
 many interesting experiments in solvents other than water, re- 
 gards the electrolytic dissociation theory as unsatisfactory, and 
 considers that the process of solution is one of chemical com- 
 bination between solvent and solute. Armstrong has also 
 attempted to account for the properties of aqueous solutions 
 on the basis of association between solvent and solute. 3 
 
 Although, as we have seen, cases are known in which a 
 maximum or minimum or a change in the direction of a curve 
 may correspond more or less completely with the formation of 
 a compound between the two components of a homogeneous 
 solution, this does not by any means always hold. It has 
 already been pointed out that the curve representing the varia- 
 tion of the electrical conductivity of mixtures of sulphuric acid 
 and water with the composition has a maximum at 30 per cent, 
 of acid (p. 2^9). As the pure liquids are practically non- 
 conductors, whilst the mixtures conduct, there must necessarily 
 be a concentration, between o and 100 per cent, acid, at which 
 the conductivity attains a maximum value. This maximum will 
 clearly have no reference to the formation of a chemical 
 compound between sulphuric acid and water, since this would 
 tend to diminish the conductivity. 
 
 Similar considerations appear to apply for other physical 
 
 l Loc. cit. z Loc. cit. 
 
 3 Loc. cit., also Encyc. Britannica, loth Edition, vol. xxvi., p. 741. 
 
342 OUTLINES OF PHYSICAL CHEMISTRY 
 
 properties which attain a maximum value for binary mixtures. 
 The curve representing the variation of the viscosity (internal 
 friction) of mixtures of alcohol and water with composition 
 shows a maximum at o for a mixture containing 36 per cent, 
 of alcohol, corresponding with the composition (C 2 H 5 OH) 2 , 
 9H 2 O, and it has therefore been suggested that the solution 
 consists mainly of this hydrate. At 1 7, however, the mixture 
 of maximum viscosity contains 42 per cent., and at 55 rather 
 more than 50 per cent, of alcohol. The last-mentioned mixture 
 corresponds with the composition (C 2 H 5 OH) 2 , 5H 2 O. If we 
 accept the association view of this phenomenon, it must be 
 assumed that at o the solution contains a hydrate (C 2 H 5 OH) 2 , 
 9H 2 O, and at 55 a hydrate (C 2 H 5 OH) 2 , 5H 2 O, and that at 
 intermediate temperatures the hydrates with 6, 7 and 8 H 2 O 
 exist which does not appear very probable. 
 
 Now, Arrhenius has shown that as a general rule the addition 
 of a non-electrolyte raises the viscosity of water. Therefore, if 
 the viscosity of the non-electrolyte is less than, or only slightly 
 exceeds that of water, the curve obtained by plotting viscosity 
 against the composition of the mixture must necessarily attain 
 a maximum at some intermediate point. Why mixtures of two 
 liquids have often a higher viscosity than either of the pure 
 liquids is not known, no general agreement having yet been 
 reached on this and allied questions. 1 
 
 Meohanism of Electrolytic Dissociation. Function 
 of the Solvent The fundamental difference between associa- 
 tion theories of solution, as discussed in the last section, and 
 the electrolytic dissociation theory is that the advocates of 
 association entirely reject the postulate of the independent 
 existence of the ions. As, however, the different theories of 
 association unaccompanied by ionisation have so far proved 
 quite inadequate to account quantitatively for the behaviour 
 of aqueous solutions, whilst the electrolytic dissociation theory 
 not only affords a satisfactory quantitative interpretation of the 
 
 1 Compare Senter, Proc. Chem. Soc., 1909, 292. For evidence in 
 favour of the association view of this phenomenon, compare Dunstan, 
 Trans. Chem. Society, 1907, 91, 83; Dunstan and Thole, ibid., 1909, 95, 
 
THEORIES OF SOLUTION 343 
 
 more important phenomena observed in solutions of electrolytes 
 (chap, x.), but has led to discoveries of the most fundamental 
 importance for chemistry, it is not surprising that the electro- 
 lytic dissociation theory has now met with practically universal 
 acceptance. It is very likely that as a result of further investi- 
 gation the theory may require modification in some subsidiary 
 respects, but its general validity appears no longer doubtful. 
 
 We are now in a position to discuss the possible mechanism 
 of electrolytic dissociation, /.<?., the factors concerned in the 
 conversion of a non-ionised salt into ions when dissolved in 
 water or other ionising solvent. In this connection two points 
 of importance must be kept in mind: (i) There is a more 
 or less complete parallelism between the ionising power of a 
 solvent and certain of its properties, such as the magnitude of 
 the dielectric constant and its degree of polymerization ; (2) the 
 sum of the energy associated with the ions is usually not very 
 different from that associated with the non-ionised molecules 
 in solution. The first point has already been fully dealt with. 
 The energy relations in the process of ionisation will now be 
 briefly considered. 
 
 As such elements as potassium and chlorine give out a large 
 amount of heat when they combine to form potassium chloride, 
 it seems at first sight as if the splitting up of potassium chloride 
 into K' and Cl' ions must absorb a large amount of heat. 
 Although it has to be borne in mind that the salt is not split 
 up into its component elements (a process which certainly 
 absorbs a large amount of heat) but into K- and Cl' ions, of 
 the energy content of which little is known, it is nevertheless 
 probable that the process of ionisation does involve a taking up 
 of energy, as the following considerations show. 
 
 It has already been pointed out that the heat of ionisation 
 
 (the heat given out when a mol of a non-ionised salt in 
 
 solution is completely ionised) may be calculated by the van't 
 
 Hoff formula connecting heat given out with displacement of 
 
 'equilibrium, when the equilibrium at two near temperatures is 
 
344 OUTLINES OF PHYSICAL CHEMISTRY 
 
 known (p. 285). The available data indicate that the heat of 
 ionisation is small, and may be positive or negative. According 
 to Arrhenius, the energy equation for the ionisation of dissolved 
 potassium chloride is as follows 
 
 KC1 = K- + Cr + 362 cal, 
 
 the heat of ionisation being thus exceedingly small. On the 
 other hand, when potassium chloride is formed from its elements, 
 105,600 cal. are given out for gram-equivalent quantities, hence 
 the energy equation for the decomposition of potassium chloride 
 into metallic potassium and free chlorine is 
 
 KC1 = ?* + Q - 105,600 cal. 
 
 The further separation of metallic potassium and free chlorine 
 into uncharged atoms would probably absorb still more heat, so 
 that the reaction 
 
 KC1 = K + Cl 
 
 must absorb a very large amount of heat. On the other hand, 
 we have just seen that in the splitting up of the salt into charged 
 atoms 1 the ions a little heat is actually given out. It seems, 
 therefore, that the process of ionisation must be attended by 
 some exothermic reaction, which more than compensates for 
 the heat presumably absorbed in splitting up the molecules. 
 
 Up to the present no definite conclusion has been arrived at 
 as to the source of the energy in question. Some of the possi- 
 bilities are as follows : 
 
 (a) As the molecule splits up into particles charged with 
 electricity, the energy in question may come from some inter- 
 action between electricity and matter. On this point nothing 
 can be said with certainty in the present state of our knowledge. 1 
 
 (b) The energy may result mainly from combination between 
 the ions and water. Van der Waals 2 (1891) has suggested 
 that ionisation in aqueous solution is essentially a hydration 
 
 1 Cf. Ostwald, Fitzgerald, British Association Reports, 1890. 
 *Zeitsch. physikal Ghent., 1891, 8, 215. Cf. Bousfield and Lowry^ 
 Trans. Faraday Society, 1907, 3, i. 
 
THEORIES OF SOLUTION 345 
 
 process ; he considers that it is the affinity of the ions for the 
 solute which effects the break-up of the molecule, and that the 
 energy required for ionisation comes from the heat of hydration 
 of the ions. 
 
 (c) Dutoit, arguing from the parallelism between the poly- 
 merization of the solute and its ionising power, and assuming 
 that the process of solution is attended with partial simplification 
 of the solute molecules, has suggested that the energy is ob- 
 tained from the depolymerization in question. 
 
 The suggestion of van der Waals finds a certain amount of 
 support from recent investigations on solvents other than water. 
 Carrara l considers that the ions are produced as a consequence 
 of the chemical affinity between the two parts of the molecule 
 and the solute, and that in consequence of the high dielectric 
 constant in ionising media, they, along with their associated 
 solvent molecules, are kept in the ionic condition once they 
 are formed. 
 
 It should, however, be remembered that this suggestion as 
 to the mechanism of ionisation, although interesting and 
 suggestive, is little more than a plausible speculation, and that 
 there are many points about this process which are very im- 
 perfectly understood. 
 
 Hydration in Solution 2 In recent years the view that 
 the ions are hydrated to a greater or less extent in aqueous 
 solutions of electrolytes has gained ground very considerably, 
 but it cannot be said that the evidence is very conclusive. 
 Perhaps the most definite positive result is the observation of 
 Kohlrausch, that the influence of temperature on the electro- 
 lytic resistance of dilute solutions of salts approximates to the 
 
 1 Loc. dt. t p. 417. 
 
 2 For a general discussion on hydration in solution, see Trans. Faraday 
 Soc., 1907, 3, i. For summaries of the methods which have been suggested 
 for determining the degree of hydration in solution, see Baur, Ahrens' 
 Sammlung, 1903, 8, 466 ; Senter, Science Progress, 1907, i, 386 ; Trans. 
 Faraday Soc., 1907, 3, 24. Cf. also Philip, Trans. Chem, Soc., 1907, 
 9L 7". 
 
346 OUTLINES OF PHYSICAL CHEMISTRY 
 
 temperature coefficient of the mechanical friction of water. 
 This observation at once becomes intelligible if it is assumed 
 that the ions are surrounded by water molecules, as the resist- 
 ance to movement of the water-coated ions would then be the 
 frictional resistance of water against water. 
 
 Many attempts have been made to determine the degree 
 of hydration both in solutions of electrolytes and of non- 
 electrolytes, but the results are very uncertain. If each ion 
 becomes associated with a number of water- molecules, the 
 total number of solute particles will not be altered, and the 
 osmotic pressure, lowering of freezing-point, etc., will not be 
 directly affected. If, however, it is assumed that only the un- 
 bound part of the solvent can really act as solvent, the effect of 
 association will be to diminish the "free " solvent, and there- 
 fore the effective concentration of the solution the ratio of 
 solute to free solvent will be increased, and the lowering 
 of freezing-point will be greater than the calculated value. 
 Unless, however, the average hydration of the solute is very 
 great, the effect in question will only be considerable in con- 
 centrated solutions, under which circumstances freezing-point 
 and other determinations often give unsatisfactory results 
 (p. 122). 
 
 For our present purpose, it will be sufficient to refer to the 
 method for determining the degree of hydration of electrolytes 
 suggested by H. C. Jones, as this author and his co-workers 
 have collected much valuable experimental material which will 
 be serviceable in attacking some of the yet unsolved problems 
 of solution. The method depends on the observation that in 
 many cases the experimental freezing-point depression is greater 
 than the value to be expected from the degree of dissociation 
 of the solute, and Jones assumes that this is due to the associa- 
 tion of part of the solvent with the solute, so that the depression 
 actually observed is that corresponding with a more concen- 
 trated solution, as described in the previous paragraph. If the 
 true depression in the absence of association (when the whole 
 
THEORIES OF SOLUTION 347 
 
 of the solute is " free ") can be calculated, the mode of esti- 
 mating the hydration will be evident, but this is just where the 
 weakness of the method lies. Jones calculates the theoretical 
 depression for a particular dilution from conductivity measure- 
 ments at o, the values of /* and /* a being determined directly. 
 To do this it has to be assumed that the exact degree of 
 dissociation can be obtained from conductivity measurements, 
 which is not the case for concentrated solutions, 1 although very 
 nearly so for dilute solutions (p. 261), and further, that the 
 van't Hoff-Raoult's law holds strictly for concentrated solu- 
 tions, which is highly improbable (p. 122). Jones himself, 
 while admitting that several of his assumptions are not strictly 
 valid, considers that a nearer approximation to the true values 
 for the degree of hydration is obtained by his method than by 
 any other method so far suggested. While this may be the 
 case, we have no means of knowing the magnitude of the error 
 thus introduced, and the true values for the degree of hydration 
 may be very different from those given by Jones and his co- 
 workers. Moreover, the observed irregularities in freezing- 
 point depression may be due, in part at least, to some cause 
 other than hydration. The present position of the subject 
 may be summed up in the statement that while there is a 
 good deal of evidence in favour of hydration in solution, very 
 little is known on the subject with certainty from a quantitative 
 point of view. 
 
 1 The uncertainty in the case of concentrated solutions arises from 
 the difficulty of distinguishing between the effects of alteration of 
 viscosity and of alteration of the degree of dissociation on the electrical 
 conductivity (p. 261). 
 
CHAPTER XIV 
 ELECTROMOTIVE FORCE 
 
 The Daniell Cell. Electrical Energy The Darnell 
 cell consists essentially of a large beaker or other vessel, con- 
 taining a small porous cell. In the outer vessel, surrounding 
 the porous cell, is a solution of zinc sulphate in which dips a 
 rod of zinc, and the porous pot contains a solution of copper 
 sulphate, in which dips a plate of copper. When the copper 
 plate is connected to the zinc by a wire, an electric current 
 flows through the wire. As positive electricity passes in the 
 wire from copper to zinc, the copper is termed the positive, 
 the zinc the negative pole. An examination of the cell will 
 show that when the current is flowing, zinc is being dissolved at 
 the negative pole, and copper is being deposited on the positive 
 pole. If, instead of copper sulphate, the porous pot contains 
 dilute sulphuric acid, hydrogen is given off at the surface of 
 the copper when the zinc and copper are connected. It is, 
 therefore, clear that the chemical change which gives rise to 
 the current is the dissolving of zinc in sulphuric acid, repre- 
 sented by the equation Zn + H 2 SO 4 = ZnSO 4 + H 2 , and that 
 in the Daniell cell the hydrogen is got rid of by reducing 
 copper sulphate in the porous pot, the change being repre- 
 sented by the equation CuSO 4 H 2 = H 2 SO 4 + Cu. Com- 
 bining the two equations, the chemical changes taking place 
 while a Daniell cell is working are represented by the equation 
 
 Zn + CuSO 4 = ZnSO 4 + Cu, 
 which may be written 
 
 Zn + Cu- + SO 4 " = Zn- + SO 4 " + Cu, 
 348 
 
ELECTROMOTIVE FORCE 349 
 
 or more concisely, 
 
 Zn + Or- = Zn- + Cu, 
 
 that is, zinc changes from the metallic to the ionic form, and at 
 the same time copper changes from the ionic to the metallic 
 form. 
 
 The Daniell cell may be looked upon as a machine for the 
 conversion of chemical to electrical energy, and it is of great 
 interest to inquire whether chemical energy is transformed into 
 the equivalent amount of electrical energy, or whether other 
 kinds of energy, such as heat, are also produced in the cell. 
 This question can at once be settled by determining the heat 
 equivalent, in calories, of the chemical energy transformed 
 when i mol of copper in solution is displaced by zinc, and 
 comparing it with the electrical energy (measured in calories) 
 obtained during the process. 
 
 From Thomsen's measurements (p. 144) the displacement of 
 i mol of copper in dilute solution by zinc is associated with a 
 heat development of 50,130 calories. The electrical energy in 
 volt-coulombs is, as has already been pointed out, the product 
 of the electromotive force E in volts and the quantity of elec- 
 tricity in coulombs. E for the Daniell cell is about i '09 volts 
 at room temperature. From Faraday's law we know further 
 that the liberation of a gram-equivalent of positive or negative 
 ions is associated with the passage of 96,540 coulombs, hence, 
 as zinc and copper are bivalent, the reaction 
 
 Zn + Or* = Zn" + Cu 
 
 is associated with the passage of 2 x 96,540 coulombs. The 
 electrical energy, in calories (i volt-coulomb = 0-2391 calories), 
 is therefore given by 1 
 
 ECt = 1-09 x 2 x 96,540 x 0-2391 = 50,380 cal. 
 
 as compared with 50,130 calories, the heat equivalent of the 
 chemical change which has taken place in the cell. The excel- 
 lent agreement between observed and calculated values shows 
 that in this case the assumption that the chemical energy is 
 
 1 C is the current, / the time and Ct therefore the quantity of elec- 
 tricity. 
 
350 OUTLINES OF PHYSICAL CHEMISTRY 
 
 transformed completely to electrical energy is approximately 
 fulfilled. 
 
 It should be remembered that in an actual cell other chemical 
 changes may accompany the primary ones ; for example, those 
 giving rise to the so-called E.M.F. of polarization (p. 373). 
 In comparing the heat equivalent of the main chemical action 
 taking place in the cell with the electrical energy obtained, 
 such changes must, of course, be allowed for. In trie Daniell 
 cell, however, there is no polarization, and the only reaction 
 which takes place to any appreciable extent is the displacement 
 of copper by zinc, as has already been assumed. In this chapter 
 we are mainly concerned with such non-polarizable cells. 
 
 On account of the excellent agreement between the amount 
 of chemical energy expended and the electrical energy generated 
 in the Daniell cell, it was at first thought that the transformation 
 of chemical to electrical energy in all cells is practically com- 
 plete, but further investigation showed that this is by no means 
 always the case. As indicated above, the question can readily 
 be investigated for a non-polarizable element by comparing the 
 heat equivalent, Q, of the chemical change taking place in 
 the element with the electrical energy generated while known 
 quantities of the reacting substances are being transformed. 
 The electrical energy may conveniently be measured as follows. 
 The poles of the cell are connected by a thin wire of high re- 
 sistance, R. If r is the internal resistance of the cell, the 
 electrical energy generated in unit time is given by 
 
 EC = C 2 (R + r\ 
 
 Now R may easily be made so large compared with r that the 
 amount of electrical energy converted to heat in the cell is 
 negligible in comparison with that transformed in the outer wire. 
 If then the current is measured and R is known, the quantity of 
 electrical energy produced by the cell in unit time is known, and 
 hence that produced when n x 96,540 coulombs have passed 
 round the circuit. This quantity of electricity energy is obtained 
 during the chemical transformation of n equivalents of the cell 
 
ELECTROMOTIVE FORCE 351 
 
 material, and may be compared with Q, the heat given out 
 when the chemical change takes place in a calorimeter. 
 
 When a Daniell cell is examined in this way, it is found, as has 
 already been pointed out, that the electrical energy is practically 
 equal to Q, the heat equivalent of the chemical energy. There 
 are, however, elements in which the electrical energy thus ex- 
 pended in the outer wire is less than the heat of the reaction ; 
 in this case part of the chemical energy is expended as heat in 
 the cell. In other elements, however, the electrical energy, 
 ECt, is actually greater than the heat of reaction, Q, so that the 
 element, while working, takes heat from its surroundings and 
 transforms it into electrical energy. 
 
 Relation between Chemical and Electrical Energy 
 The exact relationship between the chemical energy transformed 
 and the maximum energy obtainable electrically in a reversible 
 galvanic element can be obtained on thermodynamical principles 
 by means of a cyclic process. The formula expressing the 
 relationship (Willard Gibbs, 1878; Helmholtz, 1882), usually 
 known as the Helmholtz equation, is as follows : 
 
 nFE~Q = nFT~ . . . (i) 
 or K-Q.Tg ...(a) 
 
 where E is the E.M.F. of the cell, Q is the heat equivalent of 
 the chemical change for molar quantities, expressed in electrical 
 units, F is 96,540 coulombs, T is the absolute temperature at 
 which the cell is working, and n is the valency, or the number 
 of charges carried by a mol of the substances undergoing 
 change. By dE/dT is meant the rate of change of the E.M.F. 
 of the cell with temperature' in other words, the temperature 
 coefficient of the E.M.F. 
 
 The Helmholtz formula is simply a direct application of the 
 free energy equation 
 
 A-U-T . ... (3) 
 
352 OUTLINES OF PHYSICAL CHEMISTRY 
 
 to a chemical change taking place in a galvanic cell In the 
 latter formula, U represents the total diminution of energy in a 
 reacting system, and A is the available or free energy, that is, 
 the maximum proportion of the total change of energy, U, which 
 can be transformed into work (p. 151). In order to obtain 
 formula (i) Q, the heat of reaction measured calorimetrically, 
 is substituted for U in the free energy equation, and for A, the 
 free energy, is substituted FE, the maximum energy obtain- 
 able electrically for molar quantities. 1 dA/dT in the free 
 energy equation then becomes nFTdE/dT (n and F being 
 constants), equation (i) being thus obtained. The equation 
 should be remembered in the second form. 
 
 From the first form of the Helmholtz equation, some very 
 important conclusions can be drawn as to the behaviour of 
 reversible elements : 
 
 (a) If dE/dT is positive, that is, if the E.M.F. of the 
 element increases with temperature, the electrical energy, FE 
 is greater than the heat of reaction, Q, and the cell takes heat 
 from its surroundings while working. 
 
 (b) If dE/dT is negative, that is, if the E.M.F. of the 
 element diminishes as the temperature rises, the heat of re- 
 action, Q, is greater than the electrical energy, #FE, and the 
 cell warms while working. 
 
 (c) If the E.M.F. does not alter with change of temperature 
 (as is approximately the case in a Daniell cell), the heat of 
 reaction, Q, is equal to the electrical energy, and the cell does 
 not alter in temperature while working. 
 
 (d) At absolute zero, that is, T = o, the right-hand side, and 
 therefore also the left-hand side, of equation (i) becomes zero, 
 hence at the absolute zero the heat of reaction (diminution of 
 chemical energy) is always equal to the maximum electrical 
 energy obtainable (or in general to the free energy, cf. p. 152). 
 
 1 When the change taking place in a galvanic cell is reversible, then, 
 according to the second law of thermodynamics, the electrical energy 
 obtained, nFE, is the maximum work obtainable from the change and is 
 therefore a measure of the free energy of the reaction (p. 150). 
 
ELECTROMOTIVE FORCE 353 
 
 The applicability of the Helmholtz formula to chemical 
 changes which can be brought about reversibly in a galvanic 
 element has been proved experimentally by Jahn and others. 
 As an example, we will consider a cell made up of zinc and 
 silver in contact with solutions of the respective chlorides. The 
 observed E.M.F. of such a cell at o is 1*015 volts, and the 
 electrical energy in calories for the transformation of i mol of 
 the reacting substances is given by nE = 1-015 x 96,540 x 
 .2 x 0*239 = 46,840 calories. The heat of reaction (due to 
 the displacement of the equivalent amount of silver from solu- 
 tion by i mol of zinc) is 52,046 cal. Hence, as the heat of 
 reaction is greater than the electrical energy, the temperature co- 
 efficient of the E.M.F. is negative, and is given by the formula 
 
 dE nFE - Q (46,840-52,046)4-184 
 
 - = =^- = ^ = - 0-00041 (volts 
 
 di nFL 2 x 96,540 x 273 
 
 per degree) in excellent agreement with the experimental value, 
 - 0-00040. As the value of dE/dT is required in volts, the 
 energy in the above equation is necessarily represented in volt- 
 coulombs (calories x 4-184). 
 
 In order further to illustrate the above statements, which are 
 of fundamental importance for our work, they will now be 
 repeated in a somewhat different form. The quantities of 
 energy given in the following paragraphs refer to molar 
 quantities throughout. 
 
 When a chemical change takes place without the performance 
 of external work, the total change of energy, U, is equal to the 
 heat of reaction, Q, measured in a calorimeter. 
 
 When the same reaction takes place in a battery the poles 
 of which are connected by a thin wire, the whole apparatus, 
 including the connecting wire, being enclosed in a calorimeter, 
 the heat developed in the latter is again Q for molar quantities 
 transformed, the electrical energy in the connecting wire being 
 degraded to heat. 
 
 When the cell is in the calorimeter, and the thin connecting 
 wire outside, there are three possible cases : , " 
 
 \ (a) If the heat of reaction, Q, is equal to the maximum 
 
 23 
 
354 OUTLINES OF PHYSICAL CHEMISTRY 
 
 electrical energy obtainable, nE, and if further the resistance 
 of the cell is zero (in practice, very low) no heat will be de- 
 veloped in the calorimeter, and the equivalent of Q calories 
 of electrical energy will pass round the external wire. 
 
 (b) If the heat of reaction is greater than the maximum 
 electrical energy obtainable, heat will be developed in the 
 calorimeter as well as in the external circuit. 
 
 (c) If the heat of reaction is less than the maximum electrical 
 energy heat will be absorbed in the calorimeter, and the heat 
 equivalent of the electrical energy in the outer circuit will be 
 greater than Q (provided that the resistance of the cell is 
 negligible). 
 
 It has already been pointed out that the maximum work can 
 only be obtained by carrying out a process reversibly, and 
 therefore the Helmholtz formula only applies to elements in 
 which the process can be effected reversibly. This means that 
 if an element such as the Daniell is giving out electrical energy 
 at an E.M.F. represented by E, it can be restored to its initial 
 condition by passing through it, at an E.M.F. only just exceeding 
 E, a quantity of electrical energy equal to that given out. In 
 order that an electrochemical process should be completely 
 reversible, the energy should be given out and taken in at the 
 same potential, which in practice is impossible. A completely 
 reversible process is therefore only an ideal case, which can be 
 more or less completely attained to in actual practice. 
 
 Many electrochemical processes are irreversible, more par- 
 ticularly those in which gases are evolved. For example, a 
 Daniell cell in which the copper sulphate solution is replaced 
 by sulphuric acid is irreversible, as, owing to the escape of 
 hydrogen, it cannot be restored to its former condition by 
 passing a current in the contrary direction. 
 
 Measurement of Electromotive Force The difference 
 of potential between the two poles of a cell may have very 
 different values according to the conditions of measurement, 
 If the poles are connected by a wire of very high resistance^ 
 
ELECTROMOTIVE FORCE 355 
 
 R, and the resistance of the cell is r, the fall of potential in 
 the wire is CR and in the cell Cr. Since E = C(R + r), it is 
 clear that the fall of potential CR in the outer circuit is only 
 equal to E when the resistance R is infinitely large compared 
 with r. It is therefore easy to understand why the resistance 
 of volt-meters (instruments on which the difference of potential 
 between two points can be read off directly in volts) is always 
 very high. A further complication in measuring the E.M.F. 
 of a cell while working is that in many cases the products of 
 electrolysis accumulate on the electrodes, and set up an electro- 
 motive force in the opposite direction to the E.M.F. of the cell 
 itself a phenomenon which is termed polarization. 
 
 These difficulties are avoided by measuring the difference of 
 potential between the poles of a cell when the circuit is open, 
 that is, when the cell is not sending a current. In the present 
 chapter, the symbol E stands for the E.M.F. of the cell on open 
 circuit. The effect of completing the circuit will be dealt with 
 more fully at a later stage. 
 
 The E.M.F. of a cell on open circuit is most conveniently 
 measured by the Poggendorff compensation method. The 
 principle of the method is that the E.M.F. of the cell is just 
 compensated by an equal and opposite E.M.F. so that no current 
 passes, and the measurement consists in altering an adjustable 
 E.M.F. till the above condition is fulfilled. The arrangement 
 of the apparatus for this purpose is shown in Fig. 37. A is a 
 lead accumulator, or other source of constant E.M.F., which is 
 connected to the two ends of a uniform wire BC, which may 
 conveniently be a metre in length. The cell E, the E.M.F. of 
 which is to be measured, is connected through a sensitive gal- 
 vanometer G and a tapping key to one end of the bridge wire 
 at B, and on the other to a sliding contact. The slider is moved 
 along the wire until a position D is found at which, when con- 
 tact is made, no current passes through the galvanometer. It 
 is then clear that the following relation holds (cf. p. 255) : E.M.F. 
 of accumulator : E.M.F. of cell : : length BC : length BD, so that 
 
356 OUTLINES OF PHYSICAL CHEMISTRY 
 
 the E.M.F. of the cell can readily be calculated. As the E.M.F. 
 of the lead accumulator is not quite constant, it is preferable 
 in accurate determinations to standardize the apparatus by finding 
 the position on the bridge wire corresponding with the E.M.F. 
 of a constant cell, such as the Clark or Cadmium cell (p. 357),' 
 when the latter is substituted for the cell E in the diagram. As 
 null instrument, for showing that the point of balance has been 
 reached, by the fact that a current no longer passes when 
 contact is made, a high resistance galvanometer or a capillary 
 electrometer (p. 378) may be used; the former maybe made 
 the more sensitive, but the latter is in many respects very 
 convenient. 
 
 FIG. 37- 
 
 The measurement of the E.M.F. of a cell on open circuit 
 may also be made by means of the quadrant electrometer, the 
 deflection of the needle being proportional to E, but this method 
 is in some respects less advantageous than the compensation 
 method. 
 
 In general, the observed E.M.F. of a cell on closed circuit is less 
 than that on open circuit ; this is due, as already indicated, to 
 polarization effects and to the effect of the internal resistance 
 of the cell. 
 
 Standard of Electromotive Force. The Cadmium 
 Element The importance of obtaining a cell, the E.M.F. 
 of which is accurately known, has already been pointed out. 
 
ELECTROMOTIVE FORCE 357 
 
 The cadmium element, sometimes called the Weston element, 1 
 which is most largely in use for this purpose, will now be 
 described. 
 
 The cell itself consists of an H-shaped glass vessel, the side 
 tubes are closed at their lower ends, and platinum wires, sealed 
 in the glass, pass through near the lowest points of the side 
 tubes. In the bottom of one of the tubes is a layer of mercury 
 about i cm. deep, above which is a paste (5 mm.) of mercurous 
 sulphate, which has been carefully freed from traces of mercuric 
 salt. In the bottom of the other side tube is a layer of 
 cadmium amalgam (about 13 per cent, of cadmium quite free 
 from zinc) above which is a paste (5 mm.) of cadmium sulphate, 
 prepared by rubbing together in a mortar water and crystallized 
 cadmium sulphate, and pouring off the clear saturated solution. 
 The remainder of the cell on both sides, as well as the 
 connecting tube, contains a saturated solution of cadmium 
 sulphate, in which are moderately large crystals of the solid 
 salt, and both side tubes are hermetically closed at the top, 
 a small air space being left to allow of expansion. When 
 required as a standard of E.M.F., connection is made by means 
 of the platinum wires, which are in contact with the mercury 
 and amalgam respectively. 
 
 The great advantage of the cadmium element is that its 
 E.M.F. is practically independent of temperature. The E.M.F. 
 at temperatures from o to 30 is as follows : 
 
 o 5 10 15 20 25 30 
 
 1*0189 1*0189 1*0189 1*0188 1*0186 1*0184 1*0181 volts, 
 
 the mercury being positive. 
 
 When in use the element should only be allowed to send a 
 very small current for a very short time; if this condition is 
 not observed the E.M.F. soon alters owing to polarization. 
 
 The mode of action of the element will be understood when 
 the section on the calomel electrode has been read. 
 
 1 The true Weston element differs slightly from that here described. 
 
358 OUTLINES OF PHYSICAL CHEMISTRY 
 
 The corresponding cell with zinc amalgam and zinc sulphate 
 instead of cadmium amalgam and cadmium sulphate, was 
 formerly in general use as a standard element under the name 
 of the Clark cell. Its chief drawback is that the E.M.F. alters 
 considerably with temperature. The E.M.F. of the Clark cell 
 is given by the equation 
 
 E = 1*4328 0*00119 (/ 15) 0^000007 (/ 15)2. 
 
 Solution Pressure In a former section it has been shown 
 that the relation between chemical and electrical energy in a 
 voltaic cell is given by the Helmholtz formula 
 
 F Q + T ** 
 = nF + T ft ' 
 
 This formula has been deduced from considerations which are 
 quite independent of any assumption as to the mechanism of 
 the establishment of differences of potential in a cell, and 
 therefore holds quite independently of any theory as to the 
 origin of differences of potential. A much deeper insight into 
 this problem is gained on the basis of a theory due to Nernst, 
 based on the theory of electrolytic dissociation, and this theory 
 will now be considered. 
 
 Every substance has a tendency to change from the form in 
 which it actually exists to another form. Water, for example, 
 has a tendency to pass into vapour, and if the vapour be con- 
 tinually removed from its surface, a definite quantity of water 
 will change completely to vapour. The tendency in question 
 is measured by the vapour pressure of the water, and is con- 
 stant at constant temperature. Further, a solid, such as sugar, 
 when brought in contact with water, tends to pass into solution, 
 and from the analogy with water and water vapour we may say 
 that sugar has a definite solution pressure, which is constant at 
 constant temperature, since the active mass of a solid, such as 
 sugar, is constant (p. 173). On the other hand, the dissolved 
 sugar has a tendency to separate in the solid form, which is the 
 greater the higher the concentration, and when the solution is 
 
ELECTROMOTIVE FORCE 359 
 
 supersaturated the tendency to the separation of solid sugar is 
 greater than the tendency of the latter to pass into solution. 
 From the considerations advanced on p. 98, it is clear that 
 the pressure of the sugar in solution is its osmotic pressure, and 
 under, definite conditions sugar will enter into or separate from 
 solution according as its solution pressure is greater or less than 
 its osmotic pressure. 
 
 These considerations, in conjunction with the ionic theory, 
 enable us to express the E.M.F. at a junction metal/solution 
 in terms of solution pressure and osmotic pressure. If a metal 
 is dipped into water it tends to dissolve, in consequence of its 
 solution pressure, P, and as it can only do so in the ionic form, 
 it sends a certain number of positive ions into solution. The 
 solution thus becomes positively charged, and the metal, which 
 was previously neutral, becomes negatively charged in conse- 
 quence of the loss of positive ions. This process will proceed 
 until, by the accumulation of positive electricity in the solution, 
 the latter becomes so strongly positive that it prevents the pas- 
 sage of more positive ions into solution. As the charge on 
 the ions is so great, this process comes to a standstill when the 
 amount of ions gone into solution is still excessively small, too 
 small to be detected by analytical means. 
 
 The state of affairs is rather different when a metal is dipped 
 into a solution of one of its salts, e.g., zinc in a solution of zinc 
 sulphate. In this case there are already positive metallic ions 
 in the solution, which tend to resist the entrance of further 
 positive ions, and what actually occurs will clearly depend upon 
 the relative values of the solution pressure, P, of the metal and the 
 osmotic pressure, p, of the ions in solution. There are, in fact, 
 three possible cases, which are represented diagram matically in 
 the accompanying figure : 
 
 (a) If P > /, the metal sends ions into the solution until the 
 accumulated electrostatic charges prevent further action ; the 
 metal is then negatively and the solution positively charged. 
 
 (b) If P < /, the positive ions from the solution deposit on 
 
360 OUTLINES OF PHYSICAL CHEMISTRY 
 
 the metal until the electrostatic charges prevent further action ; 
 the metal is then positively and the solution negatively charged. 
 
 (c) \f P = /, no change occurs, and there is no difference of 
 potential between metal and solution. 
 
 As will be shown later, the solution pressures of the different 
 metals are very different. Those of the alkali metals, zinc, iron, 
 etc., are so great that they always exceed the osmotic pressures 
 of their respective solutions- (which cannot be increased beyond 
 a certain point owing to the limited solubility of the salts), and 
 these metals are, therefore, always negatively charged with 
 
 
 - 
 
 Metal 
 
 -h 
 
 4- 
 
 + + 
 
 Solution. 
 
 
 Solu 
 
 Metal 
 
 
 tion. 
 
 
 f 
 
 Metal 
 
 - 
 
 - 
 
 . Solution. 
 
 p=p. p-=p. 
 
 FIG. 38. 
 
 reference to their solutions. On the other hand, the solution 
 pressure of mercury, silver, copper, etc., is so small that they 
 become positively charged, even in very dilute solutions of their 
 respective salts. 
 
 Calculation of Electromotive Forces at a Junction 
 Metal/Salt Solution Provided that the changes at the 
 junction of an electrode with a solution are reversible, the 
 E.M.F. at the junction can readily be calculated in terms of 
 the solution pressure, P, of the metal and the osmotic pressure, 
 p, of the solution. This can perhaps be done most simply by 
 calculating the maximum work obtainable when a mol of 
 
ELECTROMOTIVE FORCE 361 
 
 the electrode metal is brought from the pressure P to the lower 
 pressure /, (i) osmotically, (2) electrically. If a mol of a 
 dissolved substance is brought reversibly from the pressure P 
 to/ the work gained (in this case the osmotic work) is (cf. p. 
 135) 
 
 Further, the dissolving of i equivalent of a metal is associated 
 with 96,540 coulombs, and that of a mol of a metal of valency 
 n with 96,540 n coulombs. The work done is the product of 
 the E.M.F. E in volts and the quantity of electricity, 96,540 n 
 coulombs. Equating the osmotic and electrical work, we have 
 n 96,540 E = RTlog e P// 
 
 RT . P , . 
 
 or E = - log*-; (i) 
 
 96,540^ 6 V 
 
 In order to obtain E in volts, R must be expressed in electrical 
 units (volt-coulombs). If, at the same time, the change is made 
 to ordinary logarithms (by multiplying by 2*3026) the above 
 equation becomes 
 
 2*3026 x 1*99 x 4*183 T P 0*0001983 T P 
 
 96,540* gl 7 = ~^~ logl< 7 
 
 The numerical values of 2*3026 RT/Fat o, 18, 25 and 30 
 are as follows : 
 
 Absolute temperature 273 273 + 18 273 + 25 273 + 30 
 Value of 2*3026 RT/F o'o54i o'577 '59i 0-0601 
 
 At room temperature (15-20) the value of the expression in 
 question is about 0*058, and the general formula becomes 
 
 which should be remembered. It is clear from the form of 
 the above equation that a tenfold increase or decrease in the 
 osmotic pressure of the ions of the metal will produce a change 
 of E.M.F. of 0*058 volts for a univalent metal, and o'o$8/n 
 volts for a w-valent metal, at room temperature. 
 
362 OUTLINES OF PHYSICAL CHEMISTRY 
 
 Differences of Potential in a Yoltaic Cell Two such 
 electrodes as have just been described may be combined 
 together to form a voltaic cell. This may be done in many 
 ways, but a convenient arrangement is that for the Daniell 
 cell represented in Fig. 39. in which the solutions are separated 
 by a porous partition, A, which prevents convection, but allows 
 the current to pass. When the poles are placed in the respec- 
 tive solutions, the zinc becomes negatively charged, since P x > 
 p l ; on the other hand, the copper becomes positively charged 
 as/ 2 >P 2 . As already explained, the solution and precipita- 
 tion soon come to a standstill because of the accumulation 
 
 of electrostatic charges. 
 If, however, the elec- 
 trodes are connected 
 by a wire, the contrary 
 charges neutralize each 
 other through the wire, 
 and in the solution 
 more metal can then 
 be dissolved and de- 
 posited respectively (as 
 there are no longer any 
 opposing forces), the 
 corresponding charges are again neutralized, and so on. The 
 neutralization of charges through a conductor corresponds with 
 the passage of a current. 
 
 The general question as to the seat of the E.M.F. in such 
 a cell as the Daniell has now to be considered. If the poles 
 of the cell are connected by a wire of metal M, there are no less 
 than five junctions at which there may be contact differences of 
 potential ; two metallic junctions, Zn/M and M/Cu, two metal/ 
 solution junctions, Cu/CuSO 4 and Zn/ZnSO 4 , and one liquid 
 junction, ZnSO 4 /CuSO 4 . The question as to whether there 
 are contact differences of potential at the junction of two metals 
 gave rise to great difference of opinion, and the controversy 
 
 
 Zh 
 
 / 
 
 X 
 
 Cu 
 
 
 
 
 
 3d 
 
 ^f 
 
 
 
 + 
 
 
 
 
 
 
 
 
 
 
 
 
 + 
 
 ^ mm 
 
 +~ 
 
 
 
 
 
 _. 
 
 _ _=: 
 
 -r 
 
 
 
 ~z, 
 
 ^SO 4 " 
 
 +- 
 
 C 
 
 ~^0 4 
 
 
 
 FIG. 39. 
 
ELECTROMOTIVE FORCE 
 
 363 
 
 lasted the greater part of last century. It is now generally 
 agreed, however, that if there are such differences they are 
 exceedingly small in comparison with those of the junctions 
 metal/salt solution. The difference of potential at the liquid 
 junction is of much more importance and can be calculated by 
 Nernst's theory (p. 384). It also is small in comparison with 
 those at the liquid/metal junctions, and mav therefore, be left 
 out of account for the present. 
 
 The distribution of differences of potential in the Daniell cell 
 with open circuit is represented in Fig. 40 (a), the ordi nates 
 
 Cu 
 
 f 
 
 o 
 
 n-Cu SQ n-Zn 
 
 FIG. 40. 
 
 representing the potentials of the different parts of the circuit. 
 The horizontal lines, AB, CD, DE and FG, illustrate the very 
 important fact that the copper, the zinc and the solutions are 
 each of a definite constant potential, and the ordinates, BC and 
 EF, that there are sudden alterations of potential at the junc- 
 tions metal/solution. For simplicity the solutions of zinc 
 sulphate and copper sulphate are represented as being at the 
 same potential, which is only approximately true. It is as- 
 sumed for the present that the difference of potential be- 
 tween copper and N copper sulphate solution is 0-585 volts, 
 the copper being positive, and that the potential difference, 
 Zn/ZnSO 4 , is 0^52 volts, the metal being negative. The 
 
364 OUTLINES OF PHYSICAL CHEMISTRY 
 
 total difference of potential between zinc and copper on open 
 circuit is thus 0-585 + 0*52 = 1*105 volts. 
 
 When the circuit is closed by connecting the copper and zinc 
 by a wire of fairly high resistance, R, the distribution of poten- 
 tial in the cell is as shown in Fig. 40 (ft). The sudden changes 
 of potential at the junctions ZnSO 4 /Zn and CuSO 4 /Cu are 
 of the same magnitude as before, but the difference of potential 
 between the zinc and copper, measured by the vertical height, 
 AG, is much less than on open circuit. This is owing to the 
 fall of potential in the cell owing to the resistance of the elec- 
 trolyte, so that the solution in contact with the zinc is at a 
 higher potential than that in contact with the copper, as repre- 
 sented by EDC. If C is the current passing through the cell, 
 and r is the resistance of the electrolyte, the E.M.F. of the cell 
 on closed circuit is given by E = CR + Cr, and CR, the fall of 
 potential in the external wire (represented in the figure by the 
 vertical distance AG), approaches the more nearly to the E.M.F. 
 of the same cell on open circuit the greater R is compared with 
 r (compare p. 355). 
 
 The E.M.F. of such a combination as the Daniell cell is the 
 algebraic sum of the E.M.F.s at the two junctions, and is 
 represented by the formula 
 
 Where P x and P 2 are the solution pressures of zinc and copper 
 respectively, p^ represents the osmotic pressure of the zinc ions 
 in the solution, and / 2 that of the copper ions. The values of 
 pi and / 2 are therefore known, but the absolute values of the 
 solution pressures P x and P 2 are unknown. The sign of 
 E 2 is due to the fact that at that junction ions are leaving the 
 solution. 
 
 In obtaining E as the algebraic sum of the differences of 
 potential E l and E 2 at the two junctions, it is naturally of the 
 utmost importance to take the values of E x and E 2 with their 
 proper sign. Perhaps the best method of avoiding errors in this 
 
ELECTROMOTIVE FORCE S 6 5 
 
 connection is to consider the tendency of one kind of electricity, 
 say positive electricity, to pass round the circuit. In going 
 round the circuit in the Daniell cell, starting with the zinc, 
 the different junctions are met with in the order 
 Zn | ZnSO 4 | CuSO 4 | Cu 
 
 0-52 0-585 
 
 > 
 
 1*105 
 
 and this is a very convenient method of representing the Daniell 
 or any other cell. 
 
 Now at the junction Zn/ZnSO 4 positive electricity tends to 
 pass from zinc to solution at a potential (pressure) of 0*52 
 volts, as indicated by the arrow. Further, as the osmotic pres- 
 sure of Cu" ions in copper sulphate solution is greater than the 
 solution pressure of copper, positive electricity tends to pass 
 across the junction CuSO 4 /Cu, in the direction of the arrow at 
 an E.M.F. of 0-585 volts. It is clear that the forces at the 
 two poles are in the same direction, and therefore positive 
 electricity tends to pass through the solution in the direction 
 indicated by the lower arrow at a total E.M.F. of 
 
 0*520 + 0-585 = 1-105 volts. 
 
 Further illustrations are given at a later stage. 
 
 Influence of Change of Concentration of Salt Solution 
 on the E.M.F. of a Cell The general equation just given 
 may be written in a slightly different form by substituting for 
 the pressures the corresponding concentrations. Considering 
 first the solution pressure, P lf of the zinc, it is theoretically 
 possible to choose a Zn" ion concentration, C lf such that its 
 osmotic pressure will just balance the solution pressure of the 
 metal ; this may be substituted for P 1 in the general equation. 
 Similarly, for p v the osmotic pressure of the zinc ions in the 
 solution, we may substitute the corresponding concentration, 
 c^ Dealing in the same way with the copper side of the cell, 
 the equation for the Daniell cell (or any other cell of similar 
 type) becomes 
 
366 OUTLINES OF PHYSICAL CHEMISTRY 
 
 C C 
 
 v^i i V-/r 
 
 In this form the general equation may be employed to in- 
 vestigate the question as to how the E.M.F. of the cell is 
 affected by varying the concentration of the salt solutions. 
 For the zinc side, since C x is greater than c lt it is clear that 
 the quotient Cj/^, and therefore E lf is increased by diminish- 
 ing c lt the concentration of the Zn" ions. For the copper 
 side, however, as C 2 is less than c% (p. 342), the quotient 
 C 2 /^, therefore E 2 , will evidently be diminished by diminishing 
 the copper sulphate concentration. The general rule with re- 
 gard to the influence of change of ionic concentration on the 
 E.M.F. of a cell may be expressed as follows : Diminishing th& 
 concentration of a solution from which ions are separating lowers, 
 and diminishing the concentration of a solution into which new 
 ions are going increases, the E.M.F. of a cell. It is evident 
 from general principles that the effect must be as described ; 
 in the first case, the tendency to the separation of ions is 
 lessened, and the E.M.F. falls ; in the second case, the 
 entrance of new ions is facilitated, and the E.M.F. increases. 
 
 If the concentration of the Or* ions in the solution is pro- 
 gressively diminished, a point must be reached at which the 
 solution pressure of the metal is just balanced by the osmotic 
 pressure of the Cu- ions. If the concentration is still further 
 diminished, the tendency for copper - to pass into solution will 
 steadily increase, and ultimately may become greater than the 
 tendency of zinc to pass into solution. It should therefore be 
 theoretically possible to reverse the direction of the current in 
 the Daniell cell by sufficiently diminishing the Cu" ion con- 
 centration, and this state of affairs can be realised experimentally 
 by adding potassium cyanide to the copper sulphate solution. 
 
 A further important deduction can also be drawn from the 
 general equation. As c and c^ stand for the concentration 
 of the positive ions in the solution, the E.M.F. of the cell 
 should be independent of the nature of the negative ion, pro- 
 
ELECTROMOTIVE FORCE 367 
 
 vided that the salts are equally ionised. This consequence of 
 the theory is completely borne out by experiment. For twenty- 
 one different thallium salts, in N/5o solution, the difference of 
 potential between metal and solution varied only from 0*7040 
 to 0-7055 volts, the slight variations being readily accounted 
 for by differences in the degree of ionisationr. 
 
 Concentration Cells We have now to consider what are 
 termed " concentration cells," cells in which the E.M.F. de- 
 pends essentially on differences of concentration. In some 
 respects, concentration cells are simpler than those of the 
 Daniell type, which have so far been considered. 
 
 Concentration cells may be divided into two main classes 
 
 (a) Those in which the solutions (and therefore the active 
 
 ions) are of different concentrations. 
 
 (b) Those in which the electrode materials yielding the ions 
 
 are of different concentrations. 
 
 (a) Concentration Cells with Solutions of Different Concen- 
 trations As a type of the elements in question, we will con- 
 sider a cell in which silver electrodes dip in solutions of silver 
 nitrate of different concentrations, c l and c The arrange- 
 ment for the practical determination of the total E.M.F. of 
 such a combination is shown in Fig. 41, where A and B repre- 
 sent the cells containing the silver nitrate solutions and the 
 vessel C contains an indifferent electrolyte. As this form of 
 cell is largely employed in measurements of E.M.F., it may be 
 well to describe it fully. It consists of a glass tube 3-4 cm. 
 wide, with a straight side-tube D on one side and a bent side- 
 tube E on the other, the latter being employed for making con- 
 nection with the indifferent electrolyte in C as shown. Into the 
 lower end of a glass tube, F, is cemented a thick rod of silver 
 covered with the finely-divided metal by electrolysis, and the 
 glass tube is held by a cork closing the cell. The cell is filled 
 with a solution of silver nitrate of definite strength through the 
 bent tube by suction through the straight side-tube, D, which 
 is then closed by a clip. The other " half-cell," B, is prepared 
 
368 OUTLINES OF PHYSICAL CHEMISTRY 
 
 in exactly the same way, but contains a solution of silver nitrate 
 of different concentration. The ends of the bent tubes are 
 then dipped into an indifferent electrolyte in the vessel, C, as 
 shown, and the total E.M.F. of the combination determined by 
 the potentiometer method in the usual way, connection with 
 the silver electrodes being made by wires passing down the 
 interior of the glass tubes. In this case, the general equation 
 for an electrolytic cell, 
 
 simplifies to 
 
 E = 
 
 0- 
 
 (0 
 
 FIG. 41. 
 
 
 
 since C, the solution pressure of the metal, is the same on both 
 sides, and is therefore eliminated. A cell of the type 
 
 Ag I AgNO 3 dil AgNO 3 conc Ag 
 
 works in such a way that silver is deposited from the more 
 concentrated solution, in which the osmotic pressure is higher, 
 and is dissolved at the pole in contact with the weaker solution, 
 which offers less resistance to the entrance of Ag 4 ions. The 
 
ELECTROMOTIVE FORCE 369 
 
 change, therefore, proceeds in such a way that the differences 
 of concentration tend to equalize, and when the solutions have 
 reached the same concentration, the current stops. Positive 
 electricity therefore passes in the element from the weak to the 
 strong solution, as indicated by the arrow, and in the connecting 
 wire from the strong to the weak solution ; the electrode in 
 contact with the strong solution becomes positively charged, 
 the other electrode negatively charged. 
 
 The equation shows that the E.M.F. of such a concentra- 
 tion cell depends only on the respective concentrations of the 
 positive ions in the two solutions and their valency, and not on 
 the nature of the electrodes or on the nature of the anions, and 
 the experimental results are in full accord with this deduction. 
 Otherwise expressed, the E.M.F. of any element made up of a 
 univalent metal M dipping in solutions of one of its salts of 
 different concentration is of the same absolute value as that of 
 the silver concentration cell, provided that the solutions are 
 of corresponding concentration, and ionised to the same extent. 
 Further, if the solutions are dilute, and electrolytic dissociation 
 therefore fairly complete, the ratio of the ionic concentrations in 
 different dilutions will be approximately the same as the ratio of 
 the concentrations themselves. Thus, in the example under 
 consideration, the ratio c^c^ for i/ioo molar, and i/iooo molar 
 solutions, will be approximately 10 : i ; log^/^ is therefore i, 
 and the value of E for the cell 
 
 Ag I AgN0 3 | AgN0 3 I Ag 
 j m/iooo j m/ioo \ 
 
 is 0*05 S/n = 0*058 volts, since n, the valency of the ions con- 
 cerned, is unity. 
 
 If, however, the solutions are more concentrated, the fact 
 that ionisation is incomplete must be taken into account in 
 calculating the E.M.F. of a cell. Suppose, for instance, it is 
 required to calculate the E.M.F. of the cell 
 
 Ag I AgNO 3 m/ioo \ AgNO 3 tn/io \ Ag. 
 24 
 
370 OUTLINES OF PHYSICAL CHEMISTRY 
 
 N/io silver nitrate solution is ionised to the extent of 82 
 per cent, at 18, whence c^ = 0*082, and N/ioo silver nitrate 
 to the extent of 94 per cent., whence q = 0*0094. We have 
 therefore c^c^ = 0*082/0*0094 = 872, and E = 0*054 volts, in 
 excellent agreement with the experimental value. 
 
 Strictly speaking, it is not justifiable in cells of this type to 
 neglect the contact difference of potential between the two 
 solutions, which may amount to a considerable fraction of the 
 total E.M.F. The accurate formula for the calculation of the 
 E.M.F. of cells of this type is given in a succeeding section (p. 
 384). If, however, both solutions contain an indifferent electro- 
 lyte in equivalent concentration great in comparison with those 
 of the active salt, the difference of potential at the liquid 
 junction becomes negligible, and the above formula (i) holds 
 accurately (cf. p. 383). 
 
 It is evident from the formula that the E.M.F. of a con- 
 centration cell cannot be greatly altered by increasing the con- 
 centration on one side, owing to the limited solubility of the 
 salts used as electrolytes. On the other hand, the E.M.F. 
 may be greatly altered by diminishing the ionic concentra- 
 tion on one side. Conversely, when a cell is made up with 
 a solution of silver nitrate of known Ag* ion concentration, t lt 
 and one of unknown concentration, c , and the E.M.F. of the 
 cell is measured, c can readily be calculated. This principle 
 has been applied more particularly for the determination of 
 very small ion concentrations, and may be illustrated by the 
 determination of the Ag* ion concentration in a saturated 
 solution of silver iodide. When the concentration on one side 
 is very small, it is usual to add some salt, with or without a 
 common ion, to eliminate the potential difference at the liquid 
 junction, and also to increase the conductivity in the cell, so 
 as to render the measurements more accurate. In this case 
 potassium nitrate may conveniently be used. The observed 
 E.M.F. of the cell 
 
 Ag | KNO 3 + Agl | AgNO 8 o'ooim + KNO 3 1 Ag 
 
ELECTROMOTIVE FORCE 371 
 
 is 0*22 volts. Since c^ = 0*001, we have 
 
 E = 0-22 = 0-058 Iog 10 (o -oo i /<:<,), 
 
 whence c = 1*6 x io~ 8 . In other words, a litre of a saturated 
 solution of silver iodide contains i'6 x io~ 8 mol of silver 
 iodide, in excellent agreement with the value, 1-5 x io~ 8 mol, 
 obtained from conductivity measurements (p. 301). 
 
 (b) Cells with Different Concentrations of the Elec- 
 trode Materials (Substances Producing Ions) Not only 
 can concentration cells be obtained by employing different con- 
 centrations of an electrolyte, but also by using different con- 
 centrations of metals or other electrode materials yielding ions. 
 The concentration of metals can for our present purpose be 
 most satisfactorily varied by employing their solutions in 
 mercury, the so-called amalgams. For example, a concentra- 
 tion cell can readily be built up as follows : 
 Zinc amalgam conc./zinc sulphate solution/ zinc amalgam dilute 
 
 which differs from the cells of the first type in that the osmotic 
 pressure of the zinc ions in contact with the two poles is the 
 same, but the concentration, and therefore the solution pressure 
 of the metal on the two sides, is different. 
 
 The E.M.F. of a cell of this type is represented by the 
 general formula (p. 361) 
 
 Since the same solution (in this case zinc sulphate) is in contact 
 with both electrodes, p =/ 2 , and the formula becomes 
 
 On the assumption that Pj and P 2 , the solution pressure of the 
 zinc in the concentrated and dilute amalgams respectively, are 
 proportional to the respective concentrations, we obtain 
 
 EJ 
 
372 OUTLINES OF PHYSICAL CHEMISTRY 
 
 which is exactly the same form' of equation as that for cells 
 with different electrolyte concentrations. 
 
 As the solution pressure of the zinc is higher in the con- 
 centrated amalgam, it passes into solution from the latter and 
 is deposited in the less concentrated amalgam, so that the con- 
 centrations tend to become equal. It follows that positive 
 electricity passes in the cell from the concentrated to the dilute 
 amalgam, as shown by the arrow. 
 
 As an illustration, the E.M.F. of a cell for which Cj = 0*14 
 mol and C 2 = 0*00214 mol of zinc per litre of amalgam at 23 
 may be calculated. If it be assumed that zinc is unimolecular 
 when dissolved in mercury, n = 2, and 
 
 -TV o'oooioSi x 296. 0*14 
 
 E = - Iog 10 = 0-053 volts, 
 
 in excellent agreement with the observed value, 0-052 volts. 
 
 So far, the possible effect of mercury on the potential has 
 been disregarded, and this is justified by the excellent agreement 
 between observed and calculated values for the E.M.F. on the 
 assumption that mercury simply acts as an indifferent solvent. 
 The explanation is that for a mixture of two metals it is the 
 metal with the higher solution pressure that goes into solution, 
 and mercury can consequently be used as solvent in potential 
 measurements for any metal which is "less noble," /.*., which 
 has a higher solution pressure than mercury itself. 
 
 As indicated above, the E.M.F. of a metal dissolved in mer- 
 cury depends on the concentration. The difference of potential 
 between a saturated solution of a metal in mercury and an 
 aqueous solution of one of its salts is the same as that between 
 the salt solution and the pure metal, and even for dilute amal- 
 gams the E.M.F. is not very different from that of the pure 
 metal, as the example shows. On the other hand, the potential 
 of a metal in chemical combination with a more noble metal 
 may be quite different from that of the pure metal. 
 
 The energy relations in concentration cells in which very 
 dilute solutions are employed are remarkable. In the silver 
 
ELECTROMOTIVE FORCE 373 
 
 nitrate concentration cell described above, the change con- 
 sisted simply in bringing Ag ions from the pressure /j to 
 the lower pressure / 2 - When a perfect gas expands from the 
 pressure / x to / 2 , no internal work is done, and this is the 
 more nearly the case for ordinary gases the lower the pressures. 
 In an exactly corresponding way no internal work will be done 
 when a salt is further diluted in sufficiently dilute solution ; in 
 other words, the heat of dilution will be zero. This means 
 that the change of chemical energy' (Q in the Helmholtz 
 formula), also termed the heat of reaction, is zero, so that the 
 electrical energy obtained from a concentration cell with suffi- 
 ciently dilute solutions does not come from a chemical change 
 at all, but entirely from the surroundings. Under these circum- 
 stances, as Q is zero, the Helmholtz formula simplifies to 
 
 F T 
 T J?T 
 
 Electrodes of the First and Second Kind. The Calomel 
 Electrode So far only electrodes which are reversible with 
 regard to the positive ion have been considered; these are 
 termed electrodes of the first kind. In an exactly similar way it 
 is possible to construct electrodes which are reversible with 
 regard to the negative ion these are termed electrodes of the 
 second kind. They are prepared by immersing a metal in a 
 saturated solution of one of its difficultly soluble salts, which 
 solution also contains a salt with the same anion as the in- 
 soluble salt. The E.M.F. of the electrode depends only on 
 the concentration of the anion, since the concentration of all 
 the other substances is constant. The most important 
 electrode of this type is the calomel electrode, which consists 
 of mercury in contact with solid mercurous chloride and a 
 saturated solution of the latter salt in potassium chloride solu- 
 tion as electrolyte. The calomel electrode is reversible with 
 regard to Cl' ions, just as the Cu/CuSO 4 electrode is reversible 
 with regard to Cu- ions. If positive electricity passes from 
 
374 OUTLINES OF PHYSICAL CHEMISTRY 
 
 metal to solution, the mercury combines with Cl' ions and 
 calomel is formed; if passed in the reverse direction chlorine 
 goes into solution and solid calomel disappears. The electrode, 
 therefore, acts like a plate of solid chlorine, which gives up or 
 absorbs the element depending on the direction of the current. 
 As the tendency of Cl' ions to enter or leave the solution 
 depends on the concentration of Cl' ions already present, the 
 difference of potential between mercury and the solution must 
 depend on the concentration of the potassium chloride solution 
 used, as already pointed out. A normal solution is mostly 
 largely employed. 
 
 The calomel electrode is largely used as a normal electrode 
 
 by means of which 
 the E.M.F.s of other 
 electrodes may be 
 compared ; its chief 
 advantage for this 
 purpose is that it can 
 readily be reproduced 
 with an accuracy of 
 about i millivolt. A 
 convenient form of 
 the electrode is shown 
 in Fig. 42. A vessel 
 of the type already 
 described in connec- 
 tion with concentra- 
 tion cells (p. 347) 
 may conveniently be 
 used. A layer of dry 
 mercury is first placed 
 in the bottom of the 
 
 FIG. 42. 
 
 vessel, then a paste made by rubbing in a mortar mercury and 
 calomel with some of the potassium chloride solution, and the 
 vessel is then filled up with ^-potassium chloride solution 
 
ELECTROMOTIVE FORCE 375 
 
 which has previously been saturated with calomel by shaking 
 with excess of the latter. Connection with the mercury may 
 conveniently be made by means of a platinum wire sealed at 
 the bottom into a glass tube A, the latter passing up through 
 the rubber stopper closing the vessel. In making measure- 
 ments, the bent side-tube, C, must also be filled with the 
 potassium chloride solution. This is done by suction at the 
 straight side-tube, B, which is then closed by a clip. 
 
 For measuring the potential of another electrode by means 
 of the calomel electrode, the arrangement already shown in 
 Fig. 41 is used. 
 
 Neglecting for the present the differences of potential at the 
 liquid junctions, the E.M.F. of the combination in question is 
 the algebraic sum of the differences of potential at the two 
 metal/solution junctions. It follows that if the single potential 
 difference between mercury and solution is known, the single 
 potential difference at the other electrode can readily be cal- 
 culated. Unfortunately no single potential difference is known 
 with certainty (see next section), and it is, therefore, necessary to 
 refer them to an arbitrary standard. Two such standards are 
 in general use, (a) the so-called " absolute " standard ; (S) the 
 hydrogen standard. As regards the first standard, Ostwald as- 
 sumes that the potential difference between mercury and the 
 solution in the normal calomel electrode is 0-560 volts at 18, 
 the mercury being positive, and differences of potential referred 
 to this standard are termed " absolute potentials," e c (see next 
 section). On the other hand, Nernst refers E.M.F.s to the 
 hydrogen standard, c h the difference of potential between a 
 platinum electrode saturated with hydrogen and a solution of 
 an acid normal with regard to H- ions being put equal to zero. 
 The "absolute" potentials, referred to the calomel electrode, 
 have a theoretical basis, and there is reason to suppose that 
 the real, but at present unknown, single potential differences 
 are not very different from the " absolute " potentials. Inde- 
 pendently of this, however, the use of the calomel electrode in 
 actual measurements is justified by the fact that it can be re- 
 
376 OUTLINES OF PHYSICAL CHEMISTRY 
 
 produced with a high degree of accuracy. The use of the 
 hydrogen electrode as standard is purely arbitrary, as it is not 
 pretended that the difference of potential between electrode 
 and solution is actually zero, but the reference of potential 
 differences to this standard has certain advantages. It is, in 
 fact, usual to make the actual measurements with the calomel 
 electrode, and then to refer them to the hydrogen standard, 
 on the basis that when the hydrogen electrode is taken as zero 
 the E.M.F. of the normal calomel electrode is + 0*283 volts; 
 that of the electrode with N/ioKCl + 0-336 volts, at 18. 
 
 In order to illustrate the use of the calomel electrode for 
 potential measurements, the separate determination of the dif- 
 ferences of potential metal/solution for the two parts of the 
 Daniell cell will be considered. When the zinc electrode is 
 combined with the calomel electrode, as shown in Fig. 41, to 
 form the cell 
 
 Zn 
 
 i -080 
 
 the E.M.F. of the combination, as shown by potentiometer 
 measurements, is 1-080 volts, the zinc being negative with 
 regard to mercury, so that positive electricity flows in the cell 
 from zinc to mercury, as indicated by the lower arrow. In 
 order to obtain the potential difference Zn/ZnSO 4 , we proceed 
 as follows (p. 365). It is known that mercury in contact with 
 a solution of calomel becomes positively charged, and that for 
 the calomel electrode the tendency for positive electricity to 
 pass across the junction towards the mercury is 0*560 volts. 
 The tendency for positive electricity to pass round the circuit is 
 equivalent to 1-080 volts, hence the E.M.F. at the Zn/ZnSO 4 
 junction must act in the direction shown by the upper left-hancj 
 
 -ZnSO 4 
 
 KC1 
 (in vessel C) 
 
 Hg 2 Cl 2 in 
 KC1 
 
 0*520 0-560 
 
ELECTROMOTIVE FORCE 377 
 
 arrow, and is 1*080 -0*5 60 = '5 20 volts. In other words, the 
 E.M.F. at the junction Zn/ZnSO 4 is 0-520 volts, the zinc being 
 negatively charged. 
 
 Similarly, the observed E.M.F. of the cell 
 
 Cu | -CuS0 4 | KC1 | Hg 2 Cl 2 in -KCl | Hg 
 
 < > 
 
 0-585 0-560 
 
 0*025 
 
 is 0-025 volts, the copper being positive with regard to mercury, 
 hence positive electricity flows from mercury to copper in the 
 cell, as indicated by the lower arrow. As far as the calomel 
 junction is concerned, the tendency for positive electricity to 
 flow round the circuit is equivalent to 0-560 volts towards the 
 right, as indicated by the arrow. Hence in order that for the 
 whole cell the tendency of positive electricity may be to flow 
 towards the left at a potential of 0-025 v l ts tne E.M.F. at the 
 Cu/CuSO 4 junction must act in the opposite direction to that 
 at the calomel junction and exceed it by 0-025 volts. The 
 E.M.F. at the junction Cu/CuSO 4 is therefore 0-585 volts and 
 positive electricity flows from solution to copper, as indicated 
 by the arrow. 
 
 The total E.M.F. of the cell Zn/ZnSO 4 /CuSO 4 /Cu is, there- 
 fore, -0-520 + ( 0-585)= 1*105 volts, which agrees with the 
 value obtained by direct measurement (p. 34^). It is evident 
 from the above that although the single potential differences at 
 the junctions depend upon the value of the potential assumed 
 for the standard, the E.M.F. of the complete cell does not 
 depend upon the E.M.F. of the standard, which is eliminated. 
 
 If referred to the hydrogen electrode as standard, the poten- 
 tial difference Zn//rZnSO 4 is 0*520 + ( - 0-283) = ~~ '83 
 volts, and that for Cu/?rCuSO 4 is + 0-585 + ( - 0*283) 
 = + 0-302 volts, the E.M.F. of the Daniell cell being as before 
 j= (-0-803) + ( - 0-302) = 1*105 volts, 
 
378 OUTLINES OF PHYSICAL CHEMISTRY 
 
 As reference electrodes for alkaline solutions, the mercuric ox- 
 ide electrodes Hg/HgO in .NaOH and Hg/HgO in /ioNaOH 
 are most convenient. The same kind of vessel may be used 
 as for the calomel electrode. The potentials, which become 
 constant after 2-3 days, are for the normal electrode Eh = + 
 0-114 volt and for the N/io electrode + 0-169 volt at 18. 
 
 As reference electrode for acid solutions the hydrogen 
 electrode (p. 375) or the mercurous sulphate electrode 
 Hg/Hg 2 SO 4 in .H 2 SO 4 may be used. For the latte e A = + 
 0-689 volt at 1 8. 
 
 Single Potential Differences. The Capillary Electro- 
 meter. When mercury and sulphuric acid are in contact in 
 a capillary tube, and the arrangement is connected with a 
 source of E.M.F. in such a way that the mercury is in contact 
 with the negative pole and the acid with the positive pole, the 
 area of the surface of separation between acid and mercury 
 tends to diminish. The following out of this observation of 
 Lippman's has led to an approximate estimate of the absolute 
 differences of potential at metal/solution junctions. 
 
 When mercury and sulphuric acid have been in contact for 
 some time, it is probable that there is a constant difference of 
 potential between them, brought about in a rather complicated 
 way. We have already learnt that well-defined differences of 
 potential are established when a metal is in contact with a 
 solution of one of its salts of definite concentration, and that 
 is probably the state of affairs in the present case. We may 
 suppose that some of the mercury dissolves in the sulphuric 
 acid to form mercurous sulphate, and that the solution im- 
 mediately in contact with the mercury is saturated with regard 
 to the salt. As, however, the osmotic pressure of solutions of 
 mercury salts is in general greater than the solution pressure 
 of mercury, H g 2 " ions deposit on the mercury and the latter 
 becomes positively charged with regard to the solution. The 
 two kinds of electricity attract each other, and we will assume 
 with Helmholtz that the effect of this attraction is that there is 
 a layer of positive electricity near the surface of the mercury 
 
ELECTROMOTIVE FORCE 379 
 
 holding a corresponding layer of negative electricity near the 
 surface of the acid (" Helmholtz double layer ") (cf. Fig. 38). 
 
 Now there will be* a certain surface-tension at the junction 
 mercury /solution in the capillary tube, and, as is well known, 
 the effect of surface tension is to make the areas of the surfaces 
 in contact as small as possible. This tendency will, however, 
 be counteracted by the electric layers; the positive charges 
 will repel each other and tend to enlarge the surface, and 
 the same is true of the negative charges. The effect of the 
 difference of potential is, therefore, to diminish the surface 
 tension. The fact that a contrary E.M.F. applied to the 
 junction tends to diminish the surface of separation between 
 acid and mercury will now be readily understood. The con- 
 trary E.M.F. diminishes the difference of potential between 
 acid and mercury, part of the force diminishing the surface 
 tension is removed, and the latter attains more nearly its true 
 value when undisturbed by electrical forces. When the con- 
 trary E.M.F. is gradually increased, the surface tension increases 
 at first, attains a maximum value, beyond which it gradually 
 diminishes. It is plausible to suppose that the surface tension 
 increases as the difference of potential between mercury and 
 acid gets slnaller and smaller, that it attains its maximum value 
 when the contact E.M.F. at the junction is just neutralized by 
 the contrary E.M.F., and that it again diminishes as the latter 
 is further increased and the surfaces become charged with 
 electricity of opposite sign to the original charges. This at 
 once gives us a method of determining single differences of 
 potential. It is only necessary to note when the surface tension 
 attains its maximum value ; under these circumstances the 
 applied E.M.F. is clearly equal to the single difference of 
 potential at the junction mercury/solution and the problem 
 as to the value of a single potential difference is solved. In 
 this way Ostwald estimated the E.M.F. of the normal calomel 
 electrode at 0*560 volts. 
 
 Unfortunately the matter is not quite so simple as the above 
 
380 OUTLINES OF PHYSICAL CHEMISTRY 
 
 considerations would lead us to suppose, and it is fairly certain 
 that the absolute potentials arrived at in this way may differ 
 to some extent from the true values. It has'already been pointed 
 out that two standards are in use, and that the use of the 
 calomel electrode for measuring differences of potential has 
 certain practical advantages. 
 
 Before considering another method which has been suggested 
 for measuring single differences of potential, it should be 
 mentioned that the phenomena just described have been 
 utilized in the construction of an electrometer the so-called 
 capillary electrometer which, as already mentioned (p. 356) 
 can be used as a null instrument in E.M.F. measurements by 
 the compensation method. A convenient form of the capillary 
 electrometer is represented in Fig. 43. 
 A tube C, 4*5 mm. internal diameter 
 and a bulb-tube A are connected by a 
 vertical capillary tube B 0-5 mm. in- 
 ternal diameter. So much mercury is 
 poured into C that it stands at a con- 
 venient height in the capillary tube. A 
 quantity of mercury is also placed in A 
 and the latter and the capillary are then 
 filled up with dilute sulphuric acid. As 
 indicated in the figure the two quantities 
 of mercury can be connected with the 
 positive and negative poles of a source 
 of E.M.F. when required. 
 
 The wire D connected with the 
 mercury in the bulb-tube is sealed 
 i n a gl ass tu be so as not to come in 
 contact with the sulphuric acid. When 
 the apparatus is so arranged that the 
 mercury and acid are in equilibrium at a point in the capillary 
 tube and the two quantities of mercury are then connected with 
 a source of E.M.F. the potential at the junction in the capillary 
 tube will alter owing to alteration in the concentration of mer- 
 Qurous salt produced by the current. The surface tension be- 
 
 FIG- 43- 
 
ELECTROMOTIVE FORCE 381 
 
 tween acid and mercury must therefore also change (p. 379) 
 and also the position of the junction in the capillary, since the 
 position of the meniscus depends to some extent on the surface 
 tension between mercury and acid. 
 
 The use of the apparatus as an electrometer will now be 
 evident. It is best so to arrange matters that the mercury at the 
 narrow surface is connected with the negative pole of the external 
 source of E.M.F. through a tapping key, and the junction is 
 observed through a small microscope. If an external E.M.F. is 
 applied, the surface will move when the key is momentarily 
 depressed, and for small differences of potential (up to 0*01 
 volt) the movement of the meniscus is proportional to the applied 
 E.M.F., so that the name electrometer is justified. When the 
 applied E.M.F. is zero, no movement of the meniscus occurs on 
 making contact, and the electrometer may therefore be used as 
 a null instrument. When not in use, the electrometer should 
 be connected up with a cell of E.M.F. not exceeding i volt. 
 
 An alternative very instructive method of determining single 
 potential differences, the theory of which is due mainly to 
 Nernst and the practical realization to Palmaer, will now be de- 
 scribed. When mercury in a fine stream is allowed to flow into 
 an electrolyte containing a definite concentration of mercurous 
 salt (e.g., mercurous chloride), Hg 2 " ions from the solution 
 deposit on the drops as they enter (the osmotic pressure of 
 Hg 2 "ionsin the solution being greater than the solution pres- 
 sure of the mercury), the drops thus become positively charged, 
 and further become surrounded with a layer of the liberated 
 Cl' ions. When the drops reach the bottom of the vessel 
 containing the electrolyte, the positive ions are given up and 
 reunite with the Cl' ions to form more calomel. The net result 
 of this process is that the solution gets poorer in calomel where 
 the drops enter, and richer where they unite with the mercury. 
 A concentration cell is thus formed, and it is evident that 
 positive electricity must flow from the weak to the strong 
 solution, that is, from top to bottom of the vessel, a deduction 
 which is borne out by experiment. 
 
382 OUTLINES OF PHYSICAL CHEMISTRY 
 
 Now it must be possible to reduce the concentration of Hg 2 " 
 ions to such a point that the osmotic pressure of the Hg 2 " 
 ions is just equal to the solution pressure of the mercury ; there 
 is then no deposition of Hg 2 " ions on the entering drops, and no 
 current flows. Conversely ', when no current results when mercury 
 is dropped into an electrolyte containing Hg^- ions, the difference 
 of potential between mercury and the solution must be zero. If 
 the Hg 2 - ion concentration is still further reduced, the solution 
 pressure of the mercury is greater than its osmotic pressure 
 in the solution, and the current flows in the opposite direction. 
 
 The Hg 2 " ion concentration was reduced by adding potassium 
 cyanide till the point of no current and therefore zero difference 
 of potential was reached. The solution in equilibrium with 
 mercury under these conditions may be termed the null solu- 
 tion. If then a cell is built up of the type 
 
 Hg | null solution | solution of salt of metal M | M 
 o ^ <? 2 
 
 * 2 , the difference of potential between metal and solution, can 
 be determined directly if the E.M.F. e 1 at the junction of the 
 solution is known or can be made negligible. 
 
 In this way Palmaer has found that the E.M.F. at the junction 
 Hg//ioKCl saturated with Hg 2 Cl 2 is 0-573 volts at 18, cor- 
 responding with about 0*520 volts for /iKCl, as compared 
 with Ostwald's value of 0-560 volts. 
 
 Some writers consider that the problem of the determination 
 of single potential differences is thus finally settled, but Palmaer 
 himself does not consider that all the difficulties of the measure- 
 ments have been overcome, so that the above results should 
 only be taken as provisional. 1 
 
 Potential Differences at Junction of Two Liquids Up 
 to the present, we have left out of account the possible dif- 
 ferences of potential at the junction of two solutions. When 
 the E.M.F. of a cell is considerable, the error thus arising is 
 only slight, but if the E.M.F. is small, as for many concentra- 
 1 Cf. Palmaer, Zeitsch. physikal Chern., 1907, 59, 129. 
 
ELECTROMOTIVE FORCE 383 
 
 tion cells, the potential difference at the liquid contact be- 
 comes of importance. 
 
 It has been shown by Nernst that in many cases these dif- 
 ferences of potential can be calculated according to his theory 
 of electromotive force, and the results obtained in this way 
 have been fully confirmed by experiment. The calculation 
 is effected most readily for solutions of the same electrolyte 
 in different concentrations, for example, solutions of hydro- 
 chloric acid. When the solutions are brought in contact the 
 acid will tend to diffuse from the more concentrated to the more 
 dilute solution. As, however, the acid is highly ionized, the 
 H- and Cl' ions will diffuse independently, and, as the former 
 move the more rapidly, the dilute solution will soon contain an 
 excess of H* ions and the strong solution an excess of Cl' ions. 
 Owing to the electric charges conveyed by the moving ions, 
 the dilute solution will become positively charged, and the 
 strong solution negatively charged. However, the excess of 
 positive electricity in the dilute solution will retard the entrance 
 of H* ions and accelerate the Cl' ions, so that in a short time 
 the ions will be moving at the same rate. The difference of 
 potential thus produced will persist until both solutions attain 
 the same concentration. The above considerations show that 
 the contact difference of potential between two solutions is due to 
 the different migration velocities of the two tons, and the dilute 
 solution takes the potential corresponding with that of the more 
 rapid ion. The contact difference of potential between dif- 
 ferent solutions of the same salt will be the smaller the more 
 nearly the speed of the two ions agrees, and this explains why 
 solutions of potassium chloride and of ammonium nitrate are 
 used as connecting solutions in potential measurements (com- 
 pare p. 370). 
 
 It can easily be shown that the potential difference E be- 
 tween two solutions of a binary electrolyte with univalent 
 ions (for example, hydrochloric acid) is represented by the 
 formula 
 
384 OUTLINES OF PHYSICAL CHEMISTRY 
 u ~ v 
 
 where ^ and 2 represent the ionic concentrations of the two 
 solutions, and u and v the migration velocities of the anion 
 and cation respectively. From the above equation it can be 
 calculated that the contact E.M.F. between N/io and N/ioo 
 hydrochloric acid is 0*036 volt, a result which is fully con- 
 firmed by experiment. 
 
 From the above result, the value of E may be calculated for 
 the cell 
 
 Ag | AgN0 3 /io | AgN0 3 /*/ioo | Ag 
 
 when the contact difference of potential between the two 
 solutions is not practically eliminated by the use of potassium 
 nitrate. Taking the junctions in order, we have 
 
 E. 
 
 RT lo glo ' - RT Iog 10 & 
 
 mm? S81ogl0 ., 
 
 for room temperature. 
 
 For silver nitrate = 0-522 (p. 233), ^ = 0-082, c z = 
 
 0*0094. Hence 
 
 E = 0-522 x 2 x 0-058 x 0-945 = 0-057 volts, 
 
 in excellent agreement with the value found experimentally, 
 0-055 volts* 
 
 Gas Cells So far we have dealt only with solid substances 
 and amalgams as electrode materials, but it is interesting to 
 note that gases may be used in the same way. This is made 
 possible by using metallic electrodes, usually of platinum coated 
 with the finely-divided metal, as absorbents for the gases. The 
 
ELECTROMOTIVE FORCE 385 
 
 prepared platinum electrode is partially immersed in a solution 
 of an electrolyte containing ions derived from the gas, and the 
 gas is bubbled through till the potential difference between 
 electrode and solution becomes constant. 
 
 As an example of a gas electrode, the hydrogen electrode, 
 already referred to, will be described. The form of cell repre- 
 sented in Fig. 4 1 may be used ; it is half filled with normal 
 acid, the platinum pole, held by a well-fitting cork, is partially 
 immersed in the acid, and hydrogen gas is passed in by the 
 bent side-tube and allowed to bubble through the acid for ten 
 or fifteen minutes till the electrode is saturated. The straight 
 side-tube, which has been open, is now closed by a clip, and 
 the electrode is ready for use. The platinum pole itself usually 
 consists of a piece of platinum foil joined by hammering to a 
 platinum wire, the latter being sealed into the bottom of the 
 glass tube carried by the cork closing the cell. Electrical con- 
 nection may be made by a copper wire passing down through 
 the glass tube and dipping into a little mercury at the bottom, 
 the mercury being also in contact with the upper part of 
 the platinum wire which projects into the interior of the glass 
 tube. 
 
 The electrode is completely reversible and behaves like a 
 plate of metallic hydrogen. When positive electricity passes 
 from solution to metal, hydrogen ions are discharged accord- 
 ing to the equation 2H> -> H 2 ; when it goes in the contrary 
 direction gaseous hydrogen becomes ionised according to the 
 converse equation, H 2 -> 2H\ 
 
 A hydrogen concentration cell is obtained when two hy- 
 drogen electrodes, containing the gas at different pressures, are 
 combined in the usual way. Such cells correspond exactly with 
 those made up with amalgams of different concentrations. The 
 direction of the current is such that the pressures on the two 
 sides tend to become equal, so that hydrogen becomes ionised 
 at the high pressure sJ4e and is discharged as gas at the low 
 pressure side. 
 25 
 
386 OUTLINES OF PHYSICAL CHEMISTRY 
 
 In calculating the E.M.F.'s of such cells by the general formula 
 (p. 371) it has to be renumbered that, since the hydrogen 
 molecule contains two atoms the work gained in bringing a 
 mol of the gas reversibly from the pressure P l to the lower 
 pressure P 2 is RT log e PJ/P.J whilst if the same change is 
 carried out electrically, H 2 + 2F~ > 2H 4 , the energy concerned is 
 2F coulombs. Hence, since 2F = RT log e PJP 2 the E.M.F. 
 of the cell is 
 
 T7 RT 1 P l 
 
 E= JprlOg.gJ, 
 
 The same formula applies to gas cells in which chlorine and 
 other divalent gases are used (see also p. 398). On the other 
 hand, since 4F coulombs are associated with the solution of i 
 mol of oxygen (O 2 + 2H 2 O - 4F^4OH') the E.M.F. of an 
 oxygen concentration cell is 
 
 v RT , P, 
 E = -p !og p 1 . 
 
 4r P2 
 
 Another type of hydrogen concentration cell is obtained 
 when the gas concentration in the electrodes is constant and 
 the H' concentration in contact with the two electrodes is 
 different. An interesting cell of this type is built up as 
 follows: 
 
 H 2 (Pt) | N/io alkali | N/ioacid | H 2 (Pt) 
 
 Since the equilibrium H + OH'2H 2 O always holds, there 
 must be a minute concentration of H' even in alkaline solution 
 and therefore the above represents a hydrogen concentration 
 cell. From the E.M.F. of the above cell which, after ap- 
 plying a correction for the contact difference of potential, 
 amounts to o 6951 volt at 18, the product of the ionic 
 concentration for water, [H-] [OH'] = k can be calculated as 
 follows. From the general equation E = 0-0577 Iog 10 cjc^ 
 we have 
 
 0-6951 = 0-0577 Iog 10 ^ 2 
 
 f lt the H- concentration in N/io acid at 1 8, is 0*0888 
 hence 
 
ELECTROMOTIVE FORCE 387 
 
 and 2 , the H- concentration in N/io alkali = 0-0888 x lo" 12 ' 045 . 
 The OH/ concentration in the same solution, allowing for in- 
 complete dissociation, is 0*0892. Therefore 
 
 [H-] x [OH'] = 0-0888 x 10 - 12 ' 045 x 0-0892 = 0-7 x 10 ~ 14 
 at 1 8, which is in ex client agreement with the value found 
 by other methods (p. 294). 
 
 Cells in which the electrodes are in contact with different 
 gases, for example, the hydrogen-oxygen cell, are referred to 
 below. 
 
 Potential Series of the Elements During the considera- 
 tion of the Daniell cell (p. 360), it was pointed out that metals 
 differ greatly with regard to their solution pressures. Zinc, for 
 example, has a very high solution pressure, whilst that of copper 
 is very small. 
 
 The difference of potential between a metal and a solution of 
 one of its salts at room temperature is represented by the 
 formula 
 
 T , 0-058 , P 
 
 t = _1_ log,.-, 
 
 and if/, the osmotic pressure of the positive ions of the salt, is 
 the same for all the electrodes, say that represented by a solu- 
 tion containing a gram-ion per litre, it is evident that the value 
 of E is proportional to the solution pressure of the metal. As 
 regards the standard to which the E.M.F.s are to be referred, 
 the hydrogen standard has in this case certain advantages. The 
 potential of metals with regard to normal-ionic solutions of 
 their salts is therefore obtained by measuring the E.M.F. of 
 cells of the type 
 
 H 2 (Pt) | H- | normal ionic solution of the metallic salt | metal, 
 the difference of potential H 2 (Pt) | #H- being taken as zero. 
 The E.M.F. of the combination 
 
 H 2 (Pt) | H- | ;*-ZnSO 4 | Zn 
 0-770 
 
388 OUTLINES OF PHYSICAL CHEMISTRY 
 
 measured with the potentiometer in the usual way, is 0-770 
 volts, the hydrogen being positive with regard to the zinc. 
 The value of E Zn804 _ Z n is therefore + 0770 volts, the poten- 
 tial difference at the other junction being zero by definition, 
 and positive electricity goes in the cell in the direction indicated 
 by the arrow, that is, the solution tension of zinc is greater than 
 that ofhydrogen> so that the former displaces the latter (indirectly) 
 from solution. 
 
 On the other hand, the E.M.F. of the cell 
 
 H 2 (Pt) | nH' | CuSO 4 | Cu 
 
 is 0*329 volts, copper being positive ; positive electricity goes in 
 the solution in the direction represented by the arrow. Hydro- 
 gen therefore goes into solution and copper is deposited, so that 
 the solution pressure of hydrogen is greater than that of copper. 
 
 The numbers obtained as above indicated, that is the dif- 
 ference of potential between a metal and a normal-ionic solution 
 of one of its salts (solutions which contain a gram of the corres- 
 ponding ion per litre) are termed the normal potentials or electro- 
 lytic potentials of the metals in question and are usually indicated 
 by the symbol e . In a similar way the normal potentials of 
 electrodes which yield negative ions (such as oxygen, chlorine, 
 and bromine) may be measured. 
 
 The following table contains the normal potentials e oh for a 
 large number of elements referred to the hydrogen electrode 
 as standard. The normal potentials referred to the normal 
 calomel electrode, e^ (its potential being taken as zero), are 
 obtained from the normal hydrogen potentials by means of 
 the formula e oh = *. + 0-283 volt. 
 
 The numbers for chlorine, bromide and iodine, are comparable 
 with the others, and are obtained in a somewhat similar way. 
 The value for chlorine, for example, may be obtained by 
 measuring the E.M.F. of a cell of the type. 
 
ELECTROMOTIVE FORCE 389 
 
 H 2 (Pt) | n ' H | n * Cl | Cl 2 (Pt), 
 
 the right-hand electrode being reversible for chlorine just as the 
 left-hand one is reversible for hydrogen (see below). 
 
 Normal Potentials, e oh . 
 
 Na Mg Al Mn Zn Fe Cd 
 
 - 2-7 - 1*48 - 1-28 - 1-07 - 0-770 - 0-46 - 0-42 
 
 Tl Co Ni Sn Pb K 2 Cu 
 
 - 0-32 - 0*3 - 0-25 - 0*19 - 0-12 o + 0-329 
 Hg/Hg 2 - Hg/Hg- Ag Pt Au O I Br Cl F 
 0-775 '^35 '8 '86 i'o8 0-393 0*63 1-09 1*40 2'o 
 By means of this table, the E.M.F. of a cell made up of two 
 metals in contact with normal solutions of their salts can at 
 once be calculated. As the following schemes show, a zinc- 
 nickel element has the E.M.F. 0-770 - 0*25 = 0*520 volts, 
 and a zinc-silver element the E.M.F. 0*770 - ( - 0-80) = 
 1*570 volts. 
 
 Zn I nZn" I Ni" I Ni 
 
 0-770 0-25 
 
 Zn | nZn" \ #Ag- | Ag 
 
 0*770 o'8o 
 
 0*520 v. 
 
 positive electricity flowing in the respective cells, in the direc- 
 tions indicated by the lower arrows. The student should have 
 no difficulty in understanding these schemes in the light of the 
 considerations advanced on p. 365. Both in the case of zinc 
 and of nickel the solution pressure of the metal is greater than 
 the osmotic pressure of the metallic ions in normal solution, and, 
 therefore, when arranged to form a cell, the tendency for posi- 
 tive electricity to pass round the circuit is in the opposite direc- 
 tion at the two junctions. Positive electricity, therefore, flows 
 in the direction in which the acting force is the greater, and the 
 total E.M.F. is the difference of the forces at the two junctions. 
 
390 OUTLINES OF PHYSICAL CHEMISTRY 
 
 In the zinc-silver cell, on the other hand, the forces act in the 
 same direction, and the total E.M.F. is therefore the sum of the 
 forces at the junctions. 
 
 According to Nernst's formula, and in agreement with the 
 convention now widely used that the potential difference has 
 the positive sign if the electrode is positively charged with 
 respect to the solution and the negative sign if the electrode is 
 negatively charged, the normal potential is represented by the 
 formula : 
 
 RT, 
 '- - inr"*^ 
 
 since c, the concentration of the ions, in the above 
 measurements is unity. Therefore the general formula repre- 
 senting the P.D. between an electrode and an electrolyte of 
 the ionic concentration c is 
 
 when the electrode gives positive ions, and 
 
 nF 
 when the electrode gives negative ions. 
 
 In order to illustrate the use of these formulae we may cal- 
 culate the P.D. at each electrode and the total E.M.F. of the 
 cell 
 
 Ag | AgCl sat. sol. | Ni SO 4 o-i molar | Ni 
 at 25, assuming that the P.D. at the liquid contact is eliminated 
 ( and that the nickel salt is 60 per cent ionised. [ for nickel 
 - 0-25 volts ; for silver o'8o volts.] 
 The formula is as follows : 
 
 RT 
 ' n 2 F 
 
 n 1 for nickel is 2 and Cj is 0-06; n 2 for silver is i and c 2 is 
 1-25 x 10 ~ 5 gram-ions per litre at 25. Hence 
 E= - 0*25 + 0-029 Iog 10 (0-06)- 078- 0-058 log 10 (i-25 x'icr 5 ) 
 - 0-25 - 0-035 ~ ' 8 + 0-289 
 = - 0-796 
 
ELECTROMOTIVE FORCE 391 
 
 The total E.M.F. of the cell is - 0796 volts; the P.O. at 
 the silver electrode is + o^n volts, and at the nickel electrode 
 - 0*25 - '035 = - 0*285 volts. 
 
 In order to avoid errors of sign, it is well to check results 
 such as the above from the point of view of general principles. 
 Since diminishing the concentration of a solution from which 
 ions are separating lowers, and diminishing the concentration 
 of a solution into which new ions are going increases the E.M.F. 
 at a junction (p. 366) it is evident that the effect on the 
 E.M.F. of alteration of the concentrations of the solutions 
 must be as shown above. 
 
 From the above considerations it would appear that metals 
 which stand higher than hydrogen in the tension series can 
 liberate hydrogen from acids anfl that,the numbers in the table 
 afford an approximate measure of the energy of the change 
 On the other hand, hydrogen at atmospheric pressure should 
 displace the metals which stand below it in the tension series. 
 This has been shown to hold in some cases at least with 
 hydrogen occluded in platinized platinum electrodes, the plat- 
 inum presumably acting as a catalyst for reactions which under 
 ordinary conditions are extremely slow. Finally each metal 
 should be able to displace from combination any metal below 
 it in the tension series, the difference of potential between the 
 metals being a measure of the free energy of the change. On 
 the whole these conclusions are borne out by the experi- 
 mental results except in so far as the phenomenon of over- 
 voltage comes into play. This subject is briefly discussed in 
 a later section (p. 403). 
 
 Cells with Different Gases The simplest example of 
 these cells is the hydrogen-chlorine cell, already referred to. 
 One-half of the cell consists of a hydrogen electrode in acid, 
 the other of a similar electrode saturated with chlorine, and 
 the two electrodes are combined as represented in Fig. 41, the 
 intermediate vessel containing acid of the same strength as that 
 in the cell. The chemical change which takes place in the cell 
 
392 OUTLINES OF PHYSICAL CHEMISTRY 
 
 is the combination of hydrogen and chlorine to form hydro- 
 chloric acid. Representing the cell as usual 
 
 H 2 (Pt) | H- | C1' I Cl 2 (Pt), 
 
 1-40 
 
 it is clear that positive electricity flows in the cell from hydrogen 
 to chlorine in the direction represented by the arrow, the 
 chlorine becoming the positive and the hydrogen the negative 
 pole. The E.M.F. of the cell in normal acid at the ordinary 
 temperature is about i '40 volts. 
 
 The most important cell of this type is the hydrogen-oxygen 
 or Grove's cell, the two poles being saturated with hydrogen 
 and oxygen respectively. Wh$n connection is made the gases 
 gradually disappear, hydrogen becoming ionized at one pole 
 and oxygen uniting with water to form hydro xyl ions at the 
 other pole. The cell may therefore be represented by the 
 following scheme 
 
 (acid, alkali or salt) 
 
 1-23 
 
 and positive electricity flows through the cell from hydrogen to 
 oxygen as represented by the arrow, so that hydrogen is the 
 negative pole and oxygen the positive pole. The hydrogen 
 electrode is reversible with regard to hydrogen, as follows : 
 H 2 ^2H*, the reaction taking place at the oxygen electrode 
 is as follows : H 2 O + iO 2 ^2OH'. When employed as indicated 
 above, the change is that represented by the two upper 
 arrows and 2F passes through the wire ; when, on the other 
 hand, 2F is sent through the cell in the opposite direction, 
 the changes at the two poles are represented by the two lower 
 arrows. 
 
 If absolutely indifferent electrodes were used for absorbing 
 the gases, and the changes at the electrodes were fully rever- 
 
ELECTROMOTIVE FORCE . 393 
 
 sible, the calculated E.M.F. of the cell is 1-23 volts. 1 The 
 values actually observed are smaller, probably owing to the 
 formation of an oxide of platinum, which has an oxygen 
 potential different from that of free oxygen. 
 
 Theoretically, only pure water is necessary as electrolyte, 
 but, in order to increase the conductivity, dilute acid or alkali 
 or a dilute salt solution is employed as electrolyte. The 
 E.M.F. of the cell is independent of the nature of the electro- 
 lyte since the product [H-][OH'] is the same in acid, alkaline, 
 or neutral salt solution, but this is not the case for the single 
 potential differences at the electrodes. 
 
 Oxidation-Reduction Cells The gas cell just described 
 is a typical oxidation-reduction cell, as when working hydrogen 
 is being oxidized at the negative pole and oxygen reduced at 
 the positive pole. 
 
 As may be anticipated, corresponding cells can be con- 
 structed in which instead of hydrogen another reducing agent 
 is used, and instead of gaseous oxygen another oxidizing agent. 
 Indifferent metals, such as platinum or iridium, are used as 
 electrodes in all cases. 
 
 We will first consider a cell built up of a hydrogen electrode 
 on one side and a platinized platinum electrode dipping in a 
 solution of a ferrous and a ferric salt on the other. When the 
 two electrodes are connected up, a current flows in the cell 
 from the hydrogen to the other electrode. Hence at the 
 hydrogen electrode gaseous hydrogen is going into solution 
 as hydrogen ions according to the equation H 2 + 2F = 2H*, 2 
 and at the other electrode Fe"* ions are being reduced to Fe** 
 ions according to the equation 2Fe*** - 2F = 2Fe**, the charges 
 neutralizing each other through the wire and thus producing 
 
 1 Corresponding with the free energy of formation of water from its 
 elements. The calculation is rather complicated. (Compare Nernst 
 and von Wartenberg, Zeitsch. physikal. Chem., 1906, 56, 544.) 
 
 a aF or 2 x 96,540 coulombs converts a mol of hydrogen to H- 
 ions. 
 
394 OUTLINES OF PHYSICAL CHEMISTRY 
 
 a current. When the same quantity of electricity is passed 
 through the cell in the opposite direction, Fe" ions are con- 
 verted to Fe ions, and hydrogen gas is liberated at the other 
 pole ; the cell therefore works reversibly, and the measurement 
 of the E.M.F. gives a measure of the free energy or affinity of 
 the reaction. The total change is, of course, expressed by the 
 equation 2Fe + H 2 = 2Fe" + 2H\ 
 
 Other oxidizing agents can be measured in the same way 
 against the hydrogen electrode, and from the results a table 
 of various solutions, arranged in the order of their oxidizing 
 potentials, can be obtained. Some of the values obtained in 
 this way may be given 
 
 SnCl 2 in HC1 0-23 volts FeCl a in HC1 0-98 volts 
 
 NH 2 OH in HC1 0-38 volts KMnO 4 in H 2 SO 4 i -50 volts 
 
 The above are only meant .to indicate the order of the results, 
 as the accurate values depend greatly on the concentration and 
 composition of the solutions. 
 
 The four solutions mentioned, even stannous chloride, in 
 acid solution exert an oxidizing action on gaseous hydrogen, 
 and therefore the direction of the current is the same as in the 
 ferric chloride cell. As might be anticipated, potassium per- 
 manganate has the highest oxidation potential. 
 
 When, on the other hand, a platinum electrode dipping into 
 a solution of stannous chloride in potassium hydroxide is con- 
 nected with a hydrogen electrode so as to form a cell 
 
 Sn" in 
 
 H 2 (Pt) 
 
 (Pt) 
 
 0-560 
 
 hydrogen ions are discharged and the stannous salt becomes 
 oxidized, positive electricity, therefore, flowing in the cell in the 
 direction of the arrow. The change which takes place in the 
 cell may be represented by the equation 2li- + Sn" =Sn-"- + H 2 , 
 the hydrogen acting as the oxidizing agent. In this case we 
 
ELECTROMOTIVE FORCE 395 
 
 may say that the stannous chloride solution has a certain reduc- 
 tion potential. 
 
 The above considerations are sufficient to show that the 
 terms " oxidizing agent " and " reducing agent " are relative and 
 not absolute ; whether a substance acts as an oxidizing or a 
 reducing agent depends on the substance with which it is 
 brought in contact. 
 
 The hydrogen electrode may be replaced by a platinum 
 electrode dipping in a solution of a reducing agent, an oxida- 
 tion-reduction cell containing only liquids being obtained. One 
 well-known cell of this type consists of platinum electrodes 
 dipping in solutions of ferric chloride and stannous chloride 
 respectively. The changes at the electrodes may be represented 
 by the equations 
 
 (j) 2Fe- - 2F = 2Fe- (2) Sn" + 2F = Sn 
 and the total change as follows 
 
 2Fe- + Sn" = 2Fe" + Sn 
 
 It is now easy to understand what at first sight appears very 
 puzzling, that a ferric salt can oxidize a stannous salt at a dis- 
 tance, the solutions being in separate cells and possibly con- 
 nected by an indifferent solution. The above equations show 
 that the essential feature of the phenomenon is the transference 
 of two positive charges from the iron to the tin ions through 
 the wire. 
 
 As a definite potential may be ascribed to every substance 
 acting as an oxidizing or reducing agent, it is clear that the 
 E.M.F. of an oxidation-reduction cell may be represented as 
 the algebraic sum of the differences of potential at the two junc- 
 tions. When a strong oxidizing solution is combined with a still 
 stronger oxidizing solution to form a cell, the former will be 
 oxidized at the expense of the latter, but the E.M.F. of the cell 
 will be small, as the solutions are acting against each other. 
 The further apart two solutions are in the oxidation-reduction 
 potential series, the greater will be the E.M.F. of the cell formed 
 by their combination. 
 
396 OUTLINES OF PHYSICAL CHEMISTRY 
 
 We are now in a position to give a clear definition of oxida- 
 tion and reduction in dilute salt solutions. An increase in the 
 number of positive charges or a diminution in the number of 
 negative charges on an ion denotes oxidation ; decrease in the 
 number of positive charges or increase in the number of negative 
 charges on an ion denotes reduction. The usual definition of 
 oxidation as consisting in an addition of oxygen to a compound 
 or the abstraction of hydrogen from it, is clearly inapplicable 
 to salt solutions, but the older definition retains its value for 
 changes in which organic compounds are concerned, and for 
 solid compounds ; these have so far been very little investigated 
 from an electro-chemical standpoint. 
 
 According to the above definition, all reactions which take 
 place electromotively are oxidation - reduction reactions, 
 oxidation taking place at one electrode and reduction at the 
 other. In the Daniell cell, for instance, oxidation takes place 
 at the cathode, Zn + 2F-Zn", and reduction at the cathode 
 Cu" - 2F = Cu. It follows that the displacement of one 
 metal by another is to be regarded as an oxidation-reduction 
 process. Elements which can only give positively charged 
 ions, e.g. the typical metals, can only act as reducing agents, 
 whilst elements such as chlorine, which only yield negatively 
 charged ions, invariably exert an oxidizing action. Solutions, 
 on the other hand, may behave according to the conditions 
 either as oxidizing or reducing agents, since one or the other 
 ion may react. Cupric bromide solution, for instance, acts as 
 an oxidizing agent towards zinc and as a reducing agent towards 
 copper. Moreover, a single ion, e.g. ferrous ion, Fe" may 
 act either as an oxidizing or as a reducing agent since it can 
 be changed into uncharged Fe or into Fe m * f . 
 
 Electromotive Force and Chemical Equilibrium. In 
 the previous section, we have considered oxidation-reduction 
 cells from the qualitative standpo : nt only. Just as in the case of 
 the Daniell cell, which indeed is a special type of oxidation- 
 
ELECTROMOTIVE FORCE 397 
 
 reduction cell, the E.M.F. at an electrode depends upon the 
 concentration of all the ions taking part in the change. Thus 
 the E.M.F. at a platinized platinum electrode immersed in a 
 solution of a ferric salt is only definite when a certain proportion 
 of Fe" ions are also present, and the E.M.F. depends on the 
 concentrations of both ferric and ferrous salt. The general 
 equation representing the dependence of the E.M.F. of such 
 cells on the concentrations of the substances taking part in the 
 reaction will now be given. 
 
 The reversible reaction n 1 A l + n 2 A 2 + . . . ^iij'Aj' + 
 n 2 'A 2 ' ... (p. 159), when it proceeds in one direction in an 
 electrolytic cell, may be represented by the equation 
 
 njAj 4- n 2 A 2 + . . . + nF-Mi/A/ + n/A 2 ' + . . . 
 which indicates that n 1 mols of Aj and n 2 mols of A 2 . . . are 
 converted into n/ mols of A/ and n 2 ' mols of A 2 ' by taking 
 up n faradays. The maximum work obtainable when the 
 substances on one side of the equation at definite concentrations 
 are transformed isothermally and reversibly into the substances 
 on the other side of the equation also in definite concentrations 
 may be derived by a non-electrical method or by carrying out 
 the reaction in a galvanic cell. In the former case the maximum 
 work may be stated in the form 
 
 where the square brackets represent concentrations and K 
 represents the equilibrium constant. When the process is 
 carried out in a galvanic cell the maximum work obtained is 
 A = nFE (p. 352) hence 
 
 When both the initial substances and the final products are 
 in unit concentration the maximum work obtainable non- 
 electrically is 
 
 A = RT log e K 
 
398 OUTLINES OF PHYSICAL CHEMISTRY 
 
 and in a galvanic cell nFe , where e is the normal potential 
 Hence the general equation representing the dependence ol 
 the E.M.F. of the cell on the concentrations of the reacting 
 substances is 
 
 E = e - ?I 10R [A 1 /]>V t A >'. 
 nF ge [A,]", [A 2 ]" 2 
 
 The concentrations of the more highly oxidized substances 
 (i.e. those formed by taking up positive charges) occur in the 
 denominator. 
 
 For the change 
 
 Fe- + 
 the above expression has the form 
 
 For the chlorine electrode 2CV + 2F^C1 2 
 
 RT 
 = <- log. LCIT/[C1 2 ]. 
 
 As regards the permanganate electrode, for which the 
 equation MnO 4 ' + 8H- 5F^ Mn~ + 4H 2 O represents the 
 probable chemical change, the expression for the E.M.F. is as 
 follows : 
 
 RT . [Mnl- [H 2 O] 4 
 K = _ _ IOP- I . . J _ t _ ? _ =L. 
 
 5F ge [Mn0 4 ]' [H-] 8 
 
 It is clear from the above equations that the effect on the 
 E.M.F. of the cell of systematic variation in the concentrations 
 of the reacting substances throw light on the nature of the 
 chemical change taking place in the cell and also on the number 
 of faradays associated with the change in question. 
 
 Electrolysis and Polarization If an external E.M.F. of 
 i volt is applied to two platinum electrodes dipping in a con. 
 centrated solution of hydrochloric acid, it will be found that 
 the large current which at first passes when connection is made 
 rapidly diminishes and finally falls practically to zero. The ex- 
 planation of this behaviour is that while the current is passing 
 hydrogen accumulates on the cathode and chlorine on the 
 
ELECTROMOTIVE FORCE 399 
 
 anode, thus setting up an E.M.F. which acts against the E.M.F. 
 applied to the poles of the cell. This phenomenon is termed 
 polarization. In the above case the gases go on accumulat- 
 ing in the electrodes till the back E.M.F., which we will term e t 
 is equal to the applied E.M.F., when the current ceases. If, 
 however, an E.M.F. of 1*5 volts is applied at the electrodes, a 
 continuous current passes through the solution and it is evident 
 that in this case the back E.M.F. e has not attained the value 
 of i '5 volts. The explanation is evident when it is remem- 
 bered that the E.M.F. of a cell in which platinum electrodes 
 are charged with hydrogen and chlorine respectively at atmo- 
 spheric pressure is 1-40 volts (p. 392). When an E.M.F 
 of 1*5 volts is applied, the electrodes become charged up tc 
 atmospheric pressure, but no higher, the excess of the gases 
 escaping into the atmosphere. It follows that e cannot under 
 ordinary circumstances attain a higher value than i -40 volts, so 
 that electrolysis proceeds at an E.M.F. of E - e 0*10 volts. 
 
 The E.M.F. which must just be exceeded in order that a 
 continuous current may pass through an electrolyte is termed 
 the " decomposition potential" of the electrolyte, and it is clear 
 from the above example that the decomposition potential is equal 
 to the E.M.F. of a cell in which the products of electrolysis are 
 the combining substances. As the E.M.F. of such a cell is the 
 algebraic sum of the differences of potential at the electrodes, it 
 is clear that the decomposition potential is also the sum of two 
 factors, namely, the sum of the potentials required to discharge 
 the anion and cation respectively. 
 
 The decomposition potential of an electrolyte may be deter- 
 mined in two ways. According to the first method, the external 
 E.M.F. applied to the electrodes is gradually raised and the 
 point noted at which there is a sudden increase in the current. 
 The value of the current, C, is determined by the equation 
 
 E - e = CR, 
 where R is the resistance of the circuit, and will obviously 
 
400 OUTLINES OF PHYSICAL CHEMISTRY 
 
 increase rapidly as soon as E is greater than e. The second 
 method is to charge the electrodes up to atmospheric pressure 
 by using an E.M.F. greater than e t then the external circuit is 
 broken and the E.M.F. of polarization measured at once. This 
 method depends upon the fact already indicated, that the de- 
 composition potential is that E.M.F. which is just sufficient to 
 overcome the E.M.F. of polarization. 
 
 As has just been pointed out, the potential required to dis- 
 charge an ion such as Zn" must just exceed the difference of 
 potential at the junction Zn/Zn", and is, therefore, the same as 
 the potential of the metal in volts in the tension series (p. 389). 
 Further, the E.M.F. required to decompose an electrolyte is 
 clearly the sum of the separate differences of potential required 
 to discharge the anion and cation respectively, and is, there- 
 fore, obtained by addiner the values for the two ions in the 
 tension series. The matter becomes clearer when we consider 
 that the potential difference between an element and its ions 
 may conveniently be regarded as a measure of the affinity of the 
 element for electricity. Thus the affinity of zinc for positive 
 electricity is equivalent to 0*770 volts, and that of chlorine for 
 negative electricity to 1*40 volts. To convert zinc ions to 
 metallic zinc we must, therefore, apply a contrary E.M.F. which 
 just exceeds the affinity of zinc for positive electricity, in other 
 words, the decomposition potential of zinc ions is 0770 volts. 
 
 On this basis, the decomposition potential of zinc chloride 
 should be 0770 + 1-40 = 2-170 volts, of hydrochloric acid 
 1-40 volts, and of copper chloride ( - 0-329 + 1-40) = 1-071 
 volts respectively. This is fully confirmed by the experimental 
 determinations of Le Blanc, who obtained the following values : 
 ZnCl 2 = 2-15 volts, HC1 = 1-31 volts, CuCl 2 = 1*05 volts, an 
 agreement within the limits of experimental error. 
 
 Separation of Ions (particularly Metals) by Electro- 
 lysis The results just mentioned are well illustrated by the 
 phenomena observed when a mixture of electrolytes is electro- 
 lysed at different values of the applied E.M.F. The foregoing 
 
ELECTROMOTIVE FORCE 401 
 
 considerations show that on gradually raising the E.M.F. that 
 chemical change takes place most readily for which the least 
 difference of potential is required, and this may be taken 
 advantage of for the electrolytic separation of metals which are 
 discharged at different potentials. Suppose, for example, a 
 mixture of hydrochloric acid, zinc and copper chlorides is 
 subjected to electrolysis. Below i volt practically no change 
 will occur, but at n volts, a little above the decomposition 
 potential for copper chloride, copper will be deposited on the 
 cathode. When it has been almost completely removed, and 
 the potential is raised to 1-4 volts, hydrogen will be liberated 
 at the cathode. Finally, the attempt may be made to remove 
 zinc by raising the external E.M.F. above 2*2 volts, but this 
 cannot be effected in acid solution, as there is a large excess 
 of hydrogen ions, which are more easily discharged than zinc. 
 
 In an exactly corresponding way, almost all the bromine 
 may be electrolytically separated from a solution containing 
 zinc chloride and zinc bromide before the chlorine appears. 
 
 It is, therefore, clear that it is the value of the E.M.F., and 
 not the strength of the current, which is of primary importance 
 for the separation of metals, and in recent years methods based 
 on this principle have become of great commercial importance. 
 Besides the value of the applied E.M.F., the concentration of 
 the ions in contact with the cathode is of great importance, 
 as the decomposition potential necessarily depends on the ionic 
 concentration, and hence great attention is now paid to the 
 efficient stirring of the electrolyte. 1 
 
 The Electrolysis of Water. Overyoltage (Super- 
 tension) at Electrodes. When aqueous solutions of many 
 salts and strong acids and bases are electrolysed with smooth 
 platinum electrodes only hydrogen and oxygen are liberated as 
 products of electrolysis and the decomposition potential is in 
 
 1 The electrolytic separation of metals on this principle is described in 
 recent papers by Sand (Journal of the Chemical Society, 1907, 91, 373, 
 iQo8, 93, 1572), and others 
 26 
 
4 o2 OUTLINES OF PHYSICAL CHEMISTRY 
 
 all cases about i '66 volts. It was formerly supposed that one 
 or both of these gases were formed by the action of the primary 
 products of electrolysis on the solvent, but this does not account 
 for the fact that the decomposition potential is in general the 
 same for different acids and bases. It is now accepted, for 
 reasons given below, that the gases are products of the primary 
 decomposition of water. 
 
 If such is the case, and the decomposition of water proceeds 
 reversibly at the electrodes, we would expect the decomposi- 
 tion potential to be about 1*2 volts, in agreement with the 
 E.M.F. of the hydrogen-oxygen cell, whereas it is considerably 
 higher. When, however, platinized platinum electrodes are 
 used and the current is plotted against the applied E.M.F., 
 the latter being gradually increased, it is found that there is a 
 sudden increase in the current at 1*1 volts (so that water can 
 be continuously decomposed at the latter potential), but a much 
 more rapid increase at i -66 volts. Two possible explanations 
 of these remarkable facts might be suggested. Nernst was 
 formerly of opinion that at the lower potential H' and O" ions 
 are being discharged the current being veiy small because of 
 the exceedingly minute concentration of the O" ions. The 
 more rapid decomposition at 1*66 volts is due to the discharge 
 of H* and OH' ions the latter combining to form water and 
 oxygen according to the equation 4.OH -> 2H 2 O + O2. 
 
 Another mode of explaining the results is that the decom- 
 position potential depends on the nature and condition of the 
 electrode material ; at many electrodes the potential must be 
 raised above that theoretically required for reversible decom- 
 position in order to reach the point of decomposition. Thus 
 assuming that the decomposition of N/i sulphuric acid takes 
 place reversibly at a platinum cathode, the following values for 
 the cathodic decomposition potential with other metals were 
 obtained: Pd + 0-26, Pt o, Fe - 0-03, Cu - 0-19, Al - 
 0*27, Pb - 0*36, Hg - 0*44. Thus hydrogen is eliminated 
 more easily at a palladium than at a platinum electrode, per- 
 
ELECTROMOTIVE FORCE 403 
 
 haps owing to the formation of an alloy with the former metal ; 
 in all other cases a greater or less excess of E.M.F. is required 
 in order to liberate the gas. Overvoltage phenomena also oc- 
 cur- at the anode when oxygen is being liberated, but in this 
 case the order of the metals is not the same as with hydrogen. 
 The magnitude of the overvoltage increases considerably with 
 increase of current-density. 
 
 There is no doubt as to the great, importance of overvoltage 
 phenomena, although they are not ye} fully understood. They 
 appear to depend, in part at least, on supersaturation with the 
 gas. Thus when hydrogen is liberated at a platinized platinum 
 anode, the latter dissolves a large amount of the gas and facili- 
 tates its escape in bubbles, thus bringing about equilibrium 
 between the gas in solution and in the gas space. On the 
 other hand smooth platinum, and such metals as lead and 
 mercury, have very little solvent power for hydrogen, and a 
 much higher pressure is required in order to force in sufficient 
 of the gas to admit of the formation of bubbles (Nernst). Ac- 
 cording to Forster, the liberated substance forms some com- 
 pound with the electrode material, and 'the supertension is 
 determined by the concentration of the gas thus dissolved 
 in some form in the electrode. By making use of the high 
 concentration of hydrogen obtainable at electrodes showing 
 considerable supertension, reductions not readily effected by 
 other methods can be performed. 
 
 Supertension phenomena have an important bearing on the 
 dissolving of metals in acids. Pure zinc should liberate hydro- 
 gen from acids at a potential of 0770 volts but the superten- 
 sion is so great as almost to reach this value and the reaction 
 therefore proceeds very slowly. The overvoltage at impure 
 zinc is much less and therefore the metal dissolves much 
 more readily in acids. In the latter case local differences 
 of potential doubtless also play a part. 
 
 The main evidence in favour of the view that water under- 
 
404 OUTLINES OF PHYSICAL CHEMISTRY 
 
 goes primary decomposition during electrolysis is that the de- 
 composition potential is largely independent of the electrolyte, 
 whether acid, base or salt ; and further, that of the possible 
 changes which can take place at the electrodes the decomposi- 
 tion of water is usually that which can take place at the lowest 
 potential (compare previous section). In the case of hydro- 
 chloric acid the relationships are more complicated. In con- 
 centrated solution the decomposition potential is lower than 
 that of water (p. 400) and the main products are hydrogen and 
 chlorine; with progressive dilution the decomposition potential 
 rises and ultimately a mixture of oxygen and chlorine is liber- 
 ated at the anode. 
 
 The decomposition potential curve of sulphuric acid shows 
 two further points of rapid increase of current, at 1-95 and 2-6 
 volts respectively. It seems probable that the former value is 
 connected with the discharge of SO/' and the latter with the 
 discharge of HSO/ ions. 
 
 Electrolysis and Polarization (continued). The E.M.F. 
 required to bring about decomposition of an electrolyte is not 
 determined solely by the magnitude of the polarization due to 
 the products of electrolysis. The current also causes concen- 
 tration changes at the electrodes and these changes always act 
 in opposition to the E.M.F. driving current through the cell. 
 This effect is known as concentration polarization and is mini- 
 mised by stirring the electrolyte. 
 
 Any substance which tends to diminish the polarization in 
 a cell is termed a depolarizer ; It may act as a catalyst in ac- 
 celerating the changes at the electrodes, e.g., platinized 
 platinum in the liberation of hydrogen, or it may alter the 
 change taking place at the electrodes to one that takes place 
 more easily, e.g., the use of potassium dichromate in the so- 
 called Bichromate cell. The " insoluble " salt in an electrode 
 of the second kind acts as a depolarizer. 
 
 Recent investigations have shown that polarization occurs in 
 
ELECTROMOTIVE FORCE 405 
 
 many cases where it would not be anticipated, and this fact has 
 raised the question as to the exact nature of the changes taking 
 place at the electrodes during electrolysis. When, for in- 
 stance, a current is passed through the cell Cu | CuSO 4 | Cu 
 it would be anticipated according to the accepted views re- 
 garding electrolysis that Cu" ions would be discharged at the 
 cathode as metallic copper, that SO 4 " ions after discharge at 
 the anode would immediately attach the latter forming copper 
 sulphate. As a matter of fact Le Blanc l has shown that under 
 these circumstances considerable polarization occurs both at 
 anode and cathode, so that the changes taking place at the 
 poles can scarcely be as simple as those just assumed. A still 
 more striking case occurs in the electrolysis of solid silver 
 salts between silver electrodes. With silver sulphate, for in- 
 stance, a polarization E.M.F. of 0*312 volts was observed two 
 minutes after breaking the circuit and at - 80 an E.M.F. of 
 no less than 1*562 volts one minute after breaking the circuit. 2 
 The cause of these remarkable observations is still by no means 
 understood. 
 
 Accumulators As is well known, accumulators are em- 
 ployed for the storage of electrical energy. An accumulator is 
 a reversible element ; when a current is passed through it in one 
 direction the electrodes become polarized, and when the polar- 
 izing E.M.F. is removed and the poles of the accumulator are 
 connected by a wire, the products of electrolysis recombine with 
 production of a current and the cell slowly returns to its original 
 condition. 
 
 It will be clear from the above that the Grove's gas cell is a 
 typical accumulator or secondary element; when a current is 
 passed through it in one direction the electrodes become 
 charged with hydrogen and oxygen, and these gases can be 
 
 1 M. Le Blanc, Abhandlungen der Bunsen-Gesellschaft, No. 3, 1910. 
 
 2 Haber and Zawidzki, Zeitsch. physikal. Chem., 1911, 78, 228. Com- 
 pare Annual Reports Chemical Society for 1912, p. 19. 
 
406 OUTLINES OF PHYSICAL CHEMISTRY 
 
 made to recombine with production of a current. From a 
 technical point of view, however, a satisfactory accumulator 
 must retain its strength unaltered for a long time when the 
 poles are not connected, and must be easily transported. A gas 
 accumulator would be in many respects un suited for commercial 
 purposes. 
 
 The apparatus most largely used for the storage of electricity 
 is the lead accumulator, the electrodes of which in the un- 
 charged condition contain a large amount of lead sulphate 
 (obtained by the action of sulphuric acid on the porous lead of 
 which the electrodes largely consist at first) and dip in dilute 
 sulphuric acid. The accumulator is charge by sending an 
 electric current through it. At the cathode, the lead sulphate 
 is reduced by the hydrogen ions (or rather by the discharged 
 hydrogen) to metallic lead according to the equation 
 PbSO 4 + 2H- - 2 F = Pb + 2H- + SO 4 " 
 
 or more simply, PbSO 4 - 2F = Pb + SO 4 " 
 
 On the other hand, the SO 4 " ions wander towards the anode 
 
 and react with it according to the equation 
 
 PbSO 4 + SO 4 " + 2H.O + 2F = PbO 2 + 4H- + 2SO 4 " 
 
 so that the anode and cathode consist mainly of lead peroxide 
 and metallic lead respectively. 
 
 On connecting up to obtain a current (discharging), SO 4 " ions 
 are discharged at the new anode (the lead pole), and reconvert 
 it to lead sulphate, according to the equation 
 
 Pb + S0 4 " + 2 F = PbS0 4 , 
 
 and simultaneously H* ions are discharged at the new cathode 
 (the peroxide pole), the peroxide being reduced to the oxide, 
 and acted on by sulphuric acid to reform the sulphate, according 
 to the equation 
 
 PbO 2 + 2H- + H^SO 4 - 2F = PbSO 4 + 2H./). 
 
ELECTROMOTIVE FORCE 40? 
 
 The chemical changes taking place on charging and discharging 
 are summarized in the equation 
 
 Pb + PbO 2 + 2H 2 SO 4 ^ 2PbSO 4 + 2H 2 O ; 
 
 the upper arrow represents discharging, and the lower arrow 
 charging. 
 
 The E.M.F. of the lead accumulator is about 2 volts. It is 
 not strictly reversible, but under ordinary conditions of working 
 about 90 per cent, of the energy supplied and stored up in it 
 can again be obtained in the form of work. 
 
 The only other accumulator of commercial importance is that 
 developed more particularly by Edison and his co-workers, and 
 known as the Edison accumulator. In the charged condition 
 the positive plate consists of hydrated nickelic oxide, 
 
 Ni 2 3 , x H 2 
 
 and the negative plate of finely divided iron, the electro- 
 lyte being about 4 N alkali. On discharging the nickelic oxide 
 is reduced to nickelous hydroxide Ni(OH) 2 and the iron is 
 oxidized to ferrous hydroxide Fe(OH) 2 . According to Forster 
 the changes taking place on charging and discharging are re- 
 presented approximately by the equations : 
 
 Fe + Ni 2 O 3J i-2H 2 O + i -8H 2 O^Fe(OH) 2 + Ni(OH) 2 . 
 
 Both the anode and cathode consist of steel flames provided 
 with a large number of pockets (made of nickel-plated steel) in 
 which the active electrode materials are packed. The potential 
 during discharge is about 1*34 volts, and, as the above equation 
 shows, is independent (in practice only very slightly dependent) 
 on the alkali concentration. One advantage possessed by the 
 Edison accumulator is its comparative lightness. 
 
 The Electron Theory l In the previous chapters we have 
 learnt that certain atoms (or groups of atoms) can become 
 associated with definite quantities of electricity, and that certain 
 other atoms can take up twice as much, three times as much, 
 and so on. No atom is associated with less positive electricity 
 
 J Nernst, Theoretical Chemistry, chap. ix. ; Rutherford, Radio-Activity; 
 Ramsay, Presidential Address, Trans. Chem, Soc. t 1908, 93, 774. 
 
4 o8 OUTLINES OF PHYSICAL CHEMISTRY 
 
 than a hydrogen atom, and we may therefore state that a 
 hydrogen atom unites with unit quantity of electricity to form 
 an ion. A barium ion has twice as much positive electricity, 
 and a ferric ion three times as much positive electricity as 
 a hydrogen ion. Further, since quantities of hydrogen and 
 chlorine ions in the proportion of their atomic weights are 
 electrically equivalent, it follows that Cl' (and other univalent 
 negative ions) contain unit quantity of negative electricity. 
 
 This increase by steps in the amount of electricity associated 
 with atoms at once recalls the law of multiple proportions, 
 and it appears plausible to ascribe an atomistic structure to 
 electricity ; in other words, to postulate the existence of positive 
 and negative electrical particles, which under ordinary cir- 
 cumstances are associated with matter. On this view, the 
 number of dots or dashes ascribed to positive and negative 
 ions respectively indicates the number of electrical particles 
 (positive or negative) with which the atoms become associated 
 to form ions. 
 
 These views 1 (Helmholtz, 1882) have received powerful 
 support during the last few years from the results of experi- 
 ments on the passage of electricity through vacuum tubes, the 
 so-called Hittorf s or Crookes' tubes. When a current at very 
 high potential is sent through a highly evacuated tube, rays 
 from the cathode the so-called cathode rays stream across 
 the tube with great velocity, and it -has been shown that these 
 rays consist of negative electricity. The speed of the particles 
 depends on the E.M.F. between the poles of the tube, but 
 at a difference of potential of 10,000 volts is about one-fifth 
 of the velocity of light. The mass of these particles is about 
 i/i ooo of that of the hydrogen atom. They are usually 
 termed negative electrons. More recently, it has been dis- 
 covered that the ft rays given off from disintegrating radium 
 at a speed approaching that of light also consist of negative 
 electrons. 
 
 1 Helmhgltz, Faraday Lecture, Trans. Chctn. Soc., 1883, 
 
ELECTROMOTIVE FORCE 409 
 
 As negative electrons have thus been found to exist separate 
 from matter, it is natural to expect that free positive electrons 
 may also be isolated. So far, however, this has not been 
 found possible, and opinions differ somewhat as to the reason. 
 Nernst, following Helmholtz, considers that there is no ground 
 for doubting the existence of positive electrons ; the reason why 
 it has not yet been found possible to isolate them is due to 
 their great affinity for matter. Further, Nernst and others 
 assume that positive and negative electrons unite to form 
 neutral atoms or neutrons^ and that these neutrons constitute 
 the ether which is assumed to pervade all space. Other 
 investigators regard a positive ion as an atom minus one or 
 more negative electrons ; the loss of a negative electron would 
 leave the previously neutral atom positively charged. 1 From 
 observations on the effect of a magnetic field on the cathode 
 discharge, it has been calculated that the actual charge carried 
 by a univalent ion (positive or negative) is about 4 x 10 ~ 10 
 electrostatic units. 
 
 The application of the electron theory to ordinary chemical 
 changes yields interesting results. For simplicity we will assume 
 the existence of positive electrons, and designate them by the 
 symbol , negative electrons by the symbol 0. When hydro- 
 gen arid chlorine unite to form hydrochloric acid, we assume 
 that under ordinary conditions the valency of the hydrogen 
 is satisfied by that of the chlorine. We may, however, dis- 
 place a chlorine atom by a positive electron, and thus obtain 
 the saturated chemical compound H or H ; in an exactly 
 similar way, the hydrogen may be displaced by a negative 
 electron, forming the saturated compound C10. In the 
 same way, a dilute solution of copper sulphate contains the 
 
 / / 
 
 saturated compounds Cu(^ or Cu - and SO 4 <^ or SO 4 ". 
 
 The electrons, therefore, behave exactly as univalent atoms, 
 1 Ramsay, loc. cit. 
 
4 io OUTLINES OF PHYSICAL CHEMISTRY 
 
 the positive electrons enter into combination with positive 
 elements such as H, K, Na, Ba, etc., the negative electrons enter 
 into combination with negative elements or groups, such as 
 Cl, Br, I, NO 3 , SO 4 . Nothing is known as to the constitution 
 of a non-ionised salt molecule in solution. The formula for 
 non-ionised sodium chloride may perhaps be Na Cl, the 
 molecule being held together at least partly by electrical forces. 
 
 Just as there are great differences in the affinity of the 
 elements for each other, so the elements have very different 
 affinity for electrons. Zinc and the other metals have a great 
 affinity for positive electrons, the so-called non-metallic ele- 
 ments have in many cases considerable affinity for negative 
 electrons. The order of the elements in the tension series may 
 be regarded as the order of their affinity for electricity. On this 
 view the potential required to discharge the ions is simply the 
 equal and opposite E.M.F. required to overcome the attraction 
 of the element and the electron (p. 375). 
 
 Practical Illustrations. Dependence of Direction of Current 
 in Cell on Concentration of Electrolyte It has already been 
 pointed out (p. 346) that the current in a Daniell cell may be 
 reversed in direction by enormously reducing the Cu- ion con- 
 centration by the addition of potassium cyanide. The two chief 
 methods for diminishing ionic concentration are (i) the forma- 
 tion of complex ions (as in the above instance) ; (2) the forma- 
 tion of insoluble salts. 
 
 When a cell of the type 
 
 Cd 
 
 CdS0 4 
 
 dilute 
 
 KNO 3 | CuSO 4 I Cu 
 dilute 
 
 is set up, and the poles are connected through an electroscope, 
 it will be found that positive electricity passes in the cell in the 
 direction of the arrow. If some ammonium sulphide solution 
 is then added to the copper sulphate solution, "insoluble" 
 copper sulphide is formed, and the concentration of the Cu" 
 
ELECTROMOTIVE FORCE 411 
 
 ions is reduced to such an extent that the current flows in the 
 reverse direction. 
 If the Daniell cell 
 
 Zn 
 
 ZnSO 4 | KNO 3 I CuSO 4 
 
 Cu 
 
 dilute I I dilute 
 
 is built up in the same way, it will not be found possible to re- 
 verse the current by the addition of ammonium sulphide, owing 
 to the greater solution pressure of the zinc as compared with 
 cadmium ; but if potassium cyanide is added, the current 
 changes in direction, owing to the fact that the Cu" ion con- 
 centration in a strong solution of potassium cyanide (in which 
 the copper is mainly present in the complex anion Cu(CN) 4 ") 
 is considerably less than in a solution of copper sulphide. 
 
 The following experiments, which are described in consider- 
 able detail in the course of the chapter, should if possible be 
 performed by the student. For further details text-books on 
 practical physical chemistry should be consulted. 
 
 (a) Preparation of a standard cadmium cell (p. 357). 
 
 (b) Measurement of the E. M . F. of a cell by the compensation 
 method (p. 355). 
 
 (c) Preparation and use of a calomel "half-cell " (p. 375). 
 
 (d) Preparation and use of a capillary electrometer (p. 380). 
 
 (e) Measurement of the E.M.F. of a concentration cell 
 (p. 368). 
 
 (/) Measurement of the E.M.F. of the hydrogen-oxygen 
 cell (p. 392). 
 
 (g) Determination of the solubility of a difficultly soluble 
 salt, <?.-., silver chloride, by E.M.F. measurements (p. 370). 
 
PROBLEMS AND QUESTIONS 
 
 With scarcely an exception, the problems set in the following pages 
 can readily be solved by means of the formulae and general considerations 
 to be found in the book. It will, however, be advisable in this connec- 
 tion to deal a little more fully than has hitherto been done with three 
 subjects which lend themselves to numerical illustration. 
 
 (1) Correction for the change of volume of gases taking fart in 
 chemical reactions. It has been pointed out (p. 144) that when gases are 
 involved in chemical changes the heat given out or absorbed may differ 
 considerably according as the reaction is carried out at constant pressure 
 or at constant volume. When a mol. of gas is generated at the absolute 
 temperature T, work is done by the system and RT calories is absorbed ; 
 when, on the other hand, a mol. of gas disappears RT calories is given out. 
 A little consideration will show that if Q r is the heat of reaction in 
 calories at constant volume (when no external work is done) and Q p the 
 heat of reaction at constant pressure (when external work may be done) 
 the relationship between Q,, and Q p is given by the formula 
 
 Qp = Q, + (! - w 2 ) RT 
 
 where n x and 2 represent the number of mols. of gas in the initial and 
 final stages of the reaction respectively. 
 
 For example, the reaction 2H 2 + O 2 - H 2 O (liquid) gives 2 x 68,400 
 calories at constant pressure, and therefore the heat of reaction at 
 constant volume at 17 is 68,400 = Q, + i$ x 580, whence Q v = 67,530 
 cal. for 18 grams of water. 
 
 (2) Formula Connecting Latent Heat of Vaporisation and Change 
 of Vapour Pressure with Temperature. The formula in question can 
 readily be obtained by means of a reversible cyclic process applied to va- 
 porisation. According to the second law of thermodynamics, in a revers- 
 ible cycle 
 
 work done change of temperature 
 
 heat absorbed temperature of absorption* 
 
 The work done is (V 2 - V,) dp where Vj is the volume of a mol. of 
 substance in the form of liquid, V 2 the corresponding volume in the 
 vapour form and dp is the change of vapour pressure for a small 
 change of temperature dT. Substituting in the above equation, we 
 obtain 
 
 ( 
 
 where q is the latent heat of vaporisation of i mol. of liquid at the 
 absolute temperature T. 
 
 413 
 
PROBLEMS AND QUESTIONS 413 
 
 As the volume of a definite quantity of a substance is so much greater 
 in the vapour than in the liquid form, no great error will be made by neg- 
 lecting V : in comparison with V 2 . Further, according to the gas laws 
 (p. 26) p V 2 = R T. Substituting the value of V a from this equation in the 
 formula V^dp/q = dT/T we obtain 
 
 1 l !L = 1 or 
 p ' dT RT a 
 
 * QPge P) _ / 2 \ 
 
 dT RT2 v '* 
 
 The above relationship can be obtained in more convenient form by 
 integration, on the assumption that q remains constant for the small 
 interval of temperature concerned. Integrating between the absolute 
 temperatures T l and T 2 (the corresponding pressures being p l and/ a ) we 
 obtain (cf. p. 166) 
 
 (3). 
 
 As an illustration of the use of formula (3) the value of q for benzene 
 may be calculated from the observation that the vapour pressures at 
 20 and 3oC. amount to 75-0 mm. and 118 mm. respectively. Hence 
 
 0-19682 = --( } 
 
 4*581 \293 x 3037 
 and q = 8004 calories. 
 
 An approximate relationship between latent heat of vaporisation and 
 the absolute temperature at which vapor'sa.ion takes place the so-called 
 Trouton's rule states that the quotient of the molecular heat of vaporisa- 
 tion by the absolute temperature, is constant in symbols q/T = constant. 
 The rule is approximately valid for non-assoc'ated liquids, but associ- 
 ated liquids, such as water and alcohol, show very considerable deviations. 
 
 Formula (i) applies also to other changes of state, for example from 
 solid to liquid. As an illustration the effect of a change of pressure of 
 one atmosphere on the melting-point of ice will be calculated. In all 
 such calculations, care must be taken to state the factors concerned in 
 corresponding units. 
 
 The available data are that the latent heat of fusion of ice is 80 
 calories or 80 x 42,650 gram-cms., that the specific volume of ice is 
 1-087 when that of water is taken as unit, and that dp is 1033 grams/cm. 2 
 Substituting in the equation 
 
 (v. 2 - v.) dp dT 
 
 *_ II L = , we have 
 
 0*087 x 1033 _ dT 
 
 80 x 42*650 ~ 273 
 
 whence dT = 0-0072 (cf. p. 181) 
 
 It can readily be deduced from the formula that if v^is greater than v^ 
 dp and dT must have the same sign, in other words increased pressure 
 will raise the melting-point, when, on the other hand, v a is less than v lt as 
 
4 i4 OUTLINES OF PHYSICAL CHEMISTRY 
 
 in the case of water, dp and dT are of opposite sign and increase 01 
 pressure will lower the temperature at which the transformation occurs. 
 
 (3) Mathematical Formulation of the Relationship between Electri- 
 cal Conductivity and the Nature of the Electrolyte. Consider the 
 transport of electric current through a cm. cube of electrolyte. As 
 already pointed out, the strength of the current depends upon the number 
 of ions, on their speed and on the charge which they carry. If c is the 
 concentration of the electrolyte in gram-equivalents per c.c. a is the 
 degree of dissociation and Uj and V 1 the speed of the cation and anion re- 
 spectively, in cm. per sec., the current passing through the cm. cube is 
 96,540 ac (U x + V x ) amperes, 96,540 coulombs being the charge 
 carried by i gram equivalent of electrolyte. The current may also be re- 
 presented as the product of the specific conductivity k and the difference 
 of potential, E, between the two sides of the cube. Therefore 
 
 E* = 96540 ac (Uj + Vj). 
 
 Now k/c or kv is the equivalent conductivity of the solution, which we will 
 term A. If further, the potential gradient is i volt per cm. E = i and the 
 abovs equation simplifies to 
 
 A = 96540 a (U + V) 
 
 where U and V represent the speed of the ions, in cm. per second, 
 when the potential gradient is i volt per cm. 
 
 When dissociation is complete a = i and the above formula, in the 
 case of a binary uni-univalent electrolyte, for which \ x = /t^ becomes 
 (cf. p. 252) 
 
 M(f) = + v = 96540 (U + V). 
 
 From this equation it follows that, as already pointed out (p. 253), the 
 speed of the ions in cm. per sec. are obtainable from the velocities ex 
 pressed in the ordinary units by dividing by 96,540. 
 
 (4) Proof of Conditions under which Two Solutions are Isohydric (cj. 
 p. 278). Suppose the electrolytes with a common ion are two monobasic 
 acids, HA and HA', both of which obey Ostwald's dilution law, and that 
 in isohydric solutions of the two acids the dilutions are v and v' and the 
 degrees of dissociation a and o' respectively. For the acid HA, according 
 to the law of mass action, the equation holds 
 
 and for the acid HA' the corresponding equation 
 
 ,rA=>-K'. . (2) 
 
 When the solutions are mixed, the volume becomes v -4- v l and the pro- 
 portion of H' ions o H- a'. For the acid HA we now have 
 
 (I - a)(v + V') 
 Dividing equation (3) by equation (i) we obtain 
 
 (o + a')v _ a + a' __ v + ' v' 
 
 ' ~~ 
 
 (v + v')a 
 whence 
 
 v a _ a 
 
 or -- 
 
 v v v 
 
PROBLEMS AND QUESTIONS 415 
 
 Now al v and a'fv' are the respective concentrations of H' ions in the 
 isohydric solutions of the acids, and have now been shown to be equal 
 in other words, the condition for isohydry is that the concentration of the 
 common ion in the two solutions before mixing must be the same. 
 
 (5) Measurement of Chemical Affinity. We have seen that there are 
 two principal methods of measuring chemical affinity (T) by means of 
 E.M.F. measurements in a galvanic cell (p. 351) and (2) by measurements 
 of the position of tquilibrium in the system under definite conditions. 
 The first method has already been fully described. A proof of the for- 
 mula used in the equilibrium method and one or two examples will now 
 be given. It will be sufficient for our present purpose to derive the 
 affinity formula for a reaction in which gases only are concerned; the 
 same formula applies to reactions in heterogeneous systems. 
 
 A vessel contains hydrogen, oxygen, and water vapour of the respec- 
 tive concentrations Cn 2 , Co 2 , and Cn 2 o at the constant temperature T. 
 It is assumed that one of the walls of the vessel is permeable for hydro- 
 gen only, another for oxygen only, and a third for water vapour only, and 
 that the walls can be displaced without friction. Outside each of these 
 walls is the particular gas for which it is permeable, at the same tempera- 
 ture and concentration as the corresponding gas inside, and the amounts 
 both outside and inside are so great that no appreciable change in con- 
 centration is caused by the passage of a mol of gas into or out of the 
 vessel. 
 
 The wall permeable for hydrogen is now moved inwards so that 2 mols 
 of hydrogen are removed from the vessel, and similarly, by moving in- 
 wards the wall permeable for oxygen, i mol of the latter gas is brought 
 outside. In these processes no work is done, as no alterations of pressure 
 are set up. The hydrogen and oxygen are now allowed to expand rever- 
 sibly at constant temperature T until they attain any desired smaller 
 concentrations C'u 2 and Co 2 The work done by a mol of a perfect gas 
 in expanding from the volume v to v l is 
 
 A = RT log e vjv = RT log e C/C' 
 
 and therefore (assuming that both hydrogen and oxygen behave as perfect 
 gases) the total work gained in the above processes is 
 
 f 
 
 The 2 mols of hydrogen (concentration Cn 2 ) and the mol of oxygen 
 (concentration Co 2 ) are now combined to form 2 mols of water vapour of 
 concentration CH 2 0> the latter concentration being so chosen that the 
 water vapour is in equilibrium with hydrogen and oxygen of the respec- 
 tive concentrations Cn 2 and Co 2 . No work is done in this combination, 
 which is carried out under equilibrium conditions. Finally the 2 mols of 
 water vapour of concentration Cn 2 o are brought isothermally and rever- 
 sibly to the initial concentration Cn 2 o and added to the contents of the 
 vessel through the wall permeable for the vapour. In the latter process 
 the work gained is 
 
 The result of these processes is that in the interior of the vessel 2 mols of 
 hydrogen of the concentration Cn 2 an d * m ol of oxygen of the concentra- 
 tion Co 2 have disappeared and a mols of water vapour of the concentration 
 
416 OUTLINES OF PHYSICAL CHEMISTRY 
 
 Cfl 2 o have been formed without any alteration of temperature or concen- 
 tration inside the system. The total work gained is 
 
 A = A, + A 2 = 2 RT log e ^ + RT log, -^ + 2 RT lo ge 
 
 Cn 2 Co 2 
 
 which is a measure of the affinity of hydrogen and oxygen at the tempera- 
 ture and concentration in question. The above equation may be written 
 in the form 
 
 Now A depends only on the initial and final states of the system and is 
 therefore independent of the arbitrarily chosen concentrations C'n 2 and 
 Co 2 to which the gases were brought after removal from the vessel. It 
 
 follows that the expression ... 2 - = constant = K, in other words, 
 Cli 2 Co 2 
 
 whatever be the concentrations Cn 2 and Co 2 , the concentration Cn 2 o of 
 water vapour in equilibrium with Cn 2 and Co 2 is such that the above 
 equation holds. 
 
 We have here a thermodynamical proof of the law of mass action, first 
 established by Guldberg and Waage from kinetic considerations. 
 
 The affinity of hydrogen to oxygen is therefore represented by the 
 formula 
 
 A = RT log e K - RT log e Il2 
 
 and can be obtained for any concentration C of the reacting substances 
 when the equilibrium constant of the action has been determined. 
 
 It can easily be shown that a formula of this type applies both to 
 homogeneous and heterogeneous reactions. The general formula is as 
 follows (p. 297) : 
 
 A -RT log. K-RT log. t*'.]'' [*.'*- 
 
 [AJ, [AJ-V- 
 
 which, when the initial substances and the products are in unit concen- 
 tration, simplifies to 
 
 A = RT log K, 
 For the combination of hydrogen and iodine the general formula becomes 
 
 A=Rriog.K-RT 1 og. [ ^ f L_. 
 
 At the temperature of boiling sulphur (T = 273 + 445 = 718) 
 K = = 50. 
 
 0-02 
 
 The work obtained by the combination of i mol of iodine and i mol of 
 hydrogen, each at unit concentration, to 2 mols of hydrogen iodide, also 
 at unit concentration, is therefore 
 
 A = 1-985 x 718 x 2-303 Iog 10 5o = + 5575 calories. 
 
 We have here a case where the affinity of a reaction and the heat of 
 reaction are of opposite sign - the latter is about - 6000 cal. 
 
 The general formula will now be used to calculate the chemical 
 affinity of a heterogeneous reaction, namely, the maximum work obtain- 
 able in the combination of i mol of carbon dioxide with calcium oxide at 
 
PROBLEMS AND QUESTIONS 417 
 
 a definite temperature to form calcium carbonate. It is usual to denote 
 the affinity of a gas as that shown towards the solid substances at atmo- 
 spheric pressure. We proceed to calculate the affinity of carbon dioxide 
 for calcium oxide at 671 C. = 944 abs., the equilibrium pressure at this 
 temperature being 13-5 mm. In this case (p. 174) 
 [CaC0 3 ] _^_ 
 ~ [CaO] [COJ /cos, 
 and A = - RT log e p. 
 
 therefore A = - 1-985 x 944 x 2-303 Iog 10 13-5/760 
 
 = 7540 calories. 
 
 The value of A obtained is the same whether concentrations or partial 
 pressures are used for calculations in which gases are concerned. In the 
 above calculation the pressure of the atmosphere is taken as unit. 
 
 (6) Deduction of the Formula Connecting Displacement of Equilibrium 
 with Change of Temperature. Van't Hoff's formula (p. 166) which is 
 usually written in the form 
 
 <*(log a K) _ - Q 
 dT " RT a 
 
 can easily be deduced from the expression A - Q = T(dA./dT) for the 
 second law of thermodynamics (p. 151) and the affinity formula 
 
 A = RT log e K 
 given in the last section. 
 
 From the latter formula we obtain by differentiation 
 
 and 
 whence 
 
 dT RT a 
 
 1. A certain quantity of a gas measures 100 c.c. at 25 and 700 mm. 
 pressure. What pressure will be required to change the volume to 50 c.c. 
 at - 10 C. ? Ans. 1236 mm. 
 
 2. What volume is occupied by (a) i gram of nitrogen, (b) i gram of 
 carbon dioxide at 20 and a pressure of 72 cm. of mercury ? 
 
 Ans. (a) 906-3 c.c. ; (b) 576-8 c.c. 
 
 3. An open vessel is heated till one-third of the air it contains at 20 is 
 expelled. What is the temperature of the vessel ? Ans. 117-6 C. 
 
 4. If 0-5 grams of a gas measure 65 c.c. at 10 and 500 mm. pressure, 
 what is its molecular weight ? Ans. 271-5. 
 
 5. If i gram of nitrogen, i gram of oxygen and 0-2 gram of hydrogen 
 are mixed in a volume of 2-24 litres at o, calculate the respective partial 
 pressures of the gases in the mixture, in grams per sq. cm. 
 
 Ans. 369, 323, and 1025 grams /cm. 2 
 
 6. The density tot benzyl alcohol, C 6 H r ,CH 2 OH, at its boiling-point is 
 1*145. Compare the observed and calculated values of the molecular 
 volume (p. 61). Ans. Obs. 123-7. Cal c. 128-8. 
 
 7. The density of a solution containing 4*1375 grams of iodine in 100 
 grams of nitrobenzene is 1*2389 at 18, the density of the solvent at the 
 same temperature being 1-20547. From these data calculate the molecu- 
 lar solution volume of iodine. Ans. 67*2 c.c. 
 
 8. The density of formic acid at 20 is 1*2205 and n D at the same tem- 
 perature is 1*3717. Calculate the molecular refractivity of formic acid by 
 
 27 
 
 A - Q = T = RT log, K + 
 
418 OUTLINES OF PHYSICAL CHEMISTRY 
 
 the Lorentz formula and compare it with the value calculated from the 
 atomic refractivities (cf. p. 65). Ans. Obs. 8-56. Calc. 8-35. 
 
 g. The value of no for a mixture of formic acid and water containing 
 62*7 per cent, of the latter was found to be 1*3625 at 19*5, and the 
 density at the same temperature 1*1462. Calculate the refraction 
 constant by the Lorentz formula and compare it with, that calculated on 
 the assumption that the components exert their effects independently. 
 [D 19 for water 0-9984 ND = i*3333-] Ans. Obs. 0-1937. Calc. 0-1936. 
 
 10. Find the relationship between the solubility, s, of a gas and its 
 absorption coefficient, a, in a liquid at t (cf. p. 83). 
 
 Ans. 5/a = (273 + *)/273. 
 
 11. Calculate the gas constant, R, in litre-atmospheres from the obser- 
 vation that a solution containing 34*2 grams of cane sugar in i litre of 
 water has an osmotic pressure of 2*522 atmospheres at 20. 
 
 Ans. 0-0860. 
 
 12. The osmotic pressure of a 2 per cent, solution of acetone in water 
 is equal to 590 cm. of mercury at 10. What is the molecular weight of 
 acetone? Ans. 60 (found), 58-0 (theor.). 
 
 13. What is the molecular concentration of an aqueous solution of 
 urea which at 20 exerts an osmotic pressure of 4*6 atmospheres ? 
 
 Ans. 0-19 molar. 
 
 14. The vapour pressure of ether (mol. wt. 74) is lowered from 38-30 
 cm. to 36-01 cm. by the addition of 11*346 grams of turpentine to 100 
 grams of ether. Calculate the molecular weight of turpentine. 
 
 Ans. 132 (theor. 138). 
 
 15. The vapour pressure of water at 50 is 92 mm. How much urea 
 (mol. wt. 60) must be added to 100 grams of water to reduce the vapour 
 pressure by 5 mm. ? Ans. 18-1 grams. 
 
 16. A current of dry air was passed in succession through a bulb con- 
 taining a solution of 30 cane sugar in 160 grams of water, through a 
 bulb, at the same temperature, containing water, and finally through a 
 tube containing concentrated sulphuric acid. The loss of weight in the 
 water bulb was 0-0315 grams and the gain in weight in the sulphuric acid 
 bulb 3-02 grams. Calculate the molecular weight of cane sugar in the 
 solution. Ans. 339. 
 
 17. The addition of 1*065 grams of iodine to 30-14 grams of ether 
 raises the boiling-point of the latter by 0-296. What is the molecular 
 weight of iodine in ether ? Ans. 251. 
 
 18. The vapour pressure of ether at o is 183-4 mm., at 20 433*3 mm. 
 Calculate the latent heat of vaporisation per mol. of ether at 10. 
 
 Ans. 6840 cal. 
 
 19. The vapour pressure of water over a mixture of CuSO 4 , 5H 2 O and 
 CuSO 4 , 3H.X) is 2-933 mm. at 13-95 and 21-701 mm. at 39*7. Calculate 
 the heat given out when i mol. of water combines with CuSO 4 , 3H 2 O to 
 form CuSO 4 , 5H ? O. Ans. - 13,730 cal. 
 
 20. 0-3 grams of camphor, C 10 H 16 O, added to 25*2 grams of chloroform 
 raise the boiling-point of the solvent by 0-299. Calculate the molecular 
 elevation constant for chloroform. Ans. 38-2. 
 
 21. From the data in the previous question calculate the heat of vapo- 
 risation of chloroform (boiling-point 61). Ans. 6931 cal. 
 
 22. 1-2 grams of a substance dissolved in 24-5 grams of water (K =s 
 
PROBLEMS AND QUESTIONS 419 
 
 18-5) caused a depression of the freezing-point of 1*05. Find the mole- 
 cular weight of the substance. Ans. 86. 
 
 23. Beckmann found that 0*0458 grams of benzoic acid in 15 grams oi 
 nitrobenzene (K = 80) caused a depression of the freezing-point of 0-099. 
 What conclusion can be drawn from this observation as to the molecular 
 condition of benzoic acid in nitro benzene ? Ans. Acid is associated. 
 
 24. At 343 the vapour pressure of ammonium bromide is 195 mm. 
 and at 356 it is 289 mm. Calculate the heat of vaporisation of 
 ammonium bromide, assuming dissociation complete. Ans. 45,000 cal. 
 
 25. From formula (i) p. 137, deduce the expression 
 
 and hence calculate the osmotic pressure of an ethereal solution the boil- 
 ing-point of which is 35*2. (Boiling-point of ether, 34*8 ; latent heat of 
 vaporisation per gram 84-5 cal. Ans. 6*5 atmos. 
 
 26. At 21 the surface tension, 7, of diethyl sulphate in 28-28 
 dynes/cm. 2 and at 62*6 7 is 24*00 dynes/cm. 2 Find the value of c, the 
 temperature coefficient of the molecular surface energy (D 21 = 1*0748 ; 
 D 62 * 6 = 1*0278). Ans. 2*17. 
 
 27. For rronochlorhydrin 7 at 17 is 47*61 dynes/cm. 2 (D = 1-3254) 
 and at 57*8 4372 dynes/cm. 2 (D = 1*2883). What conclusions can be 
 drawn from these data as to the molecular complexity of the liquid ? 
 
 Ans. c = 1*44, liquid is associated. 
 
 28. Calculate the heats of formation of ethane, ethylene and acety. 
 lene respectively from their elements at 17 (a) at constant pressure, (b) al 
 constant volume from the following data. Heats of combustion : ethane 
 370,440 cal., ethylene 333,350 cal., acetylene 310,100 cal. Heats of 
 formation : carbon dioxide 94,300 cal., liquid water 68,400 cal., all at 
 constant pressure. 
 
 Ans. Ethane: C.P. 23,360 cal., C.V. 22,200 cal. Ethylene: 
 C.P. - 7950 cal., C.V. - 8530 cal. Acetylene : C.P. 
 - 53,100 cal., C.V. - 53,100 cal. 
 
 29. Find the heat of formation of anhydrous aluminium chloride from 
 the following data (Thomsen). 
 
 2A1 + 6HClAq = 2 AlCl 3 Aq + 3H 2 + 239,760 cal. 
 
 H 2 + C1 2 = 2HC1 + 44,000 cal. 
 
 HC1 + Aq = HClAq + 17,315 cal. 
 
 A1C1 3 + Aq = AlCl 3 Aq + 76,845 cal. 
 
 Ans. 321,960 cal. (for A1 2 C1 6 ). 
 
 30. The heat of solution of anhydrous strontium chloride is 11,000 
 cal., that of the hexahydrate - 7300 cal. What is the heat of hydra- 
 tion of the anhydrous salt to hexahydrate. Ans. 18,300 cal. 
 
 31. The specific heats of diamond and graphite in the neighbourhood 
 of 10 (o-i7) are 0*1128 and 0-1604 calories per gram respectively. The 
 heats of combustion are 94,310 and 94,810 calories per 12 grams respec- 
 tively. Find the heat evolved in the transformation of graphite to 
 diamond at o C. Ans ; 490 cal. 
 
 32. In the synthesis of nitric acid and. of ammonia the primary reac- 
 tions are 
 
 N 2 + O 2 ;-;>2NO - 43,200 cal. 
 24,000 cal. 
 
4 2o OUTLINES OF PHYSICAL CHEMISTRY 
 
 Discuss fully the effect of temperature and pressure on these reactions 
 and refer, for illustration, to the manufacturing processes : why is it that 
 in these processes an elevated temperature is used, although one reaction 
 is endothermic and the other exothermic ? 
 
 At 2000 abs. K = [NO]/[N 3 ]*[O 2 ]i = 0-0153. 
 
 Assuming the heat of reaction independent of temperature, calculate 
 the equilibrium constant at 2500 abs. (Birmingham Univ.) 
 
 33. The vapour density of phosphorus pentachloride referred to air as 
 unity was found to be 5-08 at 182, 4-00 at 250, and 3-65 at 300, calcu- 
 late the degrees of dissociation at these temperatures. 
 
 Ans. 41-7 per cent. ; 80 per cent. ; 97-3 per cent. 
 
 34. From the following data for the equilibrium N 2 O 4 ^2NO 2 at 
 49-7 calculate the degree of dissociation at each pressure and show, by 
 finding the dissociation constant, that the law of mass action applies : 
 Pressure in mm. Hg 26-80 93'75 182-69 261/37 497*75 
 Density (air = i) 1-663 1-788 1*894 i*993 2 * I 44 
 The vapour density of N 2 O 4 is 3*179 (air = i). 
 
 Ans. 0-93 ; 0-789 ; 0-69 ; 0*63 ; 0-493. 
 
 35. Bodenstein found that the degree of dissociation of carbonyl 
 chloride, according to the equation COC1 2 ^> CO + C1 2 , is 67 per cent, at 
 503, 80 per cent, at 553, and 91 per cent, at 603. From these results 
 calculate the heat of dissociation of carbonyl chloride. 
 
 Ans. 19,210 cal. from 5O3-553 ; 22,880 cal. from 553-6o3. 
 
 36. The ratio of distribution of aniline between benzene and water is 
 io-i : i. When a litre of aniline hydrochloride solution, containing 
 0-0997 mols. of the salt, was shaken with 59 c.c. of benzene at 25 it was 
 found that 50 c.c. of benzene had taken up 0-0648 grams of aniline. 
 Find the amount of hydrolysis of aniline hydrochloride in the solution 
 and calculate the dissociation-constant of aniline as a base (cf. p. 291). 
 
 Ans. i'56 per cent., 4*6 x 10 10 . 
 
 37. When heated in aqueous solution at 52-4 the concentration of 
 sodium bromoacetate in solution was n-o, 9-4, 7-9 and 6-9 at times o, 26, 
 52 and 74 hours respectively from the commencement of the reaction, the 
 decomposition being ultimately complete. Find the order of the reaction 
 and calculate the times required to complete (a) one-third, (b) two-thirds 
 of the change. Ans. Unimolecular. 65-2 hours, 177 hours. 
 
 38. In an experiment on the rate of reaction between sodium thiosul- 
 phate and ethyl bromoacetate (cf. p. 232) 50 c.c. of the reaction mixture 
 required the following amounts of O-QIIO N iodine at the times from the 
 commencement of the reaction indicated in the table. 
 
 t (min.) o 5 10 15 25 40 ^ 
 
 cc.s. iodine solution 37-25 24-7 18-75 15-3 ir6 8*85 4-4 
 
 Show that the reaction is of the second order and find the velocity 
 
 constant for concentrations of i mol. per litre. Ans. 14-6. 
 
 39. From the electrolysis of hydrochloric acid in a cell with a 
 cadmium anode the following results were obtained : change in concen- 
 tration of chlorine at anode and cathode respectively + 0*00545 gram 
 silver deposited in voltameter connected in series with the cell 0*0986 
 gram. Calculate the transport numbers of hydrogen and chlorine (Cl = 
 35-46; Ag = 107*9). Ans. H = 0*832; Cl =a o'i68. 
 
PROBLEMS AND QUESTIONS 421 
 
 40. The transport number of the cation in potassium chloride was found 
 to be 0-497 and \ M = 130-1. 
 
 What is the absolute velocity of K' in cm. per second, under unit 
 potential gradient ? Ans. 0*00067 cm. /sec. 
 
 41. At 18 the velocity of migration of the Ag- ion is 0*00057 cm./sec. 
 and of the NO 3 ion 0-00063 cm./sec. What is the value of p^ for silver 
 nitrate at 18 ? Ans. jt ^ = 115-8. 
 
 42. Find the degree of ionisation of lactic acid at different dilutions 
 and. calculate the ionisation constant from the following data, valid for 
 25: 
 
 v (litres) 64 128 256 512 oo 
 
 pv 34-3 47-4 64-2 87-6 381 
 
 Ans. 0-000138. 
 
 43. If the velocity coefficient for catalysis by N/4 acetic acid is 
 0-00075 what will be the coefficient when the solution is also N/4O with 
 respect to sodium acetate, assuming that the latter is dissociated to the 
 extent of 86 per cent. ? Ans. 0-000075. 
 
 44. If an amount of base insufficient for complete saturation is added 
 to an equimolecular mixture of acetic and glycollic acid, in what propor- 
 tion will the salts be formed? (Dissociation constants at 25. Acetic 
 acid 0-000018, glycollic acid o'ooois.) Ans. i : 2*9. 
 
 45. A N/io solution of sodium acetate is ionised to the extent of 80 
 per cent, at 18. What is the osmotic pressure of the solution at this 
 temperature ? Ans. 4*28 atmos. 
 
 46. Sodium chloride in 0*2 molar solution is dissociated to the extent 
 of 80 per cent, at 18. What will be the concentration of a urea solution 
 which is isotonic with the salt solution ? Ans. 21-6 grams per litre. 
 
 47. Calculate the E. M. F. of an oxyhydrogen cell from the facts that 
 
 2H 3 + O 2 = 2H 3 O + 2 x 68,400 cal. 
 
 and that the temperature coefficient of the E. M. F. of the cell is 
 - 0-00085 volts at room temperature (17). Ans. 1*23 volts. 
 
 48. Find the E. M. F. at each electrode and the total E. M. F. of the cell 
 
 Fe I FeSO 4 o-iN | CuSO 4 o-oi N | Cu 
 
 at 25, assuming that the iron salt is 40 per cent, ionised and the copper 
 salt 60 per cent, ionised at the dilutions in question, and that the P. D. at 
 the liquid contact is eliminated. [Use the general formula on p. 390.] 
 
 Ans. Fe electrode - 0*46 - 0*041 = - 0-501 volts. 
 
 Cu + 0-329 - 0-064 = + 0-265 volts. 
 
 Total E. M. F. of cell - 0-776 volts. 
 
 49. Discuss the influence of temperature and pressure on the equilibrium 
 
 2SO 2 + O 2 ^2SO 3 + 45,200 cal. 
 
 and show how the change of the equilibrium constant with temperature 
 can be calculated. (University Coll., London.) 
 
 50. Give an account of the different methods, static and dynamic, 
 which have been used for determining the relative chemical affinities of 
 acids in dilute aqueous solution. (University Coll., London.) 
 
 51. Explain the action of the Daniell battery (copper in copper sul- 
 phate, zinc in zinc sulphate) (a) on the hypothesis of the dual nature of 
 electricity + and - , (b) on the hypothesis that negative electricity consists 
 of electrons. Draw two diagrams, one to illustrate each method of re- 
 presentation. (University Coll., London.) 
 
422 OUTLINES OF PHYSICAL CHEMISTRY 
 
 52. Criticise the various theories which have been advanced to explain 
 the mechanism of electrolytic conauction. 
 
 53. Criticise Berthelot's principle of maximum work. 
 
 54. What do you understand by the solubility product ? Discuss the 
 question of the simultaneous solubility of two salts possessing a common 
 ion. (St. Andrews Univ.) 
 
 55. Explain carefully why the ratio of the specific heats of a gas de- 
 pends on the number of atoms in the molecule. 
 
 Briefly describe the method by which for physico-chemical purposes 
 this ratio is determined. (Sheffield Univ.) 
 
 56. Define the terms " absorption coefficient," " critical solution tem- 
 perature," " solid solution " and " eutectic alloy " giving examples. 
 (Sheffield Univ.) 
 
 57. " Every chemical reaction is reversible." Discuss the statement 
 carefully so as to reveal the extent to which it is true, quoting examples. 
 (Sheffield Univ.) 
 
 58. Write a brief account of the developments in electro-chemistry 
 associated with the names of Daniell, Faraday, Hittorf and Arrhenius. 
 (Sheffield Univ.) 
 
 59. How would you determine, either directly or indirectly, the critical 
 temperature, pressure and volume (or density) of a pure substance. 
 (Dublin Univ.) 
 
 60. Discuss the conditions on which the possibility of completely 
 separating a mixture of two miscible liquids by fractional distillation de- 
 pends. (St. Andrews Univ.) 
 
 61. Discuss the associating and dissociating properties of solvents. 
 (St. Andrews Univ.) 
 
 62. Describe some experiments in support of the view that ions migrate 
 during electrolysis. How are the transport numbers of ions determined ? 
 (St. Andrews Univ.) 
 
 63. Describe the preparation and properties of some colloidal solutions. 
 What is the present view as to the nature of such solutions ? (St. 
 Andrews Univ.) 
 
 64. Explain why (a) a higher potential is necessary to electrolyse a 
 decinormal solution of hydrochloric acid than is necessary for a normal 
 solution, (b) why an E.M.F. is usually set up at the surface of contact of 
 dilute and concentrated solutions of the same electrolyte, (c) why magnesium 
 displaces hydrogen slowly from water but rapidly from hydrochloric acid. 
 (St. Andrews Univ.) 
 
 65. Describe an experiment to illustrate the migration of ions during 
 electrolysis. How would- you show that the cation and anion in a solu- 
 tion of copper sulphate move with different velocities, and how would 
 you determine the relative velocities ? (Birmingham Univ.) 
 
 66. State the phase rule and explain the terms involved. Apply the 
 rule to explain the fact that the extent of dissociation of calcium carbonate 
 depends only on the temperature. (Sheffield Univ.) 
 
 67. What class of substance have in solution molecules of larger size 
 than that calculated from the formula, and how does the phenomenon 
 depend on (a) the constitution of the solute, (b) the nature of the solvent. 
 (Sheffield Univ.) 
 
APPENDIX. 
 
 EQUIVALENT CONDUCTIVITY AT 18. 
 TABLE la. 
 
 Normal 
 Concentr'n. 
 
 KC1. 
 
 NaCl. 
 
 KNO 3 . 
 
 AgN0 3 . 
 
 NaC 2 H 3 2 . 
 
 JK.SO, 
 
 Na 2 C0 3 . 
 
 BaCl 2 . 
 
 0*0001 
 
 129-5 
 
 109*7 
 
 1247 
 
 "5*5 
 
 76-8 
 
 133 '5 
 
 _ 
 
 120-5 
 
 0-0002 
 
 129*1 
 
 109*2 
 
 124*3 
 
 115-2 
 
 76*4 
 
 1327 
 
 
 
 119-8 
 
 0*0005 
 
 128-3 
 
 108-5 
 
 123-6 
 
 "4'S 
 
 75-8 
 
 130-8 
 
 
 
 118*3 
 
 O'OOI 
 
 127-6 
 
 107-8 
 
 122-9 
 
 114-0 
 
 75 '2 
 
 129*0 
 
 II2'O 
 
 n6'9 
 
 0*002 
 
 126*6 
 
 1067 
 
 I22"O 
 
 113-0 
 
 74*3 
 
 126-3 
 
 108*5 
 
 115*0 
 
 0-005 
 
 124-6 
 
 104-8 
 
 I2O*I 
 
 III'O 
 
 72-4 
 
 121-9 
 
 102-5 
 
 111*3 
 
 O'OI 
 
 122-5 
 
 I02'8 
 
 Il8*I 
 
 108-7 
 
 70*2 
 
 117*4 
 
 96*2 
 
 1077 
 
 0*02 
 
 120*0 
 
 100*2 
 
 115*2 
 
 105*6 
 
 67*9 
 
 111*8 
 
 89-5 
 
 IO3"3 
 
 0*05 
 
 115*9 
 
 95 '9 
 
 IIO'O 
 
 lOO'I 
 
 64*2 
 
 102-5 
 
 80*3 
 
 96*8 
 
 O'l 
 
 iii*9 
 
 92*5 
 
 104*4 
 
 947 
 
 6rx 
 
 95*9 
 
 72*9 
 
 92-2 
 
 0'2 
 
 1077 
 
 88*2 
 
 98-6 
 
 88-1 
 
 57 'i 
 
 88*9 
 
 65-6 
 
 867 
 
 '5 
 
 102*3 
 
 80-9 
 
 897 
 
 77*8 
 
 49*4 
 
 787 
 
 54'5 
 
 77*6 
 
 I 
 
 98*2 
 
 74'4 
 
 80-4 
 
 67*8 
 
 41*2 
 
 71*8 
 
 45 '5 
 
 70-3 
 
 
 92*6 
 
 64*8 
 
 69*4 
 
 55-8 
 
 30*0 
 
 
 34'5 
 
 60-3 
 
 TABLE U. 
 
 Normal 
 Concentr'n. 
 
 CuS0 4 . 
 
 KOH. 
 
 NaOH. 
 
 NH 4 OH. 
 
 HC1. 
 
 HN0 3 . 
 
 H 2 S0 4 . 
 
 HC 2 H 3 2 . 
 
 O'OOOI 
 
 II3'3 
 
 _ 
 
 __ 
 
 66 
 
 _ 
 
 _ 
 
 __ 
 
 107 
 
 0*0002 
 
 in*i 
 
 
 
 
 
 53 
 
 
 
 
 
 
 
 80 
 
 0-0005 
 
 106*8 
 
 
 
 
 
 38-0 
 
 
 
 
 
 368 
 
 57 
 
 O'OOI 
 
 zoi'6 
 
 234 
 
 208 
 
 28-0 
 
 377 
 
 375 
 
 361 
 
 41 
 
 O'OO2 
 
 93 '4 
 
 233 
 
 206 
 
 20 '6 
 
 376 
 
 374 
 
 351 
 
 30*2 
 
 0*005 
 
 81*5 
 
 230 
 
 203 
 
 13*2 
 
 373 
 
 37i 
 
 33 
 
 20*0 
 
 0*01 
 
 72*2 
 
 228 
 
 200 
 
 9*6 
 
 370 
 
 368 
 
 308 
 
 I4-3 
 
 0-02 
 
 63-0 
 
 225 
 
 197 
 
 7'i 
 
 367 
 
 364 
 
 286 
 
 io"4 
 
 0*05 
 
 5 J *4 
 
 219 
 
 190 
 
 4'6 
 
 360 
 
 357 
 
 253 
 
 6*48 
 
 O'l 
 
 45 'o 
 
 213 
 
 183 
 
 3'3 
 
 35i 
 
 35 
 
 225 
 
 4*60 
 
 0-2 
 
 39-2 
 
 206 
 
 178 
 
 2'3 
 
 342 
 
 340 
 
 214 
 
 3 '24 
 
 o*5 
 
 30*8 
 
 197 
 
 172 
 
 *'35 
 
 327 
 
 324 
 
 205 
 
 2*01 
 
 i 
 
 25-8 
 
 184 
 
 160 
 
 0-89 
 
 301 
 
 310 
 
 198 
 
 I-32 
 
 2 
 
 20'I 
 
 160*8 
 
 I3I-4 
 
 '53 2 
 
 254 
 
 258 
 
 183 
 
 0*80 
 
 From Kohlrausch and Holborn, Leitvermo*en der Elektrolyte. 
 423 
 
4 2 4 OUTLINES OF PHYSICAL CHEMISTRY 
 
 IONIC MOBILITIES (VELOCITIES) IN AQUEOUS SOLUTION AT 18. 
 TABLE Ila. 
 
 Normal 
 Concentr'n. 
 
 K. 
 
 Na. 
 
 Li. 
 
 NH 4 . 
 
 Ag. 
 
 Ba. 
 
 *Sr. 
 
 *Ca. 
 
 *Mg. 
 
 $Zn. 
 
 O'OOOO 
 
 65-3 
 
 44*4 
 
 35*5 
 
 64-2 
 
 557 
 
 57*3 
 
 54 'o 
 
 53' 
 
 49 
 
 47 '5 
 
 0*0001 
 
 647 
 
 43 '8 
 
 34 '9 
 
 6 3 -6 
 
 55'4 
 
 55'o 
 
 517 
 
 50-6 
 
 47 
 
 45*i 
 
 0'0002 
 
 64-4 
 
 43'6 
 
 347 
 
 63-4 
 
 55' 1 
 
 54'3 
 
 Si 'o 
 
 50*0 
 
 46 
 
 44 '5 
 
 o'ooo5 
 
 64 'I 
 
 43'3 
 
 34'4 
 
 6 3 -o 
 
 54*9 
 
 53'3 
 
 50-0 
 
 48-9 
 
 45 
 
 43 '5 
 
 O'OOI 
 
 637 
 
 42-9 
 
 34'0 
 
 627 
 
 547 
 
 52-2 
 
 48-9 
 
 47-8 
 
 43 
 
 42-3 
 
 0'CO2 
 
 63-2 
 
 42-4 
 
 33*5 
 
 62-2 
 
 54 '2 
 
 57 
 
 47 '4 
 
 46-4 
 
 42 
 
 40-9 
 
 0*005 
 
 62-3 
 
 41-4 
 
 32-6 
 
 6l'2 
 
 53 '2 
 
 48*2 
 
 44*9- 
 
 43*9 
 
 40 
 
 38-4 
 
 O'OI 
 
 6i'3 
 
 40-5 
 
 31-6 
 
 60-2 
 
 5i'9 
 
 457 
 
 42-4 
 
 41-4 
 
 37 
 
 35*9 
 
 0*02 
 
 60 'o 
 
 39 '2 
 
 3' 3 
 
 59 'o 
 
 5' 
 
 427 
 
 39'4 
 
 38-3 
 
 34 
 
 32-9 
 
 0-03 
 
 59 '2 
 
 38-3 
 
 2 9'4 
 
 58-1 
 
 48-6 
 
 40-5 
 
 37'2 
 
 36-1 
 
 1 32 
 
 307 
 
 0*05 
 
 57 '9 
 
 37 'o 
 
 28*2 
 
 56-8 
 
 46-6 
 
 377 
 
 34 '4 
 
 33 '4 
 
 29 
 
 27-9 
 
 O'l 
 
 55'8 
 
 35'o 
 
 26-1 
 
 54'8 
 
 43 '3 
 
 33'8 
 
 30-S 
 
 29-4 
 
 25 
 
 24-0 
 
 TABLE 
 
 Normal 
 Concentr'n. 
 
 H. 
 
 Cl. 
 
 1 
 
 NO S . 
 
 C10 3 . 
 
 C 2 H 3 3 . 
 
 JS0 4 . 
 
 |C 2 4 . 
 
 C0 3 . 
 
 OH. 
 
 O'OOOO 
 
 3i8 
 
 65-9 
 
 667 
 
 60-8 
 
 56-2 
 
 337 
 
 697 
 
 6 3 
 
 _ 
 
 174 
 
 O'OOOI 
 
 318 
 
 65-3 
 
 66-1 
 
 60 '2 
 
 55'5 
 
 33'i 
 
 67*2 
 
 61 
 
 
 
 172 
 
 0'0002 
 
 316 
 
 6 5 -i 
 
 65-9 
 
 60 'o 
 
 55'2 
 
 33'o 
 
 66-6 
 
 60 
 
 
 
 172 
 
 0*0005 
 
 315 
 
 64-8 
 
 65-5 
 
 59 '6 
 
 54*6 
 
 32-8 
 
 6SH 
 
 59 
 
 
 
 171 
 
 O'OOI 
 
 3M 
 
 64-4 
 
 65-1 
 
 59*3 
 
 54 'i 
 
 32-6 
 
 64^0 
 
 58 
 
 69 
 
 171 
 
 0-002 
 
 313 
 
 63'9 
 
 64-6 
 
 58-8 
 
 53'4 
 
 3 2< 4 
 
 62 '3 
 
 56 
 
 66 
 
 170 
 
 0-005 
 
 3" 
 
 63-0 
 
 637 
 
 57-8 
 
 52'4 
 
 31-6 
 
 59'2 
 
 54 
 
 60 
 
 168 
 
 0*01 
 
 310 
 
 62 'o 
 
 627 
 
 56-8 
 
 Si'3 
 
 30-8 
 
 56-i 
 
 5i 
 
 55 
 
 167 
 
 0'02 
 
 307 
 
 607 
 
 6i'5 
 
 55-6 
 
 497 
 
 29 '8 
 
 52'3 
 
 48 
 
 5 
 
 165 
 
 0*03 
 
 305 
 
 59'8 
 
 60-6 
 
 547 
 
 48-4 
 
 29*0 
 
 497 
 
 46 
 
 47 
 
 163 
 
 0'05 
 
 302 
 
 58-6 
 
 59*3 
 
 53'4 
 
 46-4 
 
 28 'o 
 
 46-1 
 
 43 
 
 43 
 
 161 
 
 O'l 
 
 296 
 
 56-5 
 
 57 '3 
 
 5i '4 
 
 43'2 
 
 26-4 
 
 41-9 
 
 39 
 
 38 
 
 157 
 
 from Kohlrausch and Holborn, Leitvermogen der Elektrolyte. 
 
APPENDIX 
 
 TRANSPORT NUMBERS OF ANIONS. 
 
 TABLE III. 
 n = gram equivalents per litre. 
 
 425 
 
 n =3 
 
 O'OI. 
 
 0-02. 
 
 0-05. 
 
 O*I. 
 
 O*2. 
 
 0-5- 
 
 i. 
 
 2. 
 
 KC1 1 
 KBr 
 KI f 
 NH 4 ClJ 
 
 0*506 
 
 0-507 
 
 0-507 
 
 0-508 
 
 0-509 
 
 o'S^ 
 
 O'.5i4 
 
 0-5I5 
 
 NaCl 
 
 
 
 
 
 0*614 
 
 0*617 
 
 0*620 
 
 0-626 
 
 0-637 
 
 0-642 
 
 KNO, 
 
 
 
 
 
 
 
 0-497 
 
 0-496 
 
 0-492 
 
 0-487 
 
 0*479 
 
 NaN0 3 
 
 
 
 
 
 
 
 0-615 
 
 0*614 
 
 O"6l2 
 
 0*611 
 
 0-608 
 
 AgN0 3 
 KC 2 H 3 O 2 
 
 0-528 
 
 0-528 
 
 0-528 
 
 0-528 
 o'33 
 
 0-527 
 
 o*33 
 
 0-519 
 0-33 
 
 0*501 
 0*331 
 
 0-476 
 0-332 
 
 KOH 
 
 
 
 
 
 
 
 0735 
 
 0-736 
 
 0-738 
 
 0*740 
 
 
 NaOH 
 
 
 
 
 
 o'8i 
 
 0-82 
 
 0-82 
 
 0-82 
 
 0-825 
 
 
 
 HC1 
 
 
 
 
 
 0-172 
 
 0*172 
 
 0-172 
 
 0*173 
 
 0*176 
 
 0-185 
 
 iBaC! 2 
 
 0*56 
 
 0^65 
 
 o'575 
 
 0*585 
 
 o*595 
 
 0-615 
 
 0*640 
 
 0-657 
 
 ICaCL 
 
 iK 2 C0 3 
 
 0-58 
 
 0'59 
 
 o'6i 
 o*39 
 
 0*64 
 0-40 
 
 0-66 
 0-41 
 
 0-675 
 0-435 
 
 0-686 
 o'434 
 
 0700 
 0-413 
 
 |Na 2 C0 3 
 
 
 
 
 
 0*52 
 
 o-53 
 
 o'53 
 
 o*54 
 
 0-548 
 
 0-542 
 
 |MgS0 4 
 
 
 
 
 
 o"6o 
 
 0-64 
 
 0-66 
 
 0*70 
 
 0-74 
 
 076 
 
 *CuSO 4 
 
 . 
 
 0-62 
 
 0-626 
 
 0-632 
 
 0-643 
 
 0-668 
 
 0-696 
 
 0720 
 
 IH^O, 
 
 ~~ 
 
 ~ 
 
 0-193 
 
 0-191 
 
 0-188 
 
 0-182 
 
 0-174 
 
 o'i68 
 
 POTENTIAL SERIES OF THE ELEMENTS. 1 
 
 TABLE IV. 
 
 The numbers in the following table give the potentials of the substances in question 
 in contact with normal ionic solutions of their salts, referred to three different standards, 
 The numbers under * oh are referred to the potential of the hydrogen electrode as o - o 
 volis, those under e oc are referred to the normal calomel electrode as zero, and those 
 under e c to the calomel electrode = + 0-560 volts. 
 
 
 *oh 
 
 g 
 
 " Absolute 
 
 
 oh. 
 
 oc. 
 
 " Absolute 
 
 
 
 
 Potentials." 
 
 
 
 
 Potentials." 
 
 Manganese 
 
 -1-07 
 
 -1*36 
 
 0*80 
 
 Hg/Hg 2 " 
 
 0775 
 
 0-492 
 
 1-052 
 
 Zinc 
 
 - 0*770 
 
 
 -0-493 
 
 Hg/Hg" 
 
 0*835 
 
 0-552 
 
 1*112 
 
 Iron 
 
 -0*46 
 
 -0-74 
 
 -0-18 
 
 Silver 25 
 
 0-798 
 
 
 1*075 
 
 Cadmium 
 
 0*42 
 
 -0*703 
 
 -0-143 
 
 Platinum . 
 
 0-86 
 
 0-58 
 
 I-I4 
 
 Thallium 
 
 0*32 
 
 - 0*603 
 
 - 0-043 
 
 Gold 
 
 i -08 
 
 0*80 
 
 
 Cobalt 
 
 -0*30 
 
 -0-58 
 
 0-02 
 
 Fluorine 
 
 2*0 
 
 i*7 
 
 2-26 
 
 Nickel 
 
 -0*25 
 
 
 + 0*03 
 
 Oxygen 2 
 
 1-2(1-66) 
 
 0*9 
 
 1-46 
 
 Lead 
 
 0*119 
 
 0*402 
 
 + 0*158 
 
 Chlorine^ 
 
 1*400 
 
 I'I2O 
 
 1-68 
 
 Hydrogen 
 
 + 000 
 
 -0*283 
 
 + 0-277 
 
 Bromine 5-25 
 
 1-095 
 
 0*812 
 
 1*372 
 
 Copper 
 
 + 0-329 
 
 + 0*046 
 
 + 0-606 
 
 Iodine J 
 
 0*628 
 
 0-345 
 
 0*905 
 
 1 From Le Blanc, Lehrbuch der Elektrochemie. 
 
 2 Cf. p. 400. These values apply to a solution of normal H" concentration. In order 
 to liberate oxygen from a solution of normal OH' concentration 0*8 volts less are re- 
 quired, and to liberate hydrogen from the same- solution 0*8 volts more are required 
 than in the case of a normal solution of acid. 
 
INDEX. 
 
 ABNORMAL molecular weights in solu- 
 tion, 123. 
 
 vapour densities, 40-42. 
 "Absolute" potentials, 375, 378. 
 Absorption oflight, 69-73. 
 
 spectra and chemical constitution, 
 
 73- 
 
 Accumulators, 405. 
 Acetic acid, adsorption of, 325. 
 
 atomic volume of, 61. 
 
 density of vapour, 41. 
 
 dissociation of, 267. 
 
 Acids, catalytic action of, 205, 219, 
 272. 
 
 effect of substitution on strength 
 
 of, 304. 
 
 strength of, 269-274. 
 Active mass, 156. 
 
 of solids, 173. 
 
 Additive properties, 62, 251, 335. 
 Adsorption, 324. 
 
 and enzyme action, 332. 
 
 and surface tension, 330. 
 
 by charcoal, 324. 
 
 formulae, 328. 
 
 theories of, 324-328. 
 Affinity, chemical, 148, 154, 271, 415. 
 
 constant, 157, 273, 275. 
 Amalgams, cells with, 351. 
 Amlcrons, 318. 
 
 Ammonium chloride, dissociation of, 
 42, 219. 
 
 hydrosulphide, dissociation of, 
 
 176. 
 
 Amphoteric electrolytes, 307. 
 Argon, position in periodic table, 23. 
 Associated solvents, ionising power of, 
 
 339- 
 
 Associating solvents, 123. 
 Association in gases, 41. 
 
 in solution, 124, 338-342. 
 Atomic heat, 21. 
 
 hypothesis, 5. 
 
 Atomic refractions, values of, 65. 
 
 volumes, 60. 
 
 weights, determination of, 8-15. 
 standard for, 17. 
 
 table of, 19. 
 
 Attraction, molecular, 34, 58. 
 Available energy, 104, 150, 352. 
 Avidity of acids, 269-274. 
 Avogadro's hypothesis, 10, 36. 
 
 deduction of, from kinetic 
 
 theory of gases, 32. 
 
 valid for solutions, 103, 109. 
 
 BASES, catalytic action of, 220. 
 
 strength of, 274-276, 292. 
 Beckmann's methods, 116, 120. 
 Benzoic acid, distribution between 
 
 solvents, 178, 198. 
 Beryllium (glucinum), atomic weight 
 
 of, 10, ii. 
 
 Bimolecular reactions, 207, 230. 
 Binary mixtures of liquids, 84-91. 
 
 distillation of, 87. 
 
 vapour pressure of, 87. 
 
 Blood, catalysis by, 202, 229. 
 Boiling-point, elevation of, 114, 118. 
 Boron, atomic heat of, 12. 
 Brownian movement, 318. 
 
 CADMIUM standard cell, 356. 
 Calomel electrode, 373. 
 Calorimeter, 146, 147, 153. 
 Capillary electrometer, 380. 
 Cane sugar, hydrolysis of, 205. 
 Carbon, atomic heat of, 12. 
 
 dioxide, critical phenomena of, 49. 
 Catalysis, 217-224. 
 
 mechanism of, 222. 
 
 technical importance of, 219. 
 Cathode rays, 399. 
 
 Chemical affinity, 148, 154, 271. 
 Chemical equilibrium" and E.M.F., 
 396. 
 
 427 
 
428 OUTLINES OF PHYSICAL CHEMISTRY 
 
 Chemical equilibrium and pressure, 
 169. 
 
 and temperature, 166, 169. 
 
 Clark cell, 358. 
 Coagulation of colloids, 320. 
 
 adsorption theory of, 323. 
 
 Colligative properties, 62. 
 Colloidal particles, charged, 319. 
 
 size of, 318, 323. 
 
 platinum, 220, 232, 316. 
 
 solutions, 313. 
 
 coagulation of, 319. 
 
 filtration of, 323. 
 
 optical properties ot, 317. 
 
 preparation of, 315. 
 
 Colloids, 313. 
 
 diffusion of, 313. 
 
 electrical properties of, 319. 
 
 irreversible, 323. 
 
 precipitation by electrolytes, 319. 
 
 reversible, 323. 
 Combining proportions, law of, 4. 
 
 volumes of gases, law of, 9, 
 Combustion, heat of, 145. 
 Complexions, 281, 303, 310. 
 Components, definition of, 181. 
 Concentration cells, 367-373. 
 Conductivity, electrical, effect of tem- 
 perature on, 263. 
 
 equivalent, 251. 
 
 of pure substances, 258. 
 
 measurement of, 254-258. 
 
 molecular, 249, 257, 265. 
 
 specific, 235, 249. 
 Conservation of energy, law of, 140. 
 
 of mass, law of, 3. 
 Constant boiling mixtures, 89. 
 Constitutive properties, 62. 
 Continuity of gaseous and liquid states, 
 
 Copper sulphate, hydrates of, 174. 
 Corresponding states, law of, 56. 
 
 temperatures, 57. 
 Critical constants, 51, 56. 
 
 phenomena, 49-58, 69. 
 
 solution temperature, 86. 
 
 temperature, determination of, 77. 
 Carbohydrates, 188. 
 Crystallisation interval, 192. 
 Crystalloids, 313. 
 
 "Cyclic" processes, 133, 136. 
 
 DANIELL cell, 348, 362, 366. 
 reversal of current in, 366, 401 . 
 
 Decomposition potential of electrolytes, 
 
 399- 
 
 Deliquescence, 175. 
 Density of gases and vapours, 36, 43. 
 
 determination of, 37, 41, 48. 
 
 Dialysed iron, 315. 
 
 Dialysis, 315. 
 
 Dielectric constant, 225, 338. 
 
 and ionisation, 338. 
 
 Diffusion of gases, 33. 
 
 in solution, 108. 
 
 and osmotic pressure, 98, io3. 
 
 Dispersed system, 315. 
 Dispersion, 315. 
 Dissociating solvents, 123. 
 Dissociation constant, 266, 273, 275. 
 
 electrolytic, 260-263, 335-337. 
 degree of, 261. 
 
 evidence for, 335'337. 
 
 mechanism of, 342. 
 
 of water, 283, 293. 
 
 of salt hydrates, 174. 
 
 in gases, 42, 125, 163. 
 solution, 125, 260-263 
 
 thermal, 163. 
 
 Distillation of binary mixtures, 87, 9*, 
 
 steam, 90. 
 
 Distribution coefficient, 95, 178, 197, 
 
 327- 
 
 Dulong and Petit's law, n. 
 
 Dyeing, adsorption theory of, 330-331. 
 
 EDISON accumulator, 407. 
 Efflorescence, 175. 
 Electro -affinity, 410. 
 Electrode, calomel, 373. 
 
 hydrogen, 376, 385. 
 
 mercuric oxide, 378. 
 Electrodes, normal, 375, 376. 
 Electrolysis, 236-238, 393. 
 
 of water, 401. 
 
 separation of metals, etc., by, 395. 
 Electrolytes, strong, 279-282. 
 Electrolytic dissociation. See Dis- 
 sociation, electrolytic. 
 
 Electrometer, capillary, 380. 
 Electromotive force and chemical 
 
 equilibrium, 396. 
 
 Electromotive force and concentration 
 of solutions, 365. 
 
 measurement of, 354. 
 
 standards of, 356. 
 
 Electrons, 407. 
 
 and light adsorption, 72. 
 
INDEX 
 
 429 
 
 Electrons and valency, 409. 
 Elements, i. 
 
 disintegration of, 2. 
 
 periodic classification of, 20. 
 
 potential series of, 386. 
 
 table of, 21. 
 Emulsoids, 322. 
 Enantiotropic substances, 197. 
 Endothermic and exothermic com- 
 pounds, 144. 
 
 Energy, available, 104, 150, 332. 
 
 chemical, 140, 351-354- 
 
 conservation of, 140. 
 
 free, 104, 150, 352. 
 
 internal, of gases, 46. 
 
 intrinsic, 143. 
 
 kinds of, 139. 
 
 Enzyme action and adsorption, 332. 
 
 reactions, 221. 
 reversibility of, 222. 
 
 Equilibrium, effect of pressure on, 169. 
 of temperature on, 166-170. 
 
 false, 218. 
 
 in gaseous systems, 161-164. 
 
 in electrolytes, 266-312. 
 
 in non-electrolytes, 164-166. 
 
 kinetic nature of, 158, 160, 241. 
 Equivalents, chemical, 8, 15, 237. 
 
 electrochemical, 237. 
 Ester equilibrium, 156, 164. 
 Esters, hydrolysis of, 206. 
 
 sapouification of, 207, 275. 
 Eutectic point, 187, 191. 
 Exothermic and endothermic com- 
 pounds, 144. 
 
 Expansion of gases, work done in, 27. 
 
 FARADAY'S laws, 237. 
 
 Ferric chloride, hydrates of, 194. 
 
 Filtration of colloidal solutions, 323. 
 
 Fluidity, 74. 
 
 Formation of compounds, heat of, 143. 
 
 Freedom, degrees of, 179, 182. 
 
 Freezing-point, lowering of, 119-121, 
 
 194. 
 Friction, internal. See Viscosity. 
 
 GAS cells, 364. 
 
 constant, R, 26, 102. 
 
 laws, 25-26. 
 
 deduction of, 31-33. 
 
 deviations from, 28, 33-35. 
 
 Gases, 25-48. 
 
 adsorption of, 328. 
 
 Gases, behaviour of, on compression, 51. 
 
 kinetic theory of, 29-35, 46. 
 
 liquefaction of, 58. 
 
 solubility of, in liquids, 82. 
 
 specific heat of, 43-48. 
 Gay-Lussac's law of gaseous volumes, 
 
 9- 
 
 of expansion of gases, 25. 
 
 Gel (hydrogel), 323. 
 Gladstone-Dale formula, 64. 
 Grotthus' hypothesis, 264. 
 
 HAEMOGLOBIN, osmotic pressure of, 
 
 316. 
 
 Heat of combustion, 145. 
 additive character of, 148. 
 
 of ionisation, 285, 295, 344. 
 
 of solution, 146, 153. 
 Helium, liquefaction of, 59. 
 
 critical constants of, 51. 
 Helmholtz formula, 351-354, 373- 
 
 views on valency, 399. 
 Henry's law, 82, 95, 178. 
 Hess's law, 125. 
 
 Heterogeneous equilibrium, 172-199. 
 Hydrate theory, 333, 339. 
 Hydrated ions, 346. 
 Hydrates, dissociation of, 174. 
 
 in solution, 340. 
 
 Hydration in solution, 345-347. 
 
 Hydrogel, 323. 
 
 Hydrogen, adsorption of, 329. 
 
 electrode, 375, 385. 
 
 iodide, decomposition of, 159, 161. 
 Hydrogen-oxygen cell, 392. 
 
 peroxide, decomposition of, 202, 
 
 229. 
 
 Hydrogen sulphide, dissociation of, 
 167. 
 
 INDICATORS, theory of, 296, 309. 
 Intermediate compounds in catalysis, 
 
 223. 
 
 Ionic and non-ionic reactions, 307. 
 Ionisation and chemical activity, 311. 
 
 degree of, 261, 281. 
 
 energy relations in, 343. 
 
 heat of, 285, 295, 344. 
 
 mechanism of, 342, 345. 
 
 role of solvent in, 314, 338, 342- 
 
 345- 
 Ionising power of solvents, 338, 339. 
 
 and free affinity, 339. 
 
 Ions, 236. 
 
.430 OUTLINES OF PHYSICAL CHEMISTRY 
 
 Ions, complex, 281, 303. 310, 
 
 migration of, 236, 243-249. . 
 
 reactivity of, 306, 311. 
 
 velocity or mobility of, 252-254. 
 Irreversible electro-chemical processes, 
 
 354-. 
 
 Isoelectnc point, 320. 
 Isohydric solutions, 277, 414. 
 Isomorphism, 13, 14. 
 Isosmotic solutions, 105. 
 Isotonic coefficients, 106. 
 
 solutions, 105. 
 
 JOULE-THOMSON effect, 58. 
 
 KINETIC energy, 32, 46, 140. 
 
 and temperature, 32, 319, 
 
 of gas molecules, 46. 
 
 theory of gases, 29-35, 46. 
 Kohlrausch's law, 251. 
 
 LEAD accumulator, 405. 
 Le Chatelier's theorem, 169. 
 Light, absorption of, 69-73. 
 Liquefaction of gases, 58. 
 Liquids, molecular weight of, 125. 
 
 miscibility of, 84, 95. 
 
 properties of, 49-79. 
 Lorenz-Lorentz formula, 64. 
 
 MASS action, law of, 155-160, 173. 
 
 and strong electrolytes, 279- 
 
 283. 
 
 in heterogeneous systems 
 
 172. 
 
 proof of, 160. 
 
 Maxima and minima on curves, 76, 
 
 88, 339-342- 
 
 " Maximum work and chemical af- 
 finity, 150. 
 Medium, influence of, on reaction 
 
 velocity, 224. 
 Metastable phases, 184. 
 Microns, 318. 
 
 Migration of the ions, 236, 243-249. 
 Miscibility of liquids, 84. 
 Mixed crystals, 13, 95, 190, 192. 
 Molecular attraction, 34, 58. 
 
 surface energy, 125. 
 
 volume, 60, 78. 
 
 in solution, 62. 
 
 weight of colloids, 316. 
 abnormal, 40, 123. 
 
 of dissolved substances, 109- 
 
 125. 
 
 Molecular weight of gases, 36-43. n c- 
 of liquids, 125-127. 
 Molecules, velocity of gaseous, 33. 
 Monotropic substances, 197. 
 Morse and Frazer's measurements of 
 
 osmotic pressure, 104. 
 Movement, Brownian, 318. 
 Multiple proportions, law of, 4. 
 
 NATURAL law, definition of terms, 6, 7 
 Neumann's law, 13. 
 " Neutal salt action," 84. 
 Neutralization as ionic reaction, 284. 
 
 heat of, 148, 283-285, 336. 
 Neutrons, 409. 
 
 Normal electrodes, 374, 375. 
 
 potentials, 388. 
 
 OCTAVES, law of, 20. 
 Optical activity, 66-69, 229. 
 
 van't Hoff-Le Bel theory of, 67. 
 
 Order of a reaction, 213. 
 Osmotic pressure, 97-109. 
 
 and diffusion, 98, 108. 
 
 and elevation of boiling-point, 
 
 109, 138. 
 
 and gas pressure, 102, 103. 
 
 and lowering of freezing-point, 
 
 109, 136. 
 
 and lowering of vapour pres- 
 sure, 109, 131. 
 
 measurement of, 99, 105. 
 
 m^c anism of, 106. 
 
 of colloids, 316. 
 Ostwald's dilution law, 266, 307, 
 Overvoltage, 401. 
 
 Oxidation, definition of, 396. 
 Oxidation-reduction cells, 393. 
 Ozone-oxygen equilibrium, 168. 
 
 PARTIAL pressures, law of, 81. 
 Periodic law, 23. 
 
 system, 20-24. 
 
 table, 21. 
 
 Phase, definition of term, 17?. 
 
 rule, 179. 
 
 Phosphorus pentachlori 'e, dissociation 
 
 of, 163. 
 
 Plasmolysis, 105. 
 Polarisation, concentration, 404. 
 
 electrolytic, 350, 355, 398, 404. 
 
 of light, 66. 
 
 Potential differences at liquid junctions, 
 382. 
 
INDEX 
 
 43 1 
 
 otential differences, origin of, 362. 
 single, 375, 379, 381. 
 
 series of the elements, 387. 
 Potentials, "absolute," 375, 378. 
 
 normal, 388. 
 Protective colloids, 332. 
 Prout's hypothesis, 17. 
 
 QUADRUPLE point, 196. 
 
 R, value of, for gases, 27. 
 
 for solutes, 102. 
 
 Radium, 2, 379. 
 Raoult's formula, 112. 
 Reaction, order of, 213. 
 Reactions, consecutive, 216. 
 
 counter, 216. 
 
 side, 215. 
 
 Reduction, definition of, 395. 
 Refraction formulae, 64. 
 Refractivity, 63-66. 
 Reversibility in cells, 354. 
 Reversible reactions, 151, 156 222. 
 Rotatory power, 66, 229. 
 
 magnetic, 69. 
 
 SALT solutions, solubility of gases in, 
 
 84. 
 Semi-permeable membranes, 81, 97, 
 
 105, 107, 130. 
 
 Silicic acid, colloidal, 315, 323. 
 Sodium sulphate, solubility of, 93. 
 Sol (hydrosol), 323. 
 Solubilities, determination of small, 
 
 300-302, 370. 
 Solubility, coefficient of, 83. 
 
 curves, 86, 93, 196. 
 
 effect of temp?r.<ture on, 93, 96. 
 
 of gases in liquids, 82. 
 
 of liquids in liquids, 84. 
 
 of solids in liquids, 91-94. 
 
 product, 298. 
 Solution, heat of, 146, 153. 
 Solutions, boiling-point of, 114. 
 
 colloidal, 127-129. 
 
 freezing- point of, 119. 
 
 isotonic, 105. 
 
 solid, 94, 192. 
 
 supersaturated, 92. 
 
 Solvent, influence of, on ionisation, 
 
 338, 342-345- 
 Solvents, associating, 123, 338. 
 
 dissociating, 123, 338. 
 
 Specific heat of gases, 43-47- 
 
 solids, 11-13. 
 
 Spectrum, absorption, 70. 
 
 effect of dilution on, 71. 
 
 Strong electrolytes, ionisation of, 279. 
 
 Submicrons, 318. 
 
 Substitution, effect of, on ionisation, 
 
 273. 276, 34- 
 
 Sulphur, equilibrium between phases, 
 183. 
 
 vapour density of, 41. 
 Supersaturated solutions, 92. 
 Supertension at electrodes, 401. 
 Surface tension and adsorption, 330. 
 
 and molecular weight of, 
 
 liquids, 125-127. 
 
 nature of, 129. 
 
 Suspensions, 322. 
 Suspensoids, 322. 
 
 TELLURIUM, atomic weight of, 24. 
 
 position in periodic table, 23. 
 Temperature coefficient of chemical 
 
 reactions, 225-229. 
 
 of conductivity, 263. 
 
 Theory, definition of term, 7. 
 Thermoneutrality, law of, 148, 279. 
 Transition curves, 184. 
 
 points, 183, 197. 
 
 determination of, 198, 199. 
 
 Transport numbers, 246-249. 
 Trimolecular reactions, 209. 
 Triple point, 181. 
 Tyndall phenomenon, 317. 
 
 ULTRAFILTER, 324. 
 Ultramicroscope, 317. 
 Unimolecular reactions, 202, 229. 
 
 VALENCY, 15. 
 
 Van der Waal's equation, 33-35, 53-57- 
 
 Van't Hoff-Raoult formula, 112. 
 
 Hoffs factor, i, 125, 262. 
 theory of solutions, 101. 
 
 Vapour densities, 36-41. 
 at high temperatures, 40, 41. 
 
 pressure of binary mixtures, 87, 
 
 91. 
 
 of solids, 173. 
 
 lowering of, in. 
 
 measurement, 113. 
 
 Velocity of reaction, 200-233. 
 Victor Meyer's method of determining 
 vapour densities, 37. 
 
432 OUTLINES OF PHYSICAL CHEMISTRY 
 
 Viscosity and electrical conductivity, 
 
 253> 261, 347- 
 
 of liquids, 73-77, 322. 
 
 measurement of, 75. 
 
 absolute values of, 76. 
 
 of binary mixtures of, liquids, 76, 
 
 342. 
 
 WATER, catalytic action of, 219, 220, 
 
 2 33- 
 
 decomposition by, 285-293. 
 
 dissociation constant of, 283, 293, 
 
 387- 
 
 electrolysis of, 401-404. 
 
 equilibrium between phases, 179. 
 
 ionisation of, 283, 293. 
 
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