THE LIBRARY OF THE UNIVERSITY OF CALIFORNIA LOS ANGELES u The D. Van No&rand Company intend this book to be sold to the Public at the advertised price, and supply it to the Trade on terms which will not allow of reduction. A TEXT-BOOK OF THERMODYNAMICS A TEXT-BOOK OF THERMODYNAMICS (WITH SPECIAL REFERENCE TO CHEMISTRY) JAMES RIDDICK PARTINGTON M.Sc. With 91 Diagrams NEW YORK D. VAN NOSTRAND CO. TWENTY-FIVE PARK PLACE 1913 Ml-' PREFACE IN the following pages I have endeavoured to deduce the prin- ciples of Thermodynamics in the simplest possible manner from the two fundamental laws, and to illustrate their applicability by means of a selection of examples. In making the latter, I have had in view more especially the requirements of students of Physical Chemistry, to whom the work is addressed. For this reason chemical problems receive the main consideration, and other branches are either treated briefly, or (as in the case of the technical application to steam and internal combustion engines, the theories of radiation, elasticity, etc.) are not included at all. The arrangement of material adopted was the result of careful consideration, and it is hoped that the particular utility of the various methods in the treatment of special problems has been made apparent. It is of course unlikely that this arrangement will meet with agreement in all quarters, but it seemed to me to offer advantages over a strictly uniform treatment in that the reader will thereby see more clearly the connection between the various methods, and hence his subsequent study of these in the original literature will be rendered easier. On the one hand, the student has been informed by some writers that the only certain way lies in the use of the entropy-function and the thermodynamic potentials ; on the other hand, he is told with equal authority that the method used by the original investi- gators has been the consideration of cyclic processes, and that the former method is nothing but a mathematical (perhaps unnecessary) refinement of the results obtained by the latter. These extreme attitudes appear to me to be unfortunate, and more especially when one observes the physical clearness intro- duced by the use of cyclic processes, but at the same time remembers that most of the results obtained by separate investi- gators using cyclic processes had, with a great many more, previously been found by J. Willard Gibbs by means of a purely analytical method. The mathematical knowledge pre-supposed is limited to the elements of the differential and integral calculus ; for the use of those readers who possess my Higher Mathematics for vi PREFACE Chemical Student* (Methuen, 1911), references to the latter under the symbol " H. M." have been made whenever it appeared desirable. In spite of (or perhaps on account of) recent attempts to prove the contrary, I am of the opinion that no satisfactory progress can be made even in the elementary parts of thermo- dynamics without a good working knowledge of the calculus. In considering the older literature of the subject, I have not thought it necessary to give detailed references, since these will be found in any of the standard text-books of physics. The space thus made available has been utilised in an attempt to explain in greaterdetail some points which usually offer difficulties to the student. The same applies to the numerical constants, which are to be found in the tables of Landolt-Bdrnstein, and at present are also being actively revised. The modern developments of the science are considered in the last two chapters. The last chapter can be regarded only as a fragmentary sketch of the subject, but a more complete treatment, in view of the limitations of space and the very rapid way in which the theories are undergoing alteration, appeared undesirable. In the preparation of the book, most of the existing treatises have been consulted, but in the majority of cases I have preferred to use the original literature of the subject. All the diagrams were drawn specially for the book ; in this connection my best thanks are due to Mr. R. T. Hardman, M.Sc. for valuable assistance. If the present volume will help towards the comprehension of the fundamental principles on which the science of thermo- dynamics rests, and also serve to bring home the importance of a knowledge of these principles in the suggestion and inter- pretation of experimental work, the purpose which has been kept in view during its preparation will have been amply fulfilled. In any case, it is hoped that neither the extreme view that thermodynamic principles alone suffice in the construction of a systematic physical or chemical science, nor the equally mistaken opinion that they are of little practical utility to the experimental worker, can fairly result from its study. J. R, PARTINGTON. MANCHESTER, AND RERUN, 1913. TABLE OF CONTENTS CHAP. PA3E I. THERMOMETRY AND CALORIMETRY . . 1 20 II. THE FIRST LAW OF THERMODYNAMICS AND SOME APPLICATIONS .... '21 50 III. THE SECOND LAW OF THERMODYNAMICS ; ENTROPY ...... 51 89 IV. THE THERMODYNAMIC FUNCTIONS AND EQUILIBRIUM. ..... 90 116 V. FLUIDS 117 130 VI. IDEAL AND PERMANENT GASES . . . 131 108 VII. CHANGES OF PHYSICAL STATE . . . 169 2*20 VIII. VAN DER WAALS' EQUATION AND THE THEORY OF CONTINUITY OF STATES . . . 221 252 IX. THERMOCHEMISTRY . . . . . '253 261 X. GAS MIXTURES ...... 262 278 XI. THE ELEMENTARY THEORY OF DILUTE SOLU- TIONS 279321 XII. CHEMICAL EQUILIBRIUM IN GASEOUS SYSTEMS 322357 XIII. EQUILIBRIUM IN DILUTE SOLUTIONS . . 358 379 XIV. GENERAL THEORY OF MIXTURES AND SOLU- TIONS 380 428 XV. CAPILLARITY AND ADSORPTION . . . 429 449 XVI. ELECTROCHEMISTRY ..... 450 482 XVH. THE THEOREM OF NERNST .... 483 512 XVffl. KINETIC THEORIES IN THERMODYNAMICS . 513 537 INDEX 539544 CORRIGENDA AND ADDENDA Page 14, line 2 : The method of Xernst, Koref, and Lindemann, by the use of the copper-calorimeter, determines the mean specific heat over a range' of temperature. The mode of procedure is the same as in ordinary calorimetry, except that a hollow block of copper, the temperature of which is determined by means of inserted thermoelements, is used instead of a calorimetric liquid, and the method therefore made applicable to very low temperatures. "ST" "8T" Paye 66, line 2, for -- rew.l jp - Page 225, line 2 from bottom, for " TI " read " T-." Page 502, table of chemical constants. In the most recent publication (Theoretische Chemie, 7th edit., 1913) Xernst has changed the values slightly in some cases : CO 3-5 H 2 O 3-6 C1 2 3-1 I 2 3-9 The following additional values may be mentioned : HI 3-4 SO. 3-3 NO 3-5 CS, 3-1 X 2 O3-3 CeHe 3-0 Paye 535, line 1 : In Bjerrum's calculation the value : r = e/X where c = velocity of light, X = wave-length of the absorption band, is substituted for in the Xernst-Lindeniann equation. If we put : where /3c = 14,580, the mean molecular heats of hydrogen, oxygen, and steam may be expressed by the formulae : H 2 : C, = E [2-5 + (2-0)] 2 : C, = R [2-5 + (2-4)] H 2 : C r = B [ 3 + < (3-6) + 2 (1-3) + (3^5)'] THERMODYNAMICS CHAPTER I THERMOMETRY AND CALORIMETRY 1. Temperature. The volume of a given quantity of air, or other gas, contained in a flask connected with a manometer, varies from day to day, although the barometric pressures corresponding w ith two unequal volumes may be equal. Closer investigation discloses the dependence of the volume of gas on the degree of " hotness " or " coldness " of the atmosphere, the volume being greater on hot days than on cold days. The names "hot" and "cold" are m general use to denote specific sense-impressions; the physical property of the object investigated, on which these sense- impressions depend, is called its temperature. Definition. -We shall, provisionally, employ the word 'temperature" to denote the physical state of hotness or cold- ness of bodies ; hot bodies are said to have a (relatively) hi^h temperature, cold bodies a (relatively) low temperature. ' 2. Measurement of Temperature. The measurement of temperature is achieved by making use of two experimental facts : (1.) Certain states of definite substances are marked off as existing between fixed limits of temperature, and the transition from one state to another occurs, other things being equal, at a definite constant temperature. A constant temperature may be recognised by unchanging volume in the apparatus of 1, a rising or falling of tempera- ture by expansion or contraction. Used in this way the apparatus may be called a tiiermoscope. The temperature of a mass of ice rises on warming until it 2 THERMODYNAMICS reaches a point at which the ice begins to melt. The tempera- ture then remains constant until the whole of the ice has melted into water ; the latter then rises in temperature until it begins to boil, when its temperature again remains constant until the last drop has evaporated. The temperature of the steam then rises progressively. The pressure is supposed constant through- out. The constant temperatures characterising the transitions : Ice > Water Water > Steam provide fixed points of temperature which may, other things being equal, be used as points of reference. We refer to them as the ice-point and steam-point, respectively, when the transitions occur under normal atmospheric pressure (cf. 21). (ii.) Certain properties of definite substances change in a con- tinuous manner with the temperature, so that, corresponding with each value of the temperature, 0, we have (it may be within certain limits) a definite value of the property, x. If other conditions are constant, a; is a continuous and single-valued function of 6 : *=/(*) (1) Further, if the system, after undergoing such a change of pro- perties, is brought back to the initial temperature, the original value of the property is then often regained. Systems showing the property of reversion so defined, i.e., which exhibit no hysteresis of properties with respect to temperature variation, may be used to set up scales of temperature for use in measure- ment, and so to define temperatures intermediate between the fixed points. The interval is divided into an arbitrary number of " degrees," each corresponding to an equal increment of the property, and hence by definition, to an equal increment of temperature. If the change of volume of air between the ice-point and steam-point is divided into 100 equal parts, each of these is called a Centigrade degree. As examples of properties of systems satisfying the conditions of definiteness at a particular temperature and of reversion, we may refer to the electrical resistance of a metal wire ; the electro- motive force of a thermocouple with a fixed temperature at the cold junction ; the volume of a homogeneous gaseous, liquid, or THERMOMETRY AND CALORIMETRY 3 solid substance ; the length of a rod of a solid substance ; the vapour pressure of a pure liquid such as water ; the dissociation pressure of certain systems such as nitrogen tetroxide, iodine, or ammonium chloride (homogeneous), or a salt-hydrate, a carbonate or hydride, etc. (heterogeneous) ; and finally the intensity of thermal radiation from a black body, or hollow enclosure. 3. Thermometers. Any instrument which can be used for measuring temperatures is called a thermometer. Thermometers may be, and are, con- structed which utilise any property of a body such as those mentioned above. To evade the difficulty of comparison of scales, they are usually all referred to a gas thermometer, with Centigrade scale as standard. The ice and steam-points on the latter are taken as and 100 respectively. There are in consequence various kinds of thermometers, the detailed description of which belongs to books on experimental physics. We may mention the following : (1) Thermometers depending on volume (or pressure) changes : (i.) Gas thermometers, containing air, hydrogen, or helium. On account of the low condensing temperatures of hydrogen and helium, thermometers containing them are particularly useful at very low temperatures ; the regu- larity of expansion of gases makes them very suitable for thermonietric purposes. (ii.) Mercury or alcohol thermometers, used at moderate tempera- tures. (2) Electrical resistance thermometers, the most widely used of which is Callendars platinum resistance thermometer. This is probably the most convenient and accurate apparatus for measur- ing temperatures between the boiling-point of liquid air ( 190 C.) and the melting-point of platinum (1,500 C.). Lead has recently been applied at very low temperatures. (3) Thermocouples of platinum with an alloy of platinum and 10 per cent, of rhodium or iridium are used at higher tempera- tures, and of copper and constantan at lower temperatures. (4) Optical pyrometerx, depending on radiation phenomena, may be used up to the highest attainable temperatures (2,500 3,000 C.). B 2 4 THERMODYNAMICS 4. Fundamental Law of Temperature-Reciprocity. If a hot body is placed in contact with a cold body, the tem- perature of the former falls, that of the latter rises, until both have attained the same temperature. When the system reaches a steady state, the temperature is the same in all its parts. The change of temperature finds its physical interpretation in the transfer of something called heat from the body of high to the body of low temperature, in a similar manner to the passage of air from a vessel containing it under high pressure to a communi- cating vessel containing it under low pressure. Tempera- ture may therefore be regarded as something analogous to pressure in a fluid, and heat to the fluid itself. This analogy, in fact, constituted an earlier theory respecting the nature of heat, in which it was regarded as an " imponderable (i.e., weight- less) fluid " and called caloric. The fact that two bodies A and C are in thermal equilibrium when they have the same tempera- ture may appear to be equivalent to a definition of temperature. If, however, the temperatures of A and C have been found to be equal to that of a third body, B, used as a thermometer, they are found to have the same temperature when tested by the above method, i.e., if A and C are put into direct contact, there is, in the absence of chemical action, no disturbance of their thermal states. Thus, if two bodies have equal temperatures when com- pared separately with a third body, they are found to have equal temperatures when compared directly with each other. This is by no means a self-evident truth ; it is a physical law based on experience, and depends on the property of temperature equilibrium. 5. Unit of Heat. For the purpose of measuring quantities of heat we require a unit of heat, and the preceding considerations would lead us to define a quantity of heat in terms of the change of temperature which it produces in a given mass of a particular substance under specified conditions. Definition. The unit quantity of heat is that quantity of heat which raises the temperature of 1 gram of water from 15 C. to 16i C. This is called a calorie. Various similar units have been proposed and used : THERMOMETRY AND CALORLMETRY Unit. Mass of Water. Rise of Temperature. () 15 Calorie . 1 gram 16| o C . > , w , C. (6) Zero Calorie . 1 gram o c. > P C. (c) Mean Calorie . 0-01 gram C. 100 3 C. (rf) Ostwald's Calorie K. (e) 4 Calorie ( f) 20= Calorie . 1 gram 1 gram 1 gram o c. > IOO D 3J C. > 4^ : 19A C. > 20| C. C. C. The zero calorie is 0'06 per cent, larger than the 15 calorie, whilst the mean calorie is 0'03 per cent. (Behn) 0"2 per cent. (Dieterici) larger than the 15 calorie. All temperature's are supposed to be measured on the constant pressure hydrogen thermometer. The large number of heat units employed by various experi- menters has given rise to a corresponding amount of confusion in the specification of experimental results, and the name of the unit should now always be given. It is not essential, however, that the unit of heat should be defined in terms of the rise of temperature produced when heat is absorbed by a standard body, say unit mass of water. Any effect of heat absorption which is capable of measurement and numerical expression might be used, and the method of measurement would in all cases be consistent with the axiom that if two identical systems are acted upon by heat in the same way so as to produce two other identical systems, the quantities of heat supplied to the systems are equal. Lavoisier and Laplace (1780-84) took as unit that quantity of heat which must be absorbed by unit mass of ice in order to convert it completely into water. This unit is of course different from the one we adopted, but if a quantity of heat A has been found to raise from 15^ to 16^ twice as much water as another quantity of heat B, then A will also melt twice as much ice as B. 6. Heat Capacity and Specific Heat. If equal masses of lead, iron, mercury, and glass, all having a temperature of 0C., are dropped into vessels containing, say, 100 grams of water at 50 C., it will be found that the temperature falls in each case, but when the temperatures have again become steady they are all different. In accordance with our definition, we say that in each case a certain number of units of heat has passed from the water to the body, but the number so passing before the temperature of the body is equal to that of the (some- what cooled) water .is different for the different substances. We 6 THERMODYNAMICS may regard each mass as having a definite capacity for absorbing heat. The fact that the effect of a given quantity of heat in raising the temperature of a body depends not only on the mass but also on the composition of the body was clearly grasped by Joseph Black about 1768 ; this property of bodies was named Capacity for Heat by his pupil Irvine. Definition. The heat capacity of a body, under specified condi- tions, is measured by the number of heat units which must pass into that body to raise its temperature 1 C. It has been shown experimentally that if a body or system in a given state A is converted into another B, by the absorption of a definite quantity of heat, the same amount of heat is given out again if B is reconverted into A, so that all the intermediate states between A and B are retraced in the same order as they occurred in the first operation. In this connexion the fluid analogy may be useful in giving precision to our ideas. It is evident that very different volumes of water must be poured into cylinders of different diameter in order that the level of water may be the same in all. The rise of level is proportional to the volume of water poured in, and inversely proportional to the area of cross-section of the cylinder. If quantities of water are poured into the cylinders so as to produce unit rise of level in all, the volumes will be proportional to the cross-sections, and the volumes of water may be used as measures of the latter, which obviously correspond with the "capacities" of the jars for holding liquid. In the same way we have defined the capacity of a body for absorbing heat as measured by the number of heat units which must be put into the body in order to raise its temperature, or heat level, by one unit. We must remember, however, that heat capacity is not heat, any more than bulk capacity is water used in measuring it. Although the heat capacity of a body is measured in terms of the number of heat units required to produce a standard change of one property of the body, it denotes a specific property of the body itself. The use of the fluid analogy obviously does not affect the truth or otherwise of any equations derived from the definition of heat capacity, because the latter involves only measure- ments of temperatures and masses, magnitudes directly ascertainable by experiment and independent of any theoretical views on the nature of heat. If Q units of heat are required to raise the temperature of a body 1, then 2Q will be required to raise the temperature of two such bodies through 1, and so on. Hence the heat capacity of a homogeneous body is proportional to its mass. The heat capacity of unit mass of a homogeneous body may therefore be THERMOMETRY AND CALORIMETRY 7 regarded as measuring some specific property of the substance of which the body is composed. Definition. The heat capacity of unit mass of a substance is called its specific heat. The name " specific heat " was introduced by Gadolin in 1784. As unit mass we shall take 1 gram, and unit rise of temperature 1 C. on the hydrogen gas thermometer. The number of heat units which must be imparted to a mass m of a substance to raise its temperature through 1 under specified conditions will be : Q = me (1) where c is the heat capacity of unit mass, or the specific heat, of the substance, under the given conditions. The specific heat of a substance must always be defined relatively to a particular set of conditions under which heat is imparted, and it is here that the fluid analogy is very liable to lead to error. The number of heat units required to produce unit rise of temperature in a body depends in fact on the manner in which the heat is communicated. In particular, it is dif- ferent according as the volume or the pressure is kept constant during the rise of temperature, and we have to distinguish between specific heats (and also heat capacities) at constant volume and those at constant pressure, as well as other kinds to be considered later. If the temperature of a mass m of a substance rises from 61 to 02, we may represent the amount of heat absorbed by : Q = me fa - 00 . . . . (2) if we understand by c the average heat capacity of unit mass in the range of temperature considered. It must, however, be carefully noted that the amounts of heat required to raise the temperature of unit mass of a substance from to 1, from 1 to 2, and so on, are not usually equal, because after the temperature has been altered from to 1 we are not dealing with the same substance as we started with, but with a warmer substance, and it is quite possible that the specific heat of the latter is different from that of the initial substance. In many cases, however, c is nearly independent of temperature over a range of a few degrees, so that we may assume that the same number of calories will be required for the following changes of temperature : 0, to (0x + 1), (*i + 1) to (0! + 2), . . . (0 2 - 1) to 0. 2 , provided (0 2 0i) is not large. The admissible range of tempera- 8 THERMODYNAMICS ture varies very much with the nature of the substance and with the initial temperature. The variation of specific heat with temperature was discovered by Dulong and Petit in 1819. It explains why so many different heat units exist (cf. 5), and requires the definition of specific heat to be so framed as to allow for this variation. For this purpose we replace the finite changes by infinitesimal ones. If SQ units of heat are absorbed when unit mass of a substance is raised in temperature from (6 | 80) to (6-\-\ 80) under specified conditions, the true specific heat at the temperature 6 is : It must be observed that the quotient has a definite limiting value only when the conditions of heating (e.g., constant pressure) are specified. The heat Q absorbed in a finite rise of temperature from 0i to %d0 = mdo . . (4) The form of /(0) can only be determined by experiment, or by means of atomistic hypotheses (cf. Chap. XVIII.). Since c 6 is known to change continuously and only comparatively slowly with rise of temperature, we may expand /(0) in a Maclaurin's series (cf. H. M., 95) : or c e =c + a<9 + Z>0 2 + . . . . (5) where c =/(0) = specific heat for 6 = 0. Thus, if we take only terms as far as 2 into account (which is usually abundantly sufficient) : . (6) . (6a) THERMOMETRY AND CALORIMETRY If we compare equations (2) and (4) we find (m = 1) : . , . , . (7) so that the mean specific heat in the interval (0 2 #1) is the mean value of the true specific heat in the same interval (cf. H. M., H7). Corollaries. (1) To find the true specific heat when the mean specific heat is given, multiply the latter by and differentiate with respect to 0. (2) If c = Co + a0 + b0 2 then c = c v + | (6 l + 6> 2 ) + | ( and (ii.) The specific heat at constant volume, denoted by c v _ The first accurate measurements of the specific heats of gases at constant pressure were made by Eegnault (1862), who used the "constant flow" method. A slow stream, of gas from a large reservoir was allowed to pass through a pressure regulator, and then through a copper spiral immersed in an oil bath of known temperature. The hot gas then passed through 10 THERMODYNAMICS a series of thin metal boxes immersed in a water calorimeter, to which it gave up heat. E. Wiedemann (1876) replaced the heating coil and metal boxes by metal tubes filled with metal turnings, thus exposing a larger surface to the gas. Holborn and Austin (1905) have recently extended the method so as to make it applicable at high temperatures. The gas was sent through a heater consisting of a narrow, electrically heated, nickel tube containing nickel turnings, and then passed into a silver calorimeter containing water. Initial temperatures up to 800 were used. The method was also used by Holborn and Henning (1905) to determine the specific heat of steam at high temperatures. The measurement of the specific heat at constant volume is attended with considerable difficulty, because the thermal capacity of a vessel strong enough to contain the gas after heating has a value much greater than that of the thermal capacity of the enclosed gas. Mallard and Le Chatelier have measured the specific heats at constant volume at high temperatures by their so-called explosion method. In this an explosive mixture (e.g., 2CO + 2 ) is fired in a very strong metal chamber, and the pressure measured by an indicator similar to those used in recording the pressures attained in engine cylinders. By assuming that the gases conformed to the laws of Boyle and Charles, the mean specific heats between the ordinary temperature and the explosion temperature could be calculated. Berthelot and Vieille (1882-5), instead of measuring the static pressure as above, allowed the force of the explosion to deform a copper cylinder, placed beneath a steel piston exposed to the gas (" crusher manometer "). Langen (1903) and M. Pier (1908) have made the most recent determinations with the explosion method. Sarrau and Vieille (1882-7) used the crusher manometer, but instead of firing an explosive gas mixture, they ignited a charge of high explosive hanging freely in the simple gas. Measurements of c v at moderate temperatures were made by Joly in a steam calorimeter. The weight of water condensed from steam blown through a chamber containing a copper globe filled with gas was compared with the weight deposited on a similar but vacuous globe. Eegnault and Wiedemann, cf. Haber's Thermodynamics of Technical Gas Reactions, Eng. trans., lect. 6. Holborn and Austin, Sitzungsber. Konigl. preuss. Akad. (1905), p. 175. Holborn and Henning, Ann. der Phys., [4], 18, 739 (1905). Mallard and Le Chatelier, Journ. de Phys., [2], 1, 173 (1882) ; Ann. des Mines, [8], 4, 274 (1883). Berthelot and Vieille, C. E., 95, 1280 (1882); 96, 116, 1218, 1358 (1883); Ann. chim. phys., 4, 13 (1885). Langen, Mitteilungen uber Forschungsarbeiten aus dem Geibiete des Ingenieurwesens, Berlin, 1903, Heft 8. M. Pier, Zeitschr. physical. Chem., 62, 397 (1908) ; 66, 759 (1909) ; Zeitschr. Ekklrochem., 15, 536 (1909). N. Bjerrum, Zeitschr. Elektrochem., 17 (1911) ; Zeitschr. phi/sikal. Chem., 79, 513, 537 (1912). THERMOMETRY AND CALORIMETRY 11 Sarrau and Vieille, C. R., 95, 26, 133, 181 (1882); 102, 1054 (1886); 104, 1759 (1887). Cf. Heise, Sprengstojfe und Zundung der Sprengschiisse, Berlin, 1904. Joly. Proc. Boy. Soc., 55, 390 (1894); Phil. Trans., 182, 73 (1892). Measurements of the effect of pressure on c p have been made by Lussana (1895 8). With pressures of 5 150 atm. he found c p to increase consider- ably with the pressure with all the gases investigated. In the case of air and carbon dioxide the results showed that c p would reach a maximum and then decrease at higher pressures a result directly verified in the case of air by Witkowski (1895). Lussana's results are represented by: H 2 = ( c p )iatm. + ( c p )iatm. 3-4025 (P- !) 0-013300 OH 4 C0 2 0-5915 0-2013 0-003463 0-0019199 C 2 H, , 0-40387 0-0016022 N 2 0-22480 0-0018364 Air : c p = 0-23702 + 0-0015504 (p 1) 0-00000195 (p I) 2 . The more closely a gas approaches its point of liquefaction the greater is the influence of pressure on its specific heat. The effect of pressure on the specific heat of steam has been examined by Thiesen and by Lorenz. The molecular heat has a minimum, C p = 7-34, at 80 C. With diminishing pressure, steam behaves more and more like a permanent gas, the molecular heat tending to a limiting value C p = 7 -74. Lussana, Nouvo dm., [3], 36, 5, 70, 130 (1894); Fortschr. der Phijs. (1896), 345; (1897), 331 ; Witkowski, BuU. internat. de I'Acad. Sci. Cracovie (1895), 290; Fortschr. der Phys. (1896), 343; cf. Amagat, C. E., 122, 66, 121 (1896); Thiesen, Ann. der Phijs., [4], 9, 88 (1902); Tumlicz, Wein. Akad. Her. (1897), Ha, 654; (1899), Ha, 1395; Macfarlane Gray, Phil. Mag. (1882), 13,337; Amagat, C. R., 142,1120,1303 (1906); 143, 6 (1906); Journ.de Phys., [3], 9, 417 (1900); [4], 5, 637 (1906); Dalton, Phil. Mag., [6], 13, 536 (1907) ; Nernst, Vtrhl. d. d. Phys. Ges., 11, 320 (1909). According to Haber (loc. (.it., p. 131) the results of Langen are probably correct to 3 per cent, even at 2,000. Pier, in his recent explosion experi- ments, has shown, however, that the maximum pressures were not obtained by the previous observers, on account of the oscillations of their manometers, He used a steel plate with very high frequency of vibration, and registered the distortion by reflecting a beam of light from a mirror attached to the manometer disc on to a revolving drum of sensitised paper. The recorded curves show a well-defined maximum pressure, and his results are probably accurate to 1 per cent. Values of C,, : Argon 2-98 constant to 2,350 C. N 2 4-900 + 0-000450 (1,300 2,500) H 2 4 -'7 00 -4- 0-000456 HC1 4-600 + 0-0050 Steam 6'065 + 0'00050 + 0-2 X 10-0 3 12 THERMODYNAMICS Of. also F. Keutel, Inauy. Dies., Berlin, 1910 ; E. Thibaut, Inaug. Dies., Berlin, 1910; Petrini, Zeitschr. physik. Chem., 16, 97. A very complete account of the data up to 1908 will.be found in the work of Haber cited above. 8. The Specific Heats of Vapours. The results of Regnault and of E. Wiedemann (which are not in good agreement) indicate that : (1) The specific heats of vapours run parallel to those of the liquids, and the temperature coefficients a : are only slightly different in the two cases. c p vapour at, 0. c v liq. at 0. a vap. a liq. CHC1 3 . . 0-1341 0-2323 0-000135 O'OOOlOl C 6 H 6 . . 0-2237 0-3798 0*00102 0*00144 (C 2 H 5 ) 2 . 0-3725 0-5290 0'00085 0-00059 c v X molecular wt. (2) The quotient - = - ; - r is not constant, no. of atoms in molecule varying between 2*0 and 4*5, and depending on the temperature. (3) The specific heats increase with rise of temperature to different extents with different vapours. (Cf . the work of Thibaut quoted in 7.) 9. The Specific Heats of Solids. The variation of specific heat of a solid with temperature was, as has already been stated, discovered by Dulong and Petit as early as 1819, but it is only quite recently that the remarkable relations exhibited at very low temperatures have come to light (cf. Chap. XVIIL). In most cases c e is, at the ordinary temperature, approximately a linear function of 6 : c iron (15 to 320) = 0'10442 (1 -f -001290). The specific heats of solids at low temperatures are appre- ciably less than at higher temperatures. A maximum specific heat has been observed in the case of iron at 740 and nickel at 320 (Lecher, 1908). Since these are the temperatures at which recalescence and loss of magnetic properties occur, the close relation of specific heat to molecular structure is evident. THERMOMETRY AND CALORIMETRY 13 The most marked effect of change of temperature on the specific heat is, however, exhibited by the non-metals, carbon, boron, and silicon. The following values were obtained by H.F.Weber (1875): Diamond ( 50 to + 250) : c $ = '0947 + '0009940 -000000360 3 Graphite (0) : c e = 152 + '00070. Boron (0 to 250): c e = 0"22 + 0'000710. Silicon (cryst.) : 0= - 50 + 50 100 200 c 9 = 0-13 -16 -18 "195 -202 At higher temperatures than those over which the formulae hold good the rate of increase of the specific heat with tempera- ture falls off rapidly, and the specific heat tends to a limiting value which changes only slightly with further increase of temperature. These limiting values are (approximately) : Diamond 0'46 Boron 0'50 Graphite 0'47 Silicon 0'205 At the other temperature extreme we have the measurements of specific heats executed at the temperatures of liquefied gases. A known mass of the substance is dropped into liquid carbon dioxide ( 78), oxygen (- 183), or hydrogen (- 250), and the volume of gas liberated is measured. Dewar, Proc. Roy. Soc., A, 76, 325 (1905), finds the following values of the specific heats : 18 to -78 -78 to -188 - 188 to - 252.i Diamond 0*0794 0'0190 0*0043 Graphite 01341 0'0599 0*0133 Ice 0-463(-18to-78) 0-285 0146 The specific heats of diamond and graphite are reduced to ^ and -j\y respectively between the ordinary temperature and the boiling- point of liquid hydrogen ; the specific heats of the substances between the temperatures of liquid air and liquid hydrogen are in fact less than those of any other substances, even less than that of a gas at constant volume. 14 THERMODYNAMICS The method just described leads to the mean specific heats over a fairly large range. Nernst, Koref, and Lindemann (1910) have recently described a method of measuring the true specific heat at a given low temperature. The substance is contained in a block of copper cooled to the requisite temperature in liquid carbon dioxide, liquid air, etc., and energy is supplied by a heating spiral of platinum wire carrying an electric current, the measurement of the resistance of which serves at the same time to determine the temperature. Measurements of the true specific heats at low temperatures have been carried out by Eucken, who worked in such a way that to a weighed quantity of the substance in the form of a block, or in a proper isolated vessel, a known quantity of heat was added by means of an electrically heated platinum spiral, the resistance of which at the same time served to measure the temperature. The quotient of the electrical energy spent by the rise of temperature gives the specific heat. The correction for cooling (in vacuum of ^Q mm -) amounted to 20 per cent., and for heat capacity of the apparatus 5 per cent., yet the results are stated to be accurate to 1 per cent. For very low temperatures a lead wire is used as thermo- meter, with a heating coil of constantan wire (Kamerlingh Onnes). The method has been applied by Eucken to determine specific heats of gases (e.g., H 2 ) at constant volume by enclosing them in small metallic vessels. The following references include most of the recent experi- mental data : Eucken, Physik. Zeitschr., 10, 586 (1909) ; Sitzungsberichte Kgl. Akad., Berlin (Berl. Ber.) (1912), p. 141 ; Fortschritte dcrClim/., [iv.], 105 (1911). Lindemann, Berl Ber. (1910), 12, 247 ; 13, 316; (1911), 22, 492; Dissertation, Berlin, 1911; Lindemann and Nernst, Zeitxcltr. Elektrochem., 18, 817 (1911). Koref, Berl. Ber. (1910), 12, 253 ; Ann. Plnjs., [iv.], 36, 49 (1911). Nernst, Berl. Ber. (1910), 12, 247, 262 ; 13, 306 ; (1911), 22, 492; Ann. Phys., [iv.], 36, 395 (1911). Magnus, Zeitschr. Elektrochem., 16, 269 (1910) ; Ann. Phys., [iv.], 31, 597 (1910). THERMOMETRY AND CALORIMETRY 15 Pollitzer, Zeitschr. Elektrochem., 17,5 (1911). Gaede, Dissertation, Freiburg, 1902. Schimpff, Zeitschr. physik. Chem., 71, 257 (1910). Wigand, Ann. Plujs., [iv.], 22, 79 (1907). Russell, Physik. Zeitschr., 13, 59 (1912). Kamerlingh Onnes, Commun. Phys. Lab. Leiden, No. 119, 1911. Dependence on Density. If the density of a metal is increased by hammering, its specific heat is slightly decreased. The same change is observed. if the change of density is due to a change of crystalline form, or to change from an amorphous state to a crystalline state, and with different allotropic forms (Wigand, loc. cit). Density. Sp. Ht. Range. i Diamond . 3-518 0-1128 10-7 Carbon Graphite . ( Gas carbon 2-25 1-885 1604 2040 10-8 24 >68 / rhombic 2-06 1728 >54 . , \ inonoclinic . 1-96 1809 >52 bulpnur j amorph. (insol.) 1-89 1902 >53 ,, (sol.) 1-86 2483 >50 Tin ( White { Grey . . 7-30 5-85 0542 0589 >21 According to van't Hoff (1904), the form which is stable at high temperatures has the greatest specific heat ; this rule has some exceptions. (Cf. also Griineisen, Ann. Phys., [iv.], 26, 393 (1908). ) Relation of Specific Heat to Atomic Weight in the Case of Solids. (a) Dulong and Petit in 1819 arrived at the very remarkable experimental law that the product (At. Wt.) X (Sp. Ht.) = ac = atomic heat is approximately constant at the ordinary temperature for solid elements of atomic weight exceeding 30, and is approximately 6'4. The heat capacities of a gram-atom of these various elements are therefore very nearly equal. The order of approximation which obtains may be seen from the following figures (A. Magnus, 1910) : 16 THERMODYNAMICS Element. Range of Temperature. Atomic Hc;it. Lead J 18 >100 C. 16 256 6-409 6-606 ( 16 >100 5-570 Aluminium 16 304 6-097 ( 17 545 6-475 Copper .... 15 >238 15 >338 6-048 6-090 (/3) The researches of F. Neumann (1831), Regnault (1840), and H. Kopp (1864), indicated that solid elements preserve unchanged their atomic heats when they unite to form solid compounds. Thus, the product (molecular weight} X (specific heat) = (molecular heat) is composed additively of the atomic heats : MC = niajCi + 2 2C 2 + n 3 a 3 c s + . . (9) where MC = molecular,weight and specific heat of compound !, a 2 = atomic weights, and d, c z = specific heats, of the elements present in the molecule, the numbers of gram-atoms being n\, 2 , per gram molecule of the compound. The law was stated in this form by J. P. Joule in 1844 ; it is usually referred to as Woestyn's law (1848). It shows that the carriers of heat in a solid compound are not the molecules of the latter, but the atoms of its constituent elements. Joule's law enables one to calculate the molecular heats of compounds from the atomic heats of their elements, and the atomic heats of elements in the solid state when the latter are not readily directly accessible (e.g., solid oxygen, from : c(CaC0 3 ) c(Ca) c(C) = 3c(0), or 100 X 0-203 6'4 1-8 = 3 X 4'0). Specific Heats of Solid Mixtures. The specific heat of a homogeneous solid mixture of solid components is not usually additively composed of the specific heats of the latter. W. Spring (1886) found that the total heat capacity of alloys of lead and tin was always greater than the sum of those of the components, but above the melting-point the two were equal. A. Bogojawlensky and N. Winogradoff (1908) find, however, that the heat capacities of the isomorphous mixtures : ?n-chloro- + ?n-bromo-nitrobenzene a-chloro- + a-bromo-cinnamaldehyde azobenzene + dibenzyl THERMOMETEY AND CALORIMETRY 17 are, both in the liquid and solid state, composed additively of the heat capacities of the constituents. 10. The Specific Heats of Liquids. As no simple relations between the specific heats of liquids have yet been arrived at, we shall merely bring together a few generalisations from the experimental data : (a) The specific heat of the solid modification is usually less than that of the liquid : Solid. Liquid. Gaseous. H 2 -504 1-000 -477 Hg '0319 -0333 (j3) The specific heats of liquids depend on the temperature, usually increasing, but sometimes decreasing, with rise of temperature. (y) v. Reis (1887) found that the differences between the products (sp. ht. between and boiling-pt.) X (mol. ict.} were nearly constant in particular homologous series of organic liquids. R. Schiff (1886) found some simple relations between specific heat and composition in organic liquids. (8) Specific Heats of Mixtures of Liquids. With the exception of the cases of liquid alloys, and some fused isomorphous sub- stances, the heat capacity of a mixture of liquids is usually greater than the sum of the heat capacities of the components (Bussy and Buignet, 1865). This is the case with mixtures of alcohol with H 2 0, CHC1 3 , CS 2 , C 6 H 6 ; the 20 per cent, solution of alcohol in water having a specific heat (1'046) greater than that of any other liquid below 100. On the other hand, mixtures of C 6 H 6 and CHC1 3 have heat capacities the sum of those of the constituents (J. H. Schiiller, 1871). According to Ramsay and Shields, H 2 and C 2 H 5 OH are associated, C 6 H 6 and CHC1 3 are not. Further investigation with other liquids would certainly lead to interesting results, if these were taken in connexion with the heat of admixture, and the other properties of the components and mixture. (e) Specific Heats of Solutions. Dilute aqueous solutions of many salts exhibit a peculiarity which has not yet been satis- factorily explained. Let p gr. salt be dissolved in 100 gr. water ; T. c 18 THEBMODYNAMICS if c is the specific heat of the solution, c(100 + p) is the weight of water which has the same heat capacity as (100 + p) gr. of solution (" water-equivalent " of the solution). Now c(100 -f- p) is frequently less than 100, i.e., the total heat capacity of the solution is less than that of the water alone contained in it. With sodium chloride, for example, the numbers are : p 5 10 20 30 A = 100 c(100 + p) 2-29 2-01 0'36 2'66. Thus, c(100 + p) < 100 at small concentrations, = 100 at medium concentrations, > 100 at high concentrations. At very great dilution A approaches a limiting value (Schiiller, 1869 ; Thomsen, 1870 ; Marignac, 1871-6). 11. Latent Heat. Let a quantity of powdered ice be placed in a vessel along with a thermometer, and the whole placed in hot water. The tempera- ture of the latter falls, showing that heat is being absorbed from it, but the temperatures of the ice, and the water formed by its fusion, remain constant (provided the mass is efficiently stirred) until the last portion of ice has melted. The result of absorption of heat by ice is therefore the production of water without any change of temperature. In the same way, the application of heat to water at its boiling-point gives rise to the production of steam at the same temperature. There are many other examples of changes in which a solid passes into a liquid, or a liquid into a gas, with absolution of heat at constant temperature. The constant temperature may be called the transition temperature ; the heat absorbed is called the latent lieat of the transition. The latter name is due to Joseph Black, the discoverer of the phenomenon (1757); he appears to have regarded the heat as existing " latent " in the body in some sort of chemical combination, just as "fixed air" exists latent in chalk. In both cases the entity has lost its properties by chemical combination, but may be set free again in a suitable way. There is absorption of latent heat not only in " physical " changes of state (fusion, evaporation), but also in many chemical reactions which occur at a transition temperature. In all cases the transition temperature is more or less dependent upon the THERMOMETRY AND CALORIMETRY 19 pressure, and the latent heat depends on the temperature of transition. Definition. The heat absorbed in producing a change of physical state or chemical composition of a system, at constant temperature and pressure, is called the latent heat of the given transition, and is measured by the number of calories absorbed during the transition of unit mass of the substance from the initial to the final state. If referred to one gram as unit mass it will be denoted by L ; if to a molecular weight in grams, by A = ML, where M = molecular weight. According to the nature of the transition, we have different latent heats: (1) Latent Heats of Fusion, representing the heat absorbed in the transition [Solid] > [Liquid] at a standard pressure, usually one atmosphere. (2) Latent Heats of Evaporation, absorbed in the transition [Liquid] [Vapour], usually measured at the boiling-point under atmospheric pressure. (3) Latent Heats of Sublimation, referring to the change [Solid] > [Vapour] ; e.g., camphor, iodine, etc., may pass directly into vapour on heating. (4) Latent Heats of Dissociation, absorbed in such changes as CaC0 3 CaO + C0 2 (Horstmann, 1869). (5) Latent Heats of Allot ropic Change, such as S a >S^at95 c -6. (6) Latent Heats of Transition (in the narrower sense), in which a system of substances A passes over into another system B at a definite temperature and pressure with absorption of heat: Na^SOi-lOHaO > Xa 2 S0 4 +10H 2 at 35. Na 2 S0 4 .10H 2 0+ MgS0 4 .7H 2 > NaMg(S0 4 )2.4H 2 + 13H 2 0. CuCl 2 .2KC1.2H 2 CuCl 2 .KCl+KCl+2H 2 0. In all the examples (1 6) we have two (or more) substances c 2 20 THEKMODYNAMICS concerned which can be separated from each other mechanically, e.g., ice and water ; CaC0 3 , CaO, and C0 2 . Such homogeneous bodies constituting a heterogeneous complex are called phases (Gibbs, 1876). The transition from one phase to another is called phase- transition,- it takes place at a definite temperature (under a specified pressure) with the absorption of an amount of heat A for each gram-molecule, or mol, passing over. When the various phases of a system can exist side by side for an indefinite period they >re said to be in equilibrium. CHAPTER II THE FIRST LAW OF THERMODYNAMICS AND SOME APPLICATIONS 12. Force and Work. The science of Mechanics is concerned with the strict defini- tion of force and the consequences of this definition. In daily life we give the name " force " to the muscular effort called forth in supporting a weight, or in maintaining a spring or elastic cord in a condition of strain. Each of these systems exerts a force in opposition to the muscular effort, and the pair of forces is said to constitute an action and reaction or a stress. The legitimacy of this nomenclature is assured by the fact that the strained spring is able to support the weight. From this point of view the weight of a body is a force. If we regard a given extension of a spiral spring as brought about by a definite force, it is found that the weight of a "given body is not an absolute constant, but varies slightly with the latitude and the altitude above sea-level. For the purpose of comparison, all weights are reduced to latitude 45 and sea-level. It is also found that the changes of weight are in the same ratio for the unit of weight and for all other bodies, so that if the weights of two bodies are equal in one position they remain equal in all olher positions. An unsupported body at once falls towards the earth, and its velocity increases continuously as it descends. The increase of velocity in unit time was found by Galilei (1638) to be constant ; it is called the acceleration of gravity ( (2) Later Work. (a) The water-stirring experiments have been repeated, and the results show that Joule's number is certainly about 0'3 per cent, too low : H. Rowland (1879-80) .... 4*188 X 10 7 C. Miculescu (1892) .... 4181 X 10 7 0. Reynolds and W. H. Moorby (1898) . 4'1845 x 10 7 (b) The electrical-heating method is apparently the most accurate, and has been favoured by later workers : E. H. Griffiths (1893) .... 4'1856 X 10 7 A. Schuster and W. Gannon (1895) . 41859 X 10 7 H. Callendar and H. T. Barnes (1902) . 4 '1851 X 10 7 C. Dieterici (1905) .... 41938 X 10 7 Dieterici's number is much higher than the others, which are in good agreement; this experimenter used the Bunsen ice calorimeter, which is a very uncertain instrument. His results are, however, included in the mean adopted below. Rowland first observed that the value of J varies with the temperature, and if the mechanical equivalent of the 15 calorie be taken as standard, the ratio of the value of J at any tempera- ture to this gives the specific heat of water at that temperature. He found the curious result that the specific heat of water had a maximum value at about 30 ; Callendar and Barnes located this at 37 G '5. 30 THERMODYNAMICS We shall take : J = 4*188 (+ -002) X 10 7 ergs per 15 cal. which is the value computed by Luther and Scheel (Zeitschr. Elektrochem., 1903, 9, 686). On account of the inclusion of Dieterici's value, this number is probably slightly too great. Table of the amounts of heat, required to warm 1 gram of water from (9 |) to (0 + ) on the hydrogen scale. e 15 cal. Erg x 10 7 . 1-010 4-229 5 1-004 4-205 10 1-002 4-196 15 i-ooo 4-188 20 0-999 4-184 25 0-998 4-180 30 0-997 4-176 35 0-997 4-175 40 0-997 4-175 45 0-997 4-175 50 0-998 4-180 55 0-999 4-182 60 0-999 4-184 65 0-999 4-186 70 1-000 4-188 75 1-001 4-190 80 1-001 4-192 85 1-002 4-196 90 1-003 4-200 95 1-004 4-203 100 1-004 4-205 1 mean calorie = 1-0002 15 cal. 4-1886 erg. X 10 7 The entire agreement between the values of the mechanical equivalent of heat obtained by many different methods establishes the proposition that it is independent of the process in which the conversion of work into heat occurs, and depends solely on the choice of the units of these two magnitudes. This result was first established by Joule. THE FIRST LAW OF THERMODYNAMICS 31 It has recently been proposed to specify quantities of heat directly in work units in terms of the joule (j.) = 10 7 erg, and the kilojoule (kj.) = 10 10 erg (Ostwald ; T. W. Eichards). Whilst this has an advantage from the point of view of uniformity and simplicity, it suffers from the disadvantage that the whole mass of experimental data must be recalculated every time a more exact value of J is determined. With the present value 1 cal. = 4-188 j., 1 kj. = 238'S cal. 16. The First Law of Thermodynamics. The following statement is a consequence of the experimental results of Joule, that is, of the existence of a unique mechanical equivalent of heat, and is known as the First Law of Thermo- dynamics : When equal quantities of work are produced by any means from purely thermal sources, or spent in purely thermal effects, equal quantities of heat are put out of existence, or are generated. If Q units of heat appear (or disappear) in any process, and A units of w r ork disappear (or appear) simultaneously, then, provided no other forms of energy appear or disappear, i.e., if the work is produced or spent in " purely thermal " processes, the first law of thermodynamics asserts that : A = JQ . . . . (1), where J is a constant depending only on the choice of the units in which A and Q are expressed. It is particularly to be emphasised that equation (1) holds good only when the conversion of heat into work, or vice versa, occurs directly, i.e., when no other forms of energy are appearing or disappearing at the same time. No conclusion as to the ratio of transformation of two forms of energy can be drawn unless the transformation occurs directly ; a process of the latter type we shall designate an aschistic process, because the energy change occurs along an unbranched path. 17. Cyclic Processes. If, after any process, or series of processes, a system returns to its initial state, it is said to have undergone a cycle of changes, and the process is called a cyclic process (S. Carnot, 1824). Thus, if a mass of air in a cylinder is compressed or expanded, with or without simultaneous heating or cooling, in any way 32 THERMODYNAMICS whatever, but is finally restored to its initial volume and temperature, a cycle of changes, or a cyclic process, has been executed. In any cyclic process, let a quantity of heat Q be absorbed, and a quantity of work A be done, by the system. Heat emitted, or work done on the system, is to be reckoned with a negative sign. Then the heat absorbed is equivalent to the work done, in the sense explained : A = JQ, i.e., such a cyclic process is an aschistic process. If this were not the case, it would be possible to produce or destroy unlimited quantities of heat or work without giving rise to any other changes whatever, which is contrary to the First Law of Thermodynamics. 18. Schistic Processes : Intrinsic Energy. A system is said to be in equilibrium in a thermodynamic sense when the position of each of its parts relative to a fixed point, and the state of that part, remain unchanged with lapse of time. The first condition asserts that the kinetic energy of the system, relative to the fixed point, is constantly zero, and refers to mechanical equilibrium. By the term " state " we refer to such data as the chemical composition of the parts of the system (including allotropy and isomerism), their state of electrification, magnetisation, stress or strain, their state of division, temperature, and the like, and the second condition generalises the statement of equilibrium. If a system is not in equilibrium its state changes with lapse of time. A system in equilibrium may also be caused to change by means of external actions. We now suppose that a system undergoes any change, and that no energy changes occur outside the system except the disappearance of a quantity of heat Q and the performance of a quantity of work A. If the final state is the same as the initial state, the process must, by definition, have been a cyclic process, and hence A = JQ . . . . (a). If the final state is different from the initial state, no relation THE FIKST LAW OF THERMODYNAMICS 33 between A and Q can be ascertained a priori. We shall suppose, in this case, that the magnitudes A and Q have been experimentally determined ; two possibilities are now open : either (i.) A = JQ . . . . (a'), i.e., the process is aschistic but not cyclic ; or (ii.) A is not equal to JQ . . . (ft), i.e., the process is not aschistic. Since the conditions exclude the possibility of any other energy changes occurring outside the system, we conclude that there has been a change in some amount of energy which is located in the system itself. We must therefore regard the system as possessing a store of energy which may be increased or diminished by changes in the physical or chemical state of the system, and we shall call this the intrinsic energy of the system, since, as we shall prove in a moment, it depends solely on the state of the particular system in which it is located. The change of intrinsic energy we shall now define as follows : Definition of Intrinsic Energy. Let there be given any system of bodies, and let the system undergo any change whatever, so that it passes from a given initial state [1] to a final state [2], the only condition imposed on the states [1] and [2] being that they shall be consistent with the physical properties of the system. In this change there will be in general a finite amount of heat withdrawn by the system from its environment, which may be absorbed in various parts. The total absorbed heat we shall denote by 2Q. At the same time the parts of the system, exert- ing forces on external bodies, may perform mechanical work. The total mechanical work done by the system on external bodies we shall denote by 2A. Thus in the evaporation of 1 gram of water under a piston, the heat 2251 x 10 7 erg is absorbed in "raising steam," and the work 1662 x 10 7 erg is done by the piston against the atmospheric pressure. Thus J2Q - 2A = (2251 - 1662) X 10? = 589 X 10 7 erg. Then if Ui be the intrinsic energy of the system in its initial state, and U 2 the intrinsic energy of the system in its final state, we define the increase of intrinsic energy of the system by the equation : U a - Ui = J2Q - 2A . . . . (c) 84 THERMODYNAMICS If Q is measured in the same units as A (e.y., in ergj), we may omit the factor J, provided we remember always that quantities of heat are to be expressed in work units in the equa- tions which follow. Then : U 2 - L T ! = 2Q - 2A . . . . ( c ') (U 2 Ui) is a definite magnitude, since both 2Q and 2A are experimentally determinable. If (SQ 2A^< 0, then (lJ 2 - Ui)< 0, i.e., the intrinsic energy has increased, diminished, or remained constant, accord- ing as the absorbed heat is greater than, less than, or equal to, the external work done. An aschistic process implies constancy of intrinsic energy. The actual state, and absolute amount, of intrinsic energy existing in a body, or system of bodies, are things which lie quite outside the range of pure thermodynamics. This indefmiteness has, however, not the slightest influence on the stringency of the definition, since we can proceed as in the definition of electrostatic potential, and choose any convenient standard state of the body, and use the term "intrinsic energy" with reference to this standard state. If U is the absolute amount of intrinsic energy contained in a system (with reference to a state of absolute zero of energy) in an arbitrary standard state, and if in any change from a state [1] to a state [2] the total amounts of heat absorbed and work done are 2Q and SA respectively, we have : 2Q-2A = [U x + Uoi; = U 2 - Ux, where ~U X is the intrinsic energy in any intermediate state x, with reference to the standard state (Lord Kelvin, 1851). It will be observed that the definition of intrinsic energy by means of the equation (c) implies in itself no physical law, since the value of (U 2 Ui) can always be chosen so as to make the values of 2Q and 2A satisfy the equation. We shall now show that the value of (Ua Ui) is uniquely so defined, and is quite independent of the way in which the process is executed. This is a physical law, which we shall call the Principle of Conservation of Energy. It may be formally expressed as follows. THE FIRST LAW OF THERMODYNAMICS 35 19. The Principle of Conservation of Energy. The change of intrinsic energy of a system undergoing any change of state depends solely on the initial and final states of the system, and is independent of the manner in which the change from the one state to the other is effected. Proof. Let the change from state [1] to state [2] be effected in different ways, which are denoted by (a), (/J) . . ., (A); if the changes of intrinsic energy are not equal, let them be AU a , AU fl . . . , AU A , for the different paths. Let AU a , AU/s beanyivfo of these values. We shall now assume that at least one process is possible whereby a reverse change from the state [2] to the state [1] may be effected; let this process be denoted by ((/>). Now perform the following cyclic process : (i.) Pass from [1] to [2] along (a), and return along (<). The diminution of intrinsic energy = AU<^ AU tt . (ii.) Pass from [1] to [2] along (/?,) and return along (<). The diminution of intrinsic energy is AU^ AUg. The system is now in its initial state, and since no energy has been obtained from, or given to, other systems, except that derived from the given system, we see that unless the two quan- tities (AU^ AU ), (AU^, AU0) are both zero, a quantity of energy will have been gained or lost without any other change having occurred. This is in contradiction to the First Law of Thermodynamics, and hence : AU^, AU a = AU^ AUp = 0, or AU a = AU0 = AU*, and since (a), (/3) are any two possible paths, the proposition is true for all possible paths. It must be observed that the formal proof of the theorem depends on the possibility of returning to the initial state along at least one path such as ($). The extension of the theorem to vital processes, phosphorescence, and radio- active changes, which have not yet been reversed, must therefore bs regarded as inductive, although highly probable. If the given change is infinitesimal, we shall denote 2Q and 2 A by SQ and SA respectively. Corollary 1. For any two states of a system, say (1) and (2), the value of the integral : 3-*A) D 2 36 THERMODYNAMICS is independent of the manner in which the change from [1] to [2] is effected, i.e., (SQ 8A) = dU is a perfect differential (Lord Kelvin, 1851). Corollary 2. For a cyclic process : (J) SA = (j>Q, where SA, SQ denote the elements of work done and heat absorbed during any infinitesimal part of the cycle, and ( ) is taken as signifying an integration extended round the cycle. For (J)8A = (J)SQ - (f)dU, and (J)dU = 0, by reason of the identity of the initial and final states. The magnitudes (I)SA, (|)SQ, denoting the nett work done, and heat absorbed, in the cyclic process, are however, not usually zero, and may have very different values (provided (I )&Q (|)8A is always zero), according to the particular way in which the process is conducted. The same applies to the magnitudes 2A and 2Q ; for the same ini- tial and final states, the work done and heat absorbed may have very different values according to the way in which the change is effected ; these pairs of values must, however, always satisfy the equation : 2Q 2A = constant, for fixed terminal states. This very important theorem was recognised by R. Clausius in 1850, although he did not at the time give the very simple interpretation, in terms of the conception of intrinsic energy, which was brought forward by Lord Kelvin a year later. 8Q, SA are what are called imperfect differentials ; (8Q &A) = dU is a perfect differential (cf. H. M., 57, 115). As a matter of fact, &Q, SA are not differentials of functions of the independent variables of state at all, and might be written q, a. The path of change is usually more or less open to arbitrary choice, and we shall next consider one or two important cases, in each of which some kind of constraint is put upon the energy changes : (1) The change is aschistic: 2A = 2Q .-. AU = orU = constant. THE FIRST LAW OF THERMODYNAMICS 37 A particular case is a cyclic process ; an example of a non- cyclic aschistic change is afforded by the expansion of an ideal gas at constant temperature ( 71). (2) The change is adiabatic, i.e., the transfer of heat to or from the system from outside is prevented, say by enclosing the system in a perfectly non-conducting envelope. Then : 2Q = .'.2A= AU, so that the external work is performed wholly at the expense of the intrinsic energy of the system. (3) The change is adi/namic, i.e., no external work is performed. Thus, we may imagine the system enclosed in a perfectly rigid envelope which, however, permits the free passage of heat. Then: 2A=: .'. AU = 2Q, so that the heat absorbed goes wholly to increase the intrinsic energy. This provides a method of determining the latter magnitude, since 2Q is experimentally measurable. (4) A system which is cut off from communication of all kinds of external energy is called an isolated system. A system of bodies contained in a vessel with perfectly rigid non-conducting walls is such a system. In this case 2Q = 2A = . * . V% = Ui .. U = constant. The Principle of Conservation of Energy is usually expressed in the form that the intrinsic energy of an absolutely isolated system of bodies is constant and independent of all changes of state which may occur subject to the condition that the system remains isolated. Since in this case we have absolutely no means of examining the energy content of the system, the statement appears somewhat indefinite. 20. Pressure. When two bodies are placed in contact there is in general a distribution of force over the area of contact. Such a distribu- tion of force over an area is called a thrust, and if the force at all points is normal to the area, the thrust per unit area is called the pressure. If the force is inclined at an angle 6 to the normal to the area, the resolved part, P cos 0, only is taken into account. Since in general the thrust is not uniformly distributed, we must 38 THERMODYNAMICS introduce the conception of the pressure at a point on an area. Let P be the force acting normally to the centre of gravity of a very small plane area da. Then if P/da tends to a finite limit as da approaches zero, this is denned as the pressure at the point which is the centre of gravity of da. We may also speak of the pressure at a point in the interior of a mass of liquid or gas, because if a very small plane area a- is drawn around that point as centre of gravity, and all the fluid removed from the immediate vicinity of one side, a definite force P must be applied to keep the area in position. From the principle of reaction we see that each of the two portions of fluid divided by an imaginary plane o- exerts a pressure P/o- on the other. Such a pair of equal and opposite forces is called a stress. Stresses may also exist in the interior of solid bodies, and are considered in the theory of elasticity. This pressure is the distribution of molecular impacts on the surface. If da is smaller than a finite but very small size the pressure loses its significance, and is replaced by a sporadic bombardment by individual molecules at intervals comparable with the time in the mean free path. This occurs, for instance, in the spinthariscope of Crookes. The condition that P/V?a approaches a finite limit, independent of the original size of da, when da > 0, is apparently in contradiction to the physical pro- perties of the system. A similar difficulty arises in the definition of the density at a point in a medium of varying density, by considering the mass of a volume element around the point. The space would sometimes be occupied by a small number of molecules, at other times by a larger number, and occasionally by none at all. The requirements of molecular physics and of the infinitesimal calculus are therefore apparently in direct opposition. The solution of the difficulty becomes apparent as soon as we remember how exceedingly minute are the individual molecules in comparison with any "finite portion" of a body. The infinitesimal element may be chosen so small that it satisfies the conditions imposed by the mathematical analysis, and yet remains sufficiently large to be physically homogeneous. Such an element of volume, or generally such an element of a molecular system, may be called a physically small element (Leathern : Volume and Surface Integrals in Matlie- THE FIRST LAW OF THERMODYNAMICS 39 matical Physics, Cambridge) or a macro-differential (Planck : Adit Vorlesungen ilber tkeoretischc Physik, Leipzig, 1910, 3 Vorles. ; Theorie der Wdrmcstrahlung, pp. 129 et seq.). We are very often concerned with magnitudes such as pressure, density, concentration, temperature, etc., which have the signifi- cance of mean values, and it must be remembered that we cannot apply these terms to systems which are so constituted as to pro- hibit the existence of such a mean value. This point is by no means merely a logical or mathematical refinement, but is of the very essence of the physical interpretation of the second law of thermodynamics (cf. Planck, loc. cit.). 21. Units of Pressure. unit force unit pressure . . = r . unit area 1 dvnG Absolute unit of pressure = a . Statical units of pressure = ^-4, ^-, etc. cm. 2 ' cm. 2 ' Standard Atmosphere = pressure of a column of 76 cm. pure mercury at 0C. in latitude 45 (variation of gravitation constant affects the result J per cent, per degree) : p = 76 X density = 76 X 13'595 = 1033-2 -^ cm. a = 1033-2 X 980-53 = 1013130 d ^? 2 e . A standard C.G.S. atmosphere of 10 6 % has also been proposed. The dimensions of pressure are : force force area (length) 2 " 22. Fluids. We may for the purposes of thermodynamics define a fluid as follows : (1) The line of action of the thrust exerted by a fluid at rest on an area is everywhere perpendicular to the area. (2) The pressure at a point in a fluid at rest is equally great in all directions. This implies that a small element of area 8a may be turned so 40 THERMODYNAMICS as to be perpendicular to every direction drawn through the selected point (x, y, z), without thereby altering the thrust upon it. (3) A pressure applied at any point on the boundary of a fluid is transmitted uniformly throughout the whole fluid (Pascal's law). In what follows we shall always consider the pressure as having a uniform value for all directions through any point. Gases and liquids at rest satisfy this condition ; under some circumstances a solid may be treated thermodynamically as a " fluid," e.g., when it is immersed in a liquid under pressure and is free from torsion or shearing stress. Conditions (2) and (3), however, very materially limit the range of applicability in such cases. 23. Elasticity of a Fluid. The modulus of elasticity of volume, or bulk modulus of elasticity, of a fluid under specified conditions, is the ratio of any small increase of pressure to the resulting relative decrease of volume. We shall refer to this simply as the elasticity. Let p and r be the initial pressure and volume, and suppose p is increased to (p + Bp), the total volume thereby increasing to (v + 8v). Then : increase of pressure = Bp, increase of volume = Sr, relative increase of volume = Bv/v, Bp Bp dp . ,, ,. .. Elasticity = e = ~- = v ~- = v -r , in the limit. Bv/v Be dv The negative sign indicates that all real fluids contract when the pressure increases. Since Bv/r is a mere number, the elasticity has the same dimensions as pressure. The reciprocal of the elasticity of volume of a fluid is called its modulus of compressibility (r?) : dv --. dp It is easily shown that, if corresponding values of v and p are represented in a rectangular co-ordinate system, the elasticity at any point on the curve is equal to the length of the _p-axis inter- cepted between the tangent at that point and the horizontal through the point (Fig. 1). tnroi THE FIRST LAW OF THERMODYNAMICS 11 The elasticity of a fluid is not completely specified unless the conditions under which it is measured are known, and a fluid has various elasticities corresponding to the different sets of conditions. Two are especially important : (1) Isothermal elasticity, f e , measured under such conditions that the temperature remains constant. (2) Adiabatic elasticity, e Q , measured under such conditions that no heat is allowed to enter, or escape from, the fluid during the volume change. In the measurement of the former, the cylinder enclosing the fluid may be supposed to be formed of a good conductor of heat, and to be immersed in a large tank of water at the particular temperature ; if the compres- sions are effected very slowly, the mass of fluid will then have the opportunity of acquir- ing, at every instant, the tem- perature of its surroundings by heat transfer, so that it will remain at a constant tempera- ture. The adiabatic elasticity, on the other hand, applies to compressions of a fluid en- closed in a perfectly non-con- ducting cylinder, or, since no actual cylinders satisfy this condition, to pressure changes which are performed very rapidly, so that there is not sufficient time allowed for equalisation of temperature by heat transfer. An example of such pressure changes is afforded by the periodic variations of pressure at a point in a gas which is transmitting a succession of sound-waves. 24. Work done by an Expanding Fluid. If a fluid contained in a cylinder expands so that its pressure remains constant (e.g., saturated steam in contact with water), the work done is that of raising the piston, of area a, which supports a weight W just sufficient to keep the expansive force indefinitely near equilibrium. If s = distance of outward motion of piston : work A = W. s = pa. s = p. as = p&u, where Ar is the increase of volume. FIG. i. 42 THERMODYNAMICS It is easy to show that this relation is perfectly general. Let a mass of fluid of any shape be represented (Fig. 2) by the full line PQRS, and let its volume change, under a uniform external pressure p acting everywhere normal to the bounding surface, so that the final volume is represented by the dotted line PQ'R'S'. We now take a small element of area on the original surface and erect upon it a cylindrical surface C, cutting the surface PQ'R'S'. Let the volume enclosed between the two elements of area on the surfaces bounding the cylindrical space be Ar ; this is positive if the surface PQ'R'S' lies outside the surface PQRS at the position considered (as at Q'), negative if inside (as at R'). The work done in the infinitesimal cylinder is therefore pAr. If the whole change of volume is divided into such cylinders, the total work done = SpAi- = jjSAr = p X (total change of volume) as before. If the pressure does not remain uniform during the expansion, we imagine the process divided into a very large number of very small changes' of volume, in each of which the pressure may be regarded as constant. Let p v be the mean pressure during an expansion from a volume (v ^Sr) to a volume (v -f- Sr) ; then the small element of work done is : BA. = _2> c 8f. For a finite change of volume under specified conditions (e.g., constant temperature) : FIG. 2. ~r 2 - = pdv = 2Wa pivi J 1-1 ^ i vdp. pi 25. Heat Function at Constant Pressure. The heat absorbed in the change of state of a system at constant volume is equal to the increase of intrinsic energy : Q^Ua-Ux . . .. (1) THE FIEST LAW OF THERMODYNAMICS 43 The heat absorbed in the change at constant pressure is this plus the external work : Q P = U 2 - Ui + A = U 2 - Ui + X'-a - t-i) . (la) The equation (la) may be written in the form : Q = (U 2 + jwr a ) - (Ui + pri) . . (16) If we put U+j>r=W .... (2), we see that the heat absorbed when a system passes from one state to another at constant pressure is equal to the difference in the values of a f unction (U -\-pi~) for the initial and final states : Q p = W a -Wi .... (3) The function W was called by Willard Gibbs the Heat Function at Constant Pressure. Corollary. f?W is a perfect differential when the pressure is constant, and Q p is independent of the path. The independence of the heat effect on the path requires that the change shall occur either at constant volume or at constant pressure. If the volume is maintained constant (dv = o) the pressure may be changed in any way ; if the pressure is maintained constant (dp = o) the volume may be altered in any manner so that the limiting con- ditions are satisfied; but if both pressure and volume change simultaneously I j>dv is no longer independent of the path. 26. Characteristic Equation. It is usual in physics and chemistry to speak of the " state " of a given body, and we may perhaps define the term by saying that two bodies are in the same state when they are identical except as regards accidental properties such as shape, position, and size. The independence of state on the size implies that when we have defined the state for unit mass, we have fixed it for any mass. If we abstract these unessentials we are left with the concept of a substance (cf. H. M., Introduction). What properties of two portions of a substance must agree in order that they shall be identical, i.e., in the same state? In the case of a fluid, the following properties of unit mass must be identical : (1) Chemical composition. (2) Temperature, 0. (3) Volume, i.e., specific volume, r. 44 THEKMODYNAMICS The pressure is then defined, and there must be some relation between p, v, and 6 : f(p,v,0)=0 . . . . (1) The magnitudes p, v, 6 are therefore determined by the state of the body, and so may be called functions of the state. The relation (1) is called the characteristic equation (" Zustands- gleichung ") of the fluid. Thus, if two of the variables p, v, 6 are given certain values, the third is fixed, and the state of a homo- geneous fluid is like the position of a point in a plane, capable of two and only two independent variations, or has two degrees of freedom. A particular point in a plane may be associated with every separate state of which the body is capable, so that states differing infinitesimally are associated with points infinitely close together, and the assemblage of points may be regarded as repre- senting the various states. Such a method of association is called a continuous association. All the points associated with states of equal volume, pressure, or temperature form lines, each corre- sponding to a particular volume, pressure, or temperature, and these are called isochores, isopiestics, and isotherms, respectively. When two of the magnitudes v,p,0 have been assigned, the value of the third is found by solving (1). Thus, if v, 6 are fixed, we have: p = T 2 . In order that finite quantities of heat may be added to or taken from these without change of their temperatures, we may suppose them to consist of FIG. 7. large reservoirs of water, or still better of reservoirs of steam and ice, which would preserve constant temperatures during with- drawal or addition of heat. Nothing more is assumed about the temperatures, and one result of Carnot's investigation is a rigorous definition of tempera- ture. Further, let there be a cylinder and piston, of an absolute non-conductor of heat, closed at the bottom by a perfect con- ductor of heat, and containing the working substance any substance, or mixture of substances, the pressure of which is uniform in all directions at all points and is a continuous function of temperature. Finally, we have a stand formed of a perfect non-conductor of heat (Fig. 7). There is now performed a reversible cycle called Carnot's cycle, and consisting of four operations : 56 THERMODYNAMICS (1) The working substance being initially at the temperature T 2 of i he refrigerator, we place the cylinder on the non-conducting stand, and compress the working substance reversibly until the temperature rises to 1\. By the conditions imposed, this is an adiabatic compression, and will be represented by a continuous curve on the indicator diagram, say AB (Fig. 8). (2) We now place the cylinder on the source, and allow the working substance to expand reversibly and isothermally at TI until any arbitrary quantity of heat Qi has been absorbed. In this process the temperature of the working substance must, it is true, be infini- tesimally less than that of the source in order that heat may pass into it, but in the limit this difference becomes vanish ingly small, -/Q, absorbed Prom and the temperatures approach equality. This step is represented by a continuous curve, the iso- therm BC. (3) The cylinder is again Q, rejected to refrig? kced Qn the non . conduct . FlG 8 ing stand, and the working substance reversibly and adiabatically expanded till its temperature falls to T 2 . The course of expansion is represented by the curve CD. (4) Finally, the cylinder is placed on the refrigerator and the working substance compressed reversibly and isothermally until it returns to its initial state A, rejecting heat Q 2 to the refrigerator. This operation is represented by the curve DA. The cycle is now completed, and the working substance is in exactly the same state as at the beginning. It will be observed that, with a given initial state, and given temperatures, all the curves are completely determined by the length of BC, which alone is arbitrary. The working substance is never in contact with bodies differing from it more than infinitesimally in temperature, and is never exposed to pressures exceeding or falling short of its own by more than infinitesimal amounts. The cycle is therefore reversible, and may be carried out in the THE SECOND LAW OF THEEMODYNAMICS- 57 reverse direction, the thermal and mechanical effects at every part being exactly reversed, so that if in any infinitesimal element of the direct cycle: the difference of pressures is + Sp, the difference of temperatures is + ST, the heat absorbed is -f- SQ, the work done is + $A, then in the reverse .cycle that infinitesimal element of path is retraced, and all the above magnitudes have negative signs, so that compressions correspond to expansions, heat evolved to heat absorbed, work spent to work done, and, to produce the opposite direction of heat transfer, the temperature differences must also be reversed _in . sign. The actual course of the reversed P cycle will therefore l^e com- posed of the four steps : (la) Starting at the tem- perature T 2 , expand rever- sibly and isothermally along AB (Fig. 9), till an amount of heat Q 2 is absorbed from the refrigerator. (2a) Compress reversibly and adiabatically along BC until the temperature rises to Ti. (3a) Compress reversibly and isothermally along CD until heat Qi is given up to the source. (4a) Expand reversibly and adiabatically along DA, finishing at the initial state A. In the direct circle the loop ABCD is described in a clockwise sense ; in the reversed circle it is described in a counter-clockwise sense. We will now consider the changes produced in the direct and reversed cycles. (a) In both cases the working substance is unchanged. (b) . In the direct cycle a quantity of heat Qi is absorbed from the source, and a quantity of heat Q 2 is given up to the refrigerator. In the reversed cycle a quantity of heat Q 2 is B bsorbed Prom rePrig!" FIG. 9. 58 THERMODYNAMICS absorbed from the refrigerator, and a quantity of heat Qi is rejected to the source. (c) In the direct cycle an amount of work A, represented by the area ABCD, is done by the system ; in the reversed cycle an amount of work A is done by the system, i.e., the work + A is spent on the system, 35. Carnot's Theorem. We shall now apply the two laws of thermodynamics to the energy changes occurring in the Carnot's cycle. (1) The cyclic process being aschistic, we have, as a consequence of the first law : Qi-Qa^A . . . . (1) (2) For the application of the second law we establish the following propositions : (a) Of all possible heat engines working with fixed temperatures of source and refrigerator, a reversible engine is the most efficient. Let [a], [/3] be two engines having the same source and refrigerator ; let [a] be a reversible Carnot's engine, [/8] some other engine which (if possible) is more efficient than [a]. By suitable adjustment of the working parts (e.g., length of piston stroke) each engine may be arranged so as to absorb heat Qi from the source per complete cycle. Now let [ft] work [a] backwards, so that [a] converts the work it receives into heat. This operation is possible, because [a] is a reversible engine. In a complete cycle : [takes up heat Qi from the source, [/3] j does an amount of work, say A', (gives up heat Qa' to the refrigerator, /returns heat Qi to the source, [a] j absorbs work A, I takes up heat Q 2 from the refrigerator. The efficiency of [/?] is, by definition, N, = A'/Qi, and that of [a] is similarly N a = A/QL By hypothesis, Np > N a , .-. A' > A . . . . But A' + Q a ' = A + Q 2 = Qt, by the first law, .'. Qa' < Qa - . - - THE SECOND LAW OF THERMODYNAMICS 59 Equations (a) and (b) show that the compound engine [a -f 0] is capable of producing an amount of work (A' A), which could be used to raise a weight, and that it leaves no other change in the surroundings except that the refrigerator is cooled by withdrawal of an amount of heat (Qa Qa')- This result is in contradiction to the second law, hence we conclude that the hypothesis entertained is inadmissible, so that [/3] is an impossible engine, which establishes the proposition. (b) All reversible engines -working in cycles with the same temperatures of source and refrigerator are equally efficient. For if [a] is a Carnot's reversible engine with any specified working substance, and [/3] another Carnot's engine with a different working substance, or any other reversible heat engine whatever, the preceding reasoning shows that [/3] cannot be more efficient than [a]. But since [/3] is itself a reversible engine, the functions of the two engines may be interchanged, and the same reasoning shows that [a] cannot be more efficient than ~3~ . Hence, since if/8] cannot be more efficient, or less efficient, than [a], it must Be equally as efficient as [a], so that N a = X^. From this we deduce the following important corollari/ : If a quantity of heat Q is absorbed in a reversible cycle, with given temperatures of source and refrigerator, the quantity of work A obtained from it is independent of the arrangement used in performing the cycle. This we shall call Carnot's Theorem. Beginners are usually surprised when they are informed that the work done by a reversible engine performing a cycle between fixed temperatures is always the same, for a given quantity of heat absorbed from the source, no matter what is the working substance in the engine. This may in a fluid engiue be a gas (such as air), a vapour (such as steam), a liquid (such as water), a solid (such as ice or copper), or a mixture of any of these ; the only condition imposed being that the volume and pressure shall change continuously with alteration of temperature. Xow it would appear that an engine working with a very volatile liquid, say ether and its vapour, should yield, fur a given expansion, much more work than a similar engine working with water and its vapour, by reason of the greater vapour pressure of the former. "Whilst this is quite true under the condition italicised, yet it must be borne in mind that Carnot's theorem applies only to an engine which works in agreement with two very important conditions, viz., that it works in a cycle, and works reversibly. Thus, although more work is obtained in the single forward stroke of the ether engine than in the similar stroke of the steam engine, yet, for the same reason, proportionally more work must 60 THERMODYNAMICS be given back again in the reverse stroke which completes the cycle, and the greater loss exactly compensates the greater gain. 36. Isothermal Cycles. Theorem. The work done in any isothermal reversible cyclic process is zero (J. Moutier, 1875). For if a cyclic process could be performed in a heat reservoir of uniform temperature so as to give out work, it would consti- tute a perpetuum mobile of the second kind, the existence of which is denied by the second law. And if the cyclic process absorbed work when performed at a uniform temperature, it would, by reason of its reversibility, give out an equal amount of work when reversed ; this would, however, be the case first considered. Hence the production of work in either cycle is impossible, which establishes the theorem. Corollary 1. The area enclosed by the circuit representing an isothermal reversible cycle on the indicator diagram is zero ; if, therefore, the curve is not a segment of a line transversed from A to B and then from B to A (Fig. 4), it must form two loops of equal areas but traversed in opposite senses, or else such a system of positive and negative loops that the total area is zero. Corollary 2. The algebraic sum of the quantities of heat with- drawn from or given to the constant temperature reservoir in an isothermal reversible cycle is zero. 37. Absolute Temperature. It is an immediate consequence of Carnot's theorem that the ratio of the quantities of heat absorbed and rejected by a per- fectly reversible engine working in a complete cycle, depends only on the temperatures of the bodies which serve as source and refrigerator. . . For if Qi, Q 2 are the quantities of heat absorbed and rejected, respectively, in localities at temperatures TI, T 2 , then the Efficiency (N) = ^ ~ ^ 2 = 1 9?, Vi ^i and since N depends solely on the temperatures/ by Carnot's theorem, \ "~ Q ) is a function of T I> T a alone, n 2 is a function of TI, T 2 alone, THE SECOND LAW OF THEEMODYNAMICS 61 or = ^(T a ,T!) ..... (1) We shall now suppose that nothing more is known about the temperatures TI, T 2 except that : (i.) A definite temperature may be assigned to any material system which is in thermal equilibrium ; (u.) The temperature of the source is greater than the tempera- ture of the refrigerator, i.e., TI > T 2 . These statements are implied in the definition of temperature given in 4. Equation (1) now gives, as Lord Kelvin pointed out in 1848, a quantitative definition of temperature, and this definition, being framed in terms of the efficiency of a reversible engine, is indepen- dent of the methods adopted for its measurement, and is therefore " absolute" On the other hand, a definition of the temperature of a body in terms of the volume of a given mass of air, mercury, etc., which is in thermal equilibrium with the body, will be depen- dent on the properties of some other system than the one of which we require to know the temperature : in fact the tempera- ture so measured will depend on the particular thermometric substance selected. If two exactly similar thermometer tubes are taken, one being filled with mercury and the other with alcohol, and if the levels at which the liquids stand when the tubes are placed in melting ice and in steam are marked and 100 z respectively, each of the 100 equal intervals between these marks is defined as a Centigrade degree. But if both thermometers are now placed in thermal contact with some body at a temperature inter- mediate between and 100, the readings of the two thermo- meters will not agree. In order that the temperature of a body may be defined without ambiguity it is necessary, therefore, to select a particular thermometric substance as standard, and to use this exclusively in all measurements ; hydrogen gas is the substance at present used (cf. 3). The temperature so measured still suffers from the defect that it is not solely dependent on the state of the body submitted to examination, whereas the tempera- ture of a body is undoubtedly a function of the state of that body alone. In the same way the mass of a body is a property belong- ing solely to that body : if, however, it is estimated by a spring- balance, the measure of the mass will appear to depend on the position of the balance relative to the earth's centre of gravity. 62 THERMODYNAMICS The ordinary balance, on the ..other hand, with suitable adjust- ment of its parts, gives a measure of the mass of a body, in terms of an arbitrarily selected standard, which depends only on the particular body ; the same measure would be found if the balance were transported to any part of the earth's surface, or even to another planet, such as Mars or Jupiter. Such a measure may be called "absolute." The exact form of the function $ (T 2 , TI) being to some extent arbitrary, we might give several definitions of " absolute tempera- ture," all drawn up, however, in terms of the efficiency of the reversible engine. Lord Kelvin, in 1854, adopted the following form : Definition of Absolute Temperature. " The temperatures of two bodies are proportional to the quantities of heat respectively taken in and given out in localities at one temperature and at the other, respectively, by a material system subjected to a complete cycle of perfectly reversible thermodynamic operations, and not allowed to part with or take in heat at any other temperature : or, the absolute values of two temperatures are to one another in the proportion of the heat taken in to the heat rejected in a perfect thermodynamic engine working with a source and refri- gerator at the higher and lower of the temperatures respectively." Hence: 4 (T 2 , T x ) = ^ . . . . . (2), where T denotes the measure of the absolute temperature defined as above. Equation (2) fixes the ratio of two temperatures ; for T 2 /Ti = We shall now define what is to be understood by equal intervals of temperature. Let us imagine that we have a system of rever- sible engines [1,2], [2,8], [3,4], . . . , working between constant temperature reservoirs (1), (2), (3), (4), . . . , so that the refrige- rator of any engine (except the last) forms the source of the next engine. Let each perform a cycle so that [1, 2] takes heat Q l from (1) and gives out heat Q 2 to (2), [2,3] Q 2 (2) Q 3 ,, (3), [3,4] Q 3 (3) Q 4f , (4), and so on. The quantities of heat taken from the sources are Qi, Q a> Q* . . . . , THE SECOND LAW OF THERMODYNAMICS 63 the quantities of heat given to the refrigerators are Q 2 , Qa, Q* - - - - , and therefore the amounts of work done by the engines are (Qi - Qa), (Q 2 - Qa), (Qa - QO, By Carnot's theorem these depend only on the temperatures of the heat reservoirs, i.e., on TI, T 2 , T 3 , T 4 , . . . and since the latter are supposed to be arbitrary, we may arrange them so that each engine does exactly the same amount oj work per cycle ; then (Q t - Qa) = (Q 2 -Q 3 ) = (Q 3 - Q 4 ) = . J . . But Qa/Qi = T 2 /Ti, Q 3 /Q 2 = T 3 /T 2 , .... and so on. Divide (a) by (6) : (Qi - Qa) : (Qa - Qa) = But (Q! - - T a ) : (T a - T 8 ), etc. = (Q 2 - Q 3 ), etc., so that the differences between the temperatures of the successive heat reservoirs are equal when all the members of a series of reversible engines, so arranged as to annul all thermal changes in reservoirs between the first and last, do equal amounts of work. The action of this series of engines may be represented on the indicator diagram (Fig. 10) by taking an iso- therm AA', correspond- ing to TI, and crossing it by adiabatics Aia 1} A 2 a 2 , .... If isotherms BB', CC', ... are now drawn, corresponding to temperatures T 2 , T 3 , . . . so that all the areas ABi, BCi, ... are FIG. 10. 64 THERMODYNAMICS equal, the differences between the temperatures of successive isotherms are also equal. Corollary. If AiA 2 , A 2 A 3 , ... are portions of the upper isotherm, along each of which the same amount of heat is absorbed as along AAi, show by Garnet's theorem that area ABi : heat absorbed along AAi= area AB 3 : heat absorbed along AA 3 , and thence that the efficiency of a reversible engine working between fixed temperatures is independent of the quantity of heat absorbed from the source. The definition also fixes the zero of absolute temperature. Let Qi, Q 2 be the quantities of heat absorbed from the source and rejected to the refrigerator at absolute temperatures TI, T 2 respectively by a reversible engine. We have proved that Qi - Q 2 _ T! - Ta. Qi T! Now put T 2 = 0, . Qi - Q 2 _ -. ~~ Hence the temperature of the refrigerator is zero when all the heat absorbed from the source is converted into work by the reversible engine. Corolla n/ 1. Absolute temperatures are essentially positive magnitudes. Corollary 2. A body cooled to the zero of absolute temperature (" absolute zero ") cannot be made to part with more heat. [Its intrinsic energy may, however, have any value, including zero, at this temperature.] The size of the degree alone remains to be fixed, and is quite arbitrary. To produce as little change as possible from the ordi- nary scale, Lord Kelvin divided the range of temperature between the absolute temperature of melting ice T , and that of boiling water, TI, into 100 equal parts, each of which is defined as one By special experiments, to be considered later on, he found that : C = T366 very approximately, = 1 f = 0-366, o lo . TO = 273 C. very approximately. THE SECOND LAW OF THERMODYNAMICS 65 The most recent determinations lead to the value : To = 273-09 C. Thus, if 6 is any Centigrade temperature, T = + 273-09. In what follows, the symbol T is always to be understood as referring to the absolute scale. The boiling-point of liquid helium is 4'20 abs. (Kamerlingh Onnes, Commun. Pht/s. Lab. Leiden, No. 119, 1911). A tempera- ture lower than 1*5 abs. has recently been obtained by the rapid evaporation of solid helium. 38. Analytical Expression of Carnot's Theorem. Theorem. The work obtained from a given quantity of heat absorbed from the source by a reversible engine is the greatest amount which can possibly be obtained with given temperatures of source and refrigerator. This may be called the maximum work for a given quantity of heat and a given distribution of temperatures ; the rest of the heat is irrecoverably lost so far as its availability for producing work is concerned, and is therefore " wasted," but not destroyed (Lord Kelvin, 1849). Carnot (loc. cit.), who accepted (with forcibly expressed doubt) the caloric theory of heat, according to which heat is material and therefore indestructible, explained the production of work from heat as brought about by a " re-establishment of equili- brium in the caloric," the equilibrium being disturbed by difference of temperature. The motive effect of heat was therefore due to its "fall " from a hot to a cold body, i.e., down a precipice of temperature ; it was analogous to the work done by water falling, through a water-wheel, from a high to a low level. The above theorem, deduced by Carnot from this false premise is how- ever strictly true when the convertibility of heat into work with a fixed rate of exchange is postulated, as was proved by Clausius (1850) and by Lord Kelvin (1851), who at the same time pointed out that Carnot's original theorem is a limiting case, approached when the temperature difference is infinitesimal. For : Qi - Qs _ TX - T 2 Qi Tx .-. if we put (Ti T 2 ) = ST, and (Qi Q 2 ) = work done = (SA), the brackets denoting that the enclosed magnitude refers to a cycle, then T. F 66 THERMODYNAMICS (8 A) _ ST gr : ~ TI ^m ... (SA^Q^ . .... (a) Since (8A) is infinitesimal, (Qi 8A) = Q 2 , the heat discharged into the refrigerator, is very nearly equal to Qi, so that if we put Qi = Q, TI = T, and write (a) (SA) = Q ^ . ' - (') we see that the quantity of work obtained when an amount of heat Q passes by a reversible engine, from a temperature T to a temperature (T BT) is (i.) proportional to the fall of temperature, (ii.) inversely proportional to the temperature of the source. Carnot assumed that (a') was true with a finite temperature difference ; this, however, would imply that no heat is destroyed even in a cycle. 39. The Principle of Dissipation of Energy. The different forms of energy may be classified according to their practical value as regards adaptability, or availability, for the performance of useful work. We assume, as a matter of definition, that a raised weight represents a distribution of energy most useful in this respect, as it merely requires to fall in order to perform, on an appro- priately connected mechanism, an amount of work equal to the whole of the potential energy of the system composed of the earth and weight in the initial configuration. Any distribution of energy which can be completely converted into the potential energy of a raised weight is called available energy. If a part only of the given distribution of energy is so convertible, we speak of this as the available part, and the rest as the unavailable part, of the total quantity of energy. It is an immediate consequence of the second law that all heat energy in a medium of constant temperature is (in the absence of all other utilieable distributions of temperature) completely unavailable energy. Any process whereby available energy is converted into unavailable energy is called Dissipation (or degradation) of Energy. THE SECOND LAW OF THERMODYNAMICS 67 Examples of such processes are the equalisation of tempera- ture differences by conduction or radiation, the production of heat by friction, the expansion of gases into vacuous spaces, and the mixing of chemically different substances. Tln'orem. A process yields the maximum amount of available energy irhe-n it is conducted reversibly. Proof. If the change is isothermal, this is a consequence of Moutier's theorem, for the system could be brought back to the initial state by a reversible process, and, by the second law, no work must be obtained in the whole cycle. If it is non-isothermal, we may suppose it to be constructed of a very large number of very small isothermal and adiabatic processes, which may be combined with another corresponding set of perfectly reversible isothermal and adiabatic processes, so that a complete cycle is formed out of a very large number of infinitesimal Carnot's cycles (Fig 11). The work done in such a cvcle is a maxi- mum when all the operations are conducted Fltt . u reversibly, and, since all the auxiliary processes are reversible, it follows that the given process must also be conducted reversibly to obtain the maximum work. It follows that all irreversible processes must yield less than the maximum amount of available energy, or that all irreversible processes are attended by dissipation of energy. The amount of energy dissipated in any process is equal to the difference between the maximum available energy for reversible execution and the actual available energy for the specified execution of the process. A distinction must be drawn between available energy unnecessarily dissipitfd into unavailable energy, by reason of some irreversibility inherent to some part of the process, and the necessary balance of unavailable energy left in the refrigerator of a Carnot's engine which is working in a perfectly reversible manner. The preceding considerations are summarised in a very general principle, enunciated by Lord Kelvin in 1852, and called the Principle of Dissipation of Energy: Every irreversible process leads to dissipation of energy. F 2 68 THERMODYNAMICS The amount of energy dissipated may be taken as a measure of the amount of irreversibility inherent in the process. Further, since every natural transformation, or transference, of energy is associated with irreversibility (at least in unorganised nature), it follows that the whole store of energy in that part of the universe where the two laws of thermodynamics hold good I'M toto, is constantly sinking lower and lower in the scale of avail- ability. A little consideration shows that the last stage must be the reduction of the whole of the energy into heat, diffused through bodies at a uniform temperature. Whether this temperature is high or low is immaterial, and all change, i.e., the occurrence of phenomena, must then cease, because such energy is completely unavailable. It is assumed, of course, that all chemical affinities are satisfied, and that all radio-active matter has decayed to its ultimate inactive stage. 1 40. Physical Basis of the Second Law of Thermodynamics. Throughout this book the two fundamental laws of thermo- dynamics are regarded as inductive generalisations from experi- ence. They are verified to a very great degree of probability, no single pertinent case among the many thousands which have been and are being examined has yet been found in contradiction to them. In this sense the laws, like all inductions from experience, are empirical, as distinguished from laws deduced by logical or mathematical reasoning from fundamental hypotheses. These latter are usually statements of the prevailing views on the " structure " of the system concerned, and if the law can be shown to be a consequence of the configuration and motion of the parts of a mechanical system, it is regarded as " explained." Thus Maxwell, when speaking of the deduction of Boyle's law from the mechanical kinetic theory of gases, says : " This is Boyle's law, which is now raised from the rank of an experimental fact to that of a deduction from the kinetic theory of gases." The identifica- tion of the various forms of energy with mechanical energies of hypothetical systems is another example. 1 Arrhenius (Worlds in The Making, 1907) has recently adduced evidence for the view that, although dissipation of energy occurs in planetary masses, there may be restoration owing to processes occurring in the nebulse, and that ' ' the development of the universe moves on in a progressive cycle, in which we can assume neither beginning nor end." THE SECOND LAW OF THERMODYNAMICS 69 The second law as it left the hands of Carnot required no explanation. On the caloric theory then prevalent, it was a necessary consequence of a hydrodynamical analogy the mechanical explanation was in fact, as Carnot' s words show, the source of the principle. When the caloric theory was thrown down, the analogy and explanation fell with it, and the reconstruction of Carnofs principle by Clausius and Kelvin resulted in a law of experience. A new explanation had to be found, and the fertile genius of Bankine (1851) supplied it in a peculiar hypothesis of " molecular vortices." This representation of the structure of material systems being now obsolete, it is clear that what was satisfactorily explained to Bankine would now be incomprehensible. Boltzmann (1866), Clausius (1871), and SzQy (1876), next showed that some special types of dynamical systems, involving so-called "stationary motions," could be regarded as simulating reversible thenno- dynamic systems, provided the heat entering a body was interpreted as the increase of kinetic energy of its particles, and the temperature as the mean kinetic energy. A very exhaustive investigation was carried out by Helmholtz (1884), in which an attempt was made to interpret the second law, as applied to reversible processes, on the basis of the fundamental theorem of dynamics the principle of Least Action. A similar type of investigation is contained in the work of J. J. Thomson : " Applications of Dynamics to Physics and Chemistry," where it is >ho\rn that, with the ordinary kinetic interpretations of thermal magnitudes, the general equation of dynamics may without further assumptions be applied to thermodynamic systems and leads to conclusions in harmony with the results of pure thermodynamics. Both Helmholtz and Thomson adopted Maxwell's view that irrecersibility has its physical explanation in the impossibility of controlling individual molecular motions. Starting from the kinetic interpretation of temperature, and the conception of the distribution of molecular velocities created by Boltzmann and himself, Maxwell showed that the second law is not valid for some particular systems. In these the number of molecules is too small to form what has been called a " physically small " element of a material body, or else the molecular motions are supposed to be directly controllable by an intelligent being of molecular dimensions. The law therefore appears as a statistical and not as a mathematical truth. For if we consider a mass of gas enclosed in a vessel, divided into two parts by a partition which has a number of small doors, and if we station, at each of these doors, a being of dimensions so small that he can deal with the individual molecules, then we can imagine a process which, although not violating the first law, is in absolute contradiction to the second. The temperature of the gas we have to interpret as pro- 7 THERMODYNAMICS portional to the mean square velocity of the molecules, i.e., T a H*. Owing to repeated collisions, the velocity of any Delected mole- cule will fluctuate within limits which, as Maxwell's Distribution Law shows, are all the less probable the further they are removed from the value v^. If we imagine that the beings stationed at the doors called sorting demons by Maxwell allow all rapidly moving molecules to pass to one side of the partition and all slowly moving molecules to pass to the other side, the doors being clapped to when unsuitable molecules approach, the result would be that the gas on the first side becomes hotter, that on the other side colder. From this arrangement of hot and cold reservoirs a finite amount of work could be obtained, say by means of a thermo- couple, and since this would have been produced at the expense of the heat in a body initially at a uniform temperature (i.e., such that a thermometer shows no difference between the temperatures of any two parts) we have in this arrangement a perpetnum mobile of the second class. The physical basis of the second law, which asserts that such an arrangement is in contradiction to experience, is therefore to be found in the exceedingly large number of molecules of exceedingly small size which are present in a body of finite size, and the consequent impossibility of controlling their individual motions by any actual mechanism. It is as though a human being were to attempt to direct the operations, at every moment, of the individual members of a large colony of ants. The production of small temperature differences must, however, occur in every gas. In the rapid motions of small particles floating about in a liquid " Brownian movements " we have an example of motions pro- duced, and maintained, in a medium of uniform temperature. This is probably a case in which the simplicity of the system is, comparatively speaking, too great to allow of the legitimate application of the statistical method, which lies at the basis of the second law. A mean value of the kinetic energy cannot be found. It is also quite an open question whether the second law is applicable to living organisms ; the fineness of the cell-structure, and the comparatively enormous almost microscopically visible molecules of the colloidal substances occurring in the latter, make it not impossible that there are processes going on there which are quite outside the consideration of thermodynamics. THE SECOND LAW OF THERMODYNAMICS 71 We have already shown that the first law cannot be demonstrated in this case on the basis of the impossibility of a perpetuum mobile, 41. Entropy. In a simple Carnot's cycle, in which heat Q A is absorbed from the source at temperature T A , and heat Q B is emitted to the refrigerator at temperature T B , we have : QA - QB _ T A -T B QA T A 'ft-fe = (1) Let us now fix our attention on the working substance, i.e., on the material system undergoing the cyclic process. If Qi, Q 2 are the quantities of heat absorbed by the system from the source and refrigerator respectively : Qi = QA, Q 2 = - QB . (2) and if TI, T 2 are the temperatures of the system when it is absorb- ing the quantities of heat Qi, Q 2 respectively, the condition of reversibility requires that : Ti = T A , T a = T B .... (8) /. substituting from (2) and (3) in (1) we get in which every magnitude refers to the working substance, and not to the heat reservoirs. In the operations constituting a Carnot's cycle, changes of Q and T occur separately. In the majority of cases both these changes occur together, so that the temperature of the working sub- stance may be regarded as a function of the time. Equation (4) therefore requires extension, and this was effected by Lord Kelvin in May, 1854, in the following way : If a material system experiences a continuous action, or a complete cycle of operations, of a perfectly reversible kind, the quantities of heat which it takes in at different temperatures are subject to a homogeneous linear equation, of which the coefficients are the reciprocals of these temperatures. If Q,. be 72 THERMODYNAMICS the heat absorbed at temperature T r , this is expressed by the formula : org = . . (5) IVoo/. Let there be taken, in addition to this given system, a series of (71 2) reversible engines working in the following way: ^ [1] rejects heat Qi at T b and absorbs heat Qi rjl 2 at T 2 ; [2] rejects heat Qi 2 + Q 2 at T 2 , and absorbs heat [3] rejects T 3 - 1 + + Qs at T 8> and absorbs [n - 2] rejects T n _ 2 + 2 + . . + + Q (i _ 2 at T H . a and absorbs T^ + + . . + -=| at T..,. These (w 2) auxiliary engines constitute a material system evolving heat Qi at TI, Q 2 at T 2 , . . Q H . 2 at T B . a , and absorbing heat T n , (^ + ^ 2 + + sM at T n . r This system, taken \ll 1 2 J-n-2' along with the given one, constitutes a complex system causing on the whole neither absorption nor emission of heat at the temperatures TI, T 2 , ... or at any other temperatures than T nJl and T, but giving rise to an absorption or emission at T-i, and to an emission or absorption + Q n at T R . This, having only two temperatures where heat is absorbed, is subject to (4) ; hence : Qn _ i r T /Q! Q 2 Q n ,. 2 \ i p - ~ T~ 7 l,,-i rrr Tlf + + rrr- + V i 1 I' !-! L \ll J 2 A-"/ J THE SECOND LAW OF THERMODYNAMICS 73 This may be considered as a general expression of the Second Law, the First Law taking the form 2A - (Q! + Q 2 + . . Q) = 0, or 2A - 2Q = 0, where 2 A is the sum of the amounts of external work performed, 2Q the sum of the amounts of heat absorbed, in a reversible cycle. If the temperatures of different parts of the working substance alter gradually during the process, the sign of summation must obviously be replaced by the integral sign, or : ? = . . . . (6) in which BQ denotes an element of heat absorbed at tempera- ture T in any reversible cycle, and the integration is extended round the cycle. Equation (6) was obtained in a much less direct manner by Clausius in December, 1854, and is usually known as the Equality of Clausius. It applies only to reversible cycles. If the equality of Clausius is applied to a reversible isothermal cycle (T = constant) we obtain : .'. (J) dQ = 0, which is one form of Moutier's theorem. Now consider any reversible change which is not a cyclic change, as, for example, the expansion of a gas, or the evaporation of a liquid. Theorem. If A and B ,are two different states of a system, then the value of the integral B SQ T is the same for all possible reversible processes in which the state A is converted into the state B. For if AMB, ANB are any two such reversible paths (Fig. 12), these taken together constitute a reversible cycle AMBN, for which 74 THERMODYNAMICS where the suffixes denote the paths of integration. A /B 'Q\ - . */* by the properties of definite integrals (H. M., 103), which establishes the theorem. As a particular case we may instance the Caniot's cycle ABCD, Fig. 8. c Tr along ABC = ^, Q 7p- along ADC = 7^, 'A and these two quantities have been shown to be equal. /2 Thus, for reversible changes, the value of the integral -^ A depends solely on the initial and final states, and is independent of the path, -jf is therefore a perfect differential of some function of the variables defining the state of the system (H. M., 115). Thus, in the case of a fluid passing from the state (p\, Vi) to the state (Pu, ^2), r* I .r\ /(Pl> l 'l) = (#2, ''2) (01, ^l) = ^ (fla, P'i) ty (Oi, pi). This integral may therefore be regarded as measuring the change of some magnitude which depends, like the intrinsic energy, entirely on the actual state of a material system, and is independent of the previous history of the system. If S A , S B THE SECOND LAW OF THERMODYNAMICS 75 are the values of this magnitude for the states A and B, we may write : ^ = S B -S A . . . . (2) A the integral referring to reversible changes. S is called the entropy. Definition. If an element of heat, SQ, is added to a system ly a reversible process, at a mean temperature T, then : &Q 7 Q /Q\ Pp = ab . . . . . (o) is called the increase of the entropy of the system. If two states of a system, A and B, can be connected by any I /"v reversible path of change, the integral ^ taken along this f path measures the difference of the entropies of the system in the two states, or : ... (4) A JA It follows that 1/T is the integrating factor of SQ. Now since SQ is a function of two variables (in the simple case of a homo- geneous fluid), and since the integrating factor of such a magni- tude is usually also a function of the same two variables, we must regard the proposition that the integrating factor of SQ is a function of one variable only as expressing a physical, not a mathematical, truth. Corollary. In all reversible adiabatic changes the entropy remains constant; such changes are therefore isentropic changes. It must be emphasised that this holds good only for reversible changes. To give three instances of increase of entropy in adiabatic irreversible changes we may cite : (1) The production of heat in the system itself when a mass of viscous fluid is set in motion by stirring, and then allowed to come to rest by friction in a vessel impervious to heat. (2) The mixing, by diffusion, of two gases or liquids in an adiabatic enclosure, in w r hich case there is no absorption or production of heat, but, nevertheless, an increase of entropy. (3) A chemical reaction in an adiabatic enclosure. These cases will be taken up later. 76 THERMODYNAMICS 42. Specification of Entropy. Just as the intrinsic energy of a body is defined only up to an arbitrary constant, so also the entropy of the body cannot, from the considerations of pure thermodynamics, be specified in abso- lute amount. We therefore select any convenient arbitrary stan- dard state a, in which the entropy is taken as zero, and estimate the entropy in another state /3 as follows : The change of entropy being the same along all reversible paths linking the states a and y8, and equal to the difference of the entropies of the two states, we may imagine the process conducted in the following two steps : (i.) Take the body from a to y along the adiabatic containing a, where y is the intersection of this with the isotherm containing /3. The change of entropy is zero. (ii.) Take the body from y to /3 along the isotherm. If Q^ is the heat absorbed and T^ the constant temperature, the entropy in the state 3 is ^. This divided by the mass of the body is the entropy per unit mass, which we shall call the specific entropy. If there are a number of separate masses, mi, m 2 , . . ?n if the total entropy of the system is equal to the sum of the entropies of the i separate masses plus the entropy S, M of any medium in which they are contained : If the masses are homogeneous, and * b * a , . . * f are their specific entropies : Si i I O Wll^l -j- 7??2^2 i '"j^i T O . Any entropy which the system may possess in virtue of the mutual actions of the masses is taken as included in S m . A similar expression holds for the intrinsic energy : U = niiii i -j- w 2 2 -f- . . -j- m-iUi -\- U m . 43. The Entropy-Temperature Diagram. If we take rectangular axes and put x ~ S, y= T, the Carnot's cycle will be represented by a rectangle ABCD, consisting of two isotherms BC, DA and two isentropics (adiabatics) AB CD (Fig. 13). THE SECOND LAW OF THERMODYNAMICS 77 The heat absorbed in the cycle = the heat absorbed along BC heat rejected along DA = (T a -TO(S8-SO = area of cycle. The area is positive if traced out clockwise. Since the heat absorbed in the cycle is equal to the work done, the areas of the Carnot's cycle on the (p, v) and (S, T) diagrams are equal. This may be generalised to apply to any reversible cycle where the only external work is done by expansion. For a small reversible change : dU = 6Q - 8A = TtfS - FIG. 13. Corollary 1. The area included in the pv plane between any two adiabatics (Si, 82) and any two isotherms (Ti, T 2 ) is equal to (T 2 TI) (S 2 Si), and this represents the heat absorbed in the cycle. Corollary 2. If we could draw on the j>r plane the isothermal line of absolute zero (T = 0) the area included between it, any two adiabatics, and an isotherm T would represent the heat absorbed in passing along the upper isotherm from one adiabatic to the other. Corollary 3. If any path of reversible change is drawn on the pi- plane between two adiabatics, the area between it and the absolute zero isotherm represents the heat absorbed in the change (Zeuner). 44. Entropy and Unavailable Energy. If a quantity of heat Qi is taken from a body at temperature the maximum amount of work obtainable from it is : - Q 2 = Ql l - 78 THERMODYNAMICS whilst the balance : is wholly unavailable energy given up as heat to the refrigerator at the lowest temperature T 2 . This follows from the result established for the Carnot's cycle : QL_O? T! ~ T 2 ' and the proposition that it is impossible to get more work from a given quantity of heat than we can get from it in a Carnot's cycle. The maximum amount of work obtainable from a given quantity of heat, called its motivity by Lord Kelvin (1852), is thus always less than the mechanical equivalent of the quantity of heat, except in the limiting case when the refrigerator is at absolute zero (T 2 = 0). It cannot be specified in terms of the condition of the body from which the heat is taken, or into which the heat passes, but requires in addition a knowledge of the lowest available temperature, T 2 . For if we had another body at temperature T , where T < T 2 , which could be used as a refrigerator, the amount of work : ^-\ r\ /^ 2 T ^- T 2 J -~~ Ql ITT - Ti could be obtained further from Q 2 , and the unavailable energy : Qo = Q 2 X - = Qi X . . . (4) la li would go to the refrigerator, the final result being the same as if the cycle had been performed directly between the temperatures TI and T . Equations (2) and (4) show that the unavailable part of Q is directly proportional to the lowest available temperature and inversely proportional to the temperature of the body from which Q is taken. Again, since Qo _ Qi _ Qi To - T 2 ~ IV we see that Q/T is independent of the lowest available tempera- ture, and from (2) and (4) that Q/T only requires to be multiplied by the lowest available temperature to give the unavailable part, T Q X r -p of the heat taken from a body at temperature T. But Q/T is the entropy of the quantity of heat Q at T ; hence : THE SECOND LAW OF THERMODYNAMICS 79 unavailable energy = entropy X lowest available temperature, which is an alternative definition of entropy. 45. Inequality of Clausius. If a body absorbs an amount of heat Q from a reservoir at temperature T, and at the same time does work A, / T \ the gain of available energy = Q M ^J , the loss of available energy = A, To being the lowest available temperature, . ' . the nett gain of available energy (*) is = AU - Q, this being the amount by which the capacity of the body for doing work is increased. We have assumed that the temperatures remain constant during the transference of a finite amount of heat Q, which implies that the heat reservoirs have very large heat capacities. To remove this restriction, we suppose that the amount of heat absorbed is infinitesimal, 6Q. Then, for the gain of available energy we have : = rfU - 26Q TJ?, where -5Q, 2SA are the sums of the amounts of heat absorbed from, and external work done upon, all the bodies outside the system, and rfU = 25Q - 2SA is the increase of intrinsic energy of the system. The increase of available energy in a finite change is therefore : = I T - 2 j 8Q ^ r r TS f 28Q = 1-2 Ul J-O^ I -jjf J\ . (1) 80 THERMODYNAMICS If the changes constitute a closed cycle : U 2 - Ui = >'$ (2) Now if any irreversible changes occur in the system itself during the execution of the cycle, the principle of dissipation of energy shows that the available energy will be diminished in virtue of these, and since the available energy of the system must be the same after as it was before the execution of the cycle, because the state of the system is unaltered, it follows that some available energy must have been absorbed from outside in con- nection with the absorption of heat; hence (A*) and therefore also r $\r\ r ^o T 2( )-^must be positive, and hence (])-TJT must be negative, or v/MQ^n P ~^ " ' w) This is called the Inequality of Clausius, who, however, estab- lished it in a different way. If the cycle is reversible, there is no dissipation of energy, and . (Sa) which is the result established in 41. The integral of (3) must be interpreted as follows : T refers to the temperature of the body from which the element of heat 8Q is taken, and the integral sums up all the quantities SQ/T for that body. The symbol 2 further extends this to all the external bodies con- cerned. Thence the sum of all the magnitudes SQ/T is negative. Now SQ/T represents the entropy lost by the external body during the small change, because SQ, being the heat absorbed by the system, will be heat lost by the external body, and the relations (3) and (3a) may therefore be expressed in words as follows : If any cyclic process is performed with a given material system, the entropy of all the surrounding bodies which have in any way been involved in the process, either as emitters or absorbers of heat, either remains unchanged, if the cycle is reversible, or else increases, if the cycle is performed irreversibly. Now let us consider any process which is not a cyclic process, and in which the system is taken from an initial state [1] to a final state [2]. We shall prove that if Si, S 2 are the entropies of THE SECOND LAW OF THERMODYNAMICS 81 the system in its initial and final states, and -^ has the signifi- cance of the preceding paragraph, then /* 2 5 ^0 . . . . . (9) so that in irreversible processes which are adiabatic the entropy increases whilst the energy may either increase or diminish, according to the sign of 28 i. This explains the statement of 41 that an adiabatic change is necessarily an isentropic change only when it is reversible, for then the lower sign is taken in (9) and (8S) Q =0. Further, let us suppose the system completely isolated, by enclosing it in a vessel with perfectly rigid walls, which are THE SECOND LAW OF THERMODYNAMICS 83 perfect non-conductors of heat. Then 28A = 2SQ = dU =0, so that or (SS^ >0 .... (10) In this case there is an increase of entropy in an irreversible process, whilst the energy remains constant. This result brings out clearly the independence of the two fundamental principles of thermodynamics, the first law dealing with the energy of a system of bodies, and the second law with the entropy. 46. The Aphorism of Clausius. If the system is not isolated, its entropy may either increase or decrease. Thus, if a mass of gas is compressed in a cylinder impervious to heat, its entropy increases, but if heat is allowed to pass out into a medium, the entropy of the gas may decrease. By including the gas and medium in a larger isolated system, we can apply (10) of 45, and hence show that the medium gains more entropy than the gas loses. An extended assimilation of this kind shows that, if every body affected in a change is taken into account, the entropy of the whole must increase by reason of irreversible changes occurring in it. This is evidently what Clausius (1854) had in mind in the formulation of his famous aphorism : " The entropy of the universe strives towards a maximum." The word " universe " is to be understood in the sense of an ultimately isolated system. 47. Compensating Changes. In a Carnot's cycle, the entropy Qi/Ti is taken from the hot reservoir, and the entropy Q 2 /T 2 is given up to the cold reservoir, and no other entropy change occurs anywhere else. Since these two quantities of entropy are equal and opposite, the entropy change in the hot reservoir is exactly balanced, or, to use an expression of Clausius, is compensated by an equivalent change in the cold reservoir. Again, in any reversible cycle there is on the whole no production of entropy so that all the changes are compensated. If now we have any reversible change which is not a cycle, there will be a change of entropy in the system, but this will have a compensating change outside the system. For suppose 84 THERMODYNAMICS the entropy of the system decreases, then the entropy of the external bodies might increase, decrease, or remain constant. It cannot, however, either decrease or remain constant, for then we should have the entropy of the whole system decreasing, which is impossible. It must therefore increase, and this increase must be exactly equal to the decrease in the given system ; for in virtue of the assumed reversibility of the change, we could reverse the sign of every entropy change by taking the system back to its initial state, and there would be a decrease of entropy if the external increase in the first case had been greater than the decrease in the system. The change is therefore compen- sated. Again, if we had supposed the entropy of the system to increase, we should simply have to reverse all the processes to arrive at the first case, and hence the process is compensated. Thus every reversible process admits of a compensating process. But if the given process is conducted irreversibly, we have proved that there is always more entropy generated in the system than is Lost by bodies outside the system, and the excess is called the non-compensated entropy. It may happen, and frequently does, that the entropy of the system itself decreases in a par- ticular change, but we have proved that there is an increase outside the system which is greater than the decrease in the system, or at best equal to it in the case of reversible changes. The application of the principle of entropy to irreversible processes has given rise to much discussion and controversy. The exposition here adopted is based on the investigations of Lord Kelvin (1852) in connexion with Dissipation of Energy. 48. Examples of Irreversible Changes. We shall conclude this chapter by considering a few typical cases of systems undergoing changes attended by intrinsic or conditional irreversibility. (1) Conduction of Heat. Let the quantity of the heat 8Q pass by conduction from a body at temperature T x to a body at tem- perature T2. Suppose that an auxiliary medium at temperature TO is available. m rp [ . The motivity of SQ at 1\ is SQ . ^ , and its motivity at m $ n Ta TO 1 2 is 6(J . pp . -La . - THE SECOND LAW OF THERMODYNAMICS 85 The loss of motivity is therefore Tl - T Ta ~ T 1 - T SO ( l M -~ -- - : T 8Q - Since this is positive, by the principle of dissipation of energy, 1/T, > 1/Ti .'. T! > T fc so that heat passes by conduction from a hotter to a colder body. The loss of entropy of the hotter body is 7^, and the gain of entropy of the colder body is ~, '2 .'.the increase of entropy of the whole system consequent upon the occurrence of the irreversible change is From what precedes we see that : (1) The gain of entropy is positive. (2) The gain of entropy is equal to the dissipated energy pro- duced (or the available energy lost) divided by the temperature of the auxiliary medium. It is easy to generalise this result for all processes. Loss of motivity (dissipation of energy) is therefore accom- panied by increase of entropy, but the two changes are not wholly co-extensive, because the former is less the lower the tempera- ture TO of the auxiliary medium, whilst the latter is independent of TO, and depends only on the temperature of the parts of the system. If T = 0, i.e., the temperature of the surroundings is absolute zero, there is no loss of motivity, whilst the entropy goes on increasing without limit as the heat is gradually conducted to colder bodies. Similar considerations apply to passage of heat from one body to another by radiation. In this case the energy, in its transition from one body to the other, exists as radiant energy in the ether. We have therefore to suppose, when the energy leaves the hot body and so reduces its entropy, that it must carry entropy into the ether. (2) Expansion of a Gas into a Vacuum. If a gas is allowed to rush into a vacuous space, or into a space containing a gas under a less pressure, we have an example of a process attended by conditional irreversibility. Let a volume i\ of an ideal gas be put into communication with 86 THERMODYNAMICS a vacuous vessel of volume r 2 . It rushes into the latter, and occupies a volume ri -f- r 2 . No work has been done, hence the energy dissipated is equal to the work which could have been obtained had the expansion been performed isothermally and reversibly. The gas may be restored to its initial state by compressing, and removing the heat generated by conducting it away to the medium. Since it is an experimental fact that no heat is emitted or absorbed in the process of free expansion, this heat is the exact equivalent of the work spent in the reversible compression, since no energy change occurred during the previous expansion. There is no change of quantity of the energy in allowing a gas to expand irreversibly, and then bringing it back reversibly to its initial state. There is, however, a change in the quality of the energy, because from a quantity of useful work we obtain an equivalent of useless heat in a reservoir at a uniform tempera- ture in other words, there has been a dissipation of energy. We could imagine the process reversed without dissipation if we supposed the expanded gas enclosed in a cylinder closed by a piston which is pushed in bit by bit by an army of Maxwell's demons, each element of the piston being advanced when free from, and kept rigid when exposed to, molecular bombardment. Thus the availability lost in the expansion could be restored without any compensating dissipation. Such a restoration would naturally contradict both the principle of dissipation of energy and the principle of increasing entropy, and their basis the Second Law of Thermodynamics. This violation is, however, purely imaginary, because Maxwell's demons do not exist. (3) Mia-iny of Gases by Dl/mion. Exactly similar considera- tions apply to the spontaneous intermingling of two gases by diffu- sion, the increase of entropy being calculable from the isothermal absorption of heat when the process is carried out reversibly by means of semi-permeable septa, as described in 123. The pro- cess could be reversed in . imagination, and the lost availability restored, by demons which would allow molecule s of one gas to pass in one direction, those of the other gas in the opposite direction, across a partition dividing the mixture into two parts equal respectively to the initial volumes of the unmixed gases, but, of course, such a process is physically impossible. (4) Collision. If a mass moving in any direction is suddenly THE SECOND LAW OF THERMODYNAMICS 87 arrested in its course by striking against a non-conducting wall, the kinetic energy is converted into heat in the body, and this implies a corresponding increase of entropy o! the latter. A similar result follows if two or more imperfectly elastic spheres come into collision, and if we are given a system of such spheres in motion, they must ultimately, by mutual collision, be brought to rest, their kinetic energy being converted into heat. (5) Viscosity. If a mass of viscous liquid is set in rotation in a rough vessel, the kinetic energy is gradually dissipated by friction, and the liquid comes to rest in a slightly warmed con- dition. The increase of entropy is in this case due to heat generated in the system itself. Among the causes producing irreversibility we may instance the forces depending on friction in solids, viscosity of liquids ; imperfect elasticity of solids ; inequalities of temperature (leading to heat conduction) set up by stresses in solids and fluids ; generation of heat by electric currents ; diffusion ; chemical and radio-active changes ; and absorption of radiant energy. The presence of any type of irreversibility inevitably leads to dissipation of energy, and therefore to increase of entropy. The physical (or, rather, the mechanical) interpretation of entropy is identical with the problem of the interpretation of the Second Law of Thermodynamics, and the attempts at its solution by Boltzmann, Clausius, etc., have already been referred to. In this connexion the treatment of irreversible processes offers con- siderable difficulty. The first step in this direction was Lord Kelvin's " Kinetic Theory of Dissipation of Energy" (1874), in which the relation between irreversibility and the technical impossibility of getting to grips with the individual molecules and controlling their motions was explained. In this we find the first application of the theory of probabilities to problems of the kind contemplated. As Kelvin observes : " If, then, the motion of every particle of matter in the universe were precisely reversed at any instant, the course of nature would be simply reversed for ever after. The bursting bubble of foam at the foot of a waterfall would reunite and descend into the water ; the thermal motions would reconcentrate their energy, and throw the mass up the fall in drops re-forming into a close column of ascending water. Heat which had been generated by the friction of solids and dissipated by conduction, and radiation with 88 THERMODYNAMICS absorption, would come again to the place of contact, and throw the moving body back against the force to which it had previously yielded. Boulders would recover from the mud the materials required to rebuild them into their previous jagged forms, and would become reunited to the mountain peak from which they had formerly broken away. And if also the materialistic hypo- thesis of life were true, living creatures would grow backwards, with conscious knowledge of the future, but no memory of the past, and would become again unborn. But the real phenomena of life infinitely transcend human science; and speculation regarding consequences of their imagined reversal is utterly unprofitable." If every natural process could be represented by a dynamical equation, the substitution of t for t in the equation (t being time) would lead to an equation describing the exactly reversed process. If we could represent the actual process on a cinemato- graph film, the reversed process would be seen when the film was put backwards through the machine, and events like those just described would unfold themselves to our view. That such phenomena do not appear in nature is a consequence of the irreversibility of every process. A great advance was made in the direction of the physical inter- pretation of entropy, and in the systernatisation of irreversible pro- cesses, when L. Boltzmann (1877) showed that the definition of the entropy could be regarded as a problem in the theory of proba- bilities (" Ueber die Beziehung zwischen dem zweiten Hauptsatz der mechanischen Warmetheorie und der Wahrscheinlich- keitsrechnung respektive den Sa'tzen iiber das Warmegleich- gewicht," Wiss. AbltL, II., 164). The second law, and the pheno- mena of irreversibility, depend not on any peculiarity of the motions constituting heat themselves, but on the statistical pro- perties of systems with an enormous number of degrees of free- dom, which are realised in bodies composed of an exceedingly large number of atoms. Whereas to an army of Maxwell's demons the " state " of a gas would be completely defined by the aggregate of the position and velocity co-ordinates of each separate mole- cule, i.e., by 6 variables, when there are molecules, and every change of state, being a change in a dynamical system, would be completely reversible, yet to an ordinary observer the state would be defined by two variables, the temperature and density. Both THE SECOND LAW OF THERMODYNAMICS 89 of these, however, have only a statistical significance ( 20). The conception of entropy as measuring the probability of a state has been extended by Planck to radiation, and no doubt will lie at the basis of future developments of thermodynamics. (Cf. Planck, Theorie dcr Warmestrahlung, Leipzig, 1906, pp. 129 ct seq.) CHAPTER IV THE THERMODYNAMIC FUNCTIONS AND EQUILIBRIUM 49. Equilibrium and Stability. A system is in equilibrium when the position of each of its parts, and the state of that part, remain unchanged with lapse of time. If any one of the conditions maintaining the system in its equilibrium state (r.(j. temperature or pressure) is changed, the state of the system is also changed and the equilibrium is dis- placed. If the change of external conditions is very small, the displacement of the equilibrium may also be correspondingly small, and may be reversed when the displacing change is reversed. The state of equilibrium is then called a state of true equilibrium. A dynamical illustration is afforded by a spring extended by a weight. If the weight is changed by an infinitesi- mal amount, there is a correspondingly small alteration of the length of the spring, and if the original weight is restored, so also is the original length. A physical illustration is afforded by a mixture of water and steam in a cylinder at 100 C. under an external pressure of 1 atm., and subjected to small evaporative or condensative changes consequent on small changes of tem- perature and pressure. If the spring and weight had been immersed in oil or treacle, a viscous reaction would have been opposed to the motion, but since this is proportional to the velocity, it can be made as small as we please by executing the process very slowly and vanishes in the limit. Viscous reactions do not, therefore, prevent a system from existing in a state of true equilibrium provided all changes are made infinitely slowly ; they merely retard change, and the retardation vanishes in the limit. A state of equilibrium which does not satisfy the conditions for true equilibrium is called a state of false equilibrium. A system may remain in a given state for a long period of time, and thus appear to be in an equilibrium state. A small change THEEMODYNAMIC FUNCTIONS AND EQUILIBRIUM 91 in the external conditions produces, however, either no change at all, or else a very large change which is not reversed when the external conditions are restored to their original values. The system was then in a state of false equilibrium. Dynamical illustrations are afforded by a weight maintained by friction on a rough inclined plane, which may be tilted through a small angle without producing any effect, and by a weight hanging from a wire infinitely near its breaking point, when a small additional weight, instead of producing a small reversible elonga- tion, snaps the wire. A physical illustration is afforded by a superheated liquid, which boils explosively on a slight elevation of temperature. According to Gibbs, systems in states of false equilibrium are maintained by forces analogous to friction, called l>a$sii-e resistances}, and are to be distinguished from states of true equilibrium, in which the active forces are so balanced that the slightest change of force will produce motion in either direction, as in a frictionless machine. The distinction is to be recognised by appeal to experience, and it is only in cases where the limits of operation of the passive resistances are closely approached that there will be any difficulty in recognising to which type a given equilibrium state belongs. States of equilibrium may also be classified into states of stable, unstable and neutral equilibrium, according as the system tends to return to its initial state, or to move further away from this state, or simply to remain in the altered state, when the displacing force is removed. Dynamical illustrations are afforded by a sphere resting at the bottom of a bowl, on the top of the inverted bowl, and on a smooth table respectively. A consideration of the same example also illustrates the result established in treatises on dynamics that the condition for stable, unstable, or neutral equilibrium of a mechanical system is that, for any small displacement which does not violate the constraints, the change of potential energy shall vanish to the first order, and be positive, negative, or zero respectively to the second order. When the system is in stable, unstable, or neutral equilibrium, the potential energy is a minimum, a maximum, or stationary respectively (Theorem of Dirichlet). Thus the work done by the system in any infinitesimal displacement is zero to the first order, and negative, positive, or zero to the second order, for the three cases. All these conditions refer only to a par- 92 THERMODYNAMICS ticular state, since the displacements are infinitesimal, and it does not follow that any maximum or minimum value is the greatest or least value of the potential energy respectively but only greater or less than all other values in the immediate neighbourhood (cf. H. M., 22). Physical examples of the three types are afforded by a gas contained in a cylinder under an external pressure equal to the gas pressure, by a superheated liquid, and by a mixture of water and saturated steam, under the same conditions respectively. 50. Conditions for Equilibrium, and for Stability of Equi- librium. In the investigation of the equilibrium states of therrnodynarnic systems there are two points of departure, which are really more or less equivalent. The first is Lord Kelvin's principle of Dissipation of Energy (1852) which was generalised and applied to chemical reactions by Lord Rayleigh (1875). According to this, any change (and in particular a chemical reaction) is impossible if it leads to the reverse of dissipation of energy, or, as we may call it, to motivation of energy. In the application of this criterion we have to determine how the available energy of the system depends on the variables defining its state, and we can then find whether any imaginary change in the state of the system which does not violate the given conditions {e.g., of constant volume, or constant temperature, or constant pressure, etc.) leads to an increase, or to a decrease, of the available energy. If to the former, the imaginary change (which we shall call a virtual change) cannot be realised, and we can certainly infer that the system is in equilibrium ; but if to the latter, we cannot say whether the system will, or will not, undergo the change leading to dissipa- tion of energy, because it may be in a state of false equilibrium. Thus we should find that gunpowder ought not to exist at all under ordinary conditions, because the explosion of gunpowder leads to dissipation of energy ; the fact that gunpowder and such materials do exist is a consequence of the existence of states of false equilibrium. The same holds good with respect to hundreds of organic compounds (e.a., diazo-compounds). The second general principle is based on the properties of the entropy function, and is contained in the aphorism of Clausius THERMODYNAMIC FUNCTIONS AND EQUILIBRIUM 93 (1865) that "the entropy of the universe tends to a maximum." This is closely connected with the principle of dissipation of energy, but, as we saw in 48, is not wholly co-extensive with it. By " universe " we are to understand an isolated system, and Clausius's theorem is more sharply expressed in the statement ( 46) that, in all real changes occurring in such a system, the entropy can only increase. If the system has reached such a state that any virtual change, which does not violate the con- dition of constant energy, leads to a decrease of entropy, the system is certainly in equilibrium, for then no change is possible. This method was first applied to chemical problems by A. Horstmann (1873). 51. Gibbs's Two Criteria of Equilibrium. (1) For the equilibrium of an isolated system it is necessary and sufficient that in all possible variations of the state of the system which do not alter its energy the variation of its entropy shall either vanish or be negative. For in all actual changes of such a system the entropy can only increase, so that if we consider a virtual change, and put (5S)u for the resulting change of entropy, then : if (6S)u > the change is possible and irreversible, if (S)u < the change is impossible, whilst if (SS)(j = the change is reversible. The condition for equilibrium is, therefore, (8S) F <0 . . (1) for the change represented by the inequality is impossible, whilst that represented by the equality is reversible, and reversible changes can only occur, as a limiting case, when the system is in equilibrium. The equilibrium is evidently stable when the entropy is a maximum, for then every possible change would diminish the entropy. The equilibrium will be unstable when the entropy is a minimum for a given value of the energy. This implies that if there are several conceivable neighbouring states with the same energy, that with the least entropy will correspond with a state of unstable equilibrium, whilst the others with more entropy will be essentially unstable states, except the one with the greatest amount of entropy, which will be the state of stable 94 THERMODYNAMICS equilibrium. Thus, if water vapour is cooled without admission of liquid or nuclei on which liquid can condense, there is formed homogeneous vapour which at a given temperature exists under a pressure greater than the pressure of saturated vapour. If this is now isolated we have a state of unstable equilibrium. The entropy in this state is less than that in any of the hetero- geneous states produced by separation of liquid, and all these (except the one in which liquid is in contact with saturated vapour, which has the maximum entropy and is the stable state) are essentially unstable states. The validity of the condition for unstable equilibrium may be rendered apparent as follows. Suppose that, besides the state of unstable equilibrium a, there could be some other state ,8, which had less entropy than a, for the same energy. Then we could arrive at a by starting with /3, but the system would not remain in the state a, because the latter is not a state of maximum entropy, and all the changes which carry the system further from that state are possible. Hence /3 cannot have less entropy than a, for a has been assumed to be an equilibrium state, and so a is a state of minimum entropy, with respect to states in its immediate neighbourhood. Again, if all the states in the immediate vicinity of the equi- librium state have the same entropy as the latter, the equilibrium is neutral, since there is no reversible direction in which the entropy can increase, and hence no tendency to pass from any one of these states to any other. The condition that the equilibrium shall be stable, unstable, or neutral is that the entropy shall be a maximum, a minimum or stationary respectively : if (& 2 S)u < the equilibrium is stable, if (S 2 S)u > the equilibrium is unstable, whilst if (8 2 S)u = the equilibrium is neutral. The values of 8 2 S are to be calculated in the usual way by Taylor's theorem. (2) For the equilibrium of any isolated system it is necessary and sufficient that in all possible variations of the state of the system which do not alter its entropy, the variation of its energy shall either vanish or be positive : (aU) s >0 .". . . :. , . (2) Criterion (2) is the antithesis of criterion (1), and the validity of the one implies that of the other. For it is always possible to THEPvMODYNAMIC FUNCTIONS AND EQUILTBEIUM 95 increase or decrease the entropy and energy of a system together by the addition or abstraction of heat. Then if criterion (1) is not satisfied, there will be some variation for which : 5S > and 8U = ; hence, by diminishing both the entropy and the energy of the system in its altered state, we shall obtain a state for which, considered as a variation of the initial state : SS = and 8U < 0, which does not satisfy criterion (2). Conversely, if (2) is not satisfied, there must be some variation for which : SU < and SS - 0, which does not satisfy criterion (1). Further, the equilibrium will be stable, unstable, or neutral, according as the energy is a minimum, a maximum, or stationary respectively, that is : if (e> 2 U) s > the equilibrium is stable, if (8 2 U) S is the uncompensated increase of entropy of the system ( 47). //' the change is isothermal, T = constant, and '.-. Q T = T(Sa-Si)-T . . . (8) The magnitude on the left is the heat absorbed in the isothermal change, and of the two expressions on the right the first is dependent only on the initial and final states, and may be called the compensated heat, whilst the second depends on the path, is always negative, except in the limiting case of reversibility, and may be called the uncompensated heat From (3) we can derive the necessary and sufficient condition of equilibrium in a system at constant temperature. Then, either no change at all can occur, or all possible changes are reversible. Hence, if we imagine any isothermal change in the state of the system, and calculate the value of Too for that change, this value will be positive or zero if the former state is an equilibrium state. Now Q T = U 2 Ui + A T . . . (4) /. (U a - Ui) - T (S a - SO + A T = - To> . . (5) Since the condition for the equilibrium state involves neither U nor S separately, but only the magnitude (U TS), we may put: U TS = * . . . . (6) where * is a continuous and uniform function of the state of the system, and was called by Helmholtz the Free Energy. That * has the dimensions of energy is evident from (5), and the appropriateness of the epithet " free " will appear immediately. Then (5) can be written : *a *i + AT = To> . . . (7) since * x = (Ui - TSi), and * 2 = (U 2 TS 2 ). U and S contain arbitrary terms, say a and /3, depending on the choice of the initial states of zero energy and entropy respectively ; hence * will contain an arbitrary linear function of THERMODYNAMIC FUNCTIONS AND EQUILIBRIUM 97 temperature, a /3T, which does not, however, enter (5) or (7), these involving only differences of 4>. Since o> is essentially positive for all real changes, (7) shows that in all real isothermal changes the magnitude % ^i + A T must diminish, and that in any small virtual change : (d * + 8A)r = Trfo> . . . (8), so that if the system is imagined to undergo such a change and if (dV + 8A)r < the change is possible and irreversible, (9) if (d* + SA)r > the change is impossible, whilst if (rf*+ SA) T =: the change is reversible. The criterion of equilibrium of a system maintained at constant temperature is therefore : (d* + 6A>r > . . . . (10) for all virtual isothermal changes. If 8A T >0, i.e., work is done by the system, then d* < in all real changes ; if 8A T < 0, i.e., work is spent upon the system, then d* is either > or < ; whilst if 8A T = 0, i.e., the system is mechanically isolated and the virtual change is adynainic as well as isothermal, then d* < in all real changes and <7* = in the limiting case of reversible changes. The necessary and sufficient criterion of equilibrium in a mechanically isolated system at a given temperature is : W T ,,>0 .... (11) for all virtual isothermal adynamic changes. The suffix x indicates that besides T, all the variables ?\, r*, . . . during the change of which external work is done, are maintained constant (adynamic condition). Thus, if the only external force is a normal and uniform pressure p, then x = v, the volume of the system, and (11) is the condition of equilibrium at constant temperature and volume. The equilibrium is stable, unstable, or neutral, according as * is a minimum, a maximum, or stationary respectively ; hence : if (o 2 *)r,;r > the equilibrium is stable } if (o 2 *)^ < the equilibrium is unstable r . (12) whilst if (oH)^ = the equilibrium is neutral ; The equation for reversible isothermal changes : (d* + 2oA)r = shows that d* T = 28A T .... (13), so that in such changes the external work is done wholly at the cost of the free energy of the system. The appropriateness of the 98 THERMODYNAMICS latter name is thus apparent ; * represents that part of the intrinsic energy which is freely available for conversion into exter- nal work in isothermal processes ; it is the available energy at con- stant temperature, and corresponds exactly with the potential energy of mechanical systems. We may therefore speak of a system as possessing a charge of free energy, readily capable of being realised as useful external work in isothermal changes, just as a bent bow possesses a store of potential energy realisable as the kinetic energy of a discharged arrow. The free energy is assigned to a system on the condition that the changes in which it is realised are isothermal and reversible. We can of course say that in irreversible changes that part of the free energy which is not rendered available has been lost by dissipation, e.g., con- verted into heat ; and equations (9) show that in irreversible changes d^> T > 25A T , i.e., the external work is less than the diminution of free energy. Equation (13) gives on integration : Now (6) can also be written in the form : U = * + TS . . . . (14), so that in the sense explained we may regard the total intrinsic energy as made up of two parts : (i.) The free energy SP, which is available energy in reversible isothermal changes ; (ii.) The part TS, which is unavailable energy in reversible isothermal changes, and was called by Helmholtz the Bound Energy B, of the system. The equation (Ua - Ui) = (* a - *0 + (Ba - BI) ) or (Ua - Ui) = (* a - *i) + T(S 2 - Si)' shows that the total loss of intrinsic energy in any isothermal reversible change in the system is the sum of the changes of the freely available part, or * x * 2 , and of the unavailable part, viz., BI B 2 , which latter is, as we see from the relation, B = TS, or Ba-Bi = T(Sa-Si) . . . (16), given off as heat to the constant temperature medium surround- ing the system. (* : - * 2 ) and (Bi - B 2 ) are called by Haber (Thermodynamics of Technical Gas Reactions) the reaction energy and latent heat respectively, when they refer to chemical changes. THERMODYNAMIC FUNCTIONS AND EQUILIBRIUM 99 The equation (16) shows that the increase of hound energy in a reversihle isothermal change is equal to the increase of entropy multiplied hy the absolute temperature, so that the entropy may be regarded as the capacity for bound energy in such changes. B will evidently contain the arbitrary term /3T. As soon as we had shown that is an available energy, from the defini- tion of 39 and equation (13a), we could at once have inferred the relations (10) (12) from the principle of dissipation of energy, for must be a mini- mum in stable equilibrium. In the investigation of the properties of the free energy ^ no assumption has been made as to the nature of the external work A T . Let us now assume that there is some function > of the variables defining the physical and chemical state of the system, such that : > : -il 2 = A T , so that the external forces exerted by the system on exterior bodies have & force-potential Q. (cf. H. M., 114). Thus (7) becomes : (*a - *i) + (Si - 2) = - To, . . (18), or if we put * = . . . . (19), we shall have : 4> 2 *! = Tco . . . . (20), so that 4> can only diminish in real isothermal changes. 4> is called by Duhem (Traite de Mecanique chimique, I., 90) " the thermodynamic potential for a given force-function," or " the total thermodynamic potential " (in contrast to *, which he calls " the internal thermodynamie potential "). A very important case is that in which the sole external force is a uniform normal and constant pressure j>. Then A T = - p(r 2 i'i) .-. Q. = - pr . . . . (21) and 4> P , T = * + pr = U TS + pr . . (22 ) We shall often denote ^,. T simply by <, so that : = U - TS + pr = * + pv . . . (23) and call < the thermodynamic potential for constant pressure, or simply the potential, of the system. Thus 4> 2 <#>!= Tco . . . . (24), so that in any real isothermal and isopiestic change can only diminish. From this and (20) we can deduce the criteria for equilibrium, H 2 100 THERMODYNAMICS and for stability of equilibrium, of a system maintained at con- stant temperature and constant external forces (X), in particular at constant pressure (p). For, if the system undergoes a virtual isothermal change with constant external forces, and (8) T ,x is the change of *, /'then if (8*)r,x < the change is possible and irreversible, if (8*)r, x > the change is impossible, (whilst if (84>) T ,x = the change is reversible. Hence, for equilibrium in a system maintained at constant temperature and with constant external forces, it is necessary and sufficient that for all virtual changes : (S4>) T ,x>0 .... (25) The equilibrium is : stable if (S 2 d>) T) x > 0, unstable if (8 2 4>) T ' )X < 0, . . . (26) neutral if (S 2 ) Tl x = 0. We observe that these criteria could at once have been deduced from the principle of dissipation of energy after we had established that * is an available energy, from (13), (17), and (19). Similarly for the conditions referred to $, which follow. If the only external force is a normal, uniform, and constant pressure p, the necessary and sufficient condition for equilibrium is that for all virtual isothermal-isopiestic changes : W T)3) >0 .... (27) whilst the equilibrium is stable, unstable, or neutral, according as : (S 2 ) /()T > 0, (S 2 another for which 5$. = 0, and yet another for which 8< < 0, then if the system is given an " initial impulse " it will pass along the series of changes for which 5 < 0, so that the determined equilibrium state is not absolute, but only relative to particular changes. Thus, if a system is in stable equilibrium with respect to isothermal changes, it can be shown to be in stable equilibrium with respect to isentropic changes, but the converse is not necessarily true. THERMODYNAMIC FUNCTIONS AND EQUILIBRIUM 101 53. Characteristic Functions. The functions : = U-TS ....... (1) * = U - TS + pv = V + 2 , v . . . (2) have an application in thermodynamics which far transcends their utility in the study of isothermal changes. They have been used by various authors under different names, and it seems desirable to mention these, so that the reader will have no difficulty in recognising the functions in the literature of the subject. F. Massieu (Journ. de Plnjs. 4, 216, 1869) first showed that all the characteristic properties of fluids can be expressed in terms of one or other of two functions, H and H' defined as follows : _ - U + TS _ y , > _ _ _ T T ' ' T T' and he called these, referred to unit mass, the characteristic functions of the fluid. J. Clerk Maxwell (" Theory of Heat," 1871) observed that the maximum work of an isothermal process could be represented as a diminution of a function U TS, which he at first called the " entropy," but afterwards (1875) the available energy. This latter name had been proposed by J. "\Vil- lard Gibbs (1873) in a very important memoir on "A Method of Geometrical Eepresentation of the Thermodynamic Properties of Substances by Means of Surfaces" (Scientific Papers, I,, 33). In this memoir Gibbs shows that the conditions of equilibrium of two parts of a substance in contact can be expressed geometrically in terms of the position of the tangent planes to the volume-entropy-energy surface of the substance, and he finds that the analyti- cal expression of this property is that the value of the function (U TS -f- pv) shall be the same for the two states at the same temperature and pressure. In his later memoir "On the Equilibrium of Heterogeneous Substances " (Trans. Connecticut Acad., HL, 108 248; 343524 ; Silliman'sJoitrn., 16,441, 1878) the three functions i^ = U Ts (" force function for constant temperature "), = IT Is -\-pv (" force function for constant pressure "), X = U + pv (" heat function for constant pressure ") are constantly used, and they are frequently referred to as the psi, zeta, and chi functions of Gibbs. The zeta function is identical with our , or the potential, the psi function with the free energy, whilst the x function is the heat function at constant pressure of 25. In 1879 Lord Kelvin introduced the term motivity for " the possession, the waste of which is called dissipation " ; at constant temperature this is identical with Maxwell's available energy. He showed in a paper " On Thermodynamics founded on Motivity and Energy" (Phil. Mag., 1898), that all the thermodynamic equations could be derived from the properties of motivity which follow directly from Carnot's theorem, without any explicit introduction of the entropy. Helmholtz (18-82) generalised the potential energy function of mechanics so as to obtain a function which, at a given temperature, should represent 102 THERMODYNAMICS the maximum obtainable work for a given change of configuration, and could be transferred to other temperatures by means of the second law of thermodynamics. This function was the free energy, V. Planck (Thermodynamics, trans. Ogg) has used the second of Massieu's functions : TT' V + pv * -T~ ~r which is now usually called " Planck's Potential." In some cases this makes the equations more symmetrical than those with , and in addition has the same properties at constant temperature and pressure in a non-isolated system as the entropy function in an isolated system. It must be remembered that all these functions were introduced for the purpose of simplifying the mathematical operations, just as were the energy and entropy functions in the earlier stages of thermodynamics. It is only their changes which admit of physical measurement ; these changes can be represented as quantities of heat and external work. In what follows we shall denote the energy, volume, and entropy per unit mass by the small letters u, r, s. 54. The Fundamental Differential Equations. We shall in the first place assume that : (i.) The only external force is a normal and uniform pressure p ; (ii.) The state of unit mass is completely denned in terms of two independent variables x and y. (This does not require that the system be homogeneous.) We have then the equations : SQ = du + SA (1) 8A=pdv (2) SQ = Tds (reversible change) . . . (3) /. du = Tds pdv (4) Since du and ds are perfect differentials : ***&* + * .... (5) ^-|^ + |^ .... (6) hence ^ = ( T * - p |) dx + But du is also a perfect differential, from the first law : THEKMODYXAMIC EQUATIONS AND EQUILIBRIUM 103 y cy f dy By Euler's criterion (H. M., 57) : oy cT cs^ _ cT QJ _ c/^cr _ fy> cr ' ex cy oy dx d.c cj/ c# cr This is the fundamental differential equation. The reader who is ac- quainted with the rules for transforming the variables in a surface integral will observe that it has the geometrical interpretation that corresponding elements of area on the (r, p] and (s, T) diagrams are equal (cf. 43). Of the five magnitudes p, v, t, s, u, any two may be chosen for the independent variables x and y, and for each pair we shall, by means of (11), obtain a relation between the differential coefficients of two other magnitudes with respect to the chosen variables. These are deduced as follows : (1) Let x=s,y=v: cT cs ?T _ cp op cr cs cv cr cs cr ' cs But = and = 0, because r and s are by hypothesis cr cs independent variables, .-.-() = () . . . CD \csJc \cr/s (2) Let x = v, y = T : 3T cs cs _ cp cr cp *" aF ' er ~ ?F "~ ?7 ' ?T ~ er* But ^ = 0, and ^' = 0, (3) Let x = s, y=p: 1^ fl _ ^ P fi" _ i" * *- cs c/> cji cs ' cp cs But |^ = 0, and -? = CJJ CS .-. ( f . T ) = (?) ... (III.) \Cpls \CS/p (4) Letj; = T,?/=7j: cs 8T 3s _ cjj cr cr " 3p 8p * 9T 3T * 8p 3T 104 THERMODYNAMICS But = 0. and = 0, ' -(*-) -(-) (TV) \dpJT~ V3T/, The four relations (I.) (IV.) are usually known as MaxweWt Relations, or the Reciprocal Relations ; they were deduced by Maxwell by means of an ingenious geometrical method (Theory of Heat, Chap. 9). Exercises on the Reciprocial Relations. (1) The fall of temperature per unit increase of volume in adiabatic expan- sion is equal to the increase of pressure per mechanical unit of heat supplied at constant volume, multiplied by the absolute temperature. [Multiply and divide the left hand member of (I.)byT, and put 'Ids = 80.] (2) The increase of pressure per one degree rise of temperature at constant volume, multiplied by the absolute temperature, is equal to the heat absorbed per unit increase of volume at constant temperature. (3) The rate of increase of volume per mechanical unit of heat absorbed at constant pressure is equal to the adiabatic rate of rise of temperature with pressure, divided by the absolute temperature. (4) The heat evolved per unit isothermal increase of pressure is equal to the continued product of the absolute temperature, the specific volume, and the coefficient of expansion o. (-HIX-) (5) Show that : (6) Show that : (dp\ (dv\ /a A pA W*var/, VsT/Aas (7) Prove the relations : /ar\ /aA /ar (8) The diminution of energy per unit increase of volume at constant entropy (i.e., in adiabatic changes) is measured by the pressure. (9) The increase of energy per unit increase of entropy at constant volume is measured by the absolute temperature. Corollary. At constant volume, the gain of energy is measured by the heat absorbed. THERMODYNAMIC EQUATIONS AND EQUILIBRIUM 105 55. Free Energy and Potential. Still considering the system of the preceding section, we intro- duce the free energy ^ and potential of unit mass into the list of variables : p, r, T, s, , Vr, <. If we put T = const, in the fundamental equation : rfU = SQ - SA we have for isothermal changes : dtt = d(Ts) SA so that 8 A is in this case a perfect differential, d(u Ts) (cf. 36). Now 8 A = pdr, whence it follows that in isothermal changes,^ can be expressed as a function of v alone, and F(c)dc is a perfect differential. From the equation dn = Tds pdr we have, on taking r and T as independent variables : u> In isothermal changes T = const. . T (^\ - " L ~ , V&r/ T V dv / T 9f \?r / T which give the entropy and pressure in terms of differential coefficients of the free energy. Now take p and T as independent variables : 106 THERMODYNAMICS *) + (<>) _ T (?) =o 9p/ T \C^/ T VpJ T 8p/ T /7r\ /d(?H-)\ and _p ( ^- = -5*-J ) 1 \8p/T V op /T ._ /3(T) .. _ _ P = p = - (5) ^ O ,_ TS +^) T E(^) =r . (6), op v \9p/ T which give the entropy and specific volume as differential coefficients of the potential. Since the entropy is the partial differential coefficient of ^ with r constant, or $ with p constant, with respect to T, the magnitudes ^ and are often called the thermodynamic potentials at constant volume and at constant pressure respec- tively. By combining the equations : u = + + T we find the very important equation : which is called the Free Energy Equation (Helmholtz, 1882). Similarly, from the equations u = < + Ts pv we obtain the equally important equations THERMODYNAMIC EQUATIONS AND EQUILIBRIUM 107 Equations (7) and (8) may be multiplied through by the mass m to give the relations U = *-T -PV (&>), where the large symbols refer to the total energy, free energy, potential, and volume of the given mass. If SQ is the small amount of heat absorbed in a small rever- sible igothermal-isopiestic change, we have, if W is the heat function at constant pressure : SQ = rf(U + p \) = dW = d * - T = . (10), an equation which is often useful. 56. Generalisation ; Relations for Systems with Several Degrees of Freedom. We shall now consider the properties of systems the state of which is determined by the values of the absolute tempera- ture T, and n other independent variables Xi, x. 2 , x 3 , . . . x n . If the latter are chosen in such a way that no external work is done when the temperature changes provided all the a-'s are maintained constant, they, along with T, are called the normal variables, and the state so defined is said to be normally defined (Duhem: Mecaniquechimique, I., 33). The or's may be said to define the configuration of the system, and the normal variables therefore define the state of the system in terms of its configuration and temperature. Changes of state may then be changes of configuration at constant temperature, or changes of temperature at constant configura- tion, or changes of configuration and temperature together. Taking the simple case of a homogeneous fluid of unchanging composition we see that its state may be defined in terms of any pair of the three variables : temperature T, specific volume i; and pressure p. If the state is to be normally defined, T must be taken as one variable, and v must be taken as the other, because there is the condition to be satisfied that no external work is done when the temperature changes whilst the variable remains constant. This condition is satisfied by v, but not by p. The normal variables may, according to the nature of the system considered, be the temperature and either geometrical 108 THERMODYNAMICS magnitudes such as lengths, areas, or volumes, or else physical qualities such as the relative proportion of two interconver- tible phases, or a quantity of electricity, or the concentration of a solution. We now consider the thermodynamic relations for a system which is normally denned. A system having n independent variables is said to have n decrees of freedom. For a small change of all the independent variables we have : SA = X x dx 1 + X 2 Jx- 2 + . . . = SX f . . T = /^., 2i . . T - T^, ia , . . T . . (6) and =-T ^ .... (9) The magnitude (^J^, or for any mass (^) , is the heat capacity of the system at constant configuration, which we shall denote by T x , or j x for unit mass : which determines r^ without ambiguity, the two arbitrary con- stants in * having been eliminated by differentiation. The bound energy is determined by the equation : B = U - * = - T (^) = TS . . (11) THERMODYNAMIC EQUATIONS AND EQUILIBRIUM 109 If the temperature is changed as well as the normal variables, the free and bound energies alter as follows : "*> = - S '' T - * A ' < ia > = SrfT + SQ . . . . . . . . (13) Thus, during the change, the- free energy diminishes by the amount of the external work &A and SrfT, whilst the bound energy increases by the amount of the heat absorbed 8Q and + S T , X] X2 , . - = U T . Xl . x 2 . - TS T , Xl , x 2 , + SX^ . (1) and We observe that the expression for the external work now involves the temperature, since the variables are no longer normal. If we substitute (2), (3) and (4) in the general equation : dU = SQ - SA THERMODYNAMIC EQUATIONS AND EQUILIBRIUM 111 and proceed as in the deduction for the simple case when Xi = p, X 2 = X 3 = . . . = 0, we find : ex . . . . (5) . . . . (6) and U = *-T(^( -SXirfri ... (7) which are identical in form with the equations deduced for the simpler case. 57. Intensity and Capacity Factors : Energetics. Clerk Maxwell (South Kensington Conferences, 1876), in discussing the work of "Willard Gibbs, remarked that : "the existence of a system depends on the magnitudes of the system, which are : the quantities of the components, the volumes, the entropies, as well as on the intensities of the system, viz., the temperature and the potentials of the components " (cf. 143). In his Theory of Heat he also refers to a separation of the variables in terms of which the state can be denned into two classes, one of which includes what are called intensities (pressure, temperature), and the other magnitudes (volume, entropy). A school of physicists has arisen the chief doctrine of which is contained in the assertion that the measure of every form of energy can be expressed as the product of two factors, one of which is an intensity (pressure, mecha- nical force, surface tension, electromotive force, chemical intensity, tempera- ture) and the other a capacity (volume, distance, surface, quantity of electricity, mass, entropy). The capacities determine what may be called the material properties of the system its extension in space, etc., and are additive, i.e., if two systems the capacities of which are Xi and x- 2 are com- bined into a larger system, the capacity of the latter is Xj -j- a-- 2 . Thus, if two masses HI na are placed together, the resulting mass is mi -f- J 2 ; two volumes r 1} v* produce a volume i'i + r s . The intensities, however, deter- mine the equilibrium of the various forms of energy in the system ; the hitter tend to pass from places of higher to places of lower intensity, and are in equilibrium when the intensity has the same value throughout. They are not additive ; thus, if two vessels containing a gas under the same pressure are put in communication, the capacities (viz., the volumes) are added, whilst the intensity (viz., pressure) remains unchanged. 112 THERMODYNAMICS The further step was then taken in the assumption that in reality nothing exists except energy, and all phenomena are energy transformations. Instead of considering the two worlds of matter and energy, each with its law of conservation, the disciples of the school of " Energetics "as this doctrine is called prefer to regard matter as simply a collection of energies. The so-called properties of matter ai % e certainly really those of energy, since we have no cognisance of the existence of material objects except through the medium of the senses, and this transmission is really a trans- mission of energy. I have not adopted this point of view because, although apparently very simple and plausible, it is not capable of taking us much further in the physical interpretation of phenomena. The Second Law of Thermodynamics, so far from being a particular case of a general "intensity law," according to which the availability of a charge of energy is proportional to the difference of the available intensity -factors, is a law which places in a clear light the essential difference between heat energy and the other forms of energy, a difference which certainly has a deep physical significance, some indications of which have appeared in the interpretation given to the law by Boltzmann, and applied with such signal success to the theory of radiant heat by Planck. That this interpretation, which I take it is wholly foreign to the system of energetics, is not superfluous, is abundantly confirmed by the many new fields of research which it has opened out, and the character of the harvest which has been accumulated in the short time between the enunciation of the new theory of " energiequanta " (cf. Chap. XVIII.) and the present day. 58. Kirchhoff's Equation and the Equation of Maximum Work. Our problem is to determine how the changes of total and free energy, AU and A*, or, what are the same, the heat absorp- tion at constant configuration and the maximum work, Q^. and A T , of an isothermal and reversible process, alter with the temperature of execution of the process. For this purpose we suppose the following reversible cyclic process executed. This process, it will be seen, is not a Carnot's cyclic process, but is of another type. (1) Let the system pass isothermally and reversibly from the initial state : (a) to the final state : This is a change of configuration at constant temperature, and the line traced out on an indicator diagram if there were THERMODYNAMIC EQUATIONS AND EQUILIBRIUM 113 only one x variable as abscissa (e.g., x = r), would be an isotherm. (2) Let the temperature now be raised from T to T -f- ST whilst all the normal variables remain unchanged with the values (fc). This is a change of temperature at constant configuration, and the line traced out would be an adynamic. (3) Let the change of configuration be annulled at the infinitesimally higher temperature T + oT by an isothermal reversible process so that all the normal configuration variables recover the initial values (a). This is a second isothermal process. (4) Let the temperature be lowered to T whilst the configura- tion variables remain unchanged. The system is, by this second adynamic process, brought back to the initial state. The cycle process is reversible, since the system can, with suitable adjustment of the external forces, be kept always infinitesimally near its equilibrium state at every instant. Such a cyclic process, consisting of two isotherms and two adynamics, was thoroughly studied by Eankine, and may there- fore be called a Rankine's cycle. Let AU denote the change of intrinsic energy, and let Q be the amount of heat absorbed at the temperature T, in any part of the process. Then, according to the first law : 2AU = 2Q-2A = . . . (1), whether the process is reversible or not ; and, according to the second law : 2 = ..... (2), but only when the process is reversible. Let AU, AU + <7AU be the amounts by which the intrinsic energy increases during the changes of configuration from the initial to the final state at the temperatures T and T + &T respectively, and let r,, T, be the heat capacities at constant configuration of the initial and final states at the mean tempera- ture T + |8T. The sum of the changes of intrinsic energy in the four processes is then : AU + I>/T - (AU + A T , i.e., SA T >0 since the pressure of the gas increases with rise of temperature. In some cases, however, we can have SA T < 0, as when the electromotive force of a galvanic cell decreases with rise of tem- perature ( 200). The difference SA T is the amount by which the maximum work of an isothermal process of given constant amplitude increases 1 It was first deduced by Lord Kelvin in 1855 (Math, and Phys. Papers, I., 296). I 2 116 THERMODYNAMICS when the temperature of execution is raised by ST. The increase /3A T \ /dA T \ of the maximum work per degree is ("^rj = \~jjj*) > tne suffix indicating that a constant change of configuration, or a constant amplitude, is maintained the volume ranging always between the two isochores. This can, however, be interpreted in another way, because A T is equal to the area ab'a'b, i.e., to the area of a circuit including the two processes described, but in opposite directions, and two other connecting processes which entail no further expenditure of work, viz., the parts ab' and a'b', in which the temperature changes are effected at constant configuration by raising the temperature of the medium continuously so that the pressure of the gas always corresponds with its temperature as given by the characteristic equation pr = RT (r const.) at every instant. If now the cycle is executed between the same isochores, and with the same difference of temperature ST, but at different initial temperatures, we shall have the lines aa', bb', cc', dd', . . . indicating corresponding changes at different temperatures, and the circuits abb'a', bcc'b', cdd'c', . . . indicating corresponding cycles at different mean temperatures of execution. With change of temperature the cyclic area moves up or down between the iso- chores, and its magnitude also changes in a continuous manner, the rate of increase of area per 1 ascent being -=-f for the given initial temperature. This rate of increase will also, in general, change with the temperature, since -~ is also a function of temperature. The interpretation of the equation of maximum work, although very simple, has not always been clearly expressed. CHAPTER V FLUIDS 59. The Thermal Coefficients. LET us consider unit mass of a fluid in a given state. Since the equations which \ve shall deduce in this paragraph do not depend on any particular thermonietric scale, we shall represent the temperature by 6, where 6 may be the Centigrade tempera- ture, or may be measured on any other temperature scale. The state of the fluid is therefore represented b} T (r, p, 0). If one of these variables increases by an infinitesimal amount there will, in general, be a corresponding increment in the value of each of the others, and there could be an infinite number of corresponding pairs of values of the latter for one value of the former. But if two variables are fixed, the state of the fluid is completely defined, for it has only two degrees of freedom ( 26). The heat of the path, 8Q, may be represented by any one of the following equations, according to the pair of independent variables chosen : SQ^csW + lJc . . . . (1) = c p d0+l lt dp .... (2) = y/0> + vA .... (3) The physical interpretations of the coefficients are found by setting one or other of the magnitudes dv, d0, dp, equal to zero. Thus, if in (1) we put dr = 0, d6 = successively, we obtain : Similarly : Thus, c r , c p are the amounts of heat absorbed per unit increase of temperature at constant volume and at constant pressure respectively. They are the specific heats at constant volume and at constant pressure respectively. / r , /p are the amounts of heat absorbed per unit increase of volume or pressure respectively, at constant temperature. They 118 THERMODYNAMICS are called the latent heat of volume change (or the latent heat of expansion) and the latent heat of pressure change respectively. 7,., 7^ are the amounts of heat absorbed per unit increase of volume at constant pressure and unit increase of pressure at constant volume respectively. They have received no special names. The suffixes of the r 's refer to the variable maintained constant during the change ; those of the V& and -y's to the independent variable, the small increase of which must be multiplied by the corresponding coefficient, to give the heat absorbed consequent on the change of that variable. All the coefficients will, in general, be functions of both inde- pendent variables, and since we know that the heat absorbed depends on the path of change, it follows that the coefficients are not, in general, partial derivatives of a function of the two independent variables, for &Q would then be a perfect differential (cf. H. M., 115). 60. Theorem of Reech. For an adiabatic chanae we have : SQ = . . . . . (1) hence from (1) and (2) of the last section : dv = - C ~dd tfp = ) = C 4 ar/ Q c^j, For an isothermal c/mnr/c : ' . . . . (8) From (2) and (4) we find : ' \dre~c,. where K is the ratio of the specific heats. FLUIDS 119 But 9 *8 where e Q , c e are the adiabatic and isothermal elasticities ; hence : so that the ratio of the adiabatic to the isothermal elasticity of a fluid is quite generally equal to the ratio of the specific heat at constant pressure to the specific heat at constant volume (Reech, 1854). 61. Relations between the Thermal Coefficients. In the equations for the change of state of a fluid : Q = c^ + W \ dp< . (1) SQ denotes the same element of heat ; hence the coefficients (e's, Vs, and y's) are not independent, but are related. The relations are obtained from the equations (1) and the rules for the change of the independent variable in the calculus. For the transformation of the differentials we have : ~- dp -4- ~- dr dp * ' dr dv 7/1 , dc , in which each differential is expressed as a function of the other two variables. From equations (2) alone some useful relations may be derived. Thus, if in the first we put d0 = 0, i.e., 6 is constant, or the change is isothermal, we have : " ?)A rfr = ( const.) 120 THERMODYNAMICS Similarly, from the second and third : dv 90 d w dp dv\ dr W 0V If we put dc = 0, dp = 0, (W = in the first, second, and third equations of (2), we get -JL ^ _ 1 ^i_l W ~ W ' W ~ W ' dp ~ dp ' dp dv dv Turning next to a consideration of equations (1), we observe that, if the differentials in one equation are arbitrary, those in the other two equations are fixed by them, and since each equation contains two variables, each pair of variables must lead to four relations, so that there will be twelve relations for the three pairs. These relations are obtained as follows : Let r, 6 be taken as independent variables : SQ = Cl ,dO + l e dv^ .... (i.) Then we can, from (2), write the equations 6Q = Cv dd + l v dr = 7p f h> + 7*fr in the forms respectively. The equations (i.) (iii.) are now identical, and by equating coefficients we obtain the relations dp r, = % >^,etc. The twelve relations finally obtained are tabulated below. They are clearly not all independent. FLUIDS 121 Independent variables 0, />. <,-*+*! dv w From these equations many relations between the c's, Z's, and 7's themselves can be obtained. The equations of 59 61 are independent of the mechanical theory of heat, and would apply equally well to the caloric theory. In the latter case, however, Q is a perfect differential. They are also unchanged when T is put for 9, where T is the absolute temperature. All the twelve relations can be derived from the four in the first column, together with the equations (4). Various other relations between the thermal coefficients may easily be obtained if required. Thus : L 62. Application of the First Law to Fluids. If we have unit mass of a homogeneous fluid having a uniform temperature T, and with its surface exposed to a uniform normal pressure p, its state can be defined (if we regard the chemical nature as fixed) in terms of the specific volume r, and the absolute temperature T. Let the state pass reversibly from (v, T) to an infinitely near state (r + dr, T + dT). If SA, BQ denote the elements of work done and heat absorbed, SA = pdv (1) SQ = c v dT + l K dv .... (2) According to the First Law, SA and SQ are not usually perfect differentials, but depend on the path of change, whereas their 122 THERMODYNAMICS difference is always a perfect differential, which was defined as the increase of the intrinsic energy of the fluid : ,l,i = SQ - SA . ... (3) The condition that dit is a perfect differential requires : IT . . (5) This shows that a part of the heat absorbed depends on the change of tem- perature, and another on the change of volume. The latter is composed of the external work pdv and a part depending on the change of intrinsic energy with volume. But ( 61) /, = 1 (c f c p ) . . . . (6) 8T (7) 8T Since r and T are independent variables, the coefficients of dr and dT in (5) and (7) are identical ; hence : m = e * . (8) Equation (9) may be regarded as the general expression of the First Law as applied to fluids. The characteristic equation of the fluid is of the form p = F (r, T) . (10), and since this fixes the external conditions (viz., the one that the pressure on the system must have a given value) in order that the system may be in equilibrium, with chosen values of the independent variables specific volume and temperature, it may be called the equation of equilibrium of the fluid. The two equations (9) and (10) contain all that the First Law can teach us as to the properties of the fluid. The form of (10) must, from the thermodynamic standpoint, be regarded as known from the results of special experiments with the fluid. If the variables (p, T) or (v, p) are used instead of (v, T), another equation from the twelve relations of 61 is taken instead of (6), and equation (2) is modified so as to introduce the chosen variables. FLUIDS 123 If (t>, p} are taken, the state of the fluid is not in general uniquely defined. Thus, a mass of liquid exhibiting a state of maximum density ( -V7p = 0, -j^ < ) may exist in two states on opposite sides of this, having the same values of r and p, but different values of T. The equations for the three pairs of variables are given below. The deduction of the equations for (p, T) and (r, p) is effected in a similar manner to that for (r, T), and s left to the reader : ) -ri- = fdv\ where r , p Q are the specific volume and the pressure at C., V is the specific volume at 1 atm. pressure, and a, ft, f , >/ are the coefficients of expansion, of tension, of elasticity, and of com- pressibility respectively. FLUIDS 125 From Clapeyron's equation we have : B'= l jj . . . \ . (1), and if we combine this with the relations 8T . dv dv _ dp lp - lv ty' ty~ ~ar dv we find /,,= -T^=-Ta'. . . . (2); (1) and (2) enable us to find l m l p in terms of measurable quantities. Corollary. A fluid emits or absorbs heat on isothermal com- pression according as it expands or contracts, respectively, with rise of temperature at constant pressure. Thus water below 4 C. absorbs heat on compression, above 4 C. emits heat on compression, at 4 C. it neither emits nor absorbs heat on com- pression. From the equation ./ dv c, = c.+ Z,gjp and Clapeyron's equation we find : c,,- C ,= T^.|| = Ta',3' . . . (3) But =-. /,' = =: dp which give the difference of the specific heats in terms of the coefficient of expansion and the coefficient of tension or elasticity. Example. In the case of mercury : Cp = 0-0333 T = 273 126 THERMODYNAMICS r/> _ 1013250 dyne V dv ~ 0-0000039 ~^~ v = 1/13-596 ^ = 0-0001812. 31 To obtain c e in calories we must divide by J : _ r = 273 X 1013250 X (Q-Q001812) 2 C " ( "~ 0-0000039 X 13-596 X 4'19 X 10 7 .-. c v = 0-0292, c l jc o =1-1. We also readily find : c,, c, = Tr (ea)a . . . . (4) /= TaV = Tae . . . . . (5) In the application of the First Law we have : (hi - SQ SA = c r dT + (/ - 2)) tie du m dp .-. ^ = i,- P =i^- P du BT = r " 9/ ^> _ ~ da The condition that , (ST) Q = d, FLUIDS 127 then 3 = ^o> '. .' . . (9) e& where & = rise of temperature consequent on an adiabatic increase of pressure < ; p density at C. ; a = coefficient of expansion. Exercises. (1} If 5v is the diminution of volume producing o>, show that A = T 5i> = 80 . . (10). c e r,, - T6.v Equations (8) (10) were deduced by Lord Kelvin (1857), and verified by Joule (1859). In the case of water : d < if T < (273 + 4 C.), since a < 0, and S > if T > (273 + 4 C.), since a > 0. (2) Prove that : (!L\ = T>JL \ZvJ T 31. (3) Show that _p/T f(v) is the characteristic equation of a fluid the intrinsic energy of which is independent of the volume. 65. Relation between Isotherms and Adiabatics. There is a perfectly general theorem, which applies to all bodies, or systems of bodies, to the effect that an adiabatic, when it crosses an isotherm on the indicator diagram, is more inclined to the v-axis, or the adiabatic is steeper than the isotherm. We shall not attempt any general proof of the theorem at this point, but a few illustrations may be given. If a fluid is enclosed in a cylinder with walls impervious to heat, and compressed, it is either heated or cooled according as a, the coefficient of expansion , is > or < 0. In both cases the volume would have increased , and the pressure risen more rapidly than if heat were allowed to pass out, or in, through the walls. Thus, for a given diminution of volume, the rise of pressure is greater under adiabatic than under isothermal conditions. The theorem also applies to a heterogeneous system, such as a liquid in presence of its saturated vapour, or in presence of the solid. In the former case, vapour is liquefied by compression and gives out its latent heat. Under isothermal conditions this would escape as fast as produced, but if the heat is compelled to remain in the system, it raises the temperature and thereby increases the pressure. If, on the other hand, a mixture of ice and water is compressed, ice melts and the mass is cooled by abstraction of heat. If heat is allowed to enter from outside, so as to restore the original temperature, more ice melts, and the pressure falls by reason of the contraction. 128 THERMODYNAMICS The theorem under discussion is a particular case of a very general principle, which was stated by Maxwell (1871) in the form that " a force producing alteration of the state of a con- strained system is always greater than a similar force producing the same alteration in an unconstrained system." Two isotherms, isochores, adiabatics, or generally any two thermal lines of the same kind, never cut each other in a surface representing the states of a fluid with respect to the three variables of the characteristic equation taken as co-ordi- nates, for a point of intersection would imply that two identical states had some property in a different degree ((p,T), the potential of unit mass of a fluid, is known as a function of p and T, we have : T(**-\ ^-T-^-T^ + %^- dp* _ _ ^ - a$ ' a^ ' '" ~ A ar?^' dp dp Thus, all the ihermodynamic properties of a fluid can be expressed in terms of the potential ^(_p,T). Thus, all the thermodynamic properties of a fluid are known if we are in possession of a single function of the independent variables in terms of which the state is denned. If these variables are r, T the function is the free energy i/r ; if they are p, T, the function is the potential . 180 THERMODYNAMICS This theorem, which is of great importance, was established by F. Massieu in 1869 (Journ. de Phys., 6, 216, 1869 ; C.R. 69, 858, 057, 1869). It may be extended to all systems by an application of the methods employed in 58, 59 (cf. Duhem, Traite de Mecanique chimique, L, pp. 104 113). CHAPTER VI IDEAL AND PERMANENT GASES 67. The Characteristic Equation of a Gas. The state of unit mass of a gas, like that of any other fluid, is denned by any pair of the variables p, r, 0, and its characteristic equation is therefore : /Uv,0) = . (1) If the temperature is maintained constant, the volume is found experimentally to be approximately inversely proportional to the pressure (Boyle's law) : pv = (0) .... (2) If the pressure is maintained constant, the volume is found to increase by approximately the same fraction of the volume at C. for each degree rise of temperature (law of Dalton and Gay- Lussac) : r = r (l + o0) . . . (3), where a = (^J = coefficient of expansion, a is very nearly the same for all gases : a = 1/273. The change which the volume of the gas undergoes during any simultaneous change of p and 6 is the sum of the changes it undergoes when p and 6 are separately altered, since these are independent variables. For the change of p from pi to p% at #1 : p&' pii\ = For the change of 6 from V to 2 at p* : r a _ 1 + afli r' 1 + o0i .-. pin (1 + a0 2 ) = pars (1 + a0i) Put 0i = Q .'. & = po, n = r , 0-2 = 0, then pv =^or (l + a0) . . . . (4) K 2 132 THERMODYNAMICS We shall now define the absolute yas temperature by the equation . (5) , (6) (T) = + a Corollaries. (I) If = - ( } is the coefficient of tension; 7) \utj J v and p = 2) (1 + aO).. (2) The absolute gas temperatures are proportional to the volumes at constant pressure, and to the pressures at constant volume : The isotherms of a gas have the equation (2), and form a series FIG. 18. FIG. 19. of rectangular hyperbolas ; the isopiestics (p const.) and isochores (v const.) have the equations : and form series of straight lines radiating from the zero point (T) = 0, i.e., 0= I/a (Figs. 18 and 19). If we put poi-tfi = i- . . . . . (7) we may write the general gas equation (6) in the form : pr = r(T) .... (8) This is the characteristic equation of a gas (Clapeyron, 1834), IDEAL AND PERMANENT GASES 133 and for the permanent gases, such as nitrogen, oxygen, hydrogen, etc., we can put : / MO (9) ~273 ' r is called the gas constant. If instead of unit mass we consider a mass in, of volume V at p Q , j>V = mr(T) = r'{T) . . . (10) For equal masses of different gases, the constants ;' are inversely proportional to the densities. It is very important to observe that, inasmuch as a is the same for all gases, these have characteristic equations of the same form (B. P. E. Clapeyron, Jouni. de I'Ecole Polytechn., 18, 170, 1834; Pogg. Ann., 59, 446, 1843). 68. Molecular Weight. Definitions. (1) The absolute density of a gas (S) is the weight of 1 litre in grams at C. and 1 standard atmosphere pressure. (The weight is reduced to sea-level and latitude 45.) If u- = weight of 1 litre at C. and p atm. 8 = >x|x(l + 0). (1) (2) The normal density of a gas (D) is 32 times the ratio of the weights of equal volumes of the gas and of oxygen under normal conditions (1 atm., C.). D = 32~. = 32^ . . . (2) \ V \ If the general gas law held rigorously true for all gases, this ratio would be independent of temperature and pressure, by (1), and D would depend only on the chemical composition of the gas. (3) The molecular weight (M) of a gas is the normal density, corrected if necessary for the deviations, exhibited by the gas, from the gas laws. Since the permanent gases deviate only slightly from these laws, we shall at present regard them as ideal gases ; hence : M = D (3). The slight deviations may be taken into account by a method described in 80. The definition of molecular weight just given leads at once to the conclusion that equal volumes of different permanent gases, 184 THERMODYNAMICS under the same conditions of temperature and pressure, contain the same number of molecular weights. This is called Arogadro's theorem (1811) ; it appears here simply as a definition of molecular weight, and this is really the manner in which the relation is applied in chemistry. The kinetic theory of gases gives a new, and much deeper, signifi- cance to the statement by introducing the conception of the molecule ; this, however, does not concern us in thermodynamics, and since the molecular weights are purely relative numbers, the deductions made in this book are equally strict whichever stand- point is adopted. The molecular weight of a substance is, for a large number of calculations, far more convenient than unit mass. This depends on the fact that a large number of properties are independent of the nature of the substance, and depend only on the number of molecular weights present (" molar," or " colligative," properties) (cf. Chap. XI. on " Solutions "). 69. The Molar Gas Constant. It is evident from the gas equation that >' has the same value for all gases if quantities are selected which have the same pressure, volume, and temperature. Thus r' has the same value for a molecular weight of any ideal gas. If we denote this value by R, and if r is the value for unit mass, R = Mr . . . . . . (2) /. 2w==fi(T) . ." . . (3), where r, as before, denotes the specific volume. If we put M.V = v for the molecular volume of the gas at 0>,T) : If we consider any mass in of the gas, occupying a total volume V : / = -(T) Vfl. . - 15), IDEAL AND PEEMANENT GASES 135 where n = wi/M is the number of gram-molecules, or mols, of the gas in the volume V. (Goldberg, 1867X Equation (4) applies to a mol. of any gas, R having a constant value which can be calculated as soon as we know the specific volume of the gas at a given temperature and pressure, and its molecular weight. At atmospheric pressure and at the temperature of melting ice, 32 gr. of oxygen occupy a volume of 22,412 c.c. (corrected for a slight deviation from the gas laws) : .-. p = l, (T) = 273-09 (this is more accurate than the value 273 previously used), f = 22,412 .-. E = 1 X 22 ' 412 = 0-08207 L atm - . 273-09 degree C. = 0-08207 X 1013-13 X 10 J = 8'315 X 10' =-- 7 - degree C. = 0-08207 X 1033200 = 84795 = 0-08207 X 24-191 = 1'9854 Guldberg, Forhandl. Videnskabs. Selskabet, Christiania, Oct., 1867, Ostwald's Klassiker, 139, 6; A. Horstmann, Berl. Ber. U, 1243, 1881; J. H. van't Hoff , Klassiker, No. 110, 30 ; D. Berthelot, Zeitschr. ElektrocJtem. 10, 621, 1904; W. Nernst, ibid., 629. 70. Ideal Gases. The word " approximately " has been used in framing the laws of gases, because no actual gas exists which obeys these laws strictly. By " strictly " we refer to that closeness of agreement, within the limits of experimental error, observed in such laws as Newton's law of gravitational attraction, the law of conservation of mass, the law of conservation of energy, and Faraday's laws of electrolysis. The degree of approximation is, however, different for different gases, the so-called " permanent gases" approaching very closely to exact agreement, whereas easily liquefiable gases like sulphur dioxide, carbon dioxide, and ammonia deviate markedly from the gas laws. It has been found, however, that in all cases the deviations become smaller and smaller as the pressure is reduced more and more, until under very small pressures all gases approach an ideal limiting state, which would be attained when the pressure has become infinitely small. Such a limiting form of the gaseous state we shall call an Ideal, or a Perfect, gas. For the purposes of theoretical reasoning we shall further 136 THERMODYNAMICS suppose that such an ideal gas can exist under ordinary con- ditions of temperature and pressure. This is very nearly realised by hydrogen and helium, and moderately closely by the other permanent gases. It would appear at first sight necessary to define an ideal gas as one which strictly obeys all the gas laws. As a matter of fact we can prove that if it conforms to tico conditions it will conform to all the conditions we shall take as defining an ideal gas. Definition. An ideal gas is a fluid which obeys Boyle's law, and the internal energy of which is independent of the volume : pr = const.) /dU\ _ ~ ~ T constant. \dr/T~ I The latter condition follows from a very important result established experimentally by Joule (J. P. Joule, Phil. Mag. [3] , 23, 343, 435 (1843) ; 26, 369, 1845). 71. Theorem of Joule. The isothermal expansion of an ideal gas is an aschistic process. If a mass of gas expands isothermally, the heat absorbed is equal to the external work done. This result could be inferred from the agreement between the mechanical equivalent of heat calculated by J. B. Mayer (1842) and that determined experimentally by Joule (1843). If 1 gr. air is warmed through 1 and at the same time allowed to expand under atmospheric pressure, 0'2408 cal. of heat are absorbed i.e., c p = 0'2408 cal. But if the heating is carried out at constant volume, only 01713 cal. are absorbed, i.e., c e = 01713 cal. If we assume, as was tacitly done by Mayer, that the difference, <-,, - c e = 0-0695 cal., is entirely spent in doing the external work of expansion against the atmospheric pressure, which amounts to the theorem just stated, we have : (c p c r ) X J = external work. = pressure X increase of volume. 1 gr. air at C. has a volume 773-4 c.c. /. expansion = 773-4 X ,-=_ = 2-83 c.c., and 1 atm. pressure = 1033 -^ IDEAL AND PERMANENT GASES 137 .-. external work = 1033 X 2'83 = 2933'4 gr. cm. .^ eai. = = 42,060 g , cm. =J. The close agreement of this with the direct value (41,880) verifies the theorem. The assumption tacitly introduced by Mayer was called into question by Joule, who pointed out that it certainly is not true if the expanding substance is a liquid, and who in 1845 subjected "Mayer's hypothesis " to the test of a direct experiment. The principle was the following : If a volume of air (or other gas) is allowed to expand without doing external work the process is adynamic : TBS If, therefore, the process is aschistic, i.e., SA = SQ, there ought on the whole to be no heat absorbed from or given up to the surroundings : /. SQ = 0. Two copper globes, A and B, the first containing air at 22 atm. pressure, and the second vacuous, were immersed in a can of water. On opening the tap connecting the globes, the expan- sion occurred, but after the water had been stirred no change of temperature was observed. Joule therefore concluded that : " no change of temperature occurs when air is allowed to expand without developing mechanical power [i.e., doing external work] ." The globes A and B and the tap C were then placed in separate cans of water (Fig. 20) and the experiment repeated. A fall of 0'595 C. per kilogram of water occurred in A, a rise of 0'606 C. per kilogram in B, and a rise of 0*078 C. per kilogram in C. Thus, within the limits of error : (i.) The same amount of heat is lost by the gas in A com- pressing that in B as is produced in B by the compression, the total change being zero. (ii.) No heat is absorbed or evolved in the tap. Since, from the conditions of the experiment, it was found that the thermometer used would not have been affected if a change FIG. 20. 138 THERMODYNAMICS of temperature of less than 1'88 C. had occurred in the air, Lord Kelvin in 1851 suggested modifications in the method with a view to increasing its sensitiveness, and in 1852 he and Joule carried out the classical " Porous Plug experiment," forcing the gas through a plug of silk, and measuring its temperature before and after passage. The experiments, described in detail later, showed that contrary to Mayer's hypothesis there was a very slight cooling effect with air and carbon dioxide, and a very slight warming effect with hydrogen. Still, for the permanent gases we may take the result of Joule's original experiment as being very nearly true. Thus (SQ - 2A) T = U a - Ui = or the intrinsic energy of an ideal gas is independent of the volume : The fact the changes of temperature occur during the expansion does not affect the argument, since, as we are dealing with the First Law only, it is the initial and final states alone which are of account. 72. Difference of the Specific Heats of Air. The calculation of Mayer was thrown into a different form by Rarjkine (1850), who showed that, instead of estimating the mechanical equivalent of heat from the difference of the specific heats of air, one could take Joule's value of the mechanical equivalent and the known ratio of the specific heats, and thence determine the specific heats themselves. J (<., - c ,) = p(r, - n) = r [((T) + 1} - T] = r. r = 2933-39, J = 41,880 gr. cms. /. 41,880 (c p - c v ) = 2933-39. /. 41,880 -l =?*?. Again, cjc,, = 1*405, from experiment. .-. c e = 0-172 ; c p = 0-2417. These values differed from the results of Delaroche and Berard, available at the time, but were afterwards confirmed by the more accurate work of Regnault (Regnault, Mem. de I'Inst. France, 26, 1862). IDEAL AND PERMANENT GASES 189 73. Equations for Ideal Gases. The equation of 66 : ' (1) applies to unit mass of any homogeneous fluid. If the change is isothermal : dT = . . . . . (2) *=(*+),* l, = P~ (8) For an ideal gas, however, oule's theorem) . . . (5) ; =- = (Joule's theorem) . . . (4) /. with Clapeyron's equation (63) we get : v __ 1 //?\ . ~ T "~ T ' ' ' W ' /.by integration : / (v) taking the place of the integration constant, since we are dealing with partial differentials. Thus p = T X const, (r constant) . . . (8) But pv = const. (T constant) . . . (9) /. pv = const. X T. or pv = rT (10) which is the general gas equation. We now take the equations ',= -! .... (ID . . . . (12) \J J. of 61, 64. From (9) we get cf = ' from (5). 140 THERMODYNAMICS Thence from (12), at the temperature of melting ice : or the coefficient of expansion of an ideal gas is equal to the reciprocal of the absolute temperature of melting ice. But the gas temperature (T) has been denned as (T) = 0+1 = + To . . . (15), a and since T = 6 + T . . . (16) the size of the absolute degree being, by convention equal to that of the Centigrade degree, we have (T)= T .. \ ... . r .. r ., . (17), so that absolute thermodynamic temperatures are equal to the gas temperatures measured with an ideal gas thermometer. Relation (17) is an equality, but not an identity ; it is true only for a perfect gas. The thermometer may operate either at constant volume, or at constant pressure, since, if the gas obeys Boyle's law, /3 = a. From the general equation of 62 : SQ = c V fT + 7A ..... (18) and (5) it follows that the expression of the first law for ideal gases may be written in the form 5Q = c v dT + pclr .... (19) Corollary 1. The specific heat at constant volume of an ideal gas is a function of temperature alone. For c v = (K) , and \o -L / v u is independent of v. By means of the general gas law pv = rT ' . ' . . . (20) or ^r = RT v' ."" '% . (20a) we can eliminate p, r, or T from (19). If equation (20) is used, all the thermal magnitudes refer to unit mass, i.e., c t> , c f ; if (20a) is used they refer to a mol, i.e., C p , C M where C, = Mc v , etc. (a) Elimination of p : bQ = c v dT+ 1 ^dv . . . (21) V or 8Q' = C//T + ifo . . . (21rt) IDEAL AND PERMANENT GASES 141 (b) Elimination of r : r=rT/p. .:dv=ldT-^dp .-. 8Q = ( Cr + /) dT - jdp . ' . . . (22) TIT Similarly 6Q' = (C, + R) dT dp . . (22a) Corollary 2. < = e c + r . . . . (23) or C, - C r = R = 1-9854 g. cal. . . (23a) The difference of the specific heats referred to unit volume is the same for all gases (Clapeyron, 1834). Corollary 3. c p is a function of T alone. (c) Elimination o/'T: Differentiate (20) .-. 6Q = rdp -f ^-^ pdv . . . (24) or 8Q' = ^ rdp -\ ^ ' pdv . . (24) Exercise. Show that, for a small change of state of an ideal gas: 5Q = c^T + (c p - c v ) - I (25) 74. Integration of the Equations for Ideal Gases. (1) Isothermal Ej.yansion : dT = 0, .-. bQ=pdv=:'^-dv >:?*- il = rT/ = . (1) Pi I r, where " In " denotes the natural logarithm. 142 THERMODYNAMICS Corollary 1. If an ideal gas changes its volume reversibly without alteration of temperature, the quantities of heat absorbed or emitted form an arithmetical progression whilst the volumes form a geometrical progression (Sadi Carnot, 1824). E.g.,v = 1 2 4 8 16 . . . , Q = 1 2 3 4 . . .- . Corollary 2. Since P pdv = [pv^ 2 vdp J v i jpi and [pvf = j> 2 r 2 p\v\ = (Boyle's law), i . . . . (11) P* J - 1 . ... (Ic) Corollary 3. All gases, if equal volumes of them are taken under the same conditions of temperature and pressure, evolve or absorb equal quantities of heat if compressed or expanded by equal fractions of their volumes (Dulong's rule). Corollary 4. If (p\, r lf TI) and (p 2 , r 2 , TI) are the initial and final states (T constant) (2) Adiabatic Expansion : SQ = .-. c t ,dT + (c p - Cl )~ Divide by c c T, and put c,,/c,, = K : In the integration of (2) we must know K as a function of v and T. Now it has been shown that c pt c v are functions of T only, hence K is a function of T alone. We shall now assume, as an experimental fact, that c v is independent of temperature in the case of permanent gases. This was verified to a close degree of approximation by Regnault (cf. 7). Corollary l.c,, = C K + r is independent of temperature. Corollary %,cJc,. = K is independent of temperature. Corollary 3. The internal energy of an ideal gas, the specific IDEAL AND PERMANENT GASES 143 heat of which is independent of temperature, is represented by an expression of the form u = MO + c r T, where refers to absolute zero. We may now integrate equation (2) on the assumption that c r , and therefore K, are constant : InT + (* !) Inv = constant .-. Tt"- 1 = constant, - <"> If we eliminate r or T from (2a) by means of the equation pv = rT, we obtain Pi \ N 2 , H 2 , CO, CH 4 , NO), which had resisted the efforts of Natterer (1844), who exposed them to pressure alone. 75. Determination of the Ratio of Specific Heats of a Gas. (1) General Case. Consider a mass m of a gas enclosed in a cylinder fitted with a piston and in connexion with I a manometer capable of indicating very rapid changes of pressure (Fig. 21). Let po be the initial pressure. ^ By rapidly withdrawing the piston, let the volume be increased by AV ; the increase of specific volume is AT = AV/wt. The indicated pressure will be, the instant after the expansion, less than p ; let it be p^ Pi Po = Ap. Since the change is rapid, it may be assumed FIG. 21. to be adiabatic, (1), where a is a small magnitude which tends to zero when At? tends to zero, _ _ A ( ( M\ Ar \dvj Q . (la), since o-Av is a small quantity of the second order. Now let the gas warm up to its initial temperature, and let the recorded pressure be j> 2 - Then, as before, where I -2) refers to isothermal change. /dp\ From (la) and (2): SL^LP? _ V^: Ih Po (dp \dr/ e by Reech's theorem. This equation is due to Moutier (1880) ; the method has been (3), IDEAL AND PERMANENT GASES 145 used by Maneuvrier and Fournier (1895-97), and by Worthing (1911). Moutier's equation evidently applies to any gas, whether it obeys the laws of ideal gases or not, provided Ar is small. (2) Ideal Gases. The state of unit mass of an ideal gas, under- going adiabatic compression or expansion, is completely defined by the equations K p In the deduction of these it has been assumed that : (i.) The gas conforms to the equation pr = rT. (ii.) The specific heat at constant volume is constant over the range of temperature T 2 TI. By taking logarithms we find : lg PI _ tog T 2 log TI , ^ log vi log v z log 1-1 log r 2 . log p* log PJ PI li The experimental realisation of adiabatic conditions is difficult ; heat is always transferred between the gas and its surroundings by conduction and radiation, and the usual plan is to make the changes of volume occur so rapidly that the heat transfer is negligibly small. KxampJes. (1) A gas is contained in a vessel furnished with a stopcock, and is under a pressure pi slightly greater than atmospheric pressure P. The tap is opened, so that the gas expands rapidly to atmospheric pressure, and is at once reclosed. When the remaining gas has attained its original temperature, the pressure is pz- Show that : log pi logp. 2 (Clement and Desormes, 1819). (2) "With the same apparatus as in example (1), the absolute temperature of the gas in its initial state was TI. The tap was then opened so that the gas rapidly expanded to atmospheric pressure, and the temperature, deter- mined immediately after expansion by a platinum resistance thermometer, or a thermo-element, in the centre of the vessel, was T-2. Show that : K _ log Pi tog? (Lummer and Pringsheim, 1898). T. 146 THERMODYNAMICS 76. The Velocity of Sound in Gases. A conipressional wave is a disturbance propagated through a medium such that portions of the latter are alternately compressed and expanded in the path of the disturbance. If the wave is of such a kind that an observer who moves in its direction of pro- pagation with the same velocity as the wave sees no change in the appearance of the wave, it is said to be of permanent type. Newton (1686) first calculated the velocity of propagation of a conipressional wave of permanent type in an elastic medium, and arrived at the general formula : w 2 = e/p . . . . (1), where e, p are the elasticity and density of the medium in the unstrained condition (cf. Lamb, The Dynamical Theory of Sound, 1910, Chap. VI.). The passage of a sound wave along a tube, so that no energy is dissipated by friction, is an example of a conipressional wave of permanent type, and Newton applied his equation (1) to deter- mine the velocity of sound in air. For this purpose he took e as the isothermal elasticity of air, which is equivalent to assuming that the temperature is 'the same in all parts of the wave as that in the unstrained medium. Since air is heated by compression and cooled by expansion, the assumption implies that these temperature differences are automatically annulled by conduction. Taking the isothermal elasticity, we have : -' .... (la) We may call u n , calculated for any medium, the Newtonian relocity of sound. If the fluid obeys Boyle's law, a case approximately realised by atmospheric air, we have e fl = J>f ' H = Vp/p = A/pr. This equation gives for the velocity of sound in air at 280 metres per second instead of 331, as obtained by experiment. The discrepancy was explained by Laplace (1822), who pointed out that in the sound wave the changes of volume are so rapid that the conditions are adiabatic, and not isothermal. Hence e = f Q , IDEAL AND PERMANENT GASES 147 Prom (la) and (2) we obtain by Reach's theorem. This equation is true for any fluid, and may find application in the future. If the fluid is a gas obeying Boyle's law : *Q = K P .'. = V^ . "... (4) The ratio of the specific heats is therefore determined by (3) from a measurement of the velocity of sound in the gas at a particular temperature, provided the characteristic equation of the fluid is known. In the deduction of Newton's equation (1), it is assumed that the amplitude of the vibration is small, so that the kinetic energy of the moving gas, which is proportional to the square of the amplitude, is negligible. When this assumption is not made, the differential equations offer formidable difficulties in the way of solution. The problem has been attacked by Riemann and by Earnshaw ; the former also pointed out that discontinuities may exist in the motion. The theory has also been studied by Hugoniot, whose results have been applied to the analogous problem of the propagation of explosion waves by Jouguet (cf. Riemann-Weber, Partielle D[fferentialgleichungen, II. ; Jouguet, Journ. d<> Mathem. 1905, p. 347 ; 1906, p. 6 ; Crussard, Bull. Soc.de Vindust. mineral. ,6, 1907; Gyozo Zeniplen, P/##. Zeitschi; 13, 498, 1912). 77. External Work in Expansion. For the element of external work in an infinitesimal expansion we have generally : 8A = pdi: (1) Isopiestic Expansion : p = const. r = \ pdv = p(c 2 (1) i.e., A p is represented by a rectangular area on the indicator L 2 148 THERMODYNAMICS diagram. It follows from the characteristic equation that T must increase along with v. (2) Isothermal Expansion : T = const. J n For a mol : A' = BT /n If Q T is the heat absorbed, we see from 73 (1), that : A T = Q T , as is indicated by the theorem of Joule ( 71). A T is represented on the indicator diagram by the area enclosed by the v axis, the ordinates v = v\, v = ?- 2 , and the rectangular hyperbola. (8) Adiabatic Expansion : SQ=0, . * . dA Q = pdv. But jn* = constant = k [c v const.]. :-.A a =pii) = -^-y (T 2 TO (8) -, for a mol, A' = -^ (T 2 - TO (36) K X The work done b} 7 an ideal gas of constant specific heat in p passing from one isotherm to another is the same for all adiabatic paths, is independent of the initial or final pres- sures or volumes, and is proportional to the difference of temperature between the isotherms. If we put r = c'j, c,., and Cj,/c,. = K, in (3a), we find that the work done on o be c ~d " adiabatic expansion is c r (T 2 TO, i.e., the FIG. 22. whole of the external work is done at the expense of the intrinsic energy of the gas. IDEAL AND PERMANENT GASES 149 Example. An ideal gas of constant specific heat is taken round a reversible Carnot's cycle, represented by four curves (Fig. 22): AB an adiabatic pv c lf .-. p a v a * = c\, BO an isotherm pv = c 2 , .*. p c v e = c. 2 , CD an adiabatic pv* c s , .-. p e v c K = c 3 , DA an isotherm /w = c t , . . p aVa c . Find the work done in the cycle. Work done area of cycle = ABCD = - AaiB + ECcb + ODrfc - DrZaA. But Aa&B = CVdc, since the work done in passing from one isotherm to another is the same along all adiabatic paths, .-. work = ECcb - DdaA. But p b v b = p e v e = ca ; p a v a K = p b v b K = Cl , Vf -1 =Ci/Cg. Similarly v c K ~ l = c 3 /c 2 , . . (o b /v e ] K ~ 1 = Cl /c s . Similarly (v a /v d } "~ l = ci/c s . . Vb K 1 Ci c j n v jL = ^^ln C J, V a K 1 Cl . . work done per cycle = ^-~ In -. 78. Entropy of an Ideal Gas. Let unit mass of an ideal gas pass reversibly from the state r, T to the state v + dr, T + T + rlnv = 8 +/wTW . . (10) Similarly : S = S + C>T + ~Rlnv . . . (11) where S = M* , C, = Mc r , R = Mr . . . (12) M being the molecular weight. If we eliminate v or T from (10) by means of the equation : pv = rT we find expressions for the entropy in terms of (p, T) or (p, v) : 8 = 8 + cJnT + rlnr = * + /wTW \ = *o' + (f r + /'XT - rbip = so' + /wT f " + r p ~ r l . (13) = so" + (c e + r)lnr + c,hp = s " + /?jT f " + ')/ ) where s ' = S Q -\- rlnr, s " = s cjur. If we multiply the equations (13) by M we obtain the expressions for a mol.: S = So + C r /nT + R/nr = S + /T c Vr R = So' + (C, + R)/T - Rlnp = So' + /"T 11 . + K p ~ K . (14) = So" + (C r + R)/wr + CJnp = S " + lnr f '* + K p 1 '" J where So' = S + R/ ^ ; S " = S - C,/ ^ ; ?' = specific volume. IDEAL AND PERMANENT GASES 151 It we put I/Mr = , the volumetric molecular concentration, or the number of mols per unit volume, .-. S = So"' + CJnT - R/H = So'" + /HT C , - R . (15) where S '" = S - R/nM. Example. If a mol. of an ideal gas changes reversibly from a state of 10 litres at 15 C. to 100 litres at 50 C, show that the increase of entropy is 4.949 S- cal - if C v = 3 cal. If the true specific heat of an ideal gas is a function of temperature of the form : c v = a + 26T it can easily be shown that its specific entropy is : s = s 4- 7T 4- 26 (T 1) 4- rlnv and the entropy per mol (M) is : S = S 4- alnT + 2 (T 1) 4- Elm; where a = Ma, = M6, R = Mr. 79. Free Energy and Potential of an Ideal Gas. By definition we have, for any system, the free energy * = U TS . . . . (1) the potential 4> = * + pV = U TS 4- p\ . (2) For unit mass of an ideal gas : M = |c/?T . . . . . (3) n~ +''*' ' (4) - T - rT/- = - rflnr + (/'(T) . (5) = - rT (Inr - 1) + //' (T) . (6) where r/'(T) = - T-s + Je ( WT - TJ^ . . . . (7) is a function of temperature, u Q , s being the two arbitrary constants depending on the choice of the initial states of energy and entropy. If the specific heat c v is constant we find : . = ' 152 THERMODYNAMICS , r = (MO - o'T) + T(c v -lnT* + 'jr') . . (9), = (MO - *o"T) + T(c v - InV" + ^J where o' = *o rlnr; s" = c^wr. Similarly, for the potential : = MO - s T + T( = U - S T + T(C P - /.T% B ) ^ = U - So'T + T(C,, - lnTpp ~ R ) . (lOo) , = U - S "T + T(C J( - Inv V c ") ) where v is still the specific volume. In the case of varying specific heat, if we put : c r = a + 26T ...... (11), then u = MO + aT + 6T 2 s = s + aZnT + 2?> (T - 1) + rZnv . M/r = jt Ta = ( Ts ) T7nT iT 2 rTZra; + (a + 27>)T } T 2 - rT(Z72 v - 1) + [(12) (a + 2fe)Tj For a mol of gas : M* = (U - TSo) - aTZnT - /3T 2 - ET/w r + (o + 2j8)T 1 ^ 1 M^> = (U - TS ) oTZnT /3T 2 ET(/n; Examples. (1) If a mol of gas expands isothermally and reversihly from volume Y! to volume V 2 tae diminution of free energy is : *&- t{* Po), then, if Boyle's law were strictly obeyed, we should have iwo/pv = 1. Regnault found, however, that /Wo/pr > 1 for all gases except hydrogen, for which p&ol'pv < 1. Natterer (1852) made a series of unsuccessful attempts to liquefy the " permanent gases " by exposing them to enormous pressures (3,000 atm.), in the course of which the very important 154 THERMODYNAMICS fact came to light that at rri'i/ hitjlt ;>/v.s-.s/r.s all //a.sv.s hi-han- HI,/' Jiydt'oqcn in being less compressible than an ideal gasl It follows that, at some intermediate pressure, all gases except hydrogen will obey Boyle's law strictly ; below this pressure they are more, above it less, compressible than an ideal gas. If, therefore, values of pv are plotted against p, all the gases investigated by Natterer, except hydrogen, will give curves showing minima. It is possible that the hydrogen curve may have a minimum at a very low pressure. These results were confirmed and amplified in an extensive pv P 1000 2000 5000 Atm. FIG. 24. series of researches of Amagat (Ann. Clrim. Phys. [5], 19, 435, 1880, [4], 28, 274, 1873, 29, 246, 1873; [5], 22, 353, 1880; 28, 480, 1880) with the gases 2 , N 2 , air, H 2 , C0 2 , C 2 H 4 , CH 4 . He found curves exhibiting minima with all gases except hydrogen, the minima shifting to the right with rise of temperature. Amagat's diagrams for hydrogen, nitrogen, and carbon dioxide are shown in Figs. 23 25. The investigation of gases at very low pressures is a matter of considerable difficulty, and is exposed to various sources of error (cf, Travers, Experimental Study of Gases). IDEAL AND PERMANENT GASES 155 1-0 It was therefore not surprising that very contradictory results were obtained. Thus Mendeleeff and Kiripitscheff thought that pr steadily decreased with the pressure, whilst Amagat found that air obeyed Boyle' s 2-0 law almost exactly at very low pressures. The most accurate experiments in this region are probably those of Lord Ray- leigh (1901-2), who examined hydrogen, nitrogen, and oxygen at pressures of O'Ol mm. to 1*5 mm. in a very ingenious appar- atus. He found that if any deviation from Boyle's law existed at these low pressures it was within the limit of experimental error, remarking that " experimental errors could not well transform an apparently complex to a simple relationship." The condition for the strict validity of Boyle's law is obviously ' . . .' (),=<> (1) The following results have been obtained relating to the altera- tion of pv with pressure : (1) For different gases at and with the same range of pressures, j- - ranges from a relatively large negative value for the easily liquefiable gases to a small positive value for hydrogen. (2) For the same gas at different temperatures and the same range of pressures, ( y ^ changes from a negative value at low temperatures to a small positive value with rise of temperature. 500 Pressure in Aim. FIG. 25. 4000 156 d(pv) THERMODYNAMICS 52 100 -000571 + -000347. (8) For the " permanent " gases at constant temperature z* remains nearly constant from very low pressures up to pressures of 84 atm. The last conclusion was drawn by D. Berthelot from the results of Rayleigh and Leduc, which showed that at very small PV 450 SSO FIG. 26. pressures (1*5 O'Ol mm.) the value of p\r\lp-n'z is constant, and apparently equal to 1 within the limits of experimental error. It is probable, however, that they really prove that small and constant at low pressures : = const. = a . dp (1) (2), dp .'. pi- = ap -f- b where b is a constant. pi- is thus a linear function of p, and the curves in which p, jn- are abscissa and ordinate are straight lines inclined at angles IDEAL AND PERMANENT GASES 157 tan~ 1 (a) to the p axis. If the gas obeys Boyle's law, a = 0, .-. pv = b, and the lines are parallel to the p axis. The product pv tends to a finite limit as p is reduced indefinitely : Lim (pv) = p Q v = b . . (3), which is the ordinate of intersection with the pv axis. The extrapolation to zero pressure may be effected if two measurements of pi at small pressures have been made, provided the gas satisfies equation (1). If this is not the case {e.g., hydrogen chloride, carbon dioxide), a number of points must be fixed on the curve, especially in the low pressure region. The curves for neon, helium, and oxygen are shown in Fig. 26. At C. neon gives a straight line sloping downwards to the pc axis; it is a "gaz plus que parfait." Helium gives a perfectly horizontal line, although this strict conformity to Boyle's law may hold good only at C. ; oxygen gives a straight line sloping upwards to the_pr axis. Carbon dioxide, hydrogen chloride, and the more coercible gases, give lines showing distinct curvature, more marked in the lower pressure region. The definition of molecular weight given in 68 refers only to ideal gases ; in the case of gases which do not follow the gas laws it is obvious that Avogadro's theorem is no longer strictly applic- able. For if we suppose that equal volumes of two gases contain a molecular weight of each under specified conditions of tempera- ture and pressure, these volumes will not remain exactly equal if the temperature and pressure are altered, for each gas exhibits its own peculiar deviations from the gas laws, and, since equiuiolecular weights are now contained in different volumes under like conditions of temperature and pressure, it is evident that the theorem of Avogadro is no longer valid. Since the deviations are only small, a determination of the normal density gives a very approximate value of the molecular weight, and this method has long been in use to decide between possible multiples of numbers obtained by exact gravimetric methods. In recent years, however, a method of finding the molecular weight of a gas directly from the density with an accuracy rivalling that of the gravimetric methods has been elaborated, and will next be considered. 158 THERMODYNAMICS The compressibility coefficient is the deviation from Boyle's law per unit pressure : at the pressure p. If (1) applies, the coefficient between the limits 1 and atm. is : . . PO) The variation from Avogadro's theorem may be expressed in terms of the density per unit pressure. Let m be the weight in grams of v litres of a gas under a pressure p atm. at C. The density is w/r and the density per unit pressure mjpv. When /> = 1 the expression represents the absolute normal density, and, when p is very small, the absolute limiting density. At intermediate pressures the densities per unit pressure are : in in m P&i PsF* p&a With diminishing pressure these values remain constant, decrease, or increase, respectively, according as the gas obeys Boyle's law (He), or is more (Ne) or less (0 2 ) compressible than this requires. The ratio of the absolute limiting densities of two gases is : where mi, ? 2 are any masses, and Owo)i, Ow )a tne limiting values of pv for these. Since, however, the ratio of the absolute limiting densities is the ratio of the masses of equal volumes of the two gases when the latter are in the ideal limiting state, it follows from Avogadro's theorem that this is also equal to the ratio of the molecular weights : MI _ >ni ,~ M 2 ~~ wj ' - .But if DI is the normal absolute density (at N.T P.) : Pi where pi = 1 atm., and similarly for D 2 . D! X . Mi _ \poroj i ,o, ff '~D,T(M \poVoJt IDEAL AND PERMANENT GASES 159 If (1) applies, the equation may be written : Mj M 2 or M! : M 2 : M 3 : . . . for any number of gases. Berthelot calls this: "1'echelle des poids moleculaires." If one molecular weight is fixed by arbitrary choice, the rest are determined, and 2 = 32 is taken as standard. Then DI X 32 is the normal density, and the limiting density or molecular weight is : , , -^ normal density X The products Di(l aj), etc., are the limiting absolute densities, 32 Di(l a l \, etc., the limiting densities. Also: M; M 2 M 3 the molecular volume of an ideal gas at N.T.P. The molecular weight of a gas may therefore be determined, with respect to 2 = 32 as standard, from the data : (a) Normal density of the gas, (6) Compressibility of the gas, (c) Compressibility of oxygen. Berthelot showed that the mean compressibility between 1 and 2 atm. does not differ appreciably from that between and 1 atm. in the case of permanent gases, and either may be used within the limits of experimental error. But in the case of easily liquefiable gases the two coefficients are different. According to Berthelot and Guye the value of aj can be determined from that of aj by means of a small additive correction derived from the critical data, and the linear extrapolation then applied; Gray and Burt consider, however, that this method may lead to inaccuracies, and consider that " the true form of the isothermal can only be satisfactorily ascertained by the experimental deter- mination of a large number of points," followed by graphical extrapolation. 160 THERMODYNAMICS Examples. (I) Molecular weight of hydrogen (linear relation) : mol. \vt. = normal density X (1 - j)o 2 = 2-01413. (2) Molecular weight of hydrogen chloride (non-linear relation) : 2 d o 2 (Pivi\ (Po*>o)o,(extrap,) (<4 2 )o = do 2 X ( 1-42900 139,628 139,769 1-42756 138,959 139,087 1-42768 56,526 56,311 1-42760 mean 1-42762 HC1 (/HOI (JMI''I)HCI (P*O)HCI (^HCI)O = ^HCI X ( 1-63915 54,803 55,213 1-62698 .-. Mol. wt. HOI = 32 x ^HCUO = 32 ^1^ i= 56 . 469 The molecular weights calculated in this way agree very closely with the so-called "chemical molecular weights," derived from chemical analysis, and the method therefore rivals the latter in accuracy, without the attendant complications attaching to chemical operations with pure substances. (Lord Rayleigh, Phil. Trans. 198, 417, 1902; Proc. Roy. Hoc.. 73, 153, 1904. D. Berthelot, Journ. de phys. [3], 8, 263,' 1899; Sitr les thermometres a gaz, Trav. et Mem. dn Bureau Int. Poids ct Mesures, 1907, vol. 13 ; Zeitschr. Elektrochcm. 34, 621, 1904.) 81. Deviations from the Law of Dalton and Gay-Lussac. The fact that the coefficients 1 dv differ from each other for the same gas, and among themselves with different gases, was ascertained by Magnus (1842) and Regnault (1842). With increasing density, /3 increases to a maximum and then decreases. With increasing temperature, and the same initial density, it decreases. With increasing pressure a increases to a maximum, and then decreases. It also increases to a maximum with increasing temperature, and then decreases. IDEAL AND PERMANENT GASES 161 It has often been supposed that at very high temperatures all gases would behave normally, i.e., would approach a limiting ideal state. As a matter of fact the deviations appear to be influenced by the density of the gas, and disappear at infinitely small densities whatever the temperature may be. Thus a saturated vapour at very low temperatures may behave like a permanent gas, on account of its very small density. Regnault expressed the opinion that the coefficients of expan- sion of all gases would probably approach the same limiting value as the pressure was diminished. If pv is a linear function of p, a and /3 approach a common limit, 7, as p tends to the value zero : Lim a = Lim ft = 7. p - p -> Let Aa, B6 (Fig. 27) be the pv lines corresponding to tem- peratures 0i, 6-2 for a molecular weight of the gas. Taking any pressure p', draw O'A'B' at right angles to the j>-axis, join OA' and produce to meet B6 in B". Draw B"0" at right angles to the j?-axis. B&, Aa are lines of constant temperature (isotherms), O'B, 0"B" are lines of constant pressure (isopiestics), OA'B" is a line of constant volume (isochore), because tan O'OB" = ^ = r = const. P 1 r"-t;' p't"-pc a 7f ~, (P COnst.) = 02 01 v' u /IT' O'B' - O'A' A'B' O'A' 1 O'A' 00" - 00' OO' 1 OB" - OA' ~ 2 - 0r OA' A'B" As p diminishes indefinitely, corresponding to rotation of T. M 162 THERMODYNAMICS OA'B" to coincidence with OAB, a and /3 tend to the common limit 1 AB V - 2 - 0, OA' and since, by definition of molecular weight, A and B are the same points for all gases, 7 is independent of the nature of the gas, and is the constant for an ideal gas. Hence 1 This provides a method of determining absolute temperatures, for we have proved in 73 that 1 a where a is the coefficient of expansion of an ideal gas, i.e., the limiting value 7 determined in the manner just described. In this way Berthelot found 1/7 = 273-09 .'. T = 6 + 273-09, The method in use previous to Berthelot's depended on the results of the " Porus Plug " experiments of Joule and Kelvin. (Kelvin's Math, and Phys. Papers, Vol. I.) 82. The Joule-Kelvin Experiment. The experiments of Dr. Joule and Lord Kelvin, instituted with the object of testing the validity of Mayer's hypothesis (dU/dv = 0), were carried out in the former's brewery at Man- chester during the years 1852-62. The principle of the method is very simple, although great difficulties were encountered in its realisation. A stream of gas, under a constant pressure higher than atmospheric, was forced continuously through a porous plug of cotton-wool, or silk, supported in a boxwood tube. The temperatures of the gas before and after passing the plug were determined and after various corrections the change of tempera- ture experienced by the gas was found. By the friction of the air in passing through the plug heat is produced, and at the same time heat is absorbed by the expansion. The difference will be the heat evolved or absorbed. The heat produced by friction is equal to the work which could have been obtained if the expansion had occurred without friction. IDEAL AND PERMANENT GASES 163 Let _p A , r A , T A and p B , V B , T B be the pressures, specific volumes, and absolute temperatures of the gas before and after passing the plug respectively (Fig. 28). We may suppose the transition, as actually effected in the experiment, brought about by pass- ing a volume r A of gas through the plug by means of a piston exposed to the constant pressure p^ and then allowing it to push out, through a volume V B , the piston on the other side exposed to the constant pressure J) B . If Q is the heat absorbed in the vicinity of the plug, A , U B , the internal energies of unit mass of gas in the states A and 13, referred to some standard state as zero, then Q = ("A MB) + (We p&A- Now put p A = p B + bp B , VA = VB + Sr B , T A = T B + 8T B , and drop the suffixes : 8Q = bu + b(pv) = b(u + in-) = bw . . (1) If there is no heat exchange on passing the plug, i.e., the process is adiabatic, as Joule and Kelvin assumed, SQ=0 .... (2), .-. bit, = 6(>-) .... (3), or b(u + 2>tf =bw =0 . . . (3). But (4), /. Cj ,ST + l p bp + rbp = or Now , . and since d-v dv . (5). . (6), . (7), . (8), M 2 ] 64 THERMODYNAMICS we have ^TaCPfj-fOfc or from which (P) .i.(i*_.) . . . (9.), \dp I . e y) V 91 giving the heating or cooling effect per atmosphere difference of pressure on the two sides of the plug. [If the gas is an ideal yas, pv ?'T .*. T ^ - t = *- T = v -- = from (9a).] .op If instead of (4) we write : then, from (3 a) : an equation due to Jochmann (1859). We can also write the Joule-Kelvin equation (9a) in the form (dT),, ; is the rise in temperature on passing the plug, which may be measured on a gas thermometer, because the absolute degree has been denned as equal to the Centigrade degree. We shall call (dT) w = x, the plug effect, and hg-J the J<> nlc- Kelvin effect. It was found that in the case of oxygen, nitrogen, air, and carbon dioxide, there was a cooling effect, so that x is IDEAL AND PEEMANENT GASES 165 negative for these gases, whilst in the case of hydrogen there was a very slight warming effect, so that x is positive for this gas. Further, x was, at a constant temperature of the entering gas, proportional to the fall of pressure through the plug : * = (^\ = const. = o> C. per atm. , (10) PA.I* \dp/ Equation (9a) can be written : T^ = o JI + r .... (11) and (91) is : (11) and (12) are differential equations for T in terms of the magnitudes p, r, c v , c p , co, x, but as these cannot be directly determined as functions of T, the equations cannot be integrated. We may, however, suppose T to be some function of the Centigrade temperature 0, measured on a thermometer containing the gas under. investigation, and put : dp dp W , dv dv dO /1Q \ gf = 3-sT and 3T = 5raT 52 or ^ is now measurable by separate experiments on the con- oa off stant volume, or constant pressure, thermometer with the gas, and we shall have w If /dT c r , Cp, ^, ^, p, pv, and x are now all expressible in terms of 6, and the equations are integrable, giving T as a function of 6 (Jochmann). Joule and Kelvin proceeded differently. They found that u increased (i.e., the cooling diminished) with rise of temperature of the entering gas, and they assumed that in the equation representing a> as a function of temperature, the absolute temperature could be replaced by the approximate value (273-7 + 0). In this case the experimental results agreed with the formula : (273-7 +) where a is a constant for a particular gas. 166 THERMODYNAMICS With a mixture of gases, the cooling effect was less than the average effect for the pure gases. Equation (14) with (10) is now integrable. The results agree, however, somewhat better with an equation proposed by Kankine (1854) : In (16), A/Vp is taken as constant = a. Let us now suppose that T = AS . ..... . (18) where 3 = + - is the absolute gas temperature, for the particular gas, which corresponds with T abs. ; a is the coefficient of expansion. Then T(^)=V in which the plug effect is (y- j as actually measured with the gas thermometer. But c p \ where cj is the specific heat of the gas at constant pressure as measured by a thermometer filled with the same gas ; hence : ?T_ H = *L_ _ = ^( 1 _-!)- 0-000208 (/,* - 1), where p = initial pressure in atm. The initial temperature was O 3 C. The results were in agreement with a characteristic equation due to Kamerlingh Onnes : where A, B, C, are functions of temperature. -sm .. Exercise 1. If I - \ =/is the/rre expansion effect, referring to adiabatic expansion of a gas into a vacuous space, show that : (A. G. Worthing, 1911.) 83. Liquefaction of Gases. The cooling on free expansion, without performance of external work, is made use of on a large scale in the Linde and the Hamp- son apparatus for the liquefaction of the " permanent " gases such as oxygen, nitrogen, and air. Strongly compressed gas is allowed to escape from a fine nozzle, and the cooled gas then sweeps over the spiral metal tube conveying the gas to the nozzle. The cooling effect is thus made cumulative, and after a sufficient time drops of liquid are ejected from the nozzle. We have seen that hydrogen becomes slightly warmed in this process, so that its liquefaction by free expansion would be impossible under ordinary conditions. Dewar in 1900 showed, however, that if the hydrogen was previously cooled, it suffered a further cooling on free expansion, and in this way he obtained liquid hydrogen. Olszewski (1902) found that the inversion point of hydrogen is situated at 80'5 C. This effect of temperature is general, and implies that the ratio of the potential to the kinetic 168 THERMODYNAMICS energy of gases decreases with rise of temperature, a result which is quite intelligible if we suppose the molecular forces of attraction or repulsion to vary inversely as some power of the distance between the molecules, because the latter must necessarily separate to greater distances as the temperature increases at constant pressure. CHAPTER VII CHANGES OF PHYSICAL STATE 84. Heterogeneous Equilibria ; The Phase Rule. According as it consists of one or of more phases, a system in equilibrium is said to be homogeneous or heterogeneous. If a heterogeneous equilibrium is such that the pressure of the system depends on the temperature alone, and is unchanged when the phases alter in relative amount, it is called a completely heterogeneous equilibrium (Roozeboom), or an indifferent equili- brium (Duhem). We shall show later that this can only occur when the phases are of unvarying composition. A particular case is a pure sub- stance in different states of aggregation. For the purpose of classifying heterogeneous equilibria we shall make use of a very general law, called the Phase Rule of Willard Gibbs (1876), the proof of which is deferred to a later chapter. This is an equation which fixes the relation existing between the number of phases (?), the number of components (n), and the variance, or number of degrees of freedom (F), of a heterogeneous system in equilibrium, subject to certain conditions which are usually satisfied in practice. The rule states that F = n + 2 r. By the components of the system we are to understand the least number of independently variable constituents, in terms of which the composition of every phase in the system can be completely specified. The number of components will therefore contribute to the total number of independent variables defining the state of chemical and physical equilibrium of the system. It is not necessary that the components shall be actual constituents of the system ; all that is required is that they shall be inde- pendently variable, i.e., the least number has been chosen. Thus, in systems composed of solid fuming sulphuric acid in presence 170 THERMODYNAMICS of liquid and vapour, we might conceivably have, under various conditions, the following phases : solid : S0 3 , H 2 S 2 7 , H 2 S0 4 , H 2 S0 4 .H 2 0, etc. ; liquid : H 2 S 2 7 , H 2 S0 4 , or mixtures of these ; gaseous : H 2 S 2 7 , H 2 S0 4 , S0 3 , or mixtures of these. The composition of every possible phase could, however, be completely specified in terms of its content of S0 3 and H 2 0, and these two may be taken as components, although neither may really exist, as such, in the system. By the variance, or number of degrees of freedom of the system, we mean the number of independent variables which must be arbitrarily fixed before the state of equilibrium is com- pletely determined. According to the number of these, we have arariant, univariant, bivariant, invariant, . . . systems. Thus, a completely heterogeneous system is univariant, because its equilibrium is completely specified by fixing a single variable the temperature. But a salt solution requires two variables temperature and composition to be fixed before the equilibrium is determined, since the vapour-pressure depends on both. Heterogeneous systems may differ in respect of the number of phases and their state of aggregation and composition. I. One component : (i.) One phase : F = 2, i.e., a fluid (Chap. V.) ; (ii.) Two phases : F = 1, i.e., changes of physical state ; (iii.) Three phases : F = 0, i.e., the phases coexist at one definite temperature and pressure only (triple point) . II. Two components : (i.) One phase : F = 3, i.e., a fluid of varying composition ; (ii.) Two phases : F = 2, e.g., a salt solution in contact with vapour of the solvent, or frozen solvent ; (iii.) Three phases : F = 1, i.e., the equilibrium is completely heterogeneous, e.g., CaC0 3 in presence of CaO and C0 2 ; (iv.) Four phases : F = 0, the quadruple point, as for example chlorine hydrate, ice, solution, and vapour, or the eutectic point of a salt solution, where solid salt, solid solvent, saturated solution, and vapour coexist at a definite tem- perature and pressure. A peculiar case of II. (ii.) is that in which both phases have always the same composition (e.g., solid NH 4 C1 and the vapour composed of NH 3 -f- HCl ; CHANGES OF PHYSICAL STATE 171 mixtures of maximum and minimum boiling-point). This may be treated as case I. (ii.) . III. Three components. The interesting case is F = 0, i.e., r = 5, a quintuple point in which five phases coexist. An example is the system formed on heating a mixture of Glauber's salt (Na 2 S04 . 10H 2 0) and Epsom salt (MgS0 4 . 7H 2 0) to 22, when partial liquefaction occurs with formation of astracanite Na 2 Mg(S04) 2 . 4H 2 0, and five phases are produced : (Na 2 S0 4 . 10H 2 3 Solids -:MgS04.7H 2 saturated solution, vapour. (Na 2 Mg(S04) 2 .4H 2 85. Evaporation. If heat is supplied to a liquid, a portion of the liquid on the surface passes into the state of vapour. If the liquid is freely exposed to air, the whole gradually disappears by this process of evaporation, but if it is contained in a closed vessel the transition into vapour is limited, the formation of the latter ceasing when it has attained a certain pressure, called the vapour-pressure of the liquid. Dalton (Mem. Manchester Phil. Soc. 15, 409, 1801) established the following laws, which have been verified by later observers : (1) The vapour-pressure of a pure liquid depends on, and increases with, the temperature. (2) It is independent of the volume of the vapour-space, provided that liquid is always present. (3) It is almost independent of the presence of indifferent gases in the vapour-space (Law of Partial Pressures). These statements are only true when the liquid is a pure sub- stance, i.e., does not change in composition during evaporation. This constancy of vapour-pressure serves to distinguish pure sub- stances from solutions. The effects of surface tension, appearing when small droplets are used, and of electrification, must also be absent (cf. 100102). Let us suppose we have a mass of pure liquid confined over 172 THERMODYNAMICS mercury in a tube, and that we measure corresponding pressures and volumes for a series of different temperatures. If the isotherms are drawn on a p,v diagram (Fig. 29), each is seen to consist, in general, of three parts : (i.) A nearly vertical part AB, along which the pressure of the homogeneous liquid is diminishing very rapidly with increase of volume. At B the homogeneous liquid sepa- rates into a heterogeneous complex of liquid and vapour, the curve suddenly changes in direction, and gives : (ii.) a horizontal line BC, called the line of heterogeneous states, along which the pressure is constant, and equal to the vapour-pressure at the given temperature. When the last drop of liquid has evaporated, the curve turns sharply downwards at C, giving rise to : (iii.) a curve CD, indicating the compressibility of the vapour. This approaches more and more closely to a rectangular hyperbola with falling pressure. The portion BC is so characteristic that this procedure gives a very accurate method of measuring vapour-pressures. 86. Critical Phenomena. If the temperature is increased still further, it would appear 48-1 probable that the horizontal line of heterogeneous states, which shrinks rapidly with rise of temperature, would ultimately CHANGES OF PHYSICAL STATE 173 vanish at a definite point P. The curve would now slope down- wards from the p axis at every part, without any sharply defined horizontal portion, i.e., there should be no liquefaction at all. This remarkable result was, in fact, obtained by Andrews in 1869 with carbon dioxide. Being impressed with a curious observation of Thilorier's (1835), that liquid carbon dioxide expands on warming four tunes as rapidly as the gas between and 20, he submitted the relations between the pressures and volumes of this substance at, different tempera- tures to a very careful investigation. The gas was compressed over mercury in a strong glass tube, and the (p,v) isotherms plotted for 13'l, 21-5, 31'l, 31'o, 33'5, 48'l (Fig. 30). Below 31'3 separation into a heterogeneous system ("liquefaction") began at a definite pressure for each temperature, and the isotherms exhibited flat portions. These diminished in length with rise of temperature, and at 31*3 the curve simply changed its direction at a pressure of 75 atm. and showed no flat part at all. At the same time, no liquefaction could be observed in the tube, no matter how high the pressure was taken. Above 31 C> 3 the isotherms gradually lost the definite change of direction, which may be regarded as a lingering suggestion of the flat portion, and approached the hyperbolic form at 48'l. The results of Andrews' experiments may be summarised and generalised in the statement that : There is, for every gas or vapour, a definite temperature, above which it is impossible, by increase of pressure alone, to effect a liquefaction of that gas or vapour. This is called the critical temperature (0 ). The critical temperature is the highest temperature at which a gas may be liquefied by pressure, and, since the pressure increases with the temperature, there will correspond to the critical tem- perature a critical pressure(p K ), which is the greatest pressure which will produce liquefaction. This pressure is given by the ordinate of the critical point K, or point of inflexion, on the critical isotherm. The volume of unit mass of the substance, under the critical pressure and at the critical temperature, is called the critical volume, r K . The reciprocal of this is the critical density, Ps. = !/* A substance existing at its critical temperature and pressure 174 THERMODYNAMICS must of course have the critical density ; it is then said to be in the critical state, (PK.,V K , K ). The ratio # K /^K = k is called the critical coefficient of a substance (Guye). 87. Continuous Transition of States. It is possible, by suitable changes of pressure and temperature, to pass directly from a point a in the vapour region to a point /3 in the liquid region, without any discontinuity in the way by separation into two phases. The vapour is first heated above the critical temperature, and so brought on an isotherm AB (Fig. 31) above the critical isotherm. By raising the pres- sure, the state passes along this isotherm until the volume is reduced to the value correspond- ing to a point ft lying on a liquid isotherm. The temperature is now lowered, at constant volume, till it reaches the value corresponding to the isotherm on which ft lies. If the vessel is now opened it is found to be completely filled with homogeneous liquid. There are, therefore, two ways of passing from vapour (a) to liquid (/3), or rice versa : (i.) Along aCD/3, through the region of heterogeneous states a Heterogeneous, or Discontinuous, Path. (ii.) Along aAB/3 a Homogeneous, or Continuous, Path. The dotted curve PQR, separating the region of heterogeneous states from the regions of homogeneous states, is called the limiting carve of heterogeneous states, or the border curve. This conception of the continuity of the liquid and gaseous states, expressed by Andrews in the form that liquid and vapour are " only distinct stages of a long series of continuous physical changes," is the basis of a remarkable theory of J. D. van der Waals, which will be considered later. Andrews proposed to call a substance existing above its critical temperature, a gas, as distinguished from a vapour, existing below that temperature. Whereas vapours may be liquefied by pressure CHANGES OF PHYSICAL STATE 175 alone, gases require cooling below the critical temperature before they can be compressed to liquids. This explains the failure of Natterer's attempts to liquefy the "permanent gases": as a matter of fact their critical temperatures lie below the atmospheric temperature. Above the critical temperature, the isotherms show two curva- tures ; on the right, pr increases as p decreases, on the left, pi- and p increase together. This change of curvature, suggesting vapour and liquid states respectively, is clearly seen in Andrews' curves for 32'5 and 35*5. It is also observed in the isotherms of permanent gases, and will be discussed later. 88. Thermodynamics of Evaporation. Let a quantity of liquid and its vapour be in equilibrium, in a cylinder with a piston under a pressure (p fy>), at a tempera- ture (T ST). ( P &P) * s the vapour-pressure at (T 8T). Let PR, QS (Fig. 32) be the isotherms (T 8T) and T, where P,Q correspond to pressures (p fy>) and p. PR, QS are horizontal, since p is a function of T only, and does not depend on the volume. We now take the system round a reversible Carnot cycle, the reversibility for such a system, in the limit, having been pre- viously demonstrated. (1) Compress adiabatically along AB till the temperature reaches T. (2) Transfer to the hot reservoir and expand isothermally till a mass in of liquid has been evaporated. The heat absorbed along BC is wL e where L e = latent heat of evaporation at T. (3) Transfer to the non-conducting stand, and expand adia- batically along CD till the temperature falls to (T ST). (4) Transfer to the cold reservoir, and compress isothermally along DA till the initial state is reached. FIG. 32. 176 THERMODYNAMICS The adiabatics are small curve-elements, which, for an infinite- simal cycle, may be regarded as parallel straight lines. Draw BF, CE perpendicular to the volume axis Or. Work done in cycle = area of parallelogram ABCD = area of rectangle FBCE = FB . BC. But FB = increase of pressure at constant volume due to rise of temperature 8T = -~ r ST, and BC = change of volume due to evaporation of in gr. of liquid = (volume of m gr. of vapour) (volume of m gr. liquid) = mi's mi'i = in(i'z i'i), where i\, r 2 are the specific volumes of liquid and vapour at T, respectively. Hence, work done per cycle = -^ 8T(r 2 i'i)nt. By Carnot's theorem, and the definition of absolute tempera- ture, this magnitude = Q X -m- , where Q = heat absorbed from the hot reservoir, 5T = mL e X j This equation is known as the Clapeyron-Clausius equation. (Clapeyron 1834 ; Clausius 1851.) We need not write this ( ~ I , as would appear at first sight, since p is a function of T alone, and is independent of the volume so long as both phases are present. A given rise of temperature therefore produces the same change of vapour-pressure whether the volume is kept constant or not. L e =: latent heat of evaporation. t'2 vi EE Ar = volume change accompanying unit mass of phase transition at the pressure p. T = temperature at which the phases are in equilibrium under the vapour-pressure p. Corollary 1. Since Ar, L e are known by experiment to be CHANGES OF PHYSICAL STATE 177 positive, the vapour-pressure of a liquid always increases with rise of temperature. Corollary 2. The increase of vapour-pressure for a very small change of temperature is proportional to the change of tempera- ture ; also the rise of boiling-point of a liquid for a very small increase of external pressure is proportional to the increase of pressure. The boiling-point of a liquid is the temperature at which its vapour-pres- sure is equal to the total pressure to which the liquid is subjected, usually the atmospheric pressure, and a liquid begins to boil, i.e., emits bubbles of vapour, when its temperature is raised to the point at which its vapour- pressure is equal to the total pressure on the surface. The bubbles of vapour are always evolved at definite points, where small gas-bubbles (either evolved from the liquid, or due to air adhering to the vessel) are present. As boiling proceeds, these points diminish in number, owing to expulsion of gas, and after a sufficient time disappear altogether. The temperature then rises several degrees above the boiling-point without formation of bubbles ; at last, however, an explosive rush of vapour is evolved, usually in one large bubble from the bottom of the vessel, the temperature sinking again to the boiling-point, and the process is repeated. According to Aitken (1874), normal boiling occurs only if bubbles or cavities, and hence the vapour of the liquid, are present. This result is quite general ; the phase-transition proceeds normally only if both phases are present ; if only one phase is present, the transformation into the other lags behind the change of temperature or pressure, and then occurs violently : the phenomenon is called Suspended Transformation of States. If the vapour of the liquid is allowed to form inside a closed vessel, its pressure continually increases with rise of temperature, the maximum vapour-pressure being the critical pressure. Con- versely the boiling-point is raised if the pressure is increased by enclosing the liquid and vapour in a vessel fitted with a safety-valve, or by increasing the external (e.g., atmospheric) pressure. The rise of boiling-point was first described by the French inventor Denis Papin (1679) and applied to the construction of a " new digester, or engine for softening bones " now called an autoclave. The rise of boiling-point is calculated from the formula oT = -== ty. Example. Evaporation of water at 100 under 1 atm. pressure : T = boiling-point = 273 + 100 --=373, vi = sp. vol. of saturated vapour = 1,674 c.c. v% = sp. vol. of liquid = 1 c.c. (nearly) d P _o-i.) mm.Hg , n _ 27'12 X 1,013,250 dyne per om. g dT~ ' 1 0. * 760 1 T. N 178 THERMODYNAMICS _ ;n;j x 1,073 x 27-12 x 1,013,250 _ 538 . 8 cal / obs - 38 . 7 ^ 760 If we refer the latent heat to a mol of liquid, instead of unit mass, we have, if MI, M 2 are the molecular weights of liquid and vapour : Vi = MI?'I ; V 2 = M 2 r 2 , the molecular volumes, \ e = MiL,,, the molecular heat of evaporation, dp _ ^e . . . (2) Tf M _ M . P __A__ . (2o) Ma ' dT T(V 2 - Vi) Corollary 1. Since Vi is very small compared with Y 2 , we may neglect it as a first approximation, and write : where V = molecular volume of the vapour at (p, T). Corollary 2. If the saturated vapour obe} 7 s the gas laws : V = BT/p, . d^_ X e p ' dT ~ RT 2 1 dp dln . or = = The condition assumed is not very approximately true unless we are dealing with a substance of very small vapour-pressure (e.g., mercury at 15 has a vapour-pressure of O'OOOSl mm. according to Pfaundler (1897) ). It will therefore apply more particularly to low temperatures. Example. In the case of water : R = 1-985 cal. p = 760 mm. T = 373 dp nf _ mm. 3f = 27 ' 1! -i 5 " T A c 1-985 X373 2 X 27-12 ' Le = M = ~Wx^8- = 547 ' Cal " which is about 2 per cent, too large. The molecular volume of saturated water vapour at 100 is less than that calculated from the gas laws (Fairbairn and Tate, 1860 ; Clausius, 1861, showed CHANGES OF PHYSICAL STATE 179 that the values for r 2 obtained by these observers agreed very well with those calculated from the latent-heat equation). l/r a is called the density of saturated vapour. The methods used for the determination of the density of a saturated vapour will be found in the treatises on Physics (cf. Young, Zeitschr. Physical. Chem., 70, 620, 1910, who also finds the very simple relation : where p = vapour-pressure; A, B = constants). Corollary 3. By transposing the terms of dln A e we obtain According to Clausius the latent-heat of evaporation of a liquid is approximately a linear function of the temperature, and diminishes with rise of temperature (cf. 94): X e = A aT, which gives on integration ~R hip = ~ :p InT + constant, where A, B, C are constants for a particular substance. This equation is due to Kirchhoff (Pogg. Ann. 104, 612, 1858.) The equation is usually called Dupre's, or Rankine's, formula. The latter name might well be adopted for the equation : 7 -- (Rankine, Edin. Phil. Journ. 1849), recently put forward again by E. Bose (Physik. Zeitschr. 8, 944, 1907). At least thirty formulae representing the vapour-pressure of a liquid (usually water) as a function of temperature have appeared, and new ones are always being published. Some of the best known are due to : (1) Biot (1844) : log p = a + ba r + c,8 T a, b, a, /3 are constants, I* = 6 C. const. N 2 180 THERMODYNAMICS (2) Magnus (1844) : p = ab t+e rnn (3) Bertrand (1887) : p = "\_ , m (4) Van der Waals (1899) : log p = a ^ K + a + log j> K where T K , p K are the critical constants. (Cf. Winkelmann : Plnjsik., III., 3, 903-961.) The nature of the assumptions entertained in the deduction of KirchhofFs equation ensure the theoretical validity of the latter only at low temperatures. Notwithstanding this, Juliusburger (1900), from a review of existing data, found the equation to give results deviating by 3 per cent, at most from the observed values for 74 substances out of 80 examined. He considers that it may be regarded as an empirical formula with three constants, up to the critical point, and gives the values of A, B, C for a number of substances. (P. Juliusburger, Drude's Ann. 3, 618, 1900.) 89. Empirical Relations. Interpolation and extrapolation of the vapour-pressure curve could be carried out by means of successive applications of the Clausius equation were it not that data are usually lacking. Eamsay and Young (1886) discovered an empirical rule by which the interpolation can be effected. Let TA, T A ' be the absolute boiling-points of a substance A under pressures p and p', T B , T B ' the boiling-points of another substance B under the same pressures p and p', then |4 = ^ + c (T B ' - T B ) IB IB where c is a constant, usually very small. In the case of substances which are chemicall similar rr i rp m m^ FTT = rT^ = constant, T ' T and CftHgBr, esters, aromatic substances) c = and -f~~ = 7=^. IB IB From this we easily find T/ - T A TB' T B a rule which was put forward by Diihring in 1878. CHANGES OF PHYSICAL STATE 181 If in the equation ^ = ^ - r we put (j- 2 i'i) = and al I(r 2 1\) L = 0, simultaneously, the gradient of the vapour-pressure curve becomes indeterminate, and another condition must be given to fix the latter, viz., the mass of substance. This was verified by Cailletet and Colardeau (1891), who found that the vapour-pressure curves for various quantities of liquid fell together up to the critical point, but then diverged into a bundle of curves. Of the two conditions : the first is seen to be satisfied, at the critical point, from an inspection of Andrews' diagram, and the second is also verified by experiments of Mathias (1890). Both conditions must be satisfied simultaneously, for the con- dition that both phases become identical at the critical point certainly requires that their intrinsic energies and entropies per unit mass are equal : w 2 Ml = 0, s 2 si = . . . (i.) But ?/ 2 MI = L e X l 'a i'i) (ii.) , Le ,... ^ and s 2 si = -^ (in.) Since T is finite, 2 i can vanish only when the two con- ditions (a) are simultaneously satisfied. If the phase transition is of such a kind that either condition can be satisfied separately, but not both together, it cannot exhibit a critical point (Tammann). 90. Metastable States. States such as superheated liquid and supercooled vapour are known as metastable, they are not of themselves unstable, but become so on introduction of a small amount of the stable phase. Metastable states are represented on the indicator diagram by the prolongations of the isotherms of homogeneous states beyond the intersections with the line of heterogeneous states. Thus, the productions of the liquid and vapour portions of the : isotherm, Fig. 33, meet the heterogeneous parts of the isotherms 2 , #3 at Q, P, respectively. At each of these points there are tico con- ditions of existence possible for the system, thus : 182 THERMODYNAMICS P corresponds (a) to a stable heterogeneous system on the # 3 isotherm at P ; (b) to a metastable homogeneous state on the QI isotherm produced to P (# 3 > #1). Q corresponds to (a) a stable heterogeneous system on the 2 isotherm, (b) a metastable homogeneous state on the production of the 61 isotherm (0* < #1). James Thomson (1871) suggested that these prolongations of the di isotherm of a liquid and vapour might ultimately bend round and join, so that the discontinuous flat part of the isotherm is replaced by a continuous curve (Fig. 34). a/3 corresponds to FIG. 33. FIG. 34. superheated liquid, 78 to supercooled vapour ; po,*'o refer to A, and p,v to M. Thus : u = MO + m ^e (l yv PQI'O) + The change of U on passing to any other state M' is : u' u = m'L e mL e (p'v' pv) + . (6') ^nHT . (9') il 1 \ v +? \7rr T - (9 " } J T Corollary. If M, M' are on the same adiabatic, the external work done is given by (9"). 92. Specific Heat of Saturated Vapour. Clausius (1850), in considering Regnault's data for the latent heat of steam, introduced a new specific heat, applicable to either phase of a saturated complex of two phases, viz., the amount of heat absorbed in raising the temperature of unit mass of a saturated phase by 1, the pressure being at the same time varied so as to preserve the substance in a saturated state. In the case of a vapour, this is called the specific heat of saturated vapour (. Now at temperatures considerably less than T K , > nl changes along with T so as to maintain saturation. To find the value of ( -=- e . i.e.. the rate of change of the latent dp heat with the saturation pressure, we make use of the rule for change of independent variable. --.rfT=^ .... (7) dL^_dL e dp ' TlT "dp ' dT . dL< ^ =a _ a > + L f (8) ' dp ' dT T /. from (1) and (4) : f-w-if+^-o -**> . w If the change of pressure occurred at constant temperature, dT = 0, hence: Equations (6) and (9) are due to Planck (1897). 190 THEEMODYNAMICS In the case of water and steam at 100 C. c ' = I'Ol ; ~ = - 0-610 ; v" = 1674 ; ( ~ ) = 4'813 t> ~L e = 538-7 ; v' I'O ; (^\ O'OOl ; thence c p " c p ' = 0'51 .-. Cp " = 0-50. The mean specific heat of saturated steam at a temperature slightly higher than 100 was found by Eegnault to be 0'48. If the saturated vapour is assumed to obey the gas laws * = | 5, (W rfL e Thus, c j; " = I'Ol 0'61 = 0'40, which is considerably too small.* 94. Kirchhoff's Vapour-Pressure Equation. In the case of the evaporation of a pure liquid : P =L e -p(v"-v') . . . (1) where p = u" u' .... (2) For two different temperatures TI, T 2 , P% Pi = W Uz (iii" MI}- If we assume the specific heat of the vapour to be constant : u" = wo" + c/T . . . . (3) * -?/ "" /)/ ff ^ ft /T~1 rp \ /O \ Assume for the liquid an expression of the form : f f f /m TT\ \ //i \ where c' is a mean specific heat between T 2 and TI, .-. P2 - Pl = (T a - TO (c p " - c') . . . (5) Put p! T! (c v " c') = c and Pa = p, T 2 = T, .'. p = L e p(v" v'} = c + (c," c')T . . (6) .'. also T(v" v') ^ p(v" v') = c + (c," c')T. If we neglect /' in comparison with v", and assume the vapour obeys the gas laws, we have, for a mol : ET 2 ^ -r=ET + C + (C;'-C')T . . (7) * (Cf. Callendar, Proc. Roy. Soc., 67, 266, 1901; Dieteiici,^ln.PAy., 13, 154, 1903.) CHANGES OF PHYSICAL STATE 191 Put C/R = B', jj((V'-C') = 7 . l ^_I , B ' , T p dT ~ T "T" f* "T T . . (8) or log 10 p = A' - + C log M T . . . (8) This is Kirchhoff's vapour-pressure equation ( 88). It may be written in the form (Hertz, (1882) ) : i^-rV w where o, A are constants. It was stated (I.e.} that (8) gives very accurate results with empirical values of 7, A, and B. Bertrand found, however, that these values do not even approximately agree with those calculated from A = C/R, 7 = ( " C')/E, except in the case of steam. Graetz (1903) has proposed a slight modification of Kirchhoff's equation : . . . (10 -i. JL where k is another constant. 95. Sublimation. Many solid substances (camphor, iodine, naphthalene, etc.), are known which are appreciably volatile at ordinary temperatures. Others, such as the metals, are apparently quite fixed, but they probably possess a definite, although very small vapour-pressure, even at ordinary temperatures. Thus, if magnesium is heated to 550 for a few hours in a magnesia boat enclosed in a vacuous tube it sublimes in beautiful crystals on the cool part of the tube. The vaporisation of a solid without previous fusion is called sublimation ; the vapour -pressure (like the vapour-pressure of a liquid), is definite for each temperature, is independent of the volume of the vapour space, and increases with rise of temperature. Eegnault investigated the vapour-pressures of water and benzene, both in the liquid and solid states, and represented his results graphically. He con- cluded that the curves for the liquid and solid joined at the melting-point, and gave a continuous curve. This was shown theoretically to be incorrect by Kirchhoff ( 1 858), who proved, by a method to be described later, that the 192 THERMODYNAMICS curves must at least have different tangents at the melting-point. Later experimenters confirmed this conclusion. If a solid is heated under atmospheric pressure, its vapour-pressure increases with rise of temperature, and it may happen that the vapour-pressure of the solid becomes equal to atmospheric pressure before the melting-point is reached. In this case, the substance sublimes away without previous fusion. But if the melting-point is reached before the vapour-pressure reaches atmo- spheric pressure, the substance will melt before boiling away. JI JG 37 The two cases are represented in Fig. 37. The horizontal isopiestic cuts the vapour-pres- sure curve of the solid in the first case, that of the liquid in the second. Melting can be brought about in case (1) by an increased pressure. The vapour-pressures of ice at various temperatures have recently been carefully determined; Scheel (1905) finds that they may be represented by the interpolation formula : logj>(mm.) = 1 1-4796 -0-4 log T- 26 ^' 4 . Kirchhoff's formula is therefore applicable to sublimation ( 88, Cor. 3). The thermodynamic treatment of sublimation is exactly analogous to that of evaporation. Ramsay and Young (1884) have proved experimentally that during sublimation the tempera- ture remains constant, and heat is absorbed ; for unit mass this is the latent-heat of sublimation, L> We have, therefore, the corresponding equations : Evaporation. Sublimation. (9.\ dp e L, dp s L^ dT ~ T(A^) e dT ~ T(Arj, (3) _ dT RT 2 dT RT 2 (4) log j>. = A - ? + C log T log p. = A' - I' + C' log T. 96. Fusion. The old classification of bodies into solids, liquids, and gases, based on differences in viscosity and elasticity, is not altogether satisfactory. We shall therefore adopt a method in which bodies are divided into two classes according to the nature of their CHANGES OF PHYSICAL STATE 193 internal structure (Lord Kelvin, Ency. Britt., Art. Elasticity, 1878). Let any point be taken in the interior of a homogeneous body, and suppose lines OPi, OPa, OP, . . . drawn in different directions through (Fig. 38). If now the physical properties of the body (e.g., thermal expansion, compressibility, refractive index, electric and thermal conductivities, dielectric constant, and magnetic permeability) are measured along OP b OP 2 , OP, ... we find that all the bodies fall into one or other of two large groups : (1) Bodies in which the physical properties are identical in all directions : e.g., glass, air, water. This class of bodies includes all gases, most liquids, and the so - called " amorphous solids " such as glasses (that is, solids showing no external crystalline form, and breaking with a glassy fracture). Bodies of this type are called Isotropic Bodies. (2) Bodies in which some, or all, of the physical properties are different in different directions. This class includes crystal- line solids, and liquid crystals. Thus all crystals except those belonging to the regular system are optically different along different axes; crystals of the regular system, although optically isotropic, show differences in electrical properties along different directions. Bodies of this type, the properties of which are vectorially distributed in space are called Anisotropic Bodies (or, JEolotropic Bodies). Between isotropic and anisotropic solids there is another marked distinction, based on their behaviour when heated. Whereas the former (e.g., glass) gradually soften, and pass con- tinuously, through various grades of plasticity, into more or less mobile liquids, the anisotropic bodies exhibit a sharply denned melting-point, i.e., a definite temperature at which transition from solid to liquid occurs. The process of softening throughout the mass, characteristic of isotropic bodies, is absent, and fusion occurs only at the sharply defined boundary between the crystal and its melt. (Softening before fusion, in which distortion, but T. o 194 THERMODYNAMICS not destruction, of the crystals occurs, is exhibited by a few crystalline solids, such as iron, platinum, and sodium; it is applied in the process of welding.) Tammaun has advanced the view that "amorphous solids'" are really liquids which have been cooled far below their freezing-points, and have thereby acquired great viscosity, but have not crystallised. They are super- cooled liquids. This hypothesis is supported by the following evidence : (1) Such bodies have no definite melting-point, but pass continuously into the liquid state on heating. (2) They show signs of plasticity when submitted to prolonged stresses (pitch, glass, sealing-wax, etc., "flow" very slowly). (3) A liquid may often be rapidly supercooled into a glassy condition. (4) The amorphous state frequently passes spontaneously into the crystal- line state (plastic sulphur, "devitrification"' of glass, Gore's amorphous antimony). Although Carnelley once thought he had been able to super- heat ice ("hot ice"), it is almost certain that no solid can be main- tained alone at a temperature higher than its melting-point. Tammann (Zeitschr. plujsik. Chem., 68, 257, 1910) finds, however, that a crystal- line solid may, under certain cir- cumstances, be superheated in the presence of its melt. This occurs when the supply of heat to the crystal is sufficiently great in com- parison with the linear velocity of - crystallisation of the supercooled liquid (cf. Findlay : Phase Ride). The impossibility of realising a superheated crystal is ascribed by Tammann to the very large number of '" centres " in the crystal in which fusion can commence, in contrast to the relatively small number of isolated points in the supercooled liquid where crystal-clusters begin to form. The changes of volume, and the quantities of heat absorbed during fusion, are much less than the changes which accompany evaporation : Substance At- (fusion) Av (evap.) L f L e Water 0'125 c.c. + 1646 c.c. 80 cal. 536 cal. Acetic Acid + 0'121 c.c. + 385 c.c. 43 cal. 97 cal. 6m, FIG. 39. CHANGES OF PHYSICAL STATE 195 The changes of volume accompanying fusion were first noticed by Reaumur (1726), who observed that when a fused mass solidi- fied in a crucible, the surface was usually concave, indicating contraction on solidification, but in a few cases was convex, indicating expansion. Wax and water being common examples of the two classes, we may speak of bodies of the wax-type or ice- type, respectively, according as they expand or contract on fusion. The curves for r = / (9) in each case exhibit a discontinuity at the melting-point ; in the first case the curve rises, in the second it falls * (Fig. 39). The volume-changes accompanying fusion have been measured by the dilatometeu (Pettersson, 1881, etc.). The majority of substances belong to the wax-type ; water, bismuth, bismuth sulphide, cast-iron, nitre, and some alloys, belong to the ice-type. 97. Thermodynamics of Fusion. The effect of change of pressure on the melting-point of a substance can be predicted qualitatively in the following manner : Let a Carnot's cycle be carried out between the temperatures T and (T ST) with a mixture of solid and liquid as the work- ing substance. During the fusion process heat is invariably absorbed, hence fusion must occur along the isotherm correspond- ing to the temperature of the source. Now, there are two cases : (i.) The substance expands on fusion (wax-type) the T isotherm is therefore traced out from left to right on the indicator diagram, and since the (T ST) isotherm is necessarily traced in the opposite direction, and the cyclic area is positive and therefore traced out clockwise (by Carnot's theorem), it follows that the (T 8T) isotherm lies below the T isotherm. (ii.) The substance contracts on fusion (ice-type) the T isotherm is traced out from right to left, and the above reasoning shows that, in this case, the (T 8T) isotherm lies above the T isotherm. Along each of these isotherms solid and liquid are in equili- brium ; each corresponds to a melting-point under a given pres- sure. Thus we see (qualitatively) that the melting-point of a sub- stance of the wax-type is raised by increasing the pressure ; that of a substance of the ice-type is, on the other hand, lowered. * According to Kopp, the curve is really continuous ; Pettersson and later workers consider that this is the case only in presence of impurities. o 2 196 THERMODYNAMICS This conclusion was arrived at, from considerations based on Carnot's principle alone, by James Thomson in 1849. He also calculated the magnitude of the effect, in the case of ice, by means of a cyclic process. Since the reasoning is the same for both cases, we shall deal with both together, giving appropriate diagrams. A mixture of solid and liquid in equilibrium at a temperature (T 8T) under a pressure (p fy>), is taken round a small Carnot's cycle ABCD. Work done in cycle = FB . BC where the symbols have the same significance as in 88, except p (Wax Type) Q \B \C 5 T P V V R (T-6T) F \ E \ X Y FIG. 40. P (Ice Type) P 0\ E A\F /? , \ \ s T c ^ B x FIG. 41. that n, i- 2 , are now the specific volumes of solid and liquid. Heat absorbed from source = niL f , .'. JF 8T X Ar X m = mL f X -^, by Carnot's theorem, _ dp ~ L, Thus, for small changes, the change of melting-point, or freezing-point, with pressure is given by : . .' . . (2) There are two cases possible : (i.) Ar is > (wax-type) .'. 8T has the same sign as bp, or the melting-point is raised by increase of pressure (Fig. 40). 5T = ~ bp CHANGES OF PHYSICAL STATE 197 (ii.) Ar is < (ice-type) .*. 8T and bp have opposite signs, or the melting-point is lowered by increase of pressure (Fig. 41). In the case of water : L/ = 80-4 gr. cal. per gram = 4-2 x 10 7 X 80-4 ergs, v* = 1-000 c.c., v l = 1-091 c.c., T = 273. If Sp = I atm. = 1013130 dyne/cm. a , The prediction of James Thomson was verified experimentally by his brother, Lord Kelvin, in 1850, who found ST = 0'0072 C. per atm. Dewar (1880) found that this remained practically constant up to 700 atm. Bunsen (1850), Hopkins (1854), Batelli (1887), and de Vissier (1892) also made experiments on the effect of pressure on the melting-point of bodies of the wax-type. The latter found for acetic acid : 5T = + 0-02435 C. per atm. (obs.). Now L,- = 46-42 g. cal., T = 289-6, A v = + 0-0001595 litre . . *T = + 0-0242 J C. per atm. (oalc.). The data in this field have recently been greatly extended, especially by Tammaun, to whose monograph : Kristallisieren und ticltmehen, Leipzig, 1903, the reader is referred for a detailed account of the subject. The equation of Planck ( 93) for the dependence of the latent heat of a change of state on the temperature and pressure applies, of course, to fusion as well as evaporation : T /T T f /^ rf \ /^'\ \ In the case of ice at C. : Cp, the latent heat of fusion increases with rise of tempera- ture, in contrast with the latent heat of evaporation. 198 THERMODYNAMICS 98. Allotropic and Polymorphic Change. If two or more crystalline forms of a substance exist, they are called polymorphic forms ; if the substance is an element (C, S, P) they are called allotropic forms. The Thomson equation obviously applies to this case : rjT_ T(r a - n) dp ~ L B where T = transition temperature ; L M = latent heat of the transition ; TI, v% = specific volumes of the denser and lighter forms, respectively. Keicher (1883) applied this to the sulphur transition : S a > SjS, in which ; 2 - ri = 0-0000126 c.c. ; L H = 2'52 cal. ; T = 273 + 95-6, 7m .-. J = + 0-045. (Obs. 0-05 C. per atm.) Polymorphic transition may occur very rapidly (e.g., tetrabroin methane, boracite), but usually takes a fairly long time (e.g., sulphur). In some cases there is no apparent change in finite time (e.g., diamond to graphite). In all cases a rise of temperature, and contact with the second form, accelerate the process. 99. False Equilibrium. If the pressure or temperature of a system of two phases, a and /3, in true equilibrium, is altered, even infinitesimally, the one phase passes over completely into the other, the change a > ft, or the change ft > a, taking place according as the temperature, or pressure, is greater or less than the value corresponding with equi- librium. The rapidity of the change is, when the pressure or temperature difference is not too large, regulated by the rate at which heat is supplied to, or with- drawn from, the system. If, however, experiments are made over an equilibrium curve extending through a wide range of temperature, it is found that the velocities of transition at low temperatures are very much less than would be expected from the diminished heat-flow alone. At p kg /cm* FIG. 42. CHANGES OF PHYSICAL STATE 199 some low temperature the transition may cease altogether, and the two forms may coexist under conditions such that one form should pass completely over into the other. A very instructive case is the transition of the two crystalline varieties of phenol, phenol I and phenol II, studied by Tainmann (Fig. 42). Above 25 the transition occurs reversibly, phenol I passing into phenol II with evolution of heat and diminution of volume. If, at a given temperature, greater than 25, the pressure is changed from the equilibrium value to a value greater or less than this, the form I passes into form II, or vice versa, until, after 1020 minutes, the equilibrium pressure is recovered. Below 25= the relations are quite different. Thus at 20, a difference of 28 kg. cm.' 2 remains between a rising and a falling pressure after half an hour, and the difference increases with fall of temperature until at 21 it is no less than 600 kg. cm. 2 . The curve of transition down to 25 is one of true, or rever- sible equilibrium ; at lower temperatures it is divided into two parts, called by Tammanu " limitative curves," AC and BC, enclosing a region in which both phases coexist in intimate contact, if not indefinitely at least for a con- siderable period of time. This region is called by Taninianu the region of pseudo-equilibrium, and by Duhem the region of false equilibrium. If the pressure is changed so as to pass outside the region ABC, it slowly comes back to the value on one of the limiting curves, and then ceases to change. There are two possible interpretations : (i.) The velocity of transition, which is known to decrease very rapidly with decrease of temperature, has become so small that no appreciable change occurs in ordinary periods of time. (ii.) The transition has ceased altogether, and the system is in a state of false equilibrium. Two types are recognised (Duhem, Traite tie Mecanique chimique, I, ii) : () Those in which two or more forms coexist under conditions where only one form is theoretically stable " faux equilibres reels " (Duhem) ; " Psuedo- gleichgewichte " (Tammann). E.g., the two forms of phenol. (b) Those in which one form is existing in a metastable state under con- ditions whei'e the other form is stable, and destroyed by the introduction of a trace of the second form "faux equilibres apparent" (Duhem) meta- stable equilibria. E.g., superheated or supercooled liquids. The existence of such states is closely connected with the phenomena of capillarity (Duhem, loc. cit. II., 2, also pp. 66 et seq. ; III., 121. Gibbs, Xcientif. Papers, I. 252 et seq.}. All cases are covered by a very general theorem to the effect that whereas, when the thermodynaniic conditions of equilibrium are satisfied the system will be in equilibrium, the converse is not always true (Moutier, 1880). 100. Influence of Compression. In the preceding considerations of vapour-pressure it has been assumed that the pressure is uniform throughout the whole system. A condensed phase may, however, exist under a pressure different from that of its accom- panying vapour, as in the following cases ; 200 THERMODYNAMICS (i.) A liquid or solid phase may exist under a pressure greater than its vapour-pressure if : (a) An indifferent gas is pumped into a closed vessel containing the two phases. (#) Liquid is put in contact with vapour through a capillary tube, or collection of such tubes (" sieve "), not wetted by the liquid, and is compressed from behind. (y) A solid is held in a net of wire gauze which exerts compression on it. (ii.) A liquid may exist under a pressure less than its vapour-pressure when it wets and rises in a capillary tube placed in the liquid, and exposed above to the pressure of the vapour alone. Let p be the ordinary equilibrium pressure in a system com posed of a condensed phase and its vapour, and let, r, V be the specific volumes of these phases, respectively, under a pressure p. We now assume that these values are altered to p', r', V, when the condensed phase alone is exposed to a pressure P + p. Now r' = r r>jP, where rj is the coefficient of compressi- bility ( 23), since for all pressures the relative diminution of volume of a fluid is proportional to the increase of pressure. We assume that the following isothermal reversible cycle may be carried out : (i.) Evaporate unit mass of the condensed phase at the pressure P + p, so that the vapour produced is at the pressure p', the work done = p'V (j) -\- P) r (1 Pr?r). (ii.) Change p' , V to p, V by expansion, f" the work done = pdv =J P' if the vapour obeys Boyle's law. (iii.) Condense the vapour at the pressure p, the work done = |>(V r). (iv.) Compress the condensed phase to pressure P + p, I P\ the work done = (p + -5 j P?jr. The cycle is now completed, hence, by 36 : CHANGES OF PHYSICAL STATE 201 or, if we put pV = p'\' (Boyle's law), and neglect terms con- taining rip, .'. if P is not very great, p p~ = 1 + ~ + etc., " * ~ v Hence if 6P is the increased pressure applied to the condensed phase, and ftp the consequent increase of vapour-pressure, . This equation is due to Willard Gibbs (1876). The vapour-pressure of any liquid or solid is increased by compression of the condensed phase. If m is the molecular weight, and if the vapour obeys the gas laws, ?=ik' p v where $ is the molecular volume of the condensed phase. Thus the relative increase of vapour-pressure for a given increase of pressure on the condensed phase depends solely upon, and is proportional to, the molecular volume of the latter, at constant temperature. Example. The specific volume of ice at is 1-092 ... $P = * ^_1^92xlO-/. per atm U . 082 /. atm. x 273 deg. .-. rise of vap. -press. = 0-088 per cent, per atm. The vapour-pressure of a liquid contained in a long vertical column exposed to the action of gravity is greater in the lower part, and if p ,pi are the vapour-pressures at points distant /; ,/h from a fixed horizontal plane of reference (J>i > J> ) : where -V = ^P = .^ ... (3) in the case of a sphere ; or in the case of a surface of any form. This equation is due to Lord Kelvin (1870). If the curved surface is convex, as in the case of liquid drops or the surface of mercury depressed in a capillary ^ ^ tube, p'>p, but if it is concave, as in the case of a liquid ascending and wetting a capillary tube, r is negative and p'_ J /=_8p = p!L=_2,A^ . . . (1) an equation deduced by Blondlot (Journ. de Phys. [2], 3, 442, 1884). The effect is, however, small, because it is known that the greatest tension which can exist on an isolated conductor in air under atmospheric pres- sure is equal to a pressure of about 0'3 mm. of mercury ; this corresponds to a lowering of vapour-pressure, in the case of water, of only about 10 ~ 6 mm. of mercury. Gouy (C. R. 149, 822, 1909) has shown that Blondlot's equation is incom- plete ; the correct equation is : where K is the dielectric constant of the vapour. The effect of a magnetic field has been considered by Duhein (1890), and Koenigsberger (Ann. Phijs. 66, 709, 1898). 103. Supposed Critical Point of Fusion ; Researches of Tammann. Poynting (1881), from considerations based on equation (1) of 100, surmised that ice would melt, if the pressure on it is raised in such a way that any water produced flows away, under a pressure which is for a given temperature, only ^ that on the fusion curve (i.e., when the water remains in contact with the ice). He found -r- /- JT ; = - - = O'l, i.e.. the ordinary fusion dpi dP n curve rises ten times as fast as the so-called " second fusion curve." Tammann pointed out that the validity of the calculation stands or falls with the correctness of the assumption that fusion actually occurs in process (i.) of the cycle. He compressed ice under a loose piston, and found that although it certainly became very plastic near the melting-point, the velocity of outflow increased continuously from low temperatures, thus excluding the possibility of a second fusion curve having been intersected. Poynting also expressed the opinion that a critical point of CHANGES OF PHYSICAL STATE 205 fusion, at which solid would pass continuously throughout its mass into liquid, might exist on the fusion curve. The horizontal part of the curve of fusion changes in length with change of temperature, and at some high or low temperature (according as the substance belongs to the wax or ice-type) might shrink to a critical point. Tainrnann has, by a large amount of experimental evidence, apparently refuted the hypothesis of a critical point of fusion against which of course there is no a priori objection. In some cases (e.f/., with ice) new crystalline modifications appear when the temperature and pressure are modified, and in all cases the pair of relations : L = 0, Ar = (cf. 89) were never simultaneously satisfied. The general form of diagram for the transition : Crystalline Solid > Liquid, where the specific volumes of solid and liquid, and the latent heat are represented as functions of the equilibrium pressure (and therefore, implicitly, of the tem- perature) is shown in Fig. 44. It differs completely from the correspond- ing diagram for evaporation (cf. Fig. 30 ; the L ( curve slopes down to meet the p axis at the abscissa of K). The specific volume curves intersect at a point where Ar = 0; the latent heat, on the contrary, changes only very slightly with the temperature, and its curve is either horizontal, or exhibits a maximum, falling off slightly at higher pressures, and probably approaching the p axis. Thus, when Ar = 0, L, has a considerable positive value, and when L,- 0, Ar has (probably) a considerable negative value. The melting-point (T,j9) curve (unlike a vapour-pressure curve of a liquid) does not end abruptly at a critical point (Ar = 0, L = 0) ; it is an endless curve, probably forming a closed loop ABCD, unless it intersects some other curve or the axes of co-ordinates. At high pressures it bends round towards the;; axis, and according to Tammann, takes the shape indicated by the following con- siderations. It is known from experiment that (for substances of the wax-type) the melting-point increases with rise of pressure, FIG. 44. 206 THEEMODYNAMICS but more slowly as the pressure increases, by the numbers for naphthylamine : This is well shown p atm. rp. 1 49-75 62 50-49 81 50-54 93 50-33 143 50-01 166 49-83 173 49-65 After a certain point the melting-point begins to fall hence it must pass through a maximum (Fig. 45). again, AtB , T- = , /. Ar = 0; atC , = Liq. L, = 0. Crystal C\ Gtess Inside ABCD, the crystalline solid is stable, above ABC the liquid, whilst to the right of CD the stable form is the amorphous glass. Roozeboom, however, holds a different opinion as to the latter part of the curve (Heterogcne Gleichge- tcichtc, vol. I.). The dependence of melting-point on pressure was found to be well represented, up to several thousand atmospheres, by the equation proposed by Daniien (1891) : lt = p=l + a (p - 1) - 6 0> - I) 2 - Thus p is a maximum for FIG. 45. and pnnix* 6, IMX can be calculated from the empirical constants a and b. In some cases the constants hold good up to a certain point only, when the curve suddenly changes its direction. This is attributed to the appearance of a new crystalline solid phase, and three varieties of ice have been so discovered. On the right of the curve, in the region marked " glass," we CHANGES OF PHYSICAL STATE 207 may have states of false, or fixed, equilibrium. To the right of the dotted line CE the substance is in the amorphous state, and if we subject crystals to pressures greater than the abscissae of this line, they pass into the amorphous state, whilst at pressures less than the abscissae of CD, the amorphous substance crystal- lises. In the region DCE, both crystalline and amorphous states co-exist in a condition of false equilibrium. In conclusion, we may observe that the phrase " solid state '' is indefinite, a better classification of states is the following (Tammann) : A. Isotropic States. (1) Gaseous. (2) Liquid. (3) Amorphous solid, or Glass. B. Anisotropic States. (1) Crystalline solid ; different polymorphic crystalline varieties. (2) Crystalline liquid. Continuous transition of state is possible only between isotropic states ; it may thus occur between amorphous glass (i.e., supercooled liquid of great viscosity) and liquid (" sealing-wax type of fusion "), or between liquid and vapour, but probably never between anisotropic forms, or between these and isotropic states. This conclusion, derived from purely therniodynaniic considerations, is also supported by molecular theory. 104. Dissociation. Although Deville and Debray had established the main laws of the dissociation of substances by heat, it was reserved for A. Horstmann to show that the dissociation pressure of sal- ammoniac, for example : NHjCl 7* NH 3 + HC1 solid gas is dependent on the temperature in the same way as the vapour- pressure of a liquid, and hence the latent-heat equation ( 88) can also be applied to chemical phenomena. The dissociation pressure is a function of temperature alone : P = NH 3 +HS yas mm. T ^cal . atT 175 9-5 + 273 24-500 10-7 + 273 212 12-0 21-730 13-5 259 15-0 24-090 16-5 322 18-0 20-490 20-0 410 22-0 22-500 23-5 501 25-1 dp_ppi _ 212- 175 dT T 2 T! 12 - 9-5 - 37 2-5 *= CHANGES OF PHYSICAL STATE 209 p = $ (175 + 212) = 193-5 mm. T = \ (282-5 + 285) = 2S3'7 .-. \a = 2 X 1-985 X X 14-8 = 24,500 cal. ' A,, is nearly constant, the mean value being 22,660 cal. The value calculated from thermochemical data is 22,800 cal. When \ d is constant, (5) may be integrated : Inp = ^ + const. . . . . (6) which shows that, so far as pure thermodynamics will take us, Aj, may be calculated from the p,T curve : Pl (6a) ~ T 2 although the converse is not possible. In general, \ e depends more or less on the temperature, and if we put : \ d = a + IT + cT 2 . . . (7) we find : Inp = A + BT + CM + ? . . . (8) where A, B, C, D are constants. Thus there is found for the system : CaC0 3 ~ - CaO + C0 2 the equation : log p = + 1-1 log T - 0-0012 T + 8-882 mm. 105. Theorems of Robin and Moutier. Let the masses 1 m and m of two phases which are inter- convertible at a constant temperature and pressure with absorp- tion or emission of latent heat (e.g., water and steam, or ice and water) be contained in a cylinder under a pressure p and at a temperature T. By a slight motion of the piston let a further small mass 8j of the second phase be produced. If $i(j>,T), < 2 (p,T) are the specific potentials of the phases, the increase of potential of the system is : 210 THERMODYNAMICS This change is possible and irreversible only if (80 V T < i.e., if 02 > 0i then 5m < 0, i.e., the second phase decreases in amount (e.g., condensation occurs). if 02 < 0i, then 8w > 0, i.e., the second phase increases in amount (e.g., evaporation occurs). > if 02 = 0i, then 8m = 0, i.e., change in either direction is < possible and reversible, and the system is in equilibrium. The condition for equilibrium at a given temperature and pressure is therefore : 2 (p,T) - 0i(?>,T) = . . . (1) i.e., the specific potentials of both phases are equal. Since 0i(^),T), 2 (j>,T) depend only on T and p, and since these magnitudes remain constant during the change, it is evident that the total potential of the system has a stationary value in the equilibrium state ; the system therefore remains in equilibrium in its new state, and the state of equilibrium is neutral ( 49). For any given value of T, equation (1) gives on solution at least one value of p. If we put x = T, y = p, the assemblage of points representing the various possible solutions of (1) con- stitute a curve which is called the saturation curve. dp"' TOW dp - :t *fo T ) (2) where vi, t- 2 are the specific volumes of the two phases (V.r/., liquid and vapour). We assume, as an experimental result, that (3) < j . . . (4) If an isotherm T = TI is drawn to cut the saturation curve, the point (or points) of intersection must satisfy (1), i.e., in this case : 02(^) i(p) =L F(j>) = [T const.] . . (5) Since, however, dF(p)/dp ^ 0, the function F(p) has no maximum or minimum value, hence the saturation curve cannot cut the isotherm T = T x more than once, and since this holds for all values of T which satisfy (1) we see that there is only one value of p corresponding with a given value of T. CHANGES OF PHYSICAL STATE 211 If, at any point, ^ = r 2 , the saturation curve can exhibit a maximum or minimum (cf. 103). The saturation curve divides the p,T plane into two regions, in one of which the potential of the second form is greater, and in the other less, than that of the first form. By reason of the sign of $2 $1 ^6 call these the positive and negative regions, respectively. Let AB be the saturation curve (Fig. 46), > - 01 = 0, and let P(T,p) be any point on it. Through P draw two lines aa, ftft' parallel to the axes of p and T respectively. a is the point (T, p + bp) ft is the point (T + ST, p) and a', ft' are (T, p bp), (T 8T, p) respectively. The change of potential in passing from P to a is : FIG. 46. - g?- where i'i, r 2 are the specific volumes at the point P. If (v. 2 -- vi) > 0, (^0) T > 0, and a lies in the positive region. The only change which can spontaneously occur at a is one which makes cm < 0, or which decreases the volume (e.g., condensation of steam) when it occurs on the saturation curve. If (r 2 vi) < 0, then (5<}j. < 0, so that a lies in the negative region. The only change which can occur spontaneously at a is one which makes bm > 0, i.e., which decreases the volume (e.g., fusion of ice) when it occurs on the saturation curve. At the point a'(bp < 0) the directions of change are reversed, i.e., all spontaneous changes increase the volume. These results are summed up in the following theorem, due to G. Robin : The only possible spontaneous transition which can occur above the saturation curve (ftp > 0) is one which leads to diminution of volume when it occurs on the saturation curve ; the only possible transition which can occur spontaneously below 212 THERMODYNAMICS the saturation curve (8j> < 0) is one which leads to increase of volume when it occurs on the saturation curve. In all cases, the change within the equilibrium system which occurs spontaneously when the pressure is forcibly altered from without is one which tends to annul the alteration of pressure. The change of potential in passing from P to /S is : where s b s 2 are the specific entropies, and L the latent heat of the transition [1] [2] , at the point P. If (* 2 *j) > 0, Le., L > 0, then < 0, and /3 lies in the negative region. The only change which can occur spontaneously at /3 is one which makes fan > 0, or which absorbs heat when it occurs on the saturation curve (e.g., evaporation of water, or fusion of ice). If ( 2 i) < 0, i.e., L < 0, then 5 > 0, and lies in the positive region. The only change which can occur spontaneously at /3 is one which makes fan < 0, i.e., which absorbs heat when it occurs on the curve. At the point /3'(&T<0) the direction of the spontaneous change is reversed, i.e., it occurs with evolution of heat on the curve. These results are summed up in the following theorem, due to J. Moutier : The only possible spontaneous transition which can occur in the region to the right of the saturation curve (8T > 0) is one which leads to absorption of heat when it occurs on the satura- tion curve ; the only possible transition which can occur spon- taneously in the region on the left of the saturation curve (5T < 0) is one which leads to evolution of heat when it occurs on the saturation curve. In all cases, the change within the equilibrium system which occurs spontaneously when the temperature is forcibly altered from without is one which tends to annul the alteration of temperature. Since the latent heat alters only slowly with temperature, Moutier's theorem can be applied to changes not too far removed from the curve of transition. Corollary 1. Every spontaneous isopiestic change in a uni- variant system evolves heat if it takes place at a temperature CHANGES OF PHYSICAL STATE 213 lower than the transition temperature, and absorbs heat if it occurs at a temperature higher than the latter. Corollary 2. If there are two opposite isopiestie transforma- tions possible for a univariant system at two different tempera - ,tures, the one occurring at a lower temperature will give rise to an evolution, that at the higher temperature to an absorption of heat. The theorems of Moutier and Robin apply to evaporation, fusion, polymorphic change, or dissociation of systems in com- pletely heterogeneous equilibrium. 106. Simultaneous Equilibria of Physical States. "We have still to consider whether it is possible to have the three forms of a substance coexisting in equilibrium : [S] [L] z [G] . . . . (1.) Again, if the solid can exist in two or more forms, as sulphur in the octahedral and prismatic crystalline forms, there are the further possibilities. [S-] [Sp] ^ [G] . . C2> [SJ ~ [S*] - [L] . . . (8) There is an important law referring to such equilibria, which states that if the two phases A and B of a substance, and the two phases A and C are at a given temperature in equilibrium separately, then all three phases will be in equilibrium together at that temperature. Thus if two phases are, at a given tem- perature, separately in equilibrium with a third phase, they will be in equilibrium with each other. We shall call this the Law of Compatibility of Equilibria. This result is an immediate consequence of the potential equations. Let ^i(j*,T), ^(p,T), and $3(j>,T) be the potentials per unit mass of A, B, and C, respectively. If A and B are in equilibrium at a pressure o> and tem- perature -5 : ^,(,3) = 0a(a>,d) .... (a) If A and C are in equilibrium under the same conditions : *i(,*) = *(,*) .... (ft) .*. from (a) and (6) 214 THERMODYNAMICS 107. The Triple Point. The preceding investigation shows that the T,p curves of a substance existing in three phases exhibiting the above property, meet at a point which has the specified temperature and pressure as co-ordinates. This is called the triple point for the three forms. Example. Ice and water- vapour are in equilibrium at + 0-0077 C., under a pressure of 4'57 mm., and liquid water is in equilibrium with water-vapour at the same temperature and pressure ; ice, liquid water, and water-vapour are therefore in equilibrium under these conditions, and the equilibrium curves representing pressures as functions of temperature meet at a triple point ? (0-0077 C., 4-57 mm.). These curves FIG. 47. (i.) OA, the evaporation or "vapour-pressure" curve, along which liquid and vapour are in equilibrium ; (ii.) OB, the sublimation-curve, along which solid and vapour are in equilibrium ; (iii.) OC, the fusion-curve, along which liquid and solid are in equilibrium. is the triple point ; in the regions AOB, BOG, COA, homo- geneous vapour, solid, and liquid respectively, are stable forms. The curves AO, CO may be prolonged past for a short distance ; these prolongations represent supercooled liquid, and superheated liquid, respectively ; the prolongation of BO, which would repre- sent superheated solid, has never been realised (cf. 96). We shall prove immediately that all substances of the ice-type have curves similar to the above ; if the substance is of the wax-type, OC slopes to the right. The existence of the triple point was first indicated by James Thomson (1851). 108 Kirchhoff's Formula and the Equations of the Triple Point. Regnault (1847) instituted a series of experiments to decide whether the vapour-pressure of the solid form of a substance was the same as, or different from, that of the supercooled liquid at CHANGES OF PHYSICAL STATE 215 the same temperature. He concluded that " the passage of a body from the solid to the liquid state produces no appreciable change in the curve of elastic, force of its vapour ; this curve preserves a perfect regularity before and after the transition." This implies that, if the crystalline substance and the supercooled liquid were contained in the two branches of an inverted U-tube, the whole would be in equilibrium at a given temperature. Gernez (1888), who made the experiment with acetic acid, found, however, that the liquid slowly diminished in quantity ; the amount of solid at the same time increased. The incorrectness of Regnault's conclusion was demonstrated by Kirchhoff in 1858 ; he proved that the vapour-pressure curves of solid and liquid are not continuous through the freezing-point, but are inclined at an angle. Let [1], [2], [3] be any three modifications of a substance which can exist together in equilibrium at a triple point, and let i'i, i'2, r 3 be their specific volumes ; i, s- 2 , * 3 , their entropies pel- unit mass. The gradients of the p-T curves at the triple point are given by the latent-heat equations : (i) d $ = w^- al l(t'3 r a ) (2) ^ L,_ dT T(r 3 - n) (3) ( fy ? 3 __ L 3 f/T T(r a n) Where p- t , L, denote the pressure and latent heat of transition in the system which does not contain the i-th phase. Since the changes proceed at constant pressure ( 25) : L 2 = L! + L 3 .... (4) e.g., the latent heat of sublimation = latent heat of fusion + latent heat of evaporation. Also TI v 3 = (r 2 r 3 ) -4- (n r 2 ) identically . . (5) and LI = T ( 8 s a )| L 2 = T(* 8 - *0[ .... (6) L 3 = T ( 2 - Si)} By elimination of L and v from (1) (6) we obtain the funda- mental equations of the triple point : 216 THERMODYNAMICS (* - %) ~ + (*s - i) ~ + (i- *) f = dpi dp 2 dp 3 which may also be written in determinant form : (8) dT dT = ; and -7- Si dpi dT If 1, 2, 3 refer to solid, liquid, and vapour, and if the volumes of solid and liquid are neglected in comparison with that of the vapour : dp e ~L e t dp s L, dp, dpr\ _ L, (q} " 5fJ-W, an equation due to Kirchhoff (1858), which shows that the difference between the slopes of the sublimation and evaporation curves has a finite and positive C.P value at the triple point, so that the sublima- tion curve lies beneath the prolongation of the evaporation curve. The gradient of the fusion curve near the triple point is determined by the sign of (v t r,), and according as this is positive (wax-type), or negative (ice-type), the curve slopes from left to right, or from o to to so 40 so r ighi to l eft > upwards. The difference on the FIG. 48. left of (9) is usually very small. Thus in the case of water, Fischer (1886) found 0-0465. But L/= 80 X 0-0413 1. atm. = 3'304 1. atm. T =273 v g = 209-905 litres = sp. vol. water- vapour at C. (!- ) = 2 -f^=*<- = 0-0446 mm. This very small value had eluded Eegnault's examination of his curves, but, as Kirchhoff showed, it can be inferred from his experimental data. The sublimation and evaporation curves of solid and liquid phosphonium chloride (PH 4 C1), however, meet at a very decided angle at the triple point (Tammann, Kryst. und Schmelz., p. 291). The curves are represented in Fig. 48. CHANGES OF PHYSICAL STATE 217 Kirchhoffs investigation does not show that the sublimation and evaporation curves meet each other at the temperature at which solid and liquid are in equilibrium with vapour ; it proves that they are inclined at an angle, but the further fact that they intersect requires separate proof, which was inferred by James Thomson, and experimentally demonstrated by Ferche (1891) in the case of benzene ; the point of intersection, calculated from the vapour-pressure curves, was 5'405 C., whereas the melting- point was 5'42 C. We return to the general case. Let ri > i a > r 3 ....... (9) and write (7) in the form : Then from (5), (9), and (10) we see that -^ is intermediate in value between -^jj and ( -jj. This implies : Theorem I. (Moutier, 1876). If an isotherm T + s 2 > s 3 ....... (11) and write (8) in the form : *1 S3 Then from (4), (6), (11), and (12) we find Theorem II. : Ij an isopiestic p + dp is drawn to cut the three curves of transition (or their prolongations) meeting at a triple point, the central point of section corresponds with the transition involving the greatest change of entropy. This theorem is due to Eoozeboom (1901). The triple point divides each of the curves of transition passing through it into two parts, one of which corresponds with a stable system, and the other with an unstable system. The discrimina- tion between these is effected by means of two theorems due to Roozeboom (1887), which are analogous to the theorems of Moutier and of Robin, for two-phase systems ( 105). 218 THERMODYNAMICS If there is a reversible change which increases the entropy of the system, but leaves its volume unchanged, and another reversible change which increases its volume without alteration of the entropy, there will in both cases be an increase in the amounts of some phases and a diminution in the amounts of others. The theorems, III. and IV., in question assert that, if the mass of thet-th phase -f increases > the system from which it is absent I decreases cannot exist in stable equilibrium at : (1) temperatures { J 11 ^ 61 ', I lower or (2) pressures ( lower respectively, than those at the triple point. Proof. Let the system have the temperature, pressure, volume, entropy, and potential, T, p, Vi, Si, $1, respectively. Keeping T, v, and constant, let the i-th phase be caused to appear by ] \ r 31 . " the entropy. Then for the second state we J ( decreasing have $1 = 2 $1 $2 Si < S 2 or Si > S 2 . . . (13) Vi = V 2 Vi = V 2 Now if the temperature and pressure in the \ , state are ( second changed without altering the relative masses, the changes of potential are Sfa = - SiST + ViSP . r . . (14) Sfa = - S 2 ST + V 2 SP , . % (15) respectively, and if i , equations (13), (14), and (15) lead ( ol < at once to : 4>i + tyi > 2 + S 2 . . . (16) in both cases. Thence, since in stable equilibrium < is a minimum, it follows that, at a temperature slightly j ^ than, and under a pressure differing only slightly from that at the triple point, the system containing two phases cannot be CHANGES OF PHYSICAL STATE 219 in stable equilibrium. In a similar way, if we take the relations : ! = $2 ! = $2 Si = S 2 and Si = S 2 Vi > V a Vi < Y 2 with &p > Bp < 0, we obtain again the inequality (16), and thence deduce the second theorem. This investigation is due to P. Saurel (1902). We now return to the equations of the triple point : (r, - ! ^ + (r. - n) '& + (n - v 2 )^j = 0, (* 2 - .) ^ + (* 8 - i) ^ + (i - *a) ^ = 0, ! f//> 2 3 and suppose that 'l > ''2 > ''3 1 > 2 > S 3 , respectively. The first of the two theorems just established shows that the systems of two phases can be grouped into two classes, the stabilities of which are determined by the signs of (r 2 r 3 ), (''a t'i)i (*'i fa), positive coefficients forming one class, and negative coefficients the other. The members of one of these classes will be stable at temperatures above, those of the other at temperatures below, that of the triple point. The systems without [1] and [3] form one class, and that without [2] the other. Thus we have the Theorem V., due to Duhern (1891): The system of two 2 )nases corresponding with the transition involving the greatest change of volume is in stable equilibrium at temperatures which He on one side of the triple point, while the other two systems are in stable equilibrium at temperatures which lie on the other side of tlic triple point, The second of the two theorems shows that the systems of two phases can be grouped into two classes, the stabilities of which are determined by the signs of (2 #3), (#3 si), (*i s 2 ), positive coefficients forming one class, and negative coefficients the other. The members of one of these classes will be stable at pressures above, those of the other at pressures below, that of the triple point. The systems without [1] and [3] form one class, that without [2] the other. Thence we deduce the Theorem VI. 220 THERMODYNAMICS ( Roozeboom, 1901) that the system of two phases which corresponds with the transformation involving the greatest change of entropy is in stable equilibrium under pressures lying on one side of the triple point, while the other two systems are in stable equilibrium under jwessures lying on the other side of the triple point. Theorems I. and V., or II. and VI., lead to the Theorem VII., due to Gibbs (1876) : If a small circuit is drawn around the triple point, it cuts alternately stable and unstable branches of the curves of transition meeting at that point. Roozeboom (" Heterogcne Gleichgewichte," I., 189 (1901) ), has shown that the Theorems I., II., V., VI., and VII., together with the latent heat equations dpi _ s 2 s 3 dfo _ s 3 si (fy? 3 _ .i s 2 dT i- a v 8 ' f?T ~~ r 3 TI dT Vl r a ' furnish a complete classification of the various possible types of triple point. (B. Roozeboom : Heterogen. Gleidigewichte, I., 1901 ; P. Saurel, Journ. Phus. Chcm., 1902; P. Duhein, Zeitschr. physik. Cltem., 8, 367, 1891 ; Gibbs, Scientif. Papers, I.) CHAPTER Till VAN DEB WAALS' EQUATION AND THE THEORY OF CONTINUITY OF STATES 109. Van der Waals' Equation. In Chapter VI. it has been shown that the characteristic equation of ideal gases : pr = ET (1) although closely followed at low pressures, is deviated from more or less bj* all gases. At a certain point, the gas may pass into a liquid, and the deviation is then very drastic. The critical point is also entirely left out of consideration. The question now arises as to whether it is possible to arrive at an equation which adequately represents the behaviour of actual gases, including their liquefaction, and critical phenomena. Such a characteristic equation may be found empirically, or by the aid of theoretical considerations lying outside pure thermo- dynamics. The problem has been largely worked at from both sides; from the theoretical side the point of view has been almost exclusively that of the kinetic gas theory. It must be kept in mind, however, that it is possible that a purely mechanical theory may not be sufficient to cover the phenomena, as has recently appeared in the case of the specific heats of solids. The first characteristic equation to be proposed which gave an adequate representation of the properties of gases was the equation of van der Waals, which resulted from a revision of the deduction of the equation (1) from the kinetic theory, and the introduction of corrections in the fundamental assumptions that: (i.) the volume of the molecules themselves is negligible com- pared with the total volume ; (ii.) there are no forces acting between the separate molecules. The first correction leads to a free volume (r b) instead of v ; 222 THERMODYNAMICS the second to an active pressure uj -f- -^J instead of p the latter being diminished by the attractive forces ; thence : RT _ and afterwards : RT /, (a'T- n - b')(v-b}\ f . 2 which Battelli (1892) used in the extended form : RT f n M(a'T - m a"T ~ n ) (v 6)~| ,, J (6 constants). The need for so many constants shows, says Weinstein (Thermodynamik u. Kinetik dcr Korper, I., 369), "how really unsatisfactory is yet the condition of this branch of enquiry." Dieterici (1899) proposed to replace r 2 in van der Waals' equation by r% : ,. RT According to Boltzmann and Mache, the magnitude b is not quite independent of p and T ; at high pressures it becomes only a very small multiple of the molecular co-volume. If b ^ is the value at great rarefaction, they proposed the equation : VAN DEE WAALS' EQUATION 223 The limits of variation of b are probably not greater than the ratio 1 : 10. (O. E. Meyer : Kinetic Theory of Oases (trans. Baynes). Kuenen : Die Zustandsghichung der Gasen (1907). Weinstein : Thermodynamik and Kinetik der Korper, vols. 1 and 2. Boltzmann : Vorlesungen tiler Gastheorie. J. D. van der Waals, Die Kontinuitdt, Leipzig, 1881. Jeans : Dynamical Theory of Gases, 1904. 110. Thermal Coefficients from van der Waals' Equation. RT a * = ( 7inr)-F ...() where v = molecular volume. The values of a and b depend of course on the units of p and r. Van der Waals wrote the equation in the form : by analogy with the equation for ideal gases : pv = jWo(l + aO) (2)0 = r = 1) and hence unit pressure = 1 atru., and unit volume is taken as the volume of 1 kg. at N.T.P. The values of a and b are then called the " constants calculated according to the initial volume as unity." If the equation is written in the form (a), a and b refer to 1 mol at 1 atm. pressure. If other units are chosen such that : a' = aa, // = pb, R' = /c Units a ft P cm. 3 , atm. 1 I 1 cm. 3 , mm. Hg 760 1 760 litre, atm. 10~ io- 3 io- 3 litre, mm. Hg 0-00076 io- 3 0-760 Thus for C0 2 Sundell (1899) gives : 1 ._ 0-00525) = (1 + 0-0129) (1 0-00525) (1 + 0'003660) = 1-0076 (1 + 0-003660) = 0-00369 (6 + 273) (Ideal gas : pv = 0'00366 (6 + 273).) 224 THERMODYNAMICS In the case of methylamine CH 3 NH 2 , for 1 mol as unit : (a) v in cm. 3 , _?> in atm. : (P + 74 * 1Q5 ) (F - 61) = 82-09T (6) r in litres, p in mm. Hg : L -j_ 5 |^ ( f - 0-061) = 62-39T. (Values of a and 6 in Winkelmann : Physik, III., 5, 857.) Ml) ;=;CT< inde P ento ' toIT ) < : ,9T/ v ' 2? X v ~~ &) /9p\ / rT 2a\ \*> (3) (4) . (4a) (Ah\ !k\ ^r' )2 / _p- c T (wj v m), 1 2a 2 If a is very small : 1-6 c p c v - - 273 If 6 is very small : r If both a and 6 are small : from which a value of the mechanical equivalent J can be calculated (cf. 71). The maximum work of isothermal expansion is : . (5) The work of adiabatic expansion, if we put: p + -j' (v 6) K = const. is Att= ^_ ( T 1 -T J )-a^-^ . . (6) VAN DEE WAALS' EQUATION 225 The Joule-Kelvin effect may also be calculated from van der Waals' equation. For an expansion from TI to r 2 at constant temperature, let the change of intrinsic energy be 2 HI = A T . ' ' and a P = ^Tb~^ ()ti\ a N-)r~ v 2 (9) ' "2 - HI = A T = I ~ dv = a (---- V \Vi ?2 J v, an equation also deduced on purely kinetic grounds by Bakker (1888). In the Joule-Kelvin experiment : (MS + p*r*) (MI + jpit'i) A(w + pv) = Ait? = 0. Suppose the change from the state before the plug (p^, r b TI) to that after (p%, r 2 , T 2 ) effected in two stages : (i.) Expansion at constant temperature TI. Then if n' is the intrinsic energy in the intermediate stage : /I 1 u HI = a I -- \vi r a . (ii.) Cooling from TI to T 2 at constant volume r a u 2 = u ' + c,(T 2 TO. .*. c v (Ti T 2 ) = a ^ - r 2 > ii w which gives the cooling effect ( r l\ T 2 ). For given values of r 2 and TI the cooling is a maximum when [c c (Ti - T 2 )] = which gives the required initial volume. Since the equation is symmetrical with respect to n, r 2 , the value of v 2 for maximum cooling, with given i^ and Tgjs given by : 226 THERMODYNAMICS The cooling effect vanishes when Ti=T 2 , .'. _2a _ _ rTi&_ vir a ~ (ii 6) (v 2 b)' This depends, for given values of a, 1>, T b on both Vi and r 2 . Since r 2 is large compared with b, we can put v% / (r 2 />) = !, and obtain : If the initial volume is less or greater than the root of this equation there will be warming or cooling, respectively. We have already mentioned that the inversion temperature of hydrogen is 80 C. The equation also serves to calculate the deviation of a gas from Boyle's law : rTv a pv = ---- 7 - - v b r \ rT6 , a. du 1 T ~ (v b)* "*" v 2 and the sign of the deviation is positive, negative, or zero according as rT is less than, greater than, or equal to j- 1 1 J , in which cases the gas is less, or more, compressible than an ideal gas, or in the latter case the ' gas behaves on compression like an ideal gas, although, since a is not zero, it exhibits a Joule-Kelvin effect. This behaviour is exhibited at least once by all gases (except perhaps hydrogen), viz., at the point where the order of compressibility changes sign. The maximum value ofy (l is ,, when r = oo. If >-T is greater than, less than, or equal to ,, the gas is for all densities less compressible than an ideal gas ; or is more compressible on one side, and less compressible on the other side, of the transition point, than an ideal gas ; or, at the transition point, is equally compressible, respectively. These conclusions agree with the results of Amagat ( 80). 111. Calculation of Critical Constants from van der Waals' Equation. If we write van der Waals' equation in the form : p p VAN DEE WAALS' EQUATION 227 we see that it is a cubic equation in v, and, for every given pair of values of p and T, there will be three values of v, because a cubic equation has three roots. Further, the theory of equations shows that these roots are either : (i.) all real, or (ii.) one is real, and two imaginary. The physical interpretation of this result is that, according to the conditions of pressure and temperature, the fluid to which the equation is applied can exist either in three states with different specific volumes at the same temperature and pressure, or else in only one state (imagin- ary roots having no physical significance). Case (ii.) corre- sponds to a (jas heated above its critical temperature. In case (i.) the physical interpretation is that the smallest value of v corresponds to the liquid, the largest value of v corresponds to saturated vapour, and the intermediate value corre- sponds to an unstable state, all at the given temperature. These relations are most clearly seen by plotting p as a function of r for different values of T (cf. Fig. 49). The curves for lower temperatures resemble the theoretical isotherms of J. Thomson ( 90). The real isotherms are straight lines abc, Fig. 49, along which two phases (liquid and vapour) are present. The three values of r are the abscissae of a, I, c. The values at a and c correspond with liquid and vapour, respectively, that at b with an essentially unstable state, for there the isotherm slopes from left to right upwards, showing that the pressure would increase along with the volume, i.e., the elasticity is negative. It must be observed that the intermediate value does not refer to a heterogeneous complex, which is present on the straight line, for van der Waals' equation holds only for homogeneous sub- stances. The significance of the portions ad, ec of the theoretical isotherm have already been considered in connexion with rneta- stable states. Q 2 Fio. 49. 228 THERMODYNAMICS The points on the various curves corresponding to a and c are observed to approach more and more closely as the temperature rises, and finally they coalesce. At this point the three roots become identical r n r b = r ( . = r K , say, and the point of inflection thereby indicated is the critical point of the substance. z- K is the critical volume, and if p K , T K are the values of p and T at this point, these are the critical pressure arid temperature. To find the values of i\, p^, and T K , we imagine the equation resolved into three linear factors : = r 3 V p / ' p p where a, /3, 7 are roots, i.e., values of r which satisfy van der Waals' equation for given values of p and T. At the critical point these are equal : ,- K -|- 3 n - K 2 _ r R 3 = ! ._,, | '^ + ,,)+,:_:'' PK I PK PK a ,., 1 8a or K = 84;ft = ^ ! r = -.^. Thus, from an investigation of the compressibility of a gas we can deduce the values of its critical constants. We observe that, according to van der Waals' theory, liquid and gas are really two distant states on the same isotherm, and having therefore the same characteristic equation. Another theory supposes that each state has its own characteristic equation, with definite con- stants, which however vary with the temperature, so that both equations continuously coalesce at the critical point. The correlation of the liquid and gaseous states effected by van der Waals' theory is, however, rightly regarded as one of the greatest achievements of molecular theory. 112. Theorem of Corresponding States. The equation of van der Waals leads to an extensive gene- ralisation as to the relations between the physical properties of various substances. VAN DER WAALS' EQUATION 229 For if, in the equation the pressure, volume, and temperature are expressed as fractions of the critical values p = *PI, r = K> T = ST K then But rT K = , r K = 36, ;> K = * -l) = 8d . . . (1) an equation from which all constants characteristic of the specific nature of the substance (a, b, r) have disappeared. Equa- tion (1) is in fact true quite generally for all substances, whatever be their chemical nature, provided that there is no change of molecular complexity (association or dissociation) over the range of pressures and temperatures considered. TT, <, d are called the reduced pressure, the reduced volume, and the reduced temperature, respectively, and equation (1) ma}' be stated in the form that if we know the critical volume, critical pressure, and critical temperature of a substance, and divide the values of the volume, pressure, and temperature in a series of states by these, the quotients will satisfy an equation which does not contain any constants depending on the specific nature of the substance, this being in fact the equation : Definition. Any states of two substances characterised by the same values of TT, <, 3 are called Corresponding States. Many other equations of state besides van der Waals' lead to laws of corresponding states, although naturally these will not be of the form (1). G. Meslin (1893) has investigated the conditions under which an equation of state can lead to a law of corresponding states. The most general form of equation possible is : (p, r, T, d, c 2 , c 9 , . . .) = . . . (2) where c if c 2 , c 3 , . . are constants characteristic of the particular substance. 230 THERMODYNAMICS For the critical point we must have ( 115) : By means of these equations we can eliminate three constants from (2). But, if the equation (2) is now " reduced," as is required by the law of corresponding states, it must not contain any constants characteristic of the substance, hence (2) can con- tain only three independent characteristic constants. In this case (2) can always be written in the form : V2 - i =/(*, 3 2 )-/(Oi) Divide both sides by KrK a ; b = l *, from 111, 232 THERMODYNAMICS in (2), and divide both sides by ?r0, we get, on substituting the reduced magnitudes , = r/i-x, * = T/TK, 8 *, 30 2 1 ,, o 3in ZT-, . . . . lu i in which TT = reduced pressure of saturated vapour, r = reduced specific volume of liquid, < 2 = ,, ,, ,, saturated vapour. This, with the equations of 112, gives us three equations of corresponding states, one for the liquid, one for the saturated vapour, and one for the heterogeneous complex : (1) Liquid. TT + ( 3 X _ 1) = 8$ (2) Vapour. TT + - 2 (30 2 - 1) = 8* (3) Heterogeneous complex. O \ Q Q/4 1 0102' 3 30i 1- from which we obtain The reduced pressures, and specific volumes of liquid and saturated vapour, are the same for all substances at equal reduced temperatures. The equation TT = /(&) may be interpreted thus : If for two different liquids the reduced temperatures are equal, so also are the reduced vapour-pressures, or for two liquids the ratios of the vapour-pressure to the critical pressure are the same if the ratios of the temperature to the critical temperature are the same, PK\ PK.I IKI lK2 As an illustration we will take Sajontschewski's values for sulphur dioxide and ether : SO a EtaO p K = 78-9 atm. ; T K = 428-4 p K = 36'9 atm. ; T K = 463. For S0 2 : 1' = 60 atm. when T = 412'9 = 0-964. 428-4 TAN DER WAALS* EQUATION 233 For Et 2 O : P = j> K = '7605 X 36'9 = 28-4 atm. which requires a temperature T = 445 -8 445-8 463 = 0-963. The theorem just stated may be written in the form / \ where f ( ~- ) is a function of temperature which is independent HK/ of the nature of the substance. Van der Waals adopted as an approximate equation . . . (7) PropylAc PhF T or log P = a ~Y + a + lg />K in which a is a constant for all substances and is approximately equal to 3. Guye (1894) has calculated the values of a for various substances, the mean is : a = 3'06. In the case of alcohols, acetic acid, and water, a > 3'2, indi- cating polymerisation in the liquid state. Bingham (1906) has plotted ( K 1 J as abscissa? and log^ as ordinates for the substances H 2 , A, Kr, 2 , CS 2 , C 6 H 5 F, Et 2 0, CTT rTinPTT TTfOTT 3ii6^v/L'^l3, HiUJXl. According to (6) these Log S. should give coincident f0h straight lines ; Bing- ham found, however, that the lines not only did not coincide, but also diverged more and more as the absolute zero was approached. Nevertheless, we may still conclude that the vapour-pressure curves are similar, and do not intersect. The curve for helium lies below that of hydrogen. It is at once evident that although the deviations from the theory are FIG. 50. 234 THERMODYNAMICS enormous the} 7 exhibit striking regularities. The higher the molecular weight of substances belonging to the same series (e.g. t He, A, Kr) the more inclined is the curve to the axis of (rp \ -~ 1 j , and the same result also follows from increasing mole- cular complexity. Thus hydrogen, although it has a molecular weight less than that of helium, is more complex (H 2 ), and its curve is steeper. Valuable conclusions may therefore be drawn from such curves. Corollary 3. The Clausius-Clapeyron equation, in terms of reduced magnitudes, may be written PK d _ ^e . d"jr _ L, TK ' ~dd ~ T K 3r K FO) ' ' d$~~ 2> K r K dF(d)' OT> rri But psV K = Q K , and hence 8 8 L e M. 3Wr K > and since the left-hand member is the same function for all substances, it follows that i.e., the quotient of the molecular heat of evaporation at any given temperature by the critical temperature is a constant, or A ( ./T is the same at equal reduced temperatures for all substances. H 2 Et 2 (CH 3 ) 2 CO CHC1 3 CC1 4 CS 2 TT 7'5 1 1-41 1'49 1-57 2'03 L,, 489 90 126-5 60 45 82 1-35 1-31 1-44 1-35 1-34 T15 Lemma. The boiling-points under atmospheric pressure are approximately reduced temperatures (& = ) for all substances (Guldberg, 1890). Kurbatow, for carbon compounds with more than 5 carbon atoms, finds that the quotient has a mean value of 0'666. In homologous series it varies from 0'58 for the initial members to 0'70 for the final members. Corollary 4. The quotient of the molecular heat of evapora- tion by the absolute boiling-point under atmospheric pressure, T , is a constant for all substances. This rule was published by Trouton ; it appears to have been VAN DER WAALS' EQUATION 235 known to Despretz and to Pictet at an earlier date. The va'ue of the constant is about 21; according to Kurbatow (1903), the mean value is 20*7 0*8. If, however, the constants from van der Waals' equation are substituted in the equation for ^, ID the result is 1O8, which is widely different from 21. Xernst (1906) has proposed a modified rule of Trouton : ^= 9-5 log TO 0-OOTTo. The values of A e /T vary from 12*03 for helium, to about 22 for substances of higher molecular weight (aniline), and agree very well with the observed. According to Louguinin (1900) substances associated in the liquid state, but having normal vapour densities (alcohol, water), exhibit large values of A,/T , whereas those associated in the vapour (acetic acid) give low values. If the molecular weight used in estimating \ e = ML,, is taken as that of the associated vapour (e.g., (Ca'H^Oa^), the quotient is again normal if the liquid is associated to the same extent. This is the case with acetic acid. (Guye, Arch. Geneve [3], 31, 163, 463, 1894 ; Estreicher, Phil Mwj. 40, 454, 1895; Bingham, Journ. Amer. Chem. Soc. 28, 717, 1906; Xernst: Recent Applications o/ Thermodynamics to Chemistry ; Brill. Ann. Phys. 21, 170,1906; Guldberg. Zntschr. physik. Chem. 5, 374, 1890.) By analogy with Trouton's rule we may expect that : (a) The molecular heat of sublimation of a substance, divided by the absolute temperature at which the sublimation pressure is equal to atmospheric pressure, is approximately constant : it is found that A,/T = 30 (approx.). (b) The molecular heat of dissociation of a compound, divided by the absolute temperature at which the dissociation pressure is equal to atmospheric pressure, is approximately constant ; it is equal to 32. These rules were given by de Forcrand (1903). Here again the agreement is only approximate, and a revised rule has been proposed by Nerrist (1906). ^- = 4-571(1-75 log To + 3-2). In the middle of a fairly extensive range of temperature ~- will be approximately equal to 32. 236 THERMODYNAMICS Trouton, Phil. Mag. [5], 18, 54, 1884; de Forcrand, Ann. Chim. Phys. [7] 88, 384, 1903 ; Kurbatow, BeibL 28, 967, 1904 ; Louguinine, Ann. Chim. Phys. 13, 289, 1898; Arch, de Geneve 9, 5, 1900; 0. . 132, 88, 1901; Linebarger, Sill. Journ. 49, 380, 1895 ; H. Crompton, Trans. Chem. Soc. 17, 365, 1895, D. Berthelot, Ann. Chim. Phys. [7], 4, 133, 1895; J. Traube, Bar. 31, 1562, 1898; G. Bakker, Ztitschr. physik. Chem. 18, 519, 1895; 47, 231, 1904 ; Batschinski, ibid. 43, 369, 1903 ; H. von Jiiptner, ibid. 63, 355, 579, 1908; 64, 709, 1908; 73, 173, 1910. Corollary 5. The equation of state of an ideal gas is pv = RT, where ? is the molecular volume, R = 8'26 X 10 7 inC.G.S. units. If we assume that this holds up to the critical point, v = v', T = T K , and V = ?^. PK But, on van der Waals' theory 01 a r 1 8a _ 3 RT K ' " K "8 j* If d,b are the densities of the vapour at the critical point calculated on the laws of ideal gases and van der Waals' equation, respectively, and M is the molecular weight, d = Mr K ', a = Mr K ^ Q .'.-7=5 = 2'67 for all substances. li O Young has found, however, that the ratio of the observed critical density to that calculated on the laws of ideal gases is approximately 3*75 for all substances except those, like acetic acid, which are polymerised. Dieterici (1895) has observed that this ratio would not be less than 3 if 6 were assumed to be a function of the volume ; on the assumption of the constancy of l> he has given two new equations of state which give values of bjd closely approximating to the observed numbers (i.) ( p + J ) (r - fc) = RT ; 8/rf = 3-75 (ii.) p(v b} = RT . . . e ~ ^ ; 8/rf = 3 "695. D. Berthelot (Mem. Bureau des poids et mesures, 13, Paris, 1907), has proposed a modification of Clausius's equation which VAN DER WAALS' EQUATION 237 agrees very closely with the compressibility data for gases up to about 5 atm. : -I- J!U (v 6) = RT, where I = v ~ o a ~ 64 R2 ~pl' Planck (Be/7. .Be;-., 1908, 633) has, from the fundamental statistical definition of entropy, deduced the equation : RT , {. Bt\ a where fi = 26, a = const, which differs from van der Waals' only in terms of the second order. 114. Testing the Theorem of Corresponding States. Some consequences of the theorem of corresponding states have already been considered with reference to experimental results ; it has been shown that there is a good general agree- ment, but this is not strict. The question arises as to whether the deviations observed are due to the errors of experiment, or are indications of an inherent fault in the equation itself. Amagat (1896) made an ingenious test of the theorem in the following manner. If the isotherms for various substances are drawn in a diagram in which reduced volumes (V/L- K = ) are taken as abscissae and reduced pressures O>/J>K = TT) as ordinates, then isotherms having the same reduced temperature must coincide, and the whole series of isotherms must appear as if they represented a single substance, i.e., they must be similar and must not intersect. Instead of using reduced values, Amagat took simply the ordinary values of p and r for two substances, drawing one set of curves on transparent glass, and the other on paper, and then, by means of a beam of parallel light, projected the former on the latter. By a suitable rotation of the trans- parent diagram about either axis, the relative proportions of abscissae and ordinates could be reduced directly, and it was possible to determine if the curves could be made to form a non- intersecting series. This was the case for air, ether, carbon- dioxide, ethylene, and isopentane. Raveau (1897) adopted an even simpler method. The logarithms 238 THERMODYNAMICS of the volumes and pressures were used as co-ordinates, and log = log v log r K = log r const., % and similarly for p ; hence the curves can be made to correspond by merely changing the position of the origin, i.e., by motions of one diagram over the other parallel to the axes. Again the theorem was found to hold good. Raveau now calculated the values of p, v from van der Waals' equation, plotted the logarithms, and compared the diagram with a similar one drawn from the experimental results. The results showed that the diagrams could not be made to fit in the case of carbon-dioxide and acetylene, the divergencies being very marked near the cribical point. The results of Amagat's and Raveau's work may be summed up in the statement that, whereas the theorem of corresponding states holds good very approximately, the equation of van der Waals gives results quite inconsistent with the experimental values, especially near the critical point. There still remains for consideration the question whether the theorem of corresponding states, which we have seen is at least approximately true, is in fact rigorously exact, or is only a more or less close approximation. This problem is, thanks to the now classical investigations of S. Young and his students, quite satis- factorily solved. Very careful measurements have shown that there are small deviations, the magnitude of which is much greater than the experimental errors, and the theorem of corre- sponding states, in the form previously employed : /(,$, 3) =? 6, where /is the same function for all substances, and * = P!P*> = '/'* a = T/T K , is not rigorously, but only very approximately, true. Madame Kirstine Meyer (1900) has shown that the discrepancies are not to be explained by errors in the critical data ; the law of corresponding states can be tested without making use of these constants, and differences between the observed and calculated magnitudes are still apparent. D. Berthelot (Journ. de Phys., 1903) has deduced some new equations. The results of Young, and others, have shown that substances may be divided into two large groups according as they do or do not agree closely with the theorem of corresponding states. These may be called " normal " and " abnormal " substances, VAN DEE WAALS' EQUATION 239 respectively. To the first group belong the hydrocarbons, esters, ketones, ethers, etc. ; to the latter belong water, alcohols, and fatt}' acids. The monatomic gases such as helium and argon occupy a special group, the members of which agree amongst themselves but not with other normal substances. The normal substances, however, really exhibit small deviations which are all the greater the more complex is the molecule of the substance. The theory of van der Waals, or in fact any hypothesis from which a theorem of corresponding states could be derived, assumes however that the transition from the gaseous to the liquid state, as well as the changes of density in either state, result from alterations in the propinquity of molecules which otherwise remain unaltered. Any association or dissociation of the substance would therefore give rise to abnormalities, and in fact the substances which deviate most from the normal relations (e.g., water, acetic acid) are those which appear, on other grounds, to be associated in the liquid state. In the case of acetic acid the commencement of polymerisation, even in the state of vapour, is evident from the abnormal densities. A similar explanation may account for the slight deviations exhibited by " normal " substances, but fails to explain the anomalous behaviour of the monatomic gases. A mechanical interpretation of the theorem of corresponding states has, how- ever, been advanced by Kamerlingh Oniies (" Principle of Uniformity ") which appears to embrace all known cases. In short, we may say that there is evidence that the deviations from the relations predicted by the characteristic equations may be due to chemical changes in the substances, which are not taken into consideration in the kinetic deduction of the equations. Weinstein (loc. cit.) considers that it is possible to deduce an equa- tion which takes account of these chemical changes as well. It will be sufficient here to re-emphasise the fact that there is at present no characteristic equation known which agrees accurately with the behaviour of a single substance, let alone various substances, over a wide range of temperature. Amagat, O.K. 123, 30, 83, 1896; Journ. de Phys. [3], 6, 1, 1897. Eaveau, C.R. 123, 109, 1896 ; Journ. de Phys. [3], 6, 432, 1897. K. Meyer, Zeitvchr. physik. Chem. 32, 1, 1900. D. Berthelot, C.E. 130, 565, 713, 1900; 131, 175, 1900 ; Journ. de Phys. [3], 10, 611, 1901 ; [4], 2, 186, 1906. Kamerlingh Dimes, Arch. Ne'erl. 30, 128, 1896. 240 THERMODYNAMICS 115. Gibbs's Thermodynamic Model. If in a rectangular left-handed co-ordinate system with the ^-axis upwards we put : x = r, y = S, z = U for the whole volume, entropy, and energy of a substance in its different states, the aggregate of states will form a surface which, since it was first described by Willard Gibbs (1873, Scient. Papers, L, 32), is called Gibbs's ThermoJijnamic Model. If we draw three planes perpendicular to the axes of r, S, U, respectively, these will be the loci of all states which have a constant value of these magnitudes, respectively. The plane v = is evidently fixed, but the planes S = 0, U = can only be fixed by arbitrary choice of the initial states of zero entropy and energy. The origin may therefore be chosen anywhere in the plane of zero volume. If the energy is taken as a function of the volume and entropy, we have : rfU=T TSS - p&r . . . . (4) in which the variations are to be construed strictly, i.e., quantities of order higher than the first are not to be neglected. This is equivalent to (2) of 51. Now if we pass from one point (r, S, U) on the surface to an adjacent point (r + 6r, S + 5S, U + 8U) we shall have : + (5) Thus, from (3), (4) and (5) we have, as the condition for stable equilibrium of a homogeneous phase : VAN DER WAALS' EQUATION 241 the conditions for which inequality are : I ^F ^P TO ?U !>< 6 >:ggJ> ' < 7 > and hence ^ > (8) Inequalities (6) and (7) show that the surface is, at every point which corresponds with a homogeneous phase in stable equilibrium, convex downwards in every direction. Let us next consider the equilibrium of a heterogeneous complex of two phases, numbered 1 and 2, of a single component. The conditions for equilibrium are : . ..(11) Now the equation of the tangent plane at the point 1 can, as we know from solid geometry, be obtained by writing the sub- script 1 after each quantity in equation (1) and then putting : dUi = U - Ux dSi = S - S! " From (9) (12) we deduce at once that the tangent planes at the points 1 and 2 are coincident. Hence the theorem : If two different states can exist permanently in contact, the points representing these states on the thermodynamic model have a common tangent plane. The converse is also readily proved : If two points on the surface have a common tangent plane, the states represented by them are such as can exist permanently in contact (cf. 53). We can therefore tell beforehand whether two given states of a T. R 242 THERMODYNAMICS fluid will be in stable equilibrium when placed in contact, because U and S can be determined by some process in which the two states never appear simultaneously, e.g., by a continuous transition of state ( 87). Let Mi, Ui, ri, Si, and M 2 , U 2 , r 2 , S 2 be the masses, specific energies, volumes, and entropies, of the two phases, respectively, then, if the letters without suffixes refer to the whole values for the complex : M = M! + M 2 MU = MiUi + M 2 Ua MS = MA + M 2 S a M.V = Mii?i + M 2 r 2 -j- MaUa \ MI + M 2 _ MiSi+M 2 S 2 -TTF T^TF MI -f- M 2 M a r a -- Tjjf r~Tur .... MI + M 2 so that the representative point of the complex lies on the join of the points 1 and 2, and divides this line into two segments which are inversely as the masses. Thence, if the two phases exist in equilibrium at a given temperature, and pressure, there is possible a continuous series of states of equilibrium, characterised by constancy of the specific volumes, entropies and energies of the separate phases, whilst the masses of these, and hence the specific volume, entropy, and energy of the whole system, change in a continuous manner. If the change proceeds from one extremity of the line to the other, we shall have, from (2) : U 2 - Ui = T(S 2 - SO - X'-a - n) - - - - (16) Now suppose the representative point had passed from 1 to 2, not along the line of heterogeneous states, but along some path on the surface itself; this will be a continuous transition of state. Then, from (1) : If the path is an isotherm : au /8u\ /au VAX DER WAALS' EQUATION 243 .-. U 2 - Ui = T( S 2 - SO + and if the path is an isopiestic : 8U_ /am _ /am _ dr ~ \dv ) i ~~ \af/2~ r 2 then U 2 Ui = I ^ rfS X r a r i) | jj^ j i Thence, from (16), (19), and (20) : const. = p . 9U Q = T(S 2 - Si) = (19) (20) . (21) (22) . (23) which may be regarded as the analytical expressions of the two forms of Maxwell's theorem, concerning the theoretical isotherm of James Thomson ( 90). Since the energies are single-valued functions of the volumes and entropies of the substance in the two states, and since the equations (9), (10), (11) give three relations between the four quantities r lf Si, t' 2 , S 2 , it is evident that if one is fixed the others are determined, so that the two points lie on two definite curves on the model : Ui =,/iO'i, SO, and U 2 =/ 2 (r 2 , S 2 ) and to every point on the one there is a single corresponding point on the other. If now we imagine the doubly-tangent plane to roll along the surface, the locus of its points of contact with the A surface will trace out two curves, called together the connodal curve, and the tangent plane being everywhere tangent to the surface representing heterogeneous states along a line, it follows that the latter (called by Gibbs the derived surface, as distinguished from the primitive surface representing homo- geneous states) will be a developable surface, and will form a part of the envelope of the successive positions of the rolling tangent plane. Let P b P 2 (Fig. 51) be the points of contact of the tangent K 2 FIG. 51. 244 THERMODYNAMICS plane, and let planes through these points perpendicular to the axes of v and S respectively intersect in the line AB, which will be parallel to the axis of U. Let the tangent plane cut this line in A, and let PiB, P 2 C be drawn perpendicular to AB and parallel to the axes of v and S. . and if we roll the tangent plane through an infinitesimal angle about the instantaneous axis PiP 2 , so that it meets AB in A', then : dp _ BP 2 _ S 2 Si _ L " dT ~ CPl - i- 2 1-1 ~ T(r a - n) which is the Clapeyron-Clausius equation. As the tangent plane rolls on the primitive surface, it may happen that the two branches of the connodal curve traced out by its motion ultimately coincide. The point of ultimate coinci- dence is called a plait point, and the corresponding homogeneous state, the critical state. The conditions which must be satisfied at the plait point may be deduced as follows : Expand by Taylor's theorem the expressions on the right of (9) and (10), omitting terms of higher orders than the second : f <\ _L /95A / q _ o , / 8 2 U \ \8s 2 / 1 ( 2 ' l} + UsaJ ! (ra ~ ri) a a u\ , /a 2 u N and hence, when r 2 ri, S 2 Si become very small with approach to the plait point : (S,- SO and at the plait point itself: a 2 u 8 2 u T . S 2 - Si . VAN DEE WAALS' EQUATION 245 or, if we put : cPU 3*0 as 2 ( ' 25) the equation of the plait point is A = ..... (26) If we imagine a line drawn on the primitive surface dividing all parts of the surface which are convex downwards in all directions from those which are concave downwards in one or both direc- tions of principal curvature, this curve will have the equation (26), and is known as the spi nodal curve. It divides the surface into two parts, which represent respectively states of stable and unstable equilibrium. For on one side A is positive, and on the other it is negative. If we assume that the tie-line of corresponding points on the connodal curve is ultimately tangent to that curve at the plait point, it follows that the direction of this tangent is given by either of the two equations : Now the plait point is on the spiuodal curve, and any two corresponding points of the coimodal curve adjacent to the plait point are on a part of the surface which is convex in every direction, and for which therefore A>0. Thus the spinodal curve does not cut the connodal curve at the plait point, and it is simplest to assume the two curves to be tangent at that point. From (26) it follows that the direction of the tangent at any point of the spinodal curve is given by : At the plait point this equation will give the same value for as equations (27) and (28), and hence at that point the 246 THERMODYNAMICS three determinants of the second order formed from the matrix : eW a 2 U as 2 asa<- a 2 u a^u , ' ' will be zero. Again, the limiting position of the line joining corresponding points of the connodal curve and the direction of the common tangent to the connodal and spinodal curves at the plait point is given by : a 2 !! a a u aA T . S a -Si rfS aS8r 3r 2 dv Lim 7^7 - rff - - w = ~W = ~ aA ' (31) as 2 asar as Conditions (30) and (31) are sufficient to discuss the principal properties of the critical state of a one-component system. We observe that the existence of a critical state for such a system cannot be inferred from a priori considerations, because it is not necessary that the two branches of the connodal curve should ultimately coalesce ; that such is the case must be regarded as established for systems containing liquid and vapour by the experiments of Andrews ( 86), and the following discussion is limited to such systems (cf. 103). With motion along the connodal curve towards the plait point the magnitudes Ui and Ua, Si and 82, and TI and r 2 , approach limits which may be called the energy, entropy, and volume in the critical state. The temperature and pressure similarly tend to limits which may be called the critical temperature and the critical pressure. Hence, in evaporation, the change of volume, the change of entropy, the external work, and the heat of evaporation per unit mass, all tend to zero as the system approaches the critical state : r. 2 TI = S 2 - Si = P K (VZ - i) = T K (S 2 -Si)=0 Q _ O The ratio -- -- - will, however, in general tend to a finite 'a i- i limit, because there is no reason why any of the differential VAN DEB WAALS' EQUATION 247 coefficients in (31) should be zero or infinite. Thence, from the Clapeyron-Clausius equation : dp_ S 2 Si p_ dT ~ r a - n we see that the vapour-pressure curve has a finite limiting gradient at the critical point. This has been verified experimentally by Cailletet and Colardeau (1892) who find, for water : Lim ^ = 2 '22 (p in atm.) carbon-dioxide : Lim -xL = 1'60. a I In the system considered the suffixes 1 and 2 refer to liquid and vapour respectively, and in this case it is known from experiment that: S 2 > Si, i.e., L is positive i and r 2 >n I ' and hence : Lim ^ is positive at the critical point, and the vapour-pressure curve slopes constantly from left to right upwards to the critical point. This has been verified by experimenters who have followed the curves as far as the critical point. The plait point is an ordinary point on the connodal curve, and hence it is immediately evident that the specific volume and entropy in the critical state are intermediate between those of adjectent liquid and vapour phases. From (32) we see that the fractions in (31) are all positive, and since the conditions of stability at points 1 and 2 require that : 8 2 U 8 2 U 8S 8S8r 8 2 U 8^U 8S8t: 8r 2 ' ^S 2 it follows that, at the plait point : ..- (33) Equations (3), (9) and (10) give, for the changes in tempera- ture and pressure for motion along the connodal curve : /8 2 U\ . /8 2 U 248 THEEMODYNAMICS and hence, from (31), the right-hand members of (34) and (35) are zero at the plait point. Hence at that point : . . . (86) -a = 0. , . (38) f = 0,^=0 . . . . (89) dri- c?r 2 If we take rectangular axes, and put x = S, y = T x x = r, y = p > respectively, we see that the two curves in these planes, representing the two states, meet at a point corresponding with the critical point, and have a common tangent parallel to the axis of x. Further, since values of x for the critical point are intermediate between those on the two curves, it follows that the critical temperature and the critical pressure are maximum or minimum values, and we shall assume they are the former. The inequalities : v% > i~i, 8-2 > Si now lead to the following inequali- ties, referring to the immediate vicinity of the critical point : >*<0. . - . (48) Now L = T(S 2 Si) . . , ;. (44) and for motion along the connodal curve we have : L = (Sa - SiMT + TO/Sa - rfSi) VAN DEE WAALS' EQUATION ; 2 rfSi 249 . (45) From conditions (36) and (40) we see that as the critical state is approached, ^ approaches the limit oo . If therefore L is plotted against T, the curve bends round and turns its concave side towards the T axis, meeting that axis normally at the critical temperature. This has been verified experimentally by Mathias ( 92). Again, since A = X r a *"i) (46) it can easily be shown that ^- and ^ tend to oc in the critical state. Let us suppose that T and p are always given by equation (3), even on parts of the primitive surface which represent essentially unstable states. Then for motion from one point on that surface to another point we have : dT = ^ a a u + m v dl ' 8 2 U , The solutions of (47) and (48) are : (48) (49) as 2 where A has the value defined in (25). For motion along an isopiestic : dp = 0, . (50) . (51) (52) 250 THERMODYNAMICS For motion along an isotherm : dT = 0, dSSe 8S 2 and since A is zero at the plait point, the following relations hold for the critical state of a one component system : \dS/ P ' \dr / ~ ' \ (/S/ ^~ ' \dc/ ~ Again, for a point in the immediate neighbourhood of the plait point : and if we suppose that A is positive at that point, equations (51) (52), (53), and (54) give : Since (TQ)* \j) are positive for any homogeneous phase in stable equilibrium, the first and fourth inequalities of (56) are verified for any such phase, but the second and third are not necessarily verified except in the immediate vicinity of the critical state. From (51) we have : 8A , A , 8 2 U TT- dv A . d as 2 and since at the plait point A = (57) 8A rfQ A_ : ?^_! 8t ' (58) 8 2 U 8S 2 " But 5Q5- dS -f- 5-2- dr = dp '= . . . (59) VAN DER WAALS' EQUATION 251 /. on elimination of dv from (58) and (59) we have : 8A _8_A 88 9r 8 2 U 8 2 U (60) Now, from (30) we know that the determinant is zero, hence at the plait point : the second, third, and fourth relations being obtained similarly to the first. Hence if we take the rectangular axes specified in (A), we see that the curve representing y as a function of x has a point of inflexion at the value of x corresponding with the critical state, for equations (36) (39) show that ^ = at that point, and we d 2 ii have just proved that y^ = 0. These relations, and inequalities (40) (43) show that the tangent at the point of inflexion is parallel to the axis of x, and the curve everywhere else turns its convex side to the axis of x. The most important case is the critical isotherm on the p, r diagram. This has a point of inflexion at the critical point, there becoming parallel to the volume axis, and everywhere else slopes constantly from right to left upwards (Rule of Sarrau, 1882). The investigation above is due initially to Gibbs (ticient. Papers, I., 43 46 ; 100 134), although in many parts we have followed the exposition of P. Saurel (Journ. Phi/s. Chem., 1902, 6, 474491). It is chiefly note- worthy on account of the ease with which it permits of the deduction, from purely thermodynamic considerations, of all the principal properties of the critical point, many of which were rediscovered by van der Waals on the basis of molecular hypotheses. A different treatment is given by Duhem (Traits de Mecanique chimique, II., 129 191), who makes use of the thermodynamic potential. Although this has been introduced in equation (11) as the con- dition for equilibrium, we could have deduced the second part of that equation directly from the properties of the tangent plane, as was done by Gibbs (cf. 53). 252 THERMODYNAMICS The properties of the triple point could be deduced in a similar manner, since if we imagine three states of a substance coexist- ing in equilibrium, it is evident that they will be represented by three points on the primitive surface which are at the vertices of a plane triangle which is a triply tangent plane to the surface. If this plane is rolled about any of its sides, it traces out three developable surfaces, bounded by connodal curves, and represent- ing regions in which the three phases are in equilibrium taken in pairs. The parts of the primitive surface between these develop- able surfaces represent stable homogeneous states. The connodal curves for liquid and vapour ultimately coalesce in a plait point, which is the critical point for the transition of liquid and vapour, but there is no evidence that the other two pairs of connodal curves are ultimately coincident ; in fact there is much evidence to show that with those of solid and liquid, this is not the case. The thermodynainic investigation of course leaves this question quite open. The equations : .. may be used to determine the values of the critical constants from any characteristic equation. Thus, Dieterici's equation : I'+.ji which is of the form : x * 4- aj& + fa* has only three real roots, and gives : a and -5^ = 3-75. Clausius' equation : RT c f -a T(f + gives : %= 3a + 2/3, cR T K= \/_ 8c 'V 27R(a -I- 8)' CHAPTER IX THERMOCHEMISTRY 116. Chemical Reactions. If 2 grains of hydrogen and 16 grains of oxygen, mixed in the gaseous state, are converted into 18 grams of steam at constant temperature and atmospheric pressure, the following changes of energy occur (0 = 100, p = 1 atm.) : (i.) An amount of heat 58,000 calories = 243 X 10 10 ergs, is evolved. This is the calorinietrically measured heat of reaction, which in our notation we must write negative for heat erolred .- therefore Q = 243 x 10 10 ergs. (ii.) An amount of work 6'25 X 10 10 ergs is done on the system by the contraction under atmospheric pressure, or external work = A = 6"25 X 10 10 ergs. We therefore have, for the increase of intrinsic energy, AU = Q A = 243 X 10 10 6-25 X 10 10 ergs .-. AU = 236-75 X 10 10 ergs. This denotes the difference between the intrinsic energy of the sy stem (H 2 + i0 2 ) T = 373 and of the system (H 2 0)r = s 7 3 p = I atm. j) = 1 atm. In connection with such reactions we may note : (i.) If the process is conducted adynamically, as is the case when a substance is burnt in a Berthelot calorimeter, i.e., without change of volume of the reacting system, then A = 0, and Q = AU, so that (heat of reaction at constant volume) = (increase of intrinsic energy). (ii.) If the reaction occurs in a condensed system, i.e., a system composed of liquids and solids only, the external work is negli- gibly small in comparison with the heat of reaction, and Q may be taken as equal to the diminution of intrinsic energy. Thus, in the neutralisation of 1 litre of normal KOH by 1 litre of normal HN0 3 , A is only about 0'0035 per cent, of Q. 254 THERMODYNAMICS 117. Thermochemistry. The importance of the energy changes accompanying chemical reactions, although dimly perceived by the phlogistonists, was first clearly recognised by their great opponent Lavoisier, who, in an investigation with Laplace (Ostwald's Klassiker, No. 40) stated as a self-evident truth that as much heat is required to decompose a compound as is liberated when the compound is produced from its elements. This is a special case of the first law of thermodynamics if the heat changes are those at constant volume. The law was first stated with reference to thermo- chemistry, by G. H. Hess in 1840 (Ostwald's Klassiker, No. 9). Hess's Principle of Constant Heat- Summation : The quantity of heat evolved in a chemical reaction is the same whether the change occurs directly, or in several stages. This is equivalent to the statement that the evolution of heat is dependent solely on the initial and final states, and independent of the intermediate states. In this form, however, the principle is indefinite, because the evolution of heat will depend on the conditions of the system whilst the reaction is occurring. As a matter of fact Hess's principle is strictly true only when by " quantity of heat evolved " we understand that evolved when the reaction progresses at constant volume, or when it occurs under a constant pressure* For the former is the diminution of the intrinsic energy : _ Q ( , = L\ - U 2 and the latter is the diminution of the heat function at constant pressure : - Q p = Wi - W a and both are dependent only on the initial and final stages ( 25). The heat of reaction at constant pressure is : Q lt = U 2 Ui + RT(w a ??i) = Q p + 1-985T( 2 - Hl ) cal. where i, ?? 2 are the numbers of mols of gases present before and after the reaction. The adoption of Hess's principle "without qualification, as Dtihem (Mecaniqne chimiqne, 1, 50) remarks, is not legitimate, and the success which has attended the application of the principle is due to the fact that the majority of systems studied by its aid have conformed to one or other of the necessary conditions : v = constant ...... () p = constant ...... (ft) THERMOCHEMISTRY 255 Consequences of Hess 's Law : If a reaction may be instituted in separate stages, satisfying condition (a) or (b), then q a + q b + = Q, where q p = heat absorbed in the p-ih stage, Q = total heat of reaction. This was experimentally verified by Hess, in the neutralisation of sulphuric acid by ammonia. The acid (H 2 S04) was first neutralised directly with aqueous ammonia (NH 3 aq.), and then acids diluted with successively increasing quantities of water were also brought to the state of neutrality. In every case (heat of dilution) + (heat of neutralisation of dilute acid) = (heat of neutralisation of strong acid). H 2 S04 + 2NH 3 aq. 595'8 Sum = 595*8 H 2 S04 + H 2 77-8 518-9 596'7 H 2 S0 4 + 2H 2 116-7 480-5 597"2 H 2 S0 4 + 5 H 2 155-6 446'5 601'8 Definitions : (a) The quantity of heat evolved in the formation of a mol of a compound under specified conditions is called the Heat of Formation. As variable conditions may be mentioned the temperature at which the combination occurs, the pressure, and the states of aggregation of the substances. The same quantity of heat is absorbed if the compound is decomposed into its elements, the conditions being the same at every part of the reverse process as they were during the formation ; hence (heat of formation) =. (heat of decomposition). The majority of reactions studied by thermochemists have been carried out at room-temperature (18 C.), and the heats of formation are referred to this temperature unless otherwise specified. (6) If a reaction is more complex than a formation or decom- position of a compound, e.g., if it is a double decomposition : AB + CD AC + BD, the quantity of heat evolved during the complete interaction of the stoichiometric quantities of the substances is called the Heat of Reaction. If the reaction is incomplete, coming to a standstill when a 256 THERMODYNAMICS fraction x of the stoichiometrically possible change has occurred, the heat evolved is x X (heat of reaction). Hess (1840) already suspected that the heat of combustion of a compound (e.g., HaS, CSa, CO) must be less than the sum of the heats of combustion of its components, and not equal to that sum, as was previously supposed. The difference is, as his principle indicates, the heat of formation of the compound. Example (Hess, 1840) : Heat of Formation of Carbon Monoxide. (Generally one uses square brackets to indicate that the formula- weight of the substance, the symbol of which they enclose, is taken in the solid state, round brackets to show that it is in the gaseous state, and no brackets when it is liquid.) [C] + 2(0) = (C0 2 ) + 94,300 cal. (CO) + (0) = (C0 a ) + 68,000 cal. By subtraction : [C] + 2 (0) - (CO) - (0) = 26,300 cal. /. [C] + (0) = (CO) + 26,300 cal. The substance indicated by the same symbol in two or more equations is in exactly the same state in the reactions represented by those equations. In particular, the different allotropic modi- fications of a solid element (e.g., charcoal, graphite, diamond ; or yellow and red phosphorus) have different heats of combustion, and the particular form used must be specified in every case. Favre and Silbermann (1852) measured the heat of combustion of carbon in nitrous oxide, and found : [C] + 2(N 2 0) = 4(N) + (C0 a ) + 133,900 cal. But [C] + 2(0) = (COa) + 96,900 cal. .-. 4(N) + 2(0) = 2(N 2 0) - 37,000 cal. or 2(N) + (0) = (N 2 0) - 18,500 cal. Reactions which occur with evolution of heat are called exothermic reactions ; those which give rise to absorption of heat if they proceed directly, are called cndotliermic reactions. These names are due to Berthelot, 1879. The author has found the following method of applying Hess's law very simple : Rule : To find any proposed heat of reaction write down the chemical equations of the component reactions so that each symbol appears equally often on both sides of the sign of equality. If the heats of reaction (with proper signs) have been inserted, thejinknown heat of reaction being denoted by x, then the latter THERMOCHEMISTBY 257 may be found by adding both sides and solving the simple equation which results. Heat of formation of hydriodic acid. Let (H) + [I] = (HI) + xK (HI) + aq. = HI aq. + 192'01 K KOH aq. + HI aq. = KI aq. + 135'67 K KI aq. + (Cl) = [I] + KC1 aq. + 262'09 K 137-4 K + KC1 aq. = KOH aq. + HC1 aq. 220 K + (HC1) = (H) + (Cl) 173-15 K + HC1 aq. = (HOI) + aq. By addition we find 220 + 137-4 + 173-5 = x + 262-09 + 135'67 + 192-01 .-. x = 59-31 K (K = Ostwald's calorie.) [Note : "aq." is used to denote a large amount of water.] 118. The Development of Thermochemistry. After the work of Lavoisier and Laplace, and of Hess (1840), thermochemistry was developed mainly by the simultaneous labours of Julius Thomsen in Copenhagen (Thermochemische Untersuchungen, Leipzig, 1882-6), and Marcellin Berthelot in Paris (Mecaniqne chimiqtie, 1879). The former in 1853 made the first application of the mechanical theory of heat to chemistry, setting out from the fundamental assumption that the intrinsic energy of a body under the same conditions is constant, and showing that one aspect of the principle of Hess is a consequence of this postulate. To bring thermal magnitudes into relation with chemical energy, the heat evolved in a reaction was taken as the difference of the energies of the substances before and after the reaction, a proposition which is strictly correct only if the change occurs adynamically. To this consequence of the First Law, Thomsen added a new "principle" which he developed from the views on chemical affinity then in vogue. He assumed that the heat evolved in a reaction was a measure of the work done by the "chemical forces," and so was a measure of the " chemical affinity." Thus, in the decomposition of an exothermic compound a great expenditure of energy is necessary, and only such processes can bring about a decomposition which themselves develop more heat than is absorbed in the decomposition. Metals such as zinc, iron, and magnesium, the oxides of which T. s 258 THEKMODYNAMICS are formed with evolution of more heat than is developed in the formation of water vapour from the same amount of oxygen, are known to decompose steam, but if the heat of formation of the oxide is less than the heat of formation of steam, the metal (e.g., copper, silver, gold) is not oxidised by steam. In this way Thomsen arrived at the following : Hypothesis: Every simple or complex action "of a purely chemical nature" is accompanied by the evolution of heat. The criterion of the possibility of any reaction was, on this hypothesis : 2Q,-SQ, = a, where 2Q/, SQ ( are the algebraic sums of the heats of formation (heat evolved taken positive) of the final products and initial substances, respectively, and a is a positive magnitude. Similar considerations were advanced by M. Berthelot (1865), and this so-called " Principle of Maximum Work " found its way in some form or other, into all chemical treatises. The following, among other, objections were later brought against it : (i.) It implies that a reaction can proceed in one direction only, viz., that in which heat is evolved, and therefore that reversible reactions are impossible. In this sense it is a retreat to the old doctrine of affinity due to Bergmann. (ii.) There are hundreds of reactions which occur spontaneously with absorption of heat ; thus most hydrated salts dissolve in water with absorption of heat. This was got over by saying that in such cases there were "physical changes " in which solid salt became liquid, as well as " chemical changes " in which the salt combined with the water. The absorption of heat attending the first change exceeded the evolution in the second. To all such exceptions it was thought sufficient to answer that they were not " of a purely chemical nature." In spite of the fact that the general statement of this " principle " has been shown to be false from all standpoints, it must be admitted that its enunciation was quite in harmony with the spirit of the times ; the great physicists Lord Kelvin (1851) and Helmholtz (1847) had previously formulated an identical principle in connection with galvanic cells. Thomsen and Berthelot went wrong, not in their enunciation of the so- called " theorem " as a working hypothesis, but rather in their THERMOCHEMISTRY 259 persistent and blind retention of a dogma which had been proved by their own work to be incorrect. J. Willard Gibbs (1876) first pointed out the correct principle, but his statement was expressed in terms of differential equations quite meaningless to the chemists. Helmholtz (1882) (Ostwald's Klassiker No. 124) showed in a more intelligible way that the heat evolved in a chemical reaction is not usually a measure of the work done by the chemical forces (Arleitsicerth der chemischcn Verwandtschaftkrqfte), and in some cases the two can be opposite in sign. In spite of its deposition from the rank of a natural law, the Thomsen-Berthelot principle holds good in too many cases to be entirely false ; Nernst (1906) has recently given a new interpretation of it which will be considered later. 119. Heat of Reaction and Temperature. For the dependence of the heat of reaction at constant volume, we have Kirchhoffs equation ( 58) : where F t ,, r/ are the total heat capacities of the initial and final systems at constant volume : If the reaction occurs at constant pressure we have a similar equation : Q p = W 2 - Wi = (U 2 + 2>V 2 ) - (Ui + /VO .-. H" = ~ [u + pV] ! 2 = f^l 2 = r/ - r p . (2) from 62 (12), where r p , T p ' are the total heat capacities of the initial and final systems at constant pressure : r,j = ^nG p ; iy = s/j'cy. These equations enable one to calculate the heat of reaction at any temperature from its value at one temperature Qr 2 = QT, + (F' T) dT (v, or p, const.) f- J Ti In thermochemistry one usually takes heat evolved as positive ; we shall denote this by H=-Q. 8 2 260 THEBMODYNAMICS Example. Formation of steani at constant volume : H 2 + i0 2 = H 2 C

e calculated additively, from the values of the pure components, and the relative amounts of these present in the solution : (l>i + 7*2 + Pa + ) T = 7i a i + 7-22 + 7a3 -f where pi, p* are the amounts of the components ; a\, a- 2 the specific values of the property. This is called the Mixture Rule, and solutions which satisfy it are often called " mixtures." 121. Concentration. It is convenient to have some uniform method of representing the composition of a solution, and for this purpose use is made of the so-called concentration of a component, which may be referred to the total mass, or to the total volume, or to the molecular constitution. Let u~i, ic-2, ... be the weights of the various components ; MI, n-2, ... the numbers of uiols of the various com- ponents ; V the total volume, TV the total weight, and X the total number of mols. Then: (ci) = ?, . . . are the weight concentration*, [ci\ = =i, ... volume concentrations, Ci = ^, numerical concentration*, $i = -i, . . -. rolu metric molecular concentration*. If one component is present in such small amount that its ic, n, or r, may be omitted in forming the sums W, N, or V, the solution is called dilute with respect to that component. 122. Mixtures of Ideal Gases. The phenomena accompanying the admixture of gases were first accurately studied experimentally by John Dalton (1801), who arrived at the following conclusions : (1) If two or more gases are placed in contact, either by 264 THERMODYNAMICS stratifying in layers, or by connecting together the vessels con- taining the gases, then after the lapse of a certain time, which depends to some extent on the nature of the gases, a homogeneous mixture is produced which never spontaneously separates into its components. This spontaneous admixture of gases, called diffusion, was thus established as a real property of gaseous substances, in opposition to the views of Priestley and others, that if the gases were placed together in layers, the lighter ones being uppermost, they would remain separate, like oil and water, unless mechanically agitated. (2) If different gases are mixed by diffusion, at the same tempera- ture and pressure, then (provided no chemical action occurs) : (a) The volume of the mixture is the sum of the volumes of the constituent gases ; (6) The pressure remains unchanged throughout the process. Definition of an Ideal Gas Mixture. It a mixture formed of the masses MI, ?n 2 , . . >n n of the ideal gases GI, G 2 , . . G ?i has a free energy equal to the sum of the free energies of these masses of the separate gases, at the same temperature and each occupying a volume equal to the total volume V of the mixture, it is called an ideal gas mixture. The justification of this definition will be considered later; at present we shall show that it leads to consequences in agreement with experience. The equations are greatly simplified if we refer everything to molecular quantities. Let : nil, viz, w>i be the masses, i'i, r 2 , . . r h the specific volumes, MI, M 2 , . . M,, the molecular weights, V'l* ^2, $h ^e free energies per unit mass, C^, Ci 2) , . . C ( r\ the molecular heats at constant volume, MI = /MI/MI, n a = wia/Ma, . . n, = w,-/M ; , the number of mols, of the gases GI, G 2 , G,-, in the mixture. The free energy $, of unit mass of the i-th gas is ( 79) : - felJP - T fe where <7j'(T) = u Ts -f | <- v j. | and the free energy of the mass m- t is, since m, = w,M ; : GAS MIXTURES 265 But if V is the total volume : r< = V/ra, = V/n,.M, where // the mixture, and with a free enerni/ equal to that which it possesses in the mixture. Pi is called the partial pressure of the i-th gas in the mixture, and hence (9) states that the total pressure is the sum of the partial pressures. This is nothing else than the well-known Law of Mixed Gases, or Law of Partial Pressures, discovered experimentally by John Dal ton in 1801, and expressed by him in the somewhat vague 266 THERMODYNAMICS statement that " one gas acts as vacuum towards another " (Manchester Memoirs, 5, 550, 1802). Let us put M = ^MH where M is called, for convenience, the mean molecular ireinlit of the gas mixture. In assigning a molecular weight to a mixture we merely state what weight of that mixture occupies, under normal conditions, the same volume as 32 grams of oxygen ; a mixture of course has no " molecular weight " at all in a purely chemical sense. If we put n = iii -\- n z + + n i then ?*M = n^ + 2 M 2 + . . + n^ . . . (11) which is simply an expression for the law of conservation of mass. The density p of the mixture is obviously equal to -^- and the density p t of the i-ih constituent when in solitary confinement in the same volume is - J ^T :I , hence, from (9) and (11) : P = Pi + P* + +Pi = 2p, (12) But if we put p f = , where , C p are the molecular heats at constant pressure, and at constant volume, respectively, of the mixture, and Cp 1 ', C;?', those of the first, CJ?, Cf, those of the second, component, and so on, (16) = C, + R and it is easy to show from this and the preceding equations that : c = "i + + III! + so that the specific heats of the mixture are calculable from those of the constituents, at an assigned temperature, by the mixture rule. Now suppose we have the gases GI, G-2, . . G,-, in the amounts specified at the beginning of this section, all at the same tempera- ture T, separately confined in vessels of volumes Vi, V->, . . V , respectively. If we add together the free energies of the separate gases, the sum *o may be called the free energy of the unmixed gases. Thus : . . (18) Now let all the vessels be put in communication, and let the gases mix so as to form a homogeneous gas mixture of volume : . . (19) 268 THERMODYNAMICS The free energy of the mixed gases, *, has already been calcu- lated ; from (3) it is : + n . . (20) Thence, the diminution of free energy incurred by the mixing of the gases is : * - * = ET (21) which is evidently positive. Hence diffusion is a genuine spon- taneous phenomenon exhibited l>y gases, which result was experi- mentally established by Dalton. Further, if Uo, U are the total intrinsic energies of the unmixed and mixed gases, respectively then ( 58) : = (* - >IO - T p . . . (22) But we see by differentiating (16) and multiplying the result by T, that the expression on the right of (22) vanishes, so that : Uo-U = 0) or U = U ) Corollary. If different ideal gases mix by diffusion so that the total volume of the mixture is equal to the sum of the volumes of the constituents, there is no evolution or absorption of heat. This result, which may be regarded as an extension of the theorem of Joule, was also experimentally discovered by Dalton. For the purpose of throwing the equations into convenient forms as required, we may deduce a few simple relations. GAS MIXTURES 269 We have Vi/Vg = wi/a, by Avogadro's theorem, hence in virtue of a well-known algebraical theorem : Vi + V 2 + ^ + V, /n + na + . . + MJ "Vi ttl or generally, SV,- _ S, V; ~~ ; If Pi, lh, J^; are the partial, and p the total, pressures : * = v, + v,+ .. + v, "=v, + v,+ i . . +v,^ and generally ^,. = ' ^> . . . . (25) If Ci, c 2 , .... c ; are the numerical concentrations : Cj = W../S/I,- = Vf/SV,-. . . . . (26) It is also evident that the numerical concentrations are pro- portional to the partial pressures : Pi = ^-P = c i p .... (27) If ii &i ' & are the volumetric molecular concentrations, 6 = n,/V = w,/2Vi But V = . (28) Equation (23) shows that the energy of the mixture is the sum of the energies of the constituents, and is independent of the volume. Thus if : I C J rfT . . . (29) is the energy of a mol of the i'-th gas, nU = Ma = S/iiUj = SwjUi 4 ' + Sn { CydT . . (30) 270 THERMODYNAMICS or, if C[f is assumed independent of temperature : U = 2,-H,-[Uo + HiC'j.'T] . - v . (31) where ,-, M;, C' r ", M, have the usual significance, and n~M.u is the whole energy of the mixture, U its energy per mol, and W is the arbitrary term in the expression for the energy of the i-th gas. For the free energy of the gas mixture we have : wM^ = 2i [~BTJn J - gi(T)~\ ^> - L f] . . (32) - Kin *j + Btocf] . (33) + Binfi] . . . (34) from (27), and (28). If the specific heats are independent of temperature, we obtain //;(T) from (4), and hence : nlVty = T2w; fcii'Cl - ZwT) - B/u - - M^S^ - B?w ^ + M^ L Pi M, 1- - iwT) - B/w - - M { Sy - R fo* - P . (35) BiwM,. + B/f ( - . (36) The potential is obtained simply by adding 2w,-BT to the expres- sions for the free energy (cf. 79) : (37) s , = T2, [^ - Bfot ^ + R(l + toe,)] . . (88) If the specific heats are independent of temperature : - B/-H -^ + R 8^ - B/M ~ TTl^-i 4-- - (39) GAS MIXTURES 271 = T2n f [c r "'(l - /T) - R/ii -^ + B - SJf' (40) The entropy is obtained from the free energy by means of the equation S ^~: /I - R(l -f- /we,-) - If the specific heats are assumed to be constant : /*M* = 2w,- c^/wT + R/H + S " + R/ - fact (42) Examples. il) Prove that the entropy of an ideal gas mixture is the sum of the eutropies of the components at the same temperature, each occupying the whole volume of the mixture. (2) Show that the potential of an ideal gas mixture is the sum of the potentials of the components at the same temperature, each occupying the whole volume of the mixture. (This result is very important.) (3) Show that the diminution of free energy on mixing i. n. 2 , . . niols of the gases GI, G*, . . so that the total volume remains constant is : V *i = BT2H.ZH _^ = BTStUn = - KTSn./nc i Pi = - BTSi^n*. + 2ft7n JL- (4) For two gases, with HI = a = 1, the maximum loss is incurred with equal volumes, and is BTf2. 123. Verification. All the expressions for^o *i evidently represent the di$si2>a- tion of energy which occurs when the gases are allowed to mix by diffusion in the specified manner. It follows from the principle of dissipation of energy that work will have to be spent in sepa- rating the mixture into its constituents, and, conversely, work should be obtained if the gases are allowed to mix in a suitable manner. The first quantity of work will be a minimum, the latter a maximum, and both equal and opposite, when the pro- cesses are conducted reversibly. The definition of an Ideal Gas Mixture given in 122, although it leads to results in entire accord with those established bv 272 THERMODYNAMICS experiment, cannot be regarded as entirely justified unless we can show, by some physical method, that the free energy of a gas mixture is less than the sum of the free energies of its components in the free state at the same temperature and^ pressure by the amount of work which is obtained in producing the mixture isothermally and reversibly from its components. This has been calculated from the definition, and the result is contained in (21), 122. The solution of the problem depends solely on the possibility of finding a process by which a gas mixture can be formed reversibly from its components, or the mixture separated reversibly into the latter. Passing over such special methods as separation by differences (I) (2) A B FIG. 54. of solubility in a liquid, or by different condensation tempera- tures, we come to a general method of separation of gaseous mixtures introduced into the theory of the subject by Rayleigh (1875), and Boltzmann (1878). This method has the great advantage that, by its aid, the separation may be effected isothcr- mally and reversibly. It depends on the existence of substances exhibiting a selective permeability to gases; such as palladium, platinum, and iron at high temperatures, which are freely permeated by hydrogen, but not by nitrogen. It is therefore legitimate to postulate, for the purposes of thermodynamic reasoning, ideal septa each of which is permeable to one gas but quite impervious to all others. Such septa are called semipermeable septa. We now suppose that we have (Fig. 54) an impervious cylinder fitted with two semipermeable pistons A and B, connected to some outside sources or receivers of work by rigid piston rods, passing gas-tight through the cylinder ends. Let the pistons be first placed in contact, and let A be freely GAS MIXTURES 273 permeable to an ideal gas [1], contained in the space (l)of volume r i ; let B be freely permeable to an ideal gas [2] , contained in the space (2) of volume r 2 . Each piston is impervious to the gas which penetrates the other. Then it is evident that no pressure is exerted on either piston by the gas which freely permeates it, and the pressure to which a piston is exposed is therefore always that exerted upon it by the gas to which it is impervious. The pistons therefore tend to separate, A under the pressure exerted by the gas [2], B under the pressure exerted by the gas [1], and motion ceases only when the pistons are in contact with the cylinder ends, and the gases are uniformly mixed. The mixing can be performed isothernially and reversibly by sinking the apparatus in a constant temperature bath, and opposing the expansive forces of the gases by forces differing only infinitesimally from them, and applied to the piston rods. The slightest increase of applied force will reverse the direction of the process, and the mixture is separated reversibly into its constituents. If there are HI inols of gas [1] and n- 2 mols of gas ~2~, the work done during the mixing is equal to the sum of the amounts of work done by each gas hi expanding from its initial volume to the final volume of the mixture, for, as is evident from the conditions, each gas performs work as if the other were absent. H! mols of [1~ expand isothernially and reversibly from a volume t~i to a volume (i'i -\- r 2 ) : .-. A, = Ml RT7it oVo P~V == ^1 )X 1 . (8) from (5). By addition and substitution from (1) : PoVo-PV p_ VpPo/P - V - \' p - , p - (A _ v ;a? which gives the concentration in the gas space after equilibrium is attained. The absolute values of the solubilities of gases are not at present calculable from any general law, although W. M. Tate (1906) finds in the case of aqueous solutions a relation with the viscosities of the solution (ju. fl ), and water (^o), the critical temperatures of the gas (T f/ ), and of water (T !( .), and the absorp- tion coefficients : (Meddd. fran Vetensk. Abaci. Nobel hist. 1, 4, 1906, Centralbl , 1908, i. 1659.) CHAPTER XI THE ELEMENTARY THEORY OF DILUTE SOLUTIONS 127. Osmotic Pressure. The relations of liquids to seinipermeable septa were observed by the French natural philosopher, the Abbe Nollet (1748), who tied a piece of bladder over the mouth of a jar containing alcohol, and placed the whole in water. The bladder swelled up, and ultimately burst from the internal pressure. The same observer gave a correct interpretation of the phenomenon, which arises from the much more marked permeability of the septum to water as compared with alcohol. Similar phenomena are met with in organised nature, where two liquids, such as cell-sap and water, are separated by a membrane, and they received the name of osmotic phenomena (oxrpios an impulse). M. Traube (1867) showed that artificial semipermeable septa could be produced from such slimy precipitates as gelatine tannate, cupric and zinc ferro- cyanides, and W. Pfeffer (1877) succeeded in depositing these in the walls of a porous jar, which when filled with a solution of salt, or sugar, etc., closed by a cork through which passed a mercury manometer, and plunged into pure water, furnished a means of measuring the osmotic pressures. By the use of such an apparatus (" osmorneter "), or modifications of it, quantitative measurements have been made by Pfeffer, Adie, Morse, Lord Berkeley and others. These are described in treatises on physical chemistry ; we shall here confine our attention to the theory of the subject. Definition of Osmotic Pressure. Let a given liquid solution (e.g., a solution of sugar in water) be separated from the pure liquid solvent by a fixed rigid diaphragm, permeable only to the latter. If -, ->' are the pressures which must be applied to solvent and solution, respec- tively, to maintain equilibrium, then : 280 THERMODYNAMICS TT is defined as the osmotic pressure of the solution at the (uniform) temperature of the system, and with the solvent under the pressure TT. If the septum is not rigidly fixed to the cylinder bounding the liquids, it will be necessary, in order to maintain equilibrium, to apply to it a pressure equal to TT' IT, or P. in a direction from left to right. For the pressure TT is transmitted unchanged through the septum to the fluid on the right, and if an additional pressure TT' TT is put on the latter by means of the septum, the right hand piston remains in equilibrium. If the septum is fixed, this force is exerted on it by reaction from the fluid, and becomes evident in Nollet's experiment. It is a consequence of the principle of Dissipatien of Energy that P is positive for all concen- trations of the solution and all temperatures. For if we suppose the end pistons fixed, and the septum moved towards the solu- tion, there will be a separation of the latter i nto : (solvent) + (more concentrated solution), and the work (TT' 77)8 V will be spent on the system, where 8V is the volume through which the piston advances. Since this process involves the reversal of a spontaneously occurring process, viz., the mixing of solvent with solution by diffusion, the work spent is positive, hence TT' TT, or P, is always positive. We shall now prove that P, for fixed values of TT and the temperature, is definite for a given solution. For this purpose we have first of all to show that the dilution or concentration of the solution can be effected isothermally and reversibly. If the above apparatus is constructed of some good conductor of heat, placed in a large constant-temperature reservoir, and if all pro- cesses are carried out very slowly, the isothermal condition is satisfied. Further, suppose the end pistons fixed, and then apply to the septum an additional small pressure ^5P towards the solution. There will be a slight motion of the septum, through a small volume 8V, and work (P- ELEMENTARY THEORY OF DILUTE SOLUTIONS 281 will be spent on the system in separating pure solvent. But if the pressure on the septum is reduced by an infinitesimal amount 8P, there will be a slight motion in the opposite direction, and work (P - i8P)8V will be done by the system. Since SP can be made as small as we please, the operation becomes in the limit reversible. It is a necessary consequence of the reversibility of osmotic processes that the osmotic pressure is independent of the nature of the septum used to measure it. For, suppose there are two semipermeable septa [a] and [/3], and let the osmotic pressures of a solution when separated from pure solvent under a given pressure by these septa be P a and P^. Then if we separate a volume 8V of solvent through [a], the work P a 8V is spent on the system, and if the solvent is readmitted through [3] the work P08V is done by the system. The isothermal cycle being now completed, we have : - P.8V = Leakage of solute through an imperfect septum would constitute an irreversible phenomenon (diffusion), and no equality of P a , P^ need then result. It was formerly believed that the osmotic pressure of a solution depended on the nature of the septum, but this was merely a consequence of the use of imperfect septa in the experiments. The experimental investigation of such septa is a matter of great practical interest and importance, but is of no more significance in the theory of the subject than is leakage of steam in the cylinders of actual engines in the consideration of the expansion of gases. We shall now pass on to a study of the laws of osmotic pressure, taking up in the first instance the very important case of dilute solutions. In this section it is assumed that there is no change of total volume when a solution is diluted by further addition of pure solvent, and that solution and solvent are practically incom- pressible. The reader will then easily see that the osmotic pressure in such a case is independent of the pressure supported by the pure solvent ; the complete investigation is taken up by A. W. Porter, Proc. Roy. Soc., A, 79, 519, 1907 ; 80, 457, 1908. 282 THERMODYNAMICS 128. The Laws of Osmotic Pressure for Dilute Solutions. We assume, on the basis of experimental results, that : (1) The volume of the solution obtained by mixing a volume TI of pure solvent with a volume r 2 of a dilute solution is (i\ + y 2 ). This is usually expressed by saying that the volume of the solute in a dilute solution is independent of the total volume of the solution. We might equally well say that the volume of the solute is either zero, or equal to the total volume of the solution. (2) If a dilute solution is mixed with pure solvent, without performance of external work, there is no evolution or absorption of heat, so that AU = 0. Corollary. The work done in the isothermal and reversible dilution of a dilute solution is equal to the heat absorbed from the constant temperature reservoir. The above result is often expressed by saying that the intrinsic energy of the solute in a dilute solution is independent of the volume. (3). If a gas is in isothermal equilibrium with its solution in a . , liquid, the concentration in the solution is proportional to the pressure of the gas (Henry's law). In the original investigation of van't Hoff (Zeitschr. physikal. Chem., 1, 1880; Phil. May., 26, 1888), the laws of dilute solutions are arrived at separately, but the deduction of the proportionality between osmotic pressure and concentration, as given by van't Iloff, is rather an analogy than a stringent proof, since it makes use of hypothetical considerations as to the cause of osmotic pressure. The following proof, due to Lord Bayleigh (1897), is quite strict, and has the advantage of leading directly to the whole theory. (Rayleigh, Nature, 55, 253, 1897 ; Donnan, ibid. ; cf. Larmor, Phil. Trans. A, 190, 205, 1897.) We shall suppose the solute to be a mol of an ideal gas, occupying a volume r at the pressure p$, and the solvent a volume V of liquid just sufficient to dissolve all the gas under the pressure j?o- If the gas is brought directly into contact with the liquid, an irreversible process of solution occurs, but if it is first of all expanded to a very large volume, the dissolution may be made reversible, except for the first trace of gas entering the V 6 / 1 ' '',' ' ' ' ////y////' IP^ 7 FIG. 56. ELEMENTARY THEORY OF DILUTE SOLUTIONS 283 liquid, because there is a gas pressure corresponding with every concentration of the solution, however small, which is determined by Henry's law. The gas can then be slowly compressed so that dissolution proceeds isothermally and reversibly under con- tinuously increasing pressure, until the last trace goes into solution under the pressure p Q . Let the gas and liquid be contained in a cylinder of unit cross-section, fitted with pistons and fixed diaphragms as shown in Fig. 56. a is an impermeable piston, /3 a fixed diaphragm permeable to gas but not to liquid (the upper surface of a non- volatile liquid satisfies this condition), 7 is a piston permeable to liquid, but not to dissolved gas, and 8 is an impermeable diaphragm which can be put over @ or 7 as desired. Now carry out the following isothermal and reversible cycle : (1) Expand the gas to a volume x, which is very large compared with r, keeping 8 over yS. The pressure at any stage (x) of the expansion is and the total work done by the gas on expansion is (2) Remove 8, and let the rarefied gas begin to dissolve in the liquid, pressing down a reversibly. The pressure on a is, in any given position, less than before, because some gas is now in solution. It follows from assumption (1) at the beginning of this section, that the pistons ft and 7 may be kept fixed, and no work is done by them. The value of the gas pressure for the position a- has been calculated in 126, (5), to be : A, being the solubility of the gas. p is therefore a function of .r. The work done by the system during the whole second operation is : r ,1 r _ p(h . = _ Pov ^ Jo + __ _ o ___. o The total work spent during operations (1) and (2) is therefore x + AV ri f x + AY , 284 THERMODYNAMICS x 4- AY By hypothesis, r = AV, and since - becomes more and more nearly equal to unity as .r becomes larger and larger, it follows that the (/as has been dissolved rcrersibli/ without loss or gain of work. (3) To complete the cycle, we must get the gas out of solution, and restore it to its initial state, by an osmotic process. Raise a and 7 simultaneously through the spaces r and V respec- tively, so that the solution maintains a constant composition throughout, and the gas and osmotic pressures are constant. The work done is }w PY, and since the whole work in the cycle vanishes : Po r = PY. But 2W = RT /. PY = RT (1) where T is the temperature at which the isothermal process is executed. If we start with n mols of gas we have p v = nRT /. PV = RT .... (2) We may suppose the cycle carried out at various temperatures, TI, T 2 , . . T.,, and so obtain P!\ T = RT 1( P 2 Y = RT 2 , P r V = RT,, so that (1) and (2) apply quite generally. It therefore appears that the osmotic pressure is : (1) proportional to /V, i.e., to the concentration f, at constant temperature ; (2) proportional to the absolute temperature at constant concentration ; (3) the same for equimolecular concentrations of all solutes, and is independent of the nature of the solvent ; (4) equal to the pressure which would be exerted if the solute occupied the space Y in the condition of an ideal gas. We may therefore sum up the results in the statement that the laws of osmotic pressure of a dilute solution are formally identical with tJie laws of qas pressure of an ideal gas (van't HofTs Gaseous Theory of Solution). ELEMENTARY THEORY OF DILUTE SOLUTIONS 285 129. Remarks on the Theory of Solution. It has been assumed in the deduction of (1) that the solute is an ideal gas, or at least a volatile substance. The extension of the result to solutions of substances like sugar, or metallic salts, must therefore be regarded as depending on the supposition that the distinction between " volatile " and " non-volatile " substances is one of degree rather than of kind, because a finite (possibly exceedingly small) vapour pressure may be attributed to every substance at any temperature above absolute zero. This assumption is justified by the known continuity of pressure in measurable regions, and by the kinetic theory of gases. It is also assumed that the solute does not change its molecular weight on passing into solution, that is, does not polymerise or dissociate. If this were the case, n in the two states would be different, and if 1 mol of gas on passing into solution gave rise to t mols of solute, we should have PV = iRT (3) where i is known as van't Hop's factor. Lord Kelvin has pointed out that the similarity between the laws of gases and of dilute solutions carries with it no inference as to physical similarity between the states, although Boltzmann has developed a theory of osmotic phenomena which regards the pressure as due to a bombardment of the semipermeable wall by the molecules of dissolved solute, whilst it is subjected to equal and opposite forces by bombardment from the solvent molecules inside and outside. This kinetic explanation of osmotic pressure has recently received experi- mental support in the beautiful researches of Perrin on the Brownian move- ment (Brcwnian Movement and Molecular Jieality, J. Perrin, trans. F. Soddy, 1910). The kinetic theory of gases shows that the pressure p exerted by a gas is given by : pv = fnE . (1) where H is the number of molecules in a volume v, and E is the mean kinetic energy of a molecule. If we now assume that the osmotic pressure of a solution has its urigiu in the bombardment of the semipermeable diaphragm by the molecules of the solute, we shall, if the same reasoning is applied which led to (1), obtain : Pr = fnE' where E' is is the mean kinetic energy of a molecule of solute. If is the number of molecules per unit volume : P = fE' (2) Now it is a consequence of the experiments of Pfeffer, which proved that the osmotic pressure was equal to the pressure which would be exerted by the same number of molecules of an ideal gas occupying the volume of the solution, that : E = E' (3) so that the ilissolveil molecule has the same mean kinelif energy as if it existed as 286 THERMODYNAMICS a yas at the same temperature. This kinetic energy being proportional to p at constant volume, is proportional to the absolute temperature. Perrin found that, if an emulsion of gamboge were allowed to settle, the granules did not all fall flat to the bottom of the vessel, but remained per- manently forming a kind of atmospheric haze extending to a short distance into the liquid. The suspended particles were seen under the microscope to be in Brownian motion. Imagine an emulsion formed of suspended particles all exactly alike, and contained in a vertical cylinder of unit cross-section. The state of a horizontal slice contained between the levels h arid h -f- dh would not be changed if we enclosed it between two semipermeable pistons which allow the molecules of water to pass, but stop those of the gamboge. Each piston experiences an osmotic pressure, by reason of its bombardment by gamboge particles, and if there are /( particles per unit volume at height /< l\ = %nW and P h+ T. We now assume that : (i.) The vapour of the solvent obeys the gas laws. (ii.) The osmotic pressure obeys the laws of dilute solutions. Then pr B BT . .. . . (3) where n = number of mols of vapour in the volume r B ; and PV = ?iET .... (4) where n = number of mols of solute in the volume V of solution. From (1), (3), and (4) we obtain the relation : But In , - lnl + = - (6) P \ P I P approximately, if p p' is small compared with p. Thence or '- - = V- (7) p -no + Corollary 1. The relative lowering of vapour pressure i' is independent of temperature (Law of von Babo). Thus the vapour-pressure curves of the solution and solvent are similar and similarly situated, i.e., if we know the form of the vapour -pressure curve of the pure solvent, those of all the solutions are also known. Corollary 2. The relative lowering of vapour pressure is pro- portional to the concentration c n/ (//o + ) at all temperatures, so that if equimolecular amounts of different substances are dis- solved in equal weights of solvent, the solutions all have the same vapour pressure, independently of the natureof the solute. Corollary 3. If solutions are prepared with different solvents and solutes so that in all cases number of mols of solute total number of mols in solution has the same value, the relative lowering of vapour pressure will ELEMENTARY THEORY OF DILUTE SOLUTIONS 291 be the same in all cases, and numerically equal to this fraction. These results were all experimentally verified by Raoult, who found the value O0104 for the mean molecular lowering with 12 solvents and a variety of organic solutes, in a solution with u 4. ,} ~ -mo' w ^ ereas (7) gi yes O'Ol. The following figures for cane sugar at in aqueous solution, by Dieterici (1897), and Srnits (1904), show how good is the agreement when we take account of the fact that (_/; p') was always less than 0*05 mm. Mols per l.OOOgr. H-A n mols per 100inolsH 2 O. (P ~ P'} 4- n. (-*' n \ \ p molec. lowering. 0-0509 0-0916 0-0467 0-0102 0-1506 0-2711 0-0467 0-0102 0-1723 0-3101 0-0477 0-0104 0-2653 0-4775 0-0467 0-0102 0-4541 0-8174 0-0483 0-0105 0-4993 0-8987 0-0483 0-0105 1-0098 1-8134 0-0500 0-0109 1-0122 1-8220 0-0500 0-0109 It will be noticed that we make no assumption as to the mole- cular weight of the solvent in the liquid state. Equation (3) refers to the vapour only. It is to be expected, therefore, that when the solvent does not yield a vapour having the normal density, the value of the molecular lowering will be abnormal. Eaoult found that when acetic acid was used as solvent the observed molecular lowering was 0'0163. Acetic acid, however, is known to be polymerised in the state of vapour ; at the boiling- point the molecular weight as determined by the vapour density is 1'64 times the normal (C 2 H40 2 = 60). The number of mols per unit volume will be reduced in the same ratio, and hence we must write (3) : u 2 292 and (7) : THERMODYNAMICS . X :r - A = 0-0164 + it 1'64 very approximately, since n is very small compared with n . This agrees almost exactly with the observed value. In the case of aqueous salt solutions, the observed molecular lowering was invariably greater than the calculated. In this case we put instead of (4) : p\ = inET and hence 7 P H In * -, = i - P o or .P approximately . (8) The following numbers were obtained by Smits (1902) : Sodium chloride in water at C. Mols per l,000gr. H 2 0. Mols per 100molsH 2 O(7i). 1 np i 0-0591 0-1064 0-01743 1-79 0-0643 0-1157 0-01741 1-76 0-1077 0-1939 0-01727 1-72 0-1426 0-2567 0-01727 1-72 0-4527 0-8149 0-0170 1-70 0-4976 0-8957 0-0170 T70 1-0808 1-9454 0-01727 1-723 1-2521 2-2538 0-01740 1-730 Sulphuric acid in water at C. Mols per l,000gr. H 2 O. Mols per 100molsH. 2 0(n). P-P' i np 0-0951 0-1712 0-0204 2-03 0-1208 0-2174 0-0188 1-87 0-4215 0-7587 0-0193 1-93 0-9762 1-7572 0-0207 2-06 ELEMENTARY THEORY OF DILUTE SOLUTIONS 293 The relation between osmotic pressure and vapour pressure was deduced by Gouy and Chaperon (1888), and independently by Arrhenius (1889). Equation (7) is true for volatile as well as involatile solutes, pro- vided n denotes the number of niols of solute in the liquid phase, and p' is the partial pressure of the vapour of the solvent, the latter being independent of the presence of other gases in the vapour space. The sole remaining problem is therefore the determina- tion of the partial pressure of the solute, or, what will lead to this, the total pressure in the vapour space. The partial pressure of the solvent is, from Raoult's law : and if TT is the partial pressure of the solute, the total pressure is : n = P i^fT7, + T (9) If the solute is a gas, -n is known, for a fixed total volume, from the solubility law of Dalton, hence FI is determined. Put ii/(ii -f- M) = e, TT/U = c' . . (10) for the concentrations of solute in the liquid and vapour phases, then: which reduces to Raoult's equation as a special case for a non- volatile solute (TT = 0, c' = 0). Equation (11) is due to Nernst (1891). Corollary 1. The vapour pressure of the solution is greater than that of the pure solvent when the concentration of the solute is greater in the vapour than in the liquid phase. Corollary 2. The vapour pressure of the solution is equal to that of the pure solvent when e = c'. Since, by Henry's law, c/c' depends only on temperature, and since distillation of liquid cannot alter its composition in this case, the solution will distil unchanged at a constant temperature exactly like a pure substance. This holds only within the limits of applicability of Henry's law. If several different solutes are present together : If we assume, with Nernst (1893), that the osmotic pressure of a solute (being independent of the nature of the solvent provided '294 THERMODYNAMICS no molecular change results) is the same in a mixture of solvents as in a pure solvent, then, since for NX mols of solvent [1] and N 2 mols of solvent [2] per mol of solute, we have by Raoult's law : In&r = w ;>> JNo therefore 1 = NI In 1>l t + N 2 /n *~ , '. . (13) Pi PS This was verified by Roloff (1893). Some of the terms on the right of (13) may be negative. Thus, if potassium chloride is added to a mixture of water and acetic acid, the partial pressure of the acetic acid is raised : p% > ^2, ' In, < 0, which implies that the solubility, A, of acetic acid in water is reduced by the addition of the salt (Nernst : Thcoretische Chemie). 131. Boiling-Points. Let AA', BB' represent portions of the vapour-pressure curves of pure solvent and solution, respec- tively. Draw PQ, QR parallel to the axes, then PQ = RQ tonPRQ .'. p p' = ST tawPRQ . (1) p p' is the lowering of vapour pressure, ST the elevation of boiling- point. I If RQ is small, we have by the mean value theorem (H.M. 69) : i = TTp for the pure solvent. /. p-p' =5T^ . . . . (la) But ^ = - *- = p \ . . . (i/,) with the assumptions of 88. ^ ' = RW 8T - < 2 > If p = 1 atm., 5T is the elevation of boiling-point, T is the boiling-point of the pure solvent. ELEMENTARY THEORY OF DILUTE SOLUTIONS 295 Corollary. The molecular elevation of boiling-point is indepen- dent of the nature of the solute. A e refers to a mol of solvent, /. R = T985 g. cal., or 2 g. cal., approximately, But * - = , very nearly, 1> n .-. for 1 mol of solute in 100 niols of solvent, T = ^W . . . . (4) in which the molecular elevation of boiling-point, ST, is given entirely in terms of quantities which depend only on the properties of the pure solvent. Equation (4), deduced by van't Hoff (1886), was verified experimentally by Beckmann (1889). If p p' is too large to justify the assumption of (la), we can integrate (lb) on the assumption that A, is constant : Inp = ~ + const. where p is the vapour pressure of the pure solvent at the boiling- point T of the solution. At the boiling-point of the pure solvent : Inp' = ^ + const., since p' is atmospheric pressure, But From the results of 130, we have : ~p PV = ;*oRT ^ = oBT/w ^. J P , P P If Mo, p are the molecular weight, and density, of the solvent : y _ OM p 296 THERMODYNAMICS To express P in atmospheres, we take the litre as unit volume, aiid put R = 0-08207 1. atm. _ 0-08207 X Mo From (6) and (5) we obtain : But A,/M<, = L,,, and T - T = 5T, PT 8T = pL; ' <8) To express P in atmospheres, we put L, " 24-191 and use (6a) : 1000/>L, 5T . P = ^19T T atm ' K| ' " (8a) 132. Thermodynamic Theory of Freezing-Points of Solutions. The freezing-point of a solution is the temperature at which the solution is in equilibrium with ice, the latter term being used in its general significance of frozen, or solid, solvent. The vapour pressure of the solution at the freezing-point is equal to that of pure ice at the same temperature (Guldberg, 1870). For if we take the system : ice, solution, vapour, at the freezing- point, and suppose that p 1 ', p" are the vapour pressures of solution and ice, r', v" the specific volumes of the vapour under these pressures, and V, Y" the specific volumes of solution and ice, we may execute the following isothermal cycle : (1) Evaporate unit mass of solvent from a large quantity of solution. (2) Compress or expand the vapour until it is in equilibrium with the ice. (3) Condense the vapour on the ice. (4) Allow unit mass of ice to melt in contact with the solution. ELEMENTARY THEORY OF DILUTE SOLUTIONS 297 The net work in the cycle is zero, hence : p'(v' - V) + pdv + p" (V" - v") + p"(T - V") = V dp + (p" - j/)V" = .'.p" = p' . (1) Thus the vapour -pressure curves of solution and of ice must intersect at the freezing-point. But the vapour-pressure curve of the solution lies below that of pure solvent throughout its length, and since the latter cuts the vapour- pressure curve of ice at the freez- ing-point of the pure solvent, the former must cut the vapour-pressure curve of ice at a lower temperature, or in other words the freezing- point of the solvent is depressed by addition of solute. It is, of course, assumed that the solid which separates is pure ice, for otherwise the pressure curve of the solution would not cut the pressure curve of ice at the freezing-point, but some other curve, belonging to the separating solid. Let OA, AS represent the vapour-pressure curves of the ice and liquid solvent respectively, BS' that of the dilute solution. AC is the vapour-pressure curve of supercooled liquid. T freezing-point of pure solvent, T that of the solution. CS, OA we have : dp, A,, A,, cW = T(r, - r,) = dp, A . A, To FIG. 60. is the Along \ 1 V,) LVg If the vapour obeys the gas laws : dlnp e \ e dT~ = RT* dlnp, _ \s_ and if \ e , A, do not vary appreciably with temperature over a 298 THERMODYNAMICS small range, which presupposes (T T) to be small, or the solution very dilute, we have : lnp e = < + c', and lnp x = - ^ + c" . . . . (2) At the temperature T , we have Pe=P*=Po - (3) where p Q is the common pressure of ice and solvent at A, and /7>o = ^?jr + " I ' -a - _ ^ P- 1 PQ R LT ToJ and In = : so that ^^^. . . (4 ) But p e = p = vapour pressure of supercooled liquid at C, 2> s =p' = ,, solution at its freezing-point. Also A, A,, = Ay, the molecular heat of fusion ( 117), Equation (5), in a slightly more general form, was deduced by Guldberg (1870). The latter allowed for the change of A, with temperature, which has been neglected above on account of its very small magnitude. If A is in calories, K = T985. To get the relation with osmotic pressure, we have : P = .4. BT//I >- Mo p . P _ f - MO LT~TJ = To" 8 where L f = A 7 /M = latent heat of fusion per unit mass, ST = TO T = depression of freezing-point. To express P in atmospheres, we multiply L y by 24*191 to ELEMENTARY THEORY OF DILUTE SOLUTIONS 299 reduce to litre-atmospheres, and take 1000/3, so as to refer to the litre as unit volume : 100QpL,ST " 24-191 To If there are mols of solute per litre, ..... (8) - - ' 24-191 ' To' and since R = 0'08207 But if R in (8) is taken as 1"985 g. cal., L/must be in g. cal., and we still obtain (9). In the case of water : L, = 80, p = 1, T = 273 .-. 8T = l-86 .... (10) which gives the depression for mols of solute per litre in terms of quantities depending on the pure solvent only. Raoult expressed his results in terms of the molecular depres- sion (J3, for a mol. of solute in 100 grams of solvent. The volume of the solvent is 100/p, and this may be taken as the volume of the dilute solution. The corresponding osmotic pressure P', on the assumption that the law of proportionality holds good at this concentration (which is only a fictitious extrapolation) is given by : P:F = 10 :1000 P .'. P' = 10pP .... (11) .-. P'V = flOpRTo from (8). Thence 8T = o-oonTO xl(v ;jV. ; .;; : ^ : =^5 ..... (12) very approximately, where 5 denotes the number of mols of solute per 100 gr. solvent. If 5 = 1, 8T = <, the molecular depression, .'.*=5y .... (18) This equation was deduced by van't Hoff in 1885, and provides a simple method of determining the latent heat of fusion of a 300 THERMODYNAMICS substance. All that is necessary is to find the freezing- point (T ) of the pure substance, and the molecular depression by means of a substance of known molecular weight used as solute. Eykman has verified van't Hoff's equation with a large number of substances used as solvents. The values of L f calculated from (13), and those observed directly, are given below in a few cases : - <*> 0-02 T 2 /L/ obs. TO - 273. L/ obs. L/ calc. Water 18-5 18-54 80 ! 78-75 Formic acid 28 27-5 8 57-38 56-4 Acetic acid 39 38-5 17 43-66 43-1 Stearic acid . 45 47-7 64 47-6 50-5 Benzene . ;" 51-2 51-9 5-5 29-9 30-3 2>toluidine 53 51 42-1 39 37-5 Phenol 727 78-6 40 24-93 27*0 Phenylacetic acid 90 97 79 25-4 27-5 It was found that the law of proportionality (12) holds good .only if the solution is dilute, and in the determination of one usually adopts Eykman's method by finding the molecular depressions for a few different concentrations in dilute solutions, plotting these against the concentrations, and extrapolating to zero concentration, or infinite dilution (H.M., 67). Raoult observed that many substances dissolved in benzene, nitrobenzene, and ethylene dibromide, gave depressions only half the normal, and this he explained as due to a polymerisation of the solute to double molecules : 2A = A 2 . In confirmation it was observed that such substances (e.g., acetic acid) gave abnormally high vapour densities. But when solutions of salts in water were found to give depres- sions considerably in excess of the normal, usually approaching double that amount at high dilution, the interpretation was by no ELEMENTARY THEORY OF DILUTE SOLUTIONS 801 means clear. We might suppose that all the so-called normal depressions produced by organic solutes are really due to double molecules, and that the salts are normal, but the whole body of external evidence is unanimously against this hypothesis. The only other explanation possible, if we suppose the law of propor- tionality to be valid in all cases, is to suppose that the salts are broken up in solution, and Arrhenius (1887) showed that this is also in agreement with the electrical properties of the solutions. He supposed that the salt molecules break down, to a greater or less extent, into sub-molecules, or ions, which carry electrical charges : KC1 = K- + C1-, and the increased number of molecules of solute so produced accounted for the abnormally large depressions. In this case, if we put : P=iRT . . . . (8a) instead of (8), where i is the ratio of the observed osmotic pres- sure to that calculated from van't Hoff's theory, we shall obtain instead of (12) : 6T = ow "/ and i is also the ratio of the observed to the calculated depression of freezing-point. A similar method was used in connexion with the lowering of vapour pressure in 130. It is evident that, since the factor / was introduced in the same connexion in both investigations, the values of i obtained by both methods, viz., by measurements of vapour pressure and of freezing-point, are necessarily the same, and their agreement is therefore independent of any theory which we may adopt to explain the anomalous behaviour of aqueous salt solutions. The test of the validity of the theory of Arrhenius is not therefore to be found in the agreement between the values of obtained from measurements of any properties of solutions which are conditioned by the osmotic pressure ; it is in quite another field that of electrochemistry that a comparison of known relations with the deductions from the theory may be instituted. 302 THERMODYNAMICS The following numbers are given by Dieterici (1891) : Solute. i (vapour pressure). i (freezing-point). NaCl 92 1-90 KC1 78 1-82 KBr 74 1-90 KI 82 1-90 LiCl 92 1-99 NaN0 3 65 1-82 There is fair agreement between the two sets, considering the difficulty of the vapour-pressure measurements. . 133. Heat of Solution ; Effect of Temperature on Solubility. A liquid solution may be separated into its constituents by crystallising out either pure solvent or pure solute, the latter pro- cess occurring only with saturated solutions. (At one special temperature, called the eryohydnc tempera- ture, both solvent and solute crystallise out side by side in unchanging proportions.) We now consider what happens when a small quantity of solute is separated from or taken up by the saturated solution by reversible processes. Let the saturated solution, with excess of solute, be placed in a cylinder closed below by a semipermeable septum, and the whole immersed in pure solvent. The system is in equilibrium if a Vl{i - (il - pressure P, equal to the osmotic pressure of the saturated solution when the free surface of the pure solvent is under atmospheric pressure, is applied to the solution. Dis- solution or precipitation of solute can now be brought about by an infinitesimal decrease or increase of the external pressure, and the processes are therefore reversible. If the infinitesimal pressure difference is maintained, and the process conducted so slowly that all changes are isothermal, the heat absorbed when a mol of solute passes into a solution kept always infinitely ELEMENTARY THEORY OF DILUTE SOLUTIONS 303 near saturation, whilst at the same time the maximum amount of osmotic work is done, is called the latent heat of reversible dissolution in a saturated solution, A'. If the changes of volume are executed very rapidly, they may be made adiabatic, and a Carnot's cycle may be performed with the apparatus. We take V, the total volume of the system in the cylinder, as the abscissa, and P, the osmotic pressure, as ordinate. Let V = increase of volume which occurs when one mol of solute is dissolved by entering solvent to produce saturated solution, at constant temperature T ; A' = the heat absorbed during this process. The isotherms T, and T ST are parallel to the Y axis, and the whole cycle is exactly similar to that investigated in connexion with change of state. The reader will easily prove that : ^ - - (1) dT ~ TV ' 4m need not be written (^J , since P is a function of T alone provided the solution always remains saturated, further increase of volume, i.e., entrance of solvent, simply increasing the amount of solution without altering its concentration. Equation (1) was deduced, independently, by Le Chatelier (1885) and by van't Hoff (1886); it applies generally to all solutions, whether concentrated or dilute. If the solution is dilute (which restricts the theory to sparingly soluble substances) we have : /V = RT . . . . (2) But 1/V = , the volumetric molecular concentration, .-. P = RT . . . . (3) If the solution had been prepared by simply mixing solvent and solute in a calorimeter, without any performance of external osmotic work, the heat A would have been absorbed, where A = A' - PY = A' RT . . . (4) A is called the calorimetric heat of solution in a saturated solution. From (1), (3), and (4) we get : * = BT-** (5) If the calorimetric heat of solution in a saturated solution 304 THERMODYNAMICS is | Dative' * n w ^ c ^ case tne addition of a small quantity of solute to a large volume of solution infinitely near saturation gives rise to j g^^ 11 of heat > the effect of rise of temperature >*>** This is an example of the application of a very general theorem, formulated somewhat imperfectly by Maupertius, and called the Principle of Least Action. We can state it in the form that, if the system is in stable equilibrium, and if anything is done so as to alter this state, then something occurs in the system itself which tends to resist the change, by partially annulling the action imposed on the system. In particular, if we raise the temperature, there will occur some change which will give rise to absorption of heat. If a saturated solution of potassium nitrate is in equilibrium with crystals of solid salt at a particular temperature, and if we now raise the tem- perature, a change must occur which absorbs heat and so tends to cool the system. This is the dissolution of more solid, because the heat of solution is positive, that is, heat is absorbed when salt goes into solution. But if we have a saturated solution of calcium sulphate, there will occur a precipitation of solid on warming, because the heat of solution is negative, and heat is absorbed when salt comes out of solution. The principle has been enunciated, more especially in con- nexion with chemical reactions, by van't Hoff, under the name of the Principle of Mobile Equilibrium, and by Le Chatelier, as the Principle of Reaction. In the integration of (5) we must know A as a function of temperature : / \ (6) Case 1. If the interval of temperature in which the change of solubility is required is small, A may be without sensible error assumed to be constant : *dT A = ~~ RT ~*~ C0ns ' ' ^ ELEMENTARY THEORY OF DILUTE SOLUTIONS 305 It is evident that, although A may be deduced from a know ledge of the solubility curve : A =XT) we cannot deduce the solubility from a knowledge of A. R/II |? A = - *LT- ... (8) tf, Example. Solubilities of succinic acid in water : T! = 273 To = 273 + 8-5 2 _ 4-22 fi ~" 2-88 j. cal. R = 1-985 , degree. 1-985 X log ^|| X 2-3026 = 6857 cal. fj M V273 281-5/ (A obs. = 6700 cal.). Equation (8) was found to give results in good agreement with experiment by van't Hoff (1885), van Deventer and van der Stadt (1892), and Noyes and Sammet (1903). An agreement between the observed and calculated values of A implies that the solute has a normal molecular weight in solution. Case 2. If equation (6) is to be integrated over a moderate range of temperature, A cannot be considered as constant. Suppose that s grams of solute saturate 100 grams of solvent, and let c 1} c 2 , c be the specific heats of solute, solvent, and solution, respectively; then if Q' is the heat absorbed in the process, Kirchhoffs theorem ( 58) shows that !/T~ = (S + 10 )C ~ ^ Tl + 10 C ' 2) ' ' (9) The heat of solution may increase, remain constant, or decrease, with rise of temperature, according as (s + 100)c = (*C! < In all cases investigated, the upper inequality obtains. 306 THERMODYNAMICS If I is the heat absorbed when 1 gram of salt forms a saturated solution, Q' = 8l . T . . . (11) Now it has been shown that the difference of the specific heats of the solution and solvent is proportional to the concentration, in the case of aqueous solutions ( 10). Assuming, therefore, in general c = c 2 + ks . . . . (12) where k may be positive or negative, we have I = 1 + aiT + aaT = lo + aT . . (13) where a\ = c ci, a 2 = (c c%)/s, are constants. It follows from what has been said that a is negative. For a molecular weight of solute, we have A = ml = Ao aT . . . (14) where a = ma, is a constant. We now substitute in (6) : A n ^Jp .5 /nT + const. . . (15) From (13) we have : RT R or logs = A-?- Clog T . . . .' . (17) where B = ^ X 2'3026, C = m ^ ai ^ ^ X 2'3026, A =: const.' The analogy between (17) and the Kirchhoff vapour-pressure equation ( 88) is evident. R. T. Hardman and the author have shown that (17) enables one to calculate the solubility when the " solubility parameters " A, B, C have been obtained, and this ELEMENTARY THEOEY OF DILUTE SOLUTIONS 307 even with very concentrated solutions. Thus, with aqueous solutions of cane sugar : A = 32-285 B = - 1283-65 C = 12-2267 T sobs. s calc. s calc. s obs. 273 179-2 179-2 283 190-5 189-8 -0-7 293 203-9 203-2 -0-7 303 219-5 219-5 313 238-1 238-9 + 1-7 323 260-4 262-0 + 1-6 333 287-3 288-4 + 1-1 343 320-5 320-5 It is evident that the laws of dilute solution, assumed in the deduction of (17), cannot apply to solutions, or rather syrups, containing more than three times as much sugar as water. The analogy between (17), and KirchhofFs vapour-pressure equation is therefore surprisingly extensive. A. Findlay (1902) found that Eamsay and Young's rule for the vapour pressures of pure liquids ( 89) has an analogue in the case of solutions. If T A , T A ' are two temperatures at which the substance A has the solubilities s, s' , and T B , T B ' two tempera- tures at which another substance has the same solubilities in the given solvent then : T ' T ^ = ^ + C (T B ' - T B ) A B ^B where c is a constant. The rule is, however, not very closely followed. Le Chatelier (1888) has discussed the general form of the solubility curve in the light of equation (5). If dX/dT is negative (which is usually the case) the curve begins asymptotically to the T axis, and is convex to it. It then passes through a point of inflexion, and is concave up to the maximum where A = 0, d^/dT = 0. If A then becomes negative, the solubility x 2 308 THERMODYNAMICS decreases again with rise of temperature. The greater part of the theoretical curve has been realised with calcium sulphate. The equation shows that the solubility curve must be continuous ; all breaks indicate that the solid phase in contact with the saturated solution has altered in character, and we really have to do with two distinct solubility curves meeting at an angle. This occurs, for example, with Glauber's salt at 32'6, for this is the transition temperature for the reaction Na 2 S0 4 . 10H 2 ^ Na 2 S0 4 + 10H 2 0. The curve below 32'6 is the solubility curve of Na 2 S0 4 . 10H 2 ; that above 32'6 is the solubility curve of Na 2 S0 4 . The idea that such breaks correspond with changes of " hydration " in the solution is quite unfounded, because all the properties of the homogeneous solution pass continuously through the transition temperature. Case 3. The solute changes its molecular state when the con- centration of the solution is altered. Let us suppose that instead of the equation : which" holds for a mol of solute in a volume V, or mols of solute in unit volume, we have : P = 5^ = #BT .... (18) where i is van't HofF s factor, and is a function of concentration and temperature. Then : x = A' PV = A' I.BT . . . (19) BT ,. ',. . (20) If the range of temperature is very small, we can integrate (20) on the assumption that i is constant : ln t=-fifi + const (21) But if the range is at all considerable, we can no longer regard i as independent of temperature, in other words we must take ELEMENTARY THEORY OF DILUTE SOLUTIONS 309 account of the alteration of the extent of ionisation with temperature. It is shown later that i and are related by an equation : = K (22) 1 where a = i 1, for a binary electrolyte, K = a constant for a particular temperature. (22) is called Ostwald's Dilution Law. Now (20) can be written : and if we substitute from (22) we find : \_ 2 # a(l-a)dlnK RT 2 ~ 2 - a dT ~* 2 - a dT an equation deduced by Noyes and Sammet (1903). Examples : o-nitrobenzoic acid : A calc. by (6) = 6,480 cal. A obs. = 6,025 cal. Potassium perchlorate : = 12,270 cal. = 1,213 cal. Similar equations may of course be deduced for a polymerised solute ; in this case i < 1 in (18). An interesting calculation due to J. Meyer (1911) enables the transition temperature of two forms of a substance to be derived from two measurements of the proportions of the forms in the saturated solution at two temperatures. If accented symbols are used for one form, and unaccented for the other, we have : A = Kin & > and A' = R/ % ^\ 1 la li Ci la li .-. A A' - Q(heat of transition) = R/w 1 , 2 ' . CiCa la li At the transition temperature 3, both forms have the same solubility, Cl * -M the ratio of the concentrations being independent of the solvent. f ' T & At the second temperature, Q = Eln f 2 - 2 ^ 2 * ^2 6. rp rp ''^Cl %2 . . -J lila 310 THERMODYNAMICS 134. Heats of Solution and Dilution. The exact significance of A in the equations of the preceding section must be remembered. Different " heats of solution " have been used, and among these we have : (1) The Differential Heat of Solution, L. (2) The Integral Heat of Solution, A. (3) The Reversible Heat of Solution A', in a saturated solution. Let us consider a mass m of solid solute, and a mass M of sol- vent, brought together in a calorimeter. When the whole has passed into a homogeneous solution at the original temperature, a quantity of heat Q will have been absorbed. We now set, by way of definition : A(w,M>,T) = . . . . (i) Let us now consider what happens when, during the above pro- cess, the system contains a mass M of solvent and a mass p of solute and a further small quantity dp of the latter goes into solution. The concentration is o- =. ~, and if BO is the element of heat M absorbed we put, by way of definition : SQ = L(,T) . . . (6) If the integral heat of solution is independent of concentration, i.e., the same amount of heat is absorbed when unit mass of solute dissolves in any quantity of solute : \(s,p,T) = L(s,p,T) . . . (7) A(s,^,T) may be determined directly by calorinietry ; if its dependence on s is found by carrying out the process with different values of s, ~L(s,p,T) may be obtained by differentiation. A simple graphical method of effecting the calculation is described by B. Roozeboom (Heterogcn. Gleichgeic. II.). We take AB as unit length on the axis of abscissae, and let the point a, where Ba = x, A = 1 x, denote the composition of a mixture of x parts of B with (1 x) parts of A. The corresponding ordinate denotes the heat absorbed (positive or negative) in the formation of the mixture. A The latter will not usually change sign with change of concentration of the mixture, although cases in which this occurs (e.g., cupric chloride, the hydrates of ferric chloride, and trichloracetic acid, in water) are known. The summits of the ordinates will therefore lie on a continuous curve, either wholly above (Q > 0), or wholly below (Q < 0) the com- position axis, as for example, A6B or A/3B. ab = heat absorbed in the formation of unit mass of the mixture containing Ba = x parts 312 THERMODYNAMICS of B. The heat for 1 part of B is - times this = BC. If x becomes greater and greater, the point C moves upwards, and finally attains the limiting position F. BF represents the heat for solution of 1 part of B in an infinite amount of solvent, i.e., the differential heat of solution for infinite dilution. BC is the integral heat of solution. Similarly, if IG is drawn tangent to the curve, BG represents the differential heat of solution of 1 part of B in an infinite amount of the solution represented by a. If a solution of concentration s, containing unit mass of dissolved substance, is mixed with an infinite amount of pure solvent, the heat absorbed is called the integral heat of dilution, A(s,T,p). The reason for the use of an infinitely large amount of solvent is that A depends on *, but after the dilution has pro- gressed to a greater or less degree, depending on the character of the solute, any further addition of solvent gives rise to no further heat effect, or at least one which is altogether too small to detect. The heat absorbed when unit mass of solute is dissolved in an infinite amount of solvent is the differential heat of solution for zero concentration, LO, and this is evidently equal to the integral heat of solution for concentration s plus the integral heat of dilution for concentration s : Lo = A. + A g , . . . . (8) .,^-+^ = ^0 =0 . . . (9) 3s cs ds .'. from (4) A, = L.-^ . . . . . (10) Also ^ = r s -r . . . . . . (ii) where F,, F are the total heat capacities of the solution and of its , 8L, dA s . a 2 A, . components, and ~ = ^ + s -^^ from (4) ELEMENTARY THEORY OF DILUTE SOLUTIONS 313 135. The Distribution Law. If twopartially miscible liquids are shaken together, two saturated solutions, one of A in B, and the other of B in A, will in general result. If now a third substance is added, which dissolves in either pure liquid, it will go into solution in both layers. Thus, if ether and water are agitated together, the lower layer will consist mainly of water, and the upper mainly of ether. If iodine is now added, it will nearly all go to the ethereal layer, but a little dissolves in the aqueous layer, as is evident from the colour. It was experimentally established by Berthelot and Jungfleisch (187*2), that : " a body brought in contact with two liquids, in each of which it is soluble, always divides itself between them in a simple ratio, however great maybe its solubility in one of them, and the excess of the volume of this same solvent. The quantities dissolved simultaneously by the two liquids stand to one another in a constant ratio which is independent of the relative volumes of the two liquids." This ratio is called the coefficient of distribu- tion, or the partition coefficient, k. Thus, succinic acid is shared between ether and water with the following values of /,- : 0-024 0-0046 5-2 0-070 0-013 5-3 0-121 0-022 5-5 TI, c-2 are the grams of solute in 10 c.c. water and ether respectively. It was also found experimentally that two solutes were dis- tributed between a pair of solvents as if each were present alone. (This is analogous to Dalton's law of partial pressures.) Berthelot and Jungfleisch considered, from their experimental results, that for any given trio of substances, A- depended on the temperature, and on the concentration of the shared substance. It was shown, however, on theoretical grounds, by Aulich, and Xernst, that, provided the solute does not alter its molecular state in passing from one solvent to another, A- should be indepen- dent of concentration, but will depend on the temperature. 314 THERMODYNAMICS We assume that an equilibrium can subsist with a specified pair of solvents in contact and a solute distributed between them in any one ratio of concentrations : We have now to prove that, if 1 is increased by di, then will increase by <7 2 , such that : This will establish the law for all concentrations within the range to which the laws of dilute solutions apply, because the same argument could be used with the second pair of concentrations, fi -f- dfi, and 2 + ^ 2 , a s initial concentrations, and so on, hence !/ 2 = constant = k . . . (a) This means that if one of the magnitudes 1 or 2 has been fixed by arbitrary choice, the other assumes the value given by (a), A- being a fixed constant for all values of . Let there be two cylinders containing the pairs of solvents in FIG. 63. which the same solute exists under the osmotic pressures PI, P 2 in the first, PI + <^Pi> ?2 + ^P 2 in the second, and let Vi, V 2 and Vi rfVi, Va rfV a be the corresponding volumes of solution con- taining a mol of solute. The suffixes refer to the first and second solvent, respectively. ELEMENTAEY THEORY OF DILUTE SOLUTIONS 815 We now press in the piston 1 so that a mol of solute goes out of the Yi space into the Y 2 space across the plane of contact, at the same time allowing the piston 2 to move out*so that the requisite volume of solvent enters and the concentrations are unaltered. The work done is d]=--P x Vi + P,V. That volume of the second solution which contains a mol of solute, viz., V2, is now isolated, and the volume f/V 2 of solvent removed through the semipermeable piston. The work done is [2] = - P 2 r/V 2 , The resulting solution is now mixed with the identical solution in the other cylinder, and the mol of solute sent into the YI d\ r i space, whereby the work [3] = (P! + rfPO (Vi - dVO - (P a + dP a ) (Y 2 - . Now let the following reversible isothermal cycle be per- formed : (1) Increase p to p + p. (2) Allow solvent to enter until a mol of solute has been dissolved inside the osmotic apparatus. The work done is (3) Decrease the external pressure to p. (4) Express solvent until a mol of solute is precipitated from the saturated solution, and the work - PV, - pto, is done. Everything is now in the initial state, and the total work done must vanish. By reason of the slight compressibility of the solution and solvent, the amounts of work done in (1) and (3) may be taken as equal and opposite. /. (P + 8P)V, + *, + ( p + S;>)A 0) + S]> - PV, - 7>Ar p = . (1) But V^ = Y,+^, and Ar, + Sp Ar p -f- -g " 8p, whence, if we neglect small quantities of the second order, .... (2) 318 THERMODYNAMICS Now 1/Vp = p , the volumetric molecular concentration, TfP and 8P = |r^ i ap ... ap, OT -I- -m - <8) or ~- = ^J5 . . . ... (3) "1 ~\T Now (133): V p = ~ . . . . . . (4) ap a_T _ TAg, 8f " a^ ~ " A' " ap ~ " A' aT This equation is general, and applies to solutions of all concentrations. From equation (3) we see that, since ~ is essentially positive, -is of opposite sign to Ar (Sorby's rule). The quantitative relations have been tested by Braun, by Stackelberg (1896), and by Cohen, Inouye, and Euwen (1911). Ammonium chloride passes into solution with increase of total volume, and hence its solubility should be diminished by increase of pressure. Sodium chloride, on the contrary, dissolves with contraction, and its solubility should be increased by rise of pressure. Above 1,530 atm., however, the latter salt dissolves with expansion, and its solubility then decreases with pressure. These deductions from the equation have been confirmed. We may assume, as a close approximation, that the osmotic pressure is defined by the equation : P = RT .... (6) ELEMENTARY THEORY OF DILUTE SOLUTIONS 319 dln thence an equation due to Planck (1897). It will be noticed that in the deduction of (6) it was assumed that no change of volume occurred when solute passed into solution. If the change is small, however, which is always the case, we can neglect the work done by the osmotic pressure against this change of volume in comparison with that done against the change of volume V,,. If the solute dissociates with increasing dilution, the equation (7) requires modification ; thus, van Laar (1893) deduced for a binary electrolyte : AB=A + + B- the equation : Ar 2-q ' BT ' 2 where a is the degree of electrolytic dissociation (cf. 133). This is readily obtained by putting : P = iRT .... (6a) t = 1 + a . . . . (9) and substituting in (3). The following results have been obtained : NaCl : s = 35-898 + 0'001647p 0'0000003286^ 2 Mannitol : s = 20'65 + 0'000931p O'OOOOOOISOG/ (Cohen, Inouye, and Euwen, Zeitschr. physik. Chem., 67, 432, 1909; 75,257,1910.) 137. Other Causes Modifying Solubility. Besides the effect of temperature and pressure, the mechanical pressure exerted on the solid phase, and its state of division, influence (although only to a slight extent) its solubility in the liquid. Thus, if a moist precipitate is exposed to pressure in a filter-press, it usually aggregates together, and Hulett (1901) showed that the effect of division (i.e., of surface tension) becomes 320 THERMODYNAMICS appreciable in the case of calcium and barium sulphates when the diameter of the grains falls below 1'9 /x(> = 10~*crn.) CaS0 4 (ordinary) at 25 : f = 0-01533 CaC0 4 grains ' < 0"2 /x : = 0'01869. This effect appears to be of importance in the case of normal galvanic cells, the electromotive forces of which depend on the concentration of solutions in equilibrium with depolarising solids such as calomel or mercurous sulphate. The exact relationships are, unfortunately, not yet wholly elucidated. G. Hulett, Zeitschr. physik. Chem., 37, 385, 1901 ; Hulett and Allen, Joimt. Amcr., Chem. Soc., 24, 667, 1902. 138. Dilute Solid Solutions. The theory of the depression of freezing-point of a solvent by addition of a soluble substance considered in 132 is based on FIG. 64. the assumption that the solid separating is the pure frozen solvent. In this case the equilibrium temperature is a function of the composition of the liquid phase, but obviously not of the composition of the solid, since the latter remains invariable. In certain cases, however, the solid which separates is a homogeneous mixture of both components, and hence may be referred to as a solid solution. These are often called " mixed crystals," but the name is clearly unsuitable in view of the ELEMENTARY THEORY OF DILUTE SOLUTIONS 321 homogeneous character of the phase ; a mixed crystal is a mixture in a crystal, not a mixture of crystals. The effect on the equilibrium temperature was investigated by van't Hoff (1890) by means of the vapour-pressure diagram. Instead of the diagram of 132, in which the vapour-pressure curves of the solid are confined to a single curve OA for the pure frozen solvent, we have now Fig. 64. AB, BC are the curves for pure solid and liquid solvent respectively, B(T ) being the freezing-point of pure solvent. B 2 C 2 is the curve for a liquid solution, and if the solid does not dissolve the second component, the freezing-point is T 2 , and the depression T T 2 . If, how- ever, the solid also dissolves some of the second component, its vapour pressure is lowered, and the curve AiBi is obtained for the solid phase instead of AB. The freezing-point is now T 3 , the abscissa of the intersection of the curves of the solid and liquid. The depression T T 3 is obviously weaker than before, and in some cases the freezing-point may be actually raised, the intersection lying to the right of T lt as at T If we assume that the lowering of vapour pressure is pro- portional to the concentration of solute, in both the liquid and solid solutions, and that the solute is involatile, then : Tj - T 3 _ BB : T! - T 2 ~ BB 2 This applies to what may be called dilute solid solutions and has been confirmed by Bijlert (1891) for iodine, and for thiophene, dissolved in benzene, and by Bruni for iodoforin in bromoform. The form of the curves also leads to an important rule deduced by Roozeboom. It is obvious that the abscissa of the inter- section of the curves of the solid and liquid solutions will lie to the left (T 3 ) or to the right (T 4 ) of TI, according as BB 2 is greater or less than BBx respectively. These lengths will, by Raoult's law, be proportional to the concentrations of the solute in the liquid and solid phases, respectively, and hence the concentration of that component, by the addition of which the freezing-point is depressed, will be greater in the liquid than in the solid phase, and the concentration of that component, by addition of which the freezing-point is raised, will be greater in the solid than in the liquid phase. (Bruni, Feste Losungen, Ahrens Sammlung. CHAPTEK XII CHEMICAL EQUILIBBIUM IN GASEOUS SYSTEMS 139. Chemical Equilibrium. If vi, r 2 , . . i'; mols of the substances A lf A 2 , . . A,- are mixed together, so as to form either a homogeneous phase, or a hetero- geneous system of two or more phases, the system may behave in one of two ways : (i.) The substances disappear as such, and v , v% . . v? mols of the new substances A/, A 2 ' . . A/ appear in their place. The change is called a chemical reaction, and may proceed either very rapidly, as is the case with explosive reactions, or reactions of neutralisation in solution, or else may go on with a finite and measureable velocity. The determination of the velocity with which a reaction progresses, and of the influence of various conditions, such as the concentration of the reacting substances, the temperature, and catalysts, on the velocity, forms the subject of that branch of chemistry called chemical kinetics, or chemical dynamics. (ii.) The initial substances persist in unchanging amount for an indefinite period of time, so that the composition of the system is independent of time. The system is then said to be in a state of chemical equilibrium, and the complete study of states of chemical equilibrium is the aim of that branch of chemistry which is called chemical statics. By means of experimental investigation, states of chemical equilibrium may be divided into two groups. For this purpose we change the amounts of some of the components in the system, and see if any change in the amounts of the other components ensues. (a) If, when the amount of any one component is changed by any quantity, however small, a corresponding change in the amounts of one or more of the other components ensues, the state is said to be one of true chemical equilibrium. A mixture CHEMICAL EQUILIBRIUM IN GASEOUS SYSTEMS 323 of acetic acid, alcohol, water, and acetic ester, in the proportions : iCH 3 COOH + |C 2 H 5 OH + CH 3 COOC. 2 H5 + f H 2 satisfies this condition at the ordinary temperature. Berthelot and Saint-Gilles could detect no change in the composition of such a mixture after it had stood for seventeen years, but if the slightest change in the amount of one component is produced by adding a little of it to the mixture, the amounts of the other components very soon alter accordingly. (b) If, when the relative amounts of some of the components are changed, no change in the proportions of the rest ensues, the state is certainly not one of true chemical equilibrium. In this case it may be possible to get the reaction to proceed by the introduction into the system of a so-called catalyst, which remains chemically unchanged after the reaction, as in the case of the catalytic influence of platinum sponge on a mixture of hydrogen and oxygen gases ; or else the reaction cannot be so instituted, as in the case of metallic gold and oxygen. The first case is an example of false chemical equilibrium, the second of a system of chemically indifferent substances. It may be that these distinctions are only arbitrary, and that all substances may really react when this is possible, but sometimes the reaction is either so slow, or proceeds to such a limited extent, that it is imper- ceptible. In practice, however, there is usually not the slightest difficulty in making out to which class a given system belongs. In what follows, " chemical equilibrium " is to be taken as meaning " true chemical equilibrium." 140. Reversible Reactions. If acetic acid and alcohol are mixed in equimolecular proportions, a reaction ensues leading to the formation of an equilibrium mixture of the two substances with the ester and water. If water and the ester are mixed in equimolecular proportions, a reaction ensues leading to the equilibrium mixture of these two substances with the acid and alcohol. Thus the reaction : acid + alcohol = water + ester may proceed in either direction ; it is called a reversible reaction, and formulated : acid + alcohol ^ water -\- ester. T 2 324 THERMODYNAMICS The fact that the same equilibrium mixture is attained whether we start with the substances on the left, or those on the right, follows from the condition that this shall be a state of true equilibrium. For if two equilibrium mixtures, say a and a', could result in the two cases, one would of necessity contain the components in proportions different from those in the other. The one, say a', which contains any specified component in excess over the other mixture, could be produced from the latter by adding to it the requisite excess of that component. But if a is a state of true equilibrium this will necessarily give rise to some chemical change in the system, and hence a' cannot be a state of true equilibrium if it differs at all from a. In general, the state of true equilibrium is represented by the symbol : i'lAi + r 2 A 2 + . . ^ i-i'Ai' + 2- 2 ' A 2 ' + . . where r- t , v{ are the numbers of mols of the components A ( , A/ taking part in the reversible reaction. 141. Chemical Equilibrium in Gaseous Systems. The thermodynamic theory of equilibrium was first stated, in a general way, by Horstmann in 1873 (cf. 50), who also obtained explicit equations of equilibrium in the case where it is established in a gas, and showed that these were in agreement with the data available at that time, and with his own experiments. Since in the majority of cases we have to deal with constant pressure, it is most useful to express the conditions of equilibrium in terms of the potential . When several gases are mixed together, or have resulted from chemical change, so as to form a homogeneous gas, it is possible to assign to the system a total potential <. If any change of com- position occurs, there will be simultaneously a variation of , and if the changes occur reversibly we shall assume that this variation takes place continuously. The condition of equilibrium at a constant temperature and pressure can then be expressed in the statement that all possible virtual isothermal-isopiestic changes in composition of the system will, when the latter is in equilibrium, leave unchanged to the first order : fy = (1) CHEMICAL EQUILIBRIUM IN GASEOUS SYSTEMS 325 subject to the conditions : ty = 0. .... .. (2) 5T = ..... (3) The equilibrium is stable if < is an absolute minimum : .*. S 2 <>0 . . . .- (4) subject to the conditions (1), (2), and (3). These relations are perfectly general. 142. Gas Mixtures with Convertible Components. A mixture of gases which can undergo chemical change is distinguished from mixtures of inert gases ( 122) by the name = .-. T2(9s + R/wc,-)8; + T2w38Znc-j = . . (P) Now /??f j = /??; Inn, where n = S; f. $H ^ fill -, ^oii; z,n- = on n = 0. n n The term with 89; vanishes, since 9,- does not alter with change of composition. The second part of (6) therefore vanishes, and : This equation applies to nearly all practical cases of gaseous equilibria, and it may be called the Canonical Potential Equation for Gaseous Equilibrium. Corollary 1. If, for any gaseous component, we have : b?i { = . . , . . (8) the term relating to that component vanishes from the canonical equation, and we have to take account of the presence of that component only in estimating the concentrations of the other gases. Equation (8) shows that this particular gas takes no part in the chemical change, and we shall therefore call it an inert gas. Corollary 2. Equation (7) shows that the equilibrium is independent of the absolute masses of the components, and depends only on their relative amounts, that is, on their concen- trations. Thus, if a gas mixture in equilibrium is divided into two or more parts by diaphragms, these parts remain in equilibrium. Example If the specific heats are constant, show that : T "R TT (i) 9i = C', ; '(l - ZwT) R/w - - S;" - R/H ~ + R + ~r - (9) P M, Notice that this can be split into three parts : T (i.) Eln 1- R, which depends solely on external conditions. TJ<* (ii.) Bi* + R/M, + ~, which depends- on the molecular CHEMICAL EQUILIBEIUM IN GASEOUS SYSTEMS 327 weight, the initial states of entropy and energy, and the temperature. (iii.) Cl-'^l /ttT), which is a function of temperature alone. If, besides the temperature, the total volume V is maintained constant, instead of the pressure, the equilibrium condition is that the free energy must be a minimum : S* = ..... (10) subject to the conditions : From equation (34) of 121 we find : /.* = T2w, T) + R/w& (12) Again, we put everything in brackets, except B/ /,-, equal to//, where / is a function which does not depend on the composition of the mixture : Thus f = ^? .... (18) m* = T2w,-(/ f + R/n) . . . (14) For a small virtual change conforming to (11) : .'. for equilibrium, conditions (10) and (11) give : 2(f { + B/6)5, + 2HiR8te- = . . (15) Now tiiblnf; = n,bln =+ = n, - = bn ; V n, since SV = /. the condition of equilibrium becomes : 2,-(f f + R + B/^)5,. = or if we put : / f + R=.7i . . . . (16) where f it like /-, does not depend on the composition of the mixture, we have finally : 2(7; + R/&)8w, = . . . (17) The conditions of constant volume and constant pressure coincide when there is no change of total volume during the reaction at constant pressure. The condition 8V = is, however, equivalent to the condition that there is no change in the total 328 THERMODYNAMICS number of molecules. For if V t is the volume of the i-th gas under the pressure p, v = ,RT P "RT .-. 8V,- = 8; (p constant) .-. if SV; = then 5;i f = 0, and if 28V, = 8V = 0, then 3fin, = bn = 0. A compound gas (HC1, HI, NO) produced from its gaseous components without change of volume is called a //as without condensation. In this case, equation (17) can be derived from the canonical potential equation by putting : RT 143. Dissociation and Mass Action in Gases. In the applications of the thermodynamic equations of equilibrium to gaseous systems we shall take 8n ( in : S(9i + Rlnc^bn.; = . , . (1) positive when it refers to a substance produced in the chemical reaction considered, and negative when it refers to a substance consumed. Further, if the absolute amount of the change of any one com- ponent is fixed, those of all the others are determined by the stoichiometric coefficients v lt v 2 , . . . in the equation : . . . ^f r 3 A 3 + r 4 A 4 -j- . . . where A is a positive magnitude independent of the *>'s. Thus 8ni : 8 2 :....= v l : v 2 :..... . . . (2) and (1) can be written : 2(9, + R//ic,>i = . . . (3) where the convention as to the sign of v i is the same as that for 5/t ; . If we compare (3) with the chemical equation of the reaction we arrive at the simple rule that the concentrations in the equilibrium state at a given temperature and pressure must have CHEMICAL EQUILIBRIUM IN GASEOUS SYSTEMS 329 such values that whenever the magnitude (9, + Rf'ic,) is sub- stituted for the chemical symbol A, in the reaction equation, the resulting thermodynamic equation is verified. We shall call T(9 f + R'H<",) the molecular chemical potential of the j-th gas in the mixture, and denote it by /Z, : ,. = T( 9( R/iie,-) = /2,+ RT/HC, . . (4) where p.'- is a function of T and ^> alone, and is independent of the concentrations. The equation of equilibrium is then : A*i* f i + W'2 + = 2/Zji'j = . . (5) If we separate the terms of (3) we have : 2^c,= -^,9, .... (6) .Sri-pi or c^ca" 2 . . . = e~ R . . . . (7) From the properties of 9, we see that the exponential factor is constant at a fixed temperature and pressure : <-* = K,t .... (8) K piT is called the equilibrium constant, and the equation : Cl cT . . . . (9) which states that the product of the concentrations of the various constituents of the equilibrium mixture, each raised to the power of the (positive or negative) coefficient of the symbol of that molecular species in the chemical equation, is equal to a constant at a fixed temperature and pressure, is the well-known Law of Chemical Mass Action, deduced on kinetic grounds by Guldberg and Waage in 1864 (Ostwald's Klassiker, Xo. 104). The deduction from the principles of thermodynamics was effected by A. Horst- mann (Ostwald's Klassiker, No. 137) in 1873, J. TViilard Gibbs in 1876 (loc. cit), J. H. van't Hoff in 1886 (Ostwald's Klassiker, No. 110), and M. Planck in 1887 (H'ied. Ann. 1887). Thus, if we consider the reaction : we have, if the suffixes 1, 2, 3 refer to H 2 SO 4 , SO 3 , H 2 O : Vl = 1, i* = + 1, "3 = + 1, .-. <*? = K Ci At each temperature and pressure K will have a fixed value, independent of c t , c a , f s : _ "i _ "a _ 3 330 THEKMODYNAMICS We see from equation (4) that .... (10) where * is the total potential of the gas mixture, for : ( here denotes what we previously called >). If to any homogeneous gaseous system we suppose an infini- tesimal quantity bn t mols of a specified component to be added, the mass remaining homogeneous and its temperature and pressure remaining unchanged, the increase of the thermodynamic poten- tial of the mass, divided by &n h is defined as the molecular chemical potential of that gas in the mixture considered. If the reaction occurs at constant temperature and volume we have as the condition of equilibrium : 2(/T+ Rtoi>i = (11) It can readily be shown that : ( ) ( ) = ( ) Voni/ P ,T \9i/ V,T \8%/ s,v where * is the total free energy of the mixture, and U its total intrinsic energy. Thence the equations : are three different ways of defining the chemical potential. It is also easy to deduce, from the definitions of c and ( 120), the relation : /T}T\ v K AT = K V)T x(^) . . . (14) where 2?', = v . -. . . (15) is the increase of the number of rnols in the reaction. 144. Maximum Work of a Gas Reaction. If a chemical reaction occurs spontaneously, the available energy of the system necessarily diminishes by an amount equal to the work which could be done by the system if the given change were executed reversibly. If the reaction occurs at con- stant temperature, this is equal to the diminution of free energy of the system, this being the energy available at constant tempera- ture. It is usual to refer to the work available at constant CHEMICAL EQUILIBRIUM IN GASEOUS SYSTEMS 331 temperature in a reversible change as the maximum ivork of the change, A T : A T = *! - * 2 = - A* . . . . (1) Its value depends only on the initial and final states and has a definite value for a given temperature at which a rever- sible isothermal process is executed, and given terminal states. Thus if 2 mols of hydrogen gas at a given volumetric molecular concentration HI, and 1 rnol of oxygen gas at a given concentra- tion H 2 , are converted into 2 mols of steam at a given concen- tration Ha, all the gases having the same temperature, the maximum work has one definite value, no matter how the process FIG. 65. is executed provided only that it is isothermal and reversible. A very instructive way of carrying out this imaginary process was used by J. H. van't Hoff ; it depends on the properties of semi- permeable membranes. Let us take the example just considered, and calculate the maximum work of the process. At the given temperature there can exist an equilibrium state between the three gases : ' 2H 2 + 2 ^2H 2 which is definite if either the total pressure or the total volume of the system is specified. We will consider the latter fixed by enclosing arbitrary amounts of the three gases in a rigid box (Fig. 65) at the given temperature. The system then settles down to a perfectly definite state in which the equilibrium con- centrations of the three gases are, say, &, 2 , 3- Now let us suppose the sides of the box fitted with three 332 THERMODYNAMICS cylinders, communicating with the interior through three semi- permeable diaphragms, which can be covered with impervious diaphragms when required. We cover the semipermeable dia- phragms with the impervious ones and put into two cylinders the gases H 2 and 2 in the initial state. These are now expanded (or compressed) isothermally and reversibly until their concen- trations are equal to those in the box. The amounts of work done are - ZRTln &/E and RT In &/E, respectively ( 79, 122). All the impervious diaphragms are now removed, and the two gases slowly compressed into the box under the constant pres- sures, the steam produced being removed as fast as it is formed through the third semipermeable diaphragm , so that the equilibrium mixture remains unchanged. The amounts of work done are 2RT, RT and + 2RT for the 2H 2 , 2 , and 2H 2 0, respectively. The membrane of the steam cylinder is now closed by the impervious shutter and the steam expanded (or compressed) isothermally and reversibly until it has the concentration H 3 ; the work done is 2RT^t ^. 3 The total amount of work done in the process is : A T = RT a/w + In - 2to =? - RT Ci C2 C/ - 2 - RT/w - 2 - RT . . . (1) ~3 3 If we keep HI, S 2 and H 3 constant, but alter the amounts of the gases in the equilibrium box, say by pumping in more hydrogen, the value of A T cannot alter (since the equilibrium box is left in the same state after the process), and it depends only on the initial and final states of the reacting gases. Hence at a given temperature FF. must be a constant, i.e., the composi- tion of the mixture in the equilibrium box readjusts itself so that this product maintains a constant value, independent of the absolute amounts of the substances present. In accordance with a previous convention( 143) we shall put : RTln = RT/wK ti Ca CHEMICAL EQUILIBRIUM IN GASEOUS SYSTEMS 333 i.e., in forming the sum 2r,- we take the terms relating to the products of the reaction considered as positive. Then A T = RT/nK RT2,7nE, + RT2v, . . (2) Thus, for the maximum work of the formation of hydrogen chloride we have : H 2 + Cl a = 2HC1 (A T ) HC1 = RT/-- - RT/n If A T is known (say from measurements of electromotive force) we can calculate K, and vice versd. If the gases are taken with initial concentrations equal to those in the equilibrium state, the maiimurn work due to the chemical change vanishes, and there remains only the part due to the change of volume : RTSi-i/wH,- = RTSi'M, = RT/K /. A T = RT2i> When the gases are contained in large reservoirs of fixed volume, so large that the extraction or addition of the reacting amounts does not appreciably alter the concentrations H, the maximum work is AT = RTtoK SV.-/HS, . . . (2) since the work done in withdrawing v mols from the reservoir is rRT, and that done in passing this into the equilibrium box is rRT. The work due to change of volume, RT5> (! therefore vanishes, as is otherwise obvious since the total volume remains constant (Xernst, Theoretical Chemistry, Eng. trans., 1904, p. 646). This value is sometimes called the maximum work for isochoral execution of the process. Again, if we consider the initial substances in the state of liquids or solids, these will have a definite vapour pressure, and the free energy changes, i.e., the maximum work of an isothermal reaction between the condensed forms, may be calculated by supposing the requisite amounts drawn off in the form of saturated vapours, these expanded or compressed to the concen- trations in the equilibrium box, passed into the latter, and the products then abstracted from the box, expanded to the con- centrations of the saturated vapours, and finally condensed on the solids or liquids. Since the changes of volume of the condensed phases are negligibly small, the maximum work is again : A T = RTSvjteS, + RT/nK 334 THERMODYNAMICS where HI, H 2 , . . are the concentrations of the saturated vapours, and InK = Si^w, where is the concentration in the equilibrium gas mixture. In this way we can calculate the maximum work of a condensed reaction when we know the vapour pressures of the various substances, and K, for : Pi = S,RT (Cf. Nernst, Recent Ap2)lications of Thermodynamics to Chemistry, 1907.) The same equations may be derived from the consideration of the free energy *. For by definition : ' A T =-A* = 2* Ila + * ( ^-2* H>0 where ^j^ denotes the free energy of a mol of free gaseous hydrogen at a given temperature T and concentration H H2 : Thus A T = BT/n + T[2/ H2 + J 02 - 2/ H2 o] . (3) ~ H 2 or generally : A T = (BTS^/nE,- + T2v,) .' <* ( 3 ") with the previous convention as to 2^. The term in brackets in (3) may be calculated if we use the relation between the chemical potentials obtaining ichen a mixture of the three qases is in equilibrium, viz., 2/i H2 o = 2/Z H2 + ySo 2 where /Zn 2 is the chemical potential per mol of gaseous hydrogen in the mixture at the temperature T and equilibrium concentration H2 , i.e. ( 143) : H2 = Tj H2 + RT/n^ H2 where / Ha = /H 2 + R /. T[2/ H2 + /o 2 - 2/ H2 o] = BT - BT/w %^* = RT + RT/K H 2 where K is the equilibrium constant. Generally, T2^/7 = BTS^ + BT/nK. Thus A T = RT/n "J** * + BTtoK + RT * H 2 O or generally A T .= RTS^S, + BT/nK - BT2i-, which are the equations just obtained. The maximum work of a chemical reaction is therefore largely dependent on the initial concentrations of the gases ; the heat of CHEMICAL EQUILIBRIUM IN GASEOUS SYSTEMS 335 reaction at constant volume is independent of the concentrations, since the intrinsic energy of an ideal gas is independent of the volume. It may be remarked that a certain amount of confusion exists in the literature respecting the magnitude we have called the maximum work. Thus Nernst and Haber always omit the term ET2^,- which belongs to the work done on the system by the change of volume. Although the latter author (Thermodynamics of Technical Gas Reactions, Eng. trans., p. 54) observes that this term should be included, he decides to omit it from his equations on the ground of " simplicity." It is not, however, a matter open for definition in this way, since the maximum work of a change must always be calculated for a definite choice of conditions. Thus, the work done by the galvanic cell consuming hydrogen and oxygen gases at its electrodes, and producing liquid water, will include the external work done by the atmosphere in compressing the gases 2H 2 and 2 , and this term must be included if we wish to calculate the electromotive force of the cell (cf. Haber, loc. cit. ; also later 205). Nernst's equation (2) is deduced for the gases taken from large reservoirs (i.e., at constant volume), and is therefore not general, since no actual gas reactions with a change in the total number of molecules, and none whatever when the products are liquids or solids, or remain dissolved, occur in this way. The " simplicity " is far outweighed by the uncertainty as to the actual conditions implied. 145. Influence of Temperature and Pressure on the Equi- librium in a Gaseous System. The equations of equilibrium in a gaseous system are : 2v ( /e, = =^ = /nK p>T Q>,T const.) . . (1) 2^n6 = - ?^ = /wK VtT (V,T const.) . . (2) If, for brevity, we omit the suffix T which is common to both K's we find, by partial differentiation : dlnK\ I 3 _ (5) 336 THEEMODYNAM1CS where as before : ,,= J T2 ~ = ^ =-^5-. (7) since U^ + JCi/'dT = U (0 by Kirchhoff's theorem ( 58), and U (ii + pf = U ( " + RT = W ( " by definition ( 25) - (8) We now substitute the partial differential coefficients in (3), (4), (5) : ai|< = -= _ ? T P P P dlnK\ 1 / W since 2i>jW w = Q p = heat of reaction at constant pressure, and 2^-U 1 '' = Q. F ^ heat of reaction at constant volume. The influence of a simultaneous change of temperature and pressure is now calculable : f2 f? T . . . (12) 146. Influence of Pressure on Gaseous Equilibrium. The influence of pressure increases numerically with the CHEMICAL EQUILIBRIUM IN GASEOUS SYSTEMS 337 contraction, and decreases numerically with the initial pressure. If V, V are the total volumes of the initial and final systems, 2i> and 2i/ the total numbers of mols in these respectively : pV = 2i . RT p\' = 2 f . RT .-. p(\' - V) = (2/ - RT ~ RT We can also write (1) in the form : If we put Inp = x, /wKi = i/, the curve /nK =f(lnj)) will be a straight line having a positive, zero, or negative gradient according as v = 0. (1) There is expansion, V > V ; then J/wK < 0, i.e., increase of pressure favours the production of system V with the smaller volume. (2) There is contraction, \' < V ; then rf/K > 0, i.e., increase of pressure favours the production of system V with the smaller volume. (3) There is wo change of volume, V = V, then r//wK = 0, i.e., change of pressure has no influence on the equilibrium. If p is in atm., V in litres, then R = 0'08207 1. atm. The equation representing the influence of pressure on gaseous equilibrium is due to Planck. 147. Influence of Temperature on Gaseous Equilibrium. The integration of the two equations : = ^^2 (Reaction Isochore) .... (1) = TT^ (Reaction Isopiestic) .... (2) leads to some of the most important applications of thermo- dynamics to chemistry. We take them in order. 338 THERMODYNAMICS (1) The Reaction Isochorc. \^r) v = RT^ where /K V = *'i/i + r 2 /i 2 + . . = "// . . (3) Thence foKy = g | jj|j dT . . . . (4) The integration is possible when Q t . is known as a function of temperature. But, according to Kirchhoff's theorem : Q^Q-'+^T/-^)^! . . . (5) V r,.WT - " RT ~ RT 2 (a) If IV = T . . -. . . (7) then //K, = - 1^ + const. -. . . (8) where Q r is the heat of reaction at all temperatures. Assumption (7) implies that the molecular heat of each com- pound gas is the sum of the atomic heats of its constituents (a hypothesis introduced on the basis of experiment by Delaroche and Berard (1813), and afterwards defended by Buff (1860), and Clausius (1861) ), and also that the molecular heats are indepen- dent of temperature. This rule is only very approximate, the deviations sometimes exceeding 30 per cent., although (8) is often useful over a small range of temperature. If we put " const." a, and Q,./R = const. = 1>, __b_ K v = ae T . : . . . (9) For two temperatures T lf T 2 (T 2 > TI) : . '.. (10) (/>) if r/ - r v = const. . . . (ii) i.e., the specific heats are independent of temperature (hypothesis of Clausius), then : /K ( . = - !' + r <-' ~ F( ;/,T + const. . . (12) or Iv=aXa, + 2ftT) .... (14) then Q r = Qj, r ' + (Sy/o/ 2r,a,)T + (2r//8/ 2^)1* = Q; r) + aT + T 2 . . . (15) and /K r =-^+|///&' 2r,-/8,-. (2) Reaction Isopiestic. f-dT . . . (17) Now Q p = Q r + rRT .... (18) ".*. Q p and Q r are equal in two cases: (i.) When v = 0, i>., the reaction proceeds without change of volume, (ii.) When T = 0, i.e., at the absolute zero, Hence Q p = Q; r ' + I (F/ - r p )dT .-. Q p = Qirt + f (IV - r r )dT + t-KT . since C p = C r -+R and F r = S^-C* ; r p = 2^'. (o) if r; - r p = o /K p = -^ + const. . . . . (21) Now T p ' T p can be zero only if we have simultaneously TV - T r = and v = in which case, K p = K r If, however, F/ - F r = 0, but v > then lnK p = - ^ + rZwT + const. . . (22) (6) If T p ' - T p = const. (28; z -2 340 THERMODYNAMICS If r/ F^ = 0, this passes over into (22). (c) If r e = Si;, (a, + 2&T) T+ const. . . (24) where the symbols have the usual significance. If v = 0, this coincides with the isochore equation (16). If we consider both K r and K,, together, and drop the suffixes, we see that : if Q > 0, dlnK > 0, or K increases with T, if Q < 0, dlnK < 0, or K decreases with T, if Q = 0, dlnK = 0, or K is independent of T. Thus K changes with rise of temperature in such a icay as to favour the reaction which proceeds with absolution of heat. (Yan't Hoff's Law of Mobile Equilibrium, 1886.) To avoid the confusion which sometimes arises we may remark that the sign of Q is fixed in accordance with the convention as to which substances are taken as forming the initial system. Thus, if we consider the reactions : a) 2H 2 + 2 = 2H 2 0, then K a = 2 H *> , and Q a < ; ' /3) 2H 2 = 2H 2 + 2 , then K^ = *' an d Q > 0. C H 2 In both cases rise of temperature (at constant volume or con- stant pressure) favours the dissociation of steam, since this reaction tends to cool the system down to the initial temperature. 148. Gibbs's Dissociation Equation. For a simultaneous change of temperature and pressure : dlnK >} " /. InKp = vlnp -f- p The integration is possible when Q /; is known as a function of temperature, i.e., when Q r for one temperature and the specific heats at all temperatures in the range considered, are known. We see at once that the desired result is obtained simply by CHEMICAL EQUILIBRIUM IN GASEOUS SYSTEMS 341 adding vlnp to the equations (17) (24) of the preceding section. If the specific heats are independent of temperature : /nK, = - g|r+ T ' ~ R + rR toT - vlnp + const. . (8) Put F/ F r = const. = b then K p . T = a.T* +r ~ Sf P~ l ' < 4 ) where Ina is the " const." of (3). Equation (4) is the Dissociation Equation of Willard Gibbs (1876). If the specific heats are linear functions of temperature : /K, = - jL + " f T + T - rlnp + const. _ + _4t+p_ T orK p , T = A.T R e *T R p~ v . . (5) The extension to include higher powers of T is obvious. We observe that all constants in (4) or (5) can be determined by measurements of the thermal properties of the system, with the exception of a or A, which are indeterminate from the point of view of classical thermodynamics. 149. Applications of the Equations. The extent of dissociation of a gas is usually determined from the density. Let D = density when no dissociation occurs, A = observed density of partially dissociated gas, d = density of completely dissociated gas, all reduced to normal temperature and pressure. Let each molecule of the original gas break down into x molecules on dissociation, then * = i W We have : dc 2 /RT, , = where p is the total pressure, The partial pressures are : Pi pci, P* = PC* Pa = pea CHEMICAL EQUILIBRIUM IN GASEOUS SYSTEMS 343 Corollary 1. The total pressure has no influence on the equilibrium. Corollary 2. The addition of aninerb gas has no influence on the equilibrium, since 1} 2 , 3 are all altered thereby in the same ratio. The total volume remains constant, hence the partial pressures are proportional to the numbers of mols of each gas. Thus, of 2A mols HI there will be 2yA mols dissociated, and there are formed y\ mols H 2 and 7 A mols I 2 , whilst 2(1 7)A mols HI remain. The total number of mols is constantly 2A, hence : If pure HI was initially present : Pz = PS = p' say, ~ 1 + 2\/K ' lh ' 1 + 2A/K If, however, A 2 mols H 2 and A 3 mols I 2 where A 2 =J= A 3 , are intro- duced into a closed space, and heated at a constant temperature till the system is in equilibrium with 2Aj mols HI produced, we have : A 2 mols H 2 A 3 mols I 2 no HI at the start A 2 AX mols H 2 A 3 A : mols I 2 2A mols HI at the finish /. if pi, pi, 2 } s are the respective partial pressures, A 2 \i A 3 AI 2Ai p% r p ', 2*3 j 2* 5 PI , P' A 2 + A 3 A 2 -f- A 3 A 2 -(- A 3 A 2 AX is determined by measuring the volume after absorption in water ; A 3 A! and 2A are determined by titration. Then (A 2 A!) (A 3 A x ) 4A 2 - *2 A 2 + A 3 / (A 2 ~ 21 - 4K V 41 - A 2 A 3 2(1 - 4K 2 ) V 4(1 - 4K 2 ) 2 ~ 1 - 4K 2 (2) x = 2 (binary dissociation) : AB ^ A + B 123 344 THERMODYNAMICS pi where p = total pressure = pi + jp 2 + l>s- If there is no excess of the products of dissociation : c 2 = c 3 ; and p 2 = p a = / say, pi. = p 2/ In the case of nitrogen tetroxide : NA^N0 2 + NO a fche two products are identical, and if p' is the total partial pres- sure of N0 2 , p' = 2pz = 2j? 3 ' (P Pi) 2 = 4 K 2 fti p , 27 A Also PL = 1 ~ f y, where 7 = ^ A ' (D - _ " (2A - D)D ~ The equilibrium is largely influenced by the total pressure. By solving for A we find : _ K where K' = K 2 D. Gibbs's equation may be applied in the form : loci Ki = - Q^- B/wT vlnp + A where B = -Ll", and A is an arbitrary constant. (D-A/ _ (M - A) 2 i -*/!> (2A-D)D ~4(A-dX where f/ = density of N0 2 (etc.) CHEMICAL EQUILIBRIUM IN GASEOUS SYSTEMS 345 According to Gibbs the term with InT may be omitted if it is compensated in the values of A and Q , regarded as arbitrary, thus, with ordinary logarithms : (D A) 2 ._ , (2d A) 2 _ M % (2A-D)D where M and N are constants. 0. Brill (Zeitschr. physik. Chem. 5', equation : loci 721, 1907) has used the T - Gibbs also showed, on the basis of some experiments of Play- fair and Wanklyn, that the equation applies when p is reduced, not by decreasing the mechanical pressure on the system, but by admixture with an inert gas. (Cf. Dixon and Peterkin, Trans. Chem. Soc., 1899, p. 613.) (3) If x = f we have as examples the very important equilibria : 2CO-2 z 2CO + 2 2H 2 ^ 2H 2 + 2 a description of the experimental study of which will be found in Haber's Thermodynamics oj Technical Gas Reactions. The following tables of the extent of dissociation g = lOOy contain the results of the most recent investigations. (1) Halogen Hydracids. r T ' HC1 HBr HI 290 2-51 X 10 ~ 15 414 X 10~ s 6"2 500 1-92 X 10- 2-91 X 10-* 15-5 700 1-12 X 10- 9-93 X 10 - :i 22-2 900 3-98 X 10- 7-18 X 10~ 2 27'0 1,000 1-34 X 10- 0-144 29-0 1,500 6-10 X 10- 1-19 2,000 0-41 3-40 2,500 1-30 (Vogel von Falckenstein, Zeitschr. physik. Chem., 68, 3, 270.) 346 -THERMODYNAMICS (2) Nitric Oxide. 2NO Na + Oa T Per cent. N 2 Per cent. 0- 2 Per cent. NO 1,811 78-92 20-72 0-37 1,877 78-89 20-69 0-42 2,033 78-78 20-58 0-64 2,195 78-61 20-42 0-97 2,580 78-08 19-88 2-05 2,675 77-98 19-78 2-23 3,200 76-6 18'4 5-0 (Nernst, Gottinger Nachr., 1904, p. 261 ; with Jellinek and Finckh, Zeitschr. anorg. Chem., 46, 116, 1905; 49, 212, 229, 1906.) (3) Steam. 2H 2 = 2H 2 + 2 T p = 10 atrn. 1 atm. o- 1 atm. 0-01 atm. 1,000 1-39 X 10 - 5 3-00 X 10 - 5 6-46 x io- 5 1-39 X 10- ! 1,500 1-03 X 10 ~ 2 2-21 X 10- 2 4-76 x io- 2 0-103 2,000 0-273 0-588 1-26 2-70 2,500 1-98 3-98 8-16 16-6 (4) Carbon Dioxide. T p = 10 atm. 1 atm. 0-1 atm. 0-01 atm. 1,000 7'31 X 10~ 6 1-58 X 10" 5 3-40 X IO- 5 7-31 X 10~ 5 1,500 1-88 X IO- 2 4-06 X 10 - 2 8-72 X IO- 2 0-188 2,000 0-818 1-77 3-73 7-88 2,500 7-08 15-8 30-7 53-0 (Cf. Nernst and von Wartenberg, Zeitsclir. physik. Chem., 56, 513, 534, 548, 1906 ; Haber, Thermodynamics of Techn. Gas React., Eng. trans., 1 Appendix to Sect. V.) CHEMICAL EQUILIBRIUM IN GASEOUS SYSTEMS 347 From these numbers, a large number of calculations of technical interest can be made. Further, if we divide the equilibrium constant of carbon dioxide by that of steam we obtain the equilibrium constant of the water-gas equilibrium : [CO] 2 X [QJ [H 2 0] 2 Kco, = ( [CO] X [H 2 0]] 2 [C0 2 ] 2 A [H 2 ] 2 x [O 2 ] K H * ( [C0 2 ] X [H 2 ] / . K = [C01XJEW)] _ / [C0 2 ] x [Hi] V Again, if we divide the square of the equilibrium constant for hydrogen chloride by that for steam we obtain the equilibrium for the Deacon process of chlorine manufacture : 9 4HC1 + 2 2H 2 + 2Cl a Ki 2 _ [H 2 ] 2 x [C1 2 ] 2 _JSiO] a _ _ [H 2 0] 2 x [C1 2 ] 2 R K a ~ [HC1] 4 * [Ha] 2 X [O 2 ] ~" [HC1] 4 X [0 2 ] ~ Example. The dissociation of steam : - _ p 2 27 . p 27 . f P ~ RT ' 2 + 7 ' RT * 2 + 7 ' RT where 7 = extent of dissociation, p = total pressure Thermal Data : C c for H 2 : 5'61 + 0'000717T + 3'12 X 10 C e for 2 and H 2 : 4'68 + 0-00026T Q r (at T = 373) = 115300 cal. /. from Kirchhoff's equation : Q r -= 114400 + 2-74T 0'00063T 2 - 6'24 X 1Q- 7 T 3 ... log K = - ^*9 + 2-38 log ^ - 1-38 x 10 - 4 (T - 1000) - 0-685 X 10 - 7 (T 2 - 1000 2 ) + const. by integration of the reaction isochore ( 147). For T = 1000, 7 = 8'02 X 10- 7 (obs.) /. const. = 11-46 - 1-38 X 10- 4 (T-1COO) -0-685 X 10 ~ 7 (T 2 - 1CC0 2 ). 348 THERMODYNAMICS From this equation the values in the table were calculated (Nernst and Wartenberg, loc. cit.}. 150. Maximum Work at Different Temperatures. In 144 we have deduced the expression : A T = RTZwK RT2/^wSj , RT2i-, ; . . (1) for the maximum work of an isothermal gas reaction at T. If the temperature at which the process is executed is changed, whilst the amplitude of the process ( 58), i.e., the initial and final concentrations H, remain unchanged, we shall have: A T + Q r = T^ T 4 -; . . (2) where : A T = maximum work at the temperature T, Q r = heat of reaction at temperature T and constant volume, -grjr = rate of increase of maximum work with the temperature at constant amplitude. Differentiate (1) with respect to T and substitute in (2) : T = RtaK + RT ** - RSi;,toS, - RSr, (S, const.) '* 7nK = 5 f?T + (const ' ) ' ' ' (8o) If we carry out the integration, and substitute in (1) we have an equation giving the influence of temperature on the maximum work. From Kirchhoff's equation : Thus: A T = T[^| dT - RTSi^wE, - RT2 V , + (const.)T . (4) If we put r; - r, = + 2/8T (cf . 144) .-. A T = - Qo + aT/wT + /9T 2 - RTSr^wS, + (const.)T (5) where RT2^ has been included in the constant. Partial pressures may be substituted for concentrations by CHEMICAL EQUILIBRIUM IN GASEOUS SYSTEMS 349 making use of the relations of 122. (Cf. Haber, Thermodynamics of Technical Gas Reactions, Eng. trans., p. 61.) 151. External Work in Dissociation of Gases. If *, *' are the free energies of a system before and after dissociation at a constant temperature, the maximum external work obtainable is * * f . This may be calculated directly. Let us take the case of nitrogen peroxide : Let n = number of mols N 2 04 originally taken, 7 = degree of dissociation. Then if v = total volume, we have : The pressures before and after dissociation are proportional to the numbers of mols (Avagodro's theorem) : pip' = n/[(l - 7)11 + 27*1] =!/(! + 7) /.!/= XI + 7) Also SA T = p'dv = p(l + y)dv But from Boyle's law p = BT/c = BTKp( \ ~ 7) . . (I) To find dv we differentiate (a) : /. A T r p = pdv + ypdv J VI J VI =*-* 152. The Specific Heat of a Dissociating Gas. If the temperature of a mixture of a gas and its products of dissociation is raised by ST at constant pressure p the quantity of heat absorbed C y ,5T is made up of four parts : 850 THERMODYNAMICS (i.) the heat required to warm the undissociated part through ST at constant volume, (1 y)C r ; (ii.) the heat required to warm the dissociated part through 8T at constant volume, viz., (wi7C ehimique, IL, 312 ft seq., in which various possible cases are discussed, and Swart (Zeit. phfizik, Chem., 7, 120* 1891> 153. The Shape of Dissociation Curves. To get an idea of the general trend of dissociation in a gas. we shall consider the isopiestics which represent, at various constant pressures, the density A as a function of temperature. We may take the case of binary dissociation (JT = *2 as exemplified by the reactions: for which Gibbs's equation takes the form : . (-2A - D)D M . ln (D-Af =T +faP -- N ' ' ' (1) We take the reaction for which M is positive, i.e.. in which heat is absorbed. As T increases from to + x, P meanwhile remaining constant. the second member of equation (1 ) decreases from a value -f- x to a finite limiting value t/nP X), which is all the greater the larger is the value of P. If the temperature is kept constant, and P varies, the second member of (1) is all the greater the larger is the pressure, From a consideration of the first member of (1) we see that, as A increases from D (i.e., d) to D, the expression increases from x to + x (ct H. M., 45> If we combine the results of the investigation of the first member of (1) with the results of the investigation of the second member, we are led at once to the following theorems : (1) If a dissociable gas is heated, at constant pressure, from 352 THERMODYNAMICS the absolute zero to a temperature greater than any assignable temperature (T = oo ), its density, A, decreases continuously from the value D to a limiting value which is greater than the value corresponding with complete dissociation. (2) This limiting density is all the less the smaller is the value of the pressure, and tends to the limiting value D/2 as P tends to zero. (3) At a given temperature the density is all the less the smaller is the pressure. From (1) by differentiation (cf. H. M., 47) (2) also - (2A - D) = (D - A) 2 As T tends towards zero, the numerator of (2) is infinite- of the 5 t; 2 L "^'^^^x. \ v X \ ' - *0 160 200 240 280 320 3t Temp. FIG. 66. order 1/T 2 , whilst the denominator is infinite of the order hence, by a well-known theorem, Lim 3 A A Similarly, tor T -.0 : f =^= . . '. = f = . . = 0. (4) so that the curves have contact of infinite order with the line CHEMICAL EQUILIBRIUM IN GASEOUS SYSTEMS 353 Similarly, for T > oo , it can be shown that the curves have contact of infinite order with their asymptotes, i.e., the horizontals through the limiting densities. Each curve therefore consists of three parts ; an initial and a final portion which are nearly horizontal for a finite part of their lengths, and an intermediate portion which slopes down com- paratively rapidly from left to right. This means that the dissociation with rise of temperature is slow at first, then increases very rapidly, and then becomes increasingly slower as it approaches asymptotically to the limiting value for T = x . The general form of curve so predicted corresponds exactly with the experimental curves, as will be seen from Fig. 66, which was drawn by Horstmann from the results of Wiirtz with amylene hydrobromide : C 5 H 10 . HBr = C 5 H 10 + HBr. 154. Experimental Study of Gaseous Equilibria. In the application of the equations deduced in the preceding sections, it is necessary to know : (1) The heat of reaction Q c . (2) The specific heats of the various substances, to calculate the value of (T 1 T) required in the application of Kirchhoff's equation, and the evaluation of the integral for InK : / Q T = Q, + I n \u ? ?/Hht-rnrP"ir ~ ' em ? erature and T' \ A A /, A \ I L_ taming a vacuous palla- I ''/ V '/ V V V j--* ^ J dium bulb connected with a manometer, the hydrogen will enter the bulb and finally attain a pressure equal to its partial pressure in the mixture. The temperature of the bulb may be determined by a thermocouple, or optically. The method has been used by Lowenstein with H 2 0, HC1, H 2 S, and by Vogel von Falckenstein with HC1, HBr, and HI (cf . tables in 149). (?) Dens/fy/. The methods of Dumas and Victor Meyer for the determination of vapour density may be extended for use at high temperatures by forming the apparatus of refractory material. Thus Debray used the method of Dumas at the temperature of boiling zinc by making the bulbs of glazed porcelain instead of glass. Victor Meyer's method was extended to high temperatures CHEMICAL EQUILIBRIUM IN GASEOUS SYSTEMS 357 by its discoverer in the same way, and was used by Meier and Crafts to determine the vapour densities of chlorine, bromine, and iodine at very high temperatures. Nernst constructed a small Victor Meyer apparatus from iridium, heated it in an electric furnace, and measured the expansion by the movement of a drop of mercury in a horizontal side tube. The apparatus was filled with an inert gas, and a small amount of the substance, weighed into a tiny iridium tube by means of a quartz micro- balance, was dropped in. The temperature was measured optically. In this way the dissociation : Sa ^ 2S was investigated, and the vapour. densities of several metals were determined by von Wartenberg. Preuner and Schupp have made vapour-density measurements with sulphur enclosed in a quartz bulb heated electrically and connected with a manometer consisting of a spiral of silica tubing attached to a small mirror. The pressure was measured by the amount of unwinding of the spiral. The same method has been used by Bodenstein and Katavama in studying the equilibrium : H 2 S0 4 " H 2 + S0 3 (/) Electrochemical Method. In this the value of the equili- brium constant K is calculated from the maximum work measured by means of the electromotive force of a voltaic cell (cf. Chap. XVI.). Further particulars of these methods will be found in : W. Nernst, Applications of Thermodynamics to Chemistry, 190(3 ; F. Haber, Thermodynamics of Technical Gas Reactions, 2nd edit., trans. Lamb, 1909. CHAPTER XIII EQUILIBRIUM IN DILUTE SOLUTIONS 155. Chemical Potentials. IN the section on chemical equilibrium in gases we introduced a magnitude called the molecular chemical potential of a component : 8U\ , where 8, is the number of mols introduced into the mixture, whilst the magnitude indicated by the suffixes, together with the masses of all the other components, remain constant ; and d, dV, rfU, denote the resulting increase of potential, free energy, or intrinsic energy, respectively. The condition of equilibrium of the mixture was written in the form : /Mi + prf>nV * = U - TS we have : rf* = SdT + Vrfp + 2/ d* = Sf/T p(1\ + 2/A; The conditions of equilibrium ( 51, 52) (f/U) S)V = 0; (d*)^ = 0; (fWr.v = . . (8) all lead to the same general equation : 2/v7/M; = . . . . (9) If instead of the masses (/) we had taken the numbers of rnols (H) of the components, together with V and S, as the independent variables, we should have found : 2/ijdH; = ..... (10) or 2/ifi',- =0 . . . . . (11) where /*,- is defined by (1). Since (hi; = -^ where M ( - = molecular weight, we have JVlf ji; = M,/i, ...... (12) Illustration. If we have a saturated solution of a salt in contact with solid crystals of salt, the whole is in equilibrium at an assigned temperature and pressure. If is the total potential of the salt and solution together : t> =<*>' + <*>" The condition of equilibrium : (*)!> = requires that 5' + 5" = /(xi,xz, . . #), of n variables of xi, x*, . ., x n is said to be homogeneous and of the m-tli degree with respect to these variables when the identity : (f>(kxi,kx z , . . , kx n ) = k m (xi + tei, x 2 , . . , x tl ) or, proceeding to the limit, bxi : . , kxn) _ , w _j 8^(a?i, , . . , n , Let us now differentiate both sides of (1) with respect to k. If we put kxi = HI etc., we have, if = $(k.i\, kx 2 , . . , kx tl ) : d(xi, x 2 , . . , x n ) . . . (3) which is a very important theorem due to Euler, EQUILIBRIUM IN DILUTE SOLUTIONS 361 157. Potential of a Solution. The potential of a solution is $>!, 1H a , '"3, , !,-, P, T) . (1) where in,- is the mass of the i-th component. If all the masses are increased in the same ratio /.-, the potential of the resulting mixture is /,- times that of the first : is a homogeneous function of the first degree with respect to the masses (not necessarily linear), hence, from Euler's theorem, we find : $ (3) /H 2 m, But - = fix, J- = pa, etc., the chemical potentials, Similarly, with the variables HI, H 2 , . . , H,, p, T( 155) : /ti"i + A*22 + . . . + /t,-iii = . . (4) If we put /HI = 1, ;H 2 = w* 3 = . . =},- = 0, we have ^ = Mi ..... (5) so that the chemical potential of a pure substance is its therino- dynamic potential per unit mass. From (2) of the preceding section we see that the chemical potentials are homogeneous functions of zero degree with respect to the masses, hence from Euler's theorem : ,Ml + /H 2 -!- + . . . + ,H, ^=-= Again, since d is a perfect differential : g^ . (6) 362 THERMODYNAMICS and generally ~ = --'' (reciprocal relation) . . (7) - (8) oni, onii onii A very useful special case is that of a binary mi.iiiin 1 : The chemical potentials depend on the two masses only through their ratio : s =. m 2 /nii .'. (h)ii r/-s ; ^"'2 S and it is easily shown that : The stability of the equilibrium of the solution requires that, for all isothermal-isopiestic changes : 8 2 (/> > ..... (10) But ty = ?t 8m, + |^ 8 Wfl + ^ lp + ^ 8T ?/MI 9; 2 3;> 9* .'. if 5j> = 0, 6T = 0, and # = S/i 2 + 2 - > But 8 = 9s 9s > EQUILIBRIUM IN DILUTE SOLUTIONS 363 *j? (Sw 2 s&mtf > .'. 2^ < 0. and ^ 2 >0 . , (11) cs cs This simply states that the addition of a further quantity of the second component to the solution increases its own chemical potential and decreases that of the first component, when the solution is in stable equilibrium. In the present state of thermodynamics the calculation of the chemical potential of a component of a solution can be effected explicitly in two cases only : (1) Mixtures of gases ( 143). (2) Dilute solutions. In the former case we have obtained the expression : pi = T( ?l + R/TU-,), and we shall in the next section show that a similar expression holds for the chemical potential of a component of a dilute solution. 158. Equilibrium in Dilute Solutions. The determination of the chemical potentials of the components of a solution, in terms of p, T and the masses (or concentrations) is, from what precedes, equivalent to finding the conditions of equilibrium. The form of the chemical potential of a substance present in very small amount in a solution was shown by Gibbs as early as 1876 to be a logarithmic function of the concentration : li A + B/HS. The chemical potential of each component is known when the total potential 4> of the solution is determined as a function of the composition. The potential * of a solution formed of /i , MI, "-2, mols of substances of molecular weights MO, >i, w-a, ... at the temperature T and pressure p will be a function of p, T, and the 's, and the total energy, entropy, and volume are functions of the same variables : If all the n's are increased in the same ratio, U and V are also increased in the same ratio, and are therefore homogeneous functions of the first degree in those variables. U/HO and 864 THERMODYNAMICS are therefore unchanged when all the w's are changed in the same ratio, and depend on the w's only through their ratios, . . . ; and if the values of n\. Wa, . . are small compared with wo' w WQ, and U/MO, Y/w are finite, continuous and differentiate func- tions of these variables they are necessarily linear (Taylor's theorem) : HQ HQ I = ,. + ,,!' +,.,* + ...... (8) wo "o "o where the coefficients u , wi> r o, r i, are independent of the w's and depend only on the temperature, the pressure, and the composition of the components. If we put HI w 2 = . . =0 we find that u , r depend only on the properties of the solvent ; but HI, TI depend on the properties both of the first solute and of the solvent, but do not depend on the other solutes, and so on. There is therefore an interaction between the molecules of the solvent and those of the solutes, but not between the latter themselves (cf. 129). When the solution is more concentrated, the interaction between solute molecules may be included by taking terms of higher order in the Taylor's series, viz., such terms as HH(-) , and 2 12 -^-f-. u n and u i2 may be called coefficients of self, and of mutual, interaction respectively. If we write (2) and (3) in the form : U = HQllo + 11 + 22 + .... ) ,. V = >WO + >hri -)- H 3 f3 -)- ' we see that they are equivalent to the two conditions (1) and (2) of 128, viz. : (1) The change of total volume on diluting a dilute solution is zero. For if the solution and a mol of pure solvent are separate the total volume is r -f- n^i'i -)-..+ r , i.e., V + i' , whilst if they are mixed it is V = (?? + I)r + i r i + > '.<., the same as that of the initial system. (2) The heat absorbed on diluting a dilute solution is zero. For the intrinsic energy of the solution and mol of solvent EQUILIBRIA! IN DILUTE SOLUTIONS 365 separately is U + u , and that of the solution U' is (n + l)"o + i"i + > i- e -> aJso U + HO- The heat effect on mixing is : W - W = U' - (U + o) + P [V - (V + r )] = 0. These results assume that the other it's remain unchanged, i.e., no chemical action occurs on dilution. We now proceed to find an expression for the entmpy of the solution. If the composition remains constant, the solution may be treated as a simple fluid, the entropy of which is defined by the equation : .K = ' . . . of p and T such that : 7 du. 2 -\- pdi: 2 _. -- -- - - m -- 1 -- m -- a - -- m -- The integration of (6) now gives : S = HO*O + "1*1 + a*2 + + C . . (8) where C is independent of p and T, but contains that part of the expression for the entropy which depends 011 the variability of composition, i.e., is a function of the /i's. To find C we imagine the solution converted continuously (i.e., without separating into various phases) by suitable alterations of p and T but constant w's into an ideal gas mixture, the entropy of which is known, by the direct separation by means of semipermeable membranes to be : This imaginary process of evaporation, which it is true could only proceed as a succession of labile states analogous to the continuous passage of a liquid into vapour along the James Thomson isotherm, ( 90) is legitimate because the expression for the entropy applies to all states, whether states of equilibrium or not, and the w's (together with p and T) are independent variables, and may therefore be assumed to remain constant. Since expression (8) can, by change of p and T alone, assume the form of (9) only if : C = 366 THEEMODYNAMICS it follows that the entropy of the mixture must be : S = iioSo + niSi + . . K( ^'o + n-ilnci + . .) . (11) and hence the potential is : . .)] + X'Wfl + "l j 'l + ) T*i + pci) + . . -f- KT (iiolncQ + Wi^zt'i -}- ) = T[/?o are homogeneous functions of the first degree in the w's, they are not linear, as are IT and V. The condition of equilibrium at constant temperature and pressure is : 8* o, when 8T = and Sp = 0, /. Sw^i + Rliwt) + S(9/ + K/wc,)8 ( - = As in the case of a gas mixture ( 142) the first term vanishes identically : /. S(9,- + Elnc^n-i = . . . (15) It may be shown by a consideration of 8 2 4> that this is positive, hence the equilibrium is stable ( 157, and Duheni : Mecan. chim., torn. III., ch. 1 and 2). We denote a system composed of several phases, each of which is a dilute solution, by the symbol : iiQiiio, i/i, ... | n 'm ', ii'iri , ... | HO'IHQ", HI" mi", . . . \ 1st Phase 2nd Phase 3rd Phase Special cases of such phases are ideal gas mixtures, and the limiting case of a pure substance in any state of aggregation. The total potential of the system is : $ = +'+ <" + ... and the condition of equilibrium at constant }) and T is : = 8* - S 4. Sp _|_ fy" + . . i.e. 2( - by the ratios of whole numbers : Sn Q : Bill : . . : Sn ' : Siii : . . = TO : TI : . . : i'o ' v \ ' the vs being positive or negative according as the substance is produced or disappears. 91 + . . = InK,,^ (17) where K depends only on p and T, but not on the composition. At a given temperature and pressure a mixture of the various molecular species settles down to a state of stable chemical equilibrium in which the concentrations in each phase have finite, although in some cases possibly small, values. The late of mass-action applies to each phase of the system. The equilibria in the various phases having common com- ponents are not independent, but must be related by definite ratios of the concentrations of these components. The distribution la ic applies to each pair of phases of the system. The concentrations are contained in separate terms, and the equation (17) for the whole reaction can be divided into a number of separate equations expressing the equilibrium conditions for various possible partial reactions. The equilibrium is established as though each partial reaction pro- ceeded separately, and the other components containing the same elements were prevented from undergoing change. The deduction adopted is due to M. Planck (Thermodynamik, 3 Aufl., Kap. 5), and depends fundamentally on the separation of the gas mixture, resulting from continuous evaporation of the solution, into its constituents by means of semipermeable membranes. Another method, depending on such a separation applied directly to the solution, i.e., an osmotic process, is due to vau't Hoff, who arrived at the laws of equilibrium in dilute solution from the standpoint of osmotic pressure. The applications of the law of mass-action belong to treatises on chemical statics (cf. Mellor, Chemical Statics and Dynamics) ; we shall here consider only one or two cases which serve to illustrate some fundamental aspects of the theory. 159. Influence of Temperature and Pressure on the Equi- librium. For each phase we have : co -f- vilnci -f = -- (^o?o + ^i?! + ) = /K . (1) 368 THERMODYNAMICS . (dlnK\ _ _ d_ ' V dp )r~ dp" R _ 8 yp?o + PI 0, K Decreases when increase8 . 1 increases (2) If Q,, = 0, K is independent of T, ifQ p ^O,K i increases when T increases, (decreases We observe that there can be a definite state of equilibrium EQUILIBRIUM IN DILUTE SOLUTIONS 369 even when the heat of reaction is zero, in direct opposition to Berthelot's principle ( 118) which makes the state indefinite when Q = 0. (3) If any concentration vanishes, c { = 0, the expression on the left of (17), 158, becomes infinite, since lnc t = InQ GO . But ZwK cannot become infinite when p and T are finite, and hence all the possible molecular species must, in the equilibrium state, be present in finite, though possibly exceedingly small amounts. Only when T = and Q is finite does the equation (9) show that InK is infinite and one or more of the c's must then vanish, i.e., the reaction is complete in the sense of Berthelot's rule. The law of mass-action of Guldberg and Waage is therefore deducible on thermodynamic grounds for mixtures of ideal gases, and for ideal dilute solutions, but for no other cases. It is necessary to emphasise this point because there seems to be a general opinion amongst chemists that the law is one of perfect generality, and that its application to every possible type of chemical equilibrium is therefore allowable. Whilst in an em- pirical sense this may be true, we must not be surprised if divergencies are met with in regions which lie beyond the range of application sanctioned by the assumptions introduced into the derivation of the law. 160. Examples on the Application of the Equations of Equilibrium. (1) Two components in one phase, e.g., the electrolytic dissocia- tion of a salt in aqueous solution : CHsCOOH = CH 3 COO + H A large number of other reactions are also possible, e.#., hydra- tion of the various substances, or (if the solute is a salt of a weak acid or base) hydrolysis, but in all cases the concentration of each molecular species is defined by the total amount of solute in a given mass of solution, and the ionisation proceeds as if all the other reactions did not occur at all (cf. 158). The system is : + noHaO, w^HaCOOH, w 2 CH 8 COO, n 3 H The reaction is : v o = 0, z>i = I,v 2 = i> 3 = 1. 370 THERMODYNAMICS The equilibrium equation is therefore : Inci + Inc 2 + /c 3 = /nK or, since ^2 = <" 3 ; ^ = K Cl n wi n 2 "3 where e = -, Cl =- , c 2 =-, c 3 = n = HO + i + HS + wg. Wi -f- Wo 7?i -4- no Put ci + c 2 = e = ^ = approxi then c is known if the total mass of solute is known : K HI As c (i.e., the dilution) increases, the ratio increases to a limiting value of unity for complete ionisation. The transformation of numerical concentrations c, to volu- metric molecular concentrations , cannot be effected in the case of solutions so readily as with ideal gas mixtures ( 121), on account of the changes of density. We may put pci = lf etc., approximately, where p is the density of the solvent, hence : But for 1 mol originally dissolved : c I a r _a 1-^-, &- T where a = degree of ionisation, V = total volume, which is called Ostwald's Dilution Law. The measurements of a by means of the electrical conductivity show that the dilution law holds good for weak electrolytes (a small), but for strong electrolytes (a large) it fails utterly. This behaviour has given rise to a considerable amount of dis- cussion, a critical review of which will be found in a paper by the author (" Ionic Equilibrium in Solutions of Electrolytes ") in the Trans. Chem. Soc., 97, 1158, 1910. It appears that in this EQUILIBRIUM IN DILUTE SOLUTIONS 371 case the ordinary law of mass-action fails, as it usually does when electrical energy participates in establishing a state of chemical equilibrium (formation of ammonia by the electric discharge, production of ozone, etc.). The equation proposed by Larmor on the basis of kinetic molecular theory : = X (1 a)(V + fa) where e, \ are constants, has been shown by the author (loc. cit.) to give quite good results, and reduces to Ostwald's form when a is small (i.e., weak electrolytes). Thus with NaN0 3 (e = 23 X 10 5 ) : V a X X 10* K'(0stwald)xl0 5 10 7 0-9836 0-48 0-58 10 6 0-9617 0-74 2-42 167 X 10 3 0-9295 0-53 7-34 (2) Equilibrium of a gas standing over its saturated solution : HO'O, HiWi | ;?o'?o' solution gas. wo + i Put wii = anio, then the passage of a mol of dissolved gas into the gas-space is represented by : J'O =r 0, V\ =: 1, VQ = Ct, and the equilibrium equation : reduces to : Ind = /K piT The solubility ci is therefore a function of temperature and pressure. In this case, and generally where gases, as distinguished from liquids and solids, participate in a reaction, the dependence on pressure is fairly considerable : /3/K\ _ Ar te/rJr'RT RT , Now B B 2 372 THERMODYNAMICS \ dp / T \ p .-. Jnt'i = alnp + const. or cj = Cp* (T const.) so that the solubility of the gas at constant temperature is proportional to the power a of the pressure. In many cases it is found that a = 1 /. Ci = Cp, which is Henry's law ( 126). For the dependence on temperature we have : But Inci = InC + alnp . 8T/p RT 2 Qp is the heat absorbed when a mol of gas is abstracted from a large volume of saturated solution, and the solubility / increases ^decreases with rise of temperature according as Q (negative (heat evolved) p [positive (heat absorbed). In the majority of cases the solubility decreases with rise of temperature, indicating that heat is absorbed when the gas is abstracted from its saturated solution ; with the group of inactive gases, according to Estreicher ( 126), the opposite effect is observed. In Planck's investigation of equilibrium in dilute solutions, the law of Henry follows as a deduction, whereas in van't Hoff's theory, based on the laws of osmotic pressure ( 128), it must be introduced as a law of experience. The difference lies in the fact that in Planck's method the solution is converted continuously into a gas mixture of known potential, whilst in van't Hoff's method it stands in equilibrium with a gas of known potential, and the boundary eondition (Henry's law) must be known as well. Planck (Tliermotii/nainik, loc. cit.) also deduces the laws of osmotic pressure from the theory. In the same way we can investigate the equilibrium between a sparingly soluble liquid or solid substance and its solution (cf. 182). The change of molecular state which sometimes EQUILIBRIUM IN DILUTE SOLUTIONS 378 occurs when a substance passes into solution (association, or dis- sociation) can also be expressed by the appropriate values of v. Thus, in the system composed of a solution of a sparingly soluble binary electrolyte in equilibrium with solid : WH 2 0, iAB, 2 A, 3 B | Ho'AB solution solid we may divide the change : into the two partial changes : (a) the precipitation of a mol of AB : r = 0, ri = 1, r 2 = 0, v 3 = 0, J'o = *-f 1 (6) the dissociation of a mol of AB : j' = 0, i'i = 1, r a = 1, r 3 = 1, VQ = 0. Thence : Inci = InK . . . . (a) /wci + lnc z + Inc 3 = InK' . . . (6) (a} shows that the concentration of unionised solute is definite at a given temperature and pressure ; and (b) that the concentra- tion of the ionised part is determined from that of the unionised part by the ordinary equation of example (1). The total solute is ci + c 2 = ci + c 3 , and is known, and hence K, K' are deter- mined by a further measurement of 02 by the freezing-point or conductivity. If the two measurements are repeated at another temperature, the temperature coefficients of /wK and //iK' f and hence the heats of solution and dissociation, Q^ and Q'^, are found. Thence we find the heat absorbed when a mol of electrolyte is dissolved in sufficient solvent to produce a saturated solution : If we integrate the equation : ' _ Q' p ~ over a small range of T on the assumption that Q' p is constant, we find : \TI T2/ so that Q' p is determined from the temperature coefficient of the conductivity. 374 THEKMODYNAMICS The equations for the vapour pressures and freezing-points of dilute solutions are also readily deduced from Planck's equation. The linear relation between the depression of freezing-point and the concentration strictly applies only to infinite dilution although it holds good approximately up to decinormal concentration. J. B. Goebel (Zeitschr. physik. Cliem., 53, 213, 1905 ; 54, 314, 1906; 71, 652, 1910) has found an empirical equation for the depression of freezing-point in aqueous solutions of total mole- cular concentration n : & = 0-705 log (1 + A) + 0-24 A + 0'004 A 2 CANE SUGAR. A t calcd. s n t obs. 0-0532 0-2372 2-0897 0-0286 0-122 0-864 0-0283 0-122 0-837 With a binary electrolyte : NaCl z Na + Cl' 6 & the total salt concentration = -f- 2 ,%f w = li + 26 and the equation of mass-action gives : which gives K in terms of A. From this we can calculate the value of for a given concentration of solute, . Ternary electrolytes dissociate in two stages : (i.) CaCl 2 = CaCl + Cl & & & (ii.) CaCl = Ca + Cl EQUILIBRIUM IN DILUTE SOLUTIONS 375 and we have the relations : & = & + 2& = & - To each equilibrium there corresponds an equation of mass- action : aa _ V . a* _ T- f -- JM j -? -- Av 2 . Cl C3 The determination of the state of the dissolved solute is pos- sible if two measurements are made, for if we denote all the above magnitudes by dashed symbols for the second solution : &= &=*=& K! K- 2 CaCl 2 5-91 0'109 H 2 S0 4 0-45 0-017-2 The second dissociation therefore proceeds only to a very slight extent in comparison with the first. The theoretical treatment of the dissociation of electrolytes has been extended by Jahn and by Xernst, who have introduced assumptions. equivalent to replacing the linear expressions for the energy and volume given by Planck by expressions containing higher powers in the Taylor's series and supposed to take account of the self and mutual interactions between the solute molecules. The resulting equations are so extraordinarily clumsy, and difficult of application, as to prohibit anything more than a reference to them here, and in addition their physical significance is far from clear. (Cf. Partington, Tram. Chem. Soc., 97, 1158, 1910.) 161. Equilibria in Heterogeneous Systems. We shall consider the equilibria established in systems com- posed of various solid phases, each of which is a pure substance (i.e. not a solution) in contact with a gaseous phase. 376 THERMODYNAMICS The potential of the system is the sum of the potentials of the gas and of the various solids : = + n''(p,T)+n""(p, r F)+ . , = + 2n'p . . (1) The change of potential for a small isopiestic-isothermal change : Bn' : Bn" : . . : Bnj. : 8;i a : . . = v' : v" : . . : v : : v a : . . is S = fy + 'bn r + "5n" + . . . = T [(91 + RZwei)8rti + (92 + RZc 2 )6n 2 + . . ] + 2'8n' . . (2) If 8* > the process is impossible ; if S < the process is possible and irreversible ; if S<$> = the process is reversible and the system is in equilibrium : + ] + '&<' = ^'292 + ) + -^" + j^Zwca +..= - (1/191 + ^92 + . . ri + Sw'9') InK . . . (3) where K depends on p and T alone, and is the equilibrium constant. Example. The action of steam on red-hot iron : 3Fe + 4H 2 ^ Fe 3 4 + 4H 2 v' = - 3, v" + 1, v'" = . . = ; vi = 4, v z = + 4, v a = . . =0 p(49 2 491 + 9" 89') = ?K **- or ~= \,/ K = const. Partial pressures may be substituted for concentrations ( 122). We see that in determining the equilibrium the concentrations of the solids do not appear at all. This important result was first stated by Guldberg and Waage in 1867 in the form that " the active mass of a solid is constant." It is true only when the solids are of unvarying composition. If the coexisting liquid or solid phases are not pure, but solu- tions, equilibrium will be established when the chemical potential EQUILIBRIUM IN DILUTE SOLUTIONS 377 of every component is the same in all phases which contain it as an actual component. If any phases are free from any com- ponent, its chemical potential in the other phases must every- where be less than the values it would have if present in those phases in infinitesimal amount. For the change of thermo- dynamic potential when a mass bm of the component passes from a phase containing it at chemical potential p to one containing it in infinitesimal amount at chemical potential p^ is S = (fiQ /A) SHI, and for equilibrium S _ .'. /JL O > /i. The change in question is unilateral, i.e. the opposite change is excluded by the absence of the component from the second phase. When the phases are dilute solutions the general equation deduced at the beginning of the section may be applied. Example. Dissociation of fused cuprous oxide : the copper dissolving in the cuprous oxide to form a solution increasing in concentration as dissociation proceeds. The system is: HoCUgO, WiCu I H PI, respectively. The complete partial pressure curve will therefore be one of the three types OcQPi, OaQPi, O&QPi, which may be denoted by c, a, b, and called neutral, positive, and negative, respectively. The total pressure curve will be made up additively of two such curves, the possible combinations being aa, bb, cc, ab, ac, be. 100% I FIG. 74. X FIG. 75. 384 THERMODYNAMICS A complete set of diagrams of these combinations will be found in Ostwald's "Lehrbuch," II., 2, (1), 618626; the cases where the pressures of the pure liquids are equal are shown in Fig. 75. 167. The Gibbs-Konowalow Rule. The direction of change of pressure occurring in the distillation of a mixture of changing composition is fixed by a very general rule, deduced by Gibbs (1876), and used by Konowalow as a consequence of some experiments of Pliicker (1854), who found that the vapour pressure over a mixture of alcohol and water is all the less the larger the space which the vapour has to saturate. The rule may be stated as follows : During isothermal increase of volume of a vapour standing in stable equilibrium with a liquid mixture, the total vapour pressure either decreases or remains unchanged. For, if we suppose that a portion of the liquid be evaporated isothermally and reversibly (say by raising a piston enclosing it and the vapour in a cylinder), any resulting change of composi- tion may be utilised to perform work by remixing vapour and liquid through semipermeable membranes. Hence, since no work must be done on the whole ( 163), it follows that a com- pensating amount of work has been spent on the system during the evaporation, and hence the pressure must have been diminishing during that operation. Now suppose that, at a particular composition of the liquid, the total pressure increases as the liquid becomes richer in a specified component, say I. Expansion (i.e., distillation) can then, by the above rule, only diminish the concentration of I. in the liquid. The concentration of I. in the vapour is therefore, at that point, not less than its concentration in the liquid, for if it could be less, any possible evaporation would necessarily increase the concentration of I. in the liquid. If, however, the total pressure is decreased by increasing con- centration of a specified component in the liquid, compression (i.e., liquefaction) cannot increase the concentration of that com- ponent in the liquid, and hence its concentration in the vapour is not greater than that in the liquid. It therefore .follows that a transition from a rising to a falling part of a vapour pressure curve can occur only when the concen- tration of a specified component in the vapour is neither greater GENERAL THEORY OF MIXTURES AND SOLUTIONS 385 nor less than its concentration in the liquid, or in other words, it can occur only when the compositions of liquid and vapour are identical. Hence we deduce the exceedingly important Theorem of Konoicaloic : A liquid mixture corresponding to a maximum or minimum of rapour pressure at any specified temperature lias the same composition as the rapour in equilibrium irith it. (a) If one component is completely non-volatile, as in a salt solution, the vapour can never have the composition of the liquid, and the curve of vapour pressure nowhere exhibits a transition from rising to falling, i.e., it possesses no maxima or minima. (b) The converse of Konowalow's theorem states that if there are two mixtures of slightly different composition, one of which is richer in a specified component than its vapour, and the other poorer, there is an intermediate composition at which the composi- tions of liquid and vapour are identical, and there the vapour pressure has a stationary value. If the evaporation is performed under isopiestic conditions, instead of isothermal, as is usually the case in practice (/> = 1 atmosphere), the vapour pressure curves must be replaced by the boiling point curves. For this purpose a horizontal line corre- sponding to the given pressure is drawn across the vapour pressure diagram. The abscissae of the points of intersection of this line with the various isotherms will determine the composi- tions of the liquids which emit vapour of the specified pressure at the temperatures corresponding to the various isotherms ; and if the latter are plotted against the compositions one obtains the curve of boiling points under the given pressure. This may be repeated for other pressures, and a series of boiling point curves constructed. The reverse (and more practical) construc- tion is similarly effected. In particular, it is easily seen that points of maximum or minimum vapour pressure will give rise to points of minimum or maximum boiling point respectively, with the same abscissae. The gradients of the partial pressure curves at the abscissa corresponding to a maximum or a minimum must be equal but of opposite sign. For if P is the total pressure, and dx 1. 386 THERMODYNAMICS - ' ' dx ~ dx (J. W. Gibbs, Trans. Acad. Conn. (1875), 3, 155 ; Sdentij. Papers, I. ; Konowalow, Wied. Ann., 1881, 4, 48; Zawidski, Zeitschr. pliysik. Chem,, 35, 129, 1900; Meyerhoffer, ibid., 46, 379, 1903). It is a consequence of the Gibbs-Konowalow rule that the com- positions of liquid and vapour (i.e., the residue and distillate, respectively) alter in the sense of falling and rising parts of the curve, respectively. One may imagine (Ostwald, loc. cit.) the composition of the liquid to be represented by a heavy mobile point, that of the vapour by a small balloon, constrained to move along the curve. The former tends always to sink to the part of lowest 'pressure, the latter to rise to the part of highest pressure. If we repre- sent the different types of curves in one diagram, as in Fig. 76, and draw two vertical lines a and l>, to cut all the curves, it is easy to see what will be the effect of isothermal distillation on the mixtures of compositions a and b. On curves (1), (2), (4) the composition of the liquid must move to the left, that of the vapour to the right. The distillate therefore contains a greater amount of the more volatile component than does the residue. By distilling off a portion, repeating the process with the distillate, and so on, at the same time adding the residue in the retort to the former residue, we obtain distillates and residues containing increasing concentrations of the more and the less volatile constituents, respectively, and the separation may be made as nearly complete as we please (" Fractional Distillation"). On the curve (8) the distillate must be at the commencement the mixture of maximum vapour pres- sure, for otherwise the pressure would increase during distillation. The two components are therefore withdrawn from the liquid in the proportions required for the mixture of maximum vapour pressure. If the original liquid has this composition, it evapor- GENERAL THEORY OF MIXTURES AND SOLUTIONS 387 ates unchanged ; if not, the withdrawal of both components goes on until that which happens to have been in excess remains in the pure state in the retort. In the case of a, this will be A, in the case of b, it will be B, because the massive mobile point will run down the curve to A or B, respectively, whilst the light point rises to M. If the curve is of type (5), the vapour passing off will be pure A or B, respectively, according as the mixture is a or />, whilst the residue approaches the composition M 2 . When this composi- tion is reached the whole mass distils off unchanged, like a pure liquid. Mixtures -of this type have been known for a long time. Bineau (1843) found that the solution of hydrochloric acid in water which distilled off unchanged under atmospheric pressure had approximately the composition HC1.8H 2 0. The similarly behaving solution of nitric acid was represented as 2HN0 3 .3H 2 0. He therefore advanced the hypothesis, which has been obstinately maintained by some chemists, that these solutions are definite compounds. Roscoe and Dittmar (186061) found, however, that if the pressure is altered, the composition of the mixture of maximum boiling point is altered as well, so that the locus of the maxima on the boiling point curves is a Hue which is inclined to the T-axis. Since all points on this line correspond to mixtures of maximum boiling point under the correlated pressures, and since only a few of these points have abscissae corresponding to simple stoichiometric proportions, it is evident that we must either reject Bineau's hypothesis, or else return to Berthollet's view of the nature of chemical compounds. The existence of definite compounds in the solution is not, of course, in question here at all; all that is asserted is that the maximum boiling solutions are not necessarily pure compounds. The theorem of Konowalow is the basis of the remarkably interesting Faraday Lecture to the London Chemical Society given by Ostwald in 1904. He points out that the considerations which have been summarised above in connection with Konowalow's curves lead to the general law that it is possible in ever// rase to separate solutions into a finite number of hylotropic l>o = phases For each pair of phases there must be some condition (the exact nature of which is immaterial) satisfied for each component, such that this component does not pass from one phase to the other. The r phases may now be arranged in (r 1) pairs, viz., (I, 2), (2, 3), (3, 4) ... {(/ 1), r}, and there will be therefore > 1 conditions to be satisfied for each component in all the phases, that is, (r 1) equations defining the state of equilibrium. It is evident that such pairs as (1, 3), (2, 4), ... need not be con- sidered, since if phase (1) is in equilibrium with phase (2), and phase (2) with phase (3), then (1) will also be in equilibrium with (3), by the law of Mutual Compatibility of Phases. For all the components there will be /?(/ 1) equations. The number of variables is made up of : (i.) the pressure IT, (ii.) the temperature T, (iii.) the r(n 1) independent concentrations, since for each phase there are (n 1) fractions of the total mass for the n components, the last(n-th) fraction, being obtained by subtraction of the sum of all the others from the total mass, is fixed, and is not an independent variable. Hence : Total number of variables = 2 + ''(" !) The number of variables left undetermined = number of variables number of equations = 2 + r(n -1) - n(r - 1) = 2 + w r. This, however, is defined ( 84) as the number of degrees of freedom (F) of the system, hence : F + ;= + 2 . . . . (a) which is Gibbs's Phase Rule.* * Cf. Partington, Proc. Chem. Soc., 191 1. 390 THERMODYNAMICS If any component is absent from a particular phase (for example, the vapour phase, or a solid phase), there is one vari- able the less, but also one boundary condition the less, for migra- tion of that component cannot occur with respect to the phase considered. Equation (a) is therefore still true. This very simple rule has had an almost incredible influence on the development of theoretical and experimental chemistry. In the previously obscure but technically very important department of the study of mixed metals, to quote a single example, it has proved invaluable. 169. Kirchhoff's Equation. Previous to the researches of Konowalow, the vapour pressures of mixtures had been investigated theoretically by G. Kirchhoff (Pogg. Ann. (1858), 103, 104; Ostw. Khtss. No. 101), and by Gibbs (Scientific Papers, Vol. I.). The latter had established the theorem relating to mixtures with stationary vapour pressures. If two liquids are mixed together, there is in general a change of intrinsic energy (AU) and a change of free energy (A*). The heat absorbed when 1 mol of [1] and x mols of [2] are mixed in a calorimeter is the increase of intrinsic energy, and is usually denoted by Q(ar) : AU = Q(a) .... (1) Q(x) may be positive (heat absorbed, e.g., phenol + water), or negative (heat ecolved, e.g., sulphuric acid + water) ; it will in general be a function of x and of temperature, but changes only very slightly with pressure. Thus, with water and sulphuric acid, Thorn sen found at the ordinary temperature, for H 2 S0 4 + ,rH 2 : If r, r" are the heat capacities of the unmixed and mixed systems respectively ( 58) : If F' = F, the heat of admixture, Q(*'), is independent of temperature. In his original investigation, Kirchhoff proceeded to deduce an equation connecting the heat of admixture of two liquids, one of which is very sparingly volatile, with the vapour pressure of the volatile component over the mixture. GENERAL THEORY OF MIXTURES AND SOLUTIONS 391 Let us consider a system formed of a homogeneous mixture of NI mols of a non-volatile solute with N 2 mols of a volatile solvent, together with a further >? 2 mols of pure liquid solvent. The system has, at an assigned (arbitrary) pressure and tem- perature (P, T) a potential : = ,^ 2 (P, T) + $'(X lf N 2 , P, T) . . (1) where fa, <' are the potentials of a niol of the pure solvent, and a quantity of solution such that NX -f N 2 =1, respectively. It' V is the total volume of the homogeneous mixture : V = (N 1 .+ X a )f .... (2) where T may be called its mean molecular volume. Now f> ' (N %p 2 ' P? T) = V = (N, + N 2 )f(;r, P, T) . (3) where : .-Bjf*!-.- ^ W are the molecular fractions, or numerical concentrations, of the components. The chemical potential /z-2, of the solvent in the solution is, by definition : ,,= *'> _ f)N2 "" aF c>t' '^.1 2 <- **' This relation is frequently applicable. The heat absorbed in any reversible isothermal-isopiestic change is ( 55 (10) ) given by : dQ = e*($-T?*) .... (7) The heat absorbed when a niol of solvent is added reversibly to the solution at constant temperature (T) and pressure (P) is the heat of dilution, Q (x, P, T). since dtt. 2 = 392 THERMODYNAMICS The heat absorbed when a mol of the solvent is evaporated at a constant temperature T from a volume of solution so large that no perceptible change of concentration occurs during the process, is called the heat of volatilisation \ (x, T). From (7) : A (r rm _ i, * i _ T _ -8N2~~ where p is the pressure corresponding to a concentration x, and 92 is the potential of a mol of the vapour of the solvent under the given conditions. The heat of volatilisation of the pure solvent under its saturated vapour pressure, ir, at the temperature T is similarly : A(T) = fc (,T) - *(,,T) - T - . (10) Then E, > > - T rd-P T) T 1- A ~ n ' ' 8T . . . (11) We shall now assume that r is negligibly small, then EI is independent of the pressure P .. .. . (a) If 2 (P, T) is the molecular volume of the pure liquid solvent : (P, T) - T = , (P, T) - T = E 2 , say. If 2 is negligibly small, E 2 is independent of the pres- sure P . . ...... . . . (b) Further, 1 ( 92 (p, T) - T 8 ? 2 ^' T ^) = V 2 (p, T) where 2 is the molecular volume of the vapour. But if the vapour obeys the gas laws, E 3 = , (c) Thence we can write : GENERAL THEORY OF MIXTURES AND SOLUTIONS 393 as we see from the expressions for EI, E 2 , E 3 and the condition that these are all independent of the pressure. From (a), 08), (7) and (8), (9), (10) it follows that : Q (.r, T) = A (x, T) - A(T) . . . (13) This equation, it must be observed, is not true generally, but only on the assumptions introduced : (1) r, 2 are negligible ; (2) The saturated vapour obeys the gas laws. We shall now find an expression for A (r, T). Let the system consist of the solution as before, in contact with 2 mols of the vapour of the solvent, under a pressure j>. The increase of entropy when SN- 2 niols of solvent are removed from the solution in the form of vapour at the constant tempera- ture T and pressure p is : cj A \tl' 9 -L ) v" -f^ But S = -* = - ~ _ _ L f>T ?T , T) = - T We shall next examine the quantity enclosed in the brackets. The solution is in equilibrium with its vapour when the condensa- tion of an infinitesimal amount of the latter leaves the whole potential of the system unchanged, i.e., the changes of potential of solution and vapour are equal and opposite ; 394 THERMODYNAMICS . (15) i.e., the chemical potentials of solvent in solution and vapour are equal. If Ma ^ 92, evaporation, or condensation, respectively, will occur. Put ft (x, p, T) - ?2 (p, T) = F (x, p, T) d(x,p,T) a/i (./-,_/>, T) tnen ^ = J dp ST Thence X (j?, T) = T v 2 + x - p . . . (16) This is a very important general equation. Again we assume that r is negligible in comparison with V 2 , and that the saturated vapour obeys the gas laws. /. A (.r, T) = RT* 1 e >^l> = BT ^fa^>T) (ly) which is analogous to the Clapeyron-Clausius equation and is due to L. Natanson (1892). We have also A(T) = BT 2 . . - . . (18) for the pure liquid solvent ( 88). Thus, from (13), (17), and (18) we have : ^ '.. . (19) This equation was deduced by G. Kirchfaqff(P 0, i.e., heat is absorbed on dilution (as is the case with a large number of salt solutions) then In , and hence the relative lowering of vapour pressure - , decreases with rise of temperature. (2) If Q (x, T) < 0, i.e., heat is erolretl on dilution (H 2 S0 4 . CaCla, NaOH). the relative lowering increases with rise of tem- perature. (3) If Q (x, T) = 0, i.e., the heat of dilution is zero, the rela- tive lowering is independent of temperature. Thus, the necessary and sufficient condition for the validity of the law of von Babo ( 130) is that the heat of dilution is zero at all temperatures and concentrations in the range considered. In the above investigation Q (,r, T) has the significance of a heat of dilution, i.e., it denotes the heat absorbed when more solvent is added to a solution of concentration ,r. If we consider a solid salt which is dissolved in a solvent, Q (,r, T) has the signi- ficance of a heat of solution. If we consider a saturated solution we recover the case treated in 132. The vapour pressure p of the solution is now a function of temperature alone, since the concentration of a saturated solution is, at a given pressure, completely defined by the temperature, and it alters only very slightly with change of pressure. 170. The Duhem-Margules Equation. The equation of Kirchhoff applies to the case of a volatile solvent and an involatile solute ; we shall now consider the case of a mixture of two volatile substances, i.e., a binary mixture in the sense of 163. We use the following notation : 396 THERMODYNAMICS NI, N 2 = numbers of mols of the two components, [1] and [2], in the liquid phase ; n lt 2 = numbers of mols in the vapour phase ; P lf P 2 = vapour pressures of pure [1] and [2] at a given temperature T ; p lt p 2 = partial pressures of [1] and [2] in the vapour of the mixture at the temperature T. The molecular fractions, or the numerical concentrations in the liquid, are: Ni . (l . _ --' - N 2 We shall now calculate the diminution of free energy which results from the admixture of NI mols of [1] and N 2 mols of [2], both in the liquid state. The simplest method is an application of equation (13) of 52, which states that the work done in the isothermal and reversible execution of a process is equal to the diminution of free energy : SA T = tl * .-. A T = - f rf* = * * where ^o, * refer to the initial and final states. It follows from Moutier's theorem ( 36 ; cf. 58) that SPo * is equal to minus the least possible amount of work which must be spent in separating the mixture into the two liquid components, by any isothermal reversible process. We shall select the process of isothermal distillation, introduced by Kirchhoff (1858). Let each component be removed in the state of vapour through two semi- permeable partitions, so that the operation proceeds isothermally and reversibly, and with unchanging composition of the liquid. Each component is therefore removed under a constant pressure, equal to its partial pressure over the mixture. It is further assumed that : (i.) Both vapours obey the gas laws ; (ii.) The volume of the liquid is small in comparison with that of its vapour. The operations are as follows : (1) Evaporate out NI mols of (1) and N 2 mols of (2). The amount of work done is : m + p& 9 = (Nj + N 2 ) BT. GENERAL THEORY OF MIXTURES AND SOLUTIONS 39? (2) Compress the vapours to PI, P 2 . The work done is : -RT Ni Zni + N 2 ln^ . V ' pi pi> (3) Condense on the liquids. The work done is : - (Pi Vi + P 2 Va) = - (Nx + N 2 ) RT. The total expenditure of work = loss of free energy *. To obtain the required relation between the partial pressures and the concentrations in the liquid, we suppose a very small quantity SNi mols of (1) is distilled isothermally and reversibly from the pure liquid (1) to the mixture of NX mols (1) + N 2 mols (2). The work done is : p S [A *] E S* = SNi. RT In - 1 (N a const.) Pi .'. in the limit, with unaltered concentrations : 7} & P ^ = RT In (N 2 const.) dJNi _/>i Similarly - |^ = RT In ^ (Ni const.) But Ni = N 2 l x ... d Ni = N 2 ^' 2 (N 2 const.) ... _iy? = B?y? ! .... (2) Differentiate (1) with respect to a-, and compare with (2) : _. rnvr ~- R1Na i Inpi djnpfi ~ (i-*f~ fa J (3) Thence +(l_*) = . . . (4) 398 THERMODYNAMICS Equation (4) is the fundamental differential equation in the theory of binary mixtures; it was obtained by Duhem in 1887 ( Traite de Mecaii. Chim. IV.). Duhem's equation was integrated by Margules (Sitzungsber. Wien Akad., 1895) by means of the substitutions : The a's aud/3's are constants fora fixed temperature and given components. If one curve, say pi = (.*), is known, the other, and hence the total pressure curve, is known from (4) for all concentrations, and conversely if the total pressure curve, II = F (x), is known, both partial pressure curves are determined when (4) is written in the form : (xfl 7^) -^ + (1 x)p, TT- (6) ox * dx The coefficients of (5) are related as follows : /JO #o ^1 > Pi ~ #1 j Pa * #2 ~\~ &3 ~T" ) r &= -a 3 -2a 4 -aa 5 -... \ ' (/) The equations of the tangents to the partial pressure curves are : ', / O \ ~ ' At their intersections with the pressure axes, x = and x = 1 : f ]M _ I * * ' ^ 9) But if we assume Raoult's law for these limiting dilutions : .'. ao = ^o = 1 ; a\ = fti = . . . (11) GENERAL THEORY OF MIXTURES AND SOLUTIONS 399 From (5) and (11) : Differentiate (12) with respect to a?, and put ,r = 0, then a; = 1 : /i'M ^f- \?7/.r=i~~ \i which are the equations of the tangents of the angles of contact, and are analytical expressions of Henry's law, as we see on putting : = const. = a e ~ = const. = I where a, b have the significance of solubility coefficients tZawidski, Zeitschr. physik. Chem., 69, 1909 ; cf. Story, ibid., 71, 129, 1910 ; Rosanoff and Easley, ibid., 641 ; R. Plank, Phys. Zeitschr., 11,, 49, 1910; Konowalow, Journ. de Phys., [iv], 7, 207, 1907). Let S\, Si' be the volumetric conceatratious of pure [1] iu liquid and vapour, 3- 2 ', S 2 " similar vahies for pure [2]. Also let &', |i", and &, | 2 ", be the concentrations in the liquid and vapour phases of the solution. In general >,., >|; and if then <'."4<'- But li"/fi' = and &",'&' = '' These relative solubih'ty coefficients completely define the character of the partial and total pressure curves of binary mixtures, 400 THEBMODYNAMICS (i.) If all the as and /3's are zero : a = b = 1 . fcl'^fc" ' ' fi' ft' i.e., the concentrations in both phases are equal to those of the pure components : li" Si" fe" S 2 " TT = =rrand -r-r = ,^-r ll *! 1-2 S 2 In this case equation (12) leads to the relations Pi = Pi* J>g = P a (l -.X) so that both partial pressure curves, and the total pressure curve, are neutral curves. where k is a constant : * = Pi /P 8 This equation is similar to that of F. D. Brown (1879), PI p2 = k fir7> where .' = concentration in the vapour. It holds good for mixtures of benzene with ethylene dichloride. For other non -associated and closely related liquid pairs, equation (lo) applies, but A- is not usually equal to Pi/P a and has to be determined by experiment. (iL)If 3+1+ >0,and| + | ! +. .. >0, a > l.and 6 > 1, $>%"*&>$ Both partial pressure curves Pi =/i(-''). Pi =./i ('), lie over the lines joining the points ( 2h = for x = (^1 = ?! for,r=l, I ^ = P 2 for ao = (/* 2 = forx=-r. They, and the corresponding total pressure curve, are positive. .(iii)If ?+!+ ..- <0,and^+f +. . <0 a < 1, and 6 < 1, GENERAL THEORY OF MIXTURES AND SOLUTIONS 401 <-<. so that both partial pressure curves, and the total pressure curve, are negative. (iv.) If f-+y + > 0, butJ+^+ - - < a>l, but&IV,andfV = - A* + VP [31,6, ?! + NV^ 9T rT L PI j> 2 If AU = Q (Mi,N a ,T) is the heat of admixture at T. we have generally ( 58) : = - RT 2 [NX/W L+ NV" ? . . . (16) Now suppose N- 2 mols of pure liquid [2] are isothermally and reversibly distilled into NI mols of pure liquid [!". The change of free energy for distillation of SN 2 mols of [2] into a mixture over which the partial pressure is p% is, as we have shown : .*. for the complete process, in which the partial pressure p of [2] increases from to p* t and its concentration from to .r, the change of free energy is : f* P A * = RT /it -? rfN a (Ni const.) , Jo ' Similarly, for distillation of [1] into [2] : . (18) - A* = RT ! In ?i dNi (N 2 const.) D D 402 THERMODYNAMICS / Na ... RT I H 1? dN a = RT / J I H 1? dN a = Pi Pz n p 2 r o These relations were deduced by Nernst (1893). 171. Dolezalek 's Theory of Binary Mixtures. The vapour-pressure curves of binary liquid mixtures have been considered from another point of view by Dolezalek (Zeitscher. physik. Chem. 64, 727, (1908)), who starts out with the very simple assumption that the partial pressure of each component is pro- portional to its concentration in the liquid phase, provided no chemical change occurs when the liquids are mixed, and that neither component is polymerised in the liquid state. Thus : P v ^A _ p v N] The neutral curves are therefore characteristic of non-associated substances, a conjecture which is in accord with the results of independent branches of investigation. A partial pressure curve which is concave to the concentration axis, i.e., a positive curve, indicates the dissociation of a poly- merised component, whilst a curve which is convex to the same axis, i.e., a negative curve, indicates the formation of a chemical compound of the two components. In the first case the con- centration of the constituent passing into the vapour would be increased, in the second case reduced, by the assumed change. As examples, Dolezalek quotes : (1) No association or combination neutral curves : benzene + chloroform. (2) Association positive curves : carbon tetrachloride + benzene (CC1 4 ) 2 ^ 2CC1 4 (15 per cent, double molecules at 50). (3) Combination negative curves : chloroform -f- acetone CHC1 3 + (CH 3 ) 2 CO^:CHC1 3 (CH3) 2 . CO. GENERAL THEORY OF MIXTURES AND SOLUTIONS 403 The proportion of polyniersed substance, or of the compound, may be calculated by the law of mass action. Emil Bose (1910) maintains that Zawidski's calculations, with Margules' solution with only a few coefficients, are not satisfactory, and proposes to find the partial pressures by a graphical method which consists in drawing the two partial pressure curves so that the sum of their ordinates is everywhere equal to the ordinate of the (known) total pressure curves. The Duhem equation shows that />!,#> are positive, continuous, and single- valued functions of x, so that only one decomposition of the total pressure curve has any physical significance, and for every value of . : .* _
  • ! 'fa ~ i^I7:' TT^r 172. Dolezalek's Rule. Dolezalek had previously (1903) proposed a very simple relation between the vapour pressures of concentrated salt solutions and their composition ; the logarithm of the vapour pressure of the solvent is nearly a linear function of the number of mols of salt (x) per rnol of water : Inp ax + b ...... (1) /. _ a a characteristic constant . . (la) dx Now according to Nernat and Roloff ( 130) : ( l^P + x ( !^ = Q (2) dx dx where p,P are the vapour pressures of the solvent and of a volatile solute, respectively. Thence : dlnP x i = a dx or /P = alnx -f- const. . . . (3) so that the logarithm of the partial pressure of the solute is also a linear function of the number of mols per rnol of water. Equation (3) was verified with aqueous solutions of hydro- chloric acid. 173. Thermal Relations of Binary Liquid Mixtures. We have, so far, in considering the evaporation of binary mixtures, taken account only of the partial pressure relations. D D 2 404 THE RMOD YN AM 1C S These, however, are intimately related to the quantities of heat absorbed in the formation of the mixture, and in its evaporation. In considering the thermal magnitudes of importance in the study of binary liquid mixtures, we shall confine our attention to the simplest case, in which it can be assumed that : (1) The volume of the liquid is negligibly small in comparison with that of its vapour. (2) The vapour is a mixture of ideal gases (cf. 122). From a very large volume of the mixture of NI mols of [1] and N 2 mols of [2], let a very small amount of vapour be taken. The concentration may be assumed to be constant during the process, and hence also the total pressure : n=pi + p* . (1) If iii, "2 are the numbers of mols in the vapour : j>i = ^-n ; P , = _^_n . . (2) MI -j- M 2 "i + "a Let A lt A 2 be the quantities of heat absorbed when a mol of each component is evaporated, under its own partial pressure, from a large volume of liquid into a large volume of vapour. Then, from hypothesis (1) : ... (3) We put -^ - A! + _-!?*_ A a = A. , (5) Ml + 2 MI + M 2 the heat of evaporation of a mol of the mixture. Then : /am AH ' ' ' ' ' which is formally identical with the Clapeyron-Clausius equation. If AI', A 2 ' are the molecular latent heats of evaporation of the pure liquids at the same temperature, then from hypothesis (2) : A = NiAj' + N 2 A 2 ' - W . . . (7) where W is the heat absorbed in mixing NI mols of the first liquid with N 2 mols of the second. The amounts of heat absorbed GENERAL THEORY OF MIXTURES AND SOLUTIONS 405 on adding a niol of each liquid component to the mixture are %, oW/3N 2 , respectively. Thus : /?W\ S ' \9N 2 / T J But, if PI, P 2 are the vapour pressures of the pure liquids : (9) in which only the vapour pressures of the unmixed liquids appear. Also, from (1) and (5) : Pi _ "i _ A. A 2 _ II l + 2 ^1 ^2 ' 11 "l + "2 A l Corollary. The change of composition of the vapour with temperature is determined in magnitude and sign by the difference of the heats of evaporation from the mixture. From (3) and (10) we find : ... (13) ar \Pzl = K ' RT* ' VaNai i.e., the partial pressures may be calculated if we know the pressures of the unmixed components, and the heat of admixture as a function of temperature and of the composition of the mixture : W = W(N 1 ,N 2 ,T) (R. Luther, 1898 ; P. Duhem, 1899.) 406 THERMODYNAMICS The heat of admixture of a mol of solution, w, is defined by the equation : W=(N + N 2 )w . , ' ." . (14) and since N^Ni + N 2 ) = ./ . . . . (15) Wehave -:/'..:' _ _ ^ x fa From (13) and (16) we find : + V.-*)@y\ \ox / T so that we have the equations : w = RT 2 a:Zn -+ (1 - *) / . . (17) which are due to Nernst (1893). (Of. Duhem, Traite, IV., 215 et seq.) Equation (18) was applied by Margules (1895) to calculate the heats of admixture of water and alcohol from the vapour-pressure data of Regnault ; the results agreed with the direct deter- minations of Winkelmann (1873). 174. Partially Miscible Liquids. There are two very important generalisations which may be established in connexion with the second type of liquid mixtures investigated by Regnault, e.g., mixtures of ether and water. (1) So long as tivo layers are present in contact with vapour, the composition of each is, at a specified temperature and pressure, independent of the absolute or relative amounts of the layers. Further addition of either component therefore leads to the growth of one phase at the expense of the other, but the concentration of each remains unchanged. GENERAL THEORY OF MIXTURES AND SOLUTIONS 407 This is a consequence of the phase rule ; there are two com- ponents in three phases, hence the number of degrees of freedom is 2, so that when the temperature and pressure are fixed, the composition of each layer is also defined. (2) If two liquid solutions are in equilibrium with each other, their vapour pressures, and the partial pressures of the components in the vapour, are equal. Konowalow established this important rule by means of the following reasoning. The two liquid layers a and /3 are con- tained in a ring-shaped tube, and above them is the vapour. The liquids are in equilibrium across the interface A. Then if the pressure of either component in the vapour were greater over a than over (3, diffusion of vapour would cause that part lying over ft to have a higher partial pressure of the given component than is compatible with equilibrium. Con- densation occurs and ft is enriched in the specified component. By reason of the changed composition of /?, however, the equilibrium across the interface is disturbed, and the component deposited by the vapour ivill pass into the liquid a. The whole process now commences anew, and the result is a never- ending circulation of matter round the tube, i.e., a perpetual motion, ivhich is impossible. Hence the partial pressures of both components are equal over a and /3, and therefore also is their sum, i.e., the total vapour pressure. This is evidently a special case of the Law of Mutual Com- patability of Phases ( 168). For the application of the theory of chemical potential to such systems cf. 155. If the temperature is changed the miscibility of the liquids alters, and at a particular temperature the miscibility may become total; this is called the critical solution temperature. With rise of temperature the surface of separation between the liquid and vapour phases also vanishes at a definite temperature, and we have the phenomenon of a critical point in the ordinary sense. According to Pawlewski (1883) the critical temperature 3 of the 408 THERMODYNAMICS mixture may be calculated from those of its components by the mixture rule ( 120) : + This equation is only approximate (Kuenen, 1895), but usually gives very good results (Schmidt, 1891). In some cases, how- ever the critical point may be even lower than that of either component. 175. Vapour-Pressure Curves of Partially Miscible Liquids. The vapour-pressure curves for such mixtures as ether and water consist of three parts : (1) and (2) The end curves corresponding with the homo- geneous solutions of B in A, and of A in B. (3) The central curve, corresponding with the two layers. There are three types formally possible, viz., those in which the end curves both ascend or descend, or one ascends and the other descends, to the central curve. Konowalow found, how- ever, that the second type does not actually occur, so that the two known forms are a and /3. The non-existence of the other type is also evident from the consideration that the pressure of the mean curve lies above those of the components in a, between them in /?, and would lie Icncath them in the other case. Now on a rising part of the curve the concentration of the compo- nent on the other side of the central curve must be greater in the vapour than in the liquid phase, for a falling part the reverse is true. For, taking the curves starting from the A axis, we see that addition of B increases the pressure. But we have already proved that compression, i.e., condensation of vapour, always increases the pressure. Hence condensation of vapour increases the concentration of B in the liquid, hence the vapour is richer in B than the liquid. If the curve descends, as in the case of the 100% A 100% B FIG. 78. GENERAL THEORY OF MIXTURES AND SOLUTIONS 409 part of a coming from the B-axis, we see that addition of A raises the pressure. But compression, i.e., condensation of vapour, increases the pressure, hence the vapour is richer in A than the liquid, and hence poorer in B. If now we have two end curves descending to a mean curve, in the point where the first cuts the mean curve the vapour is poorer in B than the liquid. The second point of intersection, on the other hand, is on an ascending portion of the curve, and hence the vapour is richer in B than the liquid. At hoth points, however, the vapour has the same com- position (because two layers are present), and in the second point the liquid is much more concentrated than in the first. Thus the vapour is more dilute than the most dilute, and more concentrated than the most concentrated liquid, which is impossible. One end curve must therefore lie below the central curve. 176. Immiscible Liquids. If two liquids, such as benzene and water, do not mix appre- ciably, the vapour-pressure relations are very simple, for each constituent emits vapour under the same pressure as if it alone were present. The total pressure is the sum of the vapour pressures of the pure liquids : If the temperature is raised, both partial pressures increase, and when their sum reaches the value of the total pressure to which the free surface of the mixture is exposed (usually the pressure of the atmosphere) the liquid boils. It is at once evident that the boiling-point will be lower than that of either pure liquid. The partial pressures in the vapour are in the ratio of the molecular concentrations : MI + n a HI + 2 If mi, / 2 are the molecular weights, the ratio of the actual weights of the components distilling over is ici : w z = mini : / 2 2 = PII : Paa (1) an equation due to A. Naumann. We observe that, although one component may be only slightly 410 THERMODYNAMICS volatile alone, it will distil over 'freely if it happens to have a molecular weight which is large in comparison with that of the second component. At a pressure of 760 mm. a mixture of nitrobenzene and water boils at 99. The vapour pressure of water at this temperature being 733 mm., that of nitrobenzene will be 760 733 = 27 mm. Now wi/> 2 = 18/123 .'. wi : - a = 18 X 733 : 123 X 27 = 4:1, approximately. Thus one-fifth of the distillate is nitrobenzene. Since the com- plex organic liquids one wishes to purify are usually of high molecular weight, the method of distillation in steam is very valuable. If nil is known, and PI is a known function of temperature, equation (1) serves to determine m 2 , the molecular weight of the second liquid, for P 2 is determined by the total pressure (usually 1 atm.) : 177. General Theory of Binary Systems. In 163 167 we have deduced some properties of systems of two components in two phases (" binary systems ") directly from the fundamental principles, and in 169 173 we have obtained quantitative relations in certain special cases. Here we shall obtain some general equations relating to such systems with the help of the thermodynamic potential (cf. 155). We shall first establish some relations which apply generally to systems of n components in r phases. Let iiti, ))iz, nia . . . , m,,' mi", w a ", !", . . . , m n " be the masses of the first, second, . . . n-tli, components in the first, second, . . . r-th phases. The state of physical and chemical equilibrium of the system, at a constant temperature and pressure in all parts, may now be completely characterised by two sets of relations : GENEEAL THEORY OF MIXTURES AND SOLUTIONS 411 (i.) In each separate phase the chemical potentials of the components A b A 2 , A 3 , . . . A re must satisfy the equation : V\P\ + V-2p2 + + v nPn = (1) where j-iAi -f i- 2 A 2 + . . + z'A w = is the chemical equation representing a possible reaction between the various components (cf. 143, 155). (ii.) The chemical potentials of each component must be the same in all the r phases of the system : Pi = Pi" = pi" = = pi r \ p2 = p2 n ' = PS" = = p2 r \ _ (2) Pn = Pn" = Pn" = . . . = Pn' where the indices denote the various phases. If the system undergoes a virtual change of composition, B, at constant temperature and pressure, it is transferred to another state, which need not however be an equilibrium state, since the changes of the masses are quite arbitrary. They must, however, satisfy the equations : = Snii -j- Bmi" -\- = 8/ 2 ' 8;w a " . . . (6) since the total mass of each component, no matter how it is distributed, must remain constant. We have also : ST = ; Bp= ... (4) Since the initial system was by hypothesis an equilibrium state the variation of potential in the virtual change must vanish to the first order : S = ...... (5) and be positive to the second order if the equilibrium is stable : 8 2 <>0 ..... (6) Now suppose that an actual change, in which the system is transferred from one equilibrium state to another infinitely close equilibrium state, occurs, and let us denote this real change by d to distinguish it from the virtual change B. The temperature 412 THEKMODYNAMICS and pressure will also change along with the composition so as to keep the system in equilibrium. Since the displaced system is also in equilibrium, we have, for a virtual displacement from it : 8(0 + , T, s', s". Now 5/V = l S/in' + 9^1 Sm/| . . (12) and similarly for the second phase. For the given system there are two kinds of virtual change possible, according as the first or second component passes from the first to the second phase. In the first case : Snii = Bmi" ; Sw. 2 ' = & a " = and in the second case 8ni i Bmi" = ; 8)11-2' = 81)1-2". Hence, in the first case : W and in the second case : 8 W2 "' - S*a" = . ._ 77 = l 1 5^ - T/ * 2 414 THERMODYNAMICS for the amounts of heat absorbed, and the increases of volume per unit transfer of the first or second component, respectively, from the first to the second phase, .'. 4 l dl - rrfp + (V - S/ii") = 0) ... (13) y rfT - vydp + (S/*a' - S/V') = OJ We now calculate the values of the expressions in brackets. From (12) : To change the independent variables ? x ', ??ia', ?n/', 2 " to 5', " we have the relations (11) and the following, derived from them by partial differentiation : from which we readily find : **"* ! = ^-^ & (U) In 157 we have obtained the equations : + 8 '==+ 8 "= and if the equilibrium is stable, the inequalities : --<>or3ap-.-5> s 9s" s" . (16) . . (17) GENERAL THEORY OF MIXTURES AND SOLUTIONS 415 Equations (13) now go over into : 2i dT vjdp + = r - - - > or T- ~~ jfr' - / - (80) ds s n + *"r a dp To- *' " ^.= -^! *' ~ *" ->r d *" = J- ri + ' V M To-" s' s" by a similar process of elimination. At constant temperature, p is a function of s' or s" alone. In all cases we have the inequalities : a-' < 0, a" < Qi>0, Q 2 >0 hence from the above equations we can readily deduce the following theorems : (1) Constant Temperature : (a) If the composition of the liquid is altered so that the con- centration of a component is increased, the concentration of that component is also increased in the vapour. (b) If to the liquid a portion of that substance is added which the vapour contains in the greater proportion, the total vapour pressure is increased, and inversely. (c) The pressure has a maximum or minimum value when the liquid and vapour have the same composition. (Theorem of Gibbs and Konowalow.) (2) Constant Pressure : (a) If the concentration of the liquid is increased, so also is that of the vapour. (b) If to the liquid a portion of that substance is added which it contains in less proportion than the vapour, the boiling-point is lowered. (c) The boiling-point is a maximum or minimum when the liquid and vapour have the same composition. (d) An increase of temperature without alteration of composi- tion of the liquid raises the total pressure. (e) An increase of temperature without alteration of composition of the vapour raises the total pressure. The various relations have already been described ( 163, 167). GENERAL THEOEY OF MIXTURES AND SOLUTIONS 417 179. Isomorphous Mixtures; Solid Solutions. The relations apply also to the case of a liquid mixture of two substances which is solidifying to a homogeneous solid which contains the two substances in proportions depending on the composition of the melt a so-called solid solution or mixed crystal ( 138). If two salts which do not react chemically to produce a double salt (e.g., K 2 S0 4 and A1 2 (S0 4 ) 3 ), or another salt-pair (e.g., NaCl and CuS0 4 ) are brought in contact with a quantity of solvent insufficient for complete dissolution, the composition of the solution is independent of the proportions of the two solids and is definite at a fixed temperature, as we see from the phase rule : F = w + 2-r = S + 2-S = 2 .'. when T and p are fixed, so is the concentration. This was verified by Riidorff (1873) with a large number of salt-pairs ; e.g., NH 4 C1 + NH 4 N0 3 , KC1 + XaCl, NH 4 C1 + BaCL 2 , Na 2 S0 4 + CuS0 4 . With other salt-pairs, e.g., K 2 S0 4 + (NH 4 ) 2 S0 4 , Ba(N0 3 ) 2 + Pb(N0 3 ) 2 , CuS0 4 + FeS0 4 , the concentration was not uniquely determined by the temperature, but depended on the relative amounts of the two salts. It was observed that the members of all such salt-pairs were isornorphous, and it was shown by Roozeboom (1891) that the apparent contradiction vanishes when the solid is regarded as a homogeneous crystal con- taining both constituents in varying proportions an isomorplious mixture. Then r = 2 and .'. F = 3. These two processes solidification of a melt and crystallisation from a solvent, are the most important cases in wbich solid solutions appear. The theory of dilute solid solutions, in which one component is present in small amount only, has already been considered ( 138). The general theory was worked out by Roozeboom (Zeitschr. physik. Chem., 1899) from the standpoint of the theory of thermo- dynamic potential. The equations (2a, I), (3, b) of the preceding section apply equally well to the present case, and details need not be given here. The liquid solidifies at a constant tempera- ture when it has the same composition as the solid deposited the so-called eutectic point. The description of the various types of freezing-point curves 418 THEKMODYNAMICS which occur a subject of great importance in the study of metallic alloys will be found in text-books on the Phase Kule, e.g., Findlay, Phase Ride, 2nd edit, 1906, pp. 173 et seq. (cf. also Desch, Metallography, 1910). 189. Freezing-Points of Solutions. Let the system consist of a binary liquid solution in equili- brium with the solid form of one substance in the pure state. We have then, if the doubly-accented symbols refer to the solid : *" = 0, ih" = The second equation (186) of 178 then vanishes, and the first takes the form : fi dT _ i-dp + crtfo = . .' . (1) in which the suffixes and accents are omitted. According as we put <7T, dp, or d equal to zero, we have the equations representing the alteration of pressure required to keep a solution of altered concentration in equilibrium with ice at the same temperature, or the alteration of freezing-point with con- centration, or the alteration of freezing-point of a given solution with pressure, respectively. Similar equations apply when the solid is the pure solid solute, e.g., a salt along with its saturated solution. The most important case is the alteration of freezing-point with concentration at constant pressure, when : '3T\ . To- -- In general Q < 0, i.e., heat is evolved when ice crystallises out ; then T falls with increasing s. In the case of strong solutions of sulphuric acid, Q may be positive, and the freezing-point would rise with increasing concentration. If the temperature is constant : then GENERAL THEORY OF MIXTURES AND SOLUTIONS 419 ThU8 ef and c have alwavs PPOsite signs, hence if a solution is in equilibrium with ice at a given temperature and pressure, and if the pressure is increased, it will be necessary, in order to maintain equilibrium, to |^^ the concentration of the solu- tion according as the crystallisation of ice from the original solution | ^^^ the total volume. There do not appear to be any experiments on this subject. The equations just obtained, and those relating to vapour pressures, are quite general and apply to solutions of any con- centration. Unfortunately we are not yet in a position to calculate the magnitudes o-', cr" in the general case, although we have seen in 158 that the form of the chemical potential a, and hence ^ = -p as a function of concentration, is known when the latter is very small. J. J. van Laar (Seeks Vortraq? iibcr das thermodynam. Potential) has also worked out the theory of vapour-pressure and freezing-point curves on the assumption that the mixtures conform to van der Waals' theory of binary mixtures, according to which the mixture obeys the characteristic equation : in which a = (1 x where x, 1 x are the numerical molecular concentrations and the magnitudes ai, CM, 02, &i, & 2 , are constants. He makes^ however, certain assumptions with respect to the specific heats' which do not appear to be justified, and we shall not enter into detail here. In the development of the theory of freezing-points for very dilute solutions (Chap. XL) it was assumed that both the change of total volume and the heat absorption on further dilution are zero. With solutions of moderate concentration (say up to 5N), neither assumption is true, but we know that the change of volume is always very small, and_possibly negligible, whilst the heat absorption is not usually of such small magnitude. It E E 2 420 THERMODYNAMICS therefore appears that a theory developed on the assumption that the volume changes are negligible, but which takes account of the absorption of heat, will very probably agree fairly well with the relations actually observed. Such a theory was developed simultaneously and independently by Dieterici and T. Ewan (1894), and may be called the theory of the freezing points of solutions of moderate concentration. If the solution is in equilibrium with pure frozen solvent at the temperature T we have : ... . (2) where the symbols have the significance of 131. If TO is the freezing-point of the pure solvent, (1) and (2) are integrable on the assumption that A g , A e are constant only if (T T) is small. If this is not the case, i.e., the solution is of moderate concentration, we must write : A, = (A,) O + J (c, - c>rr = (A.) + cc, - c.)T . (3) A p = (A.) + J (C, - C,)rfT = (A e ) + (C, - C,)T . (4) where C s , C/, G p are the molecular heats of solid, liquid, and gaseous solvent under constant pressure, and these are assumed independent of temperature over the range (To T) (cf. below, where it is seen that the effect of temperature below T is not involved). Equations (1), (2) are again integrable : The arbitrary constants Ii, I 2 are eliminated by the relation p' = p = po when T = T GENERAL THEORY OF MIXTURES AND SOLUTIONS 421 . ' . from (5) (8) we find : Put (T T) = t the depression of freezing-point, expand T into a power series, and retain the first three terms : "'r = '"( 1 + f)=f- 2 -i + 8T3 . - . (10) From (9) and (10) : . - L r( A )o-( A *)o+(c<-c,)To _ d- c. * p' RL ToT 2 T 2 (ID But (A g ) - (A e )o + (C, - C,) To = (A r ) + (C, - C.) To = A,o. . (12) the latent heat of fusion at T , . ln P _ i r A,? _ c, -o. j c,-c. * 2 n Q3 y " R LT O T 2 T 2 ^ ~~3 f 3 J The equation for the osmotic pressure .... (14) deduced in 131 is true when there is no change of total volume on mixing the solution with extra solvent, a condition which is very approximately fulfilled even with fairly concentrated solutions, hence we find for the osmotic pressure at the freezing- point T : P _,rv d-c, * , c, -c, fi ( . [To" ~2~ "T 4 ~3~ T" a J ' * To calculate P at any other temperature we integrate KirchhofF s equation for the heat of dilution : & ..... < 16 > As a first assumption we take Q independent of temperature, i.e., we suppose that the same amount of heat is absorbed when a 422 THERMODYNAMICS mol of solvent is mixed with the given solution at all temperatures in the range T T. Then : If T = To : Add (13) to (17) and subtract (18) : whence the osmotic pressure at TO, the freezing-point of the pure solvent, is : P _ Ewan took account of the variability of Q with temperature, which introduces the specific heat of the solution ; this is usually quite negligible within the range of validity of (14). \^-\- Q is the heat absorbed when a mol of ice melts and the liquid mingles with the solution. If we take Dieterici's numbers for the vapour pressures, and Roloffs for the freezing-points, of solutions of potassium chloride, we can calculate the osmotic pressure (Po) from the two equations : (a) P = RT In -, (vapour pressures). \ 2 ) ' (b) p. = r [*A+_^=C.^+C.rC. M,-| ((l . eezingpoints) . In the table, m = grams KC1 per 100 grams solvent, t = TO T = depression of freezing-point, Q = heat of dilution, p' = vapour pressure of solution in mm. Hg at C. GENERAL THEORY OF MIXTURES AND SOLUTIONS 423 Also, p = 4*620 mm. = vapour pressure of water at C. To = 273, d C s = 18 X 0-475 = 8'55 cal., A /0 = 18 X 80'3 = 1445-4 cal. R = 1-985. . P Calories. m. t Q. P*. Gale. (a). Calc. (6). 4-620 _ _ 3-72 1-667 1-63 4-546 8-805 8-80 7-45 3-284 5-96 4-472 17-645 17-55 14-90 6-53 19-5 4-326 35-595 35-18 22-35 9-69 34-3 4-190 52-905 52-64 181. Graphical Representation. The treatment of systems in which hydrates (or compounds), double salts, etc., are deposited, or of ternary systems, proceeds on the same lines as the investigation of the simpler cases considered in the preceding sections. The equations, especially in the case of ternary systems, are necessarily more complicated, but nothing fundamentally new appears. We shall therefore omit all the analytical theory of such cases. There are, however, two graphical methods which have been largely used, arid since the relations involved are quite simple, a short description of one of them may be given here. The funda- mental theorems underlying both are contained in Gibbs's memoir, where a short account of the graphical representation was also given. The latter was then extended on the one side by van Rijn van Alkemade (1893) and Roozeboom (1899), and on the other side by van der Waals (1891). The theorems assert that a system of coexisting phases at a given temperature and pressure s ^ r j ves ^ a ^ain a state of equilibrium in which the I volume rthermodynamic potentials Qf ^ te hases haye fch \free energies ^ least possible sum. 424 THERMODYNAMICS The consideration of leads to the potential diagrams, that of i//- to the free energy surface. 182. The Potential Diagrams. We consider for simplicity systems of two components, and take as unit quantity of a phase that containing x mols of the first component A, and (1 x) mols of the second component, B, i.e., 1 mol in all. We take a horizontal axis, and erect on it two perpendicular ordinates at unit distance apart. If compositions are taken as abscissae, potentials as ordinates, the potential and composition of every possible phase will be represented by a point in the plane between the two parallel ordinates -r = and x = 1. If we join the points representing two such phases by a straight line, and if we imagine 1 mol of a hetero- geneous complex formed of p mols of the first and q mols of the second phase, the complex will be repre- sented by a point on the straight line with the co-ordinates. = p xi + qx 2 where x\, x% are the compositions, and $1, $ 2 the potentials per mol, of the first and second phases. (x, ) is the centre of gravity of masses p, q placed at the points (#i i)> (#2, 2)> respectively. Different systems composed of ,rA + (1 #)B are represented by points on a vertical ; one system can pass spontaneously into another lying below it, and the stability is indicated by the relative height of the point. The most stable system (< an absolute minimum) is represented by the lowest point. Now consider the case of two substances forming a homo- geneous solution, say water and an anhydrous salt (NaCl), and let us draw the potential curve for an assigned temperature and GENERAL THEORY OF MIXTURES AND SOLUTIONS 425 pressure. Let a, ft be the values of per mol of liquid A and B, respectively, at the given temperature and pressure (the salt must usually be regarded as existing in a supercooled state). The shape of the curve in the vicinity of a and /3 is determined by Gibbs's expression for the chemical potential of a component in a very dilute solution ( 158). = yu, B = L -f- InM. x where L, M are functions of T and p. For x = 0, ^ = oo , so that the curve touches the < axis at a. Similarly at /3. The potential is a continuous function of the composition, hence the curve must have the form shown in Fig. 80. The stability of a solution is determined by supposing it to separate into two phases represented by two points on the curve on opposite sides of the point P on the curve representing the solu- tion. The value of for the hetero- geneous complex is the ordinate of the point of intersection P' of the vertical through P with the join of the two points, and the homogeneous solution is stable or not according as P lies below or above P', i.e., according as the curve is convex or concave to the x axis in the vicinity of P. If a point A is taken on the < axis to represent the potential of a mol of solid salt, and a point B to represent the potential of a mol of ice, the ordinates of the points of intersection of straight lines from A and B to the curve, with the latter, represent the potentials of the heterogeneous systems made up of salt and solution, or ice and solution, in the proportions represented by the ab'scissse. The tangents from A and B to the curve correspond with the least values of , and the points of contact denote solutions which exist in equilibrium with solid salt or with ice, respectively, at the given temperature and pressure. If the two tangents coalesce, FlG 80 426 THERMODYNAMICS we have ice, salt, and solution in equilibrium at a given tempera- ture and pressure the so-called eutectic point. If the points A and B lie above the points a and /3 respectively, no tangents can be drawn, and the solution cannot, at the given temperature and pressure, exist in equilibrium with salt or ice, respectively. Thus, the point B lies above the point for all temperatures above 273 for aqueous solutions, since ice can never coexist with a solution if the temperature is greater than the freezing-point of the pure solvent at the assigned pressure. The points A and a, and B and ft, coincide at the melting points of the solid salt and ice respectively. The form of the curve, and the altitude of A and B change with rise of temperature. Now consider two coexisting liquid phases, such as those formed from water with ether, phenol, benzoic acid, or salicylic acid. For the homogeneous solu- tions the curve is convex, and the separa- tion into two layers therefore implies that at some intermediate part (cf. 115) the curve is concave, whilst for x = and x I the course of the curve must be that indicated above. The complete curve has therefore the form shown in Fig. 81. All heterogeneous systems lie on the double tangent PQ ; P and Q represent the two liquid phases coexisting at the given tem- perature and pressure. All points on the curve between P and Q, represent labile homogeneous states. From each of the points, A and B, two tangents may in general be drawn to the curve, so that solid salt or ice maybe in equilibrium with either of two solutions of quite different composition at a given temperature and pres- sure. With change of temperature, the positions of A and a, and of B and ft, change, and at a certain temperature the tangent from A or B may touch both P and Q simultaneously. There is then equilibrium between three phases : two solutions and solid A or B. Thus, at 98 C. solid benzoic acid, a very concentrated solution, and a solution of moderate concentration are in equili- brium. (This is the so-called "melting-point under water;" GENERAL THEORY OF MIXTURES AND SOLUTIONS 427 in reality the "fused acid" is the very concentrated solution, represented by P.) If P and Q coalesce with rise of temperature, both liquid phases become completely miscible, and we have reached the " critical solution temperature " ( 174). Lastly, consider the equilibria of solutions with a solid hydrate. The latter will have a characteristic potential represented by a point H in the plane. All points on the two tangents HRi, HR 2 , to the curve of solutions represent heterogeneous systems composed of solid hydrate in contact with solutions. If the curve between R! and R 2 is convex the heterogeneous systems are stable, and inversely. At a given temperature and pressure the hydrate can be in equilibrium with two liquid phases of dif- ferent composition, one containing relatively more, the other relatively less, salt than the hydrate. With rise of temperature the form of the curve and the altitude of H change ; H approaches the curve, and at a certain temperature meets it. When this occurs, the hydrate is in equilibrium with one solu- tion of its own composition, and the tem- perature is the melting-point of the hydrate. At higher temperatures no equilibrium exists. From these considerations, the general form of the concentration-temperature curve is easily deduced. It is cut in two points by a perpendicular to the T-axis and is bounded on the side of higher temperature by a tangent line also perpendicular to the tempera- ture axis. The abscissa of this point of contact denotes a maximum temperature which is the melting-point of the hydrate, and its ordinate represents the composition. The curves of the hydrates of ferric chloride, investigated by Roozeboom, afford an excellent illustration of this type. According to Le Chatelier (1889) the maximum on the freezing-point curve corresponding with the deposition of a chemical compound is the intersection of the two curves which represent the lowering of freezing-point of each component by addition of the other. The tangent therefore changes its inclination to the composition axis discontinuously on passing the maxi- mum, and the gradient of the tangent at that point is indefinite. Roozeboom, however (1889), insisted that the tangent at the maximum is parallel to the composition axis. Kiister and Kremann (1904) pointed out that the part of FIG. 82. 428 THERMODYNAMICS the curve in the vicinity of the separation of the compound is a flat maximum when the compound is in dissociation equilibrium with its components. Then by addition of either component the dissociation is forced back and the melting-point is lowered to an extent less than that calculated from the per- centage of added substance. The more dissociated is the compound in the fused state, the flatter and more extensive will be the maximum. If the compound is wholly undissociated in the fused state, the two curves meet each other sharply. The first case is observed with most mixtures, the latter with pyridine and methyl alcohol (Aten, 1905). From the shape of the maximum, some information can be obtained as to the extent of dissociation of the compound in the liquid state; thus the maximum on the freezing-point curve of CaCl 2 and H 2 O is very broad and flat, an indication of the almost complete dissociation of the compound CaCl2 . 6H 2 in the fused condition (Eoozeboom and Aten, 1905 ; Kremann, 1906). The same methods apply also to ternary systems, the only difference being the use of a third axis in the definition of the composition, and the consequent substitution of surfaces for curves. 183. Liquefaction and Evaporation of Mixtures. If a gaseous mixture of 0'41 vols. carbon dioxide with 0*59 vols % chloromethyl is compressed, whilst the whole mass is kept agitated by a stirrer, a peculiar series of changes can be observed (Kuenen, 1892). After a certain point, a droplet of liquid appears, and this increases in volume with the pressure, more and more of the mixture becoming liquefied. Contrary to what one would expect at first sight to find, the liquefaction does not go on progressively till all the vapour has disappeared, but the quantity of liquid reaches a maximum, then decreases, and at a still higher pressure vanishes, so that the system is again wholly gaseous. With further increase of pressure, liquid again appears, and finally the whole is liquefied. The thermodynamic theory of this so-called retrograde condensation will be found in Duhem's Traite, t. IV., pp. 109 156, and in Kuenen's valuable monograph: Vcrdampfuny und Verflussigung von Gemisclien. CHAPTER XV CAPILLARITY AND ADSORPTION 184. Surface Tension. The effect of surface energy on the properties of heterogeneous systems has been considered in 101. A surface possessing an amount of surface energy 2 per unit area behaves as though subjected to a tension which acts tangentially to the surface at every point, and tends to separate two portions of the surface meeting in a line of length I by a force o7 ; the force per unit length is a-, and this is called the surface tension. It is easily shown, by considering a displacement of one side of a rectangular element of the surface, that : or that the surface energy per unit area is numerically equal to the surface tension per unit length. The dimensions of surface tension are therefore : For the majority of liquids a- varies from 20 100 . The surface tension of a film is independent of the area of the film, so that the latter may be stretched to any extent such that its thickness does not fall below a certain limit without altering the value of a. The limit (about 10 ~ 8 centimetres) is the range of action of the molecular forces. In the older theory of capillary action (1 ', developed by Laplace T. Young (1805), Gauss (1830), and Poisson (1831), no attention was paid to the possibility of thermal changes attending the alteration of surface at constant temperature. That such changes must exist was first demonstrated by Lord Kelvin (2) (1859), and the theory of capillarity was developed more parti- cularly from the thermodynamic standpoint in the masterly treatise of Willard Gibbs (3) (1876). 430 THERMODYNAMICS Let us consider a surface film (say a soap film, which however is composed of two films) of area o> and having a surface tension a- (T) at the temperature T. If the surface is increased by do> the work done is : 8Ai = - adat (rfw > 0) since the work is positive when to decreases. If the temperature of the film is lowered to T dT at constant (o, the surface tension becomes (a -^ dT), and if the film is now allowed to shrink to its original area the work done is : SA 2 = + (a- - ~ dT) <2u ' . . . (dw < 0). The work done in the cycle is (8A) = 8A 2 + SAi = - ^ dT do> . .' . . (1) Unless -jm = 0, this will differ from zero, and by the equa- c t i tion of maximum work ( 58) if l u d-a> is the heat absorbed in stretching the film : an equation analogous to that of Clapeyron ( 63). l u is the latent heat of extension of the film. In all cases it is known from experiment that the surface tension diminishes with increase of temperature of the surface. (Bede has shown that this is the case even when the fluids in bulk preserve their original temperature.) Corollary 1. A surface film absorbs heat when extended. Example. In the case of water (soap-tiltns) it is found that at the ordinary temperature d, the element of heat absorbed is : 8Q = 1836) : :. ^ = const., or ^ = . . ( i ) /. o- = A + BT. Since o- = when T =T A = BT. A very remarkable theorem respecthig the constant B was found empirically by Eotvos (5) .(1886). Let a mol of a liquid 482 THEKMODYNAMICS be formed into a sphere in contact with the second fluid used in the determination of a- (usually air). This will have a surface S,,,, and if V MI is the molecular volume : 2 OT (1893) showed that equation (12) is not strictly accurate ; they replaced it by . y| = -ft (r -t) . . J." . . (13) where T = T K T and e is usually about 6. Even now the equation is valid only if T is greater than 35, i.e., T is fairly widely removed from the critical point. The mean value of />; for a large number of liquids is 2*12 ; liquids containing hydroxyl groups, or those which have an enolic modification, liquid chlorine, fused salts, and metals, have much smaller values of k, and this is attributed to polymerisation, since if the actual molecular weight is larger than that assumed in calculating Vl, the latter value will also be smaller. Ramsay and Aston (1894) assumed that it will be proportionally smaller, but the values of the polymerisation coefficients thereby deduced obviously depend on the formula assumed for the polymerised molecule, and are therefore arbitrary. Again, if we differentiate (2) with respect to T and compare with (6) and (10) we find, in the case of states far removed from the critical state : ^ _ L d L? + ' _ I 8( '" - o (1834), that the adsorbed gas is sometimes present in the liquid state. The adsorbed amount increases with the pressure and diminishes with rise of temperature. The first effect does not follow a law of simple proportionality, as in the case of the absorption of gases by liquids, rather the adsorbed amount does not increase so rapidly, and the equation : i = v* . . . . a) where x = total mass adsorbed on the surface m p = pressure a, n = constants holds good (Freundlich ( 10 ^ 1906). If n = 1, the adsorbed amount would be proportional to the pressure; in adsorption phenomena n > 1. In the case of solutions, if we define the extent of adsorption in the same way, the equation = <* ' r..' applies, where is the concentration of the solution (Freundlich CAPILLARITY AND ADSORPTION 435 1906). u is always greater than unity ; it may reach the value 1012. It is of course assumed that no chemical reaction occurs between the adsorbent and the substance adsorbed; if this is the case (as may happen, for example, in the taking up of a dye by a fibre), the equation (2) no longer holds good, and the solute may be practically completely withdrawn at all concentra- tions. If we suppose that a solution A of concentration is in con- tact with an immiscible substance B (say air, or petroleum) over a surface S, there will be a different concentration of the solute in the immediate vicinity of S from that in the free bulk of A. This generalisation is due to Gibbs (1874), who at the same time showed how quantitative relations could be found. We express the altered concentration in terms of the adsorption excess. If all the adsorbed substance were contained to the extent of k gr. per cm. 2 on a superficial layer of zero thickness and surface co, the total mass present in the volume Y would be m = Y + /,<>. The layer of altered concentration must, however, have a certain thickness. We will there- fore imagine a plate 2 placed in front of the surface and parallel to it, and define the adsorption excess as the concentration in the included layer minus the concentration in the free liquid. That this result is independent of the arbitrarily chosen thickness is easily proved when we remember that the problem is exactlv the same as that of finding the change of concentration around an electrode in the determination of the transport number of an ion by Hittorf's method. Let us suppose a quantity w gr. of solute to wander into a layer of thickness S parallel to the adsorbing surface. If the area of the latter is eo, the adsorption excess is defined as f , i~C; the excess per cm. 2 of surface. The layer B we shall call the true adsorption thickness ; the liquid beyond S, which has almost exactly the original concen- tration provided a large volume was present at the start and T is small, we shall call the free liquid. Now suppose a F F 2 436 THERMODYNAMICS layer of thickness d is isolated, and the liquid contained in it mixed and analysed. The layer d we shall call the arbitrary adsorption thickness. If is the concentration of the free liquid, the amount of solute in the arbitrary layer is / -f- -, whereas that in a similar layer of free liquid is */. The excess is ir, and the excess per cm. 2 is -, i.e., T. a a' 186. Gibbs's Adsorption Formula. In his original demonstration Gibbs (1874) showed that the surface layer may be considered as a third phase having specific values of density, energy, and entropy, and further that the results of the theory are quite independent of the actual extent of the capillary layer and the way in which it merges into the free fluids on either side. As a matter of fact, the transition of density, etc., probably occurs continuously but rapidly over a very small distance (van der Waals). Let us suppose that we have a solution A in contact at one side with a surface of adsorp- tion ab separating it from another phase B which, for simplicity, we shall first take to be the vapour of the solvent. At the other side the solution is in contact with pure liquid solvent C through a semipermeable piston c, exposed to an osmotic pres- sure P. Let the volume of the solution be V, its concentration , and let the area of the adsorbing surface ab be -}- do) whilst V remains constant (say by alteration of form to a'b'). The work done is &Ai = a da> where o- is the surface tension at the interface. At the same time the osmotic pressure changes from P FlG (ii.) Let the volume of the solution be increased from V to CAPILLARITY AND ADSORPTION 437 y _j_ According as is = there will be positive, zero, or negative adsorption, respectively. The method of deduction also applies when B is a liquid, or a solid, and (6) therefore holds for these cases. The equation (6) is called Gibbs's Adsorption Formula ; it was deduced independently by J. J. Thomson <> (1888). The present deduction is due to Milner W (1907). 187. Experimental Examination of Gibbs's Adsorption Formula. On account of the very great difficulty of measuring the extremely small amounts of adsorbed substance at a liquid/yets or liquid/liquid interface, very few experiments are available for testing Gibbs's equation. Zawidski (13) (1900) pointed out that the concentration of the foam of a solution should be different from that of the latter in bulk, and Miss Benson (14) (1903) by the analysis of a solution of amyl alcohol in water found in the foam 0'0394 mol per litre liquid 0-0375 .*. difference = 0'0019 A quantitative examination of this case has been made by R. Milner (1907), W. C. McC. Lewis, ^ and Donnan and Barker < 16) (1911). The latter passed air bubbles up a tube containing an CAPILLARITY AND ADSORPTION 439 aqueous solution of nonylic acid, or of saponin. Each bubble carried with it a very slight alteration of concentration, and after a time the concentrations at the top and bottom of the tube were different. F (nonylic acid) = 1 x 10~ 7 (agrees with calcd.). T (saponin) = 4 X 1(T 7 - (twice as great as calcd.). cm* In the case of liquid /liquid interfaces we have the experiments of W. C. McC. Lewis (1908), who examined the relations at the surface of separation between an aqueous solution and paraffin oil or mercury. If o-, a-' are the surface tensions between paraffin oil and pure water and the solution, respectively, it was found that o-'< o-, i.e., the substances examined, were positively adsorbed. The adsorption excesses of the ions were calculated according to 203. Substances were divided into three classes according as the adsorption excess F -^-^ was about 60 times (Na glycocholate cm. , r = 7 X 10" s calc. ; 5 X 10~ ;; obs.), or 510 times (Ag 4-5 X 10" 9 calc. ; 2'5 X 10~ 8 obs. ; in AgN0 3 ), or about equal (Caffeine 2'4 X 10~ 8 calc. ; 3'7 X 10~ s obs.) to the calculated. The deviations are all on the same side, viz., the actual adsorbed amount is too great. Further investigation will probably show what is the cause of this phenomenon. If we now enquire what sort of influence different solutes have on the surface ten- sion of a liquid in contact with its vapour, i.e., the magnitude of ~r, we find that, in dc the case of water, solutes may be divided into two classes, the members of which exert either a very strong influence, or a very slight influence, respectively, on the surface tension. The former, called active substances, include the halogens, fatty acids (especially higher members), alcohols, aldehydes, amines, and esters. ;17) Thus, in a 0*00079 normal solution of nonylic acid the surface tension of water is lowered from 75*3 to 40. The other class of inactive substances includes the salts of inorganic acids. FIG. 440 THERMODYNAMICS It is curious that active substances are usually strongly smelling. In the typical diagram w r e see that the addition of B strongly lowers the surface tension of A (Fig. 85). The adsorption equation shows that a solute may very strongly lower the surface tension of a solvent, but cannot strongly raise it, since although F may reach high values by positive adsorption (in some cases, as with solutions of some aniline dyes, the pure solute appears as a thin skin on the surface), it can never sink below that of the pure solvent by negative adsorption. In the case of inactive substances the difference between the tensions of solution and solvent is very nearly proportional to the concentration ( 18 ^ : With active substances it is proportional to some power of the concentration ^ : S W **9 . \^J where *, - are constants. The value of - is very similar for n n different substances. If we combine (2) with Gibbs's equation we obtain Freundlich's adsorption equation ( 185). The tension of a solution is lowered with rise of temperature according to a linear law : as early as 1814. According to Baerwald <-> (1907) if A, B, C . . . denote different gases, it appears that the relation 'L'WCU. (-)=(-) \a 1 4 \a 1 (3) holds as a limiting case, where a', a" are the constants for charcoal and glass respectively. The same relation has been observed in adsorption from liquid solutions, but is not absolute. It is to be observed that the value of ~ is the determining factor in adsorption ; if this is small a large increase of surface will produce very little increase of adsorbed amount a pheno- 442 THERMODYNAMICS menon which has given rise to the erroneous view that adsorption must always be a chemical combination. In the adsorption of a gas on a liquid, in which it may be more or less soluble, we may replace the concentration by the partial pressure, since the amount dissolved is proportional to the latter: _ Rf dp (4) In the adsorption of gases on solids the constants a and - decrease and increase, respectively, with rise of temperature. C0 2 on Charcoal (Travers <- 3) 1906) 6 a , n - 78 14-29 0-133 2-96 0-333 + 35 1-236 0-461 61 0-721 0-479 100 0-324 0-518 In considering the effect of temperature we have two important cases : (1) p is kept constant and the adsorbed amounts at different temperatures are compared. The curves , T ) are called \m ' Isopncuma (Ostwald). (21 x in (2) - is kept constant and the pressures at different temperatures are compared. The curves (p, T)? are called Isosteres. We first consider the isopneuma. It has been found experi- mentally that, for a given pressure p { : (5) CAPILLARITY AND ADSORPTION 443 where 5 is a decreasing function of pressure : / J W (*) = *(*)- (5 -#ni>)0. . . (6) \m/ e vTw/ or * = -e- *-** . . . (6a) \w/ \m/ o In the case of isosteres there is no such simple relation. We now differentiate the adsorption equation (2) with respect The equation of an isopneumon is -^ = Now differentiate (6) with respect to at constant p : - But In at = Ina 5# from (6) : .'. from (11) and (13) 75 Thence da /-, n \ ^ s. do) (ID _ ' ~W ~ The change of In ' is composed additively of the two changes of Ina and - with temperature. ] / X \ The equation of an isostere is ( .1" = 0. 444 THERMODYNAMICS Divide through by p, and substitute the values of ( -^ and *<*), d0 /7 7/i 111 . . . . (12) Thus -*- [ a no constant ; it increases with falling tempera- du ture or pressure. j0- is independent of the nature of the solid adsorbent over wide intervals. 189. Heat of Adsorption. It was observed by de Saussure in 1814 that heat is evolved during the adsorption of gases on charcoal, and quantitative measurements were made by Favre (24) (1874), Chappuis (1883), and Dewar < 25 > (1904). According to the circumstances attending the adsorption we may have different heats of adsorption : (1) Integral Heat of Adsorption corresponding with a heat of solution, and evolved when the gas is brought in contact with just enough adsorbent to take it up. (2) Differential Heats of Adsorption corresponding with heats of evaporation ( 173), and evolved or absorbed when one equili- brium state |jPi>(~) I is transferred to another Ipa, ( ) The change may be effected : (a) laoaterically one phase has an unaltered composition and the pressure in the other varies r const. ; p. T variable ) . The \m I heat of adsorption is analogous to the heat of evaporation of a mixture. (b) Isopneumically the composition of one phase alters whilst the pressure in the other remains practically constant p, T const. ; variable). The heat of adsorption corresponds with the heat of reaction in a condensed system. The isosteric heat of adsorption is determined by the Clapeyron- CAPILLARITY AND ADSORPTION 445 Clausius equation. If the initial conditions are T, - , p, and the final conditions T + dT, , p + dp, it is readily proved that : Heat of adsorption per mol = q = RT 2 ( j^ But ^ = ,,(5-. No researches with the proper conditions are available. The heat of adsorption is often greater than the latent heat of evapora- tion (or even than the latent heat of sublimation) of the gas in the liquid (or solid) state. Thus for 1 mol XH 3 on charcoal : heat of adsorption = 8100 cal. heat of sublimation = 5000 cal. It may be that the liquid layer is strongly compressed, when it would have a higher vapour pressure and heat of evaporation. Similar gases have similar heats of adsorption (C02, XH 3 S0 2 , CHC1 3 ); difficultly liquefiable gases have much smaller values. 190. Wetting. If a drop of liquid is placed on a solid, the condition that it spreads over and wets the latter is : (T (s _ g] > (1865) and Schwalbe < 2S > (1905) made the remarkable observation that the heat evolved on wetting a solid with water is positive below 4C. and negative 446 THERMODYNAMICS above that temperature. It is therefore probably connected with the heat of compression /, and arises from the strong attraction of the solid for the liquid. 191. Characteristics of Adsorption Phenomena. According to Freundlich ( - 9) adsorption equilibria may be dis- tinguished from chemical equilibria by the following peculiari- ties : (1) The conformity to the Adsorption Isotherm : = ag\ (2) The slight variability of with very different substances. (3) The slight variability of a with the nature of the solid phase. (4) The very large velocity of attainment of equilibrium. (5) The relatively small displacement with temperature. 192. The Phase Rule and Dispersed Systems. Modern experiment has proved beyond doubt that the so- called colloidal solutions are systems composed of two or more phases, i.e., heterogeneous, characterised by an enormously great extent of division, in which the surface of contact has, so to speak, been spread out throughout the whole mass. Capil- lary phenomena are therefore predominant here (cf. Ostwald, Kolloidchemie, Leipzig, 1909 ; Freundlich, Kapillarchemie, Leipzig, 1909). A surface of separation between two phases is called a specific surf ace of separation, and in considering the states of such systems it is evident that every specific surface constitutes a new inde- pendent variable. If there are n components in ; phases with x specific surfaces, the Phase Rule will therefore read : F = 2 + n + x - r A classification of dispersed systems on this basis has been worked out by Pawlow (30) (1910), who introduces a new variable called " the concentration of the dispersed phase," i.e., the ratio of the masses of the two constituents of an emulsion, etc. When the dispersed phase is finely divided the thermodynamic potential is a homogeneous function of zero degree in respect of this concentration. CAPILLARITY AND ADSORPTION 447 193. Surface Tension and Solubility. The influence of division on solubility has been mentioned in 137. According to Hulett if A, A w are the solubilities of the substance in lumps and in grains of radius r respectively, A pr where p = density, a- = tension at solid/liquid interface (ef. Freundlich, Kapittarchemie). 194. Influence of Capillarity on Chemical Equilibrium. It can be shown, (Gibbs, Scientific Papers, I. ; J. J. Thomson, Applications of Dynamics to Pliysics and Chemistry), that a chemical equilibrium can be modified by the action of capillary forces. Thus, a state of equilibrium in solution rua}* conceivably be modified if the latter is in the form of thin films, such as soap bubbles. Since, according to Freundlich (Kapillarchemie, 116), there is at present no direct evidence of the existence of such modification (which would no doubt be exceedingly, though possibly measurably, small) we shall not enter any further into the matter here. 195. Heat of Swelling of a Colloid. If gelatine, or starch, etc., is placed in water, it absorbs the latter and produces a flabby mass. At the same time heat is evolved. This is supposed to result from the work done by an unknown tension P, which tends to open out the structure of the colloid. At T let the contraction of total volume = A ; the work done is At T + dT the tension has the value f ^dT + ^-^ 01 0

    . jrr A ... work done is P - -^ + ^ IT A*. The difference of .the two amounts of work is equal to the heat im absorbed multiplied by -^ : ap ap a<) 448 THERMODYNAMICS The expression in brackets may be regarded as practically constant, hence the heat evolved should be proportional to the contraction. By examining the swelling of starch containing different amounts of water, in water, this result was verified by H. Rodewald (1897). (31) 196. References to Chapter XV. : (1) Cf. Lord Eayleigh, Phil. May., [5], 30, 285, 456, 33, 209. (2) Lord Kelvin, Phil: May., (iv.), 17, 61, 1859. (3) J. W. Gibbs, Scientific Papers, 1. (4) Frankenheiin, Journ. f. prakt. Chem., 23, 401, 1841; Lehre ron der Kohdsion, 86, 1836. (5) E. Eotvos, Wied. Ann., 27, 452, 1886; G. A. Einstein, Drudes Ann., 4, 513, 1901. (6) Eamsay and Shields, Zeitschr. physical. Chem., 12, 433, 1893 ; Bamsay and Aston, ibid., 15, 98, 1894; Guye and Baud, Hid., 1(2, 379, 1903; cf. Nernsfc, Jahrb. der Chem., 3, 18, 1893, van der Waals. Zeitschr. physikal. Chtm., 13, 713, 1894. (7) E. T. Whittaker, Proc. Roy. Soc., A, 81, 21, 1908; E, D. Kleeman Phil. Mag., [6], 11, 491, 901, 1909. (8) McBain, Trans. Chem. Soc., 91, 1683, 1907. (9) Faraday, Phil. Trans., 1^, 55, 1834. (10) H. Freundlich, Zeitschr. physikal. Chtm., 57, 385. 1906; cf. also Kapillarchemie ; see also the author's Higher Mathematics, 91. (11) J. J. Thomson, Applications of Dynamics to Physics and Chemistry, London, 1888, p. 191. Cf. Freundlich and Emslander, Ztitschr. physikal. Chem., 49, 317, 1904; Warburg, Weid. Ann., 1^1, 14, 1890. (12) Milner, Phil. Mag., [6], 13, 96, 1907. (13) Zawidski, Zeitschr. physikal. Chem., 35, 77, 1900; '4!, 612, 1903. (14) Miss Benson, Journ. Phys. Chtm., 7, 532, 1903. (15). "W. C. McC. Lewis, Phil. Mag., [6], 15, 498, 1908; ibid., 17, 466, 1909. (16) F. G. Donnan and J. T. Barker, Proc. Roy. Soc., A, 85, 557, 1911. (17) Duclaux, Ann. Chem. phys., [5], 13, 76, 1878; M. Iraube, Berl. Ber., 17, 2294, 1884 ; J. prakt. Chim., 3%, 292, 515, 1886 ; Lieb. Ann., 265, 27, 1891 ; Forch, Wied. Ann., 68, 801, 1899 ; Whatmough, Zeitschr. physikal. Chem., 39, 129, 1902; Drucker, ibid., 52, 641, 1905; Eitzel, Hid., 60, 319, 1907; Linebarger, Amer. Journ. Sci., [4], 2, 226, 1896; Sutherland, Phil. Mag., [5], 38, 194 ; Herzen, Arch, de Sci. phys. et nat., [4], lit, 232, 1902 ; Eontgen and Schneider, Wied. Ann., 29, 165, 1886. (18) Valson, Ann. Chem. Phys., [4], 20, 361, 1870. (19) Szyszkowski, Zeitschr. physikal. Chtm., 6$, 385, 1908. (20) Chappuis, Wied. Ann., 19, 29, 1883. (21) T. de Saussure, Gilb. Ann., 7, 113, 1814. For earlier literature cf. Ostwald, Lehrbuch, 1, 1084, 2nd. edit. 1891. (22) Baerwald, Drude's Ann., 23, 84, 1907. CAPILLARITY AND ADSORPTION 449 (23) M. Travers, Proc. Boy. Soc., A, 78, 9, 1906 ; cf . Miss Homfray, Ztitschr. physikaL Chem., 1\, 129, 1910. (24) Favre, Ann. CJiim. Phys., [5], 7, 209, 1874. (25) Dewar, Proc. Roy. Soc., 7%, 122, 127, 1904 (low temperatures). (26) Parks, Phil. Mag., [6], It, 240, 1902. (27) Jungh, Pogg. Ann., 1*5, 292, 1865. (28) Schwalbe, Drude's Ann., 16, 32, 1905. (29) Freundlich, KapiUarthemie, 115. (30) Pawlow, Zfitsrhr. physikal. Chem., 75, 48, 1911. (31) H. Rodewald, Zeitschr. i.hysikaL Chem., 2$, 193, 1897 ; cf. Eeinke, Hanstens lotan. AbhandL, 'j, 1, 1879 ; Wiedmann and Liideking, ITiW. Ann., S.J, 145, 1885 ; Pascheles, Pfliiger's Archives, 07, 225, 1897. T. G G CHAPTEK XVI. ELECTROCHEMISTRY 197. The Thermoelectric Circuit. If the two junctions of a circuit of two wires of different metals are maintained at different temperatures, TI > T 2 , an electric current flows round the circuit, its direction and magnitude depending on the nature of the metals and on the temperatures (Seebeck, 1821). If TI T 2 is small, the electromotive force acting round the circuit is approximately proportional to it. The source of the energy of thermoelectric currents is indicated by the observation of Peltier (1834) that heat is absorbed at the hot junction and evolved at the cold junction, and that if the direction of the current is re- versed by inserting a battery in the circuit, these thermal effects at the junctions are also re- versed. The heat liberated or absorbed is proportional to the quantity of electricity crossing the junction, and for unit quantity is denned as the Peltier effect, TT at the junction. Besides the reversible production of heat at the junctions, there is an evolution of heat all round the circuit due to frictional resistance, this Joule's heat being proportional to the square of the current, and hence not reversed with the latter. There is also a passage of heat by conduction from the hotter to the colder parts. But if the current strength is reduced, the Joule's heat, being proportional to its square, becomes less and less in com- parison with the Peltier heat, and with very small currents is negligible. We shall further assume that the reversible thermo- electric phenomena proceed independently of the heat conduction, so that the whole circuit may be treated as a reversible heat ELECTROCHEMISTRY 451 engine (Lord Kelvin (1854), the junctions being placed in the two heat reservoirs (Fig. 86). We first assume that the Peltier effects are the only reversible heat effects in. the circuit. Then if TTI, 7r 2 are the Peltier effects at the hot and cold junctions : TTi-TTa^E . . . . (1) the electromotive force acting round the circuit, i.e., the work done per unit quantity of electricity circulating. From the entropy equation : 771 ^ f'->\ E~E ' ..ErrCTx-Ta)-^ .... (3) so that with a fixed temperature of the cold junction the electro- motive force should be proportional to the difference of the temperatures of the junctions (Clausius, 1853). It was known even to Peltier that this result is not general ; there are circuits the electromotive force of which decreases with rise of temperature until at a definite so-called inversion tempera- ture, it vanishes. With further rise of temperature the electro- motive force and current are reversed. The existence of this thermoelectric inversion, or Cummin g effect, led Lord Kelvin to surmise that reversible heat effects other than the Peltier effects existed, and he succeeded in showing experimentally that heat may be absorbed or evolved when a current flows along a single homogeneous wire from a place at higher to one at lower tem- perature, and the effect is reversed with the current. In the case of copper heat is evolved when the current flows from a hot to a cold portion, in the case of iron it is absorbed, in the case of lead there is no change. This is referred to as the Thomson effect. The current appears to convey heat convectively from one part of the circuit to another, much as a current of water in an unequally heated tube. Let crcW be the heat developed per second in a portion of a homogeneous conductor the ends of which are at temperatures 8 and 6 + dO, when unit current passes from the warmer to the colder end. cr is called the specific heat of electricity in the metal. Let the values of cr in the arcs (1) and (2) be cr 1? o- 2 respectively. If TO is the temperature of a chosen position in the arc (1), the G G 2 452 THERMODYNAMICS total heat developed per second by unit current all round the circuit in this way is : p p p MT + <7 2 r7T + ov/T = JTI Ji 2 JT O - 0, i.e., the electromotive force increases with rue of temperature, then EF + Q > 0, hence A > 0, so that the cell absorbs heat in action ; 7TJ1 (ii.) if y = 0, i.e., the electromotive force is independent of temperature, then EF + Q 0, hence A = 0, i.e., the cell neither absorbs nor emits heat in action ; F (iii.) if -rp- < 0, i.e., the electromotive force decreases icith rise of temperature, then EF + Q < 0, hence A < 0, so that the cell emits heat in action. ELECTROCHEMISTRY 459 Cases (i.) and (iii.) indicate that there is a change of entropy in the corresponding processes, since A = T(Sa - SO . . . . (IQa) 7TP .-. if -- > 0, 82 > Si, and the entropy of the cell increases, if -7777- < 0, S 2 < Si, and the entropy of the cell decreases. The total entropy of the cell and heat reservoir remains con- stunt, since the process is reversible. If ~ = we see that EF = - Q, so that the electrical work done by the cell is exactly equal to minus the heat of reaction. Lord Kelvin (1851), who found this relation verified in the case of the Daniell cell, assumed that it held generally, i.e., the electrical work done by a cell is equal to the diminution of chemical energy of its components. The heat of reaction when one electrochemical equivalent of zinc displaces copper in sulphate solution is 2*592 cal. = q .'. E = 2-592 X 4-18 X 10 7 = 1'09 X 10 8 E.M. units. = 1'09 volt, which agrees with the observed value. As Gibbs and Helmholtz pointed out, however, this so-called Thomson Rule (which had also been proposed by Helmholtz in 1847) cannot be true in general, because the change of entropy cannot always be neglected. Cases in which a cell cools or warms itself in action had been investigated by Brauri (1878 1883), and the quantitative relation was verified in a number of cases by Jahn (1886), who measured the latent heats by placing the cell in an ice calorimeter. In the Clark cell at 0, according to E. Cohen : Q = - 340730 j. 2 EF = 1-4291 volt X 2 X 96540 cmb. = 275930 j. .-. A = 64800.;', i.e., the cell converts only 76 per cent, of its chemical into electrical energy, and emits the rest as heat. The temperature coefficient of the Daniell cell is -f 0'000034 , its E.M.F. at C. is 1*0962 volt degr. .-. 2EF = 2 X 96540 X 1*0962 = 50526 j. 460 THERMODYNAMICS The heat of reaction is Q = 50110.;. .*. the latent heat is 416 j. whereas 2FT ^ = 2 X 96540 X 273 X 0'000034 = 428 j. The direct measurements of Jahn (1888, 1893) and of Gill (1890) show that the latent heat A arises at the surfaces of contact of the electrodes and electrolyte and is fully accounted for by these Peltier heats at the junctions of conductors. The equation of 197 : *.= -T|| - , - (ID where TT is the sum of the Peltier effects round the circuit, and E is the total electromotive force, was found to apply, and hence the Gibbs-Helmholtz equation may be written in the form : E + q = 77 or q = (E + STT) . . . (12) The heat of formation of a substance in a voltaic cell may therefore be calculated from the measured Peltier effects and the electromotive force. The integral of the Gibbs-Helmholtz equation is : where C is the integration constant, since the differential equation (10) is easily transformed into : /EX d W rfT T 2 If q is independent of T, i.e., the total heat capacity of the initial reacting substances is equal to that of the products of reaction ( 58), then: . ..." ~. (14) If there exists a temperature T at which the electromotive force of the cell vanishes : . . . (15) ELECTROCHEMISTRY 461 and E = q T ~ T .... (16) an equation due to Gibbs (1886). TO is called the transition temperature, and at this temperature the chemical reaction in the cell would go on reversibly in either direction, since its progress involves no change of available energy. If two of the magnitudes E, T , T are known, the third can be calculated from a knowledge of q. Since T can be determined by non-electrical methods (e.g., by measurements of solubilities, or changes of volume, etc.) the equation serves to determine the electromotive force of a voltaic cell without actually setting up the latter, as was emphasised by Gibbs. The method has been applied by Cohen (1894) to the deter- mination of transition temperatures. Thus the electromotive force of the cell : Zn | sat. sol. ZnS0 4 . 7H 2 | sat. sol. ZnS0 4 . 6H 2 | Zn vanishes at the transition temperature of the reaction : ZnS0 4 . 7H 2 = ZnS0 4 . 6H 2 + H 2 (cf. 11) at which the two hydrates have the same solubility. If the temperature is raised above T , the polarity is reversed. The existence of a transition temperature at which E vanishes permits of a very important transformation of the equation (13), viz., it enables us to replace an indefinite integral by a definite integral, in that a fixed lower limit of integration may be assigned to the former. The equation (13) may now be written : T . . . . (17) and the electromotive force is therefore defined without ambiguity in terms of the thermal magnitudes. We shall make use of a similar transformation later ( 209). 201. Effect of Pressure on the Electromotive Force. If the change of volume occurring in the cell is taken into account, which is particularly of importance when gases partici- pate in the reaction, we may proceed as follows : For a small reversible change : rfU = 2SQ - 2SA = TrfS - Ede - pdV . . (1) 462 THERMODYNAMICS where, in addition to the electrical term ~E>dc, e being the quantity of electricity flowing round the circuit, the external work involves a term pdV due to expansion of the whole system by dV under an external (e.g., atmospheric) pressure p. Subtract rf(ST pV) from each side of (1) : .-. d(U - TS + 2>V) =d = - SrfT - E = U - TS + pV is the thermodynamic potential of the system. Since d(f> is a perfect differential we have : ^=-E,^ = V,|=-S . . . (8) de op 9T and hence such relations as : 8E\ . a /\ _ a /a\ _ /av a7 V^ - f ' i.e., the electromotive force increases with the pressure at a rate equal to the rate of decrease of the total volume at constant pressure per unit quantity of electricity passing round the circuit, the temperature in both cases being constant. According to Faraday's law the change of volume depends, at a given tempera- ture and pressure, only on the quantity of electricity passing : where YO, Y are the volumes before and after the electric transfer. For chemically equivalent amounts e = F. Thus jjy = V ~ V . . . . (6) If only condensed phases are present, YO, Y are practically independent of pressure, hence at a constant temperature the integral of (6) is, for this case : ^-E l = ^^(p 2 - Pl ) . . . (7) which has been verified by Cohen for the Clark cell, and also by other investigators for other cells. If gases participate in the reaction, we can put : : (8) ELECTROCHEMISTRY 463 where A refers to the condensed part, which suffers a change of volume practically independent of temperature and pressure, and refers to the volume of the gas phase, which is proportional to the absolute temperature and inversely proportional to the pressure. The interpretation of A and B is simple. Thus, by integration of (6) : E 2 EX = A (#, 2>i) + BT/H y (9) at constant temperature. This equation has been verified by Gilbaut (1891). 202. Concentration Cells. In cells like the Daniell, the electrical energy is derived from chemical changes occurring between the electrodes and electro- lytes ; the final system possesses less free chemical energy than the initial, and the difference goes over into free electrical energy. A voltaic cell may also derive its energy not from changes of composition, but from changes of concentration, which may occur in the electrodes or in the electrolyte. If we assume that the heat of dilution is zero, which limits the discussion to dilute solutions : ff = (1) .'. from the Gibbs-Hehnholtz equation : E = T or ^ = ^ ; .-. /wE = InT + const. /. E = CT (2) The total inapplicability of the Thomson rule to this case is at once apparent; none of the electrical energy comes from chemical change, but the cell functions as a heat engine, converting the heat of its environment into electrical work. The theory of concentration cells was first developed with great generality by Helmholtz (1878), who showed how the electro- motive force could be calculated from the vapour pressures of the solutions, and his calculations were confirmed by the experiments of Moser (1878). The simplest case is one hi which we have two reversible 464 THERMODYNAMICS electrodes in a gas at different pressures, for example, two platinum plates saturated with hydrogen, dipping into acidulated water, and surrounded by hydrogen gas at the pressures pi, p%. If these are connected, gas dissolves at the higher, and is evolved at the lower, pressure. Let us suppose the cell works against a balancing electromotive force until 1 mol of gas has passed from the higher to the lower pressure. The electrical work is : E X 2F since each molecule of hydrogen yields two ions. If we remove the niol of gas from the low pressure space, compress it isothermally and 'reversibly until its pressure rises to p%, and then introduce it into the high pressure space, the cycle will be completed. The work done with the gas is : /P2 /V>2 I pdv 2W% = \ Vt J PI J ^le's law, j r / 1 ~ J vdp If the gas obeys Boyle's law, p\r\ = pdr = - RT In . pi I This must be equal to the electrical work, since No experiments appear to have been made with such cells, although the equation has been verified with oxygen at different partial pressures in admixture with nitrogen, with platinum electrodes and hot solid glass as electrolyte (Haber and Moser). A similar case is that of two amalgams of a metal, of different concentrations, as electrodes, and a solution of a salt of the metal as electrolyte (G. Meyer, 1891). Here we must take the osmotic pressures of the metals in the amalgams, PI, P2, and, for an ?i-valent metal : But ,.E = te .... (5) ELECTROCHEMISTRY 465 It has been found, however, that this case is somewhat compli- cated by the formation of definite compounds in some amalgams ; still the general results are in agreement with the theory. Some exceptional cases found by Meyer have recently been shown to depend on the large heats of dilution of the particular amalgams (Smith, Zeitsclu: anorg. Chan., 58, 381). The equation shows that the electromotive force is proportional to the absolute temperature, which was verified by experiment. It is also independent of the nature of the electrolyte. A very important practical case of concentration cell is that in which two electrodes of the same material are immersed in solutions of an electrolyte of different concentrations. Thus, if two silver Ay, plates are immersed in solutions of silver nitrate of different concentrations, and are connected by a wire, the metal dissolves in the dilute solution and is precipitated from the strong solution, and this goes on until both solutions have the same concentration. Let us consider a cell containing the solutions of con- centration 1, 2 in two chambers A and B, separated by a porous partition, and containing silver plates. When F coulombs pass round the circuit the following changes occur, where n is the migration ratio of the anion : ir (l-n) NO, (1) Vessel A. Gains 1 equiv. Ag by dissolu- tion from the electrode. Loses (1 n) equiv. Ag by migration. Gains n equiv. N0 3 by migra- tion. /. gains n equiv. AgN0 3 . (2) Vessel B. Gains (1 ) equiv. Ag by migration. Loses 1 equiv. Ag by deposition. Loses n equiv. N0 3 by migra- tion. .'. loses n equiv. AgN0 3 . The nett result is a transference of n equiv. AgN0 3 from the strong solution to the weak solution, where n is the migration ratio of the anion. The cycle may be completed by returning the n equiv. of salt to the weak solution by a reversible osmotic process. Place T. H H 466 THERMODYNAMICS the strong solution in a cylinder under a semipermeable piston covered with pure water, and allow solvent to enter reversibly until the concentration of the solution is restored to its value (&) before the process. If the volume of solution is large, the osmotic pressure remains practically constant, and the work done is P^i, where i\ = change of volume. Now separate a portion of solution containing n mols of AgN0 3 , and compress it reversibly until its osmotic pressure rises to P 2 , corresponding with the concentration fP2 PJr. This solution is now mixel with the identical strong solution, and solvent is removed by the semipermeable piston till it regains its original volume. The work done is Pat' 2 . The cycle is isothermal and reversible : .-. 2 A = EF + fpi-l * Pile = L Js Jft /Bs or EF = vdP . . . . (6) J Pl We shall assume that the solutions are so dilute that the electrolyte can be regarded as completely ionised. Then, for a binary electrolyte, for the amount considered : Pi; = 2/iRT . . . . (a) if we regard n as the limiting value of the transport number attained at infinite dilution, or since P 2 /Pi = ?i/r 2 = 2 /i : & -g (7) 1, 2 refer to the concentrations round the kathode and anode, respectively, and the negative sign shows that the potential of the electrode in the stronger solution is less than that in the weaker solution. ELECTROCHEMISTKY 467 If the solutions are more concentrated, this formula does not apply, for two reasons : (i.) The transport number n is a function of the concentration ; (ii.) The equation (a) is no longer valid. In this case Lehfeldt writes : Pr = niET where / is simply the ratio between the actual osmotic pressure and that calculated from (a). Then .... (8) which may be regarded as giving the osmotic pressure P of a solution of moderate concentration. If we assume, with Arrhenius, that : where ft is the number of ions produced from one molecule of electrolyte, and take n as denoting a mean value between those for the two solutions : E _ _ RT " Now, according to Ostwald's dilution law : a 2 v _ K _ K (1 + a)/*RT l^a~ "I" ~^~ .'. /P = /(! + a) + //i(l a) - 'Una .-. dln=dln(l + a) + dln(l a) -Mlna(T const.) 2 (1 + a) [tottna - tUn(l + a) - cU(l-a)] L J an equation due to Xernst (1892). This is of very wide applica- bility, for by its help we can determine either ti, or , or a, for a dilute solution. H H 2 468 THERMODYNAMICS The lead accumulator has been studied from the thermo- dynamic point of view by Dolezalek. The reaction is : Pb0 2 + Pb + 2H 2 S0 4 :T2PbS0 4 + 2H 2 If two such accumulators, containing differently concentrated acids, are connected in opposition, i.e., both peroxide plates together and both lead plates together, the one with the stronger acid is discharged in the sense of the above equation from left to right, whilst the one with the weaker acid is charged in the sense of the equation from right to left. The reaction involves the passage of 2F, as we see on writing down its anodic and kathodic components : Pb + S0 4 = PbS0 4 +2 (anode) Pb0 2 + H 2 S0 4 + 2H + 2 9 = PbS0 4 + 2H 2 (kathode) The quantities of Pb, Pb0 2 , PbS0 4 remain unaltered, provided the substances produced are in the same state of aggregation, etc., as those disappearing, and the nett result is the transport (of course, apparent, since there is no material connexion between the cells) of 2 mols of acid from the stronger (I.) to the weaker (II.) solution, and 2 mols of water in the opposite direction. The weak acid is concentrated and the strong acid is diluted, the process taking place in such a direction as to equalise the concentrations as if the acids had been directly mixed, and the electrical work produced is equal to the free energy which would have been lost in the spontaneous intermingling of the two liquids. This free energy can be calculated if we can construct some process in which the mixing can be performed isothermally and reversibly, since it is equal to the maximum work of this process. The changes of concentration may be very simply brought about by Helniholtz's method of isothermal distillation. Solution I. has lost 2H 2 S0 4 and gained 2H 2 0, whilst solution II. has gained 2H 2 S0 4 and lost 2H 2 0. This change may also be carried out as follows : (i.) Take from I. an amount of solution containing 2H 2 S0 4 , say 2(H 2 S0 4 + iH 2 0), and distil water from it to the solution L, over which its vapour pressure is p, until (almost) pure acid remains. [The removal of the very last trace of water would, ELECTROCHEMISTRY 469 from the equation, require an infinitely large expenditure of work, but since the amount remaining in the acid can be made as small as we please, the integral has a finite limiting value." The work done, if p denotes the pressure at any intermediate concen- tration over the acid of varying composition, is for dn rnols transferred : p and the total work done is : Ai = 2RT | In *- dtii o (ii.) We now distil water from II. on to the acid till we arrive at an acid of the same composition as that in II., say 2(H 2 S04 + /^HoO). The work done during the distillation is : This is mixed with the acid of II., and the 2H 2 S04 restored. (iii.) The 2H-20 may now be distilled from II. to I., and every- thing is in the same state as after the electrical process. The work done in the last process is : The total work in (L), (ii.), and (iii.) is : A = 2RT>^ + I /A + /*B) J (AB = A + 7^)0 .' (/ZA + M,= /*A + /*B) / I (2) The relations (/Z A ) = (/* A ) p ; (/Z B ) = (u)0 would only by the merest chance form the solution of (2), hence there will not in general be a partition equilibrium between the ions when one is established between the neutral molecules, but one solvent, say a, will contain more A ions than corresponds with ionic partition equilibrium. These will pass through the surface of contact into /3, and similarly B ions from /3 to a. The separa- tion of the two kinds of ions will however set up an electrostatic field across the boundary, and the two kinds of ions collect there in two sheets very close together in fact, we have an electrical ELECTROCHEMISTRY 471 double layer. Let E a , E^ be the electrical potentials on the a and /S sides of the double layer, then E = E^ E a will be the change of potential across the boundary. If the kations collect on the /8 side of the latter, E is positive. Now suppose a small quantity Sc of electricity passes across from a to /8. This corresponds with the transport of ^ equiv. of ions. If the ions are univalent this is also ^ mol. The change of thennodyuaniic potential is (/i A ),3 (/*B) P er m l .-. <**>- W-Sf .... (3) for the quantity of kation transported. At the same time the electrical work E&? is done at the double layer, hence, since in equilibrium : Sty + A) = . . . . (4) F or E= w* ^w_ . . . (5) which is the equation of equilibrium of the kations. Similarly E = ^ B p -^ . . . . (5a) is the equation of equilibrium of the anions, hence or (/* A + /Z B ) = (MA + MB)^ O 2 ) which we found above. In general, for a dilute solution : p. = T( ? + Rlc} .... (6) where 9 is a function of temperature and pressure ( 158). ' E = " (} a + RZ (f A ) 1 = T f(9 A ) + R/i (c, .. (9) Similarly f = /K B = Jns . . (9tt) " \ C B)O. Thence E = - * [>K A + In g] = * [foK a + In^] . ' . . (10) The condition that the solutions in bulk are electrically neutral, the potential difference being located in the surface of separation, requires that : E=0. ,. . . . (11) . Ml_ (C B )]8 /.OX ' (c A )."~ (CB). If we add the two equations of (10) we obtain the potential difference across the double layer : which is positive when K B > K A . If we subtract the same two equations, we find : A X K B = 2fo \ C A.)P VK-xK e ... (14) The left-hand member denotes the distribution ratio of the kation in both solvents which would be found if the ions could be distributed like any ordinary solute, and condition (12) had not therefore to be fulfilled. K A and K B are called by van Laar, to whom the above formulae are due (1903), the fictitious distribution ELECTROCHEMISTRY 473 ratios (or partition coefficients) of the ions ; their geometrical mean is equal to the true distribution ratio. In the above we have assumed that no other forces than the electrical are acting at the surface of separation. In general, there will be the capillary forces as well, and we have to take account of the influence of the electrical double layer in con- sidering the adsorption of an electrolyte. If <> is the area of the surface, , done by the atmosphere in compressing the two gases as they are used up in the current- producing reaction ; (ii.) The work set free by the diminution of available energy in the chemical reaction occurring between the gases dissolved in the electrolyte or in the platinum electrodes. If we know the free energy of the reaction at any given temperature, we can at once calculate the chemical equilibrium at that temperature from the equation ( 144) : A T = RT/nK RT(2i^Ei>,- + 2i>,-) . . (1) Thus, in the Grove's oxy-hydrogen cell : HI, Ha, H 3 are the concentrations of the hydrogen, oxygen, and water vapour supplied to the cell and produced by it, respectively, in the reaction : 2H a + 2 = 2H 2 0. 478 THERMODYNAMICS The water produced is, however, liquid, hence we must take ES as the concentration of saturated water vapour at the given temperature, since the liquid water produced could be evaporated to form vapour at this concentration without additional work, except that due to the change of volume under the given pressure, viz., 2RT, which must now be omitted fron the RTSr,. This result is general ; the maximum work for the production of a gas differs from that for the production of the same sub- stance in a saturated solution only by the work done in with- drawing the gas from the solution under the constant external pressure, viz., RT per mol. The influence of temperature is given by the equation ( 150) : = T AT = T dT RTSi^wEi RTS^ + (Const.)T . . (2) or, for specific heats which are linear functions of T : A T = - Qo + aT/wT + /3T 2 - RT2i/,foE, RTX^ + (Const.)T . . (8) If E is the electromotive force of the cell, and if r faradays are transported through the cell during the change for which the maximum work is calculated, we have : A T = rEF ... V.: . . (4) since A T depends only on the initial and final states, and is independent of the particular way (osmotically, electrically, etc.) in which the process is supposed to be executed. Thus: rEF = T ^rfT RTSv^wE, RTSz;, + (Const.)T . (5) If we consider the electromotive force at a given temperature, we have from (1) : rEF = RTtoK - RT(S^H, + 2^-) - E = ~ [inK - 2^E, - 2X] . . (6) Thus E is, for fixed concentrations of the substances used up and produced in the reaction, determined by the value of the equilibrium constant K, at the given temperature. The change of E with temperature is given generally by the Gibbs-Helmholtz ELECTEOCHEMISTRY 479 equation of 200, or, in this particular case, by the change of K as determined by the reaction isochore : tUnK _Q^ dT ' ' ET 2 ' The two methods are of course equivalent, both being founded on the same equation, viz. : A T + Q e = T ^. In equations containing terms relating to electrical energy, EF, and terms relating to heat absorbed, Q c , we must not forget that both are to be measured in the same units. If E is in volts, and F in coulombs : EF = E X 96540 joule, also I cal. = 4-188 X 10 7 erg = '188 joule, '' l joule = 4~W = ' 2887 cal or EF joule = 96540 X 0'2387 = 23046E cal. Equation (6) therefore gives us a means of calculating chemical equilibria from measurements of electromotive force, and vice versa. It must be remembered that E has a sense only when it refers to a reversible cell ; if the cell is not reversible this simply means that no equilibrium can be set up at its electrodes between the reacting materials. As was pointed out by Ostwald the processes usually called oxidations and reductions generally involve the taking up of a positive charge (or loss of a negative charge), and the loss of a positive charge (or the taking up of a negative charge) by an ion, respectively : Fe + = Fe (oxidation) G = Mn0 4 = Cu (reduction) + = FeCye If the ion is that of a metal, the latter must therefore be regarded as the reduced form, or, if more than one ion exists, as the most reduced form : Fe + 3 = Fe + = Fe. 480 THERMODYNAMICS Luther has put forward a general rule relating to the potentials and equilibria between states of oxidation of a metal. If the lowestj intermediate, and highest stages of oxidation (e.g., Fe, Fe; Fe) are denoted by I, m, h, we have : the change of free energy on the conversion of a mol of the lowest stage of oxidation to the highest being independent of the path, and thus equal to the sum of the two changes via the intermediate stage. If the electrode potentials between the metal and the two ions are represented by E,_* E^^ h , and if a, b are the valencies, respectively, of the two ions : m X F ; ty,_ h = 6E,_ A X F Thus electrode potentials may be calculated from oxidation potentials, and vice versa. Example. E m >/,, measured by the potential of the oxidation electrode Pt | Fe, Fe, is 0'99 volt for a N solution of both ions (Peters, Zeitechr.physik. Chem., 26, 193, 1898); Ej_ >,, measured by the electrode potential Fe j Fe is + 0'08 volt. = 0-28 volt. .*. if we mix together Fe, Fe, and Fe. we shall arrive at Fe, Fe, + + since the mean state of oxidation Fe has a lower potential than the highest and there must always be a degradation of free energy. When the three forms are in equilibrium, it can easily be shown from the relations Si/r/_>* = T (J' ul + Eluc m ) >/, etc. and the equation of equilibrium : vilnci + vzlnc?. + ..= (i/j/j -f- i- 2 f 2 + . .) that E,_ A = E,_ w = E w _ /( . ELECTROCHEMISTRY 481 In the case of gas cells it is more convenient to substitute partial pressures for concentrations in the equation : [ / Let - x , JT S , . . . ir, be the pressures of the gases supplied to the cell, e.g., H 2 and C1 2 at the electrodes of the cell : Ft, H 2 /HC1 aq./C! 2 , Pt. Pi, P-2 Pi the partial pressures of these gases in an equilibrium mixture (e.g., H 2 + C1 2 ~ 2HC1). From the relations : we find : E = >-F * 1 "V 2 ~ Sz where ZK' = vilnpi + If with Nernst and Haber we omitted Sr, " for simplicity," we should have : which is quoted by Nernst (Berl. Bet:, 1909, p. 247), on the authority of Helmholtz (Ges. Abhl., III., 108), but is not given by the latter in this form. Similar considerations apply of course to the opposing electromotive forces of polarisation during electrolysis, when the process is executed reversibly, since an electrolytic cell is, as we early remarked, to be considered as a voltaic cell working in the reverse direction. In this way Helmholtz (ibid.) was able to explain the fluctuations of potential in the electrolysis of water as due to the variations of concentration due to diffusion of the dissolved gases. It must not be forgotten, however, that peculiar phenomena so-called supertension effects depending on the nature of the electrodes, make their appearance here, and corn- T. I I 482 THERMODYNAMICS plicate the theoretical treatment (cf. Caspari, Zeitschr. pliyslk. Chem., 30, 89, 1899; Tafel, Ber., 33, 2209, 1900). Finally, we may observe that measurements of electromotive force can often serve to distinguish which kind of ions are really present in a solution. A concentration cell containing a solution of a known ion with an electrode reversible to the latter on one side, and the given solution with a similar electrode on the other side is taken. From its electromotive force, the concentration of the par- ticular ion is calculable. In this way, for example, it was found + that only a very small amount of Ag ion exists in a solution con- taining CN ion ; the greater part is combined as a complex anion Ag(CN) 2 . CHAPTER XVII THE THEOREM OF NERNST 206. The Statement of the Problem. The system of equations based solely on the two fundamental laws constitutes what may be called the Classical Thermo- dynamics. Although perhaps different points of view may be adopted in the future in the interpretation of these equations, it is as unlikely that any fundamental change will be made in this region as that the two laws themselves will turn out to be incorrect. It must repeatedly have been remarked, however, that these equations are not in themselves sufficient to lead to a complete solution of the problems to which they have been applied. This arises from the fact that they are differential equations, in the solution of which there always appear arbitrary constants of integration (H. M., 73, 101, 121). Thus, the relation between the pressure of a saturated vapour and the temperature is expressed by the differential equation of Clausius ( 80) : (1) dT R T 2 the solution of which is : Inp = ^ y + const. . . . (2) so that, although the absolute value of A, the latent heat, can be evaluated by eliminating the integration constant at two temperatures : ln& " T, ~ the absolute value of p as a function of T cannot be calculated. A little consideration will show that this incompleteness has i i 2 484 THERMODYNAMICS its origin in the inability of the system of classical thermo- . dynamics to inform us as to the absolute values of the intrinsic energy and entropy of a system, since we have proved that if these are known, and thence the free energy, or thermo- dynamic potential, we are in possession of a complete description of the thermodynamic properties of the system. The problem which the classical thermodynamics leaves over for consideration, the solution of which would be a completion of that system, is therefore the question as to the possibility of fixing the absolute values of the energy and entropy of a system of bodies. In defining the intrinsic energy of a system, Lord Kelvin remarked that the absolute amount of energy associated with a system was not capable of direct measurement, and might be very large in fact we could not be sure that it was not infinite. The newly developed theory of Relativity shows that the energy associated with a body at rest, if we leave out of account external actions such as pressure, is equal to the mass of the body multiplied by the square of the velocity of light in vacuum an enormous magnitude. Since, as we shall see later, the ambiguity does not really involve the absolute value of the energy, we shall not encumber our equations with this colossal additive constant, and we shall continue to measure the energy from an arbitrarily selected standard state. With the entropy, however, the case is quite otherwise, and we shall now go on to show that as soon as we are in possession of a method of determining the absolute value of the entropy of a system, all the lacunae of the classical thermodynamics can be completed. The required information is furnished by a hypo- thesis put forward in 1906 by W. Nernst, and usually called by German writers " das Nemstsche W&rmeiheorem" We can refer to it without ambiguity as Nernst' s Theorem.^ 207. The Theorem of Nernst. The entropy of a condensed chemically homogeneous substance vanishes at the zero of absolute temperature: By " condensed " is meant solid or liquid ; by " chemically homogeneous " is meant a pure substance, element or com- THE THEOREM OF NEENST 485 pound, as distinguished from a mixture or solution. Terms relating to such substances will be enclosed in square brackets. 208. Specific Heats and Coefficients of Expansion. If the independent variables are p and T, we have quite generally for a homogeneous substance : = f (1) M _ fr\ (2) (3) -;* The integral does not furnish the absolute value of , the entropy, because the lower limit is undetermined. If this is regarded as fixed, the integral with various upper limits gives the values of the entropies referred to this arbitrary standard state, and the differences between these values and any one of them referred to this arbitrary standard state will be the values of the entropies referred to the new standard state (cf. 42). If we introduce Nernst's theorem : we can however transform the indefinite integral into a definite integral with the lower limit zero (cf. 192), since s is positive and vanishes for T = : JP T - (5) '0 Since for T = the integral does not become infinite, it follows that: [ to be nearly constant. c p , c r are, of course, functions of temperature, and we shall see later how the form of these functions has been determined, partly by means of atomistic considerations, partly empirically. Equation (5) holds for all values of p, and hence if we differentiate with respect to p and compare with (2) we have : T P) 7T fr>\ 4 /TfZT= ~V8TJ, But r 'do^lf' the coefficient of expansion of the substance also vanishes at absolute zero. The experimental evidence for this is not so satisfactory as that relating to specific heats, but it is at least certain that the coefficients of expansion of metals diminish rapidly at low temperatures. The empirical relation : specific heat coefficient of linear expansion = constant . . . . (9) THE THEOREM OF NERNST 487 verified for some metals by Griineisen also strengthens the conclusion. 209. Transition Points. In Chap. VII. we have considered: (i.) changes of physical state, (ii.) allotropic or polymorphic changes, (iii.) chemical reactions which occur at a transition point. A particularly good example of the second type is the tran- sition from rhombic to monoclinic sulphur. 2 The equation of equilibrium is ( 97) : /. MI si 7>ri = u. 2 or 'i T.S! = ' 2 T-s'o where n- = u + 7>r. Thence T (s 2 *0 (n- a 'i) = 0. /T /.T But /T s l= \ ^ ' /.T ... T I C>> - ~ tVl r/T A = . . . (10) where A = /r- 2 ifi = latent heat of transition. The equation, with known <- /v <>,,,> an(l A determines the transition temperature for (ft). -' ^-^ = (cf- 93,97) T- = . . . (11) an equation which will be generalised in the sequel. 488 THERMODYNAMICS According to Bronsted, for the sulphur transition : A = 1-57 + 115 X 1(T 5 T 2 |j = 2-3 X 1CT 5 T /. T = 369-5 (obs. = 869-4). We observe that in the empirical power series for A given by Bronsted there is no term with T. It can be shown that this is peculiar to all reactions covered by the theorem of Nernst. For let us assume that the heat of reaction is expres- sible by such a series : Q = A + BT + CT 2 + DT 3 + (12) where Q refers either to constant volume or constant pressure : Q, = U 2 - Ui or = W a -Wi. Then ^ = r; - T, = . . . (13) L J T = and ~jf = iy r, = . . . (14) since r, = 2vG v = T yJ = ^vC tl 0. Hence there can be no constant term in the derived series, i.e., the coefficient of T in the series (12) must be zero. The expansion of Q in a power series must therefore have the form : Q = Qo +/3T 2 + 7 T + . . . where 6 = Sr/3.- and the expressions for the molecular heats of the different con- densed phases are of the form : C c . = a, + 2&T + 3 % T 2 + . . . etc. i.e., C r . = 2&T + 3 7 /r 2 + . . . since (L. = for T=0 /. a ( = 0. THE THEOREM: or XERNST 439 If the reaction occurs at constant volume : and Q,. = U 2 - L T ! = * 2 + TS 2 - (*! + TSi) /. ^ = 5 -^t=^ + (8, - SO, and hence, ?T since [S 2 ] T = o = [Si] T = o If we put Q,. = Q r = diminution of intrinsic energy = heat of reaction (erohed) we can write (15) in the form : ~dQ = . . (15a) T = which is the form adopted by Nernst. 3 This mode of expression leaves open the question as to the value of the specific heats at T = 0, and simply requires that, at this temperature, the molecular heat of a compound shall be additively composed of the atomic heats of its elements, i.e., the Xeumann-Kopp rule is strictly followed at the absolute zero. This must, how- ever, be the case, since all the atomic and molecular heats are zero. If the pressure is constant we have similarly : where A T ' is the maximum work under constant pressure, and Q,= ~ Qr Then, from the equations : A f n - AT ' 490 we find : THERMODYNAMICS Thus if we take T as abscissa, and Q, A T , as ordinates, the two curves meet tangentially and parallel to the temperature axis at T = (Fig. 88). This simply shows that the quantities A T and Q approach more and more closely as T approaches zero ; near the absolute zero they coin- cide, and the value then remains constant down to the absolute zero. Other views were previously . held. < 4 > If we suppose the specific heats are functions of T of the form C = 2/3T + 3 7 T 2 +' then : also [Q] = [<2o] + /3T 2 + 7 T 3 + (19) [A,] = [Q] + T . [(21 _ - -- 8[A T ] r?f dT T 2 r .-. [A T = [Q ] - /3T 2 - .'. 7 T 8 H Nernst uses the two very simple equations : (20) (19a) (20a) for reactions occurring in condensed systems between chemically homogeneous substances. The possibility of calculating [A T ], and thence transition points, etc. ([A T ] = 0), from purely thermal magnitudes is then especially clear. This method may also be extended to reactions in which gases occur, e.g., in the dehydration of salts <5> : CuS0 4 . H 2 -* CuS0 4 + H 2 (vap.) provided we imagine the reaction to occur between the substance THE THEOREM OF XERNST 491 and ice ; the heat of fusion is then to be subtracted from [O], and [A T ] is the work done in transferring n mols of H 2 in the form of vapour from ice, at the vapour pressure ir , to the hydrate, at the dissociation pressure IT, viz. : - can be calculated from Scheel's formula of 95. Example 1. If the specific heats of the solid and liquid forms are linear functions of temperature, show that the melting-point is determined by dividing the latent heat of fusion by the difference between the specific heats of the solid and liquid forms at the melting-point (cf. Tammanu, Kri/st. und Schmelz., p. 4*2). Example 2. Discuss the possible forms of the >\ T ] and \Q~: > curves according as /3 = (cf. T. \V. Richards, Proc. Amer. Acad., < 1902; Zeitschr. i)hysik. Chem., 40, 169, 597, 1902 ; 42, 129, 1903). These relations are, however, certain only at low temperatures (cf., Nernst, Ber. Berl. Akad., 1909, 247). 210. Gases ; Evaporation and Sublimation When we come to deal with gases or vapours, we pass at once out of the region of direct applicability of Xernst's theorem. If we assume, approximately, that the specific heat of the gas is constant over a small range near the absolute zero, we have ( 79) : .s =: s + f r /;iT -f- rlnr and this shows that the entropy of a gas by no means vanishes for T = 0, like that of a liquid or solid, but on the contrary, would become negatively infinite : ST=O = - oo . . . . (1) Since the specific heat of a gas can always be expressed in the form : c c = a + 6T -f cT 2 + where a is finite, or c c = a + F (t), where F (o) = o, but a is finite if the law of equipartition of energy holds good for the molecules of the gas at T = o (cf. 213), this result is general. If a gas is in equilibrium with a solid or liquid of the same 492 THERMODYNAMICS composition, and both are chemically homogeneous, the chemical potentials of the substance in both phases are equal ( 106, 157) : /* = M (2) vapour condensed At very low temperatures the pressures of the saturated vapours of liquids and solids are very small, and since the devia- tion of an actual gas from the laws of ideal gases becomes all the less the smaller is its density ( 70), we can safely assume that the vapour obeys the gas laws. If [m], m are the molecular weights of the condensed form and of the vapour, respectively, and, [<], their thermodynamic potentials per mol : Also ( 79) : d> = U - TS + I CdT - T I ^ dT - ET (Inv - 1) f (VT - = U - TS + I (C, + B) dT + RTlnp - ETln 5 C C^ = Uo - TS - - T j ^ dT - ET/w ~ . . . (4) and [<^] = ([U] + p{\]) - T[S] = [W] - T \-dT . (5) Thence : U - TS + Jc/T - r ij^ dT + ET//^ - BT/w 5 THE THEOREM OF NERNST U + f C.//T = W 493 But the latent heat of volatilisation referred to 1 ruol of vapour, hence : TS T y dT + RTlnp RT? ^ J For T = : A = A = U - a [W ] R = _ aT f[C T 'O CUT - a [CJrfT /. Ao - TSo + I C,//T - T I ^ rfT + RT/i^> - RT/w ^ "T XT [CU/T - T ' 11 w or, since ( 58) : A = A + {(C,, a [C p ] )rfT -^JT-al ffiicTcU-. . (7) where J-i At low temperatures the value of = |p + fa g = const. . . (8) r T r T ! a rr i //T a ^ ,/T IT [C " ] ' /T ~R! -Y~ dT \ J Jo is negligibly small, since [C^ = = 0, hence at low temperatures : top = wr ~ (9) 494 THERMODYNAMICS All magnitudes depending on the properties of the solid have now vanished, and , = o .'C;;,: . * . (l) where the summation extends over all the components and 9^ is a function of temperature : fcv - rp w, is the chemical constant ( 210, eqn. (9) ). Thus : (2) (8) ;"+ CdT = W But U;"+ CdT = W, . . . . (4) and 2r,W, = Q, ; . . . (5) the heat of reaction at constant pressure, hence : v 7 Q* , 2^fc;v/T 1 2>./c,. = - ^L _|_ _ I _^_ . v, (6.) THE THEOREM OF NERNST 499 If we put Q p = Qo + 2* ( I CjjMT . . . . (7) Jo where Q Jt = heat of reaction at absolute zero : - ~ CWT < /T - ^ 7 "' Also 2,i>ilnCi = InK, where K ==/ (j>,T) is the equilibrium constant. Equation (8) is the most general form of Gibbs's equation (J 148), and goes over into the latter when Cjj' = const. If we compare (8) with equation (2) of 148, putting const. = Q in the latter, we see that the indeterminate constant of the equilibrium equation is now completely determined as the sum of the chemical constants of the interacting gases : and may be found from investigations of the vapour pressures at low temperatures. The chemical equilibrium at any temperature may then be calculated without a single measurement of its value. If we use the equation C, = C r + R = a + R + 2/3T for the specific heat of each gas, the equation (8) goes over into : _^ + M^B),,, T + ^ T 2,,, where a = . (9) Full details of the application of equation (9) are to be found in the monographs referred to below ; it will be sufficient here to work out one case in detail, and we shall take the dissociation of carbon dioxide : K K 2 500 THERMODYNAMICS For convenience we replace concentrations by partial pressures : T + 2 - (10) P 2 CO'2 Let g = percentage of dissociation, i.e., the number of mole- cules per 100 which are broken up at a given temperature, then /;/100 = 7, the fraction of dissociation. 27 7 _ 2 (100 7) pcc ~ 200 + 7 ' P 2 ~200 + 7 ' ^ co ' 2 ~~ 200 X 7 K ' = r-.rff-^y v 2 + ioo/ V 1 ioo/ = ^ when 7 is small .'. log K' = 3 log 7 log 2 = 3 log 7 0'3. We multiply each term by 0'4343 to transform natural to common logarithms, and obtain finally : n. Q log K' = where C, = 0*4843^ According to the assumptions of 211 : 0-4848 * y<( % +B >= 1-75* \ and yS is calculated by means of another arbitrary assumption, viz., that the molecular heat of a gas is of the form : CJ? = 8-5 + 2/3/1 THE THEOREM OF NERNST 501 - Again Q P = Qo + 8-5*T + /3T 2 . . . (12) from which Q is calculated from the observed heat of reaction at any one temperature. The data are (cf. Nernst, Recent Applications) : C0 2 C, =10-05 0,020,= 6-9 Q, = 136000 cal. at T = 290 .-. Qo = 135000, and Q p = 135000 + 3-5T - 0-0030T 2 . The chemical constants, C,-, are : CO = 3-6 ; 2 = 2-8 ; C0 2 = 3-2 /. ^d = 2X3'(5 + 2-8 2 X 3-2 = 3'6 (The value for CO given by Nernst is 3'6 ; according to Weigert, Abegg's Handbuch der anorg. Chem., Art. Kohlenstott', the value 2'6 is more probable.) Thus, finally : 29600 . t __ 3 log 7 = + 1-75 T 0-00066 T + 3-9 y = lOOy. T (obs. Nernst and Wartenberg). T (calc.). 0-00419 0-029 1300 1478 1369 1552 In some cases the variation of specific heat with temperature and the difference of specific heats, are small, and the terms with T and log T, respectively, may be omitted : log K' = log K' = - 4-571T ' 4-571 since Qo in these cases is approximately equal to 502 THEBMODYNAMICS The table of chemical constants given by Nernst (Berl. Her. 1909, p. 247) is : H 2 l-6 (2) NH 3 3-31 S 2 U) 3-15 CH 4 2-5 H 2 3-7 N 2 2-6 CC1 4 3-1 2 2-8 CHC1 3 3-2 CO 3-6 HC1 3-0 C1 2 3-2 H 2 S 3-0 C0 2 3-17 I 2 4-2 (1) Preuner and Schupp, Zeitschr. physik. Chem., 68, 2, 129, 1909. (2) Nernst, Zeitschr. Elektrochem., 15, 18, 687, 1909. 213. Solid and Liquid Solutions. The theorem of Nernst applies only to chemically homogeneous condensed phases ; the entropy of a condensed solution phase has at absolute zero a finite value, owing to the mutual presence of the different components. It may reasonably be assumed that the terms in the expression for the entropy which depend on the temperature diminish, like the entropy of a chemically homogeneous condensed phase, to zero when T approaches zero, and the entropy of a condensed solution phase at absolute zero is equal to that part of the expres- sion for the entropy which is independent of temperature, and depends on the composition (Planck, Thermodynamik, 3 Ann 1 ., 279). In the case of a dilute solution this is ( 185) : S = - RSvJm; .... (1) In the case of a solution of moderate concentration we may perhaps assume the same expression (cf. van Laar, Thermo- dynamik und Chemie ; Thermodyn. Potential ; Planck, loc. cit.), whenever the solutions can legitimately be considered as brought, by suitable changes of temperature and pressure with unchanged composition, into ideal gas mixtures ( 185). THE THEOREM OF NERXST 503 The entropy of a condensed homogeneous solution (e.g., an isoniorphous solid solution) is then : . (-2) If this expression goes over into that for a chemically homogeneous substance. Corolla rii 1. The specific heat of a condensed solution vanishes at absolute zero. Corollary 2. The coefficient of expansion of a condensed solu- tion vanishes at absolute zero. 214. Heterogeneous Equilibria. For a system composed of any number of condensed chernically homogeneous phases and a homogeneous gas phase we have the symbol : H"/o" | HO' "/Mo"' ! where all letters with zero suffix refer to condensed bodies. In the isothermal-isopiestic change : ono ' &HO" ' ono'" : . . . : &HI : S 2 : on 3 : = VQ : v " : VQ" : . . . : n : r- 2 : ^3 : we have, as the condition for equilibrium : gef> = (fy = 0, 8T = 0) . . . (1) But 4> = HO'o" -f . . . + HI/LI + 11-2^-2 + (2) where ^>o' = thermodynamic potential of 1 mol condensed homo- geneous t'-th phase = W - T ; + 7V = V - W .... (3) and p>i = chemical potential of t'-th component of gas phase = T(9,- + Elnc,) ....... (4) For a very small isothermal-isopeistic change, with changes of composition of the separate phases vanishing in the limit ( 142): . . = =0 - (5) 504 THERMODYNAMICS where the first summation extends over the ; condensed phases, and the second over the i components in the gas phase. = ^p But s (r) , i.e. [s (r) ] = dT . , . (6) by Nernst's theorem, and 9< = ^ - Sff> + * \ C)|WT - | | dT + E (inp - to *, Tin/i rnu) = T ~J T where log K' = Sr, log 7;, and the other symbols have the significance of 201. It will be observed that the chemical constants refer to the gases only. We may often omit the term in T and write simply : log K/ = ~ *OTlT + l ' 75v log T + B ' C < ' (13) and for approximate calculations may even put : Qo = Q,, (approx.) when log K' = - ^^fc + 2i'A . (14) In connection with dissociation equilibria of the type Na2HP0 4 . 12H 2 NasHPOi + 12 H 2 0, Nernst points out that the equation : log J* = ~ -4fa + 1-75 log T + C where C = 3"2 (mean for 1 mol of various gases, cf. 212) leads to a revision of the de Forcrand rule ( 113), for when p = 1 atm. : -^ = 4-57 (1-75 log T + 3-2) which, according to the rule, is approximately 33. This can only hold in the middle of an extended range of T. It must be borne in mind that the equations (11), etc., hold good only when the condensed phases are chemically homo- geneous. Thus Foote and Smith, who determined the dissocia- tion pressure of cuprous oxide : 2Cu 2 0^4Cu + 0. 2 found no agreement with Nernst's equation, which is not sur- 506 THERMODYNAMICS prising, as the liquid phase is a solution of copper in cuprous oxide. Stahl (Metallurgie, 1907, p. 682) calculated the temperatures at which the pressures of oxygen over dissociating metallic oxides : 2MOz2M + 2 reach 0'21 atm. (when they are in equilibrium with atmospheric oxygen, and begin to reduce freely), by means of the equation : log 0-21 = - 0-6778 = - L_ + 1-75 ]og T _j_ 2 -g (2-8 is the chemical constant of oxygen.) He plotted log p as abscissae against T as ordinates, and cut the curves by an ordinate log p = 0'6778. Numerical Example. Dissociation of ammonium hydrostilphide : Molecular heat of solid NH^HS = 19-1 ,, ,, gaseous NH 3 = 9'5 H 2 S = 8-5 Heat of dissociation of NH 4 HS = 22800 cal. all for ordinary temperature, T = 300 .-. Q p = 21900 + 7-OT - 0-013T* .'. I log K = log| = - ^ + 1-75 log T - 0-0014T + 3-15 (3-15 = mean chem. const. == 3 ' + 3 ' 3 for NH 3 and H 2 S.) For T = 298-1, p = 0*661 atm. (obs.). If we putjj = 0-661 in the equation we find T = 318. The approximate equation : gives T = 312. 215. Befthelot's Rule and the Measurement of Chemical Affinity. If we glance back over the various branches of application of thermodynamics to chemical problems detailed in the preceding sections of this book, looking more especially at the historical sequence, we shall find that the physical chemists have, until recently, focussed their attention on the theory of dilute solu- tions. This preference is due to the great stimulus given by the THE THEOREM OF NERNST 507 pioneering work of van't Hoff, and th'e rich harvest which was then thrown open to the workers in the new field. The older thermo- chemistry, summarised in the famous "rule" of Thomsen and Berthelot ( 118), that the heat of reaction was a measure of the work done by the chemical forces, was brought into discredit, and violently attacked by the disciples of the new school, and, so far as its theoretical foundations went, was successfully deposed. At the same time it must have been obvious to anyone acquainted with thermochemical data, and not so " purely chemical " as to be unable to carry out the requisite thermodynamic calculations, that the afore-mentioned rule was not so useless as might be supposed. Nernst, in his Theoretische Chemie, devoted a whole chapter to a critical examination of the rule of Thomsen and Berthelot, and he concluded that in many cases the heat of reaction certainly does correspond very closely with the maximum work, A T , which latter magnitude he took from van't Hoff as a measure of the chemical affinity. AVhilst pointing out that it very often gives results wholly incompatible with experience, and cannot therefore be indiscriminately applied, Nernst showed that the rule nevertheless holds good in too many cases to be wholly false ; in an appropriate metaphor he claimed that it "contains a genuine kernel of truth which has not yet been shelled from its enclosing hull." This labour of emancipation was partially effected in the newer work of the same author, Application* of Thermodynamics to Chemistry, 1907, which is an attempt to place the rule of Berthelot on a scientific basis, and to determine under what conditions its use is legitimate. He points out that the equation : A T = , + T(Ss - SO shows that it cannot be true in changes involving a considerable change of bound energy, nor those in which substances of very variable concentration participate (as in gas reactions and reactions in dilute solution). It does, however, hold good in the following cases : (1) Reactions between pure solids: e.g., Pb + 2AgCl = PbCla + 2Ag. (2) Reactions between pure liquids : e.g., Pb + Br 2 = PbBr 2 . 508 THERMODYNAMICS (3) Reactions between concentrated solutions : e.g., the electro- motive force of the lead accumulator corresponds almost exactly with the heat of dilution of the acid when the latter is concen- trated, whilst in dilute solutions the difference is very great (Nernst, Wied. Ann., 53, 57, 1894). (4) Reactions occurring between dilute solutions in some galvanic cells (Helmholtz-Thomson rule, 200). The new theorem then introduced by Nernst in the form that A T = Q in the cases specified, and its consequences, have already been considered. Nernst further remarked that the stability of a compound is measured by the quantity which can exist under given conditions, and the previous equations enable one to determine this, at least approximately, with the help of the heats of formation. A com- pound under given conditions is in general either very stable or very unstable equilibria in which all the components exist in appreciable concentrations are the exception rather than the rule. A considerable heat of formation usually corresponds with marked stability (cf. HC1, HBr, HI). Thus, in the formation of acetylene from carbon and hydrogen we have : and of benzene : 3740 Only at very high temperatures would the right-hand side of (a) have a small positive value ; i.e., C 2 H 2 is formed under such conditions (e.g., in the arc). Benzene, however, must be utterly unstable under all such circumstances. 216. Application of Nernst's Theorem to the Calculation of Electromotive Forces: Condensed Reactions. The most simple application of the equations : [Ar] = [<2o] - /ST 2 . . . . (1) [Q\ = [<2o] + T 2 .... (2) proposed by Nernst ( 209) for a reaction between pure solid and liquid substances, is the calculation of the electromotive THE THEOREM OF NERNST 509 force of a galvanic cell with the aforesaid reaction as its source of current. Then : [A T ] = 23046E cal ..... (3) where E = E.M.F. in volts. Before Nernst had put forward his theory, Bodlander (Zeitschr. physik. Chem., 27, 55, 1898) had been able to calculate the solu- bility of a salt by the measurement of its decomposition voltage, and had found that where the reaction occurring is the dissocia- tion of a solid salt into solid uncharged atoms, the work done to split up the salt into its ions, and discharge these at the electrodes, is very nearly equal to the heat of formation. E.g., with the cell Pb/PbI 2 sat. sol./I 2 (Pt) Pb + I 2 = Pbl, the E.M.F. can be calculated thus : (1) Let 1 mol PbI 2 form a saturated solution of concentration f. The ionic concentrations are P J; = / = . Now suppose this brought to unit concentration by compressing with an osmotic piston. Work done (A T )i = RT/n&b + 2RT/H&' = 3RT/ (2) Discharge each ion at a reversible electrode. Work done (A T )a = sum of decomposition voltages = e P b + */ .-. total work = (A T )i + (A T ) 2 = 8RTJn + e pi; + , . . . - log K' - 512 TH EEMOLYNAMICS for the E.M.F. of a gas element, where TTI, *-... are the pressures of the gases supplied to the electrodes, log K' = logjV pj . . . is the equilibrium constant, and v = S^i- It is evident that E is known as soon as K' is known for the current-producing reaction at various temperatures. K' may, however, be calculated from Nernst's theorem in the manner explained in 210, and hence the problem of calculating the E.M.F. of a gas element is solved. 218. References to Chap. XVII. (1) Griineisen, Ann. Phys. 26, 401 ; C. L. Lindemann, Physik. Zeitschr 1.1, 1197, 1912. (2) Bronsted, Zeitschr. physik. Chem., 55, 371, 1906. (3) Recent Applications of Thermodynamics to Chemistry, Constable, 1907 : Theoretische Chonie, 1 Aufl., 1913 ; Pollitzer, Serechnung ChemiseJur Affinitdten nach m motion is supposed to have the same energy, viz., -^-, and hence a OTjrn a diatomic molecule has -.r- for the translatory part, and ^ for the rotatory part, or ~ in all. The molecular heat is therefore C t . = -Q- = 4*96, which is in agreement with the values for 2 , N 2 , etc., at low temperatures. That of the halogens is greater, and Boltzmann supposed the connexion between the atoms was loosened with rise of temperature. Triatomic gases have the fiTJ value C. p = fr = 5 '95, which is exhibited by steam and carbon Zi dioxide at low temperatures.' 2 ' KINETIC THEORIES IN THERMODYNAMICS 517 221. Specific Heats of Solids: Kinetic Theory. Boltzmann <:5) extended the theory to solids, and was led to a result which to a certain extent is in harmony with the law of D along and Petit. An elementary solid, such as silver, is regarded as composed of atoms oscillating about fixed centres. The total energy content is therefore partly kinetic and partly potential. Since the solid has a finite compressibility, the atoms may be supposed to be maintained at small distances apart by forces they exert upon one another, and these may be resolved into two sets, one of which opposes a closer approximation of the atoms, and the other tends to draw the latter together. Both are functions of the distance between the atoms, and for a given distance are equal, since the form of the body is altered by external forces alone. The force exerted on an atom in its position of rest by the neighbouring atoms is then : (,-) - (>) = 0. The two functions have the same form, since any atom may be selected, and the system is symmetrical. If the atom is displaced from its equilibrium position through the distance Sr, the force of restitution is : ftr + Sr) - (r - 6>) = (r) + '('0^ + (} (Sr) 2 + . . . At low temperatures the vibrations are small /. elastic force = %'(r)8r = a, say, and the dynamical equation of motion is : where m = mass, x = displacement, t = time. The solution of the equation is : x = A cos (kt + e) where k = 518 THERMODYNAMICS i.e., the particle executes simple harmonic motions of frequency : - L JL /?> ~ 2w ~~ 2^ V / The velocity is -rr = A/c sin kti at the instant / = ti A 2 a .*. the kinetic energy per unit mass is T = -=- sin a kti. To calculate the potential energy we assume an expression of the form : If x is small, all terms after ex* may be neglected. Also Y must be a minimum in the equilibrium state x = and since it vanishes in the equilibrium state, a is also zero, .-. V = ex* But T + V = const. since no external forces are acting on the system, i ( dx \ 2 . .*. \ I -T- ) + cx 2 = const. dx d*x . dx r C= - = - The mean values of cos kti and sin /cfi over any interval of time including a whole number of vibrations are each ^, . T = V = . 4 The mean values of the kinetic and potential energies of an atom are therefore equal over any interval of time, and if we add together all the kinetic energies and all the potential energies of a KINETIC THEORIES IN THERMODYNAMICS 519 a large number of atoms in a solid, which are vibrating in the manner described, the two sums will, at every instant, be equal. Now let the silver be supposed to be surrounded by nionatomic gas, such as helium. The atoms of the gas, when they collide with those of the solid, will impart kinetic energy to them. According to Maxwell's Law of Equipartition of Energy, two bodies are in temperature equilibrium when the mean kinetic energy for each degree of freedom of motion of each atom is the same in both. The atoms of the solid and those of the gas have three degrees of freedom each ; to each degree corresponds the kinetic RT QT?T energy -^-, therefore the kinetic energy of the solid is - . The potential energy is equal to this, and the total energy content is therefore, per mol U = 3RT .-. C, = (?U/aT) c = 3R = 5-955 which agrees moderately well with the law of Dulong and Petit. Abnormally low atomic heats were explained by Richarz on the assumption of a diminution of freedom of oscillation consequent on a closer approximation of the atoms, which may even give rise to the formation of complexes. This is in agreement with the small atomic volume of such elements, and with the increase of atomic heat with rise of temperature to a limiting value 6'4, and the effect of propinquity is seen in the fact that the molecular heat of a solid compound is usually slightly less than the sum of the atomic heats of the elements, and the increase of specific heat with the specific volume when an element exists in different allotropic forms. There is, however, a fatal objection to the theory of Boltzmann. At very low temperatures the oscillations will be small, and should conform to the theory. But the atomic heats, instead of approaching the limit 5'955 at low temperatures, diminish very rapidly, and in the case of diamond the specific heat is already inappreciable at the temperature of liquid air. A new point of view is therefore called for, and it is a priori very probable that this will consist of a replacement of the hypothesis of Equiparti- tion of Energy adopted by Boltzmann. This supposition has been verified, and the new law of partition of energy derived 520 THERMODYNAMICS from investigations in quite another department of physics, viz., in the theory of radiation. 222. Planck's Theory of Radiation. In his theoretical investigation on the exchange of energy between ether and matter, Planck (4) considered the. absorption and emission of radiation by a linear electrodynamic resonator, i.e., a system composed of two opposite electric charges, oscillating on a fixed straight line. If a closed space contains a number of such resonators, of small damping, and at large distances apart, the effect of an exciting electromagnetic radiation of definite colour on the system may be considered. Such radiation, in so far as it is open to observation, must not be regarded as com- posed of a single vibration of absolutely definite frequency, but as covering a definite breadth of the range of frequencies, i.e., it would appear in the spectroscope as an extremely narrow band, not as a line. The law of Kirchhoff, that in a space containing bodies of any character whatever, the distribution of radiation ultimately settles down into a steady state of thermodynamic equilibrium, is interpreted by Planck as indicating that in such a space there exists a magnitude which always increases in temporal changes, and attains a maximum in thermodynamic equilibrium ; this is the electromagnetic entropy (cf. 48 (1) ). The question as to the distribution of frequencies over a small range in a radiation of definite " colour " is solved on the assumption that in natural radiation the deviations of single rapidly varying magnitudes from their mean value shall be random, and the application of the theory of probabilities is carried over from the kinetic theory of gases to the theory of radiation. The energy per unit volume, and the entropy, of radiation in equilibrium with a system of resonators of frequency v can then be calculated. The remarkable result of this investigation which is of interest to us here, is the hypothesis introduced by Planck that the energy of a resonator does not increase continuously, like the kinetic energy of a particle moving in a straight line under the action of a force, but per saltum, in whole multiples of a quantity , proportional to the frequency : < = /3* (1) KINETIC THEORIES IN THERMODYNAMICS 521 where R = gas constant = 8'315 X 10 7 r^- N = number of molecules in a mol = 60 X 10' 22 /3 = a universal constant = 4'86 X 10 ~ n .-. e = 6-551 X v = h v . . . . (2) e is called the Energiequantuin for the frequency v ; we may refer to it briefly as the ergon. Einstein < 5 > remarked that this point of view can be carried over to the theory of the energy content of a solid body if we suppose that the positive ions of Drude's theory ( 198) may be looked upon as the vibrating resonators, and the seat of the heat content of the body (KSryerwarm^), He calculated the expression : U- 3K ^ (3) U - . . . . (6) t' T - 1 for the atomic heat of a solid composed of such spacial resonators, each equivalent to three linear electrodynamic resonators, with one definite frequency v. The atomic heat is therefore : c r = au/?T (4) It may be remarked that there is no call for an atomic theory of energy, analogous to the atomic theories of matter and elec- tricity, as the discontinuity arises from the peculiar character of the system (cf. Planck, Bet:, 45, 5, 1912). 223. Specific Heats of Solids: Einstein's Equation. We consider again a monatomic solid in contact with a mon- atomic gas ( 221). The atoms of the gas, by collision with those of the solid, give up energy to them, and we have to find the way in which the energy of the system is distributed between the gas and the solid when there is equilibrium. For the distribution of velocities in the gas, in any given plane we have, according to Maxwell's distribution law : + * dN = N A' 2 <> *~~dudr . . . (1) 522 THERMODYNAMICS where (IN denotes the number of molecules in the total number NO, the velocities of which lie between u + du and r + dv. If we transform to polar co-ordinates, and put - = E for the kinetic energy of a particle m, we find : dN = N A magnitude e = ^- j3v, the energy content is given by the area i>o under the step-formed curve, which shows that a certain number of atoms are at rest, with zero energy, another set are all vibrating with the energy e, corresponding with the number of gas atoms the kinetic energy of which increases continuously from e to 2e, and so on. The larger the ergon, the less will be the total energy content of the solid in comparison with the energy of the gas atoms ; when the ergon is small (e.g., with lead) the two areas become nearly equal. If we multiply by 3 to include the three degrees of freedom, we have for the total energy content : "I Ut 2e\ / 2 is the volume density of the radiant energy), with the expression for the mean energy of the atom. For the latter we may take either Boltzmann 's expression : U = (2a) No or that of Einstein : . . . (26) 1 and arrive at the equation of Rayleigh : . . . . (8a) or of Planck : R 8*1? 0I> '^No"?" -*r- . ' ' (3b) e T 1 respectively. With regard to these we may simply quote a remark of Lorentz (Theory of Electrons, Leipzig, 1909, p. 287) : " The only equation by which the observed phenomena are satisfactorily accounted for is that of Planck, and it seems necessary to imagine that, for short waves, the connecting link between matter and ether is KINETIC THEORIES IN THERMODYNAMICS 525 formed, not by free electrons, but by a different kind of particles, like Planck's resonators, to which, for some reason, the theory of equipartition does not apply. Probably these particles must be such that their vibrations and the effects produced by them cannot be appropriately described by means of the ordinary equations of the theory of electrons ; some new assumption, like 40 60 Aba, Temperature FIG. 91. Planck's hypothesis of finite elements of energy will have to be made." (Cf. Jeans, Phil. Mag. [vi.], 10, 91, 1905 ; Planck, Acht Forlesungen, p. 95 ; Ann. d. Phys., 1912.) From his equation Planck was able to calculate the number of atoms in a gram-molecule : No = 61-75 X 10 22 526 THERMODYNAMICS which is in excellent agreement with the values deduced in other quite different ways. (2) Direct Measurements of Specific Heats at Low Temperatures. By means of the experimental methods briefly referred to in 9 a large number of specific-heat measurements have been made at very low temperatures. In Fig. 91 we have the atomic heats of some metals, and of the diamond, represented as functions of the temperature. The peculiar shape of the curves will be at once apparent. At a more or less low temperature, the atomic heat decreases with extraordinary rapidity, then apparently approaches tangentially the value zero in the vicinity of T = 0. The thin curves represent the atomic heats calculated from the equation : - a, = 3R with the values of fiv indicated alongside. Since the specific heat actually measured is C,, the value of C tf must bs calculated from the equation (3) of 64 : m 9*' dp T W 8T Griineisen (Ann. Phys., 26, 401) gives the expression : where a = coefficient of linear expansion f] = ,, compressibility a = atomic volume. There is, however, little exact data for its application. Lindemann and Magnus (6) found that C,, could be fairly well represented by adding to Einstein's equation an arbitrary term aT^ where a = const. In the case of ice, the term aT 6 was used. KINETIC THEORIES IN THERMODYNAMICS 527 In the case of compounds, the terms for the separate atoms must be added together, and : where the summation extends over all values of v. The same equation may be used when an element has more than one value of v. At very low temperatures this correction term is negligible ; according to Lindemann and Magnus, it takes account of the work done against the force of cohesion during the expansion. The influence of the heat capacity of the free electrons which on Drude's theory, together with the positive ions bearing the internal vibrational heat energy, compose the solid, appears to be negligibly small, since the atomic heats at very low temperatures (when the part due to the positive ions considered in Einstein's theory is very small) are exceedingly small (cf. Nernst, BerL Ber., 12, 247); and in addition the atomic heat of lead, with its small ergon, exceeds only very slightly the value 3R. Measurements at the low temperature of liquid hydrogen (Nernst, loc. cit.) show that the curves do not follow the Einstein equation exactly ; the reason is not yet clear, but Lindemann (Diss., p. 38) suggests a polymerisation of some 20 atoms to a molecule, or that some atoms are rigidly connected together, and vibrate more slowly. In the case of sylvine (KC1) the observed curve lies above, but almost parallel with the Einstein curve, and Nernst and Lindemann (7) used the equation : r - 3R 2 gT \T7 T '7~^iV + [> T -v which represents the relations quite well. The energy per mol according to the new formula is : U = |R i.e., consists of two parts which approach equality at high 528 THERMODYNAMICS temperatures. According to Boltzmann's theory, the energy of a solid is half kinetic and half potential ; with the new assumption this cannot be true except at high temperatures, when in fact the deductions from the theory of ergonic distribution pass into those from the older theory (cf. 267). (3) Melting -Point and Atomic Volume : Law of Dulong and Petit. Lindemann (8) has made an interesting application of the new theory in the determination of the frequency of atomic vibration, v, from the melting-point. He assumes that at the melting-point, T,, the atoms perform vibrations of such amplitude that they mutually collide, and then transfer kinetic energy like the molecules of a gas. The mean kinetic energy of the atom will then increase by fRT,, when the liquid is unpolymerised and the fusion occurs at constant volume ; this is the molecular heat of fusion. Let r = distance between centres of two neighbouring atoms, p = distance between the surfaces as a fraction of r, then the diameter of the sphere of molecular activity is ) (1) and - = distance through which an atom must swing from its a equilibrium position to hit its neighbour. The kinetic energy when passing its position of equilibrium is therefore : - (2) where a 2<'(V) i 8 ^e quasi-elastic force opposing atomic approximation ( 221). But, according to Einstein's equation, the kinetic energy of an atom in its equilibrium position at the commencement of fusion is, with assumed linear vibrations : .T., / /Q,,\ 2 p v r r _R_ "NO J n >* T = I ,*_lJ e T > -1 = | (T.-^)approx. . . (8) KINETIC THEORIES IN THERMODYNAMICS 529 Thus or which gives the value of the force. Thence : ' ~ 27r V ^~ 2^ (5) P 2 /- 2 N w where m = mass of an atom /. N m = atomic weight = M. If p is assumed alike for all solids, and the second term is neglected, we find : where V = atomic volume. The values of " v obs." were determined from the atomic heats by means of Einstein's formula, those of " v calc." were obtained from equation (6) : T* vX 10 - 12 obs. v X 10- l - calc. Pb" 600 1-44 1-4 Ag 1234 3-3 (3-3) Cu 1357 4-93 5-1 I 386 1-5 1-4 Pt 2018 31 3-1 Si 1703 10-7 7"2 (A- = 2'12 X 10 12 from the atomic heat of silver.) The deviations from the law of Dulong and Petit may therefore be quantitatively calculated, at any temperature, from the melting- point and atomic volume. Elements with high melting-point and small atomic volume (e.g., carbon) deviate from the law, the value of v being all the greater the higher the melting-point and density, and the smaller the atomic weight. In the case of lithium, the effect of the small atomic weight is compensated by the low melting-point, and the atomic heat is normal. 530 THERMODYNAMICS According to Joule's law ( 9), the molecular heat of a compound is the sum of the atomic heats of its components, and since this holds good even when the atomic heats are " irregular," i.e., not equal to 6'4, it seems that the heat content of a solid resides in its atoms, and not in the molecular complexes as such. This agrees with Einstein's theory. Hence the molecular heat of a compound should be calculable by means of the formula : - I from the atomic heats of its constituents in that one extends the summation over the ^-values of the latter. The frequency of an atom, should therefore be the same in the free state and in combination. That this is not the case follows from the experimental data discussed by A. Russell (9) , and F. Koref (10) has attempted to calculate the change of frequency of an element when it enters into combination by means of the alteration of melting-point and atomic volume. According to Lindemann's equation, for the combined atom : For two atoms in a binary compound ; B 3 AV _ r b where r a , r b are the atomic radii. For the free elementary atom ; v' /T ' s /V '"' ^ = V~T, VT If the molecular volume is the sum of the atomic volumes V' = V ^- /T? ' * " V T; KINETIC THEORIES IN THERMODYNAMICS 581 The change of melting-point can produce a very marked alteration of v (e.g., carbon and carbon dioxide). Koref calculates the v' values for a number of compounds, and the molecular heats thence obtained agree with the experimental values moderately well. (4) The Theorem of Nernst. Nenist .'/, 14(i. 1908 ; A. Wigand, Ann. Phys., [4], 22, 79, 1907. (4) M. Planck, Vorlesungen iiber die Theorie der Warme*trahlnn characteristic equation, 221 ; vapour- pressure equation, 180 Van't Hoff's boiling-point equa- tion, 295 ; freezing-point equa- tion, 299 ; theory of solution, 287 ; principle of mobile equili- brium, 304, 340 Variance, 169 Vapour, energy and entropy of, 183; pressure, 171, 180, 201 ; curves, 382, 399, 408; of dilute solutions, 288 ; equations, 179, 190, 492 544 INDEX Velocity of sound, 146 Virtual change, 92 ; work, 50 Viscosity, 87 Vital processes, 35, 70 Voltaic cell, 53, 357, 455 WATT'S rule, 184 Wax -type of fusion, 195 Weston normal element, 456 Welding, 194 Wetting, 445 Woestyn's law, 16 Work, 21, 22, 41, 45, 47, 147, 349 Working substance, 53 ., PRINTERS, LONDON ANP TOKB1UDGE. VAN NOSTRAND'S "WESTMINSTER" SERIES Bound in Uniform Style. Fully Illustrated. Price S2.OO net each. Gas Engines. By W. J. MARSHALL, Assoc. M.I.Mech.E., and CAPT. H. RIALL SANKEY, R.E. (Ret.). M.Inst.C.E., M.I.Mech.E. 300 Pages, 127 Illustrations. LIST OF CONTENTS : Theory of the Gas Engine. The Otto Cycle. The Two Stroke Cycle. Water Cooling of Gas Engine Parts. Ignition. Operating Gas Engines. The Arrangement of a Gas Engine Instal- lation. The Testing of Gas Engines. Governing. Gas and Gas Producers. Index. Textiles* By A. F. BARKER, M.Sc., with Chapters on the Mercerized and Artificial Fibres, and the Dyeing of Textile Materials by W. M. GARDNER, M.Sc., F.C.S. ; Silk Throwing and Spinning, by R. SNOW; the Cotton Industry, by W. H. COOK ; the Linen Industry, by F. BRADBURY. 370 Pages. 86 Illustrations. CONTENTS : The History of the Textile Industries ; also of Textile Inventions and Inventors. The Wool, Silk, Cotton, Flax, etc.. Growing Industries. The Mercerized and Artificial Fibres em- ployed in the Textile Industries. The Dyeing of Textile Materials. The Principles of Spinning. Processes preparatory to Spinning. The Principles of Weaving. The Principles of Designing and Colouring. The Principles of Finishing. Textile Calculations. The Woollen Industry. The Worsted Industry. The Dress Goods, Stuff, and Linings Industry. The Tapestry and Carpet Industry. Silk Throwing and Spinning, The Cotton Industry. The Linen Industry historically and commercially considered. Recent Developments and the Future of the Textile Industries. Index. Soils and Manures. By J. ALAN MURRAY, B.Sc. 367 Pages. 33 Illustrations. CONTENTS : Introductory. The Origin of Soils. Physical Proper- ties of Soils. Chemistry of Soils. Biology of Soils. Fertility. Principles of Manuring. Phosphatic Manures. Phosphonitro- genous Manures. Nitrogenous Manures. Potash Manures. Compound and Miscellaneous Manures. General Manures. Farm- yard Manure. Valuation of Manures. Composition and ManuraJ Value of Various Farm Foods. THE "WESTMINSTER" SERIES Coal. By JAMES TONGE, M.I.M.E., F.G.S., etc. (Lecturer on Mining at Victoria University, Manchester). 283 Pages. With 46 Illustrations, many of them showing the Fossils found in the Coal Measures. LIST OF CONTENTS : History. Occurrence. Mode of Formation of Coal Seams. Fossils of the Coal Measures. Botany of the Coal-Measure Plants. Coalfields of the British Isles. Foreign Coalfields. The Classification of Coals. The Valuation of Coal. Foreign Coals and their Values. Uses of Coil. The Production of Heat from Coal. Waste of Coal. The Preparation of Coal for the Market. Coaling Stations of the World. Index. Iron and Steel By J. H. STANSBIE, B.Sc. (Lond.), F.I.C. 385 Pages. With 86 Illustrations. LIST OF CONTENTS : Introductory. Iron Ores. Combustible and other materials used in Iron and Steel Manufacture. Primitive Methods of Iron and Steel Production. Pig Iron and its Manu- facture. The Refining of Pig Iron in Small Charges. Crucible and Weld Steel. The Bessemer Process. The Open Hearth Process. Mechanical Treatment of Iron and Steel. Physical and Mechanical Properties of Iron and Steel. Iron and Steel under the Microscope. Heat Treatment of Iron and Steel. Elec- tric Smelting. Special Steels. Index. Timber, By J. R. BATERDEN, Assoc.M.Inst.C.E. 334 Pages. 54 Illustrations. CONTENTS : Timber. The World's Forest Supply. Quantities of Timber used. Timber imports into Great Britain. European Timber. Timber of the United States and Canada. Timbers of South America, Central America, and West India Islands. Tim- bers of India, Burma, and Andaman Islands. Timber of the Straits Settlements, Malay Peninsula, Japan and South and West Africa. Australian Timbers. Timbers of New Zealand and Tasmania. Causes of Decay and Destruction of Timber. Seasoning and Impregnation of Timber. Defects in Timber and General Notes. Strength and Testing of Timber. " Figure " in Timber. Appendix. Bibliography. Natural Sources of Power. By ROBERT S. BALL, B.Sc., A.M.Inst.C.E. 362 Pages. With 104 Diagrams and Illustrations. CONTENTS : Preface. Units with Metric Equivalents and Abbre- viations. Length and Distance. Surface and Area. Volumes. Weights or Measures. Pressures. Linear Velocities, Angular Velocities. Acceleration. Energy. Power. Introductory Water Power and Methods of Measuring. Application of Water Power to the Propulsion of Machinery. The Hydraulic Turbine, ( 2 ) THE "WESTMINSTER" SERIES Various Types of Turbine. Construction of Water Power Plants. Water Power Installations. The Regulation of Turbines. Wind Pressure. Velocity, and Methods of Measuring. The Application of Wind Power to Industry. The Modern Windmill. Con- structional Details. Power of Modern Windmills. Appendices. A.B.C Index. Electric Lamps. By MAURICE SOLOMON, A.C.G.T., A.M.I.E.E. 339 Pages. 112 Illustrations. CONTENTS : The Principles of Artificial Illumination. The Produc- tion of Artificial Illumination. Photometry. Methods of Testing. Carbon Filament Lamps. The Nernst Lamp. Metallic Filament Lamps. The Electric Arc. The Manufacture and Testing of Arc Lamp Carbons. Arc Lamps. Miscellaneous Lamps. Compari- son of Lamps of Different Types. Liquid and Gaseous Fuels, and the Part they play in Modern Power Production. By Professor VIVIAN B. LEWES, F.I.C., F.C.S., Prof, of Chemistry, Royal Naval College, Greenwich. 350 Pages. With 54 Illustrations. LIST OF CONTENTS : Lavoisier's Discovery of the Nature of Com- bustion, etc. The Cycle of Animal and Vegetable Life. Method of determining Calorific Value. The Discovery of Petroleum in America. Oil Lamps, etc. The History of Coal Gas. Calorific Value of Coal Gas and its Constituents. The History of Water Gas. Incomplete Combustion. Comparison of the Thermal Values of our Fuels, etc. Appendix. Bibliography. Index. Electric Power and Traction. By F. H. DAVIES, A.M.T.E.E. 299 Pages. With 66 Illustrations. LIST OF CONTENTS : Introduction. The Generation and Distri- bution of Power. The Electric Motor. The Application of Electric Power. Electric Power in Collieries. Electric Power in Engineering Workshops. Electric Power in Textile Factories. Electric Power in the Printing Trade. Electric Power at Sea. Electric Power on Canals. Electric Traction. The Overhead System and Track Work. The Conduit System. The Surface Contact System. Car Building and Equipment. Electric Rail- ways. Glossary. Index. Decorative Glass Processes. By ARTHUR Louis DUTHIE. 279 Pages. 38 Illustrations. CONTENTS : Introduction. Viuious Kinds of Glass in Use : Their Characteristics, Comparative Price, etc. Leaded Lights. Stained Glass. Embossed Glass. Brilliant Cutting and Bevelling. Sand- Blast and Crystalline Giass. Gilding. Silvering and Mosa/c. Proprietary Processes. Patents. Glossary. ( 3 ) THE "WESTMINSTER'* SERIES Town Gas and its Uses for the Production of Light, Heat, and Motive Power. By W. H. Y. WEBBER, C.E. 282 Pages. With 71 Illustrations. LIST OF CONTENTS : The Nature and Properties of Town Gas. The History and Manufacture of Town Gas. The Bye-Products of Coal Gas Manufacture. Gas Lights and Lighting. Practical Gas Lighting. The Cost of Gas Lighting. Heating and Warm- ing by Gas. Cooking by Gas. The Healthfulness and Safety of Gas in all its uses. Town Gas for Power Generation, including Private Electricity Supply. The Legal Relations of Gas Sup- pliers, Consumers, and the Public. Index. Electro-Metallurgy. By J. B. C. KERSHAW, F.I.C. 318 Pages. With 61 Illustrations. CONTENTS : Introduction and Historical Survey. Aluminium. Production. Details of Processes and Works. Costs. Utiliza- tion. Future of the Metal. Bullion and Gold. Silver Refining Process. Gold Refining Processes. Gold Extraction Processes. Calcium Carbide and Acetylene Gas. The Carbide Furnace and Process. Production. Utilization. Carborundum. Details of Manufacture. Properties and Uses. Copper. Copper Refin- ing. Descriptions of Refineries. Costs. Properties and Utiliza- tion. The Elmore and similar Processes. Electrolytic Extrac- tion Processes. Electro-Metallurgical Concentration Processes. Ferro-alloys. Descriptions of Works. Utilization. Glass and Quartz Glass. Graphite. Details of Process. Utilization. Iron and Steel. Descriptions of Furnaces and Processes. Yields and Costs. Comparative Costs. Lead. The Salom Process. The Betts Refining Process. The Betts Reduction Process. White Lead Pro- cesses. Miscellaneous Products. Calcium. Carbon Bisulphide. Carbon Tetra-Chloride. Diamantine. Magnesium. Phosphorus. Silicon and its Compounds. Nickel. Wet Processes. Dry Processes. Sodium. Descriptions of Cells and Processes. Tin. Alkaline Processes for Tin Stripping. Acid Processes for Tin Stripping. Salt Processes for Tin Stripping. Zinc. Wet Pro- cesses. Dry Processes. Electro-Thermal Processes. Electro Galvanizing. Glossary. Name Index. Radio-Telegraphy. By C. C. F. MONCKTON, M.I.E.E. 389 Pages. With 173 Diagrams and Illustrations. CONTENTS : Preface. Electric Phenomena. Electric Vibrations. Electro-Magnetic Waves. Modified Hertz Waves used in Radio- Telegraphy. Apparatus used for Charging the Oscillator. The Electric Oscillator : Methods of Arrangement, Practical Details. The Receiver : Methods of Arrangement, The Detecting Ap- paratus, and other details. Measurements in Radio-Telegraphy. The Experimental Station at Elmers End : Lodge-Muirhead System. Radio - Telegraph Station at Nauen : Telefunken System. Station at Lyngby : Poulsen System. The Lodge- ( 4 ) THE "WESTMINSTER" SERIES Muirhead System, the Marconi System, Telefunken System, and Poulsen System. Portable Stations. Radio-Telephony. Ap- pendices : The Morse Alphabet. Electrical Units used in this Book. International Control of Radio-Telegraphy. Index. India-Rubber and its Manufacture, with Chapters on Gutta-Percha and Balata. By H. L. TERRY, F.I.C., Assoc.Inst.M.M. 303 Pages. With Illustrations. LIST OF CONTENTS : Preface. Introduction : Historical and General. Raw Rubber. Botanical Origin. Tapping the Trees. Coagulation. Principal Raw Rubbers of Commerce. Pseudo- Rubbers. Congo Rubber. General Considerations. Chemical and Physical Properties. Vulcanization. India-rubber Planta- tions. India-rubber Substitutes. Reclaimed Rubber. Washing and Drying of Raw Rubber. Compounding of Rubber. Rubber Solvents and their Recovery. Rubber Solution. Fine Cut Sheet and Articles made therefrom. Elastic Thread. Mechanical Rubber Goods. Sundry Rubber Articles. India-rubber Proofed Textures. Tyres. India-rubber Boots and Shoes. Rubber for Insulated Wires. Vulcanite Contracts for India-rubber Goods. The Testing of Rubber Goods. Gutta-Percha. Balata. Biblio- graphy. Index. The Railway Locomotive. What It Is, and Why It is What It Is. By VAUGHAN PENDRED, M.InstM.E., Mem.Inst.M.I. 321 Pages. 94 Illustrations. CONTENTS : The Locomotive Engine as a Vehicle Frames. Bogies. The Action of the Bogie. Centre of Gravity. Wheels. Wheel and Rail. Adhesion. Propulsion. Counter-Balancing. The Loco- motive as a Steam Generator The Boiler. The Construction of the Boiler. Stay Bolts. The Fire-Box. The Design of Boilers. Combustion. Fuel. The Front End. The Blast Pipe. Steam Water. Priming. The Quality of Steam. Superheating. Boiler Fittings. The Injector. The Locomotive as a Steam Engine Cylinders and Valves. Friction. Valve Gear. Expansion. The Stephenson Link Motion. Walschaert's and Joy's Gears. Slide Valves. Compounding. Piston Valves. The Indicator. Ten- ders. Tank Engines. Lubrication. Brakes. The Running Shed. The Work of the Locomotive. Glass Manufacture. By WALTER ROSENHAIN, Superin- tendent of the Department of Metallurgy in the National Physical Laboratory, late Scientific Adviser in the Glass Works of Messrs. Chance Bros. & Co. 280 Pages. With Illustrations. CONTENTS : Preface. Definitions. Physical and Chemical Qualities, Mechanical, Thermal, and Electrical Properties. Transparency ( 5 ) THE "WESTMINSTER" SERIES and Colour. Raw materials of manufacture. Crucibles and Furnaces for Fusion. Process of Fusion. Processes used in Working of Glass. Bottle. Blown and Pressed. Rolled or Plate. Sheet and Crown. Coloured. Optical Glass : Nature and Properties, Manufacture. Miscellaneous Products. Ap- pendix. Bibliography of Glass Manufacture. Index Precious Stones. By W. GOODCHILD, M.B., B.Ch. 319 Pages. With 42 lUustrations. With a Chapter on Artificial Stones. By ROBERT DYKES. LIST OF CONTENTS : Introductory and Historical. Genesis rf Precious Stones. Physical Properties. The Cutting and Polish- ing of Gems. Imitation Gems and the Artificial Production of Precious Stones. The Diamond. Fluor Spar and the Forms of Silica. Corundum, including Ruby and Sapphire. Spinel and Chrysoberyl. The Carbonates and the Felspars. The Pyroxene and Amphibole Groups. Beryl, Cordierite, Lapis Lazuli and the Garnets. Olivine, Topaz, Tourmaline and other Silicates. Phos- phates, Sulphates, and Carbon Compounds. INTRODUCTION TO THE Chemistry and Physics of Building Materials. By ALAN E. MUNBY, M.A. 365 Pages. Illustrated. CONTENTS : Elementary Science : Natural Laws and Scientific In- vestigations. Measurement and the Properties of Matter. Air and Combustion. Nature and Measurement of Heat and Its Effects on Materials. Chemical Signs and Calculations. Water and Its Impurities. Sulphur and the Nature of Acids and Bases. Coal and Its Products. Outlines of Geology. Building Materials : The Constituents of Stones, Clays and Cementing Materials. Clas- sification, Examination and Testing of Stones, Brick and Other Clays. Kiln Reactions and the Properties of Burnt Clays. Plasters and Limes. Cements. Theories upon the Setting of Plasters and Hydraulic Materials. Artificial Stone. Oxychloride Cement. Asphaite. General Properties of Metals. Iron and Steel. Other Metals and Alloys. Timber. Paints : Oils, Thinners and Varnishes ; Bases, Pigments and Driers. Patents, Designs and Trade Marks : The Law and Commercial Usage. By KENNETH R. SWAN, B.A. (Oxon.), of the Inner Temple, Barrister-at-Law. 402 Pages. CONTENTS : Table of Cases Cited Part I. Letters Patent. Intro- duction. General. Historical. I., II., III. Invention, Novelty, ( 6 ) THE "WESTMINSTER" SERIES Subject Matter, and Utility the Essentials of Patentable Invention. IV. Specification. V. Construction of Specification. VI. Who May Apply for a Patent. VII. Application and Grant. VIII. Opposition. IX. Patent Rights. Legal Value. Commercial Value. X. Amendment. XI. Infringement of Patent. XII. Action for Infringement. XIII. Action to Restrain Threats. XIV. Negotiation of Patents by Sale and Licence. XV. Limita- tions on Patent Right. XVI. Revocation. XVII. Prolonga- tion. XVIII. Miscellaneous. XIX. Foreign Patents. XX. Foreign Patent Laws : United States of America. Germany. France. Table of Cost, etc., of Foreign Patents. APPENDIX A. i. Table of Forms and Fees. 2. Cost of Obtaining a British Patent. 3. Convention Countries. Part II. Copyright in Design. Introduction. I. Registrable Designs. II. Registra- tion. III. Marking. IV. Infringement. APPENDIX B. I. Table of Forms and Fees. 2. Classification of Goods. Part III. Trade Marks. Introduction. I. Meaning of Trade Mark. II. Qualification for Registration. III. Restrictions on Regis- tration. IV. Registration. V. Effect of Registration. VI. Miscellaneous. APPENDIX C. Table of Forms and Fees. INDICES. I. Patents. 2. Designs. 3. Trade Marks. The Book: Its History and Development. By CYRIL DAVENPORT, V.D., F.S.A. 266 Pages. With 7 Plates and 126 Figures in the text. LIST OF CONTENTS : Early Records. Rolls, Books and Book bindings. Paper. Printing. Illustrations. Miscellanea. Leathers. The Ornamentation of Leather Bookbindings without Gold. The Ornamentation of Leather Bookbindings with Gold. Bibliography. Index. The Manufacture of Paper. By R. W. SINDALL, F.C.S., Consulting Chemist to the Wood Pulp and Paper Trades ; Lecturer on Paper-making for the Hertfordshire County Council, the Bucks County Council, the Printing and Stationery Trades at Exeter Hall (1903-4), the Institute of Printers ; Technical Adviser to the Government of India, 1905. 275 Pages. 58 Illustrations. CONTENTS : Preface. List of Illustrations. Historical Notice. Cel- lulose and Paper-making Fibres. The Manufacture of Paper from Rags, Esparto and Straw. Wood Pulp and Wood Pulp Papers. Brown Papers and Boards. Special kinds of Paper. Chemicals used in Paper-making. The Process of " Beating. The Dye- ing and Colouring of Paper Pulp. Paper Mill Machinery. The Deterioration of Paper. Bibliography. Index. ( 7 ) THE "WESTMINSTER" SERIES Wood Pulp and its Applications. By C. F. CROSS, B.Sc., F.I.C., E. J. BEVAN, F.I.C., and R. W. SINDALL, F.C.S. 266 pages. 36 Illustrations. CONTENTS: The Structural Elements of Wood. Cellulose as a Chemical. Sources of Supply. Mechanical Wood Pulp. Chemical Wood Pulp. The Bleaching of Wood Pulp. News and Printings. Wood Pulp Boards. Utilisation of Wood Waste. Testing of Wood Pulp for Moisture. Wood Pulp and the Textile Industries. Bibliography. Index. Photography: its Principles and Applications. By ALFRED WATKINS, F.R.P.S. 342 pages. 98 Illus- trations. CONTENTS : First Principles. Lenses. Exposure Influences. Prac- tical Exposure. Development Influences. Practical Develop- ment. Cameras and Dark Room. Orthochromatic Photography. Printing Processes. Hand Camera Work. Enlarging and Slide Making. Colour Photography. General Applications. Record Applications, Science Applications. Plate Speed Testing. Pro- cess Work. Addenda. Index. IN PREPARATION. Commercial Paints and Painting. By A. S. JENN- INGS, Hon. Consulting Examiner, City and Guilds of London Institute. Brewing and Distilling. By JAMES GRANT, F.S.C- I UNIVERSITY OF CALIFORNIA LIBRARY Los Angeles This book is DUE on the last date stamped below. DISCHARGE-URL 41584 3 1158 00764 0559 A 001 175 020 5