CO u< OU_1 58454 >m CO TEXT BOOK THERMODYNAMICS BY PAUL S. EPSJEIN- PROFESSOR OF THEORETICAL PHYSICS, CALIFORNIA INSTITUTE Of TECHNOLOGY NEW YORK JOHN WILEY VSONS, INC. LONDON: CHAPMAN 84 HALL, LIMITED COPYRIGHT, 1937, BY PAUL S. EPSTEIN All Rights Reserved This book or any part thereof must not be reproduced in any form without the written permission of the publisher. PRINTED IN U. 8. A. ="=*ES8 OF BRAUNV^ORTH a CO.. INC. BUIL.D*Ra OF BOOKS BRIDGEPORT. CONN, TEXTBOOK OF THERMODYNAMICS CHAPTER I THE THERMAL PROPERTIES OF MATTER 1. The concept of temperature. Heat and cold belong to the most fundamental of our sensations. From time immemorial the phenomena connected with them received a great deal of practical attention because of their vast importance in everyday life, but they were not made amenable to scientific treatment until a comparatively recent date. The possibility of doing this is due to the following empirical facts: (1) When a hot and a cold body are brought in con- tact, the former gets cooler, the latter warmer. After the lapse of a sufficiently long time a stationary state is reached in which no further changes occur. We say, then, that the two bodies are in thermal equilibrium. This equilibrium persists if the bodies are separated and then brought together a second time (provided they were not exposed, in the meantime, to the influence of other hotter or colder bodies). (2) Two bodies which are separately in thermal equi- librium with the same third are in equilibrium with each other. We say, then, that they have the same temperature. The second fact permits us to use one suitable body as a stand- ard or "thermometer" with which to compare the temperatures of others. In order to determine the degree of heat of the thermometer, one, usually, has recourse to the property of matter of changing its size or volume when heated. In this way a scale of temperatures is established. The current types of thermometers are well known and need not be described here. When brought into equilibrium with any body, the thermometer associates the degree of heat of this body with a definite reading of its scale. It replaces the vague notion of hot or cold by a precise numerical datum. Its invention was, therefore, the necessary antecedent of a scientific analysis of the phenomena of heat. (Compare section 39). 2 TEXTBOOK OF THERMODYNAMICS 1 3 2. Subject of thermodynamics. Definitions. Thermodynamics deals with systems which, in addition to mechanical and electromag- netic parameters, are described by a specifically thermal one, namely, the temperature or some equivalent of it. We have seen that for the very definition of temperature one must invoke the concept of thermal equilibrium. Thermodynamics is, therefore, of necessity, essentially a science about the conditions of equilibrium of systems and about the processes which can go on in states little different from the state of equilibrium. We introduce the following terms which, for the most part, are those in general usage: A homogeneous system is uniform in every part, with respect both to its chemical composition and to its physical condition. The fact men- tioned in the preceding section that two bodies set themselves into thermal equilibrium applies, of course, also to two adjacent parts of the same body. The statement that the temperature of a homogeneous system is uniform throughout is, therefore, equivalent to saying that it is in equilibrium. A simple homogeneous system or, for short, a simple system is one completely defined by the three parameters V (volume), p (pressure), and T (temperature). Such are gases and liquids in a static condition and solids in a state of isotropic stress removed from the action of gravitational and electromagnetic forces. The case of one or several simple systems is, by far, the most important. Simple systems offer the most interesting applications, and their treatment is sufficiently typical to bring out all the methods of thermodynamics so that a generalization is easily made when necessary. We shall, therefore, restrict ourselves in many of our discussions to simple systems, and shall take up the more general case only when this is required for special applications. A heterogeneous system is composed of a number of homogeneous ones separated either by surfaces of discontinuity or by wall partitions. Those homogeneous parts are called the phases of the system. We shall admit the existence and the use of partitions of different types of which the most important are the heat-conducting one which permits two bodies separated by it to set themselves into thermal equilibrium, and the insulating or adiabatic partition which prevents any exchange of heat. 3. The thermal equation of state. We have mentioned in section 1 that the temperature of a thermometer is usually inferred from its extension. The deeper reason why it is possible to do so lies in a very fundamental and general property of every simple system: there exists a functional relation between the temperature, the volume, and I 3 THE THERMAL PROPERTIES OF MATTER 3 the pressure which is called the equation of state, or more accurately, the thermal equation of state. This relation between T, , and 7 can be written in the form p-f(V,T). (1.01) Each of the three variables can, therefore, be regarded as a function of the other two. Subjecting the variables to small changes, we can form the following three partial derivatives: (1) We keep the pressure constant (A/> = 0) and give to 7 and T the small increments A 7 and AT. The limit of the ratio, for AT 1 = 0, is the partial derivative lim(A7/Ar) = (97/37%. (It is customary to indicate the parameter which is kept constant as a subscript.) Divided by the volume 7, this partial defines the coefficient of thermal expansion < /^TA (1.02) which represents the relative (i.e. referred to unit volume) increase of volume per unit increase of temperature, at constant pressure. (2) Keeping the temperature constant (AT" = 0), we obtain in a similar way the partial (QV/Qp)T connected with the coefficient of com- pressibility 1 /^T/\ (1.03) which gives the relative decrease of volume per unit increase of pressure, at constant temperature. (3) When the volume remains constant (A 7 = 0), the partial (dp/dT)v is obtained, related to the coefficient of tension (1.04) The inverse partials are, simply, the reciprocals, (37/97% = l/(9r/97)p, etc., and do not give us anything new, and even the three partial derivatives which we have formed are not independent as there exists a relation between them. This becomes apparent when we write down the general expression for the total infinitesimal change A/> which results when both the volume and the temperature are changed by infinitesimal amounts. It is, according to the rules of differentiation, '3P\ ,rr , (& 4 TEXTBOOK OF THERMODYNAMICS 1 4 In the special case, A = 0, the ratio AF/AT 1 is, precisely, the partial (dV/dT) p . Since the left side of the equation vanishes in this case, we can solve it with respect to this ratio and find This relation can be regarded as a differential form of the equation of state (1.01). In the general case of a homogeneous system described by other parameters (in addition to temperature and volume) there exist several equations of state as will become clear in section 7. Exercise 1. In nitrogen at t = 20 C and p = 0.134 atm, a = 0.00368 and ft = 0.00368. Calculate the compressibility 0i. Exercise 2. Suppose that one of the equations of state of a homogeneous system depending on many variables has the form The increment of y, analogous to the expression for Ap, is Show that by keeping all the variables y, Xi, . . . X n , T constant, except two, one can obtain n(n -f l)/2 new relations of the type (1.05) between triples of partials. (Remark: The set of relations so obtained includes all that are independent. From them there can be derived a great number of other (dependent) relations of a similar form.) 4. The perfect gas. Absolute temperature. The so-called perma- nent gases of nature follow, more or less closely, a number of well- known laws. The first of these is the Boyle-Mariotte law according to which, for a constant temperature, the product of pressure and volume remains constant pV - const. (1.06) When we inquire about the behavior of a gas at varying tempera- tures, it is convenient to use a gas thermometer filled with this same gas as the thermometric fluid. The pressure of the thermometer is kept constant (o), and the zero of the temperature scale (t 0) is chosen arbitrarily, say, as that corresponding to a volume Fo. I 4 THE THERMAL PROPERTIES OF MATTER 5 The temperature t is, then, defined as proportional to the relative increase of the volume over Fo, or V - 7 (1 + <**)* (1.08) a being a constant. According to the law of Charles-Gay-Lussac, the formula (1.07) is universal for all permanent gases: Used as thermometric fluids in the same thermometer, they give, within a good approximation, the same readings, or, in other words, they have all pretty closely the same coefficient of expansion a. The uniformity can be best judged in the following way. Suppose we use a centigrade scale: that is, we choose the temperature of melting ice as / = and that of boiling water as t = 100. Equation (1.08) gives, then, for the volume at t = 100, Fioo = F (l + lOOa), whence a = (Fioo - Fo)/100F . The quan- tity defined in this way is called the mean coefficient of expansion, between and 100, and is denoted by o,ioo. In general this quan- tity depends on the pressure po kept constant during the expansion. The mean coefficients of expansion <*o, 100 were investigated by Henning and Heuse who measured them with considerable accuracy in helium, hydrogen and oxygen. These measurements are well represented by the expressions of Table I, 1 where po is expressed in atmospheres. Degrees of the centigrade scale will be indicated by the symbol C. TABLE 1 Gas <>, too X 10 7 He ......... 36604 - 25p H 2 ......... 36604 - 16p Q N 2 ......... 36604 + 167/> We arrive at a more rational choice of the zero point of our scale if we displace it by I/a, defining the new temperature as T = t + I/a. (1.09) The position of the old zero point / = on the new scale is then Jo = I/a, and eq. (1.08) takes the form V TtPo F r/7o. The temperature T is proportional to the total volume of the gas. We substitute this into eq. (1.06) whidi we refer to the temperature T 9 * E. Kenning and W. Heuse, Zs. Physik 5, p. 285, 1921. 6 TEXTBOOK OF THERMODYNAMICS 1 4 writing pV = poV TtPo , and obtain a formula representing the com- bination of the Boyle-Mariotte and Charles-Gay-Lussac laws: pV CT, (1.10) where C is an abbreviation for the constant C A simple system in equilibrium is completely homogeneous and uniform in all its parts (compare section 2). The properties of our gas, in every small region of it, are, therefore, completely defined by the temperature T and pressure p and cannot depend on its total exten- sion or mass. If we compare two specimens of the same gas of dif- ferent masses, but at the same temperature and pressure, their respec- tive volumes V must be, obviously, proportional to their masses. It follows then from eq. (1.10) that the constant C must be also propor- tional to the mass of the gaseous system under consideration. We shall use as the unit of mass of a chemically homogeneous or pure substance the mol, also called the gram-molecule. If /* denotes the molecular weight of this substance, its mol contains just n grams. To simplify our terminology we shall apply the term " molecular weight " also to monatomic substances where it has the same meaning as " atomic weight/' Let us consider a homogeneous system which consists of N mols of a pure substance. The volume occupied by one mol of it is called the specific molal volume or, simply, the molal volume and is denoted by v. We can apply this to our gas system if it is chemically homogeneous. The total volume V and the constant C can be represented as V**Nv, C = NR, (1.11) where R is an abbreviation for R = poVo/To or the constant C referred to one mol of the gas. The equation of state (1.10) takes then the form pV = NRT, (1.12) or pv = RT. (1.13) The third empirical law which we have to invoke is that of Avo- gadro. It tells us that the molal volume v is, within a close approxima- tion, the same for all permanent gases, namely, under normal condi- tions (p = 1 normal atm = 1.013249 X 10 6 dyne cm- 2 , t = C) it is vo = 22414 cm 3 . Consequently, the gas constant R is also uni- versal. It is an immediate consequence of the definition of the mol, that 1 mol of any substance contains the same number n A of molecules, which is called the Avogadro number. It has been determined with I 4 THE THERMAL PROPERTIES OF MATTER 7 fair accuracy and has been found to be n A =* (6.064 =fc 0.006) X 10 23 mol" 1 . A much-used quantity is the ratio * - R/n A9 (1.14) which represents the gas constant per molecule and is called the Boltz- mann constant. With its help eq. (1.13) can be written in an alterna- tive form. If we denote by Z the total number of molecules in N mols of a gas having the volume F, and by z = Z/F the number per unit volume, we can write Z = Nn A , NR = Nn A k = Zk = Vzk, and from (1.12) p = zkT. (1.15) If, instead of a single gas, we consider a mixture, we must again have recourse to experience. Observations on mixtures of permanent gases show that, with a good approximation, each of the constituent gases behaves as if the others were not present. In the case of <r gases contained in a vessel of the volume Fin the respective quantities of Ni, . . . N 9 mols, each of them exerts a partial pressure upon the walls of the vessel which can be computed in the same way as if the gas were single, i.e. by the formula p h V = N h RT. (1.16) The total pressure of the mixture is the sum of these partial pressures P = pi + p2 + ...+p., (1.17) and its equation of state is obtained by summing the formulas (1.16) for h = 1, 2, ... <r (Ni + ... + N ff )RT. (1.18) None of the above laws is followed by any of the permanent gases quite rigorously, and the formula (1.18) which embodies all four of them is, therefore, only an approximate expression of the facts. If we compare two gas thermometers filled with different gases and having the same fixed points, the readings between the fixed points will not be strictly identical because the rate of expansion is, in reality, not constant but slightly dependent upon the temperature, in a dif- ferent way for the different gases. The same thing is true for non- gaseous thermometric fluids like mercury and alcohol. The scale of temperatures established by any particular thermometer construction is, therefore, special and arbitrary. In order to have a scale which does not depend on the individual properties of any special substance, we imagine the so-called perfect gas: an ideal fluid which strictly 8 TEXTBOOK OF THERMODYNAMICS 1 4 follows the equation of state given by formula (1.12). Let us, further, imagine a thermometer filled with a perfect gas and let us specify its scale in such a manner that between the points of freezing and of boiling water (both at 1 norm atm = 760 mm Hg) are 100 equal divisions. The temperature which we would measure with such a thermometer is called the absolute temperature. We shall, consistently, denote it by T (capital), reserving / (small) for the temperature mea- sured in an arbitrary scale. It might be thought, at first sight, that such a definition of the absolute temperature must be futile since the perfect gas, and the perfect thermometer filled with it, are ideals which do not exist in nature. We shall show, however, in section 29 that there is a simple procedure to determine the absolute temperature, in every case, by indirect measurements. Every thermometer can, therefore, be cali- brated in the absolute scale within the accuracy with which it can be read, and such calibrated thermometers are, in fact, readily available. Making use of the results of this calibration, we can state that the point of freezing water corresponds to TQ = 273.1 of the absolute scale. Combined with the value of the molal volume mentioned above (VQ 22414 cm 3 ), this leads to the following numerical values for the gas constant: R = (82.049 0.009) atm cm 3 deg" 1 mol^ 1 (if the pressure is measured in normal atmospheres) or R = (8.3136 0.001) X 10 7 erg deg-" 1 mol" 1 ) (if it is measured in dynes per cm 2 ). For Boltzmann's constant defined by (1.14) there follows the numerical value k = 1.371 X 10~ 16 erg deg" 1 . The question how closely the permanent gases approximate the behavior of a perfect gas will be more fully discussed in section 6. Here we shall only state that, according to their degree of perfection, they can be arranged in the following order: carbon dioxide, oxygen, air, nitrogen, argon, hydro- gen, helium. Carbon dioxide is the least and helium the most nearly perfect gas of this series. The deviations of helium from eq. (1.13) of perfect gases are, in fact, quite small under most conditions. Except at extremely low temperatures, helium can be used as the perfect thermometric fluid unless a very high accuracy is required. (Compare also section 29.) Exercise 3. Show that the coefficients of thermal expansion and of compres- sibility are for a perfect gas I 5 THE THERMAL PROPERTIES OF MATTER 9 and check the validity of the relation (1.05). Calculate the coefficients numerically (per 1 deg, and 1 mm Hg as units) at normal conditions: r 273M, p 760 mm Hg. 5. The equation of Van der Waals. The next step in the approxi- mation to the properties of real matter is represented by an equation given by Van der Waals. In spite of its simplicity, it comprehends both the gaseous and the liquid state and brings out, in a most remark- able way, all the phenomena pertaining to the continuity of these two states. This equation has the form where a and b are two numerically small constants and v is the molal volume. It was derived by statistical considerations of so simple a nature that, even in a book devoted to thermodynamics, we can say a few words about the interpretation of the constants. It is shown in the kinetic theory that eq. (1.13) corresponds to a gas whose molecules are material points which do not collide or otherwise interact with one another. If we consider a gas with molecules of finite size, mutually exclusive as to their extension but not interacting in any other way, the difference is that the centers of the molecules cannot spread out in the whole volume of the gas but only in that part of it which is not occupied by other molecules and not immediately adjacent to them. In the first approximation, for the molal volume v there must be substituted the covolume, v b, where the constant b is proportional to the sum of the volumes of all molecules in one mol of the gas. The equation of state becomes, therefore, p = RT/(v b). On the other hand, if the molecules of the gas do interact at a distance, say, attract one another, there must be added to the external pressure p the internal pressure due to this attraction. Whatever its law is, it is safe to assume that two volume elements of the gas ATI and Ar2 act upon each other with a force Fi2 proportional to the product of the masses in them, i.e. Fi% cc AwiAma. If we choose the mol as the unit of mass, we have Ami = ATI/IF, Aw2 = Ar2/v, so that F\2 cc ATiAT2/u 2 . This means that, when the density of the gas in a given vessel is changed by adding more gas or subtracting it, all the internal forces change in the ratio 1/v 2 . Since the pressure is defined as the force per unit area, this applies also to the internal pressure and we obtain for it the expression a/v 2 which is added to p in Van der Waals' equation. 1 1 It must be assumed, also, that the force Fit decreases rapidly with the cfistance between the volume elements An and An. Otherwise the coefficient a would not 10 TEXTBOOK OF THERMODYNAMICS. 15 Let us plot the pressure p against the molal volume v keeping the temperature T' in eq. (1.21) constant. Such a curve is called an " isothermal." As long as the molal volume v is very large, b is negli- gible compared with v, and a/v 2 compared with p. In the region of large volumes the curve is, therefore, identical with that given by the equation of perfect gases, namely, a hyperbola asymptotically approaching the horizontal axis p = for v = o . When the volume is very small and close to its lower limit 6, the second term on the right side is again negligible compared with the first and the curve becomes again hyperbolic. However, its asymptote is not the vertical V = (as in the case of the perfect gas) but the vertical v = b. The shape of the isothermals in the intermediate range is given in Fig. 1. They are curves with one minimum and one maximum which are real for low temperatures and complex for high so that, above a certain temperature T c (critical temperature), the pressure is, in them, a monotonically decreas- ing function of the volume (dp/dv <[0). Below T c the curves are not con- tinually decreasing but have a middle part (between the minimum and the maximum) in which p increases with v, (dp/dv > 0). It is obvious that this middle part, dotted in our curves, can have no physical reality. In fact, let us imagine the fluid in a state cor- responding to this part of the curve contained in a heat-conducting vertical cylindrical vessel whose top is formed by a piston. The piston can slide up and down in the cylinder, and we put on it a load exactly balancing the pressure of the gas. If we take a little weight off the piston, there will no longer be equilibrium and it will begin to move upward. However, as it moves the volume of the gas increases and with it its pressure. The resultant force upon the piston gets larger, retaining its upward direction. The piston will, therefore, continue to move and the gas to expand until it reaches the state represented by the maximum of the isothermal. Vice versa, if we add ever so little to the load of the balanced piston, the gas will collapse to the state corresponding to the minimum of the be an internal constant of the gas but would depend on the shape and size of the container. It does not follow from the above argument and is a separate assump- tion that a is independent of T and p. o FIG. 1. Isothermals according Van der Waals 1 theory v to I 5 THE THERMAL PROPERTIES OF MATTER 11 isothermal. The relation dp/dv g is, therefore, the condition of dynamical stability, and in regions where it is not satisfied the fluid is in an absolutely unstable state. Leaving out the physically unreal dotted portion, the low tempera- ture curves break up into two disconnected branches. The one of small molal volume corresponds to the liquid state of matter while that of high volume represents its gaseous state. The important fact that the same equation applies to both states of aggregation was called by Van der Waals the continuity of the liquid and gaseous states. Experiments show that the following things actually happen when we gradually increase the volume (in a heat bath of constant temperature), starting from a point A of the liquid state (Fig. 1). The expansion goes along the liquid branch of the isothermal up to a certain point B. In this state the liquid begins to evaporate so that the system is no longer homogeneous, but heterogeneous, consisting of a liquid and a gaseous phase. As the volume is further increased, more and more liquid is evaporated while the specific properties of the two phases remain unchanged. In this part of the isothermal expansion the pressure (called the boiling pressure at the temperature 7') is constant, so that it is represented by a horizontal line, and remains so until all the liquid is evaporated (point E). From then on the system is again homogeneous and expands along the gas branch EF of the isothermal. Why the system behaves in such a way, and where the point B lies, are typical problems of thermodynamics which will be answered at length in section 43. The part BC of the liquid branch (like DE of the gas branch) represents only relatively unstable states: the vapori- zation of a liquid is greatly facilitated by the presence in it (or on its surface at the walls) of little bubbles of foreign gases. If these gases are driven out by a preliminary thorough boiling of the liquid and a suitable heat treatment of the vessel, it is possible to make the system expand along the liquid branch beyond the point B, bringing it into the so-called superheated state. In this state the pressure may become negative, in other words, the liquid is able to support a certain amount of tension; but at the slightest provocation it will jump over into the thermodynamically stable state on the straight line BE. Above the critical temperature there is no discontinuity in the iso- thermal, and it is arbitrary whether to call the fluid a liquid or a gas. Usually, it is considered a gas and, then, the statement applies: // is impossible to liquefy a substance at temperatures higher than the critical. If the substance follows eq. (1.21) of Van der Waals, its critical tem- perature can be determined in the following way. Below it, the iso- 12 TEXTBOOK OF THERMODYNAMICS I 5 thermals have a maximum and a minimum which are analytically determined by the condition + 2_* o. (1.22) As the temperature rises, the maximum and the minimum draw closer and closer together, and, in the critical isothermal, they coincide, forming a point of inflection. This point corresponds to the so-called critical state of the substance characterized by the values p c , v c , T c of pressure, molal volume, and temperature which are known as the critical constants. The analytical condition for a point of inflection is _ 6a ^ ) In the critical point all three equations (1.21), (1.22), (1.23) must be satisfied. It is, therefore, easy to express the critical constants in terms of the Van der Waals constants a, b, R by solving these three equations with respect to the three variables p, v, T. The result is .-. One can use the critical constants as the units in which to express, respectively, the pressure, specific volume, and temperature. The state of the substance is then described by the so-called reduced variables p v T . -ft * = :: T = V (L25) pc V c lc If we substitute these quantities for p, v, T and, at the same time, replace the Van der Waals constants by the critical, the equation of state (1.21) takes the form Referred to the reduced variables, the equation does not contain any specific constants and is, therefore, the same for all Van der Waals substances. States of two substances in which they have the same respective values of T, <f>, and T are called corresponding states, and eq. (1.26) expresses the law of corresponding states: If two of the reduced variables, IT, ^>, T, are the same, respectively, for different sub- stances the third is also the same and their states are corresponding. The mathematical reason for the validity of the law of corresponding I 6 THE THERMAL PROPERTIES OF MATTER 13 states in connection with Van der Waals' equation is that it contains just as many constants as variables, namely three. However, the law of corresponding states is not restricted to Van der Waals sub- stances. It is often satisfied with a fair degree of precision in groups of substances which show marked deviations from eq. (1.21), as will be shown more fully in the next section. It is well to add here a few words about gas mixtures. The additiv- ity of partial pressures which exists in perfect gases, according to eq. (1.17) does not hold in mixtures of Van der Waals gases. The physical reason for that additivity is that each perfect gas behaves as if the others were not present, and this means that there is no inter- action between their molecules. On the contrary, the aim of Van der Waals' theory is, precisely, to take into account the interaction. For a binary mixture (i.e. consisting of two gases) the kinetic theory has been carried through by Lorentz. 1 The equation of state obtained by him for such a mixture is formally identical with the ordinary Van der Waals eq. (1.21), but the symbols v t a, b have a different meaning. If the mixture contains NI mols of the first gas and N 2 of the second, v = V/(N\ + N 2 ), while the quantities a and b depend on the mol fractions Xi = N\/(N\ + N 2 ) and x 2 = N 2 /(Ni + N 2 ) and have the form a = #n xi 2 + 2ai2 Xix 2 -f- 022 x$ b = bn xi 2 + 2bi2 xix 2 + b 22 (1.27) The coefficients an #22 are those of the pure gases, while #12 characterizes the mutual action ; the same applies to the coefficients b. Exercise 4. Calculate by means of eqs. (1.24) the Van der Waals constants a, b and the ratio RT c /p c v c for the following substances from the critical data (R 82.07): Methyl acetate 506.9 46.33 227.6 Methyl formate 487.2 59.25 172.0 Exercise 5. Derive the coefficients of thermal expansion (1.02) and of compres- sibility (1.03) for the equation of Van der Waals. (Remark: as this equation is readily resolved with respect to />, it is convenient first to express a and pi in terms of partials of p t by means of (1.05) or in any other way.) 6. Behavior of real substances. As far as gases are concerned, it was pointed out in the preceding section that eq. (1.13) of a perfect 1 H. A. Lorentz, Ann. Physik 12, p. 127, 1881. For a discussion of this equation, in relation to experimental data, see: J. D. Van der Waals, Continuity of the Gaseous and Liquid States. Vol. II. German edition of 1900. 14 TEXTBOOK OF THERMODYNAMICS I 6 gas is already a good approximation as long as the molal volume v is large. Although the equation of Van der Waals has its limitations (see below), the numerical values of its constants a and b are apt to give an idea how much a given substance deviates from the law of perfect gases and indicate the order of accuracy here involved. We give in Table 2 these constants and the critical data of a number of gases. STABLE 2 Gas Chemical Symbol t c *C PC (atm) b = \v c (cm') a X 10- (atm cm 6 ) Helium He -267.9 2.25 23.71 0.03415 Neon Ne -228.35 26.85 17.10 0.2120 Hydrogen H 2 -239.9 12.8 26.61 0.2446 Argon A -122.4 48.0 30.22 1.301 Nitrogen N 2 -147.13 33.5 38.52 1.346 Oxygen O 2 -118.82 49.7 32.58 1.361 Carbon monoxide Carbon dioxide CO CO 2 -138.7 31.0 34.6 72.9 39.87 42.69 1.486 3.959 Nitrous oxide N 2 O 36.5 77.66 44.18 3.788 Water vapor H 2 O 374 217.5 30.52 5.468 Chlorine C1 2 144 76.1 56.26 6.501 Sulfur dioxide SO 2 157.15 77.65 56.39 6.707 The constant ft, depending on the molecular volume, is not greatly different for the different gases. The relative correction in the magni- tude of the pressure, due to the presence of this constant in eq. (1.21), is roughly b/v. In the neighborhood of the normal conditions (/ = C, p = 1 atm), the molal volume is about 22 400 cm 3 /mol and the correction has the numerical value of 0.001 to 0.002. On the other hand, there is a considerable variation, from gas to gas, in the constant a, and the correction a/v 2 depending on it varies widely. For helium under normal conditions it is only 0.00007, for nitrogen and oxygen 0.0027. For the latter two gases, it is of the same order of magnitude as the first correction (0.0017 and 0.0015), and, since its sign is opposite, the total correction is reduced. For water steam at the boiling point (t = 100 C, p = 1 atm), it becomes -0.006. It was known already to Van der Waals himself that his equation does not give an accurate representation of the properties of a fluid in all ranges of temperatures. In fact, it is possible to determine the coefficients a and b from the data about pressures and volumes relating to a single isothermal. When the coefficients, obtained in this way for widely different isothermals, are compared, they turn out not to 16 THE THERMAL PROPERTIES OF MATTER 15 be the same, especially, a is found to increase as the temperature is lowered. Another indication that eq. (1.21) is not accurate in the vicinity of the critical state is as follows. Theoretically there exists the following relation between the critical data p e , v c , T if the expres- sions (1.24) are valid: K RT C 8 - - 2.67. (1.28) In reality, the ratio of the empirically measured critical data RT c /pcV c is nearly the same within large groups of substances but it has not the value 2.67 required by the theory. The constancy of the ratio R is an indication that the law of corresponding states still approximately holds when the Van der Waals equation fails. In Table 3 we give the critical data of a few organic substances. TABLE 3 Substance Formula To PC v e RT c /p c v Ethyl ether C 4 Hi O 467.0 35.6 281.9 3.814 Benzene CH 561.7 47.89 256.2 3.755 Bromobenzene CflH 6 Br 670.2 44.64 323.4 3.813 Chlorobenzene C 6 H 6 C1 632.4 44.62 307.9 3.776 Ethyl formate CaHeO, 508.5 46.76 229.0 3.895 Ethvl acetate C 8 H 8 O2 523.3 38.00 286.0 3.885 n-Pentane C*5l~l 1J 470.4 33.04 310.9 3.762 t-Pentane v^5i~i 12 461.0 32.92 307.3 3.734 Octane CaHig 569.4 24.65 490.9 3.864 Heptane C 7 Hi 539.4 26.88 537.8 3.945 Hexane .... (-.gH 14 507.4 29.62 366.9 3.830 Carbon tetrachloride .... C C1 4 556.3 44.98 275.9 3.677 Tin tetrachloride SnCl 4 591.9 36.95 351.2 3.755 In spite of the shortcomings just mentioned, the Van der Waals equation is extremely useful. It gives an excellent quantitative account of the behavior of gases whose density is not too high. Beyond this it offers a qualitative picture of the phenomena of condensation, critical state, etc. However, it must be always remembered that it cannot be used for quantitative predictions of phenomena involving high densities. Many other equations of state were proposed but, in part, they apply only to a narrow range of the variables, in part, they offer a gain in accuracy insufficient to compensate for the much more complicated form. Therefore, we shall mention here only two of them. 16 TEXTBOOK OF THERMODYNAMICS I 6 (A) The mean empirical, reduced equation of Kamerlingh Onnes 1 is expressed in terms of the reduced variables (1.25), K is the ratio (1.28) of the critical data, while the coefficients depend on r iii #2, . . being constants. The other coefficients have the same structure. This equation grew out of the studies of its author on the validity of the law of corresponding states. He found that there exist large groups of substances which obey the law of correspondence to the following extent. The substances of a group can be arranged in a series so that each of them has a close correspondence with the pre- ceding substance while the series, as a whole, shows progressive and considerable changes in the constants. The above equation applies to a fictitious substance representing the mean type of the series. The results of Kamerlingh Onnes show us two things: (1) how extremely complicated the conditions become when a high accuracy is aimed at; (2) that the law of corresponding states has an approximate validity beyond that of any of the simpler equations of state. (B) The equation of state of Beattie and Bridgeman 2 A * v 2 ^ ' ~ l v 2 ' [ e = c/vT*, AQ, Bo, a, i, c being constants. This equation is one of the most recent that have been proposed so that its authors had the benefit of the experience of their predecessors and of more accurate experimental data. 3 It gives good results over a wide range of the variables. 1 H. Kamerlingh Onnes and W. H. Keesom, Die Zeistandsgleichung. Encyklo- paedie math. Wiss. Vol. VI, Article 10. 2 J. A. Beattie and O. C. Bridgeman, Proc. Am. Acad. 63, p. 229, 1928. 1 Another of the more recent equations is that due to A. Wohl (Zs. physik. Chemie 87, 1, 1914; 99, p. 207, 1921). It is often quoted, but the writer has diffi- culty in understanding it, as he is not convinced that it satisfies the requirements of an equation of state. tl - e) ,. . A B _ ^ _ 16 THE THERMAL PROPERTIES OF MATTER 17 The above equations of state apply, primarily, to gases, secondarily, to liquids at temperatures not far from the critical. It is not necessary to say much about matter in the condensed state. The thermal expansion and compressibility of condensed substances are so small (compared with those of gases) that they are not important in most ther- modynamical applications. In the first approximation, they are altogether neglected ; in the second, they are taken as constant over finite ranges. The process of transition from the liquid to the gaseous state is in real substances the same as that described in connection with the Van der Waals equation and illus- trated by Fig. 1. As an example we give in Fig. 2 the actually observed isothermals in carbon dioxide. (The mp- FlG 2 .-Empirical isothermals in carbon lal volume is given here in fractions of the normal vol- ume at C and 1 atm, v, marked by the symbol X). 30 D .005 ".01. V/Vn FIG. 2. Empirical isothermals dioxide. = 22 256 cm 3 ; the critical point is Exercise 6. Calculate the critical data in terms of the constants of the equation of state / n \ - b) RT (Berthelot) by way of the formulas (1.22), (1.23), (1.24). Exercise 7. Do the same for the equation p(v -b) = RT exp (-c/RTv). (Dieterici) (Remark: when the first and second partials of p vanish, those of log p also vanish.) CHAPTER II THE FIRST LAW OF THERMODYNAMICS 7. The element of work. The concept of work is defined in mechanics as follows. When a material point moves under the action of a force F, through the infinitesimal distance dl in the direction of F, the elementary work done by this force is Fdl. When the directions of the path dl and of the force do not coincide but include an angle a only the projection of F on the direction of motion Fi = F cos a con- tributes to the work, and the element of work has the expression DW = Fidl = Fdl cos a = F x dx + F y dy + F,dz, (2.01) if we denote, respectively, by F x , F v , F, and dx, dy, dz the projections of F and of dl upon cartesian axes of coordinates. In the case of the general dynamical system of n degrees of freedom, the element of work is expressed in a similar way DW - yi dXi + y 2 dX 2 + . . . + y. dX n (2.02) where X\ 9 . . . X n are the geometrical coordinates and yi, . . . y n the generalized forces. If the work is due in part to non-mechanical forces (electric, magnetic, capillary, etc.), the form of this expression is not changed : there only appear additional terms of the same type. Of particular importance to us is the form which the element of work takes in simple systems (section 2). The only force with which such a system can act upon the outer world is the pressure upon its boundaries. The pressure is the force per unit surface of the wall and has the direction of the normal to it. Consequently, the force acting upon an element d2 (Fig. 3) of the surface is F = pd2 in the direction n. Let us suppose that, because of the pressure in the system, its surface gets slightly displaced from the position AB into A'B' so that the element dS moves into the new position d2' through the very small distance A/. The work of the force upon this element is then, according to the expression (2.01), equal to dSA/ cos(n,A/), and the work of the forces of pressure upon all the surface to DW = p f dSA/ cos(n,A/). In the limit, when A/ is infinitesimal, every element, in being displaced from dl, todS', sweeps 18 117 THE FIRST LAW OF THERMODYNAMICS 19 over an oblique cylinder whose volume is, precisely, dA/ cos (n,A/). The integral f dSA/ cos(n,A/) is, therefore, the total increase of the volume of the system dV: (2.03) We shall, in general, denote by capital letters quantities referred to the whole system (N mols) and by small letters quantities referred to 1 mol of it. w represents, therefore, the molal work (W = Nw, N being the mol number), or Dw - pdv. (2.04) It is work done by the system : We define the sign in such a way that work of this kind is counted as positive, while work done by outer FIG. 3. Element of work in simple system. FIG. 4. Graphical repre- sentation of work. o v FIG. 5. Work in cyclic process. forces against the system is considered as negative. The expression (2.04) receives a simple graphical representation in the (p, v)- diagram (Fig. 4). Let the points A, A' correspond respectively to the initial state of the system and to its state after an infinitesimal expansion dv. The product pdv = Dw is, then, represented by the infinitesimal area under the element AA' y the shaded strip of our diagram. If a finite process of expansion is such that it can be represented by a curve of the (p, fl)-diagram, for instance, by the segment AB, the total work (per mol) done in it by the system is equal to the integral of the expression (2.04) w -f * A pdv. (2.05) Its graphical representation is, obviously, the total area under the segment AB. Since the work w is, in this case, positive, Fig. 4 permits us to formulate the following rule. An area represents positive work when it lies to the right of the curve giving the process (for the observer looking in the direction of the process, i.e. from A to B). This rule is 20 TEXTBOOK OF THERMODYNAMICS II 8 borne out also in the case of a cyclic process where the system returns to its initial state (Fig. 5). While expanding (ACB), the system does positive work; while contracting (BDA), negative. The negative area (double shaded) is to the left of the curve BDA and must be sub- tracted from the positive area under A CB. The difference is the net (positive) work done by the system; it is represented by the area enclosed by the circuit ACBD and lying to the right of it. Exercise 8. (If the force is measured in dyne /cm 2 and the volume in cm 3 , the work is expressed in erg). Calculate the work of vaporization of water. When 1 mol of water (18 grams) is vaporized at 100 C, the increase of volume is VB VA = 30186 cm 8 mol" 1 . The pressure in the process is constant and equal to 1 atm: p = 1.013 X 10 6 dyne/cm 2 . Exercise 9. Calculate the work necessary to compress 1 mol of a gas from the normal state (VA = v = 22414cm 8 , PA = 1 atm) to half the normal volume (VB = fo), assuming that the process follows the law of Boyle-Mariotte: pv = 8. Linear differential expressions and exact differentials. It will be well to recall to the reader a few mathematical facts which, although simple, are very fundamental in thermodynamics since the whole of this science may be said to be built up on them. Mathematically speaking, formulas (2.01) to (2.04) are examples of linear differential expressions, whose general form is k(xi ' Xn)dXl " - For instance, in the particular case of only two variables x,y it reduces to L(x,y) = M(x,y)dx + N( Xj y)dy. (2.07) The most interesting question which arises with respect to these expressions is whether they are " exact differentials 1 ', meaning whether they can be obtained by differentiating some function f(xi, . . . x n ) of the same variables. According to the rules of differentiation the total differential of / is -dx k . (2.08) We see, therefore, that the linear differential expression (2.06) is an exact differential when Jf*-j, (*-l,2...n); (2.09) 9** II 8 THE FIRST LAW OF THERMODYNAMICS 21 in words, when the coefficients Mk are the partial derivatives of a function/(#i, . . . x n ). If we single out the two independent variables Xk, xi the second partial does not depend on the order of differentiation Fropl'the UJfmulas (2.09) there follows, therefore, ; , (ft,/ - 1, 2 ...), (2.10) the so-called reciprocity relations. It is clear that the validity of the reciprocity relations is a necessary condition for the expression (2.06) being an exact differential. Moreover, it is shown in calculus that this condition is also sufficient: Whenever the relations (2.10) are satisfied, a function can be found such that the coefficients Mk are its partial derivatives. This function is called the potential of the expression. We conclude from this that a linear differential expression of the type (2.06) is not, in general, an exact differential. If the functions Mk(xi, . . . x n ) are selected at ran- dom, they will, in general, not satisfy the reciprocity relations. The exact differential is, therefore, a rare exception. Of great importance is the concept of the p IG> line integral of a linear differential expression. To fix our ideas, we shall explain this notion for the case of expression (2.07) with only two variables x, y, which we shall interpret as the cartesian coordinates of a plane (Fig. 6). The line integral along the curve from A to B [M(x,y)dx + N( X ,y)dy} is constructed in the following way. The first element is obtained by substituting, for the arguments of the functions M and N, the values of x and y in the point A and, for dx and dy, the increments which lead from A to an infinitely close point A 1 on the curve. For the second element one takes the values of M and N at the point -4' and the increments leading to the point -4", and so one moves from point to point along the curve, until its end B is reached. The integral is the sum of all the elements so constructed. In the case of n variables they 6. Notion of line integral. 22 TEXTBOOK OF THERMODYNAMICS II 8 must be interpreted as the coordinates of an w-dimensional hyper- space, and the generalization is obvious. When the linear differential expression is an exact differential (2.08), its line integral can be given explicitly. For instance, f J A The line integral depends, then, only on the coordinates of the initial point #i A) , . . . x (A) and of the final x{ B \ . . . x ( * } and not on the path between them. It has the same value for all the curves going from the same initial to the same final point. In particular, when the curve is closed, the initial and the final points coincide and the expression (2.11) is equal to zero: = 0. (2.11) If V^ Mkdxk is an exact differential, its line integral over any closed 1-1 circuit vanishes. It is shown in mathematics that the inverse theorem- is also true : If the line integral of a linear differential expression vanisjte$ for every arbitrarily chosen closed circuit, it is an exact differential. To summarize, we can say that the following three possible prop erties of the linear differential expression (2.06) are completely equiva- lent: (1) It is an exact differential; (2) its coefficients satisfy the reciprocity relations (2.10) ; (3) its line integral vanishes for any closed curve. If the expression has one of these three properties, it has also the other two. Supposing that the linear differential expression (2.06) is not an exact differential, the further question arises whether it can be trans- formed into one by multiplying it by a suitable function X(*i, . . . x n ). If this is the case, we say that it admits an integrating multiplier or, simply, that it is integrable. After multiplication by X the coefficients of the expression become XM*, and, if it is integrable, there must be satisfied, instead of (2.10), the reciprocity relations In the case of the differential expression (2.07) with two variables, there is only one reciprocity relation, 3(XAf)/3y = 9(XW)/9#, which can be regarded as a partial differential equation with X as the depend- ent variable. Since such an equation has always a solution, a linear H9 THE FIRST LAW OF THERMODYNAMICS 23 differential expression with two variables is always integrable. This is no longer the case if n is larger than 2. For instance, when n = 3, there exist the three relations 9*3 9*2 ' 9*i 9*3 9*2 9*i of which two are independent. In general, it is impossible to satisfy two equations by one function. This is brought out by the fact that X can be eliminated from the three relations which leave as the result of this elimination ,, ,_ / Mi [ ---- ]+M 2 [ --- f+Mz I ~ -- 1 = 0. (2.13) 9*2 / \9*i 9*3 / \9*2 9*i (The simplest way of checking this is to substitute the form of the coefficients M k = (9//9**)A following from (2.12). This equation is known as the condition of integr ability. Unless it is satisfied the dif- ferential expression does not admit an integrating multiplier. In the general case of n variables, there must be satisfied a condition of the type (2.13) for each triple of variables. It should be noted that the integrating multiplier, when it exists, is not unique. In fact, suppose that the linear differential expression (2.06) is transformed by multiplication by X into the exact differential df(xi 9 . . . x n ). The further multiplication by any function of/, say <p(f), changes it into <p(f)df, which is also an exact differential. There- fore, X^>(/) is another integrating multiplier. Exercise 10. Check by means of eq. (2.12) that L(p, v) = pdv vdp admits an integrating multiplier, namely X = 1/t; 2 . Exercise 11. Decide by means of the criteria (2.10) and (2.13) whether the following linear differential expressions are exact differentials, integrable, or non- integrable: (a) (2xydx + x*dy (b) zdx + xdy +]ydz, (c) yzdx + zxdy -f xydz, (d) (y + z)dx + (z+ x)dy + dz 9. Application to the element of work. In order to apply these mathematical theorems to the physical concepts of section 7, we have to find out whether in a simple system the element of work, Dw = pdv, is an exact differential. As pointed out in section 3, the state of a simple system is completely determined by two of the three variables 24 TEXTBOOK OF THERMODYNAMICS II 10 p, v, T. Let us use for its description T and v: The fact that dT does not appear in the linear differential expression Dw means, then, that the coefficient of dT is zero. The expression should be written Dw SB pdv + Q.dT, and it is easy to see that it does not satisfy the reciprocity relation (2.10) or dp/$T = 30/90, since the left side is equal to the coefficient /3i (1.04) and is, generally, finite and the right is 0. We conclude, therefore, that the element of work is not an exact differential. This is confirmed by considering the line integral of Dw over a closed circuit: we found in section 7 that it is represented by the enclosed area while, in the case of an exact differential, it would be equal to nothing. The same applies to the general case of a system with n degrees of freedom. The difference between the element of work (2.02) in mechanics and in thermodynamics is that, in mechanics, the generalized forces y k are supposed to depend only on the coordi- nates of position Xi, . . . X n , whereas, in thermodynamics, they are, in addition, functions of the temperature T. y k = y k (Xi, X 2 ,... X n , T), k = 1, 2, . . . n. (2.14) The term with the differential dT has the coefficient zero, and, therefore, it is impossible to satisfy some of the reciprocity relations, just as in the case of the simple system. This is the reason why we use for the element of work the symbol Dw: we wish to reserve the symbol d for exact differentials. A system can be considered as defined only then when the forces it exerts on its environment are known. In other words, eqs. (2.14) must be considered as given. As they are completely analogous to the equation of state, p = p(v,T), of the simple system, we conclude that a homogeneous system has one less equation of state than it has independent variables. (Compare also section 27). 10. Heat and heat capacity. As the invention of the thermometer led to the concept of temperature (section 1), so another physical instrument, the calorimeter, helped to develop the important concept of heat. We have already seen that, when two systems of different temperatures are brought in contact, the warmer cools down and the colder heats up until equilibrium is reached. A calorimeter is a standard system with which different bodies are brought in contact in order to study their respective losses of temperature in relation to its gain. To fix our ideas, let us suppose that the calorimeter consists of one kilogram of water (initial temperature T\) into which the bodies experimented upon (initial temperature Ti > 7i) can be dropped, after which the combined system assumes a final temperature T inter- II 10 THE FIRST LAW OF THERMODYNAMICS 25 mediate between the two (T% > T> 7\). The notion of heat was evolved from such measurements in the eighteenth century when the calorimeter technique was not yet very precise. Within this limited accuracy the following result was obtained: In experiments with the same body, the final temperature T adjusts itself for all variations of the initial temperatures T\ and T% in such a way that the ratio rip rr* C = 2 _ - constant. (2.15) T TI C is, therefore, a characteristic constant of the particular body or system and is called its heat capacity. Comparing different bodies of the same material, it was, further, found that their heat capacities are proportional to their masses. For a body of the mass Af, one can write C = Me', where c f is the heat capacity per unit mass or the specific heat of the material. Pouring into the calorimeter an additional kilogram of water, we can determine the heat capacity C w of this water like that of any other system. In the units implied in formula (2.15) it turns out to be 1, and this is also the heat capacity of the calorimeter itself, which was taken to consist of the same amount of water (C w = 1). Equation (2.15) can, therefore, be rewritten as follows C (T - r 2 ) + C w (T - Ti) = 0. (2.16) This relation has the form of an equation of conservation: It sug- gests that there is something that does not change in the process of calorimetric conduction. We call this invariable function or quantity heat. In fact, both terms of the left side have the form Q = C(T - T), (2.17) where T' is the initial and T the final temperature. The first term relates to the body experimented upon and is negative; it represents the heat lost by that body. The second, oppositely equal to the first, is the heat imparted to the calorimeter. The modern improved measurements have completely borne out the fact that there is conservation of heat in all processes of conduction. However, its definition in terms of the temperature had to be refined somewhat, since it was found that the heat capacity is not strictly constant but depends on the temperature interval. If Q is the heat imparted to a system when its temperature is raised from T' to T, we define C Q/(T - n (2-18) 26 TEXTBOOK OF THERMODYNAMICS II 10 as the mean heat capacity for this interval. When the interval is small, T = T - Ar, this can be written C = A^/Ar, and in the limiting case of Ar = it becomes in (2.19) the heat capacity at the temperature T. (We prefer the symbol D be- cause DQ is not an exact differential, as we shall see in section 13). Multiplying this by dT and integrating, we find the more accurate expression for the heat imparted to the system / CdT, (2.20) which takes the place of the approximation (2.17) and must be, also, substituted into the equation of conservation (2.16): L T s*T CdT+ I C w dT = 0. (2.21) J Tl The accepted unit of heat is the (gram-} calorie, the heat which must be imparted to 1 g of water in order to raise its temperature by 1C. There are several different definitions of it in use, the most important being: (a) The 15 calorie, the heat necessary to warm 1 g of water from 14.5 to 15.5 C. (b) The mean calorie, one one- hundredth of the heat which raises the temperature of the same mass of water from to 100 C. The latter is larger than the former in the ratio 1.00024. One thousand gram-calories form one kilogram- calorie. As to specific heats, we shall refer them, in the case of chemically pure substances, to 1 mol of matter and denote by c the molal heat, i.e. the heat which raises the temperature of 1 mol of the material by 1 C. This definition is incomplete because the specific heat depends also on the pressure conditions during the measurement; we shall have to say more about this in section 14. Some authors use two terms: atomic heat (heat capacity of 1 mol of a monatomic substance) and molecular heat (1 mol of a chemical compound), but we shall apply the single term, molal heat, to both cases. Joseph Black, who established the " doctrine of specific heats," discovered also another phenomenon, the latent heat of transformations. He observed (1761) that during the process of melting of ice (while the temperature of the water-ice mixture remains constant) a definite amount of heat must be imparted to it, to transform it from the solid II 11 THE FIRST LAW OF THERMODYNAMICS 27 state into the liquid, which he called the latent heat of fusion. Later he also proved the existence of a latent heat of vaporization and approx- imately measured it. We shall denote the total latent heat by L and the molal latent heat by /. Chemical reactions are also accompanied by a positive or negative heat development called the heat of reaction. If heat must be imparted to the system in order to make the reaction proceed, it is called endo- thermic; in the opposite case, exothermic. We shall adhere to the con- vention that Q is positive when it is imparted to the system. Therefore, the heat of an exothermic reaction will be counted as negative, of an endothermic as positive. In modern usage, the phenomena of change of volume and pressure with temperature (treated in Chapter I) are spoken of as the thermal properties of matter, while heat capacities, latent heats, etc., belong to the caloric properties of matter. Exercise 12. Into a calorimeter consisting of 200 g of water (initial temperature /i =s 20 C) are thrown pieces of metal weighing 300 g and having the initial tempera- ture /2 = 100 C. The temperature of equilibrium / is for (a) aluminum (ju = 27.0) 39.5C, (b) iron (/z = 55.8) 31.9C, (c) zinc (ju - 65.4) 29.8 C, (d) silver ( - 107.9) 26.2 C. Find the specific heats per 1 g and the molal heats of these four metals, (/z denotes the atomic weight.) Exercise 13. The molal latent heat of fusion of water (/z 18) is / 1440 cal mol" 1 . We drop into a mixture of 1000 g ice and 1000 g water 300 g of copper (/u = 63.6, molal heat c = 5.73 cal mol"" 1 ). How much ice will melt before the system comes to equilibrium at C? 11. History of the first law. The notion of energy and of its con- servation was, first, established as a theorem restricted to the science of mechanics. Leibnitz showed in 1693 that, in an isolated mechanical system, there remains constant the sum of the kinetic and the potential energies, called by him the live and the dead forces (vis viva and vis mortua). If the system is not isolated, these energies are convertible into work done against outer forces. About seventy years later was formulated the law of the invariability of heat in all processes of con- duction, as stated in the preceding section. The first law of thermo- dynamics asserts that mechanical energy and heat, the two quantities subject to laws of conservation, are not generically different, but equivalent to each other and interconvertible. A vague suspicion that mechanical action can have heating effects seems to have existed since early times, but the first precise statement that mechanical work can be converted into heat and the first attempts to put this theory upon an experimental foundation were due to Benjamin Thompson, a native of Connecticut, better known under 28 TEXTBOOK OF THERMODYNAMICS II 11 his Bavarian title of Count Rumford (1798). Supervising the boring of cannons in the arsenal of Munich, Rumford was struck by the high temperature which the brass shavings attained and by the general large development of heat in the process. He could think only of two causes, other than the conversion of work into heat, which, conceivably, could have produced this effect: On the one hand, there may exist, he thought, a latent heat of breaking up solid brass into chips,; on the other, the borer may induce some chemical action of the air upon brass attended by heat development. The first possibility was dis- proved by special measurements of two different kinds. One should expect that the surmised latent heat is due, largely, to a difference in the heat capacities of the solid metal and of the shavings. However, no such difference could be detected by calorimetric determinations especially undertaken for the purpose. Still more conclusive was a second line of reasoning: When a blunt borer rotating with much friction was substituted for the sharp one, the production of chips decreased many times while the yield of heat remained about the same. As to the possible action of air, Rumford tried to meet this objection by immersing the whole apparatus in water so that the air had no access to any of the heated parts. This change of conditions had no appreciable effect on the rate of the heat development. Rumford gives very explicit data how much heat the power of a live horse produces per hour. Translated into modern units, they are estimated to mean that 1 (gram)-calorie is equivalent to 5.45 X 10 7 erg, a result which is 31% above the value at present accepted. A sequel to Rumford's work was a paper by his associate in the Royal Institution of London, Sir Humphry Davy (1799). In order to refute the objection that the heat, in the above experiments, could have come from the water (which might have exercised upon brass a chem- ical influence similar to that of air), Davy studied the conversion of work into heat in vacuo. By means of a clockwork he rubbed two pieces of ice against each other under the jar of an air pump. The effect was unmistakable, inasmuch as the ice melted faster with rubbing than without, but no quantitative determinations were attempted. The brilliant lead of Rumford and Davy was not followed up by other investigators. Forty years passed as a period of quiescence dur- ing which these problems were, practically, never mentioned as far as the literature of physics and chemistry is concerned. Not until the decade from 1840 to 1850 was the principle of conservation of energy established and accepted, owing to the independent work of several men belonging to different nations. The French engineer S6guin II 11 THE FIRST LAW OF THERMODYNAMICS 29 (1839) formulated the theory of the equivalence of work and heat with great clarity and carried out much experimental work trying to prove it. The idea of these experiments was excellent and amounted to a reinterpretation of the formula of adiabatic expansion from the point of view of conservation of energy (compare section 18). Their execu- tion and analysis, however, were imperfect, and Seguin himself admitted that his results succeeded only in making his contention probable and did not constitute a convincing proof. Quite independ- ently and with equal lucidity the principle was stated by the German physician Robert Julius Mayer (1842), who also computed the ratio between the units of heat and work (mechanical equivalent of heat) from data on the specific heat of gases then available (compare section 15). This method, although not quite free from objection, disclosed an unusual insight into the nature of the phenomena involved in it. The result at which he arrived is 1 calorie = 3.57 X 10 7 ergs, or 15% below the correct value. However, Mayer seems to have overlooked that even the most ingenious hypothesis needs an experimental demonstration in order to become a scientific law: no experiments were either planned or suggested by him. The credit of having set the principle of conservation of energy upon a firm experimental foundation is, entirely, due to James Prescott Joule of Manchester (England). Owing to delicate health, Joule did not receive a regular course of instruction but was tutored at home; among his teachers was the chemist, John Dalton. Joule conceived the idea of the equivalence of heat and work at an early date and began his experi- ments upon it in 1840 at the age of 20. They were carried through in his private laboratory built for him by his father, a retired brewer. He was led to the problems of conservation of energy from a study of electric currents, having ascertained (in 1840) that there is a constant relation between the work necessary to maintain a current and the heat generated by it (Joule's heat). In his first determination of the mechanical equivalent of heat (1843) he utilized just this fact: The current was supplied by an electromagnetic generator and passed through a resistance immersed in a calorimeter. While this circuit operated at a stationary rate, he determined, on one hand, the work put into the generator, on the other, the heat developed in the calorimeter (result: 1 cal = 4.6 X 10 7 erg). Best known is his direct determina- tion of 1849 in which he produced heat by churning water and other liquids with paddle-wheels, and which gave 1 cal = 4.154 X 10 7 erg (for a long time, the accepted standard value, unsurpassed for several decades), but his work included the study of a great variety of processes relating to the interconvertibility of mechanical, thermal, electrical, 30 TEXTBOOK OF THERMODYNAMICS II 11 and chemical energies. As one of the founders of the principle of conservation of energy must be also regarded Hermann Helmholtz, at that time a young surgeon in the Prussian army. He at once recog- nized its importance and scope and was instrumental in bringing it to the attention and acceptance of scientists by publishing a brilliant paper (1848) in which he followed its workings through numerous applications to the sciences of physics, chemistry, and physiology. 1 It is very remarkable that none of these four men was by training a physicist. Sguin was brought up as an engineer, Mayer and Helmholtz as physicians, while Joule having received an irregular edu- cation cannot be classified with any professional group. The significant fact that the contemporary physicists had no part in the development of a most important phase of their science cannot be denied, even if Helmholtz and Joule later became ornaments of their profession. The reason for this has been pointed out by Ostwald and others: It was the time of reaction after the collapse of Schelling's " Natur- philosophie". All the physicists had been, for a longer or shorter period, under the influence of that movement. Now they were tired of fruitless generalities and anxious to make amends for the wasted time by conscientious factual research. They became suspicious of anything that was reminiscent of a speculative anticipation of nature. This frame of mind accounts for the hostility with which the mechan- ical theory of heat was received in physical circles. On the other hand, the active interest developed by engineers on its behalf hardly needs any explanation : The production of power from fuel is the engineer's everyday pursuit which compels him to think continuously about the connection between heat and work. These facts help us to understand the aloofness of the physicists and the active participation of an engineer in investigating the equiva- lence of mechanical and caloric energy. It remains, however, unex- plained how this principle could enlist the interest of two distinguished medical men. The writer began to feel that there might lie a historical problem in this and that it would be worth his while to inquire into the interests and the scientific background of physicians and physiologists of the period between 1820 and 1840. These suspicions were confirmed 1 Posthumous notes of S. Cannot disclose that this author, who wrote his " Reflex- ions sur la puissance motrice du feu " strictly from the point of view of the indestruc- tible caloric fluid, in later years (before his death in 1832) became more friendly to the idea of the equivalence of heat and work. However, it is hardly possible to assign him a share in the discovery of the first law, on the strength of this, because he did not possess either a rigorous formulation of it or any new empirical evidence to support it. Besides, his notes were published for the first time in 1878, a quarter of a century after the law had ceased to be controversial. II 11 THE FIRST LAW OF THERMODYNAMICS 31 by the following passage in one of Joule's papers: " On conversing a few days ago with my friend Mr. John Davies, he told me that he had himself, a few years ago, attempted to account for that part of animal heat which Crawford's theory had left unexplained, by the friction of the blood in the veins and arteries, but that finding a similar hypothesis in Haller's 'Physiology 1 , he had not pursued the subject farther " (Postscript to the paper in Phil. Mag. 23, 1843). These lines tell us about a third physician who was familiar with the idea of the equivalence of heat and work. Moreover, they give us a hint that the source of his interest in the subject lay in the theory of animal heat, with a reference to Haller's physiology as a clue. At present, the name of Albert Haller may not be generally known, but in the eighteenth century he was considered a scientific colossus and enjoyed an international prestige, as an authority in medicine and botany, which is equaled only by that of Newton, in his sphere and at his time. A native of Berne (Switzerland), he became the main scientific figure at the university of Gottingen. In German- speaking countries his reputation was further enhanced by important poetical writings. His " Outline of Physiology " appeared in 1747 and was immediately translated into French and English. It is not too much to say that it marked a new epoch in that branch of science and became the main influence in the education of several generations of physicians. The last German edition was published in 1822, and we have seen that Dr. Davies consulted this treatise even about the year 1840. The passage which Dr. Davies must have had in mind is contained in section 303 of the first volume. 1 On the preceding pages Haller analyzes the uses of the lungs and comes to the conclusion that the animal heat is produced in this organ and imparted to the blood passing through it. Of considerable interest to us is his hypothesis as to how the heat originates: In his own words, it arises " from the alternate extension and contraction, relaxation, and compression of the pulmonary vessels, by which the solid parts of the blood are per- petually rubbed together and closely compressed in the attrition that is made during expiration, as it is more rapidly moved and ground together during expiration". Very important is, also, the following sentence: " Nor is it any objection to this that water cannot be made hot by any friction. Nor in reality is this assertion true; for water by violent winds and motion, as well as milk, acquires some degree of warmth". These quotations contain Haller's theory of respiration which, for a long time, dominated the science of medicine. It is quite a surprise to find that this theory was essentially based on the idea 1 Not section 304 of Vol. II, as Joule gives the reference. 32 TEXTBOOK OF THERMODYNAMICS II 11 that heat can be generated by mechanical work, although it was enunciated 50 years prior to the work of Rumford. 1 We have found a convincing proof of the fact that, at a certain epoch, the physiologists were much concerned with the problem of the relation between work and heat, and we shall see below that they were keenly aware of the work of Rumford and Davies, seeing in it a confirmation of Mailer's hypothesis. It is surprising, at first sight, that this interest could have persisted until the middle of the nineteenth century, since the modern combustion theory of respiration, suggested already by Priestley, received strong support from the measurements of Lavoisier and Laplace (about 1781) and was rediscovered indepen- dently by the English physician Crawford (referred to by Joule as Crawford's theory). A closer investigation shows, however, that there were excellent reasons which prevented the acceptance of the combus- tion theory as a complete explanation of animal heat. Lavoisier and Laplace had found that the oxidation of carbon contained in the car- bohydrates of the blood is insufficient to account for the developed heat, and they expressed the conjecture that the balance is produced by the oxidation of hydrogen. An experimental test of this hypothesis required measurements extremely delicate and difficult for the prim- itive technique of those times. A long time passed, therefore, before they were even attempted. At length (in 1821), the Academic des Sciences announced a prize for the investigation of this problem, and this induced Dulong and Despretz to start work on it. Despretz 's essay was submitted to the academy in due time and received the prize on June 1, 1823 (published in the Ann. Chem. Phys. (2) 26, p. 337, 1824). He concluded from measurements on 200 animals that the complete heat of oxidation of both carbon and hydrogen accounts for the larger part of the animal heat but not for the whole of it. There remained a defect of about 20% for the explanation of which Despretz fell back upon Haller's theory, saying that it could be produced " by the motion of the blood and the friction of its different parts". The results obtained by Dulong were quite in agreement with those of Despretz's. He read his paper before the academy in 1822, but he did not contend with it for the prize and he resisted the urging of his friends to have it published (appeared after his death: Ann. Chim. Phys. (3) 1, p. 440, 1841). The reasons for this strange reluc- tance became clear when his posthumous papers were examined: Dulong had no faith in the thermochemical data on which his own and Despretz's conclusions were based. He intended to redetermine 1 Even Haller cannot claim priority in this matter, since similar views were held in the seventeenth century by the latro-Physical School. II 11 THE FIRST LAW OF THERMODYNAMICS 33 the heats of formation of carbon dioxide and of water in order to revise his essay. This rede termination, finally carried out, was the last scientific work done by Dulong: he obtained new and more accurate data but died before bringing his results into shape for publication (posthumously edited by Cobart, Ann. Chim. Phys. (3) 8, p. 183, 1843). While his work confirmed the previously accepted heat of formation of carbon dioxide as fairly accurate, it established an entirely new value for the heat of formation of water, showing that the old data were much too low. This correction was more than sufficient to remove the discrepancy between the animal heat and the chemical heat development of respiration and constituted a conclusive proof of the oxidation theory. However, it must be remembered that this knowledge came rather late: between 1823 and 1843 the last word of physiological science was the existence of a defect of animal heat which, apparently, was left unaccounted for by the combustion theory. There remains the question how widely this knowledge was diffused among physiologists and physicians. The writer happened to come across a very comprehensive German " Handbuch " of physiology of that time written in six volumes by leading authorities under the editorship of Karl Friedrich Burdach. The account of the theory of respiration, due to Ernst Burdach, is contained in the sixth volume, which appeared in the year 1840. Many pages are devoted to the history of the two theories, and both sides of the question are discussed at considerable length. After a lucid presentation of Haller's views (p. 544), Burdach turns to " the doctrine of the resemblance of respira- tion and combustion". It appears from his account that in the period between 1820 and 1840 the physicians interested in the theory of respiration were preoccupied, like Joule's friend Dr. Davies, by " that part of animal heat which Crawford's theory left unexplained". Most of them pointed to Haller's theory as to the most likely explana- tion and maintained that the origin of the unaccounted heat lay in the mechanical action of friction in the arteries. These mechanical effects were, in their turn, supposed to be produced by the " vital principle " (puissance vitale) invented by Barthez. Burdach gives many references to authors who advocated this view. Among them were some of the leading men of science, like Despretz and Treviranus, but also some of the rank and file. For instance, the writer had the opportunity of reading in the originals a German pamphlet by F. Lau (Bonn 1830) and an English paper by J. M. Winn (Phil. Mag. 14, p. 174, 1839). They are of great interest as an indication into how wide a circle the problem had spread. Lau and Winn were, apparently, 34 TEXTBOOK OF THERMODYNAMICS II 11 ordinary practitioners without much knowledge of the literature of the question or understanding of its deeper implications. Yet, their theories were only modifications of Haller's views. It is interesting to note that Burdach himself, who was not in sympathy with these explanations, did not question the fact that heat can be produced by friction, but regarded it as a matter of common knowledge, citing, in this respect, Rumford and Davy. He only doubted whether this cause would be sufficient to account for the desired effect quantitatively. The above facts throw a new light on the role of the medical profession in the history of the principle of conservation of energy. There was, actually, no break of tradition between the work of the eighteenth and the nineteenth centuries. However, this tradition was handed down through the unexpected channel of the science of physiology, whose representatives acted, for half a century, as the custodians of the idea of equivalence of heat and mechanical work. If this statement implies that the historical role of Mayer and Helmholtz consisted in turning back this principle to physics, the branch of science where it naturally belongs, this is in no way a detraction from the merits of these two great men: They were not merely passive transmitters of a concept, because there is an enormous gap between the vague knowledge current among physicians, that heat is generated by mechanical action, and the scientific formulation of the principle of equivalence, which could be bridged only by men of unusual ability. Helmholtz, eventually, developed into one of the greatest scientists of all times, and Mayer showed in his later physio- logical writings (1845) a clearer understanding of the nature of animal metabolism than any of his contemporaries. Nevertheless Mayer was entirely unsuccessful in his endeavors to persuade the physicists and chemists of the validity of his conception. The reason for this was, without doubt, that he could not offer them much in the way of an experimental proof at a period when they were particularly reluctant to accept anything on insufficient empirical evidence. A few years latter Helmholtz had far better success because, in the meantime, Joule's work had put the principle of conservation of energy on a secure experimental foundation. CHAPTER III THE FIRST LAW OF THERMODYNAMICS AND THE CALORIC PROPERTIES OF MATTER 12. Reversible and irreversible processes. It was pointed out in section 2 that the thermodynamical description of a system, by such variables as its temperature and pressure, implies that it is in equi- librium. In general, this equilibrium is disturbed if the system is subjected to any physical process. For instance, if we exert a mechan- ical action upon it or bring it under the influence of a source of heat, the result is a complicated transient state, of longer or shorter dura- tion, during which all the physical characteristics of the system are different in every point of it. The quantitative description of such an occurrence is, therefore, entirely outside the scope of thermodynamics. It is possible to follow through by thermodynamical methods only processes which take place so slowly that they can be regarded as a succession of states of equilibrium. Strictly speaking, the change of the system in a finite time must be even infinitesimal and the rate of change infinitely slow. Such infinitely slow processes have an important theoretical advantage in that they are reversible, that is, they can be gone through in both directions. In fact, let us take the example of a simple system: If we change two of the three variables, say v and T, so slowly that the system can be regarded as in equilibrium at any moment, the equation of state applies all the time and the third variable p is also completely determined. When we go through the same values of v and T in the opposite direction, the succession of states of the system is, simply, reversed. For every infinitesimal step, the work done by the system is expressed by Dw = pdv. When the process-is reversed, dv changes its sign while p has the same values so that the work retains the same absolute value as in the direct process,, but has the opposite sign, i.e. is done by outer forces against those of the system. We shall see in the next section that the heat received by the system in every infinitesimal part of the process is also a linear differential expression so that the same conclusions hold with respect to it: If a certain amount of heat is imparted to a system in an infinitely slow process, 35 36 TEXTBOOK OF THERMODYNAMICS III 12 the same amount is withdrawn from it when the process is reversed. We have restricted ourselves to simple systems, only, to fix our ideas : All our results apply also to the most general thermodynamical system because, for every phase of it, there exist equations of state of the type (2.14) and the elements of work and heat have always the form of linear differential expressions. When we turn to processes of a finite speed, we can no longer assert that they are reversible. As an example, we consider a gas in a vertical cylindrical vessel with a piston as its top. The reversible way of compressing such a gas would be to increase the load on the piston infinitely slowly by adding infinitesimal weights. When a finite weight is added at a time, the compression goes at a finite rate, but owing to the lag in the propagation of stresses, the compression and the pressure immediately under the piston will be higher than in the remainder of the gas and higher than in the reversible process. To produce the same reduction of volume, the outer forces have, there- fore, to do more work than in the infinitely slow case : A part of the work is consumed by elastic waves, currents, and other wasteful processes in the gas, the so-called phenomena of internal friction. On the other hand, if we expand the gas back to the original volume, proceeding at a finite speed, the same lag will cause the pressure at the piston to be lower than in an infinitely slow expansion. The gas will do, therefore, less work than in the latter process. It follows from this that the compression and expansion, at a rate which is not infinitely slow, are irreversible processes since the work which must be applied in the compression is larger than that received from the system in the expansion. The extreme case of rapid expansion is that when there is no outer force 1 all, for instance, when the piston is suddenly removed. Such an expansion into the void takes place without the system doing any work. It is quite obviously irreversible, since it is necessary to apply work of outer forces to compress the system back to the initial volume. Another example of an irreversible process is the phenomenon of thermal conduction of which we have made use in defining the con- cepts of temperature (section 2) and of heat (section 10) : If we bring in contact two bodies having a finite difference of temperatures, heat flows, spontaneously, from the warmer to the cooler, but never in the opposite direction. All processes of nature proceed at a finite speed, and experience tells us that they are all accompanied by effects of the nature of friction or conduction which make them irreversible. The engineer would call these effects " wasteful' ', since they do not produce any work, and the irreversible process " less efficient " than the Ill 13 CALORIC PROPERTIES OF MATTER 37 reversible. Strictly speaking, the reversible processes are an ideal which exists only in our imagination. However, this is enough to make its discussion very useful; besides, many real processes are con- ducted at so slow a rate that they can be regarded as reversible within the limits of experimental error. Only such processes can be mathe- matically described in thermodynamics, or represented by means of curves in diagrams. If the system is not in equilibrium no unique values of temperature and pressure can be assigned to it. 13. Analytical formulation of the first law. Internal energy. The purport of the first law has been made sufficiently clear in Chapter II. Its gist is that heat is convertible into mechanical work and other forms of energy, and vice versa, in the ratio of the mechanical equiva- lent of heat: 1 15-gram-calorie = 4.185 X 10 7 erg. By means of this conversion factor we shall always express heat and work in the same units. What is, then, the fate of an infinitesimal amount of heat DQ, imparted to a system by bringing it in contact with a suit- able heat reservoir? Since thermodynamics deals only with states which are infinitely close to equilibrium we need not consider its con- version into kinetic energy: The heat DQ will, in part, cause the system to do the work DW against outer forces; in part, it will be used to raise the temperature and, perhaps, to overcome the resistance of inner forces attending a change in volume and in other inner prop- erties. We denote the portion utilized for the latter purpose by DU and obtain the equation : = DU + DW. (3.61) Wejmow that the element of work DW is not an exact differential, but we do not yet know what the elements DQ and DU a^e, in a mathematical sense, and it will be our next objective to find this out. This is the reason why we use the symbol D: According to the con- vention made in section 8, the symbol d is reserved for expressions which, we are sure, are exact differentials. Let us now consider a reversible process in which the heat Q is imparted to our system by bringing it, successively, in contact with an infinity of heat reservoirs differing in temperature by infinitesimal amounts. For every step of this process eq. (3.01) holds, and we have, simply, to integrate it from the initial state of the system (1) to its final state (2) : Q = f* DQ, and, according to section 7, W =/f DW. We find, therefore, DU = Q - W. (3.02) f J 38 TEXTBOOK OF THERMODYNAMICS III 13 Of particular interest is the case when the process is cyclic so that at the end of it the system is, precisely, in the same state as in the beginning. The system itself does not, then, contribute anything to the energy balance. If we compare the conditions before and after the cycle is carried through, the only changes are outside the system. Namely, the heat Q is withdrawn from the reservoirs and the work W done against outer forces. The law of conservation of energy leads, therefore, to Q W 0, instead of (3.02), whence DU=Q, (3.03) for any reversible cyclic process. As we have seen in section 8, eq. (2.11), it follows from the property (3.03) of the integral that the expression DU is an exact differential. This means that there exists a certain function U of the variables defining our system whose total differential d U is identical with the differential expression D U. This function U is known as the internal energy, Every function which is completely determined by the variables of the system we shall call a characteristic function of it. The internal energy is, therefore, one of the " characteristic functions " of thermodynamical systems. Instead of eq. (3.01) we have to write, therefore, DQ = dU 4- BW. (3.04) The element DQ being the sum of an exact and a non-exact dif- ferential is itself not an exact differential according to the criteria of section 8. The integral (3.02) over a reversible process takes now the form Q - U 2 - Ui + W. (3.05) In words: the heat imparted to the system is used, partly, to increase its internal energy from the initial value U\ to the final Z/2, and partly to do outer work, Though derived for a reversible process, this equation has a wider range of application. In fact, suppose that in an initial state of equilibrium the system has the internal energy Ui. We subject it, then, to an irreversible process and, after it is over, wait until the system and its environment are again in equilibrium. ^Suppose it is now found that the internal energy has become C/2, that outer sources have lost the heat Q, and that the work W has been done against outer forces. The law of conservation of energy leads again to the relation (3.05). In this sense it can be applied also to an Ill 14 CALORIC PROPERTIES OF MATTER 39 irreversible process. In the particular case when the process is cyclic, the initiaT and the final state are identical, J7 2 Ui = 0, and Q - W. (3.06) The gist of the first law of thermodynamics is that work can be produced only at the expense of heat or some other form of energy. Therefore, it can be formulated as the impossibility of a perpetual motion machine (perpetuum mobile), as those delusive devices are called which claim to do useful work, continually, without drawing on any energy source. In its application to purely mechanical engines, this principle had already been recognized by Stevinus, Newton, and Leibnitz. The discoveries of the middle of the nineteenth century (section 11) extended it to engines of any kind. Exercise 14. 300 g of mercury are dropped from a height of 600 cm into a calorimeter consisting of 500 g of water at 10 C. In this process the potential energy of the mercury is, first, converted into kinetic energy and then into heat. What will be the final temperature of the water and mercury? (The specific heat of mercury is 0.0334 cal g" 1 deg" 1 , the acceleration of gravity g - 980.7 cm sec" 2 ). Exercise 15. The latent heat of vaporization of water is / - 9730 cal mol~ l (per 18 g). What part of it is spent on the work of expanding the steam, as calcu- lated in exercise 8? 14. Molal heat at constant volume and at constant pressure. We are going to apply the first law, as formulated in eq. (3.05), to the special case of a simple, chemically pure system. We shall refer Q, U, W to one mol of the material of the system, denoting the molaj quantities, by the small letters q, , w. Moreover, according to (2.04), the element of work is, for a simple system, Dw = pdv, whence Dq = du + pdv. (3.07) The molal heat is, according to the definition (2.19) of specific heats, given by c - Dq/dT (3.08) and depends upon the conditions under which the element of heat Dq is imparted to 1 mol of the system. Two cases are of special prac- tical interest: (1) the volume of the system is kept constant (dv 0) while heat is added to it; (2) its pressure is kept constant (dp ** 0) The internal energy u depends on the variables which define the system: in the case of a simple system, we can regard it as a function of any two of the three parameters p, v 9 T. To treat the case of 40 TEXTBOOK OF THERMODYNAMICS III 14 constant volume it will be useful to describe it in terms of temperature and molal volume: u = u(T, v), (3.09) a relation which is often called the caloric equation of state. Its total differential has, then, the expression which, substituted into (3.07), gives If we put dv = 0, we obtain, comparing this with (3.08), the molal heat at constant volume c v = On the other hand, to treat the case of constant pressure, we must find an expression of Dq in terms of the differentials dT and dp. The simplest way of doing this is to recall that, because of the equation of state, v itself can be regarded as a function of T and p so that Substituting this into (3.11) Putting dp = 0, we find by comparison with (3.08) the molal heat at constant pressure The subtraction of (3.12) from (3.14) leads to the important relation The reason why c p is larger than c 9 is obvious: If the system is per- mitted to expand while the element of heat Dq (per mol) is imparted to it, this heat is used, not only to raise the temperature of the systen| f Ill 15 CALORIC PROPERTIES OF MATTER 41 but also to do work, namely, work against outer forces to the extent p(dv/dT) p dT and against inner forces of the system itself in the amount (3u/3) r - (dv/dT) p dT. In the case of condensed systems (liquids and solids) the coefficient of expansion (3^/3 T) p is so small that it can be neglected for most purposes. Within this approximation it is unnecessary to distinguish between molal heats at constant volume and at constant pressure (c p c v = c). We postpone a detailed discussion of the molal heats, as to their experimental values as functions of temperature, until Chapter XVIII, but it will be necessary to say something about the approximate laws for permanent gases and for solids in the next two sections. Exercise 16. The approximate value of Ow/dtOr for N 2 , at C and 1 atm, is 0.0047. Estimate (dv/dT) p from the data of Table 1, and say by what percentage c p c v deviates from R. IS. The internal energy of perfect gases. The question in what measure the internal energy of gases depends on their volume can be decided in an experimental way. A very suitable experiment for this purpose is the expansion of a gas into the void mentioned in section 12. Provided that the system is adiabatically isolated, neither heat is imparted to it in this process nor does it do any external work, so that the terms Q and W in eq. (3.05) of conservation of energy vanish, and it is reduced to /2 U\ = or, referring it to 1 mol of the gas, Au = U2 ui = 0. If the expansion is small, we can write with the help of (3.10) and (3.12) or If the internal energy is independent of the volume, (du/dv)r = 0, there follows Ar = 0: the expansion into the void is not attended by any change of temperature. It is possible to test this point by conducting the process while the system is in contact with a calorim- eter> instead of being adiabatically isolated. Joule carried out the experiment (1845) in the following form: two large, closed copper vessels were connected by a short copper pipe containing a stop cock. While the stop cock was shut, one of them was filled with gas at high pressure, the other evacuated and both were immersed into the same water calorimeter. When the stop cock was opened, the gas rushed into the empty vessel without doing work, but no change of temper- 42 TEXTBOOK OF THERMODYNAMICS III 15 ature of the calorimeter could be discovered for any of the gases used. (A somewhat similar experiment, tried by Gay-Lussac as early as 1807, had a quite different purpose). It may be concluded from this that the dependence of the internal energy of permanent gases upon their volume is small, but the large heat capacity of the calorimeter reduced the accuracy of the experiment too much to say more. To increase the accuracy, it is desirable to remove the calorimeter alto- gether and to measure directly the temperature of the expanded gas. However, this is impossible in the original form of the experiment: after the stop cock is opened, the gas is in a turbulent state with no uniformity of pressure or temperature. It comes to rest only after a period of time long enough to vitiate the results by heat losses through convection. A modification which avoids these difficulties was sug- gested by William Thomson (later Lord Kelvin) and carried through, jointly, by Joule and Thomson in a MI NI M 2 Nz series of investigations extending from - l p i' T i! B ! P 2' T 2 ["> 1852 to 1862 - The y maintained a sta- hi C A 2 tionary stream of gas through a tube FIG. 7. Joule-Thomson A \A^ (Fig. 7), obstructed in the middle process. by a cotton plug C. Owing to the friction in the plug, there was no turbulence in the flow and the gas was in a homogeneous state on either side of the obstruction. Before passing it, its pressure, molal volume, temper- ature, and internal energy were pi, vi, Ti, u\, afterwards p2, 22, T%, #2. The difference of temperatures T% T\ was measured by a thermo- electric couple. Heat exchange with the environment was prevented by protecting the tube with heat-insulating material. As to the kinetic energy of the gas, and the friction losses in the plug, they are proportional to the square of the velocity of the flow, which was chosen so low that they were entirely negligible, compared with the other energy items. Let us set up the energy balance for a portion of the gas which is contained (at a certain moment) between the cross-sections MI and Af2, for the time during which 1 mol of the gas flows through the tube. At the end of this time it will occupy the new position between NI and N2 and we can apply to this process the eq. (3.05) of conservation of energy which reduces here to C/2 U\ + W = 0, since no heat is imparted to the gas (Q = 0). As a result of the dis- placement, 1 mol of the gas (with the internal energy #2) appears on the right side of the plug and 1 mol (of the energy ui) disappears on the left: Uz U\ = u% u\. On the right side, work is done by this part of the gas in pushing forward the adjacent portions through the volume v% at the constant pressure pz\ on the left, work is done Ill 15 CALORIC PROPERTIES OF MATTER 43 against it while it is displaced through the volume v\ (pressure pi): W = p2V2 p\v\. Our equation assumes, therefore, the form 2 + p2V2 = U\ + piVl. (3.16) We shall postpone the analysis of this equation and of the results of Joule and Thomson until section 28. Suffice it to say here that, generally, the partial (du/Qv)T turned out to be small, but not equal to zero. Comparing the different permanent gases at room temper- ature, it decreases in the order in which the gases become more perfect (section 6) and is almost negligible for helium. It is, therefore, logical to define the perfect gas as characterized by an internal energy entirely independent of its volume: (3w/3z>)r = 0, so that, from (3.12) and (3.10), du - c v dT. (3.17) This implies, of course, that c v is independent of v, because of the condition (2.10). As to its dependence upon the temperature, the experiments with real gases show, again, that it is the smaller, the nearer the gas approaches in its equation of state the behavior of perfect gases. In the definition of the perfect gas c v is, therefore, taken to be constant. Integrating eq. (3.17), there follows the expres- sion of the internal energy of perfect gases u - c v T + u , (3.18) where u (the so-called " zero point energy ") is an integration con- stant. This formula gives a good approximation to the internal energy of real monatomic gases, although it is not safe to extrapolate it to extremely low temperatures (section 105). In the case of diatomic and polyatomic gases, the approximation is fair only for a limited range of temperatures (on both sides of the room temperature): At very low temperatures the molal heat decreases because the rotational motions of the molecules require less energy; at very high temperatures it increases because the molecular vibrations begin to come into play. In view of this situation it is well to give the expression also for the case when the gas is thermally perfect (i.e. obeys the equation pv = RT) but calorically imperfect, to the extent that its specific heat, being in- dependent of v, is a function of T c, dT + ,. (3.19) In the special case c, = const, this becomes identical with (3.18). 44 TEXTBOOK OF THERMODYNAMICS III 15 In either case, eqs. (dv/dT) p = R/p and (9w/9^)r = hold, whence from (3.15): c p - c v = R. (3.20) This is the equation which J. R. Mayer used for the determination of the mechanical equivalent of heat, taking, for c p and c v , data expressed in calories (per degree and mol) and, for R, in mechanical units (compare section 12). Theoretically, his method was not free from objection, since, at the time, there was no foundation for the hypothesis that in the expansion of gases no energy is needed to overcome internal forces. In practice, however, the later work of Joule and Thomson showed that the formula is satisfied fairly well: Mayer would have obtained a pretty accurate value of the equivalent if good data of c v and c p had been available. The modern value of the gas constant in caloric units is R = 1.9864 cal deg mol" 1 . In the case of monatomic gases the molal heats are very close to c v = 3R/2, c p = 5-R/2, so that the ratio is 7 = c p /c v = 5/3. The data for the more important gases at 15 C are given in Table 4. TABLE 4 MOLAL HEATS OF GASES Gas 7 = C P /C V c P /R (c p - c v )/R Gas 7 = C P /CV Cp/R (c p - c v )/R A .668 2.52 1.00 C0 2 1.304 4.41 1.03 Ne .66 N 2 O 1.303 4.44 1.03 Xe .66 NH 8 1.31 4.49 1.06 rig .67 CH 2 1.31 4.27 1.01 H 2 .410 3.44 1.00 H 2 S 1.32 4.35 1.05 N 2 .404 3.50 1.00 SO 2 1.29 4.89 1.10 2 1.401 3.52 1.00 CN 1.26 5.36 1.09 CO 1.404 3.50 1.00 C 2 H 2 1.26 5.02 1.03 NO 1.400 3.52 1.00 C 2 H 4 1.255 5.08 1.03 HC1 1.41 3.56 1.00 C 2 H 1.22 5.84 1.05 cu 1.355 4.10 1.08 A more detailed account of specific heats of gases will be found in Chapter XVIII. Exercise 17. For water vapor in the range between t = C and * = 650 C Nernst gives the empirical formula c p - 8.62 + 0,002 / + 7.2 X 10- / 2 cal mol" 1 deg- 1 . Assuming that eq. (3.20) is satisfied, find c v and determine the increase of internal energy c*o ~ o. Ill 16 CALORIC PROPERTIES OF MATTER 45 Exercise 18. For CO 8 between -75 C and +20 C the empirical formula is c p - 8.71 + 66 X 10-* * - 22 X 10-' 1* cal mol'i deg~ l . Find uto M-75. 16. Molal heats of solids. We shall postpone the detailed theory and discussion of specific heats till Chapter XVIII and shall give here only two laws pertaining to solids. Though not rigorous, they are satisfied with a fair approximation at ordinary temperatures and are very useful because of their simplicity. (A) The law of Dulong and Petit (1819). The molal heats of all elementary solids have approximately the same value, close to 6 cal deg" 1 mol^ 1 . (Compare Table 5). TABLE 5 MOLAL HEATS OF ELEMENTARY SOLID SUBSTANCES Element /C C Element t C C Element /C c Ag 6.00 Fe 5.85 Zn 5.99 Au 6.07 Mg 5.83 C (graph) 1.82 Bi 25 6.07 Ni 6.05 C (diam) 1.25 Cd 6.15 Pb 6.25 B 1.9 Co 5.83 Pt 6.13 Cr 5.35 Sb 6.00 Cu 5.81 Sn 6.33 (B) Neumann's law (1831). The components of solid chemical compounds have, approximately, the same heat capacity in the compound as in the free solid state. In other words, the molal heat, divided by the number of atoms n in the molecule, is roughly equal to six (Table 6). TABLE 6 MOLAL HEATS OF SOLID COMPOUNDS Substance /C c n c/n Substance /C c n c/n AgCl 28 12.53 2 6.27 PbS 12.12 2 6.06 CuO 22 10.39 2 5.20 SnS 12.63 2 6.32 CuS 25 11.89 2 5.95 ZnO 16-99 10.15 2 5.08 KC1 23 12.4 2 6.2 ZnS 11.17 2 5.58 LiCl 11.96 2 5.98 CaF 2 15-99 16.82 3 5.61 MnS 12.11 2 6.06 Cu 2 S 19-52 19.1 3 6.37 NaBr 0-20 12.12 2 6.06 PbF 2 0-34 17.69 3 5.90 NaCl 24 12.14 2 6.07 PbCl 2 0-20 18.25 3 6.08 PbO 23 11.57 2 5.79 Pb0 2 24 15.50 3 5.17 46 TEXTBOOK OF THERMODYNAMICS HI 17 17. The heat function and the isobaric process. On both sides of eq. (3.14) of Joule-Thomson appears the expression X - u + pv. (3.21) The quantity x is known as the (molal) heat function, also called heat content and enthalpy. Like the internal energy u, the heat function is completely determined by the variables of the system: It is another example of a * 'characteristic function" (section 13). Differentiating eq. (3.21), we find du = dx vdp pdv and, substituting this into the expression (3.07) of the element of heat imparted to a simple system, Dq = d x - vdp. (3.22) In a similar way, referring all quantities not to 1 mol of a substance but to the whole simple system, we define as the total heat function X SB JJ + pV and find DQ = dX Vdp. The heat function acquires a particular importance in the so-called isobaric process, i.e. a process which takes place without change of pressure (p = const, dp = 0). For an infinitesimal isobaric change the last equation gives DQ = dX. In the case of a finite reversible change of the system, at constant />, from the state (1) to the state (2), we obtain by integration Q = X 2 - Xi. (3.23) The heat imparted to the system in a reversible isobaric process is equal to the difference of its heat functions in the final and initial states. Under usual laboratory conditions, processes of change of the physical state (melting, vaporization) are conducted at constant pressure, as well as most chemical reactions. The formula (3.23) has, therefore, a wide range of applications. For instance, the latent heat per mol is, obviously, equal to the difference of the molal heat functions of the material in the two states of aggregation / . X 2 ~ Xi. (3.24) If we choose T and p as the variables describing simple systems, we can write eq. (3.22) in the form and comparing with (3.13), (3.14), (3.26) HI 18 CALORIC PROPERTIES OF MATTER 47 In the case of a perfect gas, the heat function can be given explicitly: According to (1.13), x = " + RT, whence from (3.19) and (3.20) x - / c p dT + u , (3.27) A) and if c p = const, X = c p T + u . (3.28) Exercise 10. One mol of water is vaporized at 100 C and the vapor heated to 650 C. Calculate the increase of x in this process, using the data given in exercises 15 and 17. 18. The isothermal and adiabatic processes. It will be useful to discuss here the simplest thermodynamical processes, especially, with respect to the amounts of heat received and work done. Since the perfect gas is, so far, the only system whose behavior we know com- pletely, both as to its equation of state and its internal energy, we can give explicit analytical formulas only for perfect gases. The internal energy being a ' 'characteristic function" (compare section 13), its change is determined by the initial (1) and final (2) state of the system and is independent of the particular process. In the case of perfect gases it is, according to eq. (3.18), / rj-\ rr< \ /^ OQ\ or if the gas is only thermally perfect, and calorically imperfect, according to (3.19), /Tj c,dT. -, (3.30) On the other hand, the work w depends on the special process and, consequently, also the heat q which can be determined from the first law of thermodynamics (referred to 1 mol of a chemically pure system), q = U2 #1 + w. (3.31) We have already discussed the isobaric process (p const) in the preceding section. The process consisting in heating a simple system without change of volume (V = const) is so trivial that little need be said about it. It is called "isochoric" and, obviously, takes place without external work: w = and q = #2 i- Of great importance is the isothermal process (T const). We have already used the concept of isothermal curves (section 5), regarding them as the loci geometrici of all possible states of a system corresponding to the same temperature. However, we can interpret 48 TEXTBOOK OF THERMODYNAMICS III 18 them as the graphical representation of the following reversible process: The system is contained in a vessel with heat-conducting walls immersed in a large heat bath of the temperature T. The geometrical variables of the system (e.g. the volume V, if it is simple) are then infinitely slowly changed by external forces through their whole range of variability. If the compression or expansion takes place with finite velocity the temperature of the system will not remain strictly uniform, so that a process cannot be isothermal unless it is conducted in a reversible way. In the case of a thermally perfect gas, eqs. (3.29) and (3.30) give both u 2 u\ = 0, so that q = w. Moreover, p = RT/v, Dw = pdv = RTdv/v, and since T is constant, /2 = RT log - - RT log - (3.32) . V V\ p2 Equally interesting is the adiabatic process: This is the name for anything that can happen in a system surrounded by a heat-insulating or "adiabatic" (compare section 2) envelope which cuts off any inter- change of heat with the environment. This cover does not, in general, prevent the system from doing work against outer forces, as is illus- trated by the example of a gas in a cylindrical vessel closed by a slid- ing piston, both of adiabatic material. The gas could do work expand- ing and lifting by its pressure a load on the piston. Unlike the isother- mal, the adiabatic process can be conducted either reversibly or irreversibly. Since no heat is imparted in it to the system (0 = 0), the first law of thermodynamics (3.05) takes the form W = Ui - U*. (3.33) The system can do work only at the expense of its internal energy. In particular, the work of adiabatic expansion of a perfect gas from the state (1) to the state (2) is (per 1 mol), according to (3.29), w c v (Ti - T 2 ). (3.34) This is the formula which Sguin was trying to verify in his attempt to establish the first law experimentally (section 11). It implies the same tacit assumption which was made by Mayer: he took it for granted that the expansion of real gases proceeds without internal work. When the adiabatic process is conducted reversibly, to every infini- tesimal step of it applies the differential equation Q, (3.35) Ill 18 CALORIC PROPERTIES OF MATTER 49 and in the particular case of the compression or expansion of a perfect gas, c v dT + Dw - 0. We substitute for Dw as above, and obtain c v dT + (RT/v)dv = 0. Dividing by c v T and using the nota- tion of section IS, 7 1 = R/c V9 y + (7~ 1)*-0. (3,36) In the derivation of this formula only the thermal equation of state was used. It is valid also for the case where c v and 7 depend on temperature. However, when the specific heats are constant (calori- cally perfect gas) its integral is particularlv simple. The equation is, then, easily integrable and gives TV 1 = const. (3.37) We can substitute T = pv/R from the equation of state: pv y = const. (3.38) These two formulas are equivalent forms of the equation of the adiabatic for perfect gases, first derived by Poisson (1823) on the basis of the old ideas of a caloric fluid. Like the isothermals, the adiabatics can be represented as a family of curves in the (,p)-diagram. These curves have the same asymptotes as the iso- thermals pv = const, namely the axes p = 0, v = 0; but the slope of the adiabatic (dp/dv = yp/v) is always steeper than the slope of the isothermal (dp/dv = p/v) passing through the same point. One of the most accurate ways of determining the ratio 7 ** c p /c v experimentally is the measurement of the velocity of sound in the gas in question. The theory of elasticity gives for the velocity of sound a the formula (already derived by Newton) a2=s "T- M dv where /* is the molecular weight of the gas. The rate of compression and expansion in a sound wave is so fast that there is no time for any appreciable heat interchange by conduction and the process must be considered as adiabatic. 1 If the medium is a perfect gas, we find, therefore a 2 ypo/n. (3.40) 1 Compare section 122 with respect to the limitations of the method of sound velocities. *4 50 TEXTBOOK OF THERMODYNAMICS HI 19 Exercise 20. Calculate numerically the work done by a perfect gas in its isothermal expansion from i * V Q to v t - $t>o at C. Express it in ergs and in calories. Exercise 21. Give the expression for the work of adiabatic expansion in the case of a Van der Waals gas. Taking the coefficients a and b from Table 2 on p. 14, calculate it numerically for H* and N 8 and v\ - v 0t v* = Jt> and obtain the difference between the work done by these two real gases and the perfect gas. Exercise 22. Give the general form of the integral of the eq. (3.36) of adiabatic expansion, when c v (and c p ) is a function of T. Apply it to the empirical formula for water vapor given in exercise 17. Exercise 23. Calculating the work of adiabatic expansion from the formula (3.38), check the result (3.34). 19. The cyclic process of Carnot. The French engineer S. Carnot devised (in 1824) a process which represents a schematic approximation of the essential features in the operation of a heat engine. Suppose that we have two heat reservoirs of so large a capacity that their respective temperatures Ti and T 2 (<Ti) are not appreciably changed if we withdraw or add to them a finite amount of heat. They corre- spond in Carnot's scheme to the boiler and the cooler of a steam engine. We shall explain Carnot's process, giving to the operating part of his ideal engine, at first, a special construction: Let it consist of a cylin- drical vessel, with a sliding piston as its top, filled with N mols of a perfect gas. The cylindrical side walls and the piston are made of adiabatically insulating material while the flat bottom is heat con- ducting but can be overspread with a suitable adiabatic cover. In the beginning the gas is in a state represented by the point A (pi, vi, T\) of the (/>,iO-diagram (Fig. 8), and the process consists of four divisions: (1) The bottom is heat conducting and in contact with the first heat reservoir; by an infinitely slow reduction of the outer pressure on the piston the gas is expanded in a reversible isothermal way at the tem- perature Ti to the point A'(p'\> v'i, Ti). (2) The bottom is made adiabatic by applying the cover and the expansion continued, reversi- bly and adiabatically , to the point B'(p f 2, 1/2, T 2 ). (3) The bottom is again made conducting and brought in contact with the second reservoir; the gas is reversibly compressed along the isothermal T T* to the state B(fa t i> 2 , T*). (4) Finally, the bottom is adia- batically covered a second time and the compression conducted, reversibly, along the adiabatic BA. The point B is chosen in such a way that this adiabatic brings the gas back to its initial state A, thus completing the cycle. The general Carnot process has in common with that just described that it operates between the same two heat reservoirs and consists of four analogous divisions: Two of them are an expansion and com- Ill 19 CALORIC PROPERTIES OF MATTER SI pression of the working part of the engine, respectively, in contact with the two heat reservoirs (boiler and cooler) ; the other two are an adiabatic expansion and compression. However, it is less specialized in two respects. In the first place, the working substance need not be a perfect gas but can be any real or ideal thermodynamical system. In the second place, the four divisions of the cycle may be conducted not reversibly but with a finite veloc- ity (irreversibly). Since no heat is imparted to the working system in the adiabatic ex- pansion and compression, its total heat intake Q during the cycle con- <P 2 'V <P 2 ',v 2 '> sists, in all cases, of the heat Q\ (positive) received from the reservoir of higher temperature (T\) and the v heat Q 2 (negative) received from that FIG. 8. Carnot's cyclic process. of lower (r 2 ), so that Q = Qi + Q 2 . As the final state of the working system is identical with the initial, we can apply the equation of the first law in the form (3.06) : W = Q = Qi + Q 2 . From the standpoint of the engineer, Qi is the heat conveyed to the engine to make it operate and supplied by the consumption of fuel. Not all of it is transformed into work, because Q 2 is negative. The heat given to the cooler ( Qz) is, to his mind, a wasteful loss of energy which is, however, necessary in order to bring the working part of the engine back to its initial state and to ensure a continuous operation by, periodically, repeating the cycle. The ratio rj of the useful work W to the consumed heat Qi is called the efficiency or the conversion factor of the engine W Qi + We shall compute the efficiency for the special case when the cyclic process is conducted reversibly with N mols of a perfect gas as operat- ing substance (Fig. 8). From the expression (3.32) of the heat imparted to the gas in an isothermal process Q l - NRTi log (v'M, Q 2 - - NRT 2 log (v' 2 /v 2 ). On the other hand eq. (3.37) applied to one of the adiabatic divi- sions gives 7W" 1 * T 2 v 2 y " 1 or (vi/vtf* 1 - T 2 /Ti, and to the 52 TEXTBOOK OF THERMODYNAMICS III 19 other: (i/i/fl^) 7 " 1 = T*/T\. There follows vi/v* = v'i/v' 2 or v'i/vi = v'z/V2. Consequently Qi + Q2 = NR(Ti - T 2 and QI + & Tt- T 2 V = ^ - * ^ , (3.42) Qi Ti a relation which can, also, be written in the form + -0 (3-43) 1 1 12 and will prove very important in the next chapter. Exercise 24. Use the formula of adiabatic expansion derived in exercise 22 to show that the relation v'\/v\ v'z/v^ holds also in that case. The efficiency of Carnot's process is the same when c v is a function of T. Exercise 25. What is the thermodynamical efficiency of engines having coolers at C and boilers at 100 C, 150 C, 400 C? CHAPTER IV THE SECOND LAW OF THERMODYNAMICS 20. Several formulations of the second law. The second law of thermodynamics is older than the first. It was discovered by S. Car not, who enunciated it in the same pamphlet (1824) in which he gave the theory of his cyclic process. The reinterpretation and extension of the second law on the basis of the energy concept is due to Lord Kelvin (1851) and Clausius (1850-1863). There exist several equivalent for- mulations of this principle of which we give here the following: 1 st enunciation (Kelvin) : It is impossible to transfer heat from a colder system to a warmer without other simultaneous changes occurring in the two systems or in their environment. The reader will recognize that this is a more precise and amplified statement of one of the facts which we mentioned in the very beginning of this course (section 1), where we introduced the notions of equi- librium and temperature, and again in section 10, where we defined the concept of heat. We said there that heat flows spontaneously always from the higher temperature to the lower and never in the opposite direction. The second law goes beyond this and asserts that one cannot completely undo the results of the process of heat conduc- tion even in any indirect way. We shall show in the following sections that the above formulation is completely equivalent to the 2 nd enunciation : // is impossible to take heat from a system and to convert it into work without other simultaneous changes occurring in the system or in its environment. This law does not prohibit a process like the adiabatic expansion of a system (in which work is done at the expense of heat in the form of its internal energy) because this process is attended by " another simul- taneous change", namely, the change of volume of the system. Neither does it forbid the operation of Carnot's engine (preceding section): In his cycle heat is taken from one reservoir, but its conversion into work is not complete. The simultaneous change, which makes this process permissible, is the transfer of part of the heat to another reservoir of lower temperature. The transferred heat is less available for conversion into work in a similar way because this would require a 63 54 TEXTBOOK OF THERMODYNAMICS IV 21 third heat reservoir of still lower temperature. It is, therefore, often said that this heat is "dissipated" or "degraded". In general, a difference of pressures or of temperatures is necessary to obtain work; therefore, the phenomena of internal friction and of heat conduction which tend to reduce or destroy these differences must be regarded as attended by dissipation of energy. The establishment of the first law of thermodynamics made an end to all hopes of building a perpetual- motion machine (of the first kind) which could continually do work without some other form of energy being supplied to it. However, for all practical purposes a machine which could convert into worjc the enormous stores of internal energy contained in the oceans, the atmos- phere, and the body of our earth would serve just as well. Ostwald called such a machine, which could work without temperature dif- ferences, a perpetual-motion machine of the second kind. It is obvious that its operation would be in violation of the second law, so that we arrive at the 3 d enunciation (Ostwald) : // is impossible to construct a perpetual- motion machine of the second kind. This formulation makes the second law analogous to the first, which asserts the impossibility of a perpetual-motion machine of the first kind. 41. Efficiency of heat engines. Equivalence of the formulations. We have calculated, in section 19, the efficiency y = (Q\ + (?2)/(?i of the Carnot engine when operated reversibly with a perfect gas as the working substance. We shall call this particular type of construction and operation the perfect Carnot engine. The second law of thermo- dynamics permits us to say something about the conversion factor of the general Carnot cycle, which is the schematic prototype of any heat engine. In the real case, when the working part is any thermodynami- cal system and the cycle conducted either reversibly or irreversibly, we shall denote the efficiency by vj' = (Q'\ + Q'd/Q'i- Let us suppose, for a moment, that 17' > t?. We shall prove that this supposition is impossible, by carrying through the cyclic process alternately with a real and a perfect Carnot engine which share the same two heat reser- voirs. The length of the isothermal divisions A A' and B'B (Fig. 8) of the perfect engine is at our disposal and can be adjusted in such a way as to make the work done by the two cycles equal : W W or Q'i + Q't - Qi + Q 2 . (4.01) Moreover, the perfect engine being reversible, it can be run back- ward, in which case the quantities Q\, (>2 W simply reverse their signs, so that work is consumed and heat produced in the process. What IV 21 THE SECOND LAW OF THERMODYNAMICS 55 will now be the result of a double cyclic process in which the real engine goes through one cycle forward and the perfect backward? No work will appear in the net balance, because the first engine does as much work as the second consumes. The only effect will be that the heat Q'\ Q\ will be taken from the first reservoir and Q f 2 Qz from the second. These quantities are oppositely equal, according to eq. (4.01): Q f \ Qi = (Q f 2 (?2). Let us now recall our supposi- tion if > TJ or (Q'i + Q f 2)/Q'i > (Qi + (?L>)/(?I. Since the numerators are equal, this leads to 1/Q'i > l/(?i or Q'\ < Q\ and, consequently, Q f 2 > (?2- A positive amount of heat Q'% Q% is taken from the reservoir of lower temperature T% and transferred to that of higher Ti without any other changes occurring, either in the reservoirs or in the working parts of the engines. The supposition TJ' > 17 leads, therefore, to a contradiction with the second law of thermodynamics, and we must conclude that it is impossible and that only the possibilities *' ^ * (4.02) are permissible. In the particular case, when the process in the real engine is also conducted in a reversible way, we can run the perfect engine forward and the real backward. Repeating the same argument with transposed (primed and unprimed) symbols, we can, then, prove that 17' < 17 is impossible and that 77' S must hold. This result is compatible with (4.02) only when we have the sign of equality: rf = 77. All reversible Carnot engines have the same efficiency, no matter what the construction of their working part is. The sign < in eq. (4.02) refers, on the contrary, to irreversible cycles. Summarizing these conclusions and making use of the expression (3.42), we can give the general law for the efficiency 17 of any engine carrying out the cyclic process of Carnot: (4.03) where the sign of inequality (<) refers to irreversible, that of equality { = ) to reversible, operation. In the reversible case, there holds also the relation + % = 0, (4.04) ll -12 whicb it is only another way of writing eq. (3.42). An irreversible Carnot process is always less efficient^rfhQMaocr' wasteful than a reversible inasmuch as a smaller fraction/^ tHB Be*f (? 56 TEXTBOOK OF THERMODYNAMICS IV 22 taken from the first reservoir (boiler) is converted into useful work. However, eq. (4.03) with the sign < will also apply when only one of the four divisions of the cycle is conducted irreversibly and the three others reversibly. We can conclude from this that in the expan- sion of a system in contact with a heat reservoir less heat is received by the system and less work is done when the expansion is conducted in an irreversible than in a reversible way between the same initial and final states. In fact, let these two kinds of expansion be the first divisions of two Carnot cycles, while the remaining three divisions are identical and reversible in both. We denote the heat items in the first (partially irreversible) cycle Q'\, Q% and in the second Q\, Q%. The condition 17' < 77 or (Q'i + Qd/Q'i < (Qi + Qz)/Qi leads to Q'i < Qi, since 2 is negative. The change of internal energy is in both cases the same, because it depends only on the initial and final state, so that the smaller heat intake is attended also by the smaller production of work. We have used above the first enunciation of the second law given in section 20. However, the second leads to, precisely, the same results. Their equivalence is shown, in a general way; by the following simple reasoning. Consider a process which contradicts the second enunciation, that is, one which produces the work W taking heat from a reservoir at the temperature T^ without any other simultaneous changes. We can feed this work into a reversible Carnot engine which runs in the backward direction, adjusting it so that in a cycle just the work W is consumed, and using the reservoir T<z as the cooler. The net result will be that a certain amount of heat will be transferred from this reservoir to one of higher temperature TI (the heater of the Carnot engine), ^'without any other simultaneous changes". In other words, a process which violates the second law in the sense of enunciation 2, violates it also in the sense of enunciation 1. As to the third enunciation, it is in substance identical with the second being, merely, a restatement of it in different words. Exercise 26. Give an alternative proof of the relation (4.02), for the second enunciation, analogous to that given in the text, for the first. Instead of adjusting the two engines so as to satisfy (4.01), adjust them so that the total heat received by the cooler in the double cycle vanish. 22. The concept of entropy. All that follows from the second law of thermodynamics for a reversible Carnot cycle is embodied in the formula (4.04). We now extend our considerations by asking: What follows from it in the case of any other reversible cyclic process (different from that of Carnot's) carried out with a thermodynamical system. Our arguments and formulas will refer to systems of the most general IV 22 THE SECOND LAW OF THERMODYNAMICS 57 kind, but we shall illustrate them by a drawing (Fig. 9) relating to a simple system, the only one which can be represented in a two- dimensional diagram. We repeat what was said in section 13 about carrying out such a process. We must have an infinity of heat reser- voirs differing in temperature by infinitesimal amounts. The system is brought in contact with them, successively, and at the same time subjected to infinitely slow compressions or expansions. The cycle is, then, visualized by the solid closed curve of the figure. In its infinites- imal part corresponding to the segment A A' the system receives an element of heat DQ. Let us now draw through the point A the isother- mal AC and through the point A' the adiabatic A'C, and denote by DQi the heat which the system would receive if it were subjected to the infinitesimal isothermal process AC. The relation between DQ and DQ\ is obtained by considering the little cyclic process AA'CA and setting up the equation (3.06) of the first law for it: the isothermal part AC is carried out in the reverse direction and we have DQ - DQi = DW. The ex- pressions DQ and DQ\ are infinitesimal of the first order, but the work DW done in the little cycle is represented by the area of the triangle AA'C and is, therefore, infinitesimal of the second order, infinitely small compared with the other two terms. It must be omitted from the equation, leaving DQ = DQ\: we can, therefore, change our notations, denoting the work done in the segment A A' also by DQ\. This conclusion holds for any system and not only for a simple one: Although the work done in an infinitesimal cyclic process cannot, in general, be re- presented in a two-dimensional diagram, it is, nevertheless, always infinitely small of the second order. Let us now continue the adiabatic through the point A 1 and draw one through A until they intersect the closed curve a second time in B', B. According to the result just obtained, the heat DQ^ received by the system in the part of the process B'B is the same which corre- sponds to the isothermal segment B'D passing through B f . We see, therefore, that the two heat items DQi and DQ* are precisely the same which the system would receive from the boiler and cooler if it formed the operating part of a Carnot engine carrying out the reversible Carnot cycle ACB'DA. Denoting by T\ and T* the temperatures which the \ B FIG. 9. General cyclic process. 58 TEXTBOOK OF THERMODYNAMICS IV 22 system has in the points A and B', we can apply to DQ\ and DQ2 the relation (4.04) for a reversible cycle of Carnot (4.05) JL I J. 2 If we cut up into strips the whole area of the closed curve repre- senting the cycle, by a system of adiabatics drawn at infinitesimal distances from one another (the dotted lines of Fig. 9), eq. (4.05) will hold for the pair of segments of the cycle in every strip. Integrating over all strips C no* C nn* = o. The first integral represents the summation over all elements of the cycle between the points M and N, the second term the summa- tion over the other half of the curve from N to M. We can, therefore, replace the two terms by the single integral 0, (4.06) extended over the whole cycle. This result applies to any cyclic process, so that the integral of the linear differential expression DQ/T taken over any closed path vanishes. We know from section 8 that, in this case, the expression DQ/T is the exact differential of some function 5 of the variables of the system : DQ dU + DW J0 -j - - J> - - dS. (4.07) In the language of that section we say that DQ, while not an exact differential, is integrable. The integrating multiplier is 1/T, the reciprocal of the absolute temperature as measured by a perfect gas thermometer. Clausius who first introduced the function 5 called it the entropy of the system (derived from the Greek word Ivrpcn-ofuu = turn inside, an allusion to its one-sided character). If we integrate eq. (4.07) over a reversible process leading from state (1) to state (2), we find - Si - . (4.08) J\ T The integral depends only on the initial and final value and is independent of the path, provided that this path is reversible (compare IV 22 THE SECOND LAW OF THERMODYNAMICS 59 section 8), and the entropy is a "characteristic function", i.e. com- pletely defined when the state of the system is defined. We can take as state (2) in formula (4.08) any state of the system and as (1) some state (n) which we arbitrarily choose as normal (4.09) (n) Owing to the fact that the entropy is defined by the differential equation (4.07), it contains a constant of integration. As such ap- pears in the expression (4.09) the entropy S n of the normal state. As .far as the first and second laws of thermodynamics are concerned, it cannot be determined by any theoretical reasoning. In the case of a homogeneous system, the heat DQ received in every element of a reversible process is distributed uniformly, so that the entropy S also acquires a uniform distribution over the system. We can speak, therefore, of the specific entropy and, when the system is chemically pure, of the entropy per mol or molal entropy which we shall denote by 5 (small). We may further ask: What is the joint entropy S of two systems which, individually, possess the entropies Si and 52? According to the defining equation (4.09), S is obtained by carrying out a reversible process with the combined system. The reversibility requires, however, the equilibrium of all parts at every stage of the process, so that our two constituent systems must have at every moment the same temperature T. If we denote the elements of heat imparted in an infinitesimal step of the process, respectively, by DQ, DQi, DQ 2 , there follows DQ - DQi + DQ 2 , DQ/T - DQi/T +DQz/T and, from eq. (4.07), dS = dSi + dS 2 , or S = Si + S 2 . (4.10) The entropy is additive. We repeat, however, that the additive property rests on the fact that two systems in thermodynamical equilibrium have the same temperature. If two systems are, initially, not at the same temperature but can be brought into thermal contact, it has a good sense to ask about their joint entropy, and we shall define it as additive. In other cases this question is without meaning. It is, of course, possible to imagine a case in which the two parts are adia- batically insulated from one another but can interact by means of movable pistons and similar devices. They could be in partial equi- librium without equality of temperature, but this is a dynamical problem rather than a thermodynamical. The notion of an entropy TEXTBOOK OF THERMODYNAMICS IV 23 of such a combination is artificial and without interest in thermo- dynamics. 1 The entropy difference (4.08) takes a very simple form when the initial and the final states are of the same temperature. We can, then, take an isothermal process as the path of integration (T = const). S2 Si = 7T (4.11) (S r T 3 ) 23. The entropy principle. Let us discuss how the entropy of a system changes in an adiabatic process. If the process is reversible, eq. (3.35) or DQ = applies to every infinitesimal step of it. It follows, then, from (4.07) dS = 0, and S = const. (4.12) The reversible adiabatic process leaves the entropy unchanged: it is, therefore, also called " isentropic". When we turn to the irreversible adiabatic process, we are on new ground. We cannot expect to get an answer to our question from the considerations of the preceding sec- tion because they were restricted to reversible processes. Therefore, we have to fall back, once more, upon the second law as enunciated in section 20. In the same way as in introducing the entropy concept we shall use a (p, V) -diagram (Fig. 10) as an illustration, although our ar- gument will be valid for the most general thermodynamical systems. Suppose that the system is sub- jected to an< irreversible adiabatic process which brings it from the state A (with the temperature T\ and entropy Si) to the state B(T%, ^2). The transition itself, being irreversible, is not capable of a graphical representation (compare section 12). What can be said about the entropy change AS = 2 ~ Si? We can bring the system 1 The only known significant case in which the entropy is not additive occurs in the theory of radiation. When a pencil of rays is split by partial reflection into two (the reflected and retracted), they have, in general, temperatures different from each other's and from that of the parent pencil. At the same time the sum of their entropies is not equal to the entropy of the parent pencil. Compare: M. Von Laue, Ann. der Phys. 20, p. 365 f 1906; P. S. Epstein, Phys. Zs. 15, p. 673, 1914. FIG. 10. Entropy change in irre- versible adiabatic process. IV 23 THE SECOND LAW OF THERMODYNAMICS 61 back into its initial state A in the following reversible way: First, we let it undergo a reversible adiabatic process EC roughly opposite to the first, until it comes to the temperature Ta, while its entropy remains constant and equal to 52. From the state C on, we conduct it in a reversible isothermal (CD) until the initial entropy Si is again reached; then we complete the cycle, bringing the system adiabatically back to the state A. The temperature T% can be chosen arbitrarily higher or lower than either T\ or 7^2, or between these values. It is important to have this freedom of choice because certain processes (as transformations) can be conducted only at special temperatures. The system receives no heat in the three adiabatic branches of the cycle, and the heat Q in the isothermal part. Applying eq. (3.06) of the first law to the cyclic process (from A to A) we write where W is the total work done by the system. If Q and W were positive, the result of the process would be that the heat Q is taken from the heat bath, maintaining the constant temperature T% in the isothermal branch CD, and converted into work " without any other simultaneous changes." This is a violation of the second law (enunci- ation 2), and we conclude Q > or Q ^ 0. The application of eq. (4.11) to the isothermal CD gives Q = Tz(Si 2), since Si is here the final and 2 the initial state. The inequality implies, therefore, Si - S 2 ^ or AS = S 2 - Si ^ 0. The same argument can be easily adapted to the more general case where the system consists, initially, of many parts which are not in equilibrium. At the end of the irreversible adiabatic process it may have the same parts in new states or, even, other parts since some of the parts may have become merged, others separated into several units. At any event, the system can be brought back to the initial condition by the following reversible procedure. The final parts are brought, separately, in a reversible adiabatic way to the same suitably chosen temperature TV At this temperature they are iso thermally rearranged into the original parts, unmixing some of them if necessary (compare section SO). Still at the same temperature T$ each of the part systems is separately brought to its initial entropy. The cycle is completed by bringing the parts back to their initial states in independent, reversible, adiabatic processes. The conclusions drawn above for the simpler case obviously apply without change in the more complicated one. They apply, of course, also to every part of the irreversible process for which a definite entropy change can be defined. Under no circum- 62 TEXTBOOK OF THERMODYNAMICS IV 23 stances can this change be negative. It follows from this that, in the case 52 Si = 0, the change must be zero for every smallest part: dS = 0. That is to say, A5 = refers to the reversible process and A5 > to the irreversible : The irreversible adiabatic process is always attended by an increase of entropy. The summary of the results of this and the preceding section is the "entropy principle", due to Clausius, which represents an analytical formulation of the second law of thermodynamics. It consists of two parts: 4 th enunciation: (a) There exists a characteristic thermodynamical function called the entropy. The difference of entropies of a system in the states (1) and (2) is given by the expression (4.13) where the integral is to be taken over any reversible path connecting the two states. (b) In an adiabatic process the entropy either increases or remains unchanged: AS ^ (4.14) where the upper sign (>) refers to the irreversible, the lower ( = ) to the reversible, case. The adiabatic processes include also the case when the system is "left to itself", meaning by this that it is completely cut off from any interaction with its surroundings and does not receive any energy from outside, either in the form of heat or of work. The parts (a) and (b) of the entropy principle contain all that can be deduced from the enunciations of section 20. Sometimes, a third general law is included in the entropy principle: the fact that all systems tend towards equilibrium so that the entropy always does increase when the conditions of the system permit it. We prefer, how- ever, to discuss this law separately in section 30. As enunciated here, the entropy principle is completely equivalent to the formulations given in section 20. In fact, their validity is a sufficient condition because we derived the principle from it. But it is also a necessary condition because the adiabatic processes prohibited in those formu- lations are attended by a decrease of entropy. (Compare exercise 27). Since a spontaneous irreversible process is one of increasing entropy, we can use the condition AS > as the criterion of irreversibility. Exercise 27. Show that, in the fictitious case when the efficiency of the Carnot engine is y f > ij, the expression on the left side of (3.43) is negative. This means that the sum of the entropies of the heater and cooler decreases. IV 24 THE SECOND LAW OF THERMODYNAMICS 63 24. Entropy of the perfect gas. In the case of a simple homo- geneous and chemically pure system, the entropy can be referred to 1 mol of its material (section 23), and eq. (4.07) takes the form (4 , s) The only system for which the expressions of u and p were given in the preceding chapters is the perfect gas: According to (1.13) and (3.17) we have du = c v dT and p = RT/v t <fe-*y + ** (4.16) This, really, is an exact differential of the function s = c v log T + R log v + s'o, (4.17) where s'o is an arbitrary constant, the so-called entropy constant of the perfect gas. Using the equation of state, we can express v by p and T, or T by p and v, and so obtain two other forms of the molai entropy s = c p log r - R log p + so, (4.18) s = c v log p + c p log v + s"o, (4.19) since c p = c v + R, according to (3.20). The constants are related in the following way: s'o = so R log R, s"o = so c p log R. It is easy to see that eqs. (3.37) and (3.38) of the adiabatic are equivalent to the condition 5 = constant. It is well to mention also the case when the gas is perfect, in thermal respects, but the molal heat c v is a function of temperature (end of section 15). Equation (4.16) leads then to the expressions (4.20) So. fcJT J ~T~~ Exercise 28. Check the statement that the process of expansion into the void (section 15) from the molal volume Vi to v 2 is irreversible (i.e. As > 0) for the perfect gas, by computing the entropy difference $ 2 SL between the final and the initial state. Exercise 29. Two vessels contain each one mol of the same perfect gas at the respective temperatures Ti and TV They are brought in contact and the two specimens allowed to come into equilibrium through the heat-conducting walls without change of volume. Compute total entropy in initial and final states and 64 TEXTBOOK OF THERMODYNAMICS IV 25 show that it increases. (Remark: The arithmetical mean of two numbers J(a + b) is always larger than the geometrical * 25. The reciprocity relation of thermodynamics. The require- ment that the expression (4.15) must be an exact differential imposes a restriction on the analytical form of the internal energy u: the caloric equation of state (3.09) is not quite independent of the thermal. In fact, the exact differential must satisfy the reciprocity relations (2.10) of section 8. In the case of a simple system, there are only two inde- pendent variables and only one reciprocity relation. Let us substitute for Dq the form (3.11) *-?() 3 T.- dT + -- n ^ Then the reciprocity relation has the form ~ir\() + T\' (4 ' 22) : oi LJ. \OV'T 1 Jt> In carrying out the differentiation, we shall remember, that u is here regarded as a function of v and T only, and drop the subscripts in the second derivative 1^2 1 O ** 1 ^, ^. + l(*P\-\(*\ + J T\dTj v T 2 l\dv/T *T The order of differentiation in the second partial can be transposed and the two terms depending on it cancel out. There remains .* which gives, substituted into (4.21), (IX- (); In the case of perfect gases, (dP/dT) v = p/T and (du/3v) T = 0. We justified the assumption c v = f(T), for them, by the measurements of Joule and Thomson on real gases (section IS). We see that this assumption is consistent with the second law, as we have already checked in the preceding section. Vice versa, accepting the second law, we can conclude theoretically that u is independent of v and a function of T only, for substances satisfying the equation pv = RT. This fact is well supported by observations on real permanent gases. When IV 25 THE SECOND LAW OF THERMODYNAMICS 65 the measurements are very accurate, the deviations from the law of perfect gases begin to tell. The incentive which caused Thomson and Joule to undertake their investigation was the fact that Regnault's data for (dp/dT) v , when substituted into (4.23), gave a finite value of (du/dv)r for real gases. In the case of gases strictly obeying the eq. (1.21) of Van der Waals, we find from (4.23) = J' < 4 ' 25 ) This partial is independent of T. Since du is also an exact differen- tial, we can apply the reciprocity relation to it and conclude that the other partial c v = (du/QT) v is independent of v. The internal energy has then the expression / c v dT - - (4.26) and the term a/v represents the inner potential energy of the forces of interaction between the molecules. Equation (4.23) can be used to simplify the expression (3.15) of the difference of molal heats c p c v , making it dependent upon easily measurable quantities This formula was extensively tested by measurements o/ the velocity of sound in gases (section 18) up to pressures of 200 atm and found to be in good agreement with observations. 1 Exercise 30. Show that the entropy of the Van der Waals gas is (4.28) Exercise 31. Derive by integrating (4.23) Exercise 32. Starting from (4.23) prove the relation P. P. Koch, Ann. Phys. 26, p. 551; 27, p. 311, 1908. 66 TEXTBOOK OF THERMODYNAMICS IV 26 Exercise 3d. A further relation can be derived by partial differentiation. Sub- stitute in u( T, v) the expression of T in terms of p and v so that u = u[T(p t v) t v]. By differentiating with respect to v derive (Q\ _ (^u\ ( ^ u \ (^T\ 9v/ p \ Qv/7 1 \97/ v \9i/ /p Transform this with the help of (3.12), (4.23), and (4.27) into - P- (4.31) 26. Measurement of entropy of differences. From the definitions of molal heat (3.08) and of entropy (4.15) there follows the relation c = Tds/dT, which is valid for all ways of imparting heat to the system. In the two simplest cases of constant volume and of constant pressure this leads to <4 - 32 > Dividing by T and integrating, we obtain the following two expres- sions for the entropy (4.33) P), (4.34) whose equivalence with (4.32) can be checked by differentiating them partially with respect to T, at constant v and p, respectively. If we wish to find the entropy increase (per mol) experienced by a substance when its temperature is raised, at constant volume, from T\ to Tg, the first equation gives /TZ T I s(r a , r) - s(Ti, v) = / ^dT. (4.35) */ Tl 1 On the other hand, the increase at constant pressure is from the second equation /T2 % - , p) - s(ri, p) = / ~ dT. (4.36) */ TI 1 The above expressions give the entropy changes, as long as the system does not undergo any transformation of state. When a trans- formation occurs, it involves a latent heat / (per mol) and an increase of entropy which is, according to (4.11), *-*i-, (4.35') IV 26 THE SECOND LAW OF THERMODYNAMICS 67 since the transformation takes always place at constant temperature. Here, too, the cases v = const and p = const must be distinguished. At constant volume no work is done and the latent heat is, simply, equal to the increase of the internal energy. At constant pressure it is equal to the increase of the heat function (compare section 17): / = Au, l p = A X . (4.360 We see from this discussion that the determination of entropy dif- ferences involves only measurements of molal and of latent heats. The same data which are necessary to find internal energies and heat contents give also the knowledge of entropies. We have tacitly assumed that the system we are dealing with is a chemically pure substance. If this is not the case, we can readily adapt eqs. (4.35) and (4.36) to any system in which no chemical reaction takes place, simply writing the total entropies and heat capacities (capital letters) instead of the molal. With respect to chemical reactions, two cases may occur: (1) The reaction is conducted at constant temperature. This case is completely analogous to that of a transformation, the entropy increase being S 2 -Si = |, (4.35") with Q v = A 7, Q p = AX. (4.36") (2) There is a continuous displacement of chemical equilibrium as the temperature is raised. Then the heat of reaction must be included in the heat capacity; we shall consider a case of this sort in section 121. Exercise 34. Calculate the entropy increase in the following process: 1 mol of liquid water (18 g) at 100 C is vaporized at the pressure of 1 atm. and the vapor is heated (at constant pressure) to 650 C. Take data from exercises 8 and 19. Exercise 35. (a) The molal heats c p for Ag, Al, and graphite are: rK 50 100 150 200 250 273.1 298.1 Ag 2.69 4.82 5.54 5.84 5.97 6.02 6.04 Al . . 90 3.13 4.44 5.13 5.54 5.68 5.831 C (graphite) . . . 0.13 0.41 0.79 1.22 1.65 1.86 2. fed Calculate in a rough graphical way the increase of the heat function as temperature is raised from 50 to 298.l K (p = const). Ot the (b) Calculate from the same data (in a rough graphical way) the in' entropy between 50 and 298 .l K. Exercise 36. Calculate the heat and the entropy imparted to a v~** (4.43) the following two reversible processes: (a) starting from the state pj is conducted first at p PI to pi, , then at v vt to pi, PI. (V 'The introduc- 68 TEXTBOOK OF THERMODYNAMICS IV 27 at v v\ to pz, Vi, then at p pi to pz t vz. Suppose c v and c p to be constant. Show that the heat imparted to the system is different in the two processes but the entropy change is the same. 27. The extended law of corresponding states. The relation (4.23) between the caloric and the thermal equations of state can be written in the integrated form = f [ T (|^ - p]dv + f(T). (4.37) We have seen in section 6 that, within certain groups of substances, the thermal equation satisfies the law of corresponding states. It may be asked, therefore, if a similar law does not apply to the caloric equation of state (4.37). As the most precise expression of the law of correspondence we may take that following from the mean reduced equation (1.29) of Kamerlingh Onnes. According to that equation, the quantity K RT c /p c v c is a constant for a given substance and, at the same time, a universal function of the reduced variables ?r, <p, r. Conse- quently, it must be a universal constant, as also ^ = ., (4.38) L c since K = R/K is another universal constant. If we substitute into the combination pv/T the expressions of the parameters in terms of the reduced variables (p = p c TT, v = v c <f>, T = T c r), we find im- mediately ^ = K^ = /i(ir,*>, r). (4.39) In words, the quantity pv/T has the same value for different substances in corresponding states. In a similar way, substituting TT, #>, r into the integral of eq. (4.37), we find that this integral, divided by T, is a universal function of the reduced variables. This fact suggests he hypothesis that the same applies to the second term and to the >ole expression (apart from the zero energy) jT- =/2(7r,^r). (4.40) iantity might be also the same for different substances in states. This hypothesis was, in fact, made by nnes and others. We shall call the simultaneous validity IV 28 THE SECOND LAW OF THERMODYNAMICS 69 of eqs. (4.39) and (4.40) the extended law of corresponding states. Though far from rigorous, this law is satisfied with a fair approximation within certain ranges of temperatures by large groups of substances (compare section 47). It may be pointed out that, in the case of perfect gases, the two functions /i and /a have the simplest possible form : they are reduced to the constants R and c v . An immediate consequence of these equations is that the expres- sion (4.15) of the entropy differential is also a universal function Fof TT, <p, T, whence As = Ffa, <p2j r 2 ) - F(TTI, ?i, n). (4.41) The molal entropy difference between corresponding states is the same for all substances (obeying the extended law). There were physicists who went even beyond this in their extension of the law and applied it to more general systems, having other variables than />, V, T. Suppose, for instance, that the element of work is, in the sense of section 7, dW = pdV + ydX, whence the entropy differential (4.07) dU + pdV + ydX dS = - - -- (4.42) The generalized force y can be treated exactly on the same footing as p (exercise 38), and it is possible to argue that the expression yX/T should have similar properties (with respect to corresponding states) as pV IT and U/T. We shall see in sectioit 86 that this idea was fruitfully applied to the theory of surface tensions. Exercise 37. Substitute for p, v, T the reduced variables into the integral of eq. (4.37), divided by T, and show that the result is a universal function of IT, <p, r. Exercise 38. Substitute in (4.42) v,x /x,T \dX/ T .v and set up the reciprocity relations in analogy with (4.22), (4.23). 28. Theory of the Joule-Thomson process. We are now in a position to subject the experiments of Joule and Thomson, described in section IS, to a closer analysis. We found there that the relation between the states of the streaming gas (Fig. 7) on both sides of the plug is given by eq. (3.16), which we can write in the form AX = X2 xi = 0, (4.43) making use of the definition (3.21) of the heat function. The introduc- 70 TEXTBOOK OF THERMODYNAMICS IV 28 tion of this function permits us to express the entropy differential in a new way: From eqs. (3.25) and (4.15) We apply the reciprocity condition (2.10) to this expression and obtain, in complete analogy with the way in which we found the for- mulas (4.23) and (4.15), , . and Of course, this is not a new relation but only another form of the reciprocity condition (4.23) into which it can be transformed by applying the formulas (1.05) and (3.21). Owing to this equation and to (3.26), the differential of the heat function becomes [v - T (j) ]dp. d x = cJT + v - T dp. (4.47) Suppose that in the Joule-Thomson experiment the difference of pressures A/> = p2 pi and of temperatures AT 1 = T% T\ is so small that it is permissible to neglect squares and higher powers and use eq. (4.47) for expressing AX in terms of &p and A7\ The condition (4.43) gives then Ar v - T(Qv/dT) p " = -^ = - 7 P - ' (4 ' 48) This change of temperature per unit pressure drop, n = AjT/A/>, is, usually, called the differential Joule-Thomson effect. With the help of this formula its sign and magnitude can be calculated theoretically if the equation of state and c p are known. We see that the sign of the effect depends on that of T(dv/dT) p v, or, transforming this by means of (1.05), upon the sign of the inequality The differential Joule-Thomson effect has the sign opposite to that IV 28 THE SECOND LAW OF THERMODYNAMICS 71 of the bracket expression (because dp/dv is negative). In the case of Van der Waals gases, this condition takes the form 2a RTb > ^ 2 2 < - v 2 (v b) 2 < When the density is low, the constant b in the denominator of the second term gives rise to corrections of the second order only, and can be neglected in the first approximation, leaving the condition T T^ Ti = 2a/Rb = 6.757;, (4.51) taking into account (1.24). Ti is called the inversion temperature: below it, the effect is positive and consists in a cooling of the gas; above, it is negative and the gas gets warmer in passing through the cotton plug. We see from (4.50) that the two Van der Waals constants have an opposite influence, so that the sign depends on which of them dominates. From the physical point of view, we have here an interplay of two phenomena: On the one hand, when a is different from zero, the expanding gas does work against inner forces (compare eq. (4.26) of preceding section) and this decreases its temperature accounting for a part (a/v 2 ) of the expression (4.50). On the other hand, the outer work p22 piv\ of the process does not vanish for Van der Waals gases, even when the temperatures remain unchanged. It can be positive or negative and contributes the remainder of (4.50). On the whole; the larger part of a positive effect is due to inner work. The interplay of the two effects is more complicated when the density is not low. The expression (4.50) with the sign of equality, determining the inversion point, is best written in terms of the reduced variables (1.25) -- < 4 - 52 ' Eliminating <f> from this expression and the reduced Van der Waals equation (1.26), we find (4.53) This formula shows us that there are two inversion points. The coefficient /* is negative above the upper inversion point and below the lower, being positive in between. The position of these points depends on the pressure : the upper of them sinks as the pressure is increased, the lower rises. 72 TEXTBOOK OF THERMODYNAMICS IV 28 In a qualitative way these predictions of the Van der Waals theory are fulfilled by real gases. The lower inversion lies in the liquid state and has been observed in several substances. It is, further, borne out by the experimental data that, the more perfect a gas is, the lower its upper inversion point lies and that it is depressed by pressure, while the lower point is raised by it. However, the Van der Waals equation cannot be used for a quantitative calculation of the Joule-Thomson effect or of the inversion point because its approximation is insufficient in the range of pressures and temperatures here involved. This is apparent from the figures given in Table 7. TABLE 7 DIFFERENTIAL INVERSION POINTS Substance p aim TiK Authority Upper He 1 23.6 Roebuck and Osterberg 1934 H 2 113 (192.7) Olszewski 1901 Air 150 553 Hoxton 1919 < t 203 523 Lower A 250 164 Roebuck and Osterberg 1933 4 4 50 125 Air 150 140 Hausen 1926 4 I 125 133 CO, 18 to 100 249 Burnett 1923 Hoxton 1 gives the following empirical formula for air, in the range from to 280 C and from 1 to 220 atm, (4.54) 1 J. ~ where p is measured in atmospheres. The cooling by means of the Joule-Thomson process has acquired a great importance in the technique of low temperatures and of liquefying gases. Mostly, it is not used as the sole agency of cooling but in com- bination with adiabatic expansion. Air, oxygen, and nitrogen can be subjected to the process at room temperature since their inversion points lie higher. For hydrogen, T> lies above the boiling points of liquid oxygen and nitrogen so that it can be made amenable to the i Hoxton, Phys, Rev. 13, p. 438, 1919. IV 28 THE SECOND LAW OF THERMODYNAMICS 73 process by being precooled by liquid air. Again 7\ of helium is above the boiling point of hydrogen (20?5 K). Whether these gases are pre- cooled in this way (cascade system) or by adiabatic expansion, the process they are subjected to in technical applications is the integral Joule-Thomson effect, i.e. an expansion from a high pressure (p\) to a considerably lower one (pz) with an attendant temperature drop from Ti to T2. These changes are still governed by eq. (4.43) or x(02, r 2 ) - x(i, Ti) = 0. (4.55) We may ask under what conditions the maximum cooling is attained, (T% = min.), if we start from a given temperature TI. This question is, in fact, of practical importance, because the temperature Ti to which the system can be precooled is usually dictated by technical considerations. Let us suppose, for a moment, that the end pressure p2 is also given (Ti = const, p2 = const). Then eq. (4.55) is a relation between only two variables T^ and pi, and the condition for T2 = min. is obviously dT^/dpi = 0. The total differential of the equation is (IlMl;)*^ < Hence we conclude that, in order to obtain the lowest end tem- perature, the pressure pi must be chosen so as to satisfy the condition r = 0. (4.57) The fact that this condition is independent of p2 shows that our assumption of a fixed end pressure was unnecessary. The choice of pi as a root of eq. (4.57) assures the maximum of cooling for any end pressure, provided the difference Ap is not very small. However, owing to eq. (4.45), this condition is precisely of the same form as that for the differential inversion point, which we transformed into (4.49) (with the sign of equality). For a given (initial) temperature, the optimum in the integral Joule-Thomson effect occurs at the same pressure as the inversion in the differential effect. If the substance follows the law of Van der Waals this pressure is determined by eq. (4.53), or resolving it with respect to TT: (4.58) Exercise 39. Starting from eq. (4.47), prove the relation (4.59) 74 TEXTBOOK OF THERMODYNAMICS IV 29 Exercise 40. Using partial differentiation (as in exercise 33) derive the relation . , dp/ v \dp/T \dT/ p \dp With the help of (3.26), (4.45), and (4.27) transform it into - Exercise 41. Calculate the differential inversion points, for the gases and pressures of Table 7 (excepting air), making use of formula (4.53) and of the data of Table 2 (on p. 14). Compare them with the measured values. 29. Realization of the absolute scale of temperatures. The abso- lute temperature T was introduced in section 4 as that which would be measured by means of an ideal thermometer filled with a perfect gas as thermometric fluid. This temperature acquired a more general significance when it was found (section 21) that the efficiency of the reversible Carnot cycle is independent of the nature of the working system and, being always the same as in the case of a perfect gas, can be very simply expressed in terms of T. This fact led, in turn, to the possibility of defining the concept of entropy by its differential dS = dQ/T, in which the reciprocal of the absolute temperature appears as the integrating multiplier. The role which T plays in the analytical formulation of the second law makes it a quantity of general thermodynamical importance and gives a practical way of indirectly determining the absolute tempera- ture, although the perfect thermometer is only an abstraction. In fact, any thermodynamical equation derived from the second law represents a relation between T and the other variables of the system to which it applies. Theoretically it is, therefore, possible to express the absolute temperature T in terms of measurable characteristics of any system of nature. On the other hand, one can read directly, under the same conditions, the arbitrary temperature / on a real thermometer and, comparing it with T, graduate this thermometer in the absolute scale. This procedure was conceived and carried out by Lord Kelvin who used, for this purpose, the equation of the Joule- Thomson process. Let us take the form (4.48) which refers to the case of small pressure differences Ap. Suppose that we have estab- lished a correspondence between the scales of T and /, so that we can regard them as functions of each other. We wish to introduce into the equation the quantities A<, (9v/3/)pi C'P measured by means of the real thermometer. We obtain immediately, AT" = At*dT/dt, (dv/dT) p (9v/3/) p dt/dT; moreover, from the definition of molal IV29 THE SECOND LAW OF THERMODYNAMICS 75 heats we have, in general, c Dq/dT = (Dq/dt)dt/dT = c'dt/dT and, in particular, c p = c' p dt/dT. The substitution into (4.48) gives Ap dt whence dT __ Qp/30 T The right side of this expression contains only readily measurable quantities, so that the absolute temperature can be found by integrat- ing it. In practice it is convenient to measure the temperature t with a gas thermometer filled with the same gas which is used in the Joule- Thomson process. We have then, according to (1.08), v = i>o(l + a/), and this is the same v as in the denominator of (4.61), since p is there considered as constant. Further, (dv/Qf) p = VQOI, so that the integral of the right side becomes adt f Suppose that to the point / = C corresponds T = TO, then we can adjust the scales so (compare section 4) that to t = 100 C corre- sponds T = To + 100. The integral of (4.61). is, therefore, log (r/7o) = K(t), (4.63) and for t = 100 one obtains an expression for To in terms of measur- able quantities To = 100[exp X(100) - 1]. (4.64) The number To indicates how many degrees of the absolute scale the zero point of absolute temperature (T = 0) lies below the point of freezing water (/ = 0). It is remarkable that the measurements of A//A/> need not be carried out at very low temperatures, in order to obtain it, but only between / = and t = 100. In this range the numerical value of the term c' p &t/vo&p is pretty small. In the case of helium, it is smaller than 0.001 in any part of the interval, so that neglecting it altogether would already give an approximation of better than 0.1%. If this is done, eq. (4.62) reduces to K(f) = log (1 + a/), whence To = I/a, as if helium were a perfect gas. With the experi- mental a = 0.003659 this gives To = 273.32, which is, in fact, pretty close to the true value. The most accurate determinations of TO were carried out by Roebuck (by this method *) and by Henning 1 J. R. Roebuck, Proc. Am. Soc. Arts, Sci. 60, p. 537, 1925. 76 TEXTBOOK OF THERMODYNAMICS IV 30 and Heuse (by a different one *). The weighted mean of their results 2 is To = (273.18 0.03) K, where the symbol K indicates degrees in the absolute or Kelvin scale. Nevertheless, we shall have to adopt here the old value of To = 273M on which all current tables are based. A change in the definition of the zero point has far-reaching consequences and can be made only by international agreement. As stated in the beginning of this section, any phenomenon which satisfies the following two conditions can be used for establishing the absolute temperature scale: it must be (1) expressible by a theoretical formula as a function of T, and (2) measurable in terms of a relative scale /. Temperatures in the vicinity of K are being established by means of the magneto-caloric effect (compare section 136), the same effect by means of which these extremely low temperatures are reached. Exercise 42. In the range from to 100 C (at 1 atm) the Van der Waals equation gives a sufficient approximation for a rough estimate of the third term in the denominator of the integral (4.62). Calculate its maximum value in this interval from eq. (4.49) and the data of Table 2, for H 2 and N 2 . Estimate the percentual error in To, if this term is neglected. Exercise 43. How can the formula O/a)r = T(dp/dT) v - p be used to establish the absolute scale of temperatures? Suppose that (d/dt>)*, (dp/dt) v and p can be measured as functions of the relative temperature t. 30. Statistical interpretation of the entropy concept. Matter con- sists of discrete particles (atoms, molecules, etc.), and the actions exercised by material systems upon one another can be traced to the effects of these ultimate parts. Thus the pressure of a gas upon a wall is due to its bombardment by the moving molecules of the gas which hit it at very short, but irregular, intervals and transfer to it varying amounts of momentum. From this point of view, the pressure is a statistical average which is automatically taken and presented to us by the integrating action of our measuring instrument, the manometer. The same is true with respect to other thermodynamical variables and functions. The temperature, in particular, can be brought in con- nection with the average of the kinetic energy which is distributed over the individual particles according to the laws of chance or the laws of probability. The mathematical theory of probabilities had its origin in the analysis of games of chance. Suppose that a deck containing an * F. Henning and W. Heuse, Zs. Physik 5, p. 285, 1921 ; W. Heuse, 37, p. 157, 1926. * R. T. Birge, Phys. Rev. Suppl. 1, p. 18, 1929. IV 30 THE SECOND LAW OF THERMODYNAMICS 77 equal number of black and red cards is manufactured and shuffled so perfectly that the chance of drawing from it a black or a red card is exactly equal. In other words, if the drawing is repeated a large number of times Z, the card being returned and the deck reshuffled every time, the number of red and black cards drawn will be, relatively, the closer to Z/2 the larger Z is. We say then that the probability of drawing a red card is one half: PR = ^, and, in the same way, the probability of drawing a black card PB = ^- What is now the prob- ability PRB of obtaining, in two successive drawings, first a red and then a black card? There are four possible cases for the result of this double drawing: (1) red, red; (2) red, black; (3) black, red; (4) black, black. Since the chance of each of the four cases is the same, we conclude that the probability P RB = J or P RB = P/ePz*. This is an example of the multiplicative law of probabilities. In general, the probability Pi2 of a complex event, which consists of two simple events, is equal to the product of their respective probabilities PI, P% Pi2 = PiP 2 . (4.65) The multiplicative law holds only when the simple events are statistically independent, meaning by this that the outcome of the first in no way influences the chances of the second. The application of the laws of probability to the molecular structure of matter is treated in the branches of physics called statistical mechanics and kinetic theory. Here we shall give only a brief refer- ence to some notions about the entropy which are, in part, postulated, in part, deduced in these disciplines. A striking property of the entropy is its onesidedness. In any spontaneous process of nature it can only increase, and the state of equilibrium is the state of maximum entropy (compare also next section). On the other hand, when we regard a physical system as an assembly of interacting atoms and molecules, we say that it will change in the direction of a more probable state, as to positions and energy distribution of its ultimate units: The final state will be that of the highest probability. The question arises, therefore, whether it is possible to give a definition of the probability of a state which would bring out its parallelism with the entropy and permit to follow it through in a mathematical way. This is, pre- cisely, what is accomplished in statistical mechanics: The probability of a state is defined there as the number of ways in which this state can be realized. Let us illustrate this by the example of pure and mixed crystals. Consider a crystal consisting of Zi identical atoms in a reg- ular arrangement. We shall discuss here the probabilities, only in so 78 TEXTBOOK OF THERMODYNAMICS IV 30 far as they depend on the normal position of the atoms, and leave out of consideration the deviations and velocities due to their thermo- kinetic motions. If we could label and individually distinguish the Z\ atoms, every permutation of them would lead to a new way of realizing the crystal, and the total number of ways would be equal to the number of possible permutations P s i = Z\ !. This is called by Gibbs the specific definition of the probability. In modern statistical mechanics it is recognized, however, that it is not possible to tell the atoms apart so that all the permutations must be considered as one single realization Ppi = 1: the generic probability definition which was preferred by Gibbs himself. On the other hand, let us take the case of a mixed crystal, that is Z\ atoms of one kind and Zi of another arranged in a joint lattice. If we assume that every distribution of atoms of the second kind between those of the first represents still the same macro- scopic state of the mixed crystal, we find that the number of possible realizations is, in the specific definition, again equal to the permutation number P s i 2 = (Z\ + Z%)\. In the generic definition it is smaller because the permutations of atoms of each kind among themselves (Z\\ and Z2 ! .) do not count. The previous number, therefore, must be divided by the product Z\ \ Z* \ Let us consider the process of combining two pure crystals (Z\ and Z2) into a mixed crystal. The initial probability of the system, according to the multiplicative law (4.65), is P\Pi\ We see that it is increased in the process in the ratio given by the expression (4.66), no matter which of the two definitions we use. We repeat that the pre- ceding discussion implies that all the realizations which we counted belong to the same macroscopic state of the crystal. To what extent this is the case in reality, will be considered in sections 98 and 99. What is now the mathematical relation between the probability and the entropy function? The probability of complex systems follows the multiplicative law, the entropy the additive. We cannot, there- fore, make them simply proportional. But if we take the logarithm of the probability, the difficulty is removed, since eq. (4.65) then becomes log Pi2 = log PI + log P%. In view of this, Boltzmann postulated S = * log P, (4.67) as a general principle which now bears his name. Here k = R/n A is the Boltzmann constant as defined by eq. (1.14). Applying this to IV 30 THE SECOND LAW OP THERMODYNAMICS 7$ the above example, we find that the entropy increase in the process of producing a mixed crystal out of two pure ones is AS = k [log (Zi + Z 2 )! - log Zi! - log Z 2 !]. (4.68) As stated above, the multiplicative law of complex probabilities applies only when the simple events are statistically independent. On the other hand, the entropy of a complex system is additive, only, when the equilibrium of its constituent parts is characterized by a uniform temperature (section 22). It follows from this that the thermodynamical criterion of two statistical systems not being inde- pendent is that they come in equilibrium with different temperatures (compare footnote on p. 60). This is not the place to enter into the applications of Boltzmann's principle. We shall only mention one or two of its immediate conse- quences which will be useful in interpreting and rounding out the thermodynamical treatment. We restrict ourselves to the case when the events are considered as statistically independent. (A) There is an alternative formulation of Boltzmann's principle which we give only for two special cases as we shall not need it in its generality. (1) If the particles of a system move in a field of conserva- tive forces, the probability of finding a particle in the space element Ar is Pdr = Cexp (-e po t/&7>r, (4.69) where e po t is the potential energy of the particle, when in the space element dr, and C is independent of the coordinates of the particles. (2) A similar relation exists for the total energy, its formulation is particularly simple in the quantum theory. If a system consists of identical elements (atoms, molecules, linear oscillators, etc.), each capable of assuming the quantum states 0, 1, 2, ... with the respective energies eo, ei, 62, . . ., the probability of finding an element in the state (j) is P,= C'exp(-e,/tr). (4.70) (B) In classical statistics (when quantum restrictions do not exist and all energy levels are permissible) Boltzmann's principle leads to the equipartition of energy. The mean kinetic energy of an element is equal to kT/2 per degree of freedom: Sun = nkT/2, (4.71) n being the number of (translational and rotational) degrees of freedom. As to the potential energy, this law applies to it only in 80 TEXTBOOK OF THERMODYNAMICS IV 30 the case when the forces producing it are elastic forces (i.e. are pro- portional to the distance). Exercise 44. A coin is tossed three times; what are the probabilities of (1) head being up all three times, (2) head being up twice and tail once? Exercise 45. Calculate from the formula (4.69) the relative number of mole- cules of air in a space element dr of the earth atmosphere at the height h above the surface of the sea. Use a mean molecular mass (m), and, substituting k R/n At show that one obtains in this way the barometric formula of Laplace. CHAPTER V THE TREND OF THERMODYNAMICAL PROCESSES TOWARDS EQUILIBRIUM 31. The direction of spontaneous processes. Let us consider a complex thermodynamical system whose parts may or may not be in equilibrium. We assume it prevented from heat exchange with the outer world by an adiabatic envelope but capable of doing work. If all its microscopic dynamical, electrical, etc., parameters were given, as well as the outer forces acting on it, its future states would be deter- mined and predictable. Can we foresee anything about the spon- taneous processes in it and their direction, knowing only its thermo- dynamical characteristics? The two laws of thermodynamics as stated in the preceding chapters will not lead us very far in this endeavor : the first law only states that one type of energy is converti- ble into another, but, as far as this principle is concerned, the con- version can go either way, provided the geometrical constraints of the system permit it. The second law tells us a little more about the direction of spontaneous processes: only such changes are permissible in which the entropy does not decrease. This excludes, at once, a large number of conceivable occurrences as incompatible with this law. Still, the second law is incomplete in that it leaves the question open whether the entropy actually will increase when conditions per- mit it. In fact, in the enunciations of the preceding chapter the fun- damental fact is not contained that there is in every thermodynamical system a tendency towards a definite state of equilibrium. Fortunately, the empirical evidence relating to these questions can be summarized in the form of two simple rules supplementing the first and the second law of thermodynamics. In the first place, the internal energy U has the nature of a potential energy. A dynamical system which is at rest, at a certain moment, will set itself in motion if its constraints permit a decrease of the potential energy. The potential energy has a tendency to diminish and to be converted into kinetic energy, so that equilibrium can ensue only when it has reached its minimum. In a similar way, the internal energy of an isolated thermo- dynamical system tends to decrease when the existing conditions give 81 82 TEXTBOOK OF THERMODYNAMICS V 31 it a chance of being converted into forms of energy of non-potential nature. U tends towards its minimum, and the system will not be in equilibrium until this minimum is reached. U->U min . (5.01) In the second place, the entropy of isolated systems tends to increase. Processes attended by an augmentation of the entropy are not only permissible, as the second law states, but one of them will necessarily take place spontaneously. Equilibrium will be reached only when S has attained its maximum S-S ma *. (5.02) This condition can be regarded as a portion of the second law, a third part of the entropy principle, of equal importance with the first two enunciated in section 23. We speak of it separately in this place, in order to bring out its analogy with the rule (5.01) supplementing the first law. From this discussion follow the necessary and sufficient conditions of equilibrium of a thermodynamical system : The internal energy must have its minimum and the entropy its maximum value. It is known, from mathematics, that this will be the case if, for all virtual infinitesi- mal changes of state, consistent with the nature of the system, the variations 6 U and SS vanish : 5 [7 = 0, 55 = 0. (5.03) There is a difference between the differential -and the variation. The differential dU refers to an actual infinitesimal change of state which the system undergoes in a specified real process. In thermo- dynamics, the initial and the infinitely close final state must be states of equilibrium if they are to define a differential. On the other hand, the variation has reference to all conceivable (" virtual' ') changes whether they conserve the equilibrium or not. A variation may start from a state of equilibrium and lead away from it. It may even con- nect two (infinitely close) thermodynamically unstable states. The reason for such a definition of the variation is that we do not know in advance which virtual changes are permissible, and we wish to select, a posteriori, those which are compatible with the conditions (5.03) and with other conditions, inherent in the nature of the system and in the concept of equilibrium. It is clear from this explanation that the virtual changes are the more comprehensive class: every actual change is contained among the virtual changes, but the reverse is not true. V32 THE TREND TOWARDS EQUILIBRIUM 83 32. Variation of the internal energy. Partial molal quantities. The systems and states we are going to deal with must be capable of a thermodynamical description. We have to restrict our considerations to systems which can be divided into phases with the following prop- erties: though, possibly, not in physical or chemical equilibrium, each phase must be uniform with respect to temperature, pressure, and composition so that it has a definite internal energy and entropy. For all virtual changes in which no matter is added to the system, holds the fundamental formula (4.07), or dU = T5S - pdV. (5.04) We can choose the volume V and the entropy S as the thermo- dynamical variables. In addition, the system may be characterized by other parameters specifying its mass, chemical composition, etc. To fix our ideas let us assume that the system is not chemically pure but a mixture of Ni mols of a first chemically pure component, N% mols of a second, and so on. The internal energy is then a function of the variables The variation of U is the most general change which takes place while all the variables undergo virtual changes, We shall only consider virtual changes in which no matter is added to the system so that the variations dN k represent increases of some components at the expense of other components. In this case the formulas (5.04) and (5.06) are valid simultaneously and apply to all possible virtual changes satisfying this condition. In particular, we may consider the case when the composition remains altogether unchanged, SN k = (compare, in this connection, section 39). Com- paring the two equations, we find (H) - T - (If) --> < 5 -"> \OO/V.N \OV/S.N The subscript N is usually omitted because the symbol of partial differentiation implies that all the other variables are to be kept con- stant. Substituting this into the general eq. (5.06) and subtracting (5.04), we obtain the remarkable relation 84 TEXTBOOK OF THERMODYNAMICS V32 These results can be generalized for systems with several phases (provided T and p are uniform throughout), but we need not deal with this aspect of the question here since it will occupy us in section 42, in connection with the problem of equilibrium. The partials of eq. (5.08) should not be confused with the partial molal internal ener- gies which are defined as follows : dNk/^r In the case of a chemically pure substance, U = Nu, where the ordinary molal internal energy u is a function of temperature and pres- sure only and remains constant when p and T are kept constant. Therefore, Uk is a generalization of u for mixtures and solutions, becoming identical with it in the extreme case of pure substances. Another partial molal quantity worth mentioning is the partial molal volume. When several substances are mixed or dissolved in one another, the total volume is, generally, not additive but a complicated function of the mol numbers (compare also section 57) V = V(p, T,NL.. N 9 ). The partial molal volume of each component is defined as /oT7\ (5.10) The relation of the functions (3 U/dN k ) s . v to partial molal quan- tities will become clear in the next sections. Exercise 46. The measurements by Wade (J. Chem. Soc. 75, p. 254, 1899) on the volume of aqueous solutions of NaCl can be represented by the following empiri- cal formula (due to Lewis and Randall) where N\ refers to water and Nt to NaCl. Calculate the volumes of solutions with 0.1, 0.3, 0.5 mol of NaCl per 1000 g of water and the partial molal volumes of NaCl in all these cases. Exercise 47. Prove for the perfect gas the formula Directions. Calculate &U = d(Nu) from (3.18) and eliminate ST and Sv with the help of the conditions 55 5(Ns) = and dV - 5(Nv) - 0. For s, use the formula (4.17). V33 THE TREND TOWARDS EQUILIBRIUM 85 33. Construction of new characteristic functions. Further prop- erties of the heat function. In addition to the characteristic functions U and 5, we had occasion to introduce in section 17 the heat function defined as X - U + pV, (5.11) while its differential, in the case of a simple system, is obtained from (3.22) and (4.07), dX = TdS + Vdp, (5.12) whence in an isobaric process (dp = 0) Q P = X 2 - Xi. (5.13) Instead of using the variables 5, V of the preceding section, we can describe the system also by the variables S, p, and use X as the appropriate characteristic function. The term pV in (5.11) has the effect that its differential d(pV) = pdV + Vdp cancels the term pdV of (5.04) and leaves in its place Vdp. In this way the differ- ential dp is introduced instead of dV. In mathematics the method of replacing a term ydx by the term xdy by means of subtracting the differential d(xy) is called a Legendre transformation. It is obvious that we can construct by this expedient still two more characteristic functions from the expression (5.04), namely, subjecting to Legendre transformations the first term, or both terms simultaneously. We shall discuss these functions in the next sections and reserve the remainder of this one for a few remarks about the variation of the heat function. As explained in the preceding section, we assume that the state of the system depends on the mol numbers Ni. NZ, . . . N ff in addition to the thermodynamical variables: X = X(5, p, Ni,... N ff ). (5.14) We can obtain, for the variation of X, two expressions closely analogous to those for dU in the preceding section. On the one hand, the relation (5.12) is valid also for the variations dX = TdS + Vdp. On the other hand, we find from (5.14) 86 TEXTBOOK OF THERMODYNAMICS V33 As in the preceding section, we conclude by comparing the two expressions (i) - r - (HD - v - \dS/p,N \3P/S.N (The subscript N is usually omitted). Another form of the variation of (5.11) is dX - Vdp = dU + pdV. Hence and from (5.06) and (5.15) We cannot conclude from this equation that the terms are equal individually: 9 X\ because the virtual changes 6N k are not independent: we have excluded changes in which matter is added to the system. However, the proof of the identity (5.17) can be easily supplied by partial differentiation. In fact, let us describe U by the coordinates S, p instead of 5, V. We can effect this change of variables by substituting for V its expression in terms of 5, p, N. We need not bring S in evidence since it is con- sidered as constant, and we write instead of (5.05) U - U[V(p, N), N]. By the rules of partial differentiation According to (5.07), the second term on the right is [d(pV)/dN] St p . Taking it to the left side, the equation becomes identical with (5.17). Exercise 48. Consider the general form of d U based on (2.02) Show that the number of characteristic functions which can be constructed from it by Legendre transformations is 2 n + l 1. Exercise 49. Prove that, for a perfect gas, Directions. Calculate dX = 8(N\) from (3.28) and eliminate ST with the help ol the conditions 6S d(Ns) and dp = 0. For s, use the formula (4.18). V34 THE TREND TOWARDS EQUILIBRIUM 87 34, The work function. We subject to a Legendre transformation the first term of the differential of the internal energy (4.07) dU = TdS - DW. (5.18) Subtracting from it d(TS), we find D* - - SdT - DW, (5.19) where * - U - TS. (5.20) The function ^ is useful in bringing out certain properties of iso- thermal processes. We have, in this case, dT = and d& = - DW. (5.21) Integrating this over a reversible process at constant temperature from the initial state (1) to the final (2), *! - * 2 = - A* = W. (5.22) The work done by a system in a reversible isothermal process is equal to the decrease of the function V. The case when the system remains in contact with a heat reservoir of constant temperature but the process is not conducted reversibly is sometimes, loosely, called an irreversible isothermal process. In this kind of a process the tempera- ture of the system is not strictly equal to that of the reservoir and is, in fact, not strictly uniform. We have seen in section 12 that the process is less efficient and the work done in it by the system is less than in the reversible case. We can, therefore, say that ^i 2 represents the maximum work that can be obtained from the system in any isothermal process leading from the state (1) to the state (2). Applying eq. (5.20) to the initial and final states and taking the difference, Ui - E/2 - *i - * 2 + T(Si - S a ). (5.23) The second term on the right side is, according to eq. (4.11), equal to Q, representing the heat transferred to the heat reservoir during the process. Helmholtz, who used the function ^ a great deal, called it the free energy because the difference ^f\ ^2 represents that part of the decrease of the internal energy U\ Uz in the process which is available for work while the bound energy T(S\ 2) is lost to the reservoir maintaining the constancy of temperature. However, the name cannot be regarded as very fortunate because it loses sight of the fact that there exist in nature endothermic processes in which T(S\ ^2) is negative. In this case, work is done by the system, in 88 TEXTBOOK OF THERMODYNAMICS V 34 part at the expense of the heat reservoir, which supplies heat instead of receiving it. The decrease &i ^2 can, therefore, be larger than Ui - C/ 2 , as was first pointed out by Gibbs. To make matters worse, in a part of the chemical literature of this country the name "free energy" is applied to a different function (namely, to $ defined in the next section). We think it, therefore, advisable to avoid the term "free energy" altogether and shall refer in this book to the function ^ as the work function. This name was introduced by R. H. Fowler 1 and is highly appropriate in view of the physical meaning of the quantity ^ as expressed in eq. (5.21). This relation means that, as long as the temperature is kept constant, the elements of work used in thermodynamics are exact differentials which possess the potential &. We shall discuss the very important question, under what circum- stances ^ tends to decrease, in section 36 and shall say here only a few words about its partial derivatives. In the case of a simple system, DW = pdV, and the variation becomes, in accordance with (5.19), 6* =- SdT - pdV. (5.24) The appropriate coordinates for the description of the work func- tion are, therefore, T and V in addition to the mol numbers N k , deter- mining the composition of the system: V, N l9 ... NJ. In analogy with the preceding sections, we find '), (< \dTJv, N \OV/T,N & -P. (The subscript N is usually omitted). As to the partials with respect to Nk, it is easy to prove the identity (exercise 51) y by means of which eq. (5.08) can be expressed in terms of derivatives of the function ^. It will be useful to give here the explicit expressions of the molal work function $ for perfect and Van der Waals gases. We take the more general case of the formulas (3.19) and (4.20) when the molal heat is not considered as constant. In the case of the perfect gas, * '(D - RTlogv, (5.27) 1 R, H. Fowler, Statistical Mechanics, p. 96. 1929. V35 THE TREND TOWARDS EQUILIBRIUM 89 of the Van der Waals gas, according to (4.26) and (4.28), * = co'(r) - - - RTlog(v -ft), (5.28) where o/(r) is an abbreviation for '(r) -y cjT - rj c -^ + */<>- TS' Q . (5.29) Exercise 50. Prove, for the perfect and for the Van der Waals gas, the relation Directions. Calculate 5* = d(Ntf with ST = from (5.27) and (5.28). Elimi- nate 5v by means of the condition 5 V = d(Nv) = 0. Exercise 51. Prove the identity (5.26) in the following way: In (5.05) replace 5 by a function of V, T, N, i.e. U - U[S(T, N), N], and take the partial (dU/dN) T . v in analogy with the procedure at the end of section 33. Exercise 52. Consider the case of a system with many degrees of freedom when the element of work has the general form (2.02): DW = 2y k dX k . Write out the differential of the work function * = U - TS and, by applying the reciprocity relations (2.10), prove the equation i) . (5.30) Jx 35. The thermodynamic potential. Let us consider a system which depends on some non-mechanical (e.g. electrical, magnetical, etc.) variables, in addition to p and T. The element of work done by it can, then, be represented as DW = pdV + DW ', where DW is the work done by the non-mechanical forces. Accordingly, the differen- tial (5.04) of the internal energy takes the form dU = TdS - pdV - DW. (5.31) We subject the first and second term on the right side to Legendre transformations adding - d(TS) + d(pV). The result is d$ = - SdT + Vdp - DW, (5.32) where $ = U - TS + pV (5.33) is known as the thermodynamic potential, In the laboratory, processes (like chemical reactions, etc.) are very often conducted at constant temperature and constant pressure. The temperature is maintained by a suitable heat bath while the pres- $0 TEXTBOOK OF THERMODYNAMICS V 35 sure is that of the terrestrial atmosphere. Under these conditions (dT = 0, dp = 0), eq. (5.32) reduces to ' d* - - DW, (5.34) or integrating this over a reversible path (at T =* const, p = const) from the initial state (1) to the final (2) $1 - $2 = - A$ - W. (5.35) > The non-mechanical work done by a system in a reversible isothermal, isobaric process is equal to the decrease of the function <. When the process is led between the same two states in an irreversible way (i.e. when the temperature and pressure are imperfectly maintained), it is more wasteful. The difference $1 $2 gives, therefore, the maxi- mum of the non-mechanical work that can be obtained from the system in any process leading from state (1) to state (2) of the same temperature and pressure. The great importance of the thermodynamic potential arises from the fact (which we shall prove in the next chapter) that, in the equilib- rium state of a heterogeneous system, p and, T are the same in all its phases. They are, therefore, the most convenient thermodynamical variables, and eq. (5.32) shows that & is the appropriate characteristic function for this choice of variables. In the special case of a simple system DW f = 0, and the variation becomes *$ = - S5T + V6p, (5.36) while $ has the expression In analogy with the preceding sections (5.37) dp/T.n (The subscript N is usually omitted). The partial derivative with respect to Nk must be taken at constant p and T: it is, therefore, the partial molal thermodynamic potential, according to the definition of the partial molal quantities given in section 32. (5-38) V36 THE TREND TOWARDS EQUILIBRIUM 91 It is easy to prove the identity (exercise 54) or from (5.17) and (5.26) We shall have occasion to use the expression for the molal thermo- dynamic potential of a perfect gas. From (5.33), (4.20), and (3.19) (5.41) where (5.42) In the particular case c p = const, co(r) = c p T(l - log r) + uo - TSQ. (5.43) Exercise 53. Derive the explicit expression of <f> for the Van der Waals gas: V"(r)+*r log (/>+") --+-^~ (5.44) \ V l / V V Exercise 54. Prove the identity (5.39), making in exactly the same substitu- tion for V which was made at the end of section 33 in U (to prove 5.17). Exercise 55. Consider the case of a system with many degrees of freedom when the element of work has the general form (2.02): DW HyidXk. Write out the differential of the generalized thermodynamic potential & = U TS + Sy^Yjfc, and prove, by applying the reciprocity relations (2.10), the equation 36. The decreasing tendency of the thermodynamic potential and of the work function. We have found in eqs. (5.22) and (5.35) that work is done by a system (at T = const) at the expense of the work function, and non-mechanical work (at T = const, p = const) at the expense of the thermodynamic potential. The question arises, how- ever, as to the tendency of these functions to decrease: if the differ- ences 1 ^2 or $1 $2 are available in a system, will they be spon- taneously converted into work, and under what conditions? Let us start our considerations from the work function ^ U TS. It is obvious that ^ would decrease (if possible) in a system " left to itself " or " isolated", but remaining at constant temperature. In fact, 92 TEXTBOOK OF THERMODYNAMICS V36 we know from section 31 that the positive term U has a tendency to decrease and the negative TS to increase (in absolute value). How- ever, in the processes which we wish to consider the systems are not isolated: in the first place, there is the heat bath by means of which the temperature is maintained; in the second, the environment may do work against it through the forces of pressure. Let us try to make the system isolated by including in it both these influences. With respect to the heat bath, this is easy since it does not do any work of either a mechanical or a non-mechanical nature. According to eqs. (5.21) and (5.34), the values of the functions < and ^ characterizing the heat reservoir remain constant in view of this. We may regard it, therefore, as a part of the system without changing anything in the available differences ^i ^2 and 3>i <i>2, or, in other words, we may base our discussion on ^ = U TS of the enlarged system (with inclusion of the heat bath). The conditions are simplest when other interactions with the outer world are precluded. This is the case when the volume of the system is constant, for instance, when it is surrounded by a rigid heat-conducting envelope and immersed in the heat bath: when the temperature and the volume of a system are kept constant, its work function has a tendency to decrease. Any spontaneous isothermal and isochoric process goes on until ^ reaches its possible minimum. The necessary and sufficient condition for the starting and proceeding of a process is that the work function must be larger in the initial state than in the final (^i ^2 > 0). This can be used as a criterion whether a chemical reaction will go. At constant temperature and volume the maximum possible change of V is a measure of the chemical affinity of the substances entering into the reaction. It is to be noted that the work done by this reaction, if any, is non-mechanical work: from dV = 0, there follows DW = DW. The conditions are different when the pressure is maintained con- stant, instead of the volume. The practical method of accomplishing this is to subject the system to the hydrostatic pressure either of the atmospheric air, or of another medium (liquid or gas) compressed by a suitable load. If the system expands under this pressure from the initial volume V\ to the final V^ the work of expansion p(V^ Vi) is converted into the potential energy of lifting the load or the air column resting on the system, and pV is a measure of this potential energy at any moment. In addition to 7, having the nature of a potential energy, we have here still the item pV which tends to be converted into energy of a non-potential character. We have, therefore, to use the thermodynamic potential $ = U TS + pV: including the heat bath into the system as in the former case, we see that this function has V37 THE TREND TOWARDS EQUILIBRIUM 93 a tendency to decrease (at T = const and p = const), since the two positive terms tend to decrease, the negative to increase (in absolute value). The criterion for a process taking place under these conditions is that the thermodynamic potential must be larger in the initial state than in the final ($1 $2 > 0). This applies in particular to chemical reactions: they will continue as long as they can produce non-mechani- cal work. The measure of chemical affinity in a reaction conducted in an isothermal and isobaric way is the maximum non-mechanical work W = $1 $2 which can be obtained from it. All three terms of $ tend to decrease, but it would be wrong to conclude that they always diminish separately. The functions U, S, V are all interdependent, and the reduction of one may force the growth of another. Only the sum <i> always decreases, each individual term is only then sure not to increase when the other two remain constant. From (5.35) and (5.33) W' = - A* = (Ui - U 2 ) - r(Si - 5 2 ) + p(Vi - F 2 ). (5.46) The term T(S\ 62), representing the heat imparted to the heat bath, is often negative (endotermic reaction), and the third term p(Vi 2) is also sometimes negative (reaction with expansion). In later chapters we shall have to consider the case when the sys- tem is in partial equilibrium, its phases having the same temperatures but standing under different pressures pi, p2, p a * Since U and 5 are additive, the thermodynamic potential has, then, the form * = U - TS + piVi + . . . + p a V a . (5.47) It is obvious that it tends to decrease also in this case, because the terms pjVj are the potential energies due to the expansion of the indi- vidual phases. 37. The Gibbs-Helmholtz equation. Galvanic cells. Another form of the relation (5.46) is A$ = X 2 - Xi - T(S 2 - Si), or substituting for S from eq. (5.37) This relation is known as the Gibbs-Helmholtz equation. It applies to any isothermal difference A< as it is based only on the definitions of the functions $ and X. In the special case of an isothermal-isobaric 94 TEXTBOOK OF THERMODYNAMICS V37 process, we have A$ = W, and from (4.36"), AX = Q p (isobaric heat of reaction), (s - 49) The left side of the equation is identical with T(S2 - Si) and represents the energy gained by the system from the heat reservoir, in the isothermal-isobaric process we are considering. It is interesting to note that this heat is proportional to the temperature coefficient of W. The Gibbs-Helmholtz equation has many uses; as an example we mention here its application to the electromotive forces of galvanic cells. The active substance in a galvanic cell is an electrolyte whose negative ions react with the material of the positive plate, transferring their electric charges to it. At the same time, an equal number of positive ions is deposited upon the negative plate and neutralized by the (negative) current flowing to it through the outer circuit of the cell from the positive plate. According to Faraday's law, the electric charge supplied to the circuit by 1 mol of the electrolyte is <rF, where F is Faraday's equivalent, (9648.9 0.7) abs e-m units, and o- the valency of the ion. The electric energy w f of the current (per 1 mol of the electrolyte) is obtained multiplying this by the electromotive force E, w r = 0FE, (5.50) or substituting into (5.49) r (af) - E + >/** < 5 - 51 > where q p is the heat of reaction, referred to 1 mol. When q p is expressed in calories and E in international volts, the numerical value of the Faraday becomes F = 23 046. Let us apply this to the Weston cell in which the electrodes consist of Cd-amalgam and of pure mercury. It contains the following sub- stances: Hg-Cd | CdS0 4 (solid) | CdS0 4 (sat. sol.) | Hg 2 S0 4 (paste) | Hg arranged as shown in Fig. 11. The following empirical formula has been found (and internationally adopted) for the electromotive force of the Weston cell i [1.01827 - 4.06(/ - 20) X 10" 5 - 9.5(* - 20) 2 X 10~ 7 + (20 - O 3 X 10~ 8 ] int volt. (5.52) V37 THE TREND TOWARDS EQUILIBRIUM 95 Saturated Solution of Cadmium Sulphate Mercurous Sulphate Cadmium Amalgam Platinum Wire Equation (5.51) permits to calculate the heat of the reaction (Cd + Hg 2 SO 4 = 2Hg + CdSO 4 ) in the cell. It gives, with <r - 2, q p = 47 482 cal mol"" 1 , while direct thermochemical measurements yielded 47 437 cal mol" 1 . In the Weston cell the temperature coeffi- cient of E is negative and represents an energy loss. There are, how- ever, other cells (see exercise 56) where it is positive, so that a part of their energy and electromotive force is supplied by the heat bath (surrounding air). It is clear from this discussion that in a galvanic cell the heat Q p of an exothermic reaction is converted into (electrical) work in a practically complete way. Its operation is not cyclic but can be made continuous, [since the working substances can be fed in, at the positive plate, and the waste products taken out, at the negative. If the same reaction were used for heating the boiler of a cyclic heat engine (section 21), not more than the fraction (7\ - T 2 )/Ti of the heat Q p could be utilized, ac- cording to eq. (4.03). In view of this superiority, there was a great deal of speculation whether it is feasible to construct a galvanic cell working with the most com- mon industrial fuel, coal (i.e. carbon). But it proved impossible to bring carbon into electrolytic solution. For some of the applications it is useful to integrate eq. (5.49) and to express W = A< in terms of Q p . An integrating multiplier of the equation is 1/r 2 , and the multiplication by it gives [d(W'/T)/dT] P = &/r 2 , whence by partial integration )dT y (5.53) Exercise 56. The Pb-Hg cell is constructed as follows: Pb-amalgam | PbSO 4 (solid) | NajSO 4 (solution) | HgjSO* (solid) | Hg Its electromotive force is E - [0.96466 + 1.740 - 25) X lO" 4 + 3.8(* - 25)* X 10~ 7 ] int volt Platinum Wire FIG. 11. The Weston galvanic cell. 96 TEXTBOOK OF THERMODYNAMICS V37 What part of E is supplied by the heat reservoir, and what is the heat of reaction at 25 C? Exercise 57. In the Cu-Hg cell Hg-Cu | CuSO 4 (solid) | CuSO 4 (sat. sol.) | Hg 2 SO 4 (paste) | Hg the molal heat of reaction is, at 20 C, q p = - 39 596 cal mol" 1 , the electromotive force E = 0.3500 int volt. Find the temperature coefficient of E. Exercise 58. In the same way as (5.48) is derived for A4>, derive for * the equation . Hence prove the relation Y * -- T f^ dT + Tf(P).l (5.55) Remark. In the differentiation and integration with respect to T the pressure is regarded as constant. Therefore, the integration constant may depend on p. In (5.53) the function of p cancels out when the isobaric difference $2 $1 is formed. Exercise 59. Prove from (5.20) and (5.25) the analogous equations (5.56) ~\ \J *. r y Jt I- (J J. -M f and -dT + TMv). (5.57) Forming the difference A = 2 1 leads to the relation T (^f) v =**-* u ' (5.58) which sometimes is also given the name of a Gibbs-Helmholtz equation. CHAPTER VI 1 GENERAL CONDITIONS OF EQUILIBRIUM OF THERMODYNAMICAL SYSTEMS 38. The method of virtual displacements. The theory of thermo- dynamical equilibrium was developed by Gibbs and patterned by him after the mechanical theory of statics of Lagrange. We have explained in section 31 what virtual changes are, but it will be well to say here a few words about how they are used. Let the parameters of the system be denoted by 1, fe, n , which may include as well the thermodynamical variables as those defining the composition of the system; and let the equilibrium be determined by the condition that the variation of a certain characteristic function -F(i, . . . { n ), vanish for all virtual changes dl-k consistent with the conditions of the system. If the parameters are not subject to any further restrictions, the virtual changes are entirely independent of one another and can be chosen in any arbitrary way. In particular, we could choose all but one of them equal to zero ($* ^ 0, and when i ^ k, $ = 0). Then the condition (6.01) is reduced to (3F/3{*)% - Oi or - = 0, (* - 1, 2, . . n). (6.02) 3& Since 5* was selected at random, this must hold for each, of the parameters. We have, therefore, n equations from which the equilib- rium values of the n parameters & can be determined, and this consti- tutes a complete solution of the problem. However, usually, the parameters are not quite independent but subject to constraints or subsidiary conditions which take the analyti- 1 Suggestion to teachers. Chapters V and VI are both somewhat abstract. It is, therefore, recommended to interpose between them sections 134, 135, and 136 of Chapter XX, which contain applications of the concepts introduced in Chapter V. 98 TEXTBOOK OF THERMODYNAMICS VI 38 cal form of equations imposed upon them. Suppose that there exist m (< ri) subsidiary conditions /ifti, ...) /2i, . . 6.) 0, 0, /mftl, ...*)- 0. (6.03) In this case, only n w of the f A; are independent and can be chosen arbitrarily. The remaining m parameters are determined by these through eqs. (6.03). The same is true of the virtual changes 5 fc : only n m of them are within our free choice. In order to find the final conditions of equilibrium, it is necessary to eliminate the m dependent virtual changes from the variation dF and to represent it as a linear form of the independent $ *, only. This could be done by eliminating m of the parameters from the function F, with the help of eqs. (6.03), before taking its variation. However, this is seldom convenient, and it is preferable to take 5F in the form (6.01) and to eliminate the dependent dfa from it, using a method worked out by Lagrange. The subsidiary conditions can be, also, thrown into a variational form; in fact, they apply as well to the values of the parameters & as to & Mti = 0. (6.04) The difference between (6.04) and (6.03) is, for infinitesimal changes, v< f& = 0, (* = 1,2, . . . m). (6.05) Often the subsidiary conditions are given directly in the variational form (6.06) which cannot always be integrated. However, it is irrelevant for the further treatment whether the form is (6.05) or (6.06). To fix our ideas we use eqs. (6.05) and multiply them, respectively, by Lagrangean multipliers A,(i, . . . () and add them to eq. (6.01) w i At VI 39 GENERAL CONDITIONS OF EQUILIBRIUM 99 The m functions \i can be chosen in such a way as to make the m first parentheses of this equation vanish f or k = 1,2, ... m. The remaining expression Z" ( dF + i 9 ^+ -LX 9 - \*t + Xl 5T + ' ' ' + Xm ~ *-m+l ^3** 3* contains now only n m independent $*, so that the problem is reduced to the case of independent virtual changes treated above: this equation is satisfied only then when all the coefficients in parentheses vanish. The conditions (6.01) obtain, therefore, for all values of k( 1, 2, . . . n). There are n such equations, and together with the m eqs. (6.03), we have m + n relations between the variables 1, . . . n; Xi, . . . X m , i.e. the number necessary to determine all of them and to obtain a complete solution of the problem of equilibrium. Exercise 60. A heavy material point moves in the field of gravity of the earth. It is constrained to remain in the curve (ellipse) (ax + Py + yz) 2 + ey* = 1, ax - by = 0, where x t y, z are cartesian coordinates, z being the vertical. The point comes to rest when its potential energy has its minimum (z min.) What is its position of rest? (Use Lagrangean multipliers). Exercise 61. Make the same calculation for a point which is constrained to move in the surface x* H- (y cos a + z sin a) 2 -f 2p(y sin a z cos a) = 0. 39. Auxiliary constraints. Homogeneity of the thermodynamical functions. It follows from the considerations of the preceding sections that the equations of equilibrium (6.07) hold for any system of virtual changes $& satisfying the main condition (6.01) and the subsidiary (6.05). In some cases, it is useful to treat the problem of equilibrium, not in its entirety, but only in part, and to select a special system of variations 5& compatible with those conditions. It is trivial that the requirements (6.01) and (6.05) are satisfied if all the virtual changes are taken equal to zero (3& = 0; k = 1, 2, ...). If only some arbitrarily chosen of them are equal to naught (provided the subsidiary conditions permit it), the result is quite consistent with the general solution but represents only a part of it. In fact, in this case, the 100 TEXTBOOK OF THERMODYNAMICS VI 39 corresponding terms of the expressions (6.01) and (6.05) vanish: the method of the Lagrangean multipliers leads, then, simply to an incom- plete set of eqs. (6.07). More generally, we can say that it is permissible to impose upon the parameters any additional constraints which do not disturb the equilibrium of the system. At first sight it may appear that this principle is of little practical use because, apart from such simple cases as that just mentioned, we must first know what the state of equilibrium is before we can judge whether a constraint is compatible with it or not. In practice, however, the simple cases are the most important ones, and frequently the knowledge of the nature of the system and of the physical circumstances of its equilibrium gives enough criteria to decide this question. In fact, the introduction of auxiliary constraints is a very useful and widely applied expedient. In this chapter we are going to consider a system consisting of several phases, each described as well by thermodynamic variables as by the mol numbers NI, N 2 , . . . N, defining its composition (compare section 32). We shall make use of auxiliary constraints in two con- nections. The first application consists in breaking up the problem of equilibrium into two partial problems by first considering the case when the composition remains unchanged and dealing with the ther- modynamic variables alone. Consequently, the auxiliary constraints take here the simple analytical form dN\ = $N2 . . . = dN ff = 0. It is obvious that this assumption is compatible with the notion of equilibrium because the most perfect equilibrium obtains then when no process at all is going on in the system. The conditions, the parameters Nk are subject to, are found separately as the second par- tial problem into which the treatment is broken up. In the other application the auxiliary constraint takes a geometrical form. Let us imagine that, anywhere, in the system a thin, closed layer of its substance is removed and replaced by a rigid and adiabatic envelope. Will the presence of this envelope disturb the equilibrium? Strictly speaking, it will. The state of a substance, at any point, is influenced by all the molecules in a small radius around it (radius of molecular action). Therefore, a very thin layer of matter, immediately adjacent to the imaginary envelope, will find itself in a new condition, and this will influence the system as a whole. However, when the volume within the envelope is sufficiently large the influence of the surface layer can be neglected. In fact, the volume increases with the third power of the linear dimensions, the surface with the second, so that the influence of the surface becomes relatively less and less important as the size increases. We shall treat the properties of the VI 39 GENERAL CONDITIONS OF EQUILIBRIUM 101 surface layer separately in Chapter XII, and we are going to assume here that all the phases of the system are so large that the surface effects can be neglected (as well at the boundary as at the surfaces of discontinuity between the phases). When this assumption is made, the equilibrium is not disturbed by the rigid envelope just mentioned, whether we think it surrounding the whole of the system or a part of it. It is inherent in the notion of a system in equilibrium that there is in it no relative motion of the parts and no heat transfer between them. The imaginary envelope will not interfere, therefore, with any actual process going on in the system. As to virtual changes, some of them are precluded or restricted, but no new virtual changes are made possible, by the presence of the envelope, which did not exist without it. According to the above argument, it represents a con- straint which does not change the form of the conditions of equilib- rium. This fact, immediately, leads to some physically interesting conclusions: (1) The envelope can be laid in such a way as to cut off a part of any of the phases. Therefore, the conditions of equilibrium cannot depend on the total masses and volumes of the phases but only on their specific properties like temperature, pressure, composition, etc. (see below). (2) It can be laid so as to include only two of the phases and exclude all the rest. Therefore, the equilibrium of two phases is determined only by their own properties and is not affected by the presence of other phases. (3) Since this is true for any pair of phases, the problem of equilibrium of several of them can be reduced to the simpler one of equilibrium between two. It also follows from this that two phases which are in equilibrium with a third are in equilibrium between them. (4) In general, a thermodynamical sys- tem is not completely defined unless it is known how it is delimited at its boundary. However, it is clear from the preceding discussion that any boundary conditions can be replaced, without loss of generality, by rigid, adiabatic walls enclosing the system. In this case, the con- straints introduced by the walls are not only auxiliary but necessary as forming part of the definition of the system. In the following sec- tions we shall use the expedient of the rigid, adiabatic envelope for all these purposes. A further example of an auxiliary constraint in geometrical guise is a semipermeable partition or semipermeable membrane, interposed between two phases, which lets through one kind of molecules but is impenetrable to all other kinds. This device can be also regarded as a part of the definition of the system, inasmuch as it is the thermo- dynamical criterion for the difference or identity of molecules: two particles are different if it is possible to find a semipermeable membrane 102 TEXTBOOK OF THERMODYNAMICS VI 39 which lets through the one but holds back the other. 1 It is, there- fore, perfectly legitimate to introduce semi permeable partitions which are selective not only with respect to chemical differences of mole- cules but also with respect to differences in the physical (quantum) state of otherwise identical particles. 2 The notion of specific (also called intensive) properties used above requires some amplification. It is easy to define it accurately for phases which are described by the variables />, T, N\, N 2 , . . . N ff . Any property which does not change when all the N k are increased in the same proportion e (p and T remaining constant) is a specific prop- erty. For instance, the composition is best described in a specific way by the mol fractions oc k - N k /N, | N = Nl + N 2 + . . . + N,. J (6 ' 8) It follows from the definition of the mol fraction that *i + *2 + . . . + x a = 1. (6.08') In general, a function F n (N\, . . . NJ which changes in the pro- portion e n , when all the variables N k are increased in the proportion e, F n (eNi, . . . eN.) = e"F n (Ni, . . . NJ, (6.09) is called, in mathematics, a homogeneous function of the n-th degree. It satisfies Euler's equation which is easily derived (compare exercise 62) from the definition (6.09). Mathematically speaking, the specific quantities are, therefore, homo- geneous functions of the degree zero in the mol numbers Nk, since they satisfy the condition . . . NJ. (6.11) Of course, F depends also on p and T, but we shall compare only states and systems at the same temperature and pressure and need *An interesting discussion of these matters will be found in a paper by L. Szillard, Zs. Physik 32, p. 840, 1925. 1 Einstein pointed out the paradox that a semipermeable membrane must dif- ferentiate between two particles, be the difference in their states ever so little. This difficulty disappears in the quantum theory where the states are discrete (compare: J. Von Neumann, Math. Grundlagen d. Quantentheorie, p. 197. Berlin 1932). VI 39 GENERAL CONDITIONS OF EQUILIBRIUM 103 not bring these variables in evidence. The factor e may have any value, and, in particular, we may put e = l/N, whence Fo(Ni, ...#.)- FQ(XI, . . . *.). (6.12) In a specific quantity we may replace the mol numbers Nk by the mol fractions Xk without changing it. On the other hand, the volume F, the internal energy Z7, and the entropy 5 are additive quantities. When all the mol numbers Nk are increased in the proportion e, the mass of the whole system increases in the same proportion, without change in its composition. The quan- tities F, U, S are proportional to the mass and also increase in the proportion e (p and T remaining constant). The thermodynamic potential is defined as $ = U TS + pV and has, obviously, the same property . . . N 9 ). (6.13) The thermodynamic potential, like all the other additive functions of thermodynamics, is a homogeneous function of the first degree. Hence its derivative with respect to Nk, or the partial molal thermodynamic potential, <pk = d$/dN k (compare section 35), is a homogeneous function of the degree zero, as is easily shown by differentiating (6.13) with respect t& Nk. Therefore, $ and ^* satisfy Euler's equation (6.10) with n *= 1 and n = 0. + ...+ (6-14) As 3^*/3^y = d$j/dNk, this can be also written Every term of this expression is a homogeneous function of the degree zero. We can, therefore, make use of the property (6.12) and replace the mol numbers by the mol fractions Although this relation was already given by Gibbs, it is usually called Duhem's equation. 104 TEXTBOOK OF THERMODYNAMICS VI 40 Exercise 62. Prove Euler's eq. (6.10) from (6.09). (Directions. Differentiate (6.09) with respect to e, denoting, for short, eN k - *. Then put e = 1; & = Nk)> Exercise 63. Check by direct calculation the statement that Euler's equation is correct for the homogeneous (inx t y,z) function F n - x a y*z n ~ a ~ b . Therefore, it must, be true for any sum of such terms of the degree n. 40. The temperatures and pressures of phases in equilibrium. Let us consider a heterogeneous system consisting of simple phases. Its total internal energy U is the sum of the internal energies C7 (i) of the individual phases u = u m (6 * 17) 1-1 In the same way, the total entropy S is additive with respect to the individual entropies S (i) >, (6.18) and we define as the total volume V the sum of the volumes F (0 of the phases a **. (6.19) We can choose 5 (i) and V (i) as the two thermodynamical variables characterizing the state of the phase (i). It will, further, depend on a number of parameters describing its chemical composition. In gen- eral, the phases will not be chemically pure substances but mixtures of several or many chemically pure constituents or components. Sup- pose that they contain Ni mols of the first component, N<z of the second, and so on, the number of components being <r. The internal energy of the phase (i) is, then, a function of the following variables . . #<*>), (6.20) although a part of the mol numbers N may be equal to zero in some of the phases. This is the form we have considered in section 31 : the variation of the total internal energy U is, therefore, in accordance with eqs. (5.06) and (5.07), Applying auxiliary constraints, as explained in the preceding sec- tion, we shall at first suppose that the mol numbers do not change VI 40 GENERAL CONDITIONS OF EQUILIBRIUM 105 (dN k (i} = 0) and shall find only the conditions controlling the thermo- dynamical variables. The conditions of equilibrium are, according to eqs. (5.03) : dU = 0, 8S = 0. It is mathematically convenient to regard dU = as the primary condition. It now takes the form dU = > (T (i} 5S (i) ( '>$F (i) ) = 0. (6.22) The condition Of 55 = ^ SS (i) = (6.23) <-i we regard as subsidiary. Another subsidiary restriction is obtained if we replace the border conditions of the system by the assumption that it is surrounded by a rigid adiabatic envelope (compare the pre- ceding section) : We have then F = const, or In the case when the several phases are separated by natural sur- faces of discontinuity and each of them can expand, virtually, at the expense of the others, the variations <5S (i) , 5V (i) are not subject to any other restrictions so that eqs. (6.22), (6.23), and (6.24) exhaust the problem. Multiplying eqs. (6.23) and (6.24), respectively, by \i and \2 and adding them to (6.22) a 5 = 0, (6.25) 1-1 where the dS (i) and 6F (i> must be considered as independent (compare section 38): The temperatures of all the phases are equal to the same function which has, therefore, the physical meaning of the uniform temperature T of the whole system (Xi = T). In the same way, the pressures of all the phases are equal to p p M =p. (6.26) In short, the temperature and pressure in a system in equilibrium are uniform. However, this applies only to the case when each of the phases can change its volume. In the opposite case, when the con- 106 TEXTBOOK OF THERMODYNAMICS VI 40 straints are such that the volume of each phase remains unchanged (y> = const.), the condition (6.24) must be replaced by a separate conditions 57>=0. (6.27) This will occur, for instance, when each of the phases is enclosed in rigid heat-conducting walls, but also in some cases which are less trivial, as we shall see in sections 107 and 127. In applying the method of Langrangean multipliers, we take eq. (6.23) with the factor Ao and (6.27) with the factors X (i) and add them to eq. (6.12) S[(r> = o, (6.28) whence r = - Ao = T, />> = A (i) . (6.29) The temperature is uniform throughout, as in the previous case, but the pressures of the phases are all different. It is quite obvious how these considerations are to be extended to the mixed case when the volume of some of the phases is constant while others can expand. The case when some of the phases are adiabatically enclosed and their entropies are constant (dS (i) = 0) is also possible but hardly worth our consideration, since it is one of dynamical, rather than thermodynami- cal, equilibrium. We have obtained a partial solution of the problem of equilibrium, with respect to the thermodynamical variables T and />, and can now turn to the other side of the problem relating to the virtual changes of the mol numbers. In order to deal with it, we shall again impose upon the system suitable constraints. We already know that, in equilibrium, the temperature T is uniform while the pressures p (i) of the phases may or may not be equal. We shall now consider only virtual changes in which dT = and dp (i) = 0. It was shown in sec- tion 36 that, under these circumstances (when the temperature and the local pressures are kept constant), the function <f> tends to decrease and equilibrium is possible only when it has reached its minimum (&typ. T = 0. (6.30) The total thermodynamic potential is represented by the sum over the phases VI 41 GENERAL CONDITIONS OF EQUILIBRIUM 107 In forming the variation T and p are to be kept constant, whence In general, the virtual changes 5N t (i) are not independent but sub* ject to subsidiary conditions determined by the nature of the system. Exercise 64. Consider eqs. (6.22), (6.23), and (6.24) for the case of only two phases and solve them without use of Lagranean multipliers. Show that this method can be used to obtain the general result by the reasoning of section 39. Exercise 65. Take the case of p and T being uniform throughout the system and generalize the argument of section 32 which led to the formula (5.08) for systems with several phases. Show that this is an alternative way of obtaining the condition (6.31). 41. Number of phases in equilibrium. The phase rule. In order to derive the explicit equations of equilibrium from the condition (6.31), it is necessary to eliminate from it the dependent virtual changes dN k (l) by means of the subsidiary conditions. It was pointed out in section 38 that this can be done either before taking the varia- tion (6.30) of the function <J>, or after. The former procedure consists in expressing a part of the arguments N k (i) of the function $ by the remainder and so reducing their number as far as possible. We say, then, that the thermodynamical potential is expressed in terms of the mol numbers of the independent components of the system. To illus- trate this by an example, let us take the case that N\ mols of hydrogen (H2) and N% mols of iodine (12) are brought into the same vessel. The two gases will partially react and form a certain amount (N% mols) of hydrogen iodide which will be the third constituent of the mixture but not an independent one: in the state of equilibrium N& is com- pletely determined by N\ and N%, so that hydrogen and iodine can be taken here as the independent components. We see, moreover, from this example that the method is of no help in the treatment of con- crete cases, because we must have solved the problem of equilibrium and found the expression of Na, in terms of NI and N2, before we can eliminate it from the thermodynamical potential. This is, in fact, characteristic of all thermodynamical systems: the main part of the subsidiary conditions between the virtual changes 6N k (i) are given in the variational form (6.06), and not in the integral (6.03), and often there is no practicable way of eliminating the dependent mol numbers N k (i) except by means of the final solution of the problem. However, some interesting conclusions can be drawn from the theoretical possi- bility of this elimination: we simply suppose that the system is reduced 108 TEXTBOOK OF THERMODYNAMICS VI 41 to its independent components, without attempting to carry out this reduction in any special case. Generally, those constituents of a system should be chosen as its inde- pendent components which cannot be converted into one another by any reaction going on in it. In the above example hydrogen and iodine satisfy this requirement, but, in general, the number of independent components of a system is, by no means, identical with the number of chemical elements of which it consists. For instance, if all the phases of a system consist of water in different states of aggregation (ice, steam, liquid water), hydrogen and oxygen are not independent components, because the mol number of the one is defined by that of the other, viz. stands to it in the constant ratio 2:1. We have here a system of only one independent component. Strictly speaking, the number of components depends on the accuracy with which we wish to describe the system : there is always a certain amount of disso- ciation in steam, and, because of the different solubility of hydrogen and oxygen, the phases will not contain exactly equal mol numbers of the two elements. However, these effects are entirely negligible at ordinary temperatures. An example of a system in which the num- ber of independent components is larger than the number of elements is presented by a mixture of hydrocarbons. There are many hydro- carbons which do not react under ordinary conditions, so that the mol numbers of hydrogen and carbon are quite insufficient for a description of such a system. The ultimate criterion for the choice is, of course, that a virtual change of the mol number in one of the independent com- ponents should not necessitate any virtual changes in the others. The notion of independent components is sufficiently clear from this discussion. Suppose that their number is ft and that the mol numbers of all the constituents, the independent as well as the depen- dent, are eliminated from the functions <> (i) and replaced by the masses Mi (i \ M2 (i \ . . . Af e (i) of the independent components in the several phases. Equation (6.31) becomes, then (if we drop the subscripts (6 ' 32) The only subsidiary conditions which remain are now those which apply to each independent component separately. We shall restrict ourselves here to the case that the phases are in direct contact, so that a component can pass freely from one phase into another. As was pointed out in the preceding section, the border conditions of the system as a whole can be replaced by an imaginary rigid envelope VI 41 GENERAL CONDITIONS OF EQUILIBRIUM 109 surrounding it. This implies that the total mass of each component a _ must remain constant. /, Af* (0 = const, or taking the variation, 0, (* = 1, 2, . . . ft). Multiplying these equations, respectively, by the Lagrangean mul- tipliers X* and adding them to (6.32), \ ). -ft - +X* where the virtual changes BM k are completely independent. There follows the system of equations = - X*. (6.33) The index k goes here from 1 to ft, so that this formula represents aft relations. If we eliminate from them the functions X*, there remain Zeq = aft ft equations between the partials of the functions 3> (i) . What is, on the other hand, the number of variables on which these partials depend? It was shown in section 39 that the partials of <i> (i) , with respect to N k (i \ represent specific quantities. The reason- ing used in that demonstration is in no way affected, if the masses M k (i) are regarded as the arguments of these functions, instead of the mol numbers. The partials 9* (i) /3^ (0 are, therefore, homogeneous functions of the degree zero in the masses. In analogy to the mol fractions we introduce the mass fractions (6.34) and conclude from eq. (6.12) that 3* ( * ) /9Af* (0 can be represented in terms of the variables y k (i) . Only ft 1 of these fractions are inde- pendent because there exists the identical relation The number of variables defining the composition of the phase (i) is, just ft 1, and the number of them in all the a phases is aft a. To this must be added the two thermodynamical variables, the common temperature T and pressure p of all the phases. The total number of 110 TEXTBOOK OF THERMODYNAMICS VI 41 variables by which the partial derivatives of the system (6.33) are determined is, therefore, Z var = a/3 a + 2. A system of equations is only then compatible and capable of being satisfied when their number does not exceed the number of variables on which they depend (Z eq g Z var ). This condition restricts the number of phases which can be simultaneously in equilibrium: from a/3 (I afl a + 2 there follows a ft + 2. (6.36) This inequality expresses the famous phase rule of Gibbs: The number of phases coexisting in equilibrium cannot exceed the number of independent components by more than two. Let us consider the main cases that can arise. Examples of them will be treated in the following chapters. The maximum which a can reach is a = + 2, (6.37) it will be attained when the number of variables is just equal to the number of equations. All the variables are then completely deter- mined, the equilibrium can exist only at one definite temperature and one definite pressure, and the composition of all the phases is also quite definite. This state is often called the fundamental state of the system. In the case a = /3 + 1 (6.38) the number of the variables exceeds that of the equations just by one: the value of one of the variables can be chosen arbitrarily, and this fixes the values of all the others. For instance, if we specify the tem- perature, the pressure and the compositions are completely deter- mined by it, etc. Similarly, when a = 0, (6.39) two of the variables can be assigned arbitrary values: if p and T are specified, the compositions of the phases are definite. Finally, in the cases a < /3, the compositions of the phases are not yet defined by the temperature and pressure. There are, then, so many cases of equilibrium possible that hardly any rules of general validity can be given. In conclusion, it will be well to emphasize that the phase rule deals only with the number of phases in equilibrium. We have mentioned VI 42 GENERAL CONDITIONS OF EQUILIBRIUM 111 water in its different states of aggregation as an example of a system with one independent component (ft = 1). According to the phase rule not more than 1+2=3 phases of it can coexist (viz. vapor, liquid water, ice), and the simultaneous equilibrium of all three is possible only in the fundamental state at a definite temperature and pressure (compare also section 48). However, everybody knows that he can throw a lump of ice into a bucket of water and that there will be, at all temperatures, water vapor on top of the system. In this case, the three phases are not in equilibrium, and, in the long run, one of them will disappear, but the coexistence without equilibrium may last hours. In applying the phase rule one must be quite sure that equilibrium is, actually, established. 42. Explicit equilibrium conditions. We return now to the description of the phases by the mol numbers of all their chemically distinct constituents and not only of the independent components. We shall make use of the notation (5.38) defining the partial molal thermodynamic potential > (6 - 40) It will be well to recall here that it is a generalization of the ordinary molal potential. In the special case, when the phase is chemically pure, there is only one constituent, 3> (i) = N (i) <p (i \ where <p (i) is inde- pendent of N (i \ Then ^ (i) becomes identical with <p (i) . The condition (6.31) now takes the form ' = 0, (6.41) and is valid for all possible virtual changes dN k (i) consistent with the nature of the system. The virtual processes in the system can be reduced to two simple types. In the first place, a small amount of the constituent k can pass from a phase (1) into an adjacent phase (2). The molecular weights MA: (I) and jufc <2) of this component in the two phases need not necessarily be the same because there is the possibility of association. The mass added to the first phase is, then, M* (1) &W 1} and that added to the second n k (2) 5N k (2} . These masses must be oppositely equal or W> : 6N k - - -1- : -^ - *> : ,*>. (6.42) (1) (2) 112 TEXTBOOK OF THERMODYNAMICS VI 42 The symbols v k are integral numbers with the following meaning. Suppose that j>* (1) mols of the component k are removed from phase (1): they are converted into v k (2} mols belonging to phase (2). If no association takes place in either phase, we can put 1>* (1) = v k (2) = 1. There are no conditions in the system which should prevent us from putting all the other virtual changes bN k equal to naught, in the way of an auxiliary constraint. Therefore the condition (6.41) reduces to A$ SE v k Tp k + v h *W> = 0. (6.43) A<i> can be interpreted as the change of the total thermodynamic potential which takes place in the finite virtual process of reversibly transferring v k m mols of the component k from the first phase into the second (where they are converted into v k (2) mols). The temperature and pressure are supposed to remain constant, maintained so by a suitable arrangement of heat baths and loads. Of course, the process will be attended by the system receiving (positive or negative) heat from the heat baths and doing work against the loads. Sometimes it is convenient to express the composition not in terms of the mol numbers N k (i) but in terms of the masses M k (i) = n k (i} N k (i) of the individual components. We shall denote the partial specific thermodynamic potentials (referred to unit mass) Because of (6.42), /** (1 W n = Mt (2 V 2) , and (6.43) reduces to ?Af* (1> - ?W 2) = 0. (6.45) The second simple type of process we have to consider is a chemical reaction between several constituents of the same phase. Let us take, as an example, the equation of the reaction between hydrogen and iodine 2HI = H 2 + I 2 , or 2HI - H 2 - I 2 = 0. Like the equations of all chemical reactions it is of the general type viGi + v 2 G 2 + . + v 9 G 9 = 0, (6.46) where GI, . . . G are the chemical symbols of the different substances (components) an4 ?i, . . . v 9 the numbers of mols with which they take part in the reaction. We may observe here that the process of VI 42 GENERAL CONDITIONS OF EQUILIBRIUM 113 transferring matter from one phase into another, as considered above, can be represented by the same sort of a symbolical equation >=0. (6.47) Both equations are included in the general form > = 0. (6.48) We shall usually consider a definite direction of the reaction, namely, that in which it is endothermic and the heat of reaction positive. Con- sequently, we shall attribute positive signs to the numbers vk of those components which are produced, and negative to those which are consumed, in the endothermic reaction. When eq. (6.41) refers to a process of the type (6.46), the corre- sponding virtual changes dNk must be taken proportional to the num- vers vk v%: . . . : v ff . (The numbers vk and variations dNk for the components not taking part in the reaction can be put equal to zero because of the principle of auxiliary constraints). The condition becomes A$ E= vjQt + . . . + Vj p 9 = 0. (6.49) As in the preceding case A<i> must be interpreted as the total change of the thermodynamic potential in the finite virtual process corre- sponding to the transformation of *>i, v v mols of the several com- ponents. Equations (6.43) and (6.49) represent partial solutions of the prob- lem of equilibrium. The complete solution is given by the simulta- neous system of equations corresponding to all processes of these two types possible in the thermodynamical system. Both these equations are included in the general form in which a part of the coefficients v k (i) are equal to those of (6.43) or (6.49) and the rest vanish. It will be useful to find the partial deriva- tives of A< with respect to the pressure and temperature: from eqs. (5.37) 114 TEXTBOOK OF THERMODYNAMICS VI 42 and (3A$/9r) p = AS, where AFand AS represent the changes of the total volume and the total entropy in the process. This process being isothermal (as well as isobaric), we can apply to AS the formula (4.11), AS = Qp/T, where Q p is the isobaric heat of the reaction (6.52) Moreover, taking into account the condition A3> = 0, we can write Qp (6.53) Exercise 66. What is the structure of A 7 and Q p in terms of partial molal quantities? Show that , Q, * CHAPTER VII PHASES OF A CHEMICALLY PURE SUBSTANCE 43. Equilibrium of two phases. If all the phases of a system con- sist of the same chemically pure substance, in different physical states, it contains, according to section 41, only one independent component (ft = 1). The phase rule (6.36) tells us, then, that no more than three phases can be simultaneously in equilibrium. There exist substances which are known in more than three states. For instance, sulfur occurs in the vaporized, liquid, monoclinic crystalline, and rhombic crystalline states, but only three of them can coexist in equilibrium. The problem of thermodynamics in relation to pure substances is, therefore, reduced to the consideration of the equilibrium of one phase, two phases, or three phases. The properties of a single phase were sufficiently discussed in Chapters I to IV, and we shall restrict ourselves here to the cases of two and three phases, starting with the two-phase system as the simpler problem. The only possible virtual process in such a system is that repre- sented symbolically by eq. (6.43): v (1) G (1) + v (2) G (2) =0, viz. the transformation of v (2} mols of the second phase into v (l) of the first. It will be more convenient to define the numbers v (1) and v* both as positive and to indicate the fact that the second phase is consumed in the process by writing its equation as v (l) G (l) j/ (2) G (2) with instead of (6.42). It was pointed out in the preceding section that, for a pure substance, the partial thermodynamic potential and the ordi- nary molal are identical ^ = <p. The eq. (6.31) of equilibrium takes, therefore, the form A* = (1 V (1) - v (2 V 2 >, (7.02) or, in the special case when there is no association and the molecule is the same in both phases (> (1) = v (2) = 1), <<> - > (2) . (7.03) In words, the molal thermodynamical potentials in two adjacent 115 116 TEXTBOOK OF THERMODYNAMICS VII 43 phases are equal. The molecular constitution of the condensed phases is not always known and, in some cases, even not well defined. For this reason, many authors refer the quantities <p (i) of all the phases to 1 mol of the vapor. In particular, this is unavoidable in the chemical literature which has to present results of measurement on all sorts of substances, including those whose molecular structure is not yet suf- ficiently investigated. The purpose of this book is, however, a different one: we are not so much interested in the presentation of a large amount of experimental material as in its theoretical penetration. The questions of association and dissociation play an important role in the interpretation of the experimental behavior of substances, and they can be conveniently studied if the fundamental equation of equilibrium is written in the general form (7.02). We shall, therefore, make this formula the basis of the following discussions (compare also section 60). In general, the coefficients *> (i) are integers, completely determined by the nature of the system, while the functions <p (i) depend only on the common temperature and pressure of the two phases, p and T are, therefore, the only variables which enter into eq. (7.02) or (7.03), and, if one of them is arbitrarily chosen, the other is determined by this relation. It was pointed out in section 41 that this is always the case when the number of phases is larger by one than the number of independent components (a = ft + 1). When the equilibrium of the liquid and the gaseous phase is considered, the temperature which corresponds to a given pressure, in virtue of the condition (7.02), is called the boiling temperature or the boiling point. Vice versa, if the temperature is regarded as given, the pressure corresponding to it is known as the boiling pressure. Similarly, one speaks, in the case of the coexistence of the liquid and the solid phase, of the freezing point and the freezing pressure. More general terms including all cases of a pair of pure phases are transformation temperature and transformation pressure. Let AC and DF in Fig. 12 represent, respectively, the isothermals of a pure substance in its liquid and gaseous states. Suppose we have a single phase of it corresponding to the point A of the diagram. Keeping the temperature constant we increase the volume reversibly, thus allowing the pressure slowly to drop: the representative point of the system will move along the isothermal towards C. Let the point B correspond to the boiling pressure for this isothermal at which the coexistence of the liquid and the vapor phase is possible. The question arises now: will the substance, after reaching this point, remain in a single liquid phase and continue to move along the branch VII 44 PHASES OF A CHEMICALLY PURE SUBSTANCE 117 B, or will it begin to vaporize at constant pressure and be converted into a two-phase system whose states are represented by the horizontal BE? What is the relative stability of the states M and N lying on a vertical and corresponding to the same volume? The results of the preceding chapter are insufficient to solve this problem because they regard the number of phases as given: the criteria given there are satisfied in both cases. To obtain an answer, we must fall back on the fundamental principles governing thermodynamical equi- librium as expounded in Chapter V. Since the processes BM and BN are both isothermal and lead to the same end volume, we have to use the criterion adapted to this case (section 36): The work function ^ tends to decrease, and, of two states of the same temper- ature and volume, that with the lower is more stable. Let us now evaluate the work functions &M and ^N in the states M and N by comparing them with ty B in the state B. We make use of eq. (5.22) which applies to any iso- thermal process: ^B ^M = WBM a the right sides of the equations repr system, respectively, in the processes find Vtr VM = WBM WBN. In represented by the areas under the . Isothermals in equi- lum of liquid and vapor. M lies lower than N, we have ^fN < ^M- The states on the h< work function and of higher stabi that the states represented by impossible but sometimes occ (supercooling). However, they and the system is always read' { This reasoning can easily be transformation pressure or divides into two phases. J tional conditions. :ur cor; it is VB ^N = WBN, where . the work done by the and BN. Subtracting we iiagram WBM and WBN are and BN\ and, since We find, therefore, E are the states of lower <?-have mentioned in section 5 anch BC are not absolutely ' exceptional circumstances d to an unstable equilibrium, \ver into the two-phase state, o all analogous cases: at the i a pure substance, generally, 5 a single phase only in excep- Exercise 67. When the and vapor), there are two ; represented by the curve an^ prove that the two shaded , ___ * areas 44. The auy be directly used for the call u ' at * on -V ^ o /fc . equation applies (continuity of liquid ,O/ cesses between B and E (Fig. 1, p. 10) n ' , line. Applying eq. (5.21) to both of them, 1 are equal. equation. Equation (7.02) cannot transformation temperatures 118 TEXTBOOK OF THERMODYNAMICS VII 44 because the explicit form of the functions <p (i) is known in only very few cases. However, its differential form gives a relation between easily measurable quantities. Let us consider the equilibrium of our two-phase system in two cases: at the temperatures and pressures r, p and T + dT, p + dp. The respective equations (7.02) are A$>(r, p) = and A<i>(r + dT, p + dp) = 0. According to the definition of the differential of a function, the difference between these two equations is v -(*)+&).-* <"*> We can s-X^';iite the expressions (6.51) and (6.52) for the partial derivatives. heat of the process (Q p ) is, in this case, the latent heat of trans *ion L (compare section 10). We find, therefore, . V, \ This relation was fc the discovery of the first from the modern point o* called the Clapeyron-Clau. transformed in the process we shall denote the latent he AFj (v (1) i; (1) j/< 2) z; (2) )/j/< and before the transformat (7.05) ^ ' "* ^French engineer Clapeyron before aTermodynamics (1832). Its deduction ^ ^ due to Clausius. It is, therefore, & ^ion. If the mass of the material ^ e ^s to 1 mol of the vapor phase, ^ , hile the change of volume becomes "Terence of the molal volumes after ' referred to 1 mol of the vapor) (7 - 06) This formula has been ext to be accurate within the limi* ve ^ ' we may take water at the no sf ex ?? n of volume is AF 2 - 30 186 c/ boiling mm/deg. The conversion lo1 . ested by experiment and found 'mental error. As an example point (373. 1): the change T lg Q1S ^ dp/dT = 2? 12 e . mm/deg = (1.01325/760 X 4.1852 X 10) cal deg" 1 cm ' 3 * The formula (7.06) gives, there direct measurement gives / = 972v, . t We give in Table 8 the normal ^ *** C melting points (7 of the more together with the latent heats of at these temperatures. X 10- 5 cal deg- 1 cm" 3 . '6 cal mol- 1 , while the L Ailing points (T B ) and e | c -mentary substances and of fusion (W VII 44 PHASES OF A CHEMICALLY PURE SUBSTANCE TABLE 8 119 Element T B IB (cal mor 1 ) IB/TB TF IF (cal mol- 1 ) He 4.2 24.5 5.7 Ne 27.3 416 15.3 24.5 57 A 87.5 1 540 17.6 84.0 267 Kr 121.4 2240 18.4 104 360 Xe 164.1 3200 19.5 133 490 H 2 20.5 220 10.7 14.0 28.4 N 2 77.3 1370 17.7 63.3 170 2 90.2 1640 18.2 54.8 105 8 161 3570 22.2 22 C1 2 238.6 4900 ?0.5 171.6 1620 Br 2 332.0 7150 20.9 266.0 2560 I 2 457.5 105' 23.0 386.7 4000 Na 1153 25 21.8 370.7 633 K 1033 20 19.8 335.5 518 Cu 2570 11 43.4 1356 2750 Ag 2220 ) 26.8 1233.7 2630 Au 2870 30.6 1336 3180 Zn 1180 20.7 692.6 1690 Cd 1040 JO 24.5 594.1 . 1480 Hg 630.1 )0 23.0 234.3 556 Al 2070 26.0 933.2 1910 Si 2870 ) 14.3 1690 Sn 2530 I, < 30.6 505.0 1670 Pb 1890 4610v, 24.4 600.7 1120 Bi 1720 46100 26.8 544 2600 S 717.8 2190 30.5 392.2 262 W 6170 223000 36.2 3643 Fe 3270 90700 28.6 1808 2670 Pt 4570 167000 36.6 2028 5250 .J In the case of the transformations solid liquid and liquid * vapor, the latent heat L is positive. The volume of the vapor is, moreover, always larger than the volume of the same mass of liquid, AF> 0, so that the boiling temperature rises with the pressure (dTB/dp > 0). For instance, in the case of water we have the follow- ing conditions: TABLE 9 t PS-- 6.025 X 10- 50 /' O.fi'17 100 1.0000 120 1.9594 200 15.341 370.4 C 217.72atm On the other hand, the specific volume of the liquid phase may be either larger or smaller than that of the solid in equilibrium with it. 120 TEXTBOOK OF THERMODYNAMICS VII 45 For instance water (at C and 1 atm) has a 9% smaller volume than ordinary ice. Therefore, AF and dT/dp are in this case negative, as was pointed out by James Thomson (elder brother of Lord Kelvin). The melting point falls, therefore, with increasing pressure: when ice at a temperature slightly below C is subjected to pressure, it is brought into a state in which its T is above the melting point. It is, therefore, converted into water but freezes again as soon as the pressure is released. This phenomenon is called regelation, and it accounts for the plasticity of ice which permits, among other things, the flow of glaciers. The ease of skating on ice is also usually attrib- uted to regelation : enough water ir' '.elted under the pressure of the sharp edges of the skates to k*> / In the case of vaporization, c the formula (7.06) can be simplih so low that the equation of state ( sufficient approximation, to the gasv these conditions the molal volume . neglected compared with V2. In fact, ratio V2/vi for water (e.g.) is 1676, and law of perfect gases can be used, it is obtain an approximate form of eq. (7.C /lubricated. are sometimes such that 0^ the vapor pressure is perfect gases applies, with ihase: V2 = RT/p. Under the liquid phase can be at 100 C and 1 atm, the * temperatures, when the arger. In this way, we _ RT* which is useful for many purpose* Exercise 68. Calculate the latent / of water, at 100 C and 760 mm Hg, from eq. (7.07). Use the value dp/d /> 27.12 mm/deg. Compare the results with the exact value as given in text. Exercise 69. Calculate the latent heat of mercury at its normal boiling point (tB - 357.3, p - 760 mm). Use the value dp/dt = 13.81 mm/deg and eq. (7.07). Exercise 70. Calculate the coefficient dT/dp for the fusion of ice at C and 1 atm, using the following data (per 1 g of the substance): l g = 80 cal g" 1 , v\ = 1.000 cm 8 g" 1 , j = 1.091 cm 8 g" 1 . Apply suitable conversion factor to p. Exercise 71. Naphthalene melts at 80. 1 C, its latent heat of fusion is 4563 calories per mol, and the increase in volume on fusion is 18.7 cm 8 mol" 1 . Find the change of the melting point with pressure. Exercise 72. The transition point of mercuric iodide, red > yellow, is at 127 .4 C. The latent heat absorbed in the change is .150 cal mol" 1 , and the approxi- mate change of volume 5.4 cm 8 mol" 1 . Find dT/dp for the transition point. 45. Temperature dependence of the latent heat Another rela- tion between measurable quantities can be obtained by differentiating VII 46 PHASES OF A CHEMICALLY PURE SUBSTANCE 121 eq. (3.23) of section 17, dL = d(X (1) - X (2) ) = </(*> (1) x (1) - ^ (2) x (2> ) and substituting the expressions (4.47) for dx- We find dL = AC p <*r where AC P and AF denote the changes of capacity and volume in the transformation : AC P = v (1 V 1} - v (2 V 2) > AF - iW - / 2 V 2) . (7.08) Dividing the equation by dT and replacing dp/dT by (7.06) In the special case of vaporization at comparatively low pressure we can again make the simplification of neglecting v (2) (of the con- densed phase) and applying to v (1) the equation of perfect gases: log AF/3r) p = 1/r. The relation is then reduced to - *Cr (7.10) Exercise 73. Calculate dl/dT (per 1 mol = 18.0) for boiling water (at 100 C, 1 atm) with the help of the following data: in liquid water c pz 18.0 cal, in vapor c Pl = 8.8 cal (both per 18 g, at 100 C), AF f = 30 186 cm 3 mol- 1 , (QAKj/aDp = 84.66, I = 9720 cal mol" 1 . Compare the results given by the two formulas (7.09) and (7.10). (The measured value is -11.0). 46. Approximate and empirical expressions for the transforma- tion pressure. The temperature dependence of the latent heat is, usually, slight. Especially, in the case of vaporization AC P in the for- mula (7.10) is small compared with L. As a first rough approximation one can, therefore, put L = LQ = const, or / = /o = const. Sub- stituted into eq. (7.06) this gives for the pressure of vaporization (7.11) (B being a constant), a formula due to Van der Waals, who wrote it (for liquids) in the form (7.12) which for a number of substances gives an approximate representation of the measurements even at high pressures. We shall return to this equation in the next section. 122 TEXTBOOK OF THERMODYNAMICS VII 46 It has been found since that the formula (7.11) is insufficient for an accurate representation of the vapor pressure of liquids but can be used for that of crystalline solids. Even in this case better approxima- tions are obtained by substituting into (7.07) the expansion / = / + 1 V T + faT 2 + . . . . For instance the vapor pressure of ice between 90 and C is represented by l 944.C c + 8.2312 - 0.0167006r + 1.205 X 10~ 5 r 2 - 6.757169. (7.13) As to the vapor pressure of liquids, the very accurate modern measurements cannot be adequately represented even by formulas with many terms. For the dependence of the pressure of fusion upon temperature a theoretical formula has not yet been advanced, and one has to rely on empirical expressions. According to Simon and Glatzel 2 the follow- ing formula gives satisfactory results: logio(/> + a)=c logio T F + 6. (7.14) They give the following values for the constants (Table 10), pro- vided the pressure is measured in kg/cm 2 = 0.9678 atm. TABLE 10 Substance a b c Helium* 17 1 . 5544 1.236 Sodium 12400 5.032 3.55 Potassium 3900 -8.227 4.68 Rubidium 4 100 5.59 3.65 Cesium 2450 8.38 4.75 Mercury 45000 1.11 2.40 CO 2 4000 2.564 2.64 ecu 3 100 1.498 2.08 SiCU 5 100 -0.459 1.39 Benzene 3 900 2.496 2.49 Chlorobenzene 5 100 1,970 2.41 Bromobenzene 5000 2.070 2.42 Nitrobenzene 6000 1.037 1.97 Chloroform . . ... 6000 0.466 1.83 Bromoform 5800 0.984 1.94 Aniline 5500 1,913 2.33 * The data for helium are taken from Simon, Ruhemann, and Edwards, Zs. phys. Chemie (B) 2, p. 430, 1929. 1 E. W. Washburn, Monthly Weather Review 52, p. 488, 1924. * F. Simon and A. Glatzel, Zs. anorgan. Chemie 178, p. 309, 1929. VII 47 PHASES OF A CHEMICALLY PURE SUBSTANCE 123 Exercise 74. Calculate the latent heat / of sublimation of ice and the differ- ence Ac p of specific heats (between 1 mol of vapor and 1 mol of solid), at C. Use eqs. (7.13), (7.07), and (7.10). Exercise 75. At its normal point of fusion (-38.9 C) mercury has the molal volumes vi = 14.65 (liquid) and v, = 13.90 (solid). Its latent heat is / = 556 cai mol" 1 . Check whether the formula of Simon and Glatzel is in satisfactory agree- ment with (7.05). 47. Vaporization as a corresponding state. It was stated in section 27 that certain groups of substances approximately satisfy what was called there the extended law of corresponding states. It is meant by this that within such a group the quantities (u o)/r, 5 and pv/T are universal functions of the reduced variables. Since the molal thermodynamic potential is defined as <f> = u Ts + pv, the quantity (<f> uo)/T must also be expressible as <p UQ . ^ /T i r\ ^ =/(T,T), (7.15) where / is universal for the whole group. The reduced volume does not appear as a variable, because it can be eliminated by means of the equation of state. It must be assumed that the equations of state apply equally to the gaseous and to the liquid phase, bringing out their continuity, in the sense of section 5. Therefore, the formula (7.15) is also valid for both phases. It follows then from (7.03) that, for substances which do not associate, the boiling condition is determined by the universal equation /d)( 7r , T ) =/O>(T,T). (7.16) Substances boiling at corresponding pressures have corresponding boiling points. In practice the boiling point changes but little with pressure. For instance, we see from the data for water on p. 119 that, while the pressure is increased from 1 atm to 2 atm, the temperature changes only by 20 or about 5%. For all the liquids satisfying eq. (7.16) there obtain, in this respect, similar conditions. In fact, it follows from it that the ratio dT . dP _ dir . ~T ' p ~~ IT ' T must be universal. This explains the fact noticed already by Guldberg 1 that many liquids have roughly the same reduced boiling temperatures 1 C. M. Guldberg, Zs. phys. Chem. 5, p. 374, 1890. 124 TEXTBOOK OF THERMODYNAMICS VII 47 even when they all boil at the pressure of 1 atm, as appears from Table 11. TABLE 11 Substance T c T B TB/T C Ether 467 308 66 Ethyl acetate 523 346 66 Ethylene chloride 561 357 64 Alcohol 516 351 68 Benzene 562 353 63 Hydrochloric acid .... Oxygen . 325 155 238 90 0.73 58 Carbon disulfide Ammonia 545 404 319 234 0.59 58 Ethylene 282 163 58 Methane 191 109 57 Sulfuretted hydrogen. Nitrous oxide 373 309 211 183 0.57 59 Stannic chloride 592 387 65 Water 637 373 59 Acetic acid 595 391 66 According to (4.11), the quantity l/T B represents the entropy increase on vaporization = s<i> - s< 2 >. It is an entropy change between corresponding states and must be, therefore, a universal function of TT, r according to (4.41). There follows that I/TB must be the same for substances which boil at corresponding pressures. However, both / and T B change but little with the pressure so that, in practice, l/T B measured at any pressure of the order of 1 atm, is roughly the same for many different sub- stances (Table 12). This fact is known as the rule of Deprez and Trouton. The table contains of course a group of selected substances. If picked at random, the variations of Trouton's ratio are larger. Yet a very marked deviation of the ratio IB/TB from the average value is, usually, traceable to association. For instance, in acetic acid IB/TB = 14.8, and this is due to the abnormally high molecular weight of the vapor (97 instead of 60 corresponding to 2^02). Ethyl alcohol shows the opposite deviation, IB/TB = 26.8, due to a normal vapor but partly associated liquid. Equation (7.12) of Van der Waals is VII 48 PHASES OF A CHEMICALLY PURE SUBSTANCE TABLE 12 125 Substance t B C IB (cal) h/T B Benzene 394.8 13650 20.8 Chloroform 247 10900 21.0 Aniline 434 14920 21.1 Bromobenzene 155 9 9350 21.0 Acetal 102.9 7810 20.7 Toluene 109.6 7980 20.8 Sulfuretted hydrogen. Stannic chloride -61.4 112 4490 7 900 21.2 20.5 Ethyl bromide 38.4 6500 20.9 Ethane 71 2 7 120 20.6 only a special approximate form of the general equilibrium condition between a liquid and its vapor. To the extent to which the reduced equation (7.16) is valid, the special form can claim to have universal validity. In fact Van der Waals expected the coefficient a to be a universal constant (for all substances). However, two things must be borne in mind. In the first place, (7.12) is only a rough approximation and a is, in reality, not a constant but a function of temperature. In the second place, we should expect, at best, a uni- formity of a, not generally, but only within the groups of substances satisfying the law of corresponding states. The analysis of the experi- mental material shows that the substances of such a group can be arranged in a series so that a is nearly the same for any two successive members of the series, but it changes, systematically and considerably, from one end of the series to the other. The emphasis on Trouton's rule, as on the law of correspondence in general, has changed in the course of time. Soon after the discovery of these regularities their importance and accuracy were being over- rated. Then the pendulum swung to the other extreme: the insistence was altogether on the lack of rigor of these laws and on the numerous exceptions from them. However, as long as it is borne in mind that these rules are only rough approximations of limited range, they are interesting and useful. For instance, deviations from Trouton's rule are helpful in indicating association, as was pointed out above. 48. The fundamental or triple point. There remains to say a few words about the third case mentioned in the beginning of this chapter. It follows from the general considerations of section 41 that the coexistence of three phases is possible only at a definite temperature and pressure which jointly define the fundamental or triple point. In 126 TEXTBOOK OF THERMODYNAMICS VII 48 fact, if we label the three phases by the numbers (1), (2), (3), we can apply the results of the preceding sections to each pair of them: For the equilibrium between (1) and (2), on one hand, and between (1) and (3), on the other, we have the two equations of equilibrium of the type (7.02) the third y (2 V 2) = p (3 V 3) being a consequence of these two. Corre- spondingly, of the three Clapeyron-Clausius equations following from these three relations 2V only two are independent, while the third is a consequence of them. If we plot the pressure of the system against its temperature, the three equations (7.17) will represent three curves OA, OB, OC (Fig. 13). All points of the p, T-plane which do not lie on one of these curves correspond to a A / single-phase state of the system. The points on I I .*& ^n the curves OA, OB, OC represent the equilib- rium of two phases, and the point of intersection O is the triple point. As an illustration of the (p, r)-diagram, we have taken the most com- mon case when the three phases in question are vapor, liquid, and crystalline solid. The branch OA representing equilibrium of the liquid and gaseous phases ends abruptly at the critical point A, since there is no difference between these two states at higher temperatures. 1 Suppose that we have vapor and liquid in equilibrium and slowly change the boiling pressure and temperature so as to move the representative point of the system along the curve of boiling AO towards the triple point 0. When the point is reached, a contingency may arise similar to that discussed in section 43 with respect to the expansion of a liquid. Under special conditions of great purity of the material and of very careful manipu- lation, the solid phase may fail to appear and the vapor-liquid combina- tion may get into a state of supercooled and unstable equilibrium represented by the dotted continuation of the curve A 0. The slightest shock may cause then a sudden crystallization. 1 With respect to attempts of extending the branch OA beyond the critical point, compare: Trautz and Ader, Phys. Zs. 35, p. 446, 711, 1934; Eucken, ibidem, p. 708. FIG. 13. Triple point and equilibrium of solid, liquid, and vapor. VII 48 PHASES OF A CHEMICALLY PURE SUBSTANCE 127 The triple point of most substances lies at pressures which are low compared with the normal atmospheric. When they are heated at constant atmospheric pressure, their state changes along a vertical straight line drawn to the right of the triple point. The crystal is first converted at a certain temperature (of fusion) into a liquid ; at a higher temperature the liquid is vaporized. It is different, when the pressure of the triple point is high or when the substance is heated at a low pressure: then, the vertical lies to the left of the point and the crystal is directly sublimated into vapor without passing through the liquid state. The best-known example of the coexistence of three phases is that of steam, water, and ice at p = 4.579 mm Hg and / = 0.0075 C. However, researches by Tammann l and Bridgman 2 have disclosed that, in addition to the ordinary ice (I), there exist at least four other crystalline states of water (ices II, III, V, VI) which are all denser than water and observable only at very high pressures. (The existence of a modification, formerly labeled ice IV, is considered as doubtful). The table of the several triple points, as far as they have been observed, is as follows TABLE 13 PHASES t P FIG. 14 Ice I, liquid, vapor. . . . 0.0075 C 4.579 mm Hg Ice I, liquid, Ice III. . . -22 2115 kg/cm C Ice III, liquid, Ice V. . . -17 3530 ' ' D Ice V, liquid, Ice VI ... -f OM6 6380 ' ' E Ice I, Ice II, Ice III... -37. 7 2170 " F Ice II, Ice III, IceV... -24. 3 3510 " G A graphical representation of the measurements carried out with water in its different phases is given in Fig. 14. An interesting situation was disclosed by Tammann 's investigation of the equilibrium curves of phosphonium chloride, PH4C1 (Fig. 15). The curve of fusion could be traced to temperatures considerably higher than the critical point of the liquid-vapor equilibrium. It seems paradoxical that the crystalline phase can be observed at temperatures at which the liquid does not exist or, rather, cannot be distinguished from the vapor. A similar behavior is shown by carbon dioxide (62) and by helium (He). The curve of fusion of CO2 was observed by 1 G. Tammann, Ann. d. Phys. 2, p. 422. 1900. * Bridgman, Proc. Amer. Acad. 47, p. 441, 1912. 128 TEXTBOOK OF THERMODYNAMICS VII 49 Bridgman up to / = 93.5 C and p = 12 000 kg/cm 2 , while the critical data are t c = 31.4C, p c = 72.9 atm. The case of helium is even more striking: Simon, Ruhemann, and Edwards 1 observed its curve 12000 10000 8000 4000 2000 ^0-40-30-20-10 10 rc FIG. 14. Phase equilibrium in water. 80 of fusion up to T = 42 abs and 5600 atm, i.e. solid helium can be prepared at a temperature fully eight times as high as the critical point of liquid helium (t c = 5.2 abs, pc = 2.25 atm). The question is still open whether the curve of fusion extends indefinitely to still higher temperatures or is limited, either by a solid-liquid critical point or in some other way. Its experimental investigation is very difficult as it requires the applica- 10 20 30 40 50 tion of ex tremely high pressures. 100H 80- 60 40- 20 FIG. 15. Phase equilibrium in phos- phonium chloride. 49. Phase equilibrium of higher order. 2 The usual case of two phases in contact treated in section 1 may be called equilibrium of the first order. Influenced by observations on liquid helium, Ehrenfest 3 discussed another possible case which he called equilibrium of the second 1 Simon, Ruhemann, and Edwards, Zs. phys. Chem. (B) 2, p. 340, 1929; 6, p. 62, 1930. 2 This section may be skipped without loss of continuity. 3 P. Ehrenfest, Communications Leiden, Suppl. 756, 1933. VII 49 PHASES OF A CHEMICALLY PURE SUBSTANCE 129 order. It may happen that the transformation takes place without development of latent heat and without change of volume. Supposing the molecule to be the same in the two phases, X2 Xi = 0, AF = V2 v\ = 0. (7.18) This means that the two phases have the same molal heat function and the same molal volume. Because of the fundamental relation of equilibrium, A$ = 0, 2 - <PI = 0, (7.19) snce they have then also the same entropy (AS = $2 si == 0), A5 = Qp/T, according to (4.11). Yet the phases are different because of their other properties (for instance, the molal heats) changing dis- 30 25 20 C 15 1-0 5 2.0 2.2 2.4 2.6 2.8 3.0K *~T FIG. 16. Specific heat of helium. continuously in the transformation. This seems to happen with liquid helium at the temperature of 2.2 K: the discontinuity in its molal heat is given in Fig. 16 as measured in Leiden l while changes of entropy and volume were not observed. Similar conditions are known to obtain in many other transformations: solid methane at 20.4 K, solid oxygen at 44 K, HBr at 87 K, ammonium chloride and bromide, etc. 2 The accuracy of these observations was in no case sufficient for a conclusive proof that the conditions (7.18) were strictly satisfied, but it is interesting to follow up the theoretical implications of such a contingency. 1 W. H. Keesom and A. P. Keesom, Comm. Leiden 222 d; W. H. Keesom and K. Clusius, 219 e; K. Clusius and A. Perlick, Zs. phys. Chemie 24, p. 313, 1934. 2 Compare reviews by M. Von Laue and A. Eucken, Phys. Zs. 35, pp. 945, 954, 1934. 130 TEXTBOOK OF THERMODYNAMICS VII 49 Taking account of (6.51) and (6.52), the conditions (7.18) can be also written in the form 0, A*,--0, (7.20) which converts the Clausius-Clapeyron equation (7.04) into an iden- tity. In order to obtain a relation between the differentials of tempera- ture and pressure, Ehrenfest proceeds to differentiate these two equations. where the partial differentiation with respect to T or p is indicated by writing these variables as subscripts. He further postulates that these two relations be compatible. It should be remembered that, according to (5.37), (4.32) ~~ C PI (7.22) The elimination of dp/dt from eqs. (7.21) gives Ehrenfest's relation for equilibrium of the second order - Serious objections against these conclusions were raised by Keesom 1 and by Von Laue. 2 They can be brought out most clearly by a geo- metrical interpretation of eqs. (7.19) and (7.20). If we regard T, p, <p as cartesian coordinates of a three-dimensional space, the equations <p <pi(p, T) and <p = <f>z(p, T) represent two surfaces in this space. In the ordinary first order equilibrium the condition (7.19) is satisfied along the line of intersection of these two surfaces, as represented graphically in Fig 17a, which gives the trace upon the (<p, T)-plane. The relation between the transformation values of p and T, given by the Clausius-Clapeyron equation (and represented graphically in the figures of section 48), is then the projection of this line of inter- section onto the plane p, T. The additional conditions (7.20) for the second order transformation mean that the two surfaces are in contact 1 W. H. and A. P. Keesom, Physica 1, p. 161, 1933. 2 Footnote on p. 129. VII 49 PHASES OF A CHEMICALLY PURE SUBSTANCE 131 (Fig. 176). Von Laue points out that in this case no transformation at all is possible. In fact, the theory of equilibrium requires A4> g (section 36) and, therefore, the states of the system represented by the upper branches of the curves are unstable. It will be found in the more stable state (1) both above and below the temperature of the contact. Von Laue suggests, therefore, that the phases in question may have an equilibrium of the third order, i.e. the surfaces <p = <p\ (a) (b) (0 ^ Xv fi *i FIG. 17. Equilibrium of first, second, and third order. and <p w have contact with penetration (Fig. 17c). The mathe- matical conditions for this are A^JT = A$ pr = A$ pp = 0, (7.24) in addition to (7.18) and (7.19). They represent so much of a restric- tion upon the parameters of the two phases that Von Laue's suggestion was not favorably received by the other workers in this field. We think, however, that the topic should not be dismissed without a fuller investigation of the possibilities of the second order equilibrium. Ehrenfest and Von Laue make an assumption which, in our opinion, goes too far. They differentiate eqs. (7.20), implying that the contact exists along the whole line of intersection of the surfaces <pi, w or, at least, along a finite part of it. There is nothing in the experimental observations to justify this assumption. Theoretically it is also a rather remote case, because it cannot occur, unless the properties of the two phases happen to satisfy Ehrenfest's eq. (7.23) or Von Laue's even more restrictive conditions (7.24). It is a far more common occurrence that the two surfaces are in contact in a singular point, i.e. eqs. (7.20) are satisfied just in one point whose coordinates we shall denote by po, TQ. What is then the value of A< in a neighboring point, say, PQ + dp, TQ + dT? Expanding with respect to dp, dT to terms of the second order dT, po + dp) . . . (7.25) 132 TEXTBOOK OF THERMODYNAMICS VII 49 Because of the conditions (7.19) and (7.20), equilibrium in the vicinity of p, T is then determined by the condition 0. The expression D of formula (7.23) is the discriminant of this equation. When D < 0, the equation has no real solutions so that a change of phase is impossible. The limiting case, D = 0, is that of Ehrenfest's and must be excluded because of the reasons advanced by Von Laue. But when D > 0, the equation has two real solutions. Because of (7.22), dT dp r/9AF\ lA QT J p TQ (7.26) Consequently, transformations are possible, if not in the point po, To itself, in its immediate vicinity as is illustrated in the (p, T)- diagram of Fig. 18. There are two lines in which the phases (1) and (2) (2) (a) (b) A' (2) " \] (1) FIG. 18. Vicinity of a point where equilibrium is of second order. FIG. 19. Special cases of first order equilibrium. can coexist, and they intersect in the singular point po, TQ. When the substance is cooled at a pressure exactly coinciding with that of the intersection (dotted line AB), no transformation takes place. If it is cooled at a slightly different pressure (line A'B') it is transformed first from the phase (2) into the phase (1) and, at a lower temperature, back again into (2). Some of the experimentally measured anomalies present features compatible with this picture: as the temperature is changed, the specific heat jumps discontinuously and, at a slightly lower temperature, jumps back to the initial value (Fig. 16). How- ever, it would require a more detailed experimental investigation to establish the existence of this type of transformation beyond doubt in view of the following considerations. VII 49 PHASES OF A CHEMICALLY PURE SUBSTANCE 133 More important than the direct application of the preceding results is the light they throw, indirectly, on transformations of the first order with small but finite latent heats and volume changes. Let po, To be a point in which A$r and A$ p are very small but finite. In the infinitesimal vicinity of the point po, To there will hold, of course, the Clausius-Clapeyron equation. But if we consider, instead of the differentials dp, dT t small finite increments A/>, AT, the second order terms in (7.25) may no longer be neglected and the equation becomes = 0. This represents an ellipse or a hyperbola with its vertex near the point po, To, according to whether the discriminant (D) is negative or positive (Fig. 19). The dotted parts of the curves have no reality because the approximation breaks down when A/?, AT become appreci- able since then terms of third and higher order must be taken into account. The gist of the matter is that these />, jT-lines are strongly curved. In cooling the system along the lines A'B' we meet with conditions quite similar to those of Fig. 18. In fact, the transition from the first to the second order of equilibrium is a continuous one and the two cases cannot be distinguished without a complete experi- mental investigation of the whole region around the point po, TQ. Some of the cases enumerated in the beginning of this section are, probably, examples of this kind of equilibrium of the first order. 1 Another interesting illustration will be presented in section 137. 1 The same conclusions were drawn by A. Eucken (footnote on p. 129) from the study of the experimental material. CHAPTER VIII MIXTURES OF PERFECT GASES 50. The thermodynamic potential of a gas mixture. An example in which the general theory of equilibrium can be completely carried through in detail is the mixture of perfect gases. This is largely due to the fact emphasized in section 4 that the pressure of such a mixture is equal to the sum of the partial pressures of the individual gases p = pi+ P2 + . + p., (8.01) where pi, . . . p 9 are computed for each gas as if it were filling the available volume alone and the other gases were not present. Suppose that we have in a vessel of the volume V several gaseous constituents in the respective amounts of N\, . . . N 9 mols. Let their partial pressures be expressed according to eq. (1.12) Vh being the molal volume of the gas fe. In order to apply to this system the theory of Chapter VI, we must find the thermodynamic potential $ = JJ TS + pV, where T and p are the temperature and the total pressure of the mixture. The physical inference from the formula (8.01) is that there is no interaction between the gases, so that the thermodynamic potential of each of them must be computed, as if it were alone in the vessel, and all these potentials must be added to obtain 9 *=y%*. (8 - 3) trr A simple and convincing way of seeing this is opened by an ideal experiment devised by Gibbs. We give the vessel containing the system the following construction (Fig. 20). The cylinder C fits tightly into the cylinder C 1 and can slide within it without friction. The outer ends of both cylinders are closed by tops AB and A'B' impenetrable for the gases and adiabatic like the side surfaces. They have also bottoms 134 VIII 50 MIXTURES OF PERFECT GASES 135 MN and M'N' which are kept by a suitable arrangement at constant distances / from the tops while the cylinders are sliding. However, these bottoms are semipermeable membranes: The membrane M'N' lets through freely the component h but is impenetrable for all the others. On the other hand, M'N' is impenetrable only for h, offering no resistance to the rest of the components. The volume of each cylinder (ABMN and A'B'N'M') is the same and equal to V. We start our considerations from the state when the cylinder ABMN is pushed completely in, so that MN touches A'B' and the gas mixture fills the joint volume V. We begin now to pull out this cylinder in a reversible (infinitely slow) way. Three part volumes of the system must now be distinguished: (1) the section above M'N' is filled with the pure h component because this membrane is not permeable to the other gases. (2) The middle part between M'N' and MN contains all components. (3) The section below MN is free of the component h because this gas is pushed up by the membrane MN which is impermeable to it. The gas h is, therefore, present only in sections (1) and (2) (or in the upper cylinder). We have emphasized that its partial pressure p\ is inde- pendent of the presence of other gases: It will be, therefore, the same in section (1) where it is alone and in (2) where it is one of the constitu- ents of the mixture. Only the component h A' exercises pressure upon the top AB and the FIG. 20. Device for bottom MN of the upper cylinder. The rest of unmixing gases, the gases do not produce any pressure effects on the two surfaces: on AB, because no other component is present in the part volume (1) adjacent to it; on MN, because this membrane is completely permeable to the other gases. We see, therefore, that the forces of pressure on top and bottom are oppositely equal so that the resultant force on the whole cylinder vanishes. The process of reversibly pulling out the cylinder does not involve any work (W = 0), provided there are no outer forces upon the system (e.g. air-free outer space). Since the outer envelope of the system is adiabatic (Q = 0), the reversible process goes on without change of entropy (section 23). Moreover, it proceeds at constant internal energy and temperature. In fact, in the general eq. (3.05) of the first law, Q = t/2 Ui + W, the terms Qand W vanish leaving C/2 = E/i, and in the case of perfect gases, this is equivalent to T% Ti, accord- ing to eq. (3.18) or (3.19). Continuing the process until the membranes MN and M'N' touchy 136 TEXTBOOK OF THERMODYNAMICS VIII 50 so that the middle section (2) disappears, one can entirely separate the component h from the mixture. In a similar way, the other compo- nents can be separated, one by one, so that, in the end, the system is completely unmixed and each component occupies, separately, a volume equal to V. The whole process is carried out without change of U and S. It follows from this that the internal energy and the entropy of a gas mixture are additive U = 2C/A, S = SS*, provided the entropies Sh of the individual perfect gases are computed as if each of them occupied the volume V alone. Therefore, the first two terms of the thermodynamic potential $ = U TS + pV are addi- tive, while the additivity of the third follows from the relation (8.01), thus completing the proof of eq. (8.03). If we use eq. (4.20) for the molal entropy of a perfect gas, the entropy Sh of the component h has the expression S h - N k j<ir - R log p h and the total entropy of the mixture becomes 5 - 2] N " - R log pk + s * (8 - 04) Sometimes it is preferable to express 5 in terms of the total pressure p rather than the partial pressures ph. In order to carry out the trans- formation, it is convenient to use the mol fractions defined in section 39 x h = Nk/N, (8.05) where N = Ni + . . . + N 9 represents the total number of mols (of the different constituents) in the mixture. In the case of perfect gases, equations p h V = N h RT and pV = (N\ + . . . + NJRT hold, so that ph/p = x h or ph = x h p. (8.06) Substituting this into (8.04) . (8.07) The first three terms of this expression are linear in the mol numbers while the last shows a more complicated dependence upon them. VIII 51 MIXTURES OF PERFECT GASES 137 In a similar way, there follows from eq. (5.41) for the molal thermo- dynamic potential 7k = RT log p h + a*(TO, (8.08) and , N h [RT log p k + (DL (8.09) T^T where /T s* c c ph dT - T I ~ dT + u Qh - 7*0*. (8.10) J l or an alternative form after the substitution of (8.06). We denote the molal potential (8.08) by the symbol ^ instead of <ph for reasons which will become clear in the next chapter. This is entirely permissible because, in this case, the partial thermodynamic potentials of the mixture are identical with the molal of the pure com- ponents, as a consequence of the additivity expressed in eqs. (8.03) and (8.09). In fact, as determined by eq. (8.08), <pn satisfies the definition (5.38) of the partial molal thermodynamic potential. Exercise 76. Derive the expressions of the heat function X and the work function ^ for mixtures of gases. Exercise 77. Check the statement that the derivative (3*/3^)p f r is identical with (8.08). In the differentiation the total pressure is to be kept constant. 51. The mass law. If the gases of the mixture can react chemi- cally with one another, according to the formula (6.46), S^G* = 0, the condition of equilibrium is given by eq. (6.49) or 2*>^ = 0. Substituting for <ph the expressions (8.08), we obtain log p h = - We recall that the coefficients j>& of the components consumed in the reaction have the negative sign, of those produced in it the positive. Therefore, the physical meaning of the expressions on the right side, when (8.10) is substituted, is as follows: = AC P (8.12) represents the difference of the heat capacities (at constant pressure) of the gases produced and consumed in the reaction, or the total change of the heat capacity due to it. According to eq. (3.27), 138 TEXTBOOK OF THERMODYNAMICS VIII 51 c p hdT + UOH = *h is the heat function of the component ft. For this reason Q P (8.13) represents the total change of the heat function of the mixture AX which is equal to the heat of reaction Q p , as we know from eq. (3.23). The last term depending on the entropy constants of the constituents of the mixture, we shall denote by = R log L (8.14) We further introduce for the whole right side of eq. (8.11) the notation RT log K pt so that !<* Kp = / ^ dT - Jfr + ^g L (8.15) Within the range of temperatures, where the molal heats c p h can be considered as constant, this becomes log K P = ~2 log T - QL + i og /. (8 . 16) Strictly speaking, / is well defined only in this case, unless the lower limit of the integral in (8.15) is specified. The condition of equilibrium (8.10) takes, therefore, the form KW .../' = p. (8.17) An equivalent form is obtained by replacing the partial pressures by the molal fractions by means of eq. (8.06), ph = Xhp 9 xi*xf . */* = K, (8.18) where K = K p /p" or log K = log / - j, + dT - v log p t (8.19) with the abbreviation *=Vl + 9* + ... + 9r (8.20) The relation (8.17) between the partial pressures of the constituent gases or the equivalent eq. (8.18) between the molal fractions is known as the mass law. Whatever the constitution of the mixture may be, VIII 51 MIXTURES OF PERFECT GASES 139 at a definite temperature and pressure, the function pi*p* ... (or Xi n x 2 n . . . ) is constant. The quantity K p (respectively, K) is, therefore, called the equilibrium constant of the reaction. The mass law was first deduced by the Norwegian chemists Guldberg and Waage (1867) from statistical considerations which it will be instructive to reproduce here. Let us write the equation (6.46) of the reaction in the form vid + v 2 G 2 + . . . <=* v'iG'i + v' 2 G' 2 + . . . , where the symbols, on one side, refer to the substances entering the reaction and, on the other, to those resulting from it. In order that vi, i> 2 , . . . molecules of the substances GI, G 2 , . . . enter the reaction, they must all meet, i.e. be simultaneously present in a small volume TO. What is the composite probability of such an event? The prob- ability of one molecule of the kind GI being in the volume TO is pro- portional to the total number of such molecules available, which, in its turn, is proportional to the mol number N\. The probability of vi molecules being there simultaneously is proportional to N\ l (accord- ing to the rules expounded in section 30), provided that the molecules are statistically independent. This statistical independence implies that there are no forces of interaction of any kind between them, and this is true only for perfect gases. Since we have to restrict ourselves to this case, we may make use of the property of perfect gases that the mol number is proportional to the partial pressure and conclude that the above probability is proportional to pi* 1 . In a similar way, we find for the probability of v 2 molecules of the second component being in the volume T O the expression p<P> and so on. The probability of the composite event is, therefore, where the constant of proportionality a may depend upon the prop- erties of the system, especially its temperature and pressure. The reaction may, however, go also in the opposite direction producing Pi t v 2 , . . . molecules of the substances GI, G 2t ... It is necessary for this that i/i, 1/2, molecules of the substances G'i, G' 2 , . . . enter the reaction and meet in a limited volume. The probability for this is, obviously, P' = a'fi'yS* . . . In the state of equilibrium the number of molecules of every kind 140 TEXTBOOK OF THERMODYNAMICS VIII 52 is stationary and the process must go as often one way as the other. The probabilities P and P' must be equal (K p = a'/a) an equation which is identical with (8.17). The thermodynamical derivation given above is due to Gibbs (1871) and leads a good way beyond the result of Guldberg and Waage inas- much as it contains in eqs. (8.15) and (8.19) the explicit dependence of the equilibrium constant upon p and T. It is sufficient to make one measurement at a definite pressure and temperature in order to determine the constant /. Knowing /, one can predict the equilibrium conditions for all possible states of the mixture. Exercise 78. Find the form of the mass law (8.17) for the following reactions: N 2 + 3H 2 - 2NH, = 0; 2HBr + I, - 2HI - Br a = 0; 2O - O 2 - 0; 2O 8 - 3O 2 - 0. Exercise 79. Show that, in view of eqs. (4.36") and (3.26), the expression (8.15) can be reduced to (8.21) J JKJ. " which is consistent with (6.53). 52. Examples. Degree of dissociation. We have obtained two forms of the mass law in eqs. (8.17) and (8.18). The first form is simpler inasmuch as the quantity K p is a function of the temperature only. All the quantities on which eq. (8.17) depends are completely conserved if we add neutral gases to the mixture, because the partial pressures pi, . p 9 , as well as K p , remain unaffected by this. The second form has the advantage that it can be more readily generalized to include systems which are not perfect gas mixtures. The most interesting question is, usually, as to the fraction of the original substance or substances transformed in the reaction. Let us discuss, for instance, the oldest reaction to which this theory was applied by Gibbs, the dissociation of nitrogen tetroxide 2NO 2 - N 2 O 4 = 0. (8.22) The coefficients v are in this case v Nl04 1> ^NO* 2, v = 1. Suppose that No moles of N 2 C>4 are brought into a vessel at a very low temperature when it is quite undissociated : what mol numbers shall we observe after heating the gas to the temperature T at the pressure p? If we denote by { the fraction of the N 2 C>4 molecules which are dissociated (degree of dissociation) the mol number of this VIII 52 MIXTURES OF PERFECT GASES 141 component is ^ N2 o 4 = -^o(l ) To every dissociated molecule there appear in the mixture two molecules of NC>2. The mol number of nitrogen dioxide is, therefore, jV N o, = 2No, and the sum N = # Nl04 + WNO, = N (l + ) Recalling the definition (8.05) of the mol fractions and substituting into (8.18), we find K = K p /p. (8.23) I-? 2 We see from this that the degree of dissociation is small when K is small, i.e. according to (8.19), at low temperatures or high pressures. Under the opposite conditions of high temperature or low pressure, approaches unity. The degree of dissociation being known, it is easy to compute the density, which is also accessible to direct measurement. The density of the mixture A, obviously, stands to the density 5 of the undissoci- ated component N2O4 in the inverse ratio of the numbers of molecules: A/6 = No/N, whence A = Accurate measurements on the dissociation of nitrogen tetroxide are due to Bodenstein. 1 This gas can be regarded as approximately perfect only when the pressures are not too high and the temperatures not too low. From observations under these conditions Bodenstein derives the formula logic K p =- - ~ + 1.75 logio T + 0.00483r - 7.144 X 10- 6 r 2 + 5.943, (8.25) when the pressure is expressed in mm Hg. The agreement with obser- vations can be judged from Table 14. TABLE 14 CHEMICAL EQUILIBRIUM IN N 2 O 4 T(abs) 282. 5 285. 7 289. 5 293. 305. 9 323. 9 logio K p (obs.) 1.487 1.608 1.745 1.859 2.294 2.838 logio Kp(calc.).... 1.475 1.600 1.741 1.889 2.286 2.838 T(abs) 334. 8 342. 7 352. 5 361. 9 373. 9 387. 4 logio K p (obs.) 3.135 3.326 3.577 3.784 4.027 4.293 logio K p (calc.). ... 3.135 3.341 3.577 3.794 4.032 4.323 1 M. Bodenstein, Zs. phys. Chemie 100, p. 74, 1922. 142 TEXTBOOK OF THERMODYNAMICS VIII 52 Bodenstein's formula represents also fairly well the older measure- ments by Natanson. 1 Although not very accurate, they offer us exam- ples of how the degree of dissociation varies with the pressure at constant temperature. One of the series was measured at / = 49. 7 C, for which temperature (8.25) gives K p = 643. The normal density of N 2 O4 is d = 3.180, whence the actual density (reduced to normal) is calculated by means of eq. (8.24). TABLE 15 DISSOCIATION OF N 2 O 4 Pmm Kp/tp * A (calc.) A (obs.) 1.000 1.590 26.80 5.997 0.926 1.651 1.663 93.75 1.715 0.795 1.772 1.788 182.69 0.880 0.684 1.897 1.894 261.37 0.615 0.615 1.969 1.963 497.75 0.323 0.494 2.129 2.144 A reaction of great practical interest is the dissociation of water vapor 2H 2 + O 2 - 2H 2 O = 0, with the coefficients V H , = 2, v^ = 1, v Ht0 = 2, v = 1. We suppose again that we put into the reaction vessel JVo undissociated water molecules (at a low temperature). At a higher temperature dissoci- ation ensues and the mol numbers of the three components become Equation (8.18) gives, then, 1 *> P' (8.26) 2 (i - )2 (1 + f According to Siegel 2 the heat of this reaction is Q 9 - 113 820 + 2.6Sr - 4.41 X 10~ 4 r 2 + 1.252 X 10~ 6 r 3 -9.12 X 10- 10 r 4 + 4.36 X lO-^r 6 , (8.27) 1 E. and L. Natanson, Ann. Physik u. Chemie 24, p. 454, 1885. *W. Siegel, Zs. phys. Chemie 87, p. 641, 1914. The value of the constant 1.08 is due to Nernst and Wartenberg. VIII 53 MIXTURES OP PERFECT GASES 143 and the equilibrium constant 24900 logic K P = ~ + 1.335 logio r - 0.965 X 10~ 4 r + 0.137 X 10~ 6 r 2 - 0.665 X 10- 10 r 3 + 0.191 X 10- 17 r 6 - 1.08. (8.28) Nernst 1 gives the following comparison of observed data of the degree of ionization with those calculated from this formula (at atmospheric pressure) : TABLE 16 DISSOCIATION OF WATER VAPOR T (abs) 100$ (calc.) 100 (obs.) T (abs) lOOt (calc.) 100$ (obs.) 290 4.66X10- 2 ' 4. 6 - 4. 8 X 10~ 2 1705 0.107 0.102 700 5.4 X10- 7.6X10-* 2155 1.18 1.18 1300 0.0029 0.0027 2257 1.76 1.77 1397 0.0085 0.0078 2337 2.7 3.8 1480 0.0186 0.0189 2507 4.1 4.5 1500 0.0225 0.0197 2684 6.6 6.2 1561 0.0369 0.034 2731 7.4 8.2 3092 15.4 13.0 The agreement is remarkable in view of the width of the range and of the fact that the observations were obtained by many different observers and methods. Exercise 80. Find the equation for the degree of dissociation, analogous to (8.23) and (8.26), for the reactions: 3H 2 + N 2 = 2NH,, Hg + I 2 - HgI 2 . Exercise 81. Calculate for hydrogen iodide from eq. (8.30) at the pressure p = 100 mm and at the temperatures of Table 17. 53. Influence of the excess of one component. Another inter- esting reaction is the dissociation of hydrogen iodide H 2 + I 2 - 2HI - 0, ?H, = 1, "i, = 1, v - 2, v = 0. (8.29) The equation v = means that the number of molecules is not changed in the reaction. The equilibrium constant (8.19), then, does 1 W. Nernst, Theoretische Chemie, p. 775. llth edition. Stuttgart 1926. 144 TEXTBOOK OF THERMODYNAMICS VIII 53 not contain p, so that the molal fractions and the degree of dissociation are entirely independent of the total pressure. Suppose we bring into a vessel (at a low temperature) N\ mols of HI and N% mols of Ha and heat the mixture to the temperature T. This assumption will permit us to study the influence of the excess of one component (Ha) upon the degree of dissociation . As in the preceding case, N m = Ni(l ). Each dissociated molecule of HI produces half a molecule of Ha and la, respectively: therefore, JV Ia = %Ni and -^Ha = 2^1 { + A/a, while the sum, N = NI + NZ, is independent of the degree of dissociation. The substitution into (8.17) gives + -* -*' (8 ' 30) When the temperature is low, K is small and, with it, the degree of dissociation . We can take then for the denominator the approximate value 1 and see that the numerical value of greatly depends upon the term 2Nz/N{. When there is no excess hydrogen (N% = 0), we find = 2K*\ on the other hand, when 2Nz/Ni is not small, we can neglect , compared with it, obtaining = 2KN\/N%, a number which is much smaller: The excess of one of the products of dissociation seriously depresses its degree. When K is very large (high temper- atures), approaches unity. As an approximation, we can substitute = 1 in the numerator with the result 1 = (1 + 2N 2 /Ni)*/2K*. Qualitatively the presence of the term 2Nz/N\ has still a depressing influence. However, quantitatively sinks appreciably below 1 only when 2N%/N\ begins to approach K in its order of mag- nitude. The physical reason of this depressive action is, of course, that the excess of Ha gives to the iodine, produced in the dissociation process, a larger chance of reacting with hydrogen and of being transformed back into hydrogen iodide. This simple explanation makes it clear that we have here a general phenomenon which occurs, under similar conditions, in all cases of dissociation. The equilibrium constant of the reaction (8.29) is well represented by the formula 1 540 4 logic K = logio K p = -- - + O.S03 logio T - 2.350. (8.31) 1 Nernst, Zs. Electrochcmic 18, p. 687, 1909; K. Vogcl v. Falkenstein, Zs. phys. Chcmie72, p. 113, 1910. VIII 54 MIXTURES OF PERFECT GASES Hence and from (8.30) we obtain Table 17. TABLE 17 DISSOCIATION OF HI IN THE PRESENCE OF AN EXCESS OF H s 145 Nt/Ni = Nt/Ni - 1 r(abs) logio K 100$ (calc.) 100$ (obs.) 100$ (calc.) 600 -1.856 19.1 18.9 2.6 700 -1.691 22.2 21.5 3.7 800 -1.565 24.9 24.7 4.8 1295 - 1 . 202 33.3 32.9 9.8 1490 -1.117 35.6 37.5 11.3 The depressive action is very marked, indeed. Exercise 82. Calculate from (8.31) with the help of (8.21) the heat of the reaction (8.29) at 1000 K. Do the same for the reaction (8.22) by means of the formula (8.25). Exercise 83. Derive the generalization of the formula (8.26) when there is an excess of H a , and estimate its depressive influence. 54. Influence of neutral gases. In the absence of neutral gases the mol fraction of each component is given by the expression x k = N k /(Ni + ...+NJ. (8.32) On the other hand, if there are in the mixture other gases than those taking part in the reaction and the sum of the mol numbers of these neutral gases is NQ, the molal pressure of the same component becomes Ni+...+N.). (8.33) Comparing these two relations we find *>k = *k(Ni + . + N,)/(N + Ni + ...+NJ. (8.34) The expression of the mass law in the presence of neutral gases is in accordance with eq. (8.17) K **l or substituting (8.34) 146 TEXTBOOK OF THERMODYNAMICS VIII 55 The addition of neutral gases at P = const has, therefore, the same effect as an increase of the equilibrium constant. This influence is opposite to that of the excess of an active component because to a larger constant K corresponds a higher degree of dissociation. The presence of neutral gases in general advances the degree of dissociation, provided that the total pressure p remains the same. Of course, in the particular case v = 0, neutral gases have no effect. We see from the definition (8.19) of the function K that the same result could be achieved by reducing the total pressure p of the mixture in the proportion (Ni + . . . + N ff )/(N Q + . . . + N ff ). In a certain sense, the influence of the neutral gases is, therefore, only apparent. It comes from the fact that we refer the effect to constant total pressure to which both the active and the neutral gases contribute, so that the partial pressures of the active components are reduced in precisely the same proportion. This corresponds to the usual experimental conditions in which the total pressure p is the fundamental datum. If the question were put as to the degree of ionization in a box of constant volume, the introduction of additional neutral gases into that box would have no influence whatever on it. Exercise 84. What is the increase of volume of the gas mixture in a reaction in which the sum of the mol numbers increases by v? Use the equation of perfect gases (1.18) and check that the result is consistent with (6.51). 55. Mutual influence of two simultaneous reactions. Suppose that we mix at a low temperature N\ mols of hydrogen iodide (HI) and N2 mols of hydrogen bromide (HBr) and slowly heat the mixture. What will be its composition at a temperature T? In addition to the reaction (8.29), there can take place also the reaction of dissociation of hydrogen bromide H 2 + Br 2 - 2HBr = 0, J 1. ) "Hj = 1 "Bri = 1 ^HBr = 2, Accordingly there must be simultaneously satisfied two equations of the type (8.18) (8 37) 9 HI x HBr We can use the expressions of section 53 for the mol numbers of HI and of 12, in terms of the degree of dissociation 1 of hydrogen iodide: N m = Ni(l - 1), N lt = jAih- Similarly we can write (denoting the degree of dissociation of HBr by {2) : NSB T =* JVa(l {2), , since the two reactions are precisely of the same type. VIII 55 MIXTURES OF PERFECT GASES 147 As to the mol number of hydrogen, it comes from two sources: each dissociated molecule of both HI and HBr produces half a molecule of H2, whence JV H , = i(^ii + N 2 2 ). Substituting into (&.QS) and (8.37) 4 (1 i) 2 ' 4 (1 &) 2 It is apparent from these equations that the mutual influence of the two reactions results in the decrease of the degrees of ionization of both HI and HBr. The reason of this is easy to understand: either of the reactions liberates hydrogen and creates an excess of this com- ponent for the other process, exercising upon it a depressive action (compare section S3). In general, there exists still another mutual influence: If the total pressure is kept constant, the presence of the second gas (and of the products of its dissociation) decreases the partial pressure of the first and, in this way, produces the same effect of enhancing the dissociation as a neutral gas. We have already said in the preceding section that this effect is trivial: Therefore, we have selected here two reactions in which it does not exist because they are not pressure sensitive. They permit us to study the mutual depressive influence in its pure form. The equilibrium constant of the dissociation of HBr can be repre- sented as follows 5223 logio K 2 = - - + 0.553 logic T - 2.72. (8.38) At temperatures below 2000 K, 2 is, therefore, very much smaller than ft. If NI and N 2 are of the same order of magnitude, the disso- ciation of HI is not appreciably affected by the presence of HBr, and the values of 1 remain the same as in Table 17. On the other hand, the dissociation of HBr is greatly depressed, as is illustrated by Table 18. TABLE 18 DISSOCIATION OF HBr IN THE PRESENCE OF HI Ni = tfl - Nl T (abs) logio KI 100*2 100*, 600 -9.889 0.002 1.8 X 10- 1000 -6.284 0.14 3.4 X 10-* 1490 -4.470 1.16 0.12 148 TEXTBOOK OF THERMODYNAMICS VIII 55 Exercise 85. N mols of HI are brought in a vessel and heated to a high tem- perature where the reactions (8.29) and 21 - I* = must be taken into account simultaneously. Denote the degree of dissociation of HI by 1 and that of 1 2 by 2 . The equilibrium constant for the dissociation of I 2 is (Bodenstein, Zs. Electrochem. 22, p. 338, 1916): 7550 T K p - - ^~ + 1.75 log, T - 4.09 X 10-* T + 4.726 X 10~ 8 T - 0.440. Calculate gs under the above conditions and in pure It at the temperature 2000 K. CHAPTER IX DILUTE SOLUTIONS 56. The concept of a solution. We call a solution any homogene- ous system which consists of two or more chemically pure substances, no matter whether it is solid, liquid, or gaseous. Solutions must be distinguished from chemical compounds, on one hand, and (in general) from mixtures, on the other. A chemical compound consists of mole- cules of only one kind; therefore, the masses of the constituents of which it is formed stand in a definite ratio. From the point of view of the phase rule (section 41) it is a system with one single independent component or a pure substance: if the amount of one constituent is given, those of all the others are uniquely determined by it. It is quite different with the solution; the molecules or atoms of the con- stituents do not lose their individuality in it, and their amounts can assume any arbitrary ratio, within certain limits. The composition of the solutions being continuously variable, they must be regarded as homogeneous systems with several independent components. The difference from mixtures is most clearly brought out in the case of solid solutions. We know, at present, that a solid is always crystalline, and we say that two substances are mutually soluble, in the solid state, when they form a joint crystalline lattice, or a mixed crystal in which the lattice points are occupied, in part, by atoms (or molecules) of the one substance, in part, of the other. The solu- bility of solids is an exception, although several pairs of metals are known which are mutually soluble in all proportions, e.g. Ag Au, Ag Pd, Co Ni, Cu Ni, Cu Pd, Au Pt, Bi Sb, etc. The metals of each pair are of a closely analogous chemical nature belonging to the same group of the periodic system (or sometimes to two adjacent groups). Of non -elementary solids which are mutually soluble in all proportions we could mention AgCl NaCl, PbBr2 PbCb, SnCb PbCb, etc. More common is the case of limited solubility: the crystalline structure of a metal is preserved if a fraction of the lattice points (up to a certain limit) is replaced by atoms of another metal, usually from the same or the adjacent group of the natural system. The lattice cannot accommodate more than the limiting fraction without breaking down. Such cases are Cu Ag, Au Ni, 149 ISO TEXTBOOK OF THERMODYNAMICS IX 56 Bi Pb, Sn Pb, etc. Of course, limited mutual solubility occurs also with pairs of solids of a non-metallic nature: KNOa NaNOa, KNOa TiNOa, etc. All other cases, viz. those in which the con- stituents of an alloy do not form a joint crystal, must be classed as mixtures. For instance, the microscopical examination of Ag Pb alloys shows that they are agglomerations of tiny crystals of pure silver and pure lead. Similar conditions prevail with most other alloys (e.g. Bi Cd, Pb Sb, also KC1 AgCl, ice AgCl, etc.). We have seen in section 39 that, as far as the thermodynamical applications and conclusions are concerned, the size of the phases is immaterial; there- fore, an alloy or other mixture of this sort must be regarded as a sys- tem of two or more coexisting phases, even if these phases are of microscopical dimensions. Unlike a mixture, a true solution is homo- geneous and represents a single phase. The conditions in a liquid solution are similar. A liquid can, in general, dissolve other liquids, solids, or gases, the true solution being homogeneous and characterized by a uniform distribution over its whole volume of the molecules or atoms of all the constituents. In the case of liquids mutual solubility is not an exception but a common phenomenon. At the same time there are numerous pairs of liquids which exhibit in certain ranges of temperature a complete (e.g. water carbon tetrachloride) or a limited (water ether, water phenol, etc.) mutual insolubility. The true or molecular solutions must be distin- guished from the mixtures which in the case of liquids are called sus- pensions, emulsions, and colloidal solutions. Like the solid mixtures they are inhomogeneous and represent two or more phases intermixed in a finer or coarser degree of dispersion. In the case of gases, the difference between a solution and a mix- ture disappears, since all gases are mutually soluble or miscible, with- out restriction, forming a homogeneous system in which the molecules of all constituents are uniformly distributed. The process of forming a solution is accompanied by mechanical and thermal effects which show that there is a certain degree of inter- action between the molecules of the different constituents. When we dissolve ethyl alcohol and water in equal parts, there is a considerable contraction, the volume of the solution being about 5% smaller than the sum of the volumes of the free components (compare section 32). At the same time, a certain amount of " heat of solution " is devel- oped, because the internal energy of the solution is not equal to the sum of the internal energies of the alcohol and water forming it. Both effects are characteristic of the general case although their numerical values vary widely, so that the heat of solution may be 1X57 DILUTE SOLUTIONS 151 either positive or negative. There are certain pairs of liquids known (e.g. benzene-toluene) which form solutions with very little diminution of volume and development of heat. This usually occurs in the case of a close chemical resemblance of the constituents so that the inter- action between atoms of different kinds is nearly the same as that between atoms of the same kind. We can, therefore, imagine, as an ideal case, a solution in which the volumes, as well as the internal energies, are strictly additive and no change occurs in these quantities when the solution is formed. Following G. N. Lewis 1 we shall call it a perfect solution: although the perfect solution may not occur in nature in its pure form, it is a useful fiction as it will help us to bring out, theoretically, some typical properties of real solutions. 57. Dilute solutions. One of the components of a solution, usually the most plentiful is called the solvent, the others the solutes. In this chapter we shall discuss the case when the mol number of the solvent JVo is much larger than those of the solutes (Ni, N 2 , . . . N ). The thermodynamical properties of every system are determined by the characteristic functions, especially by the thermodynamic potential which, in turn, depends on the internal energy and the volume. A theoretical treatment of the problem requires, therefore, some knowl- edge about the analytical structure of the functions U and V, par- ticularly with respect to their dependence upon the mol numbers Nk of the components. We have seen in section 39 that the functions U, V, 5, $ are homogeneous of the first degree with respect to the mol numbers. There obtains, therefore, the equation analogous to (6.13) , T, No,... N.) = U(p,T, Taking for e the special value e = I/No, we find This equation (and a similar one for V/No) is quite general and does not contain any restrictive assumptions. However, we shall introduce now a restriction limiting the generality of the function /, in that we assume that it can be expanded into a convergent multiple power series with respect to the ratios NI/NQ, . . . N ff /No* U _Ni , .N. - WO + TT 1 + + Tf U, NQ NQ NQ 1 G. N. Lewis, J. Am. Chem. Soc. 30, p. 668, 1908, 152 TEXTBOOK OF THERMODYNAMICS IX 57 We do not claim that this condition is always satisfied, and we shall treat in section 115 a notable exception to it. However, the large majority of solutions do conform to eq. (9.02), and the present chapter will be devoted to this case. The coefficients u h , u h i, ... of the expres- sion (9.02) are functions of p and T independent of the mol numbers. In the analogous expansion of V/No we denote the coefficients by Vh, Vhi, etc. We call the solution dilute if, within the desired accuracy, the terms of second and higher order of the expansion can be neglected so that U and V are expressed by the linear forms [7 = NOUO + NM + ... + N ff u ff , V = NOV O + Ni vi + . . . + N 9 v w . (9.03) These equations contain, of course, the limiting case of the pure solvent (Ni = ... = N ff = 0), when they take the form U = NQUQ, V = NQVQ. The physical meaning of the coefficients UQ, VQ is, there- fore, the molal internal energy and the molal volume of the pure solvent. The mol number NO, representing the solvent, cannot be put equal to zero: therefore, the simplest form of the equations in which N h does not vanish is U = NQUQ + NhUh- It corresponds to a solution of only the component (ti) in the solvent. Consequently, we can say that the coefficients UH must depend on the properties both of the solute h and of the solvent. They may be called the molal energy and volume of the solute in the solution, but they will be, in general, different from the same quantities characterizing the solute in its free state. When another mol of the solvent is added to the dilute solution (9.03), its internal energy and volume become U' = (No + l)o + Niui + . . . + N u ff , V = (No + l)0o + Nivi + . . . + so that U' = U + wo, V = V + VQ. These are exactly the same values which the solution and the extra mol of the solvent have before they are brought together: there is no development of heat of solution and no change of volume when the solution is diluted still further. This additivity would not obtain if the second order terms of eq. (9.02) had to be taken into consideration: it can be taken as the experimental criterion of the diluteness of a solution. According to the definition of perfect solutions given at the end of the preceding section, the expressions (9.03) apply also to them. However, these solutions need not be dilute and the distinction between solvent and solutes disappears in them. Any of the compo- 1X58 DILUTE SOLUTIONS 153 nents can be regarded as the solvent and, therefore, all the coefficients u h , Vh are identical with the molal internal energies and volumes of the pure components. This fact guarantees the additivity of the func- tions U and V which is the essential characteristic of a perfect solution. 58. Entropy of dilute and perfect solutions. The differential of the entropy is defined by the second law of thermodynamics as dS = (dU + pdV}/T. The differentials dU and dV are taken for a definite composition of the system, so that the mol numbers N h must be regarded as constants with respect to the differentiation. Substitut- ing the expressions (9.03), we obtain - N h (du h + pdv h ) = N ds > ( 9 - 04 ) ft- if we denote, for short, ds k = (du h + pdv h )/T. With respect to s h we must repeat what was said in the preceding section with respect to u h , v h - The quantities s h are functions of pressure and temperature and may be called the molal entropies of the components while in the solution. They coincide with the molal entropies in the pure state only for the solvent and for the components of perfect solutions, and are different from them in all other cases. The total entropy 5, obtained by integrating eq. (9.04), will con- tain a constant of integration C, which is constant only in so far as it is independent of the variables p, T. However, it may depend on the mol numbers Nh (which are not variables as far as the integration is concerned), and it can be interpreted as the entropy of mixing the components a N h s k (p, T) + C(N Q , Ni,... NJ. (9.05) This expression is entirely independent of the state of aggregation of the system and applies to solid and liquid solutions and to gas mixtures. If we had a complete knowledge of the equations of state and of the energies involved, including the continuity of the states, we could give the explicit analytical form of the functions s h (p, T) cover- ing all cases : to go from one state of aggregation to another, it would be sufficient to change the value of the arguments p and T, as we do it in an inaccurate way, limited to pure substances, in the Van der Waals theory. These considerations show us that the form of the function C(N , . . . NJ determined for one state of the system will be valid for all other states. We did find, however, the expression of the entropy of a mixture of perfect gases: If we admit that all substances 154 TEXTBOOK OF THERMODYNAMICS IX 59 of nature have a finite vapor pressure, be it in some cases ever so small, there is the theoretical possibility of vaporizing any solution into a mixture of gases. Moreover, by reducing the pressure, these gases can be made to obey, with any degree of approximation, the laws of perfect gases. It follows from this argument that the expression (8.07) of the entropy of a mixture of perfect gases is only a special case of the formula (9.05) and must have the same analytical structure. This is, in fact, borne out by comparing them : there are in (8.07) terms linear in the mol numbers Nh which correspond to the sum in eq. (9.05), and terms containing the factors log x h = log (N h /N). The totality of the latter terms must, obviously, be identified with the function C 9 C(N , Ni, ... NJ = - RY] N h log x* (9.06) The possibility of this identification rests, entirely, on the hypoth- esis made in the preceding section that the functions U and V can be represented as the power series (9.02) and do not contain logarithmic terms. This assumption about the analytical form of U and V is, for the purposes of the present theory, equivalent to the conclusions which we shall deduce from it in the following sections, especially to Henry's law. Van't Hoff who originated the theory of dilute solutions based it on the validity of Henry's law. We prefer the above formulation (which is due to Planck) because, on the one hand, it is more general, being capable of extension by taking into consideration higher powers of the series (9.02). On the other hand, it paves the way to the under- standing of the reasons why in certain dilute solutions the laws deduced in this chapter are not satisfied (section 115). 59. General conditions of equilibrium. Equations (9.03) for the internal energy and volume, together with the entropy formula (9.05), (9.06), are sufficient to obtain the expression of the thermodynamic potential $ = U - TS + pV, viz. 9 * = 2 Nk(vk + RT log **> (9 - 07) ft-O the quantities <ph being defined as 9h = Uh Ts h + PVK, (9.08) they are functions of p and T, independent of the mol numbers N h . Being built up from the functions A, s*, VA, they have the properties which we have pointed out as pertaining to these quantities. We may call them the molal thermodynamic potentials of the components in 1X59 DILUTE SOLUTIONS 155 the solution. However, they are identical with the potentials of the pure substances only in the cases of the components in perfect solutions and of the solvent in dilute ones. In the case of solutes in dilute solutions they depend also on the properties of the solvent and are, therefore, different from the potentials of the same substances in their free state. The form (9.07) of the thermodynamic potential applies not only to dilute and perfect solutions but also to mixtures of perfect gases and to single phases of chemically pure substances. In fact, the com- parison with the expression (8.09) shows that, in the case of perfect gases, <ph must be defined as <p h = <p h - RT log x h = RT log p + u h (T). (9.09) It differs from ^ of (8.08) in that p is substituted instead of the partial pressure ph- On the other hand, a chemically pure substance, can be regarded as the limiting case of a solution without solutes (XQ = 1, log #o = 0). We can, therefore, apply our formulas to a heterogeneous system consisting of any number (a) of phases in equi- librium, each having one of these four characters: dilute solution, per- fect solution, mixture of gases, pure substance. If we designate the phase by an upper index j, the total thermodynamic potential of the system will be izr log **<*>]. (9.10) T^T ft^O The comparison with the general expression (6.14) shows that the partial thermodynamic potentials are generally $^ = <p^ } -{-RT log x^** We can apply to the last equation the general theory of equilibrium of section 42. The most general process possible in the system is given by eq. (6.48) The equation of equilibrium corresponding to it is, according to (6.50), r- Ct 9 1 A* - RT I ]T ]T v h (f) log xP - log K\ = 0, (9.12) L j-1 ft-0 J or a ff / j / ^ vt^ logXh^ = log K, (9.13) j-l /-0 156 TEXTBOOK OF THERMODYNAMICS 1X59 where the equilibrium constant K is an abbreviation defined by V, (9.14) which, in the particular case of a mixture of perfect gases, can be reduced to the form (8.19), as we shall see below. In application to dilute solutions the formula (9.13) is due to Van't Hoff. The left side of eq. (9.13) does not explicitly depend on temper- ature and pressure, while K is a function of p and T only, being inde- pendent of the mol numbers. An indication as to the form of the func- tional dependence of the equilibrium constant upon p and T is con- tained in eqs. (6.51) and (6.53) which give the partial derivatives of A$ with respect to T and p. Differentiating the expression (9.12) partially, we find (9.16) RT 2> where AF and Q p are the change of volume and the heat of reaction in the process characterized by eq. (9.11). These equations are often called rules of displacement of equilibrium, inasmuch as a change in the equilibrium constant leads to a displace- ment in the mol fractions of the components, according to eq. (9.13). The first of them expresses the following fact: When a system is compressed at constant temperature, the equilibrium is displaced in the direction of a decrease of volume. I.e. the less volu- minous components are enhanced at the expense of the more volu- minous. The other equation contains the rule: When a system is heated at constant pressure the equilibrium is displaced in the direction of heat absorption. Le. the components of higher heat function are enhanced at the expense of those of lower. This rule was first pointed out by Van t'Hoff and called by him the principle of mobile equilibrium. Exercise 86. From the definitions (9.08), (9.04), (9.03) of the quantities <?h, Sh, Uh, Vh check the statement that they satisfy the same differential relations as the corresponding functions in the free state of the substances, in particular that they obey eqs. (5.37). Having ascertained this, express A V and Q p in terms of t% and 5% from (9.15), (9.16), and (9.14). IX (50 DILUTE SOLUTIONS 157 60. Nernst's distribution law. Let us consider the particular case when the process (9.11) consists in the transfer of j>& (2) mols of the component h from the phase (2) into the phase (1), where they may form a different number of mols v h (l \ owing to association or dissocia- tion. It will simplify matters and lessen the chances of misapprehen- sion if we define both v h (l) and ^ (2) as positive and recognize the fact that the phase (2) undergoes a loss by writing and A (I) log* A < l) -^ (2) log**< 2 >=log^ = -[^W^-^W 2 ']/^. (9.17) Moreover, we shall often drop the superscript (2) to simplify writ- ing. With the abbreviation, ^=^ (1) /^ (2) =M^ (2) /M. (1) , (9.18) denoting, according to (6.42), the ratio of molecular weights, this becomes log [(xW/Xk] = (log^)A (2) = - [W n - * h (2) ]/RT. (9.19) In particular, when the process takes place without change of molecule, we have ^ (1) = Vh (2) = gh= 1 and /** (2) ) = log K h = - fo< - ^ (2) )/#r. (9.20) In many textbooks these equations are given only in the simpler form (9.20). Indeed, this can be justified on the following grounds. Even if the vapor of the component h has a different, say dissociated, molecule, the dissociation is theoretically never complete: according to Chapter VIII, there must exist a certain fraction of undissociated molecules, be it ever so small. Equation (9.20) gives then the equi- librium of the solution with the undissociated portion of the vapor and permits to determine its mol fraction, whereas the mol fraction of the main (dissociated) part of the component h can be obtained, in a secondary way, by using the theory of Chapter VIII. However, the more general eq. (9.19) is preferable from the practical point of view, as it gives directly the equilibrium with either of the two fractions. It permits one to describe by a single formula a much wider variety of observational results, a fact which more than compensates for its slightly more complicated form. Equations (9.19) and (9.20) are the fundamental relations in the theory of equilibrium of phases of dilute and perfect solutions. A num- 158 TEXTBOOK OF THERMODYNAMICS 1X60 ber of important laws can be derived from them. The second equa- tion can be written in the form *A a) /*A (2) = K h . (9.21) The ratio of the molal fractions of any component in two phases is a function of temperature and pressure only, provided the molecule of the component is the same in both phases. Suppose, for instance, that the phases (1) and (2) mainly consist of two solvents spread one on top of the other (as would be the case with water and carbon tetra- chloride). Let the same solute h be dissolved in both solvents: the quotient ^ (1) /^ <2) is then called the coefficient of distribution in the two phases. We see from eq. (9.21) that the coefficient of distribution is independent of the concentration of the solute in either phase, a law first pointed out by Nernst in 1891. The experimental data are, usually, expressed not in mol fractions but in mols of the solute per 1000 cm 3 of the solution, denoted by A^ Because of the dilution, Ah can be regarded as proportional to XH\ therefore, the ratio K' = Ah (1) /A h (2} must be also constant. As an illustration we may take iodine (12) dissolved in carbon tetrachloride (2) and in water (1) at 25 C. 1 TABLE 19 DISTRIBUTION RATIOES A ........ 0.02 0.04 0.06 0.08 0.09 0.10 #'0.5 ____ 85.1 85.2 85.4 86.0 86.4 87.5 In the general case, when the molecular weight of the solute is dif- ferent in the two solvents, eq. (9.19) gives (acny/scn. K I/V( *\ (9.22) For instance, benzoic acid has in benzene (1) a molecular weight twice as high as in water (j> (1) = 1, j/ (2) = 2, g = J). Another example is trichloroacetic acid (C2H ClaC^) dissolved in water (2) and in ethyl ether (1) at 25 C. The measured data are as in Table 20. 2 TABLE 20 A .............. 0.001 0.005 0.010 0.015 A .............. 0.0021 0.0048 0.0068 0.0083 15.0 14.7 14.7 14.7 1 Washburn and Strachan, J. Am. Chem. Soc. 35, p. 681, 1913; Linhart, ibidem, 40, p. 158, 1918. 1 Smith, J. Phys. Chemistry 25, pp. 605, 616, 1921. 1X62 DILUTE SOLUTIONS 159 It appears from this that g | or ji (1) = ZM^. The molecule is normal in the aqueous solution but has twice the normal weight in ether as a solvent. The results are only rarely as clean cut as this: in many solvents the association is partial, resulting in an effective molecular weight which is a non-integral multiple of the normal. Exercise 87. The following values were measured in the case of Brj dissolved in bromoform (1) and in water (2) at t = 25 C. 4 (1) = 0.125 0.25 0.50 0.75 1.00 100 X A = 0.193 0.382 0.750 1.11 1.47. Show that the distribution law is approximately satisfied. 61. Influence of a neutral atmosphere on the vapor pressure. We label the condensed phase (2) and the atmosphere above it (1), and we assume that the gases of the atmosphere are but little dissolved in the condensed phase so that it can be considered as pure (x^ 1), with a sufficient approximation. The fundamental eq. (9.19), applied to the vaporization of the condensed substance, gives log *,> = fak>/ft - n]/RT. (9.23) The potential <p h (l> relating to the vapor has the form (9.09) and contains the term RT log p which can be brought to the left side and combined with x h (l \ Noticing that x h ( "p = ph (l) (the partial pressure of the vapor in the atmosphere), we have log p k - W/ft - k(T)]/RT t (9.24) where w A (r) is the remaining part of ^ (1) and a function of the temperature only. The terms on the right depend only on the properties of the com-, ponent h and not on the rest of the atmosphere. In the case when the condensed phase is in equilibrium with its pure vapor, the left side goes over into log p (l) (where p (l) is the total pressure) while the right side remains the same. We see, therefore, that the atmosphere does not influence the vapor pressure: the partial vapor pressure in the atmos- phere is identical with the total vapor pressure in its absence. 62. Henry's law. In this section we shall apply the fundamental formulas (9.19) and (9.20) to the equilibrium of a liquid and a gaseous phase labeled, respectively, (2) and (1). The question we shall ask here is about the mol fraction x h of a dissolved component (gas) in the condensed phase if its partial pressure in the adjacent atmsophere is given. We may say that the condensed phase (2) is here preferred inasmuch as we are particularly interested in what occurs in it. It 160 TEXTBOOK OF THERMODYNAMICS 1X62 will be well to bring the preferred phase into prominence by dropping the superscript (2) relating to it: we shall write, therefore, XH instead of Xk (2 \ The process to be considered is the transfer of v h mols of the component h from the liquid solution into the gaseous phase where they may form v h (l) mols (because of dis- or association). By approximation we can neglect the change (in the process) of the volume of the condensed phase by comparison with that of the vapor. The quantity AF in eq. (9.15) denotes, then, simply the increase of volume of the gaseous phase due to VK (I) additional mols of the com- ponent h. According to eq. (1.18), this volume, before the beginning of the process, is V = (W (1) + . . . + N a ( ")RT/p\ after it takes place, the mol number Nh (l) is increased by VH (I} , the difference is, therefore, AF = VK^RT/p. (9.25) Of course, we suppose that N h (l) is very large compared with v h (l) so that there is no appreciable change in concentration during the process. (Or else the mol number changes are not the integers Vh themselves but infinitesimal numbers 5N h , proportional to v hl as in section 42). From the substitution into eq. (9.15) v- <> p a relation which can be integrated. Since T is regarded as a constant in the partial derivative, the constant of integration with respect to p will be independent of p but, in general, not of the temperature T: log K = - Vh log p + v h log k k (T), (9.27) where the second term represents the constant (or, rather, function of T). Applying this to the special case of eq. (9.20), with v h (l) = 1, we find If we denote by p h (1) the partial pressure of the component h in the gaseous phase, we can write # A (1) = p h (l) /p, according to eq. (8.06), whence x k = ph (1) /k h . (9.29) When the mol fraction of a component in a liquid (dilute or perfect) solution is varied at constant temperature, the partial pressure of its vapor varies in the same proportion, and vice versa. In particular, we can obtain from this equation some information about the solubility 1X62 DILUTE SOLUTIONS 161 of permanent gases in water and in other liquids: with neglect of quan- tities of the second order in N^/No, the amount of the gas absorbed by the solvent is proportional to #* (l) and, therefore, proportional to the partial pressure of the gas in the atmosphere over the solution. This law was established empirically by the English chemist William Henry (1803) and, accordingly, is known as Henry's law. The amount of the absorbed gas (denoted anph) is usually expressed in terms of the volume which it would occupy at C and 760 mm Hg. The quantity a h is then called Bunsen's coefficient of absorption (if p is given in mm Hg). The constancy of the coefficient a can be judged from the following data referring to the absorption of nitrogen (N2) in water at 25 C. 1 TABLE 21 HENRY'S LAW />Ar 2 (mm) 270 300 400 500 601.6 700 800 830 14.38 14.37 14.31 14.26 14.20 14.28 14.36 14.38 The trend of the dependence of kh upon T can be deduced from eq. (9.16). Substituting the expression (9.27) into it, we find d log k h Q ~- The latent heat Q of driving out an absorbed gas from the solution is positive. Therefore, kh increases as the temperature is raised, so that solvents absorb gases and vapors better when they are cold than when they are heated. As an example we give the values of k for (argon-free) nitrogen dissolved in water. 2 TABLE 22 tC 5 10 15 20 25 30 35 40 45 50 X10- 7 4.084.57 5.07 5.55 6.00 6.43 6.85 7.23 7.61 7.99 8.37 In the general case, when the possibility of association (or dissocia- tion) is taken into account, eqs. (9.19) and (9.27) give *A = (kk/P)*; p h ( "/x*' OK = ** (9.31) A case in point is the solubility of hydrogen in molten copper (at / = 1123 C). Let M be the mass of hydrogen, in mg, dissolved in 100 g of copper. 3 1 Drucker and Moles, Zs. phys. Chemie 75, p. 405, 1911. 2 1. C. T. from measurements by Fox (Trans. Faraday Soc. 5, p. 68, 1909). A. Sieverts and W. Krumbhaar, Zs. phys. Chemie 74, p. 294, 1910. 162 TEXTBOOK OF THERMODYNAMICS 1X63 TABLE 23 />(mm).... 281 403 606 775 883 971 1046 M ......... 0.380 0.443 0.549 0.610 0.680 0.705 0.745 p*/M .... 43.5 44.2 43.,7 45.1 45.1 45.3 44.2 Hence g = | or M = iM (1) the hydrogen dissolved in copper must be atomic. The constancy of p*/M has been ascertained for other gases dissolved in molten metals (62 in Ag, H2 in Ni and Fe, etc.). It is in keeping with this that the absorption increases with temperature. According to (9.30) this means that the process of driving the gas out of the metal is exothermic. In many organic solvents we find association of the gas molecules. However, it is usually incomplete, leading to irrational values of g*. Exercise 88. From the data about nitrogen in water and from eq. (9.30) calcu- late the approximate heat of solution Q. (Per 1 mol, v^ = 1). Exercise 89. Suppose that the same gaseous phase (1) is adjacent to two con- densed solvents. Equation (9.31) is then valid for either of the solvents. Show that the two relations so obtained are compatible with Nernst's law of distribution for the two condensed phases. 63. Raoult's law. The symbol p in eq. (9.31) represents the joint pressure of the liquid solution and of its vapor. According to our dis- cussion in section 41, it is completely determined when the composition of the system and its temperature are given. It is called, therefore, the boiling pressure of the solution under these conditions. Equation (9.31) is true for every component: let us apply it to the solvent (h = 0) of a dilute solution. (As in the preceding section the con- densed phase is " preferred/ 1 we shall emphasize it by dropping its superscript 2 while reserving the superscript 1 for the vapor phase). rn- - . <*> - MO/*> (I) ). (9.32) This relation holds for every composition and, in particular, for the limiting case of the pure solvent characterized by N\ =* . . . = N 9 = and#o = #o (1) 1. The pressure corresponding to this limiting case is, then, the boiling pressure p$ of the pure solvent. We can write, there- fore, 1 = PQ/HQ or po = feo whence This relation takes a very simple form when the solutes are non- volatile (e.g. most salts) so that their partial pressures in the vapor 1X63 DILUTE SOLUTIONS 163 are negligible. The gaseous phase can, then, be regarded as pure (*o (1) = 1) p/po = so 17 ' . (9.34) On the other hand, the mol fraction *o = No/N can be represented as 1 (Ni+ . . . + N ff )/N, or by approximation so _ 1 _(#! + ...+ NJ/No. (9.35) Whence, within the accuracy of the present theory which neglects terms of the second order in N h /No: PO gO No The presence of non-volatile solutes lowers the vapor pressure of a liquid. This fact was known to the chemists of the eighteenth cen- tury. In 1847 it was noticed by Babo that the relative lowering, (p Po)/Po> is independent of the temperature. A few years later (1858) Wiillner found it to be roughly proportional to the concentra- tion of the solution, and his observations were amplified by Ostwald. However, the most extensive investigation of this subject is due to Raoult (1886-1887), who established experimentally the law (9.36) according to which the relative lowering of pressure is determined solely by the mol fractions of the dissolved salts, being quite indepen- dent of the nature of the solvent and of the solutes. In earlier work (1878) Raoult also formulated the laws according to which the boiling and melting temperatures are changed by the presence of solutes. They form the subject of the next section. In the case of a single solute (1) eq. (9.36) can be written in either of the two alternative forms (9.37) , m go mipo 10 where m\ represents the number of mols of the solute in 1000 g of the solvent, called its molality. The second form follows from the first by substituting x\ = N\/No and noticing that in 1000 g of the solvent noNo = 1000 and N\ = mi . In the case of aqueous solutions MO (I)S=: 18: the quantity k m should, therefore, be numerically equal to 1.80. How- ever, in the tabulated data N\ is, of necessity, not the actual mol number of the solute (which is usually unknown) but "the gram- formula weight", i.e. the mol number as it would be if the solute were neither dissociated, associated, nor hydrated. The measured values 164 TEXTBOOK OF THERMODYNAMICS 1X63 of k m at low concentrations are for many substances close to 1.8, as appears from Table 24. TABLE 24 LOWERING OF THE BOILING PRESSURE Solute mi km Solute mi km HsPO 4 .. 1 1 80 Citric acid 1 1 90 Mannitol 1 1 775 Pb(C2H 8 O 2 )2 1 1 87 Sucrose 1-0 3 1 82 H 3 AsO 4 1 1 93 Glycerol 0.5 1 78 In general, electrolytic solutes have tabulated coefficients k m larger than 1.8 because they are often dissociated. Integral and half-integral multiples of 1.8 (K 2 S 2 O 3 3.65, Na 2 WO 4 3.64, Ba(NO 3 ) 2 3.55, etc,) are fairly common in them. On the other hand, organic solutes show mostly values of k m under 1.8, and this may be due to association. For non-aqueous solutions, the data are usually given in the first form (9.37). When go = 1 (the solvent does not associate), the formula requires k x = 1. This requirement is satisfied in a large number of cases, provided the concentration is sufficiently low, and even for values of #, from 0.05 to 0.10, it is often still approximately fulfilled. TABLE 25 LOWERING OF THE BOILING PRESSURE Solvent Solute Xi k x Benzene Naphthalene 0.005 1.02 Ethyl alcohol Benzil 0.001 0.99 Mercury * Ag 0.0025 1.01 4 * Tl 0.05 1.05 4 1 Zn 0.10 0.93 < 4 Au 0.05 0.55 *Hildebrand and Eastman, J. Am. Chem. Soc. 37, p. 2452, 1915. Of course, here too, examples of association (and dissociation) of the solute are not missing, as is indicated by the figure 0.55 relating to gold : this metal must have in mercury the molecule Au2. Exercise 90. In the last column of the following table are listed the measured values of the relative lowering of vapor pressure (po - p)/p* per 1 g of solute in 1000 1 of solvent. 1X64 DILUTE SOLUTIONS 165 SOLVENT SOLUTE M. W Water Mannitol 18 i < Glycerol 18 Br 2 BrI 179.8 Benzene Naphthalene 78.05 t W(pQ - p)/pQ 20 C 0.978 1.935 8.70 4.92 Ethyl alcohol Benzil 46 . 05 15 2.14 Calculate the approximate molecular weights of the solutes from formula (9.37). The accuracy is not very high. 64. Influence of solutes on the temperature and pressure oi transformations. Let the temperature and pressure of equilibrium of two phases of the pure solvent be TO, po. If small amounts of solutes are added to the two phases, the equilibrium temperature and pressure will be slightly changed, assuming the new values T, p but little dif- ferent from To, po- The function log K depends only on p and T, and we can expand it into the double Taylor series log K(p, T) = log K(po, To) + (p- po) neglecting terms of higher order. Equation (9.17) applied to the solvent (h = 0) gives i>o (1) log *o (1) - "o (2) log * (2) = log K(p, T). When the solvent is pure in both phases, the mol fr 'ty #o (1) = # (2) = 1 and the left side vanishes, so that log f *' For the partials of log K we can substitute from eqs. (9, ' 0^55 AF 'If/ weights of Ri*$ 1 7V/fc of eq. (9.43) We shall restrict ourselves to the foil from these data the vaporization of a liquid solution (2) conti - acid (MO 60.03). utes. Second, freezing of a liquid solutic not soluble in the solid solvent. In bot 1 b V the physiologist in our formulas, will be pure (*o (1 > = T on a semipermeable or freeze out while the solutes remain i , '' through the solvent phase is the preferred one, in the sense " and is denoted by P. we shall bring it into prominence b / dr ' er P a * of a cylindrical "*. For simplicity, we 166 TEXTBOOK OF THERMODYNAMICS 1X64 0(2) SBJ-Q we wr ite, as in the preceding section, l (N\+ . . . +N v )/No t whence log *o - - (Ni + . . . + NJ/No, log *o (1> - 0. (9.40) As we know from the theory of the phase rule (section 41), only one of the two variables T and p can be chosen arbitrarily when the composition of the liquid phase is given. We have, therefore, to dis- t nguish two cases in analyzing eq. (9.39). (1) We set arbitrarily T = TQ and find the difference of the vapor pressures of the solution and of the pure solvent, at the same tempera- ture To, RTo Nl + ' + N * The sign of the difference p po is opposite to that of the change of volume AF in the transformation. In the case of vaporization, we can use for A V the expression (9.25) and obtain with (9,18) the equation (9 . 42) (go = j>o (1> Ao) which is identical with (9.36). (2) We set p = po and obtain the difference of the boiling points f the solution and of the pure solvent, at the same pressure po. The (accc Miing of the heat of reaction Q p is here the latent heat of vaporizing Th, freezing VQ mols of the pure solvent; we shall denote it Q p = volo, aratinr % is the latent heat referred to 1 mol of the liquid phase. in the, rtdy* W1T.+ +* the solvef^ *o NO where A V 1 mo i O f t j^ of vaporization the latent heat is positive, in that of p&V-dNo t>\* ^here exists, therefore, an elevation of the boiling the solution wii '*' iing &* ni ' both due to the solutes - As its contact with ' \ s much larger in absolute va lue, the since its boiling pr, *o.^:derably less than the melting point. its original position ^ e quantities N * in formula ( 9 - 43 ) re P r e- Po is - RTgndN 1 ' lumbers (taking into account association, that dN mols of the r nd nOt the ^ ram " formula weights. The then on, condenaatV computed for every solvent. Usually, remainder of the ' ^ ^^^ ^ n ^^ ^ ^ s l ven t (normal corresponds to 1 md^f numerical values for several solvents the change of volur 1X65 DILUTE SOLUTIONS TABLE 26 167 T B B.P. elevation T M M.P. lowering Water 3731 0521 C 2731 186C Ethyl ether 307 6 2 18 1 79 Ethyl alcohol 351 6 1.17 Benzene 353 35 2.62 278 5 - 5 10 Chloroform 334 3 3 82 209 6 4 80 Acetone 329.2 1.76 Acetic acid 391 2 3 11 289 7 3 90 Aniline . . .... 457 5 3 61 266 1 5 87 Phenol 455 3 54 314 7 4 Nitrobenzene 484.0 5 73 278 8 7 This effect offers a convenient and powerful method for the deter- mination of unknown molecular weights of solutes which was of great help in the development of our chemical knowledge. The theoretical treatment of these phenomena was due to Van t'Hoff. Exercise 91. In the last column of the following table are given the measured boiling-point elevations corresponding to a concentration c of the solute in g pf 1000 g of solvent given in the third column SOLVENT Ethyl alcohol Benzene Acetone Aniline Phenol Acetic acid SOLUTE I: ! HgCla Benzil Benzil Picric acid c(g/1000g) 25.4 12.7 54.3 105.0 21.0 11.45 T-Ts 0?121 0.13^ 0..* i Calculate from Table 26 the opproximate mr* the solutes. Exercise 92. What is the relation between t' and the data of Table 26 which refer to molaliti latent heats for water (MO =18), benzene (MO = 65. Osmotic pressure. It was dis Pfeffer (1877) that solutes exercise 2 membrane which is impenetrable to t! freely. This pressure is called osmoti Suppose that the liquid solution fills vessel (Fig. 21) closed above by the 168 TEXTBOOK OF THERMODYNAMICS 1X65 shall assume (at first) that the solutes are non-volatile so that the vapor, in the upper part of the vessel, is free from them; later we shall drop this restriction. Suppose that we have in the vessel a second piston consisting of a semipermeable membrane of the sort just men- tioned whose initial position coincides with the plane border surface MN between the liquid and the gaseous phases. We push this piston a little down, in a reversible isothermal way, to the new position M'N', the volume between the planes M'N' and MN being dV. As a result, there will remain only pure solvent above the membrane M'N': It is a process for recovering a certain amount of solvent, say dNo mols, from the solution. Since the volume V of the solution is a function of TVo, N^ . . . N ff , while only NQ changes, we have FlG. 21. -Calculation b of osmotic pressure. dV = (dV/dNo)dNo = VodN , where #o is the partial molal volume of the solvent (compare section 32). In pushing the membrane down, work must be done against the forces of osmotic pressure P to the extent DW = -PdV=- PVodNo, (9.44) >rding to section 7). re is, however, another reversible and isothermal way of sep- ^'^Vo mols of the solvent from the solution. This can be done * wing two steps: (1) By moving the upper piston up, infi- and at constant temperature, we vaporize dNo mols of ^he change of volume of the system is, then, AV-dNo, s the change corresponding to the vaporization of . In this expansion the system does the work r pn-ssure of the solution. (2) We cover ly impermeable membrane. Having lost the vapor is superheated at the pressure p '> p). As we push the piston down into done till the pressure assumes the value iccording to eq. (3.32), and considering ^spond to godNv mols of the gas. From the pressure stays constant, and the ck is po&VodNo, where AFo again vent. If it is permissible to neglect id, compared with that of the vapor, IX 65 DILUTE SOLUTIONS 169 and to use for the gaseous phase the laws of perfect gases, AF=oAFo: the total work done in both steps of the process becomes DW = - RTg Q dNo log (po/p). (9 AS) In the beginning and the end of this process the system is precisely in the same state as in the other method of separation carried out with the semipermeable membrane. Moreover, both processes are isother- mal, therefore, DW' = DW, since both elements of work are equal to the change of the function ^ between the initial and the final states (compare section 34). Equating the expressions (9.44) and (9.45), we obtain g*. (9.46) We have assumed, for simplicity, that the solutes are non-volatile. However, this restriction is not essential and can easily be dropped: we need only imagine from the start a semipermeable membrane (opaque to the solutes) at the surface of the solution. This device permits us to have the pure vapor of the solvent in equilibrium with a solution containing volatile solutes, since the vapors of the solutes cannot penetrate the membrane. We can apply the reasoning leading to eq. (9.46) to the general case. This equation remains valid with the slight change that for p must be substituted the partial pressure of the solvent in the gaseous phase pxv (l) : (9.47) The osmotic pressure depends on the ratio of the partial vapor pres- sures of the solvent, in its pure state and in the solution, being propor- tional to the logarithm of that ratio. Equation (9.44) permits another interpretation of the osmotic pressure: It has the form (2.02) of the element of work in the case of generalized forces. The differential dNo represents the change in mol number of the solution due to the removal of solvent. Pvo is, there- fore, the force with which the solution resists (because of the negative sign) such a removal. In other words, it can also be regarded as a tension or suction pulling the solvent back into the solution across the semipermeable membrane. Pfeffer's method of measuring the osmotic pressure is based on this interpretation (Fig. 22) : A glass tube, \xJiO8e lower end is closed by a semipermeable membrane MN (made of cop- 170 TEXTBOOK OF THERMODYNAMICS 1X65 M B H 2 FIG. 22. Measure- ment of osmotic pressure. per ferrocyanide), is partially immersed in a beaker filled with pure water. When cane sugar is added to the water inside the tube, the solution in it rises because it sucks in water from the beaker. The rise stops when the pressure of the column AB is exactly equal to the force of suction, i.e. to the osmotic pressure P. This pressure is by no means small: In the case of more concentrated solutions, it is better to connect the tube with a manometer because the column becomes incon- veniently high. In fact, pressures up to 268 atm have been measured in sugar solutions. The fol- lowing familiar demonstration experiment shows how powerful the osmotic pressure can be. A part of an ox bladder is formed into a bag, filled with a strong aqueous solution of ethyl alcohol, and tightly sealed. The bladder is semipermeable in that it holds back alcohol but lets through water. When the bag is immersed in a water-filled vessel, the water is drawn into it with such force that the bladder cannot stand the strain and bursts within a short time. Let us turn to the general case when there are solutes on both sides of the membrane permeable only for the solvent. Both solutions exercise, then, a sucking action upon the solvent, and the resultant is a tension equal to the difference of the osmotic pressures, directed towards the side of higher concentration. Two solutions of the same osmotic pressure leave the solvent in equilibrium and are called isos- motic or isotonic. On these facts is based a method for determining the osmotic pressure within living cells, whose integuments are natural semipermeable membranes. When a living cell is placed in an aqueous salt solution of higher osmotic pressure than its own, it begins to lose water and to shrink. When the osmotic pressure of the medium is lower, it swells. Only in solutions approximately isosmotic with its interior does it retain, under the microscope, its size and general appear- ance. By this method of finding a solution isosmotic with them, the osmotic pressures in many physiological cells have been determined. In plant cells they are sometimes as high as 14 atm. In general, the vegetable and animal cells are nearly isosmotic with the natural saps and animal fluids in which they live. For instance the red corpuscles of the human blood and most other cells of the human body have the same osmotic pressure as the normal blood serum, which in turn is isosmotic with a solution consisting of 0.156 mol of NaCl per 1000 g of water. The harmful effect of certain diseases is believed to be due to 1X66 DILUTE SOLUTIONS 171 the fact that they upset the osmotic equilibrium of the body fluids and cause a destruction of cells similar to the breaking of the ox bladder in the above-mentioned experiment. 66. Van t'Hoff's equation for osmotic pressure. The expression for osmotic pressure (9.47) takes a simple form in the case of dilute and perfect solutions. On one hand, the partial molal volume then becomes identical with the ordinary molal volume of the solvent: t/o = VQ. On the other, we may substitute for o/p#o (1) from (9.33) and obtain p=-^log*o, (9.48) VQ an expression which is valid both for volatile and non-volatile solutes. If the solution is dilute, log XQ takes the form (9.40) and the volume NQ VQ of .the solvent can be replaced by the volume V of the solution (V = NQVQ), whence PV (Ni + . . . + N ff )RT. (9.49) This is Van t'Hoff's equation for osmotic pressure, which is formally identical with eq. (1.18) for a mixture of perfect gases. By specializa- tion, we can obtain from it the partial osmotic pressures of the indi- vidual solutes: if only the component h is present in the solution, P h V - NiJRT, or P h x h RT/v Q . (9.50) It follows from (9.49) that the partial osmotic pressures are addi- tive P = PI + . . . + P . In other words, the solutes in a dilute solution behave like perfect gases. The experimental data are usually expressed in terms of the molality f i.e. mol number of the solute per 1000 g of the solvent, which is denoted by m h . For aqueous solutions, this reduces to NQ VQ 1000 cm 3 and, together with eq. (9.50), gives P h - RTm h /lQQQ, or numerically for C, P k - 22.4m*atm. (9.51) A solution of the molality m = 1 is called normal. We see that the normal solution has a theoretical osmotic pressure of 22.4 atm. If the molality is computed from the gram-formula weight, eq. (9.51) can give a correct result only barring association, dissociation, and hydration. A few of the experimental values for dilute aqueous solu- tions (directly measured with copper ferrocyanide semipermeable mem- branes) are given in Table 27. The values of mi listed in this table are not corrected for association, etc. 172 TEXTBOOK OF THERMODYNAMICS 1X66 TABLE 27 OSMOTIC PRESSURES Solute mi P (atm) P/mi Arnygdaline 0.0219 0.474 21.7 Antipyrine 0.0530 1.18 22.3 Resorcinol 0097 199 20 5 Saccharin 0.0029 070 24.3 Glucose 0.555 13.2 23.8 Mannitol . ... 549 13 1 23.9 When the solution is not dilute we have to fall back on the general eq. (9.48). For aqueous solutions, vo = 18.015 and the coefficient has the numerical value (at C) P = - 2865 logio * atm. (9.52) This formula can be tested for sucrose. 1 It is known that sucrose forms a hydrate, each molecule attaching five molecules of water. The molality of pure water is 1000/18.015 = 55.51; consequently, in the sucrose solution it is wo = 55.51 5wi and l/#o = l+wi/(55.51 5wi). The values thus obtained are listed in Table 28. TABLE 28 OSMOTIC PRESSURE OF SUCROSE P (calc.) . . . P (obs.) . . . 0.1 2.24 2.25 0.3 6.90 6.91 0.6 14.12 14.22 1.0 24.4 24.76 2.0 53.5 54.9 3.0 89.0 90.0 4.0 132.5 129.5 The agreement is surprisingly good and shows that the sucrose solu- tion can be regarded as perfect, as far as this calculation is concerned. \ In a perfect solution the difference between solvent and solute dis- appears. It is better, therefore, to avoid the word " solvent " and to speak instead of the " component " without making any restric- tions as to its mol number. For a binary solution the interpretation of the preceding section can, then, be stated as follows: the partial osmotic pressure PI represents the force with which the other component is held in the solution (and vice versa). According to (9.48) it is proportional to the logarithm of the mol number of that component. If we gradually remove it from the solution, by the mechanism of 1 Lord Berkeley and Hartley, Proc. Roy. Soc. (A) 92, p. 477, 1916; H. L. Cal- lendar, Proc. Roy. Soc. (A) 80, p. 466, 1908. 1X66 DILUTE SOLUTIONS 173 Fig. 21 or by some other method, the difficulty of the removal con- tinually increases and becomes prohibitive in the end, because the last traces are held with an infinite force. This impossibility of removing the last vestiges of any component from a phase is quite general, as can be seen, in the simplest way, directly from the fundamental equation (9.17) which applies to this case as the solution is then certainly dilute with respect to this com- ponent. Log K is, by its definition (9.14), a finite number: therefore, Xh (l) cannot become zero. Otherwise the left side of eq. (9.17) would become infinite, unless J0 (2) were zero at the same time. This means that a component cannot be completely absent from one phase of a system if it has a finite concentration in another phase. All parts of our earth have been in direct or indirect interaction for immense periods of time. Consequently, every sample of terrestrial material must contain minute quantities of all known substances, and an abso- lutely complete purification of it is impossible. However, the second interpretation of the osmotic pressure here used should not obscure the fact that it is not an abstraction, but, in every way, a real pressure like that of a gas. After all, the only tangible thing about any pres- sure is that it is able to resist the motion of a piston or to set it in motion, and there is no difference between the osmotic and the ordi- nary pressure in this respect. We shall gain some insight into the deeper nature of the osmotic pressure in section 146 but shall bring out in this chapter the analogies between dilute solutions and perfect gases. We have already seen that the osmotic pressure has a tendency towards uniformity: if the concentration of a solution is non-uniform, the local differences of osmotic pressure set up diffusion currents which make it uniform (barring gravitational fields, compare section 108). In this process some mass of the solute (K) may be transferred from a place with the osmotic pressure Ph to one with P' h (in an isothermal way). The work done in this transfer must have in a dilute solution the same expression (3.32) as in a perfect gas, in view of the identity of the equations of state. Thus we obtain (per mol of the solute) w = RT log (Pk/P'd = RT log (x k /x' k ). (9.53) Exercise 93. What are the osmotic pressures of the following solutions, sup- posing that they satisfy the formula (9.51): (a) antipyrine (CiiHi 2 N 2 O, n 188.11) 10 g in 1000 g of water; (b) saccharin (CyHfiNOaS, M = 183.1 1), 2 g in 1000 g of water; (c) glucose (CHi 2 O, /* = 180.09) 100 g in 1000 g of water? Exercise 94. What are the osmotic pressures of sucrose solutions (CuHtsOu, M = 342.17), according to the formula (9.52) and taking into account hydration aa in Table 28: (a) 250 g, (b) 500 g, (c) 1000 g in 1000 g of water? 174 TEXTBOOK OF THERMODYNAMICS 1X67 Exercise 95. Calculate numerically from eq. (9.53) the work (in calories) that must be expended to reduce the mol fraction of a solvent in the ratio 1:10, at 20 C. 67. Saturated solutions. Solubility. In general, a solvent can take up only a limited amount of a solute at a given temperature. For instance, when we gradually add a salt or other solid solute to water, it dissolves at first, increasing the strength of the solution. However, when a certain limiting concentration is reached, further additions do not go into the solution but form a solid precipitate at the bottom of the vessel. Such a solution is called saturated, and the solute in it is in equilibrium with its solid crystalline phase. The behavior of liquid solutes is analogous, but we shall restrict ourselves in this section to solids, postponing the discussion of liquids until the next chapter. The mol fraction x, of the solute in the saturated state is sometimes called its solubility. More often, however, the solubility is defined as the molality m, of the solute in the saturated solution. If we denote by mo the molality of the solvent (mo = 1000/Mo), , = w,/(mo + m,). (9.54) (For water mo = 55.51). The solubilities of some of the more common solutes in water, at C, are as follows: TABLE 29 SOLUBILITIES Solute m. Solute m, Solute m. NaCl 6.10 KClOs 270 CuSO 4 . 88 NaHCOs 0.82 CaCh 5.35 FeSO 4 1 030 NaOH ... 10 50 Ca(OH) 3 . 0239 Sucrose 5 36 KC1 3.76 CaSO 4 01292 Lactose . . . 345 KOH 17.3 * AgNO 8 6 65 Maltose . . 1 70 It should be mentioned that the table is meant to illustrate merely the general notion of saturation. The substances listed in it are for the most part " strong electrolytes ", they do not obey quantitatively the theory of dilute solutions of this section but need a correction which will be given in section 115, Moreover, the numbers m, listed in it present the " gram-formula weights " of the undissociated and unhy- drated solutes. 1X67 DILUTE SOLUTIONS 175 Returning to the theory, we shall assume that the solvent is not soluble in the solute (as is the case with most solids) so that the excess of the solute represents a pure phase (xh (l) = 1). In applying to this case of equilibrium the fundamental eq. (9.17), we label the excess solute (1) and drop the superscript for the liquid solution -^log*M = logK. (9.5S) Both phases are here condensed, so that the change of volume in the process of solution is small and may be neglected in the first approximation (AF = 0). According to eq. (9.15), this means that K is independent of the pressure and is a function of the temperature alone: K = K(T). At the same time the heat, Q p = Qv, developed in the process is entirely due to the difference of the internal energies in the two states Q p = A 7 because, within this approximation, the work done by the system is negligible. If we denote by Aw* the difference of internal energies per 1 mol of solute, Q p = ^Aw*, and we find, from eq. (9.16) d log X* Aw* (9 ' 56) or substituting for Xk the expression (9.50), we obtain d log P h _ Aw* + RT _ l h dT ~ RT* ~~RT* (9.57) This relation is formally identical with the approximation (7.07) of the Clapeyron-Clausius equation. The process of solution of a salt is completely analogous to the process of vaporization, and the osmotic pressure PK of the saturated solute corresponds to the boiling pressure ps of the vapor. In fact, the quantity k in eq. (9.57) has the same structure as the heat of vaporization: According to (3.21) and (3.24), / = Aw + p&v, or, when the volume of the condensed phase is neg- lected and the vapor regarded as a perfect gas, / = Aw + RT. The heat of solution Aw* is almost invariably positive. Therefore, we conclude from eq. (9.56) that the solubility of a salt increases with temperature, unlike the solubility of a gas (compare section 62). Our assumption that the volume does not change is only a first approxima- tion, in reality AF has, usually, a very small but finite value. Conse- quently the effect of added pressure consists in a very slight increase or decrease of solubility. 176 TEXTBOOK OF THERMODYNAMICS 1X68 Exercise 96. The solubility of many aqueous solutes is well represented by the formula 1 0.052234 logio- +B. FORMIC ACID ACETIC ACID GLYCEROL MANNITOL Range.. 7.Oto8.4C -27 to 10 -1.3 to 18 40 to 100 A 12090 14980 17150 2142 B -2.243 -2.70 -3.078 -2.102 Calculate from (9.56) the approximate heat of solution AA within the range, assuming that there is no dissociation or hydration. 68. Chemical equilibrium of solutes. We suppose that chemical reactions are possible between the components of a dilute solution, according to eq. (6.46) *>oGo + vid + . . . + v 9 G 9 = 0. (9.58) The general equation of equilibrium (9.12) takes then the form VQ log #o + v\ log x\ + . . . + v 9 log x ff = log K. (9.59) Since we restrict ourselves to dilute solutions, the volume is prac- tically determined by the amount of the solvent, and its change in the reaction (9.58) is very small. As in the preceding section, we conclude from this that K is very little dependent on the volume and with a good approximation can be regarded as a function of the temperature alone, K = K(T). Moreover, we can replace x h = N h /N by N h /No, because the difference would amount only to terms of the second order in the expression (9.59) which are systematically neglected in the theory of dilute solutions. In application to the solvent, this means that we can replace XQ by 1, obtaining from the last equation si-W... x." = K(T), (9.60) without regard to whether the solvent takes part in the reaction or not. Because of the relations (9.50) this is equivalent to P^Pa* 1 . . . P/ r = K P (T) = K[), (9.61) \ VQ / (v = v\ + V2 + . . + O- This equation is formally identical with eq. (8.17) for a mixture of perfect gases. However, the conditions under which the equilibrium takes place are different in the two cases: The mixture of perfect gases is usually observed at a given pressure p which is an important factor 1X68 DILUTE SOLUTIONS 177 in determining the partial pressures ph, when only the mol numbers are known. On the other hand, the partial osmotic pressures P* of the solutes are completely determined by N h /No = WA/WO and have noth- ing to do with the pressure. This is brought out clearly, if we write the condition in the third form, resulting from (9.60) mi'W 1 . . . w/' K m = K(r).m v (9.62) (We recall that WQ is the molality of the solvent: wo = 1000 /Vo) The relation between the molalities depends only on the temperature and on the atomic weight of the solvent. As in the case of gases, the most interesting question is the deter- mination of the degree of dissociation. Let us take as an example the same reaction (8.22) which we discussed in section 52, the dissociation of nitrogen tetroxide (dissolved in chloroform). The expressions for the numbers VH and Nh remain the same as those given there. Sub- stituting them into (9.60), we obtain for the degrees of dissociation 2 K**. (9.63) where m\ is the molality of N2CU originally dissolved. Measurements on the dissociation of N2O4 in chloroform were car- ried out by Cundall. 1 Lewis and Randall 2 give the following table calculated from his data (for C). TABLE 30 CHEMICAL EQUILIBIRUM OF NITROGEN TETROXIDE #N204 #N02 10 & 0.018 0.00010 49 0.037 0.00012 35 0.050 0.00015 43 0.066 0.00019 52 0.125 0.00029 66 The constancy of K is not very good owing, perhaps, to the insuffi- cient accuracy of the colorimetric determinations of # N02 . Of course, the degree of dissociation is quite different from that which gaseous nitrogen tetroxide has under the same conditions: KP comes out here about 2.00 X 10" 4 atm, while (8.25) gives 0.020 atm for the gas reaction at C. This is not surprising since the heats of reaction are 1 Cundall, J. Chem. Soc. 50, 1076, 1891; 67, p. 794, 1895. 1 Lewis and Randall, Thermodynamics, p. 303, New York, 1923. 178 TEXTBOOK OF THERMODYNAMICS 1X68 very different in the two cases: in the chloroform solution there must be added the difference of the heats of solution of the two components. For very high dilutions (mi/wo 1), approaches unity. This is a general property of dilute solutions, as we can readily see from eq. (9.64): Whenever the dissociation leads to an increase in the number of molecules (v > 0), dilution (i. e. increasing NQ) has the same effect as increasing K (compare section 52). It advances the disso- ciation which becomes complete in the limiting case of extreme dilution. This influence of the relative amount of the solvent upon the degree of dissociation is the analogue of the effect of neutral gases upon a gas mixture discussed in section 54. The addition of neutral solutes has, according to eqs. (9.61) and (9.62), no appreciable effect upon the equilibrium. The reason for this is that in a dilute solution the quan- tity of a solute is, necessarily, small and its contribution to the avail- able volume negligible. On the other hand, the remarks made in the sections 53 and 55 about the depressing influence of the excess of one active component and about the mutual interdependence of two simul- taneous reactions remain fully valid in the case of solutions. Exercise 97. Calculate the factors RT/v Q and w entering into KP and K m for the following solvents: water (MO 18, density d = 1.00), chloroform (MO - 119.38, d - 1.526), ethyl ether (/*o - 74.08, d - 0.736), benzene (MO = 78.05 d - 0.900). Exercise 98. Carbonic acid and acetic acid in aqueous solutions dissociate into ions according to the following equations: H+ + HCOa" - H 2 CO, (aq) =0, K m - 1.84 X lO" 6 , H+ + C 2 H 8 Or - C 2 H 4 O 2 (aq) - 0, ** 3.50 X 10' 7 . Calculate the degrees of dissociation for the molalities mi (of the acid originally dissolved) 0.001/0.01, 0.05, 0.10, 0.20. Exercise 98'. Show that the equation of equilibrium (9.62) can be also written in the form ' (9.64) CHAPTER X EQUILIBRIUM OF BINARY SYSTEMS 1 69. General remarks. The character of the equilibrium between different phases of a system with two independent components (binary systems) is of great importance in chemistry and metallurgy. A vast body of experimental data, pertaining to it, has been accumu- lated in these branches of science. The role of thermodynamics is to help in the explanation and classification of this material by enumerat- ing the theoretically possible types of equilibrium. Of necessity, much of the information derived from thermodynamics is qualitative: The problem consists in treating the equilibrium between solid, liquid, and gaseous binary solutions (in the sense of section 56), but we do not* possess the theoretical knowledge of the condensed phases which would be necessary for a quantitative treatment, viz. the equations of state and the energy expressions for the solid and liquid state. Fortunately there exist, if not real, at least ideal systems in which the theory can be completely carried through, namely, the simplified models which we called perfect solutions in section 56. The essence of this approximation is that the interactions of the components are taken into account only in so far as they affect the mutual solubility. Apart from this, each component is treated as if it were alone. We propose, therefore, to work out, for every type of equilibrium, the formulas and curves pertaining to the case of perfect solutions. Comparing them with the experimental curves, we shall see that this model is not oversimplified and gives a good account of the main features of real systems, giving examples of every type of diagram that occurs in experiments. According to the discussion of section 41, the maximum number of phases which can be simultaneously in equilibrium exceeds by two the number of independent components: for binary systems it is 4. Four phases can coexist only at the " quadruple point " for which all the variables of the system (viz. temperature, pressure, and composi- tion of the phases) are perfectly determinate. In the case of the 1 This chapter may be skipped without loss of continuity. 179 180 TEXTBOOK OF THERMODYNAMICS X70 coexistence of three phases the system has one degree of freedom; i.e. one of the variables can be chosen arbitrarily. For experimental rea- sons, this is, usually, the pressure : when p is given, the whole system is determined, so that equilibrium can take place only at one definite temperature and for definite compositions of all the three phases. The most interesting case is, however, that of the coexistence of two phases, because it was shown in the general theory of Chapter VI that the other cases can be reduced to it. Then, two of the variables can be chosen at random: for instance, we can prescribe the temperature and the composition of one phase, so that equilibrium is possible at only one definite pressure. Or else, we can select arbitrarily the pres- sure and the composition of one of the phases, and so completely determine the equilibrium temperature. The parameter which determines the composition of a phase is the mol fraction of either component. If we indicate the components by the subscripts 1 and 2, and the phase by the superscript (/), we find from the definition (8.05) of the mol fraction Xl <* + * a < = 1. (10.01) In most of the applications of this chapter we shall suppose that the components do not interact chemically. In this case, there is no difference between the independent components and the actual constituents. A few remarks about ternary systems will be made in section 75. 70. Vapor pressure of binary systems at constant temperature. The available experimental data have reference to the equilibrium of liquid binary solutions and their vapors, because the vapor pressures of solid solutions are too low to be conveniently measurable. There- fore, we shall now restrict ourselves to the liquid-gaseous equilibrium and we shall further assume that, in the liquid phase, the components are mutually soluble in all proportions. We shall indicate the mol numbers of the two components, in the liquid phase, by x\, #2 and, in the vapor, by x'i, #'2, and shall apply the relations derived in the pre- ceding chapter for perfect solutions. In a perfect solution either component can be regarded as the solvent and obeys Raoult's law. If the molecules remain unchanged in the process of vaporization, this law must be used in the form (9.33), with go = 1, px'i = PBI xi, px f 2 = pB2 x 2t (10.02) where PBI and pB2 denote the boiling pressures of the two components X70 EQUILIBRIUM OF BINARY SYSTEMS 181 in their pure state. Adding these two equations and taking into account the relations (10.01) we find P = + (10.03) The plot of the pressure against the mol number x\ in the liquid phase is a straight line (Fig. 23). We can express the pressure also in terms of #'i, the mol number in the gaseous phase. Dividing the eqs. (10.02) by psi, pB2 respectively, and then adding them, pBlpB2 P '- pBl(l - (10.04) This corresponds to the curve of hyperbolic character labeled 11 vapor " in Fig. 23. This figure represents the so-called px-diagram: i.o 1.0 1.0 FIG. 23. FIG. 24. FIG. 25. Boiling pressure as a function of composition. when the composition and pressure of the liquid phase are given by the point A (on the " liquidus " curve), the constitution of the vapor in equilibrium with it is obtained by drawing a horizontal to the point B of intersection with the "vapor" curve. When we turn to the experimental ^-diagrams of pairs of organic and inorganic liquids, such a large proportion of them is found to be of the general type of Fig. 23 that it is justifiable to call it the normal type. It is true, the representation is not quantitative in that the " liquidus " locus geometricus is, generally, not straight but also curved. But it is typical in that both curves are monotonic and the vapor pressure of the solution intermediate to the boiling pressures of the pure components. Especially, pairs of chemically closely related substances (benzene-toluene, hexane-octane, methyl and ethyl alcohol, etc.) belong to this type. We have mentioned in section 56 that such pairs must be expected to approximate the behavior of perfect solu- tions. 182 TEXTBOOK OF THERMODYNAMICS X70 However, there exist two other types of (p, jc)-diagrams which are illustrated by the Figs. 24 and 25. We shall call them the mrtype and the mm-type. It is often somewhat uncertain what the molecular weight in the liquid phase is. Therefore, the pressure is usually plotted not against the mol fraction but against the mass fraction yi - Mi/(Mi + M 2 ) = *i/(*i + /*2*2/Mi), (10.05) Mi, M2 being the masses of the two components and /*i, /*2 their molecular weights. This way of plotting has also some theoretical advantages as we shall see in section 74. Our simplified model (the perfect solutions) permits us to predict a behavior of this kind in certain cases, viz. when the molecular weights of the components are different in the liquid and gaseous states. The generalization (9.33) of Raoult's equation must, then, be. used, and this leads, instead of (10.03), to the equation P = PBI xi l/ai + p B *(l - *i) 1/<7a , (10.06) where gi = MI/M'I and #2 = ^2/^2- It is to be expected that the case, when both components are subject to association or dissociation of molecules, is a rare one. It will be sufficient, therefore, to assume that only the exponent g\ is different from 1, while g% = 1. It is easy to see that, under this assumption, the slope dp/dxi will, in general, go to zero for a value of x\ between and 1, provided the boiling pressures PBI and pB2 are of the same order of magnitude. This means that the pressure reaches in this interval an extreme; whether it is a maximum or a minimum depends upon the sign of the second derivative d?p/dxi 2 (1 gi)gi~ 2 si#i (1 ~ 2ai)/w . In the case of association in the liquid phase gi > 1, and the sign is negative; in the case of dissociation, g\ < 1, and it is positive. We should, therefore, expect the max-type when the molecules of one of the liquid components are associated, and the />min-type when they are dissociated. On the whole, these expectations are borne out by the facts: It is known that hydroxylated liquids have a tendency towards association when mixed with another liquid which is free of the hydroxyl group. Upon looking through the list of systems showing the m*x-type, we find that it consists largely (although not exclusively) of such pairs of liquids. On the other hand, aqueous solutions of HI, HBr, HC1, HF which are considerably dissociated belong to the pmm-type. However, there is another cause which may produce these kinds of par-diagrams: the existence of chemical compounds of the two components. If the va- por pressure of the compound is higher than of either pure component, X71 EQUILIBRIUM OF BINARY SYSTEMS 183 the px-curve must have a maximum, if it is lower, a minimum. The reason why the two curves in the Figs. 24 and 25 have a, joint maximum or minimum will become clear in section 74. This theory has an important practical application in the process of distillation. In the normal pressure curve (Fig. 26) there corresponds to the composition A of the liquid the composition B of the vapor; that is, the vapor is richer in the component (2) with the higher boiling pressure. Prolonged boiling at constant temperature, therefore, causes the liquid to become poorer in this component so that its representative point moves downward towards psi- If the purpose of distillation is to drive out the more volatile component, one has only to continue boiling and, eventually, the liquid will consist of the practically pure component (1). On the other hand, the pro- cedure is different when it is desired to prepare the pure component (2). Starting again from the state A, we let the solution boil for a time and collect its vapor which has the constitution B. Condensing this vapor we obtain a liquid of the same constitution represented in our diagram by the point C. Boiling this fraction (at the same temperature) gives a vapor of the composition Z>, w > etc. By such steps it is possible to approach FIG. 26. Distillation. closely the point pB2 representing the pure volatile component. It is clear that pressure curves of the min-type (Fig. 25) preclude the success of the first kind of distillation, and curves of the max-type (Fig. 24) that of the second. The best-known example is the water-ethyl alcohol solution having a maximum at 4.43% water by weight so that the purification stops with 95.57% alcohol (about 194 proof). Distillers have learned, in recent years, to get around this maximum by adding a third component to the solution. 71. Temperature of transformation at constant pressure. When the pressure is kept constant, the temperature of equilibrium of two binary phases is completely determined by the composition of one of them. To treat this case theoretically for perfect solutions, we have to fall back again upon the fundamental eqs. (9.17) or (9.22), but to evaluate the equilibrium constant K in a way somewhat different from that of the preceding section. We can apply to the first component the eq. (9.16) in the form , 10(m (10 ' 07) 184 TEXTBOOK OF THERMODYNAMICS X71 where l\ is the latent heat of transformation referred to 1 mol of the first component in the primed phase (v'i = 1, vi either integral or frac- tional). We have pointed out in section 45 that the latent heat is a rather slowly varying function of T which can be regarded as constant in the first approximation. This approximation is amply sufficient for the purpose of the present investigation because we have already intro- duced an element of inaccuracy by supposing the phases to be perfect solutions. Integrating, we find therefore log Xi = - h/RT + Ci(/>), (10.08) and a similar relation with the subscripts 2, for the second component. In the particular case, when the second component is absent (xi = x'\ = 1), we find from (9.17) log K\ 0, and at the same time the temperature of equilibrium is the transformation tem- perature Tbi of the pure first substance. This serves to determine the constant as Ci(p) = h/RToi. There follows h 1 (10.09) On the other hand, the fundamental equations take the form (9.22), with v\ = j/2 = 1, Oc'i/^i) 17 " 1 = xi, (x'z/Kz) 11 ** = * 2 . (10.10) We shall treat in this section the case when the components in both phases ar^jnutually soluble in all proportions. Both components are then present in either phase, and eqs. (10.10) must be treated as simul- taneous. The relations (10.01) permit us to eliminate all the variables but xi. In particular, when there is no association in either component (vi = V2 = 1), *! = - (10.11) This equation and x'i = K\x\ resulting from (10.10) give the explicit expressions of the compositions of both phases (x\ and #'i) as functions of temperature. They apply to all cases of two-phase equi- librium, of which the more important (because of the accumulated experimental material) are liquid-vapor and solid-liquid. The graph- ical representation of these formulas leads to the so-called (Tx)- X71 EQUILIBRIUM OF BINARY SYSTEMS 185 2032 exp. theor. diagram (or Jy-diagram, if the temperature of transformation is plotted against the mass fraction y). As an example we give in Fig. 27 the yT-curves computed with the help of our equations for the solid liquid equilibrium of an alloy of platinum (Toi = 2032, /i = 5270 cal mol- 1 ) and gold (7o 2 = 1338, / 2 = 3140 cal mol" 1 ) as dashed lines. For comparison the experimental curves are drawn as solid lines. The agreement is by no means quantitative, but the general trend is well accounted for by the theory. It will be noticed that the order of the curves is inverted by comparison with Fig. 23, the curve giving the composition of the phase, stable at higher temperatures, lying above the other. Apart from this, there is a complete analogy between the cases of constant temperature (preceding section) and constant pressure inasmuch as there exist, in addition to the " normal" or monotonic type of Fig. 27, also r max - and r m i n -types of curves (of the same shape as those of Figs. 24 and 25). They are, especially, a frequent occur- rence in the liquid-vapor equi- librium of binary systems com- posed of organic substances. It would be easy to generalize our formulas for the case of any numbers v and to show that association or dissociation in the liquid phase leads to these types. However, this is unnecessary because in section 74 we shall give the general proof that conditions which produce the /w-type of the ^-diagram also cause the r m i n -type of the TVgraph. In the same relation to each other stand the mln -and r max -types. We can, therefore, refer, with respect to the influence of association and dissociation, to the discussion at the end of the preceding section. Curves of r m i n -type occur also in the equilibrium of binary metal alloys with their melts, especially, when one of the components belongs to the ferromagnetic group. We did not find in literature any discus- sion of the causes responsible for the minimum, but the prevailing tacit assumption seems to be that it is due to the existence of inter- metallic compounds. It should be mentioned, however, that in the case of Co-Mn alloys, a close investigation in Tammann's laboratory revealed that compounds of these metals do not exist. 1 In view of 1 Hiege, Zs. anorgan. Chemie 83, p. 253, 1913. 0.0 0.5 1,0 FIG. 27. Temperature of fusion as a function of composition. 186 TEXTBOOK OF THERMODYNAMICS X72 this, we calculated from our formulas the Incurves (Fig. 29) which would result on the assumption that Co (T<>2 = 1762, /2 = 3950 cal mol"" 1 ) has in the solid state a diatomic molecule when in alloy with Mn (Tbi = 1523, k = 3560 cal mol- 1 ). Comparing the theoretical curves with the experimental 7>diagram (Fig. 28) by Hiege, 1 we see that the minimum occurs at the right mass fraction (30% Co) but at a temperature which is some 80 lower than that measured. Although this discrepancy lies well within the limits of the theoretical inaccura- cies, we do not feel that the evidence is strong enough to prove posi- tively the existence of association, especially as it is not quite clear what this would mean in terms of the lattice structure of the alloy. 1762-^ 1423 Co- 1762, 1.0 0.5 FIG. 28. Temperature of fusion as a function of composition 0.5 FIG. 29. 1.0 But we do wish to point out that there lies here a problem which, pre- sumably, could be settled by experimental (X-ray) methods. 72. Case of mutually insoluble components. Let us suppose that in one of the phases the components are mutually insoluble. This can be illustrated by many examples of which we mention only a few : (1) Aqueous solution of a salt which is insoluble in ice. When the liquid solution is cooled, either a part of the salt is precipitated or a part of the water congealed, conditional upon the concentration of the solu- tion. There are, therefore, two cases of two phase equilibriums: liquid solution -solid salt and liquid solution -pure ice. (2) Molten alloy of two metals which do not form mixed crystals. Upon cooling, either of the metals can freeze out partially, according to the composi- tion of the liquid phase. (3) Mixture of the vapors of two immiscible liquids. As the temperature is lowered, either one or the other liquid condenses. In all these cases we have equilibria of two kinds of pure substances with the (liquid or gaseous) solution. If we suppose that the pressure p is kept constant, eqs. (10.09) and (10.10) of the preceding section 1 Hiege, loc. cit. X72 EQUILIBRIUM OF BINARY SYSTEMS 187 still apply, but we must put in them x\ = 1 or 002 = 1, the condition that the unprimed phase is pure. Equations (10.10) become x'l = = (1 - = K 2 , (10.12) and they are no longer simultaneous but refer separately to the two possible kinds of equilibrium. To make this quite clear, we shall consider a special example, the molten alloy of cadmium (Toi = 596, /i = 1245 cal) and bismuth (r 02 = 546, / 2 = 2110 cal). The first of eqs. (10.12) gives then the composition of the liquid phase when it is in equilibrium with solid bismuth; the second, when it is in contact with solid cadmium. Com- M -600 N >>&#* 300 *Cd 0.55 1.0 FIG. 30. Solid lines experimental, dashed and dotted curves theoretical. * l.o FIG. 31. Experimental liquidus puting Ki and K% from eqs. (10.9), we find, for the first case, the theo- retical curve of A A' of Fig. 30; for the second, the curve BB'. Let us start from a composition (x'\) and temperature (T) of the liquid alloy represented by the point M. When this system is slowly cooled, the representative point first moves down vertically, until it hits the curve AA f . From then on, bismuth begins to freeze out; accordingly, the cadmium concentration of the liquid phase increases and, as the cooling proceeds, the representative point moves along the equilibrium curve toward the point C. In a similar way, when we start from the point N, the state of the cooled system first strikes the curve BB f and then begins to move along it to the left. In no case, however, does the representative point move beyond the intersection C of the two curves. The point C is a triple point in which the liquid phase is in equilibrium both with bismuth and with cadmium. As soon as it is reached both components begin to freeze out simultaneously, forming a microcrystalline mixture of the two solid phases (pure bismuth and pure cadmium). The point C corresponds, therefore, to the lowest 188 TEXTBOOK OF THERMODYNAMICS X73 temperature at which the alloy can be maintained in a liquid state at the given pressure p: for this reason, it is called the eutectic point (eutectic = Greek for readily melting). The dotted parts of the curves A A 1 and BB' have no physical reality as far as the system under consideration is concerned. For comparison we give the experimental Tjc-diagram of the cadmium-bismuth alloy 1 (solid curves in the same Fig. 30) : considering the crude approximation of the formulas (10.09), the agreement must be regarded as very good. The theoretical eutectic point lies at the same concentration as the experimental (x = 0.55) but about 25 too low. This example is quite characteristic. The microcrystalline mixture into which the liquid phase congeals at the eutectic point is called, in the case of metallic components, the eutectic alloy. It has, usually, the same content of the two components as the liquid, but this may be said to be accidental, in the sense that it is due to the kinematics of the freezing process. As we know from section 39, the equilibrium conditions do not depend on the extension of the phases. Therefore, the eutectic liquid should be in equilibrium with the solid phases taken in any proportion. This is, in fact, observed and indicated in Fig. 30 by the horizontal line passing through the empirical eutectic point. In the case of aqueous solutions of salts, the eutectic congelations were formerly called cryohydrates. It is worth pointing out that, as long as the solution can be regarded as perfect, each of the two curves given by eqs. (10.12) depends on the properties of one component only and is quite independent of the other. In application to our example (Cd-Bi) this means that the curve AA' of cadmium has exactly the same shape and position, no matter what the other component is. In order to measure a large part of it and to bring out its curvature, we must choose, as the second component, a metal with a low boiling point, e.g. mercury. Unfortunately the 7#-diagram for Cd-Hg was not available, and to illustrate this point we give in Fig. 31 the diagram for zinc-mercury. When we consider a mixture of vapors of two immiscible liquids in equilibrium with one of the pure liquid phases, the corresponding curve gives us the equi- librium of this pure phase with its own vapor. The partial vapor pressure of this component is, therefore, exactly the same as if the other component were absent. 73. Case of partial mutual solubility. Other complications. In order to complete our discussion of binary equilibrium, we shall give here a brief schematic r6sum6 of the conditions which are to be expected in the case of partial solubility in some of the phases. To fix our ideas 1 Petrenko and Fedorov, Zs. anorg. Chemie 6, p. 212, 1914. X73 EQUILIBRIUM OF BINARY SYSTEMS 189 let us consider the case of two metals which form mixed crystals of two kinds. The bulk of the kind a consists of the first metal while the content of the second is smaller and can vary continuously from noth- ing to a certain limit. In the kind ft the conditions are reversed: the second metal dominates and the proportion of the first has a continuous range with a relatively small upper limit. Crystals of intermediate composition do not exist. There are, therefore, two cases of solid- liquid equilibrium: coexistence of the liquid alloy of the two metals with the mixed crystal a and with the mixed crystal ft. As long as the mol numbers of the second metal (#2, #'2) are small in both phases, they can change continuously and the conditions are exactly the same, in this range, as in the case of complete solubility (section 71). Assuming, by way of an approximation, that the solu- tion is perfect, we can describe the equilibrium between the liquid alloy and the crystal a mathematically by the formulas (10.09), (10.10), and (10.11). The graphical representation is, therefore, also analogous to the corresponding range of Fig. 27. Two curves diverge from the boiling point A of the pure first component (Fig. 32), the one giving the mol number x\ in the liquid phase, the other x\ in the crystal a; points of the two curves lying on the same horizontal are in equilibrium. However, these curves do not extend across the whole diagram but stop when x\ reaches its limiting value. This fact has a bearing on the interpretation of the constants TOI, To2 of eqs. (10.9): For the same reason as in section 71 we conclude that TOI is the melting point of the pure first metal. However, we cannot identify To2 with the melting point of the second because it is not permissible to extrapolate our curves until this metal is pure. It is better to denote this constant by r'o2, and to bear in mind that its numerical value cannot be predicted from any general considerations. It can be larger or smaller than TOI : When T'oz > TOI the two equilibrium curves go from the point A upwards; when T'o2 < TOI, they go downwards. Similar conditions prevail in the other case of equilibrium (the coex- istence of the liquid phase with the mixed crystal 0), when the mol numbers x\, x'\ are small. The two equilibrium curves diverge from the melting point B (temperature To2) of the pure second component and are again represented by eqs. (10.9), (10.10), and (10.11) with the new constants T'oi, To2- They go up or down from the point B depending upon the numerical value of the ratio T'oi/To2- We have, therefore, to distinguish three cases: (1) The equilibrium curves go downward on both sides, from the point A and from the point B. (2) They go downward on one side and upward on the other. (3) They go upward on both sides. 190 TEXTBOOK OF THERMODYNAMICS X73 The first case, illustrated by Fig. 32, is the most common. Suppose we slowly cool the system, starting from a state represented by the pair of points NN'. Some a crystal freezes out consisting mainly of the first component; the remaining liquid therefore becomes richer in the second component. The representative points move down along the branches AD and AC, so that later-freezing crystalline fractions are of lower XL This continues until the eutectic point C is reached marking the intersection between the " liquidus " curves AC and BC. In this state the liquid alloy is simultaneously in equilibrium with the mixed crystals a and ft. As in the case of the preceding section, the alloy freezes completely when the temperature is lowered further. The only difference is that it forms a microcrystalline mix- Liq. FIG. 32. FIG. 33. FIG. 34. Equilibrium in case of partial mutual solubility. ture, not of the two pure substances, but of the two mixed crystals a and 0, having the composition indicated by the points D and E. This is the end of the motion along the branches AD, AC, and their dotted continuation below the line DE has no physical reality. The mixed crystals represented by the points D and E are not only in equilibrium with the eutectic liquid alloy C but also with each other. The question may be put, therefore, as to the equilibrium of the two crystalline phases at temperatures below the eutetic. The answer is given by the two schematic lines going downward from D and E: Points of them lying on the same horizontals are in equilibrium. It is an empirical fact that these lines slope away from each other. The second case is represented by Fig. 33, which is self-explanatory because the notations in it are the same as in Fig. 32. We have used the example of two metals only to fix our ideas. All we have said applies, mutatis mutandis, to the equilibrium of two partially miscible liquids with their vapor, and to similar cases. The cases illustrated by the Figs. 32 and 33 are, in fact, as characteristic of mixed crystals as of the binary equilibrium of liquids and vapors. However, the third case represented by Fig. 34 applies only to a rather unusual system: X74 EQUILIBRIUM OF BINARY SYSTEMS 191 Two substances form, in the solid phase, a mixed crystal in all pro- portions, but, upon being melted, they are partially miscible liquids. At temperatures lower than the eutectic (C) the crystal melts either into the solution a or the solution ft (depending upon its composition), at the eutectic point into a turbid emulsion of the two solutions. It goes without saying that the conditions considered here include the limiting case when the first component is partially soluble in the second but does not admit the second as a solute and crystallizes (condenses or melts) in its pure state. The point D in the above figures lies then on the vertical x\ = 1 passing through A. We have enumerated in this chapter the types of binary equilibrium which can be expected when the components do not form chemical FIG. 35. Liquid us curve. compounds. The presence of chemical compounds does not add much new to the discussion, from the conceptual point of view, but makes the curves look more complicated. One example will be sufficient to illustrate this: In Fig. 35 we give the 7*-diagram for the melting of Ca-Mg alloys. 1 These two metals form the compound CaaMg-r, there are three solid phases (pure Ca, pure Mg, CaaMg^ which happen to be insoluble in one another. At 721 C we have the equilibrium of the liquid and solid phase of the compound. The diagram is divided by this point into two halves: The left representing a binary system composed of the compound and magnesium, the right of the compound and calcium. 74. Maxima and minima of the equilibrium curves. We have seen in the preceding sections that the treatment of binary systems, as if their phases were perfect solutions, gives a good qualitative account of all the observed phenomena. It is even probable that the not very large quantitative discrepancies are due, primarily, to the crudity of our approximation in putting / = const (section 71) and only in rare cases to the inadequacy of the description in terms of 1 1.C.T. baaed on Baar, Za. anorg. Chemie 70, p. 352, 1911. 192 TEXTBOOK OF THERMODYNAMICS X74 perfect solutions. In particular, we could account for the appearance of maxima and minima in the equilibrium diagrams. However, it will be useful to give here a theorem (due to Gibbs) relating to these maxima and minima which is free from any assumption as to the nature of the system. Let us denote by <l>, Mi, M 2 the thermodynamical potential and the masses of the two components in the first phase, by <', M'i, M f 2 the same quantities in the second. We shall use the equations of equilibrium between the two phases in the form (6.33) It was pointed out in sections 39 and 41 that $ is a homogeneous function of the first degree in the mol numbers Ni, N% and the masses Mi, M 2 . Consequently, the quotient </(Mi + M%) is a homogeneous function of the degree zero; in other words, it depends only on the ratio Mi/M2. In particular, we can represent it as a function f of the mass fraction (10.05), and a similar conclusion holds with respect to the other phase : <*> = (Mi + M 2 ) f(yi), *' = (M'i + M' 2 ) f'(y'i), (10.14) and by partial differentiation - toi) + (1 - *) , - toO - yi . (10.15) These expressions are quite general, no matter whether the compo- nents form chemical compounds or not. The variables entering into eqs. (10.13) are, therefore, yi, y\, p, T. We give to these variables increments dyi, dy'i, dp, dT in such a way that the two phases remain in equilibrium, i.e. eqs. (10.13) continue to hold. We can, therefore, take their total differentials 3A<f> ^ "^T 1 + ^7 ^ l + ~^T d + ~^F dT = - etc ' 3^i 3^ i 3^> 3T The partials with respect to p and T are given by the formulas (6.51) and (6.52), while those with respect to yi, y'i follow from eqs. (10.15) X74 EQUILIBRIUM OF BINARY SYSTEMS 193 where AFi, AF 2 are the increases of the volume of the system and l g i, I g 2 the latent heats contingent upon the respective transfers of one gram of the first or second component from the unprimed to the primed phase. We can eliminate dy'i by multiplying the first equation by y ; i, the second by (1 y'i) and adding Ox'i - yi) y^ d y* +A'dp- B'dT = 0, dy i with the abbreviations A' = y'i&Vi + / 2 AF 2 , B f = (y'il g i + y'd&llT. (10.17) We have now to distinguish two cases: (a) the ^-diagram, at constant temperature (dT = 0), & y=iiL (10 .i8) dyi A dyi* (b) the 7>graph, at constant pressure (dp = 0), dT y'i - yi dyi B' (10.19) Obviously, the maximum or minimum of the py- and !Py-curves are at the places where the derivatives dp /dyi and dT/dyi vanish. Since A', B' do not become infinite and d 2 /dyi 2 is not known ever to be equal to zero, this condition reduces to y'i = yi. (10.20) The maxima and minima occur in the py- and Ty-curves at the same concentrations, namely, when the mass fractions in the two phases are equal. From the partial symmetry of eqs. (10.16) with respect to yit y'\ it is clear that the conditions dp/dy'\ = and dT/dy r \ = also reduce to the same eq. (10.20). This accounts for the fact that the two curves (liquidus and solidus, or liquid and vapor) have always a joint maximum or minimum. Whether the condition (10.20) leads to a maximum or to a min- imum depends on the sign of the second derivative. The second differentiation gives in the two cases (with y'\ y\ = 0) d 2 p 1 d*$ d*T 1 d*f ^ dyi 2 A'dyi 2 ' x/ dyi 2 B' dyi 2 The quantities A' and B f can be considered as positive. In fact, B' always is positive, since our notations are chosen so as to make the 194 TEXTBOOK OF THERMODYNAMICS X75 primed phase that of the higher heat a function (i.e. giving the positive sign to the latent heats /i, fe). The changes of volume A7i, and AF 2 are always positive in the case of liquid-vapor equilibrium, and posi- tive with rare exceptions in the case of solid-liquid. The second derivatives d?p/dyi 2 and d?T/dyi 2 have, therefore, generally opposite signs. To a maximum in the py-curve there corresponds a minimum in the Ty-curve, and vice versa, as has already been mentioned in section 71. It is well to point out that all these conclusions apply, independently of the physical cause of the maximum or minimum type of curves, whether they are due to association, compound formation, or other effects. 75. Remark on ternary systems. In a ternary system each phase consists of three (independent) components Gi, 62, G$ with the mol fractions #1, #2, #3 (or mass fractions yi, y^ ya) satisfying the condition XI + X2 + *3 - 1. (10.21) The scope of this book does not permit us to enter into the very extensive theory of these systems. All we wish to do here is to explain the type of diagrams used in connec- tion with them, in order to enable the reader, to understand the graphical material relating to ternary equi- librium. The triple of numbers xi, X2, #3 can be interpreted as a point within the area of an equilateral tri- angle. In fact, it is known from geometry that the sum of the dis- tances of any internal point P from the three sides (PAi + PA 2 + PA 3 , FIG. 36.-Graphical representation in Fi *' 36 > is e( * ual tO the ^\ f of the composition of ternary the triangle. If we choose the height systems. as equal to unity, we may identify the distances with the mass fractions (PA\ = xi, PA2 = X2, PA$ = #3) because the relation (10.21) is then satisfied. Every composition of the phase can, then, be repre- sented by a point of the triangle; and vice versa, every point within it corresponds to a possible composition. The mol fractions x\ = PA\ of the component G\ are, usually, marked on the side GiG2, so that one must draw the (dotted) line normal to PA\, in order to read yi in the point 61. For instance in our Fig. 36, x\ has the value of about 0.34. In a similar way the values of #2 (and #3) are listed on the sides G*Gz (and GsGi) and can be read by drawing (dotted) lines normal to PA* (and to PA*) to the points 62 (and 63). X7S EQUILIBRIUM OF BINARY SYSTEMS 195 When two ternary phases (primed and unprimed) are in equilibrium, the condition (6.43) must be satisfied for each of the three components Vk?k(p,T, xi, * 2 ) + 'k?'k(p,T, *'i, *' 2 ) = 0, (* - 1, 2, 3) because the mol fraction #3 (resp. #'3) should not be regarded as a separate variable as it is completely determined by eq. (10.21). Two of the variables, x'i, #' 2 or x\, # 2 , can be eliminated from these three equations. If the pressure is, moreover, considered as con- stant, this leaves T - r(*i, * 2 ) or T - r(*'i, *' 2 ), (10.22) the equilibrium temperature expressed in terms of the mol numbers of either phase. These equations depend on three variables (T, x\ t # 2 70, 70 60 50 40 30 20 10 FIG. 37. Isothermals of liquidus sur- face. (Temperature in C). FIG. 38. Eutectic lines. (Temperature in C). or r, x'i, #'2) and can be represented graphically only in a three- dimensional space. For instance, we could interpret the third dimen- sion, normal to the plane of the triangle in Fig. 37, as the temperature T. Equations (10.22) represent, then, two surfaces lying above the triangle. If the equilibrium is between a solid and a liquid phase, they are called, in analogy with the curves of Fig. 28, the soKdus and the liquidus surface. Naturally, one can represent in a plane drawing only the " levels " of these surfaces, i.e. their intersections with the planes of constant temperature, T = const. As an example we give in Fig. 37 1 the iso- thermal levels of the liquidus surface for the system Ni-Fe-Mn (dotted lines, the coordinates in this figure are not mol fractions but mass fractions /i, y 2 , 3/3). In the solid state, these three metals are 1 1.C.T. based on Parravano, Gaz. Chim. Ital. 42 II, p. 367, 1912. 196 TEXTBOOK OF THERMODYNAMICS X75 mutually soluble in all proportions: therefore, there exists only one solid phase with which the melt is in equilibrium and the liquidus surface extends over the whole of the triangle. This is not the general case: if the solid components are mutually insoluble or partially solu- ble, there may exist at the same temperature three (or even more) different solid phases. This case is analogous to those of Figs. 30 and 32 for binary systems. As we have there two separate liquidus curves corresponding to the equilibrium of the melt with the two solid phases, so here the area of the triangle (Fig. 38) * is divided into three regions and in each region we have a separate liquidus surface. The three liquidus surfaces are analytically represented by three different equations T = T'(y'\, y f 2) and each corresponds to the equilibrium of the melt with one of the three solid phases. The lines of intersection of two of these surfaces (eutectic lines) represent the states in which the melt is in equilibrium with two solid phases simultaneously (Fig. 38). In the point where all three intersect (ternary eutectic point) the melt is in equilibrium with all three solid phases. There may arise further complications in the number and shape of the equilibrium surfaces when the components are able to form chemical compounds. 1 1.C.T. based on Goerens, Metallurgie 6, p. 537, 1909; Stead, Iron, Steel Inst. 91, p. 140, 1915. CHAPTER XI FUGACITIES AND ACTIVITIES 1 76. Definition of fugacity. It was shown in section 42 that chemi- cal and physical equilibrium depends on the partial thermodynamic potentials We have reviewed, in Chapters VII to X, the cases in which some theoretical knowledge of the functions permits the making of com- plete or partial predictions about the details of the equilibrium. How- ever, in the large majority of cases such knowledge is lacking or in- sufficient, and the only resource is to combine the experimental and the theoretical methods: to measure and record the partial potentials from one group of observations and to use these data for the prediction of another set of phenomena. A large amount of experimental material has been collected and published in the chemical literature. However, the chemists usually record not the partial thermodynamic potential itself but another function/ which stands to it in the simple relation / = / (r) exp &/RT) (11.02) Of V = RT[logf - log/ (r)]. (11.03) The quantity / was introduced by G. N. Lewis 2 and is called the fugacity or absolute activity. fo(T) is an auxiliary function of tem- perature only, which is defined by the following requirement: in the gaseous phase, the fugacity approaches more and more the pressure, as the density is decreased, and becomes identical with it (/ = p) in the limit V = oo . The thermodynamic potentials of perfect and of Van der Waals gases were given by the formulas (5.41) and (5.44). In order to satisfy the above requirement we have to put in both cases f (T) = - w(T)/RT. For the perfect gas we obtain / = P, (11.04) 1 This chapter can be skipped without loss of continuity. 1 G. N. Lewis, Proc. Am. Acad. 37, p. 49, 1901; Zs. physik. Chemie, 38, p. 205, 1901. 197 198 TEXTBOOK OF THERMODYNAMICS XI 76 at all temperatures; for the Van der Waals gas we find 2a RTb (11.05) Once/ is defined in the gaseous phase, it is definite also in the con- densed phases because (barring associations) the thermodynamic potentials (and, therefore, the fugacities) are equal in equilibrium. More generally, we have for two different conditions of the systems (1) and (2), at the same temperature, - /? - (11.06) In general, analytical expressions for / are lacking and, even in systems with one independent component, only the partial of log / with respect to p can be given explicitly from (5.37) (11.07) The measurement of ^ can be easily carried out in pure substances (systems with one independent component): in this case s=^> = # Ts + pv, where each of the three terms can be directly measured. As an example we give in Table 31 the fugacity of liquid water at various temperatures and pressures. 1 TABLE 31 FUGACITY OF LIQUID WATER Activity a Fugacity / 25 C 37.5 50 25 C 37.5 50 1 1 1 1 0.03125 0.06372 0.1219 100 1.0757 1.0728 1.0703 0.03362 0.06836 0.13047 200 1.1576 1.1515 1.1461 0.03618 0.07337 0.13971 300 1.2454 .2356 1.2270 0.03892 0.07873 O.H957 400 1.3394 .3254 1.3132 0.04186 0.08445 0.16009 500 1.4402 .4214 1.4050 0.04501 0.09057 0.17127 600 1.5481 .5240 1.5029 0.04838 0.09711 0.18320 700 1.6637 .6336 1.6072 0.05199 0.10409 0. 19592 800 1.7874 1.7506 1.7184 0.05586 0.11155 0.20947 900 1.9200 1.8755 1.8367 0.06000 0.11951 0.22389 1000 2.0618 2.0089 1.9628 0.06443 0.12801 0.23927 1 M. Randall and B. Sosnick, J. Am. Chem. Soc. 50, p. 967, 1928. XI 77 FUGACITIES AND ACTIVITIES 199 On the other hand, in a phase (i) with many components the function <p h (f *, as is clear from its definition (5.39), is only then accessi- ble to direct measurement when the mol number JV* (i) can be changed independently from the other mol numbers. If this is impossible (owing to chemical reactions in the system), it is often still feasible to determine it in a roundabout way when this component can be isolated in another phase (j) which is in equilibrium with (i), making use of the equilibrium condition ^ ( ) = ^ A ( ^. After being determined from observations, these data can be used for theoretical purposes, for instance to test the relations (6.49) A*]ifctyk> =0, (11.08) which must obtain within the phase (i). 77. Definition of activity. In some cases it is impossible or diffi- cult to determine the absolute value of the fugacity of a component in a complex phase (for instance, in a solution), while it is feasible to measure the ratio *-///. (11.09) of the fugacities of the same component for two states of the solution, different as to pressure and concentration, but of the same temperature. This permits choosing /, as the unit in which the fugacity is to be measured and regarding the state to which /. refers as the standard state for the component and for the temperature in question. The ratio a was also introduced by Lewis 1 and is called activity or relative fugacity. Its relation to the partial thermodynamic potential follows from the formula (11.03), taking into account that/and/, are measured at the same temperature so that/o(r) drops out of the ratio: ?*-?* = RTlo%a h , (11.10) where v, h is the partial thermodynamic potential of the component h in the standard state. The* simplest example is the case of a pure substance: in Table 31 are given the activities of liquid water. As standard states at the temperatures 25, 37.5, 50 C, are chosen those corresponding to the pressure p = 1 atm (with/, equal to 0.03 125, 0.06372, 0.1219, respectively). It must be distinctly understood that, in systems with several components, the standard states are chosen for each component inde- pendently and arbitrarily. For the definition of ?*, not only 1 C. N. Lewis, Proc. Am. Acad. 43, p. 259, 1907; ZB. physik. Chemic 61 p. 129, 1907. 200 TEXTBOOK OF THERMODYNAMICS XI 77 but all the mol numbers of the other components, must be given. Therefore, the <p,& are mutually exclusive, in the sense that there does not exist any stable state of the system with these simultaneous values of the partial potentials. This implies, of course, that they do not satisfy the conditions (11.08). The expression log K., (11.11) 1 h does not vanish but is a finite function of temperature and pressure. We find from (5.37) AF, ' ajai ) -&. <"' / p Ai where AF, =X) "A ^^* (? =]C vi^TldP* (11-14) On the other hand, the partial potentials^ in eqs. (11.10) refer to actual states of equilibrium and satisfy the condition A$ = 0. There- fore, there follows the equation ]C ^ log a h u} = log X,, (11.15) which has the same analytical form as the mass law, the mol fractions being replaced by the activities. The physical meaning of the activity becomes a little clearer from the analogy which the formulas (11.10) and (11.15) bear to those valid for dilute solutions. In section 59 we found for the partial thermo- dynamic potential of a dilute solute Since log x* vanishes for x\ = 1, it may be said that the term ^ represents the molal thermodynamic potential of the fictitious state when the component is pure while having the same properties \\& in the solution. This means that the heat of solution is contained in <pn but the entropy only partially: the second term RT log XH represents its increase due to the entropy of mixing the solute (h) with the solvent. The interactions of the component (h) with the other solutes and the entropy of mixing it with them are negligible in dilute solutions. In the general case of non-dilute solutions, we have the analogous formula (11.10) >gaA 0) . (11.17) XI 78 FUGACITIES AND ACTIVITIES 201 The standard state for the component h is, practically, always chosen so that, in it, it is the only solute (xk = 0, k j& h). The term with log a represents, therefore, the increment of the thermodynamic potential due to interactions with the other solutes, to the entropy of mixing h with them, and (in a measure depending on the definition of the standard state) also to the interactions and entropy of mixing with the solvent. 78. The activity coefficient. The simplest way of choosing the standard potentials ,/ is identifying them with the functions <f>k of eq. (11.16) used in the theory of dilute solutions V* - ** w . (11.18) The equilibrium constant K, becomes then identical with K in Van t'Hoff s eq. (9.13), whence (11.15) takes the form Yi "* a) log a h (i) -logJC. (11.19) J.h The activities ah have then the meaning of effective mol numbers which one must substitute instead of XH in order to conserve the form of the mass law. The ratio between the effective and the actual mol number *h = Ok/Xk (11.20) is called the activity coefficient. 1 In dilute solutions the activity coefficient is usually defined in another way: as the ratio of activity and molality WH y h = a h /m h . (11.21) The relation of the two definitions (since in the dilute state Xh = WA/WO) is y h = CKA/WO, (11.22) where mo is the molality of the solvent (i.e. mol number per 1000 g weight). It is obvious that the choice of the standard states defined by eq. (11.18) is permissible when the solution obeys (in the limit of low concentrations) the theory of dilute solutions treated in Chapter IX. The standard states are then, really, possible states of the system, and at low concentrations the activities approach the mol fractions: lim ah = x ht lim ah 1. (11.23) 1 This term was first introduced by A. A. Noyes and W. C. Bray (Journ. Am. Chem. Soc. 33, p. 1643, 1911), but these authors defined it with respect to the molality as in eq. (11.21). 202 TEXTBOOK OF THERMODYNAMICS XI 78 However, even if this should not be so, if the values ,* = <PH cor- respond to no existing states of the system, they can still be used. They are, then, merely auxiliary mathematical quantities with respect to which the activities are defined. There is no reason why one should not use thermodynamic potentials defined with respect to fictitious states, provided there can be given an experimental way of measuring them, either separately or in combinations. Such an experi- mental procedure exists and will be described below. In the first place, it is simple to measure the activity of the solvent by the method of boiling pressures or boiling (and freezing) temper- atures. For the process of vaporizing >o (2) mols of solvent, to obtain FO (I) mols of vapor, we have from (11.19) the equation (1) vo log a (1) - vo (2) log ao (2) - log K, (11.24) closely analogous to (9.17). We assume that the solutes in the liquid phase (2) are non-volatile so that the vapor phase (1) is pure and of sufficiently low pressure to be regarded as a perfect gas. For a perfect gas, eq. (11.16) or (9.09) is rigorous and the activity coincides with the mol fraction: a* (1) = #A (I) = 1. This gives for the liquid phase (drop- ping the superscript) vo log ao = - log K. (11.25) In the particular case when the liquid phase also consists of the pure solvent we can apply to it the theory of section 64 and find log K(p,T) = 0, where p, T are the pressure and temperature of the boiling point. When the solutes are present in a concentration that is not high, the solution boils at a lower pressure p + &p (provided T is the same), and we obtain from (11.25) *o log ao - - log K(p + A,!T) - AF- A/> (11.26) neglecting higher powers of A/>. On the other hand, if the theory of section 64 were applied to this case, we would have j>o log XQ = AF- Denoting the ratio of the actually measured pressure increase A to the theoretical Ao (if the solutes behaved like perfect gases) by (11.27) we have log 00 = /3ok>g*o. (11.28) XI 79 FUGACITIES AND ACTIVITIES 203 It is obvious, by analogy with section 64, that ft> can also be measured as the ratio of the actual (#) and the theoretical (tfo) lower- ing of the freezing point, 0o = #/#() Much of the experimental work has, in fact, been carried out by the freezing method. The above formulas imply, of course, concentrations sufficiently low for higher powers of A being neglected in the expansion (11.26). A more accu- rate formula for the activity of water as a solvent, taking in second and third powers of the lowering # of the freezing point, is 1 logio ao - - (421.00 + 0.1640 2 - 0.0037tf 3 ) X 10~ 5 . (11.29) 79. The activity function of electrolytes. Equations (11.28), (11.29) give a convenient way of determining the activity of the sol- vent. As to the solutes, their activities can be measured directly if they are volatile. This is, however, a comparatively rare case. A very important field of investigation are the electrolytes whose ions are mostly non-volatile. It was shown by Lewis that valuable knowl- edge about their activities can be obtained from eq. (11.28) by means of Duhem's relation (6.16). Let us consider the case that every molecule of an electrolytic solute is completely dissociated into ions according to the formula vid + . . . + v T G r G n = 0. If the mol number of the neutral electrolyte dissolved was N n , those of the ions are N* = VhN n , the sum being v e N n , (v 6 = v\ + . . . + v r ). Denoting * - V '?\T = 1 - *o, (11-30) (where NQ and XQ refer to the solvent), the mol fractions become x k = VKX/V., (ft - 1,2.. .r). (11.31) In Duhem's equation (6.16) the partial thermodynamic potentials <ph can be replaced by log a h because, in eq. (11.10) defining the activ- ity, the function ^. is independent of the mol fractions #*. As the variable of differentiation we take x 3 log a h Now, from (11.20) and (11.31): log a* = log a* + log x + const. Substituting this for the activities of the ions, Abel, Redlich, and Lengyel, Zs. phys. Chem. 132, p. 201, 1928. 204 TEXTBOOK OF THERMODYNAMICS XI 79 The expression log 7 - ~ vh log a* (11.32) defines the quantity y called the activity function of the electrolyte (sometimes also referred to as activity coefficient or activation function). ( i-,) + i+,-a. (n.33) In the limit of zero concentration (x = 0) the solution is reduced to the pure solvent (0o = 1) and it is experimentally ascertained that lim log ah = lim log 7 = 0. The integration of (11.33) gives, there- fore, log 7 _ /Ti + * -* 9l g' /o Lx jc 3^ In the particular case of considerable dilution, the expression (11.28) may be used for log a , and, moreover, it is then permissible to neglect terms of second order in the small quantities x and PQ. Hence the limiting law for low concentrations - go 3/3o X Since in dilute solutions x cc m (molality) this can be also written The pioneer work of collecting data about activities of electrolytes by the freezing method was largely carried out by American chemists under the leadership of G. N. Lewis and A. A. Noyes. This material confirmed the division of these substances in two groups discovered earlier by other lines of study. 1 (A) Weak electrolytes (mostly organic acids and their salts) are those which form aqueous solutions obeying, in their dilute state, the theory of Chapter IX. This implies /So = 1, 7 = 1 for all low concen- trations, so that eq. (11.34) is identically satisfied (0 = 0). (B) Strong electrolytes (the common strong acids and their salts) which exhibit the electrolytic properties in their most typical form 1 There are two other methods of measuring the activities of electrolytes. (1) Electrical measurements (conductivity or electromotive force), as described in section 114. (2) Influence of other solutes on the solubility. This method is limited to electrolytes of low solubility. XI 80 FUGACITIES AND ACTIVITIES 205 do not obey the laws of dilute solutions. While 0o tends towards 1 in the limit of infinite dilution (m = 0), it appreciably deviates from unity even for very low molalities. In fact, the curve representing 0o = 0o(w) has a vertical tangent at m = 0, i.e. QPo/dm = oo . Several empirical formulas have been proposed for this function, among them /So = 1 - ^W H , (11.35) which is supported theoretically (section 115) and represents the experimental material as well as any of the others. This formula permits one to carry out the integration of (11.34) and to find the limiting law for the activity function log 7 =- km*. (11.36) It will be better to postpone the discussion about the structure of the factor k and about the agreement of this formula with the experi- mental data until section 115. 80. Activities in binary gas mixtures. The deviations from the laws of perfect gases in electrolytes (preceding section) are due, pri- marily, to electric forces. It will be instructive to mention, as another example, the deviations in gas mixtures which are caused by molecular cohesion. The activities of a number of gases in mixtures have been determined by several investigators from their chemical equilibrium with liquid or solid substances, and a summary of their results is con- tained in a paper by Randall and Sosnick. 1 The activity coefficient of a component is defined as a h = fh/fh (0) Xh where /* is the f ugacity of the gas h in the mixture, /A (O) its fugacity in the pure state, at the same pressure. Randall and Sosnick confirm for binary gas mixtures an equation which was found by Hildebrand 2 to hold for liquid solutions, namely, log ai//i (0) *i) = (Bx 2 2 + Cx 2 * + Dxz* + . . .)/RT. (11.37) The character of this function appears from the curves of Fig. 39 3 relating to mixtures of ethylene and argon at 25 C. The only theoretical equation of state for gas mixtures is Lorentz' extension of the Van der Waals formula given in section 5. Although its validity is uncertain and, presumably, restricted to a narrow range, it is interesting to note that it leads to an expression similar to (11.37). Lorentz' equation is formally identical with the ordinary Van der i M. Randall and B. Sosnick, J. Am. Chem. Soc. 50, p. 967, 1928. 'Hildebrand, Proc. Nat. Acad. Sci. 13, p. 267, 1927. 8 Randall and Sosnick, loc. cit. 206 TEXTBOOK OF THERMODYNAMICS XI 80 Waals formula (1.21); only, the "molal volume" v has here a different meaning, namely, v = V/N (where N = ffi + N*), while the constants depend on the mol fractions as follows: a b + 2ai2XiX2 + (122X2*, + 2bi2XiX2 + The thermodynamic potential is obtained by an obvious generaliza- tion of (5.44) p + W1 (D f c p idT -T 0.1 Argon 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0,9 Ethylene Mol Fraction FIG. 39. Logarithms of activity coefficients (log o&) The entropy of the mixture must contain a term depending on the mol numbers of both components which represents the entropy of mixing the two gases. It can be evaluated by noticing that, in the limit of very large v, the components become perfect. For a binary mixture of perfect gases the thermodynamic potential is known to be (section SO) Ni*i(T) + N 2 <*2(T) + NRTlogp + RT(Ni + N 2 log * 2 ), XI 80 FUGACITIES AND ACTIVITIES 207 where the subscript i stands for " ideal.*' Comparing the two expres- sions, we conclude that (11.39) Hence It is our purpose to calculate the expression (11.37) to terms of second order in the coefficients a, 6. Expanding and making use of the equation of state (1.21), we find According to (11.04), the fugacity of a perfect gas is equal to its (partial) pressure, /u = pi = px\. Consequently (11.06) leads to This relation holds for all values of the mol fractions, including X2 = 0, xi = 1, when the left side is identical with log (/i (0) /)- Tak- ing the difference of the two expressions, we obtain, in fact, an equation of the type (11.37) with the following values of the coefficients. B = p[(an - 2ai2 + a 2 ^/RT - (6u - 2b i2 + 622)] - 612) + an(a 2 2/2RT - 6 22 ),] \2 ) [" / (3an - 4ai2 + a 2 2)(ai2/2RT - 612) - & 22 )1 , 3 . -(an - (611 2&i2 + 622)]- In the case of ethylene (1) and argon (2) at 25 C, the table values are an/RT = 135, in 42, a 2 2/RT 53, & 2 2 30. The order of magnitude of the data in Fig. 39 is consistent with this if a\2 and 612 208 TEXTBOOK OF THERMODYNAMICS XI 80 have commensurate values. At pressures of 25 atm the terms with f? are still unimportant and, neglecting them, our formula is symmetrical in the subscripts (1) and (2). This is borne out by the symmetrical character of the curves for argon and ethylene at this pressure. The constants can be adjusted so as to describe also the general character of the curves at the higher pressures of 50 and 75 atm, but it remains to be seen whether the expressions are fit for a quantitative descrip- tion of the phenomena. CHAPTER XII THE CAPILLARY LAYER 1 81. Surface layer and surface tension. The equilibrium of two adjacent phases was treated in Chapter VI as if each of them were uniform in its properties up to the very surface of discontinuity sep T arating them from each other. This, however, is not strictly true: the state of a substance at any point within it is influenced, not only by the molecules nearest to this point, but also by all the others in a small radius around it called the radius of molecular action. The sur- face layer of the thickness of this radius, between the phases or at the boundary, is, therefore, in a condition different from interior points, as it stands under the action of molecules of both media. It is called the capillary layer and exhibits peculiar properties depending on the nature of the two phases. When the phases are of large size, the energy and entropy of the capillary layer become negligible, and the conditions are those discussed in Chapter VI. But when the size of the phases is small, in one or several dimensions, the influence of the capillary layer is important. A general characteristic of the capillary layer is that it tends to contract. From the point of view of mechanics, it acts like a mem- brane in a state of uniform tension. By this we mean a condition of the membrane (Fig. 40) in which every line element dl of its boundary AB experiences a normal force F n = a dl to the inward, <r being the constant tension. If we draw an imaginary line CD in the interior of the membrane it is also under the action of two oppositely equal nor- mal forces of the magnitude a per unit length of the line. When the membrane has a chance to contract, it performs work in doing so. Suppose that the boundary line is displaced from the position AB to A'B', and the element dl to dl r , through the distance AL, as a result of the contraction. The work done against the element dl is, according to (2.01), F n AL cos (w,L) = a dl&L cos (n,L), or it is equal to a multi- plied by the area which is swept over by dl in its displacement. Conse- quently, the total work is equal to <r multiplied by the area swept over 1 This chapter can be skipped without loss of continuity. 209 210 TEXTBOOK OF THERMODYNAMICS XII 81 by the whole boundary (which, in turn, equals the decrease of the area 2 of the membrane) : DW -- <r<G, (12.01) counting dS positive when it is an increase. It does not matter whether the membrane is plane or curved. \!\ KJI B' 1 'B "2 (2) FIG. 40. Work done by surface tension. FIG. 41. Capillary layer. The tension <r of the capillary layer is called the surface tension. Its numerical values for the more important substances are contained in Table 32. A few additional data for solutions and for solids will be found in sections 86 and 87. The thickness of the capillary transition layer is not well defined as it represents a continuous change of properties from one phase to the other. In the interest of the next section we shall, therefore, pro- ceed as follows. We envisage a point in this layer and lay an imagi- nary geometrical surface AB through it and through all the other points which are similarly situated in relation to the two adjacent phases (Fig. 41). We lay a second imaginary surface, A\B\, parallel to it, in the upper phase (1), at a distance d\ where that phase is already uniform, and a third parallel surface, A^B^ in the lower medium (2) where this medium is uniform. In all the variations which we shall let the phases undergo, the distances di, dz from AB to AiBi and ^2-82 shall be considered as constant and not subject to variation. We shall restrict ourselves to the case when the distances d are small compared with the radius of curvature of the layer. Imagine now a wide l rigid and adiabatic envelope which contains parts of both phases and cuts through the capillary layer. By our device the space within it is divided into three regions, as labeled in Fig. 41, and the total internal energy and entropy can be divided into three parts pertaining to these three regions U - U+ t/ (2 >+ /< 3 >, S = 5 (1 >+ S+ S< 3 >. (12.02) 1 Near the envelope the capillary layer is distorted because the walla add their own surface effects; but when the envelope is sufficiently wide, the distorted parts are negligible compared with the undistorted remainder. XII 81 THE CAPILLARY LAYER 211 TABLE 32 SURFACE TENSIONS OF LIQUIDS IN CONTACT WITH GASES OR OTHER LIQUIDS Substance Adjacent medium Temp 9 (dyne/cm) Water Air C 75 49 Water Water vapor 20 30 80 72.35 71.03 62.3 73 21 Water Benzene 20 20 70.60 33 Water Chloroform. . . 20 27 7 Water Ethyl ether 20 9 7 Mercury Air 20 471 6 Mercury Vapor 100 360 20 456.2 376.4 480 3 Mercury Ethyl ether 20 398 3 Mercury Ethyl alcohol 20 364 3 Mercury Chloroform . 20 356 6 Oxygen Vapor 70 K 18 3 Nitrogen Vapor . 90 70 13.2 10 53 Hydrogen Vapor . . 90 15 6.16 2 83 Argon Vapor 20 85 1.97 13 2 Carbon monoxide Vapor 70 12 11 Carbon dioxide Vapor 0C 9 13 Ethyl ether Vapor 20 30 20 1.16 0.06 16 49 Benzene Vapor 100 80 7.63 20 28 Methyl alcohol Vapor 100 20 18.02 23 02 Acetic acid Vapor 100 20 14.80 23 46 Carbon tetrachloride .... Vapor 20 25 68 Ethyl alcohol Vapor 100 20 16.48 23 03 100 14.67 The energy differentials in the first and second region (first phase and second phase) have the usual expressions (12.03) 212 TEXTBOOK OF THERMODYNAMICS XII 82 while that of the inhomogeneous intermediate layer is rfZ7 ( 3 ) = TdS-DW = TdS+ <rd2 - &4V*\ (12.04) where !T <3) and ps are the mean temperature and mean pressure in it. 82. Conditions of equilibrium as modified by the surface layer. The fundamental conditions (5.03), in view of the relations (12.04), take the form 0, 65 (1 >+ 6S+ 6S (3) = 0. (12.05) When treating equilibrium with neglect of the capillary layer (section 40) we found it convenient to divide the problem into two parts by suitable restrictions of the virtual variations (auxiliary con- straints). We shall now go a step farther and break up the present treatment into three partial problems. (1) We do not admit any changes in the composition and mass or in the volume and geometrical shape of the phases. The only per- missible variations are those of transfer of heat from one phase to another. Under these conditions 6/o>= r<$S<, (j = 1,2,3). We substitute this into the first eq. (12.05) and add the second, multiplied by the Lagrangean factor X o, whence jxn = r (2) = r< 3) = r = - x. (12.06) In equilibrium the temperature is uniform throughout the system. (2) We consider only variations at constant temperatures and constant total volume. As was shown in section 36, the equilibrium is then determined by the minimum of the sum of the work functions whose differentials are (5.24) Moreover, we disallow any changes of composition or mass in the phases and permit only changes in volume and shape of the three regions at the expense of one another. Since 67* = 0, the variation d becomes, according to (12.03), (12.04) v S - 0. XII 82 THE CAPILLARY LAYER 213 The variations of volume are subject to certain subsidiary condi- tions. In the first place the total volume is constant since we suppose it enclosed in a rigid envelope or (pi - - * 52 = 0. (12.07) As a further restriction, we specialize the geometrical shape of the phases in the following manner : let the capillary layer have the con- stant curvature C = - + (12.08) r and r r being its main radii, and let the walls of the vessel be normal to the layer. We consider a transfer in which mass is added to the lower phase (1) and taken from the upper phase (2), so that all three surfaces defining the layer are displaced up- ward through the distance /. If we denote the respective areas of the surfaces AB, AiBi, A 2 B 2 by 2, Si, S 2 (Fig. 42) 6 7<D = 2i $/, 5 7 (2) = - S 2 81. FIG. 42. Virtual displacement of capillary layer. On the other hand, we have from geometry S/rr' = 2i/(r - di)(r' - di), etc,, or neglecting squares of d/r, d/r', 2i = (1 - Cdi) -2, 2 2 - (1 + Cd 2 ) 2. In the same way the increase of the area S due to the displacement s 62 = CS-6Z. Substituting all this into (12.07), we find pi - p2 = [<r - (ps ~ i)di - (#3 - p2)d 2 ] C. (12.09) The terms (3 p\)d\ and (3 pz)d 2 represent the difference between the actual force of pressure on the unit length of the cross- section through the region (3), and the force which would act if the phases (1) and (2) extended to the surface AB and changed discon- tinuously at it. Because of this subtraction, any indeterminateness, 214 TEXTBOOK OF THERMODYNAMICS XII 82 due to the arbitrary choice of the thickness di and </2, cancels out. The difference itself is an intrinsic property of the capillary layer of the same nature as the surface tension. In fact, there is no experiment which would permit its measurement separately from it. It is, there- fore, appropriate to throw it together with the surface tension and to denote by the symbol <r the whole bracket expression of (12.09). In this sense, we write p l - p 2 = Co- = * + - (12.10) It should be noted that in Fig. 42 the region (1) lies on the concave side of the surface AB. We therefore make the rule that C is to be counted as positive when (1) is on the concave side and as negative when it is on the convex. Since <r is essentially positive, the pressure is always higher on the concave side of the capillary layer. The relation (12.10) is known as Lord Kelvin s formula because it was deduced by that physicist from mechanical considerations (next section). (3) We maintain constant the temperature T and the local pres- sures p\ 9 pz, p3 in all the three regions. As we know from section 36, the equilibrium is then determined by the minimum of the thermo- dynamic potential : 6$ = 0. The only permissible variations are now those of transfer of mass from one phase into another. Besides, we restrict these variations still further by prohibiting any changes of mass or composition in the region (3) and we consider only the trans- fers of mass of any component h (through the capillary layer) from phase (1) into phase (2), or vice versa. These variations are precisely the same as those considered in section 40, and the presence of the capillary layer in no way influences the form of the conditions of equilibrium. We may, therefore, refer to that section and to 42 and take over the equations (6.43) and (6.45) ^"W^+^W^O, (12.11) or = ?W 2) . (12.12) In the second of these equations the partial thermodynamic poten- tial is referred to 1 g of the component h (and not to 1 mol). We may summarize the results of this section in the following way. We derived the conditions of equilibrium for two phases separated by a capillary layer. Comparing them with those obtained neglecting the surface layer, we find all the conditions the same, except that relating to the pressures of the phases. Instead of equality of pres- XII 83 THE CAPILLARY LAYER 21$ (1) sure, we find a pressure difference (12.10) depending on the curvature of the capillary layer. When the layer is plane, C = 0, its influence on the equilibrium vanishes. 83. Direct observation of the capillary pressure difference. It was known for a long time that the level of water is higher in narrow tubes (capillaries) than in large vessels (Fig. 43). The theoretical explanation of this phenomenon was given by Laplace, who showed that the column of liquid in the capillary tube is supported by the surface layer of the meniscus. In this way the names " capillary layer " and " capillarity " originated. A particularly simple vffcy of calculating the difference / of levels in the tube and in the wide vessel (" capillary rise ") was given by Lord Kelvin. Let us consider the case that a liquid rises in a tube of circular cross-section (with the radius R) and that the meniscus includes the angle # with the walls. The vertical com- ponent of the force of tension which acts on the rim of the capillary layer is then F = 2irR<r cos $. This force supports the liquid column of height / and must be equal to its weight in the surround- ing gaseous atmosphere. Denoting the densities of the liquid and gas by P2 and pi, respectively, and by g the acceleration of gravity, F = irR 2 l g(p% pi), whence 2<rcos# = Rlg(p2 PI). For simplicity, let us assume the curvature 1(2) FIG. 43. Capillary rise. of the meniscus as constant over its whole surface, curvature is then r = R/cos &, and we find The radius of - Pi) (12.13) For water and other wetting liquids # = and, roughly, R = r . On the other hand, for completely non-wetting liquids, like mercury, & = IT: i.e. r and / are negative. This means that the liquid is on the concave side of the meniscus and its level is lower in the capillary. The formula (12.13) offers one of the most accurate methods of measur- ing surface tensions. If p is the pressure at the plane surface in the vessel, the pressure in the gas at the meniscus is smaller by the pressure of the layer of gas of the thickness I, viz. pig/: Pi (12.14) 216 TEXTBOOK OF THERMODYNAMICS XII 84 Similarly the pressure pz in the liquid at the meniscus is decreased . . 2<r P2 p 2 - p = r pi p2 Consequently Pi- p2= (12.15) This is identical with the thermodynamical formula (12.10), in the special case r = r'. 84. Influence of curvature on the pressure and temperature of transformations. Let the pressure and temperature of equilibrium between two phases be p and T when the capillary layer is flat (wide vessel). We first consider the case of a pure substance (or of a system with one independent component), <p = (p, and use the equation of equilibrium in the form (12.12), since we are not interested in the effects of association, p and T must then satisfy the relation (12.16) When the capillary layer is curved, the conditions are slightly dif- ferent. We denote the temperature T + Ar and we have two pres- sures pi = p + &p and p 2 = p + A/> <rC, according to (12.10). The relation of equilibrium still has the same form <P M (P + A/> - <rC, r + Ar) - 9M ^(p + A/>, r + Ar) = o. Restricting ourselves to quantities of the first order in the small increments, we expand into a Taylor series and subtract (12.16) This formula can be specialized in two ways: (1) Transformation pressure at given temperature. Suppose that the flat and the curved boundaries are at the same temperature: Ar = 0, so that the third term of the equation vanishes. According to section 35, (9vW9/>)T ** ^M, the specific volume referred to Ig, which is the reciprocal of the density, V M = 1/p. Hence Pl (12.17) ' PI ~ P2 In the special case of equilibrium between a liquid and its vapor this is identical with the relation (12.14) which was obtained by Lord XII 84 THE CAPILLARY LAYER 217 Kelvin's method. In fact, the liquid boils at the meniscus in the tube at a slightly different pressure from that at the plane surface of the vessel, corresponding to the difference of altitudes. (2) Transformation temperature at given pressure. Suppose the pressure of the phase (1) is the same at its plane and at its curved boundaries: pi = p or Ap = 0. We know from section 42 that \ /W>\ _ I ),~ \~&r) f - ~ \~&r,~ \~& where IM is the latent heat of transformation at the plane surface, i.e. the heat necessary to transform Ig of the substance from the phase (2) into the phase (1), AT = ~ (12.19) P2/AT In addition to the case of a pure substance, we shall be interested in another which is but slightly more general. We assume that one of the phases, say (2), is condensed and pure, containing only the com- ponent h, while the other phase (1) is either a gas mixture or a dilute solution. We consider first the case of the gas mixture and apply to it eq. (9.24), since it can be considered an immediate consequence of (6.43) or (12.11), RT log p, = fok^tor)/** - *(r)J, (12.20) where g* = ^ (1) /^ (2) . At a flat surface of discontinuity, p means the pressure in phase (2) identical with that in phase (1) where it is com- posed of the partial pressure ph (l) of the component (h) and of the pres- sures of all the other components whose sum we shall denote by po, viz. p = p h w -f- pQ. The curvature of the surface of discontinuity cannot influence the pressure p^of those components which are absent in the pure condensed phase. It remains the same, and only the partial pressure of the component h changes into ph (l} + A/> A (1) . Consequently p 1 = p + A/> A (1) and p 2 = p +' A/> A (1) - aC, according to (12.10). At the temperature T, we have, therefore, from (12.20) >- aC, T)/g h - or, as above, When the first phase is a gas mixture (ph (1> = RT/vS 1 *), this is iden- tical with eq. (12.14) since obviously (&ifc (1) Vh^/Vk* = (P2 PI)/DI. 218 TEXTBOOK OF THERMODYNAMICS XII 85 In the case of a dilute solution, (1) labels the liquid phase and (2) the solid precipitate. ph (1) must be interpreted as the osmotic pressure of the component h which is proportional to the solubility x gh a \ according to the relation (9.50). Neglecting A/>* in the parentheses beside aC, (where C is negative, since the solid particles are convex) , which expresses the influence of the curvature upon the solubility and is called the Ostwald-Freundlich formula. It is to be remembered that v h (2) is the molal volume in the condensed phase. If the substance is dissolved without change of molecule (g h = 1), the solubilities for the two negative curvatures C\ and 2 are related by the formula log*.u (1) - log*.A2 (1) = - ^ (2 V(Ci - C 2 )/RT. (12.22) 8S. Applications. The effects listed in the preceding section have an appreciable magnitude only in the case of very large curvatures. Let us take as an example drops of water at room temperature. The constants for water at 20 C are v M (2) = 1, jr (1) 5.0 X 10 4 cm 3 /g, 1 M = 2.45 X 10 l erg/g, <r = 72.3 dyne/cm. Thence and from (12.19), with C = - 2/r, (12 .23) Even for tiny droplets of the radius r = 10 ~ 5 cm the boiling point is only 0.174 degree below that at a plane surface. However, the sur- face layer is only two or three molecular layers thick, so that our for- mula is quantitative down to about r = 5 X 10~ 7 cm. Why then pay much attention to this apparently insignificant phe- nomenon? Because it plays a large role in the processes of trans- formation of state, as they actually occur in nature. Let us consider, for instance, the condensation of water vapor when its temperature falls slightly below the point of saturation. Granting that the boun- dary layers of the vapor can and do condense on the surface of the container, its main mass is spread through the volume. Here the condensation must first form a liquid nucleus before it can proceed further, a tiny droplet which has a low boiling point, according to eq. (12.23), and is, therefore, unstable. Instead of growing it is more likely to dissolve by vaporization. This is the reason why condensa- tion does not take place through the volume of a completely pure vapor. Only when dust particles are present, do they offer to the vapor a surface of reasonably low curvature and act as nuclei of con- XII 85 THE CAPILLARY LAYER 219 densation. Very effective in this respect are electrically charged particles: their field produces an electrostatic tension at the surface of the condensed droplet which is inversely proportional to the fourth power of its radius 1 . When the radius is very small, it may overcom- pensate the effect of the curvature and make the condensation easier than at the plane surface. Quite similar conditions obtain in the case of vaporization. When water boils at 100 C, the outer pressure does not prevent small vapor bubbles arising in it. However, from the point of view of the water, the surface of the bubbles has a negative curvature, so that the boiling point in them is above 100 and they are unstable. Only when bub- bles of gases already exist in the water from the start, does the boiling take place through its volume. In the same way, surface tension has an effect on transformations in condensed phases. It is well known that crystallization does not begin without suitable nuclei, and the explanation is the same as for the condensation of gases. The case of fusion is also interesting, since most solids consist of very small microcrystals. The phenomena in crystals are somewhat complicated by the fact that the different crystallographic planes have different surface tensions, but, qualita- tively, they are similar to those treated above. We should, therefore, expect that, with the rise of temperature, the smallest units and the sharp corners and edges of the larger ones would melt first (because these parts have the lowest point of fusion) while the larger crystals swim loosely in this melt. This influence of microcrystalline structure upon the process of fusion has not yet been sufficiently appreciated. However, it seems to have a bearing on the observation of Errera that glacial acetic acid and a few other organic solids show an abnormally high dielectric constant two or three degrees below their melting points. 2 1 The electrostatic tension or additional difference in pressure between the atmos- phere and the interior of the droplet, due to the charge e on it, is equal to e*/Sira 4 . This term must be added to aC in eq. (12.17). 2 J. Errera (Trans. Far. Soc. 24, p. 162, 1928) determined the dispersion curve of the dielectric constant in the vicinity of the melting point and found that the effect in acetic acid has its maximum at 3 below 7> and for a critical frequency of electro- magnetic waves v = 10 3 . Assuming that the phenomenon is due to free solid particles capable of rotating in the liquid melt, my colleague Prof. G. W. Potapenko calculated from the critical frequency the mean radius of the particles and obtained r - 3 X 10 ~ 8 cm. This result agrees well with the formula (12.19) which gives for the surface tension <r = Jrp 2 /jif &T/T. Substituting A r = 3 (as found by Errera), IM 1.9 X 10* erg/g, P2 - 1.266 we obtain (at T - 298) a - 40 dyne/cm as the surface tension of solid acetic acid. This value lies in the range found experimentally and theoretically for other solids (section 86), 220 TEXTBOOK OF THERMODYNAMICS XII 86 Attempts to use eq. (12.22) as a means for determining the surface tension of solid salts against their saturated solutions were not very successful. The solubility of barium sulfate and calcium sulfate, in several states of dispersion, was measured by determining the elec- tric conductivity of the solutions 1 which increased with the dispersion. But the very high surface tensions calculated from these data 2 have been questioned since there are possible sources of error which would produce a similar rise of the conductivity. 3 86. Formulas for surface tension. Only when the molecular forces holding a substance together are completely known, is the theoretical calculation of its surface tension possible. So far, this knowledge is available only in the case of crystalline solids and, even here, it is restricted to a few simple space lattices. For heteropolar halogen compounds of the type of NaCl (rock salt), KC1 (sylvine), etc. Born and Stern 4 calculated the following expression er = 4.022 X 10 3 . * . , (12.24) Ai + A 2 where p is the density of the crystal and A\ t A% the atomic weights. This gives: TABLE 33 Substance LiCl NaCl KC1 NaBr KI <r (dyne/cm)... 320 149 108 125 76 Although the Born-Stern formula is only an approximation, the values derived from it are, presumably, as accurate as the results of the very difficult experiments with surface tensions of plastic solids. Experiments of this kind gave for pitch, at 20 C, about 50 dyne/cm (Ignatiew), and for lead glass, at 500 C, 70 dyne/cm (Berggren). Eotvos 5 proposed a formula for the tension of liquids against their own vapor which he obtained by a partially theoretical reasoning. He starts from the assumption that the extended law of corresponding states of section 27 applies also to the surface energy <r S which must be treated on the same footing as the volume energy pV or the internal energy U. In other words, for substances which obey the law of cor- 1 Hulett, Zs. phys. Chcm. 37, p. 385, 1901; Dunadon and Mack, J. Am. Chem. Soc. 45, p. 2479, 1923. W. J. Jones, Zs. phys. Chem. 82, p. 448, 1913. Balarew, Zs. anorgan. Chem. 154, p. 170, 1926. * M. Born, Verh. Deutsch. phys. Ges. 21, pp. 13, 533, 1919, 8 R. Edtvos, Ann. Phys. 27, p. 448, 1886. XII 86 THE CAPILLARY LAYER 221 responding states, the ratio cr 2/r must be a universal function of the reduced variables TT, T. If we imagine 1 mol of the liquid substance expanding with the temperature in such a way as to retain its geometri- cal similitude, its free surface S will change proportionally to v*. Therefore, the extended law of corresponding states leads to av** _ ~r~ ^ ' (7r ' T '* When the function / is found for one substance, it is the same for all others obeying the law of correspondence. From experiments with ethyl ether between and 190 C, Eotvos inferred that the function is/(7r, r) = 2.22(l/r - 1) = 2.22(r c /r - 1). Hence <rv" = 2.22 (T e - T) (12.25) must be a universal relation between surface energy and temperature. In fact, this formula, due to Eotvos, holds with a fair approximation for numerous substances. Ramsey and Shields 1 secured a better fit with experiments by writing ov* = k(T c - T - d), (12.26) where k is slightly variable (but not greatly different from Eotvos' value) and d is a new constant (Table 34). The formula loses, of course, its theoretical significance and becomes empirical. TABLE 34 Substance t c C k d Range in C Ethyl ether 194.5 2.1716 8 5 20-160 Methyl formate 214 2.0419 5 9 20-190 Ethyl acetate 251 2.2256 6 7 20-200 Carbon tetrachloride 283 2 1052 6 80-250 Benzene 288.5 2.1043 6 5 80-250 Chlorobenzene 360 2 077 6 3 150-300 A purely empirical formula, giving the surface tension as a function of temperature, is due to Van der Waals 2 (12.27) 1 W. Ramsey and J. Shields, Zs. phys. Chem. 12, p. 433, 1893. 2 J. D. Van der Waals, Proc. Amsterdam, 1893. 222 TEXTBOOK OF THERMODYNAMICS XII 87 It has been extensively tested by Verschaffelt l and represents the data for a number of substances fairly well. TABLE 35 Ether logio A = 1.761, B = 1.270 Benzene 1.839, 1.230 Chlorobenzene 1.810, 1.214 Carbon tetrachloride 1 . 445, 1 . 185 87. Influence of temperature and of impurities. The experimen- tal data on surface tensions in solutions can be summarized as follows. All the solutes can be divided into two classes: (A) Inactive solutes produce a slight increase of o-, proportional to ttieir molality nth (provided the concentration is not high) ff = <r (l + km*), (12.28) where ox> is the surface tension of the pure solvent. Alkali salts are inactive in aqueous solutions and the coefficient k is of the same order of magnitude for all of them, namely, between 0.0174 and 0.0357. Similar conditions obtain for solutions of aromatic substances, like camphor, aniline, benzoic acid in organic solvents. (B) Active solutes have the opposite property of strongly lowering the surface tension. Such are many organic substances in aqueous solutions, for instance, alcohols, aldehydes, ethers, fatty acids, ter- penes, etc. Soaps are particularly effective: a solution of sodium oleate of the molality m^ = 0.002 has a surface tension <r = 25 dyne/cm, while that of pure water is <TO = 73. The cleaning properties of soap solutions rest on the low values of their surface tensions. In a qualitative way these facts were explained already by Gibbs. Let us write down the expression for the differential of internal energy of the surface layer. We consider the case that the mass of the system is variable and write dU in a form analogous to (5.06), taking into account (12.01) and (5.40) dt/< 3) = TdS+ <r dS + ?*W 3 >. (12.29) ft The term pdV does not appear in this expression because it is included in 0-dS (as far as the surface layer is concerned), as was explained in discussing the formula (12.09). Similarly, N h represents here the difference between the actual value which the mol number of the components h has in the whole system and that which it would 1 J. E. Verschaffelt, Mitteil. Naturk. Labor. Gent No. 2, pp. 16, 19, 1925. XII 87 THE CAPILLARY LAYER 223 have if the surface layer were replaced by a sharp discontinuity. In other words, Nh (3) is the quantity of the component h (in mols) adsorbed by the surface layer. Although not expressly stated in section 82, it follows from it that ^ in a surface layer of negligible curvature is the same as in the two adjacent volumes. For the further discussion it is convenient to introduce the generalized work function whose differential is ^(3) = - sdT - W 3) <ty k + <* <&. (12.31) h We now make the assumption (borne out by all observations) that the work function has the analytical form ^ (3) = fa S, where fa, the work function per unit area, is a specific thermodynamical characteristic of the surface layer independent of its area S. Substituting into (12.31), we obtain o- = fa. (12.32) The surface tension is identical with the specific work function per unit area of the capillary layer. Moreover, we find We apply the last formula to the special case when one of the phases, adjacent to the surface layer and containing the component (h), is either a perfect gas or a dilute solution in the sense of Chap- ter IX. According to (9.09) and (9.10), <p h = <p h + RT\ogx h , where <ph is a function of p and T only. The partial with respect to 53* in (12.33) is taken in the sense that all the other variables (including T and p) are to be kept constant: 3^ = RT'd log #&. In dilute solu- tions Xh can be replaced by WA/WO, so that This formula is the key to the behavior of solutes. (A) In the case of inactive solutes: fiff/dm*, > 0, whence n^ h < 0, i.e. the adsorp- tion (per unit area) of the capillary layer is negative. This means that the concentration of the component (h) is less in the capillary layer than in the bulk of the solution. As we deal here with dilute solutions, the solute can exercise but a small effect upon the surface tension, in agreement with the observed facts. (B) In the case of 224 TEXTBOOK OF THERMODYNAMICS XII 87 active solutes there follows n^h > 0, the adsorption is positive. The surface layer is very much smaller in volume than the adjacent phases and may build up to a high concentration in the solute (A), by adsorb- ing it even from a very dilute solution. This is the explanation why the increase of the surface tension by a small amount of solute is always slight, while the decrease may be considerable. It will be well to say here a few words about the temperature dependence of the surface tension. As the data of Table 32 indicate, <r decreases with rising T. Moreover, it may be concluded from the good fit of the formulas (12.25) and (12.26) that this drop is linear over wide ranges of temperature. In fact, the molal heat v appearing in these expressions changes but little with T and can be regarded as constant in the first approximation. The underlying cause of the linear drop can be inferred from the following considerations. Let us apply eq. (12.30) to a pure substance: the question of adsorption has, then, no interest and we may leave out the last term, writing it in the form (dividing by S) fa = w 2 - 7s s or from (12.32), (12.33) Within the range where a can be represented as a linear function (<r = cri aT) this gives u% = <7i = const. The internal energy (per unit area) of the capillary layer is constant and independent of temperature. This ceases to be true in the vicinity of the critical temperature T where the linear laws (12.25) and (12.26) are no longer valid. The understanding of the physical nature of certain surface films (oil films) has been considerably advanced by investigations of Harkins (and collaborators), Langmuir, and Adam, 1 but this subject lies out- side the scope of the present book. 1 W. D. Harkins, F. H. Brown, and E. C. H. Davies, J. Am. Chem. Soc. 39, p. 354, 1917; I. Langmuir, ibidem 39, p. 1848, 1917; N. K. Adam, Proc. Roy. Soc. (A) 99, p. 336, 1921; 101, pp. 452, 516, 1922; 103, pp. 676, 687, 1923. CHAPTER XIII NERNST'S HEAT POSTULATE OR THE THIRD LAW OF THERMODYNAMICS 88. The principle of Thomsen and Berthelot. It was emphasized in Chapter IV that the fundamental eq. (4.07) by means of which the concept of the entropy is derived from the second law is a differential equation. Because of this, the definition (4.09) of S contains an additive constant of integration so that the absolute value of the entropy cannot be determined from the first and second law. The indeter- minateness of the entropy constant has serious implications for the general character of the thermodynamical laws, in that they contain coefficients of indeterminate numerical value. As an example, we can point to the law (8.17), (8.16) of chemical equilibrium of perfect gases: The expression of the equilibrium constant K contains a factor /, which depends, according to the formula (8.14), upon the entropy constants of the gases taking part in the reaction. It is clear from this that the knowledge of the absolute entropy would be of inesti- mable value, as it would greatly enhance the usefulness of many thermodynamical formulas. To obtain this knowledge one must have recourse either to experiment, or to new theoretical principles not contained in the first and second law. Of course, it is quite suf- ficient to determine the entropy of a substance in one special case, in order to deduce the entropy constant, and it was pointed out by Nernst (1906) that experimental data were available to make a general conclusion about the behavior of the entropy of condensed systems in the vicinity of the absolute zero point of temperature. Let us start our considerations from the Gibbs-Helmholtz equa- tion (5.48) relating to an isothermal process We have seen in section 36 that, under ordinary conditions, the process (reaction) takes place spontaneously when A$>0. The dif- ference of thermodynamic potentials A* m $ 2 $1 can be regarded, therefore, as a measure of the chemical affinity. Suppose now that 226 TEXTBOOK OF THERMODYNAMICS XIII 89 the partial (d&$/dT) p does not become infinite; when T goes to zero. In this case we find, putting T = 0, AX) = 0, (13.02) or lim Tm0 (&$ Q p ) = 0, if the process is conducted also at constant pressure. In the vicinity of the absolute zero point the heat of reaction becomes identical with A< and the criterion of the reaction taking place can be replaced by AX > or Q p > 0. For a long time it was thought that this form of the criterion holds good at any tem- perature. In fact, Thomsen and, independently, Berthelot stated the principle that out of a number of possible reactions the one will take place which evolves the largest amount of heat. According to our present knowledge, this would be true if the right side of eq. (13.01) were always negligible so that we could write at all temperatures A$ - AX = 0. (13.03) As we have already mentioned in section 37, this equation is erroneous. Modern investigations have shown that the term T(d&$/dT) p must be taken into consideration. Nevertheless, the approximation given by the principle of Thomsen-Berthelot, and by eq. (13.03), is surprisingly good even at room temperature, provided the system is condensed and the affinity of the reaction not very small. A comprehensive set of measurements of A<i> = $2 3>i for reactions in condensed systems was carried out by T. W. Richards 1 with the result that the difference A< AX was very small, indeed. Exercise 99. Check that at room temperatures r(DAf>/3D p is small compared with A* for the reactions in galvanic cells given in formula (5.52) and in exercises 56, 57. 89. Nernst's own formulation of his postulate. This last fact led Nernst to assume that in condensed systems the right side of eq. (13.01) decreases more rapidly, when the temperature goes to zero, than is accounted for by the factor T. According to him, the remaining factor also decreases to naught in the limit T = : for any isothermal process The assumption expressed in this formula is known as Nernst's postulate or the third law of thermodynamics. That it is natural and * T. W. Richards, Zs. phys. Chemie 38, p. 293, 1902, XIII 89 NERNSTS HEAT POSTULATE 227 convincing is best brought out by the following considerations. Equa- tion (13.01) can be written in the following form If the partial on the left side does not become infinite, there follows from (13.01) that both AS> AX and T become zero in the limit !T = 0. It must be well understood what a partial with respect to T at constant pressure means in this case. The pressure in the two terms of A$ = $2 $1 and AX = X2 Xi is not necessarily the same: A<i> = <S>2(r, 2 ) - 3>i(r,i), etc. But the partial is taken so that it remains equal to p% in the first term and to p\ in the second. Let us now take the limiting values of both sides of eq. (13.05) for T 0, keeping the pressure constant (i.e. equal to p% in <i>2 and X2 and to pi in <J>i and Xi). The right side is indeterminate, being equal to 0/0, but the indeterminacy can be readily resolved by means of the well- known mathematical rule : instead of the numerator and the denomina- tor, one has to substitute, respectively, their partials with respect to T. /9A$\ /3AX mr - w- hmr - Var /," hmr - whence 0. (13.06) If we plot the change of the heat function AX in any isothermal, chemical or physical, process against the temperature T at which it is conducted, the curve has a horizontal tangent at the limit T = (Fig. 44). We repeat that this conclusion is independent of Nernsfs postulate'. the only assumption on which it rests is that OAS/Q^Op does not become infinite for T = 0. The difference A<J> is equal to AX in the limit T = 0, and is very little different from AX over a considerable range of temperatures, according FlG 44. to the experimental data mentioned in the preced- postulate. ing section. Nernst's heat postulate (13.04) has the purpose of insuring such behavior. The curves for A< and for AX have, if it is accepted, a common horizontal tangent, at T = 0, and remain for some distance close to each other. Owing to eq. (5.37), (d$/dT) p S, an alternative form of Nernst's postulate (13.04) is lim r .o(5 2 - Si) - 0. (13.07) 228 TEXTBOOK OF THERMODYNAMICS XIII 90 The entropy remains unchanged in any isothermal process taking place in a condensed system in the vicinity of the absolute zero. Although Nernst enunciated his postulate only for condensed systems, this restriction may now be dropped for the following reasons: (1) In sufficient proximity to the zero point of temperature every known system of nature is condensed, if the pressure is finite. (2) We shall see in Chapter XVI that even the ideal equation of perfect gases has been changed by the modern theory of statistics so as to satisfy Nernst's law. Exercise 100. Starting from eq. (5.58) prove: (a) If limr-oteA^/aTV ^ , then lim^.o (&&U/dT)v - 0. Use the same argument as that used in the text for proving (13.06). (b) From (a) and from (13.06) there follows limr-o (CV2 - CVO - 0, Hmy.o (C P2 - C p i) = 0. These results are independent of Nernst's postulate. (c) Equation (13.07) requires 0. 90. Thermal expansion in the vicinity of the absolute zero. There can be deduced from the formula (13.07) some immediate consequences about the properties which all substances have in their condensed state at temperatures close to T = 0. Let the process to which we apply this equation consist, merely, in an infinitesimal increase of pres- sure (from pi = p to p2 = p + dp) at constant temperature. The entropy of the system is changed, accordingly, from Si = S to S 2 = S + (dS/dp) T dp, and Nernst's postulate (13.07) takes the form lim r .oO5/3/>)r = 0. (13.08) Because of the relation (4.46) of section 28 this is identical with lim r . (||) =0. (13.09) The coefficient of thermal expansion vanishes in the vicinity of T = 0. This fact is important in itself and leads to some interesting inferences. If the pressure is kept constant, the volume is sure to remain constant automatically: in the limit T = 0, a partial with respect to T at constant p is, at the same time, a partial at constant v. In the formulas of the last sections we could replace the subscript p by v or drop it altogether. According to the definition (3.08) of the specific heat this applies also to the quantities c p ,c v , lim r _ C P =* lini r .o d Hm r . c. (13.10) XIII 91 NERNST'S HEAT POSTULATE 229 We can readily obtain the generalization of the formula (13.08) for the case when the system is not simple but is described by the variables Xi, J^2, X n and the generalized forces y\, 3^2, ... ^n, in addition to T (compare section 7). In fact, let us consider processes which consist in changing only one parameter, while the rest of them remain constant: for instance, Xh changes to Xh + dX^ while all the other Xj (j j h) and all the yj (including yh) stay constant; or else, y h changes to yh + dyn and y, (j 7* h) and all the Xj are constant. We find as above, respectively, oC oC lim r .o ^T - O t lim r . f- - 0. (13.11) dXh Qyh Because of (5.30) and (5.45) this is equivalent to Hm,.o - 0. fonr-o ? - 0. (13.12) The temperature gradients of all the parameters X^ yh describing the em vanish in the vicinity of T = 0. In particular, we conclude that system 91. Specific heat in the vicinity of the absolute zero. Nernst's postulate implies that the entropy is finite at all (finite) temperatures, including the limiting case T = 0. We can choose, therefore, T = as the lower limit of the integrals (4.33), (4.34) representing the entropy of a simple system *(7>) = f /0 (13.14) r 1 , JT *(T,p) These integrals must be convergent, a requirement which could not be satisfied, if c p and c v had a finite value for T = (because this would make the integrals logarithmically infinite). We must con- clude, therefore, that the limiting values of c p , c v vanish lim r- o c p = Hm r . c* * 0. (13.15) The specific heat vanishes in the vicinity of T 0: It should be noted that Nernst's postulate is not needed for this conclusion in its 230 TEXTBOOK OF THERMODYNAMICS XIII 92 entirety: it is derived merely from the assumption that the entropy is not infinite in the limit T ~ (lim r . S ^ ) As far as the second law is concerned, the terms /i(fl),/2(/>) in (13.14) may be any functions of volume and pressure (section 26). However, if we apply to them the conditions (13.08) and (13.13) flowing from Nernst's postulate we find dfi/dv = 3/2/3/> = 0. These terms turn out to be independent of v and p and are, therefore, constant T fT %dT + s , *(P,T) = -* * '0 * dT + s , *(P,T) = -*dr + s . (13.16) SQ, the limiting value of the entropy at the absolute zero point, is a constant independent of the process by which this point is reached, as may be inferred from the requirement (13.07). From these expressions can be obtained the molal thermodynamic potential, for a condensed simple system, PV-T f */n c p dlog r- TSQ. (13.17) Exercise 101. Prove eq. (13.15) in a different way. The condition lim^-oS 5^ <*> can also be written in the forms limr-o (W/dT)v ^ and limr-o (d&/dT) p Derive from the eqs. (5.56) and (5.54) by the method used in obtaining (13.06) r n ) - limr_o I - or Hmr-o Cv = Hmr-o C p = 0. 92. Planck's formulation of Nernst's postulate. It is obvious that the last result can be generalized for the case of systems which are not simple but depend on any number of variables. Let us sup- pose, in fact, that the zero point of temperature is attained at definite values of all the other variables and that the entropy is found to be equal to 5 . According to the postulate (13.07) or to eqs. (13.11), further changes in the variables (leaving the temperature constant) will fail to have any effect upon SQ. The state T = is, therefore, a state of constant entropy. Planck goes beyond Nernst in that he makes an additional assump- tion about this limiting value of the entropy: he postulates that the entropy in the state T = is equal to zero, in every system occurring in nature: lim r . 5 = 0. (13.19) XIII 93 NERNSTS HEAT POSTULATE 231 In the special case of simple systems, the entropy expressions (13.14) are reduced to /T /*T ^dT, s(T,p) = / C -*dT. (13.20) 1 A) * The modern statistical theory has borne out Planck's contention. When we refer to Nernst's postulate in the following chapters, we shall always mean Planck's formulation of it, unless expressly stated otherwise. Equation (13.19) helps to determine the constants or arbitrary functions of integration in many thermodynamical relations. Thus, the presence of the functions /(/>), f\(v) in eqs. (5.55) and (5.57) is but a consequence of the indeterminacy of the entropy inherent in the second law. This is brought out, for instance, by partial integration (taking into account the formulas 3.12 and 3.26) The parentheses are equivalent to the entropy expressions (13.14). Replacing them by (13.20) we obtain * = T / U v d(l/T), * - T I X p d(l/T). (13.21) JQ ^o Since (3^/3 F)r = p, the first of these equations gives Knowing the internal energy as a function of T and 7, down to T = 0, we can deduce the equation of state. Exercise 102. The molal heats c p (in cal mol~ l , at 1 atm) of diamond, graphite, and copper are given in the following table: T ....... 20 K 30 40 50 60 70 80 90 100 150 200 250 300 Diamond ............................ 0.01 0.06 0.33 0.60 1.00 1.50 Graphite 0.02 0.06 0.10 0.14 0.18 0.24 0.30 0.33 0.38 0.77 1.2 1.4 2.05 Copper.. 0.1 0.4 0.8 1.4 1.94 2.5 3.0 3.5 3.9 5.0 5.4 5.7 5.9 Calculate in a rough, graphical way the absolute entropies of these substances at 300 K. 93. Einstein's formula for the specific heat of solid bodies. In order to establish his postulate it became necessary for Nernst to test whether its consequences are borne out by experiment. In particular 232 TEXTBOOK OF THERMODYNAMICS XIII 93 he carried out with his pupils l an extensive program of measurements on specific heats of solid bodies at low temperatures. In all cases the law expressed in formula (13.15) was confirmed: as the substances were cooled, their specific heat decreased, and, at the lowest point attained (temperature of liquid air, later liquid hydrogen and helium), it amounted to only a small fraction of its value at room temperature. The previous knowledge of this question had been as follows. In 1819 Dulong and Petit deduced from their measurements the law that elementary substances in the solid state have all the same molal (or atomic) heat of approximately c = 6 cal/deg (section 16). How- ever, it was realized for a long time prior to Nernst's work that the law of Dulong and Petit is not accurate and that c varies with the temperature. A well-known example of this was the diamond, which exhibited, in a striking way, the type of behavior later found to be general by Nernst and his pupils. The specific heat of diamond has a value approaching the 6 cal/deg of Dulong and Petit only at temperatures which are comparatively high. It decreases rapidly upon cooling and is already pretty small (1.5 cal/deg) at room temperature, a fact which was the source of much embarrassment for the kinetic theory of matter. Boltzmann had given the following simple approach to the problems of the internal energy of elementary solid bodies: In such a body every atom has a position of rest about which it carries out thermokinetic oscillations. As a first approximation, one can regard the forces with which the atom is attached to its normal position as proportional to the distance from it. It is well known that forces of this kind (called " elastic forces ") cause the atoms to oscillate harmonically, with a constant frequency v. Since all atoms of an elementary body are equivalent, each of them is attached by the same force and has the same frequency v. What is the energy of such a model of the solid body, according to statistical mechanics? According to (4.71), the classical statistics of Boltzmann-Gibbs leads to the so-called principle of equipartition of energy. The mean kinetic energy of every particle subject to heat motion is kT/2 per each degree of freedom (where k denotes Boltzmann's constant of section 4). It was also stated in section 30 that, in the case of elastic forces, the mean potential energy 1 Monographs on Nernst 's postulate: W. Nernst, Thermodynamics and Chemistry. New York 1907; W. Nernst, Die theoretischen und praktischen Grundlagen des neuen Warmesatzes. Halle 1918; F. Politzer, Berechnung der chemise hen Gleichgewichte. Stuttgart 1912. A very useful critical study of the subject is due to W. H. Van de Sande Bakhuyzen, Het Warrate-Theorema van Nernst, Thesis. Leiden 1921. XIII 93 NERNST'S HEAT POSTULATE 233 is equal to the mean kinetic and the total energy becomes kT per degree of freedom. An atom is supposed to have only three degrees of freedom and, therefore, the mean energy 3 kT. Correspondingly, the internal energy of one mol is u = 3n A kT + UQ = 3RT + u Qt since WA* = R, according to eq. (1.14), and the specific heat c = du/dT = 3R. (13.23) Substituting R = 1.986 cal/deg, we obtain from this theory the numerical value c = 5.95, in good agreement with the law of Dulong and Petit. However, it does not offer any explanation for the devi- ations at low temperatures. This difficulty was resolved by Einstein l in one of his most impor- tant papers. He pointed out that the quantum theory provides a modification of statistics sufficient to account for these phenomena. In fact, Planck had found that, in the case of the harmonic oscillations of an electron of frequency v, the mean energy is not the equipartition value kT but kTx/(e x 1), where x is an abbreviation for x = hv/kT, and h denotes Planck's quantum of action (h = 6.55 X 10~ 27 erg sec). Einstein made the assumption that what is true for an oscillating electron must also apply to an oscillating atom. It was the first appli- cation of the quantum theory to atoms and molecules, and led directly to the result (13.24) The entropy follows from the expression (13.20) of Nernst's theorem in Planck s formulation -<->] Y - log (1 - e-*)J. (13.26) When the temperature T is high compared with 9 = hv/k (i.e. x <3C 1), we find, neglecting terms of the order x u = 3RT + UQ, c - 3-R, 1 (13.27) 5 = 35(1 -logic), .j 1 A. Einstein, Ann. Physik 22, pp. 180, 800, 1907. 234 TEXTBOOK OF THERMODYNAMICS XIII 93 The limiting values, for high temperatures, are the same as in classical statistics and in agreement with the law of Dulong and Petit. On the other hand, in the vicinity of the absolute zero point (r 0, #-*oo) the formulas (13.24), (13.25) reduce to u = UQ, c = 0, satisfying the requirements of Nernst's postulate. Einstein's formula agrees with the general trend shown by the experimental values of the specific heats but the numerical agreement is not very close. Later work on the subject, which resulted in more accurate formulas, will be treated in section 123. For theoretical uses, however (when a fictitious solid body is as good as real one), Einstein's model is very convenient because of its simplicity. Exercise 103. Find the work function (^ = Ts) of the Einstein model and calculate its approximate expressions for very high and for very low tempera- tures (x 1 and x 1). CHAPTER XIV BEARING OF NERNST'S POSTULATE ON THE ABSOLUTE ENTROPY OF GASES 94. Chemical equilibrium and Nernst's postulate. The bearing of the first and second laws of thermodynamics upon the problem of the chemical equilibrium of perfect gases was discussed in Chapter VIII. It was stated in section 51 that the first step in solving this problem was the mass law of Guldberg and Waage (1867) > log p h = log K p , (14.01) which gives a relation between the partial pressures pi, . . . p<r of the gases taking part in a reaction ^jV h G h = 0, (14.02) h conducted at a given temperature. The next big advance was due to Gibbs (1871) who determined the dependence (8.16) of the mass constant K p on temperature. Within the range where the molal heats can be regarded as constant log K p =^ log T - % + log I. (14.03) Provided the heat of the reaction Q p and the change of the heat capacity AC in it are known, the constant of integration / can be determined from the measurement of K p at one single temperature. The formula (14.03) serves, then, to calculate the mass constant for any other temperature. We have seen that the constant / depends upon the entropy constant of the gases in a way expressed by eq. (8.14) (14.04) This shows that a theoretical calculation of the constant / is beyond the scope of the first and second law of thermodynamics, which provide 235 236 TEXTBOOK OF THERMODYNAMICS XIV 94 no way of determining the absolute value of the entropy. The formula (14.03) with an empirical constant / represents, therefore, all the information which can be obtained from the older thermodynamics, with respect to the chemical equilibrium of gases. It was pointed out in section 59 how Van t'Hoff (1878) succeeded in generalizing this formula (still with the empirical constant /) for the case of the chem- ical equilibrium of solutes in a dilute solution. The problem remained in this state until 1907 when Nernst brought his postulate to bear on it. It is possible to establish a relation between the entropy constants of a substance (ft) in its gaseous state soh and its condensed state W c) . In fact, let us consider the equilibrium between the vapor and the solid phase. According to eq. (7.02), the condition of equilibrium is * = gh<Ph (c \ where <p h without upper index represents the molal thermodynamic potential of the gas and ^ (c) that of the condensed phase while g h = M*/W C) = v h (c) /v h is the ratio of the molecular weights in the two phases. We suppose that the temper- ature and pressure are such that the vapor can be considered as a perfect gas. In this case we can substitute for ^ the expression (8.08), (8.10) where we consider c ph as constant, and for <(>h c) the eq. (13.17) and obtain a relation which may be written in the form <*r + iog4, (14.05) where ), (14.06) denotes the latent heat of vaporization per 1 mol of the vapor, and (14.07) is a constant called by Nernst the chemical constant of the gas. Equation (14.05) is, of course, only the integrated form of eq. (7.05) of section 44, It is, however, important for us to know that the constant of integration ** is connected with the entropy constant by the relation (14.07) which permits us to express $o* in the form SOH = Ri h + ft** 1 *. (14.08) Substituting this into (14.04) we find R log I = #2>A*A -Z'A^W* (14.09) The second sum on the right side is of a structure familiar to us: It represents the limiting value (for T = 0) of the entropy change A5 in the reaction SJ>A (C) (?A (C> =0; in other words, it is the entropy XIV 95 ABSOLUTE ENTROPY OF GASES 237 change which would result if all the components taking part in the gas reaction (14.02) were condensed and the reaction conducted in the condensed state in the vicinity of T = 0. We know from Nernst's postulate in the form (13.07) that Z)* W W* = lim r-o AS = 0. A Therefore the relation (14.09) is reduced to log /*]*. (14-10) A This formula represents the third important advance in the problem of the chemical equilibrium of gases and is due to Nernst. While before 1907 the constant / had to be determined experimentally for every separate gas reaction, eq. (14.10) made it possible to compute it from the chemical constants i h of the components if they were known. It was sufficient to determine by measurement the constants 4 of the individual gases by investigating their equilibrium with the condensed phases in order to know the constants / for all possible reactions between them. We have derived the relation (14.10) using the more restricted form (13.07) of Nernst's postulate. If we accept Planck's formulation of it (13.19), we have to put W c) = 0, and this reduces eq. (14.07) to 4 = SM/R. (14.11) From this point of view the chemical constant of a perfect gas is, simply, proportional to its entropy constant. Nernst's relation (14.10) is then an immediate consequence of the definition of the constant / given by eq. (14.04). 95. The Sackur-Tetrode formula for the chemical constant. The experimental determination of the chemical constant i involves very difficult measurements. It was, therefore, fortunate that in 1912 Sackur l and Tetrode 2 succeeded in making an important step beyond Nernst. Working independently they gave theoretical expres- sions for the chemical constant of monatomic gases which differed but slightly. Tetrode's formula proved to be the correct one 1 0. Sackur, Nernst-Festschrift, p. 405, 1912; Ann. Physik 40, p. 67, 1913. 1 H. Tetrode, Ann. Phyaik 38, p. 434; 39, p. 255, 1912. 238 TEXTBOOK OF THERMODYNAMICS XIV 95 Of course, this result could not have been obtained by thermo- dynamical reasoning: the resources of thermodynamics, including Nernst's theorem, are exhausted with the formula (14.10). Sackur and Tetrode used statistical methods about which we shall say more in the next section. The argument of the logarithm in eq. (14.12) is not a pure number, so that the numerical value of i depends on the system of units used for the description of the gas. The reason of this lies in the structure of eq. (14.05), where in addition to two terms of dimension zero, we have two logarithmic terms. Taking into consideration that, for a monatomic gas c p /R = f, the combination of the two logarithmic terms is log (p/T^). This term depends on the units in which the pressure and the temperature of the gas are measured and, therefore, the same is true with respect to the chemical constant i. If we substitute into (14.12) the numerical values in the G.C.S. sys- tem R = 8.315 X 10 7 erg/deg, n A = 6.061 X 10 23 mol" 1 , h = 6.554 X 10~ 27 erg sec, we obtain j = i - f = 10.171 + f log M , (14.13) an expression which corresponds to pressures measured in dyne/cm. For use with decimal logarithms, this expression must be divided by log 10 = 1/0.4343 jio - no ~ 1.086 - 4.417 + f logio /* (14.14) If we wish to measure the pressure in atmospheres instead of dyne/cm and to use decimal logarithms, we have to subtract logio 1 013 249 = 6.0057, obtaining the constant jio, atm = iio, *tm ~ 1-086 = - 1.589 + f lo glo M- (14.15) It was later found that the Sackur-Tetrode expression gives the main part of the chemical constant but is incomplete: there are additional terms depending on the electronic and nuclear spins. The agreement of the theoretical values with the experimental is, on the whole, very good but it will be better to postpone the comparison until section 120 where we shall derive the missing terms. When accuracy is not required and the question is only as to the order of magnitude, it is permissible to neglect the refinements and to use the above expressions without correction. Therefore, it will be useful to write out here the explicit form which the equation (8.16), XIV 95 ABSOLUTE ENTROPY OF GASES 239 (8.17) of equilibrium of perfect gases assumes when the numerical values of the constants are substituted in the form (14.15) logio K p = ]C fc logio ph = 0.503 v f 2J vh logi lo glo The pressure p is here expressed in atmospheres, the heat of reac- tion Q p in cal/degree. 96. Theoretical derivation of the Sackur-Tetrode formula. Although Sackur and Tetrode obtained the correct expression (14.12) of the chemical constant, the method of reasoning they used did not stand the test of later criticism, and cannot be considered as altogether valid. However, a rigorous derivation of this expression was given by O. Stern 1 in 1913. As this problem lies beyond the pale of pure thermodynamics, he had to appeal to the principles of statistics and of the quantum theory. But the statistical elements in Stern's argumentation are so simple that it can be given in its completeness even in a textbook of thermodynamics. The fundamental idea is to consider a vapor obeying the law of monatomic perfect gases in equilibrium with its condensed phase, and to calculate its pressures, on one hand, thermodynamically from eq. (14.05), on the other hand, statistically. The comparison of the two results leads to the expression for the chemical constant. This constant determines the entropy of the perfect gas and, therefore, it can depend only on the properties of the gaseous phase of our system remaining the same no matter what kind of condensed phase is in equilibrium with it. It is, there- fore, permissible to consider the equilibrium even with an imaginary condensed phase which does not really exist but could exist, inas- much as it is constructed in agreement with the laws of nature. Stern takes as the condensed phase Einstein's model of the solid body described in section 93 and assumes that the temperature T of the system is sufficiently high compared with = hv/k to use the approx- imations (13.27). 2 We substitute them into eq. (14.05), taking the solid body also as monatomic (g = 1) and its volume as negligible (v 0) and recalling that for a monatomic perfect gas u 3.RZV2 + wo, c p = 5.R/2. If we drop the subscript log = f + f log T - 3[1 - log (hp/kT)] - "~^ + * . - | - log T + 3 log (hv/k) - U ^ (C) + i. (14-17) * O. Stern, Phys. Zs. 14, p. 629, 1913; Zs. Electrochemie 25, p. 99, 1919. pifferently from the preceding section, v is here the freqnency. 240 TEXTBOOK OF THERMODYNAMICS XIV 96 With this result the thermodynamic part of the investigation is concluded and we proceed to the statistical part. We have specified that the temperature is high, and this permits us to use the principles of statistics in their traditional form since, in this case, the differences between classical and quantum statistics disappear. All we need of the quantum theory is already contained in eq. (14.17), which embodies the properties of a quantized solid. In Einstein's model every atom of the solid body is regarded as a harmonic oscillator attached to the position of rest by a force F proportional to the distance r from it. The equation of motion of an atom is, then, where M is its mass and co = 2irv its angular frequency. The right side of the equation represents the force F, so that the potential energy of an atom at the distance r from its position of equilibrium is e r = - f F dr = Mu 2 r 2 /2. (14.18) /Q These assumptions were sufficient for the theory of specific heats (section 93) but, in order to calculate the equilibrium of the solid with its vapor, we must specify them a little farther. How far does the field of the force F extend? The atoms are all considered as indepen- dent in their oscillations so that it cannot reach as far as the next atom. To satisfy this requirement we imagine, around every position of rest as a center, a sphere of the radius a. Inside each sphere there exists a radial force F = u?Mr while the space outside the spheres is field- free. The model of the solid body is reduced to a number of little spheres with fields of forces in them. The atoms which are inside the spheres belong to the condensed phase, while the free atoms outside form the vapor. Let dr be an element of space in one of the spheres situated at the distance r from the center, and let Zrdr be the mean number of atoms in this element (i.e. the mean of many observations taken at different times). The only proposition from statistical mechanics which we need is Boltzmann's principle (4.69) which tells us that the mean numerical density ZT is proportional to an exponential function of the potential energy e. (14.19) C being a constant of proportionality. XIV 96 ABSOLUTE ENTROPY OF GASES 241 As the space outside the spheres is field-free, the potential energy eo of the free atoms is the same as that of an atom at the border of the sphere eo = Mw 2 a?/2. Denoting the numerical density (i.e. number of atoms per unit volume) of the free atoms by ZQ we can apply Boltzmann's principle also to them 0o = Cexp (-eo/^r), (14.20) whence eliminating the constant C from the two equations z r = 2 exp If we integrate z r over the whole volume of the sphere, we obtain the mean number of atoms in it. It is obvious that this number must be equal to 1. In fact, in a real solid the number of atoms is equal to the number of positions of rest. If a part of the solid is vaporized the positions of rest disappear with the atoms. We can write, therefore, /FO / Zrdr = 4irzo0f I (14.21) In the thermodynamic eq. (14.17) small quantities of the order x = hv/kT were neglected. To be consistent we must neglect terms of the same order also in the expressions derived from statistics. This remark shows the way of getting rid of the radius a. Since the Einstein model is not a real but an imaginary solid, the quantities v and a can be selected at our discretion and we can choose them so that y = Mu?a?/2kT will be still large at the high temperature at which x = hv/kT is very small, so that exp( y) can be made smaller than x. The integral from to a in (14.21) can then be replaced by the integral from to oo , since the two differ only by terms of that order. We obtain thus 1 = so(27r*r/Mco 2 )* exp (eo/r). The potential energy of 1 mol of the gaseous phase is n A eo. It is the energy which would have to be expended to make n A atoms free, if all the bound atoms were at rest in the centers of their respective spheres. This state of affairs prevails at the absolute zero of temper- ature when the atoms have no kinetic energy. The physical meaning of n^eo is, therefore, the difference in the zero point energies of the gaseous and the solid phases: w^eo = wo W c) . Therefore, exp [(W c) - u )/RT] t (14.22) 242 TEXTBOOK OF THERMODYNAMICS XIV 96 since 1? = kn A and o> = 2irv. The number ZQ defines the vapor pres- sure according to eq. (1.15), p = zokT. Substituting ZQ from (14.22) and taking the logarithm, log p = f log 2wM + | log k - 3 log h 2 - | log T ~ (C) . (14.23) Comparing this expression with (14.17), we find that in forming the difference the quantities p, T, v, UQ o (c) are eliminated and we obtain for the chemical constant i precisely the expression (14.12) of Sackur-Tetrode. In the preceding discussion we have not made use of the electric neutrality of atoms. All our considerations and formulas would apply, therefore, just as well to electrified particles attached to their respec- tive spheres by electric forces, provided a gas consisting of such parti- cles is subject to the law of perfect gases. The slight corrections which are necessitated by the existence of electronic and nuclear spins as well as the theory of chemical constants of diatomic and polyatomic gases will be dealt with in section 120. CHAPTER XV CRITICAL ANALYSIS OF NERNST'S POSTULATE 1 97. Unattainability of the absolute zero. In the years following the discovery of Nernst's theorem the question was much discussed whether it really represented an independent and new principle of science or was contained, in some way, in the first and second laws of thermodynamics. The relation of the isothermal !T = to the system of adiabatics, S = const, played an important role in these discussions. The second law not only states that the entropy is a function of the state but implies also that it is a unique function of the state. No system can have two different values of the entropy at the same time. In fact, if this were possible in some state of the system, this state would be common to two adiabatics (say, S = S\ and S = 2, where Si > S%) representing their intersection. This leads to a contradiction with the postulate contained in the second law that in an adiabatic process the entropy cannot decrease: starting from the entropy Si one could lead the process to the state of intersection and thence along the adiabatic 52. It follows that the system of adiabatics S = const is a family of surfaces which do not possess intersections or envelopes. However, it must be remembered that the adiabatic process is defined as one in which no heat is imparted to the system, its equation being dQ = TdS - 0. (15.01) In addition to the solution dS = 0, or S = const, this equation has also the singular solution T = which represents at the same time an isothermal and an adiabatic process. Taking the (/>, V)- diagram of a simple system as an example, one is inclined, at first sight, to expect conditions as in Fig. 45: the lines of constant entropy form a non-intersecting family (of which we give only the two members 5 = Si and 5 = 2), while the curve T = cuts across. It seems possible to lead the system by a succession of reversible adiabatic 1 With the kind permission of Yale University Press parts of this Chapter were patterned after the exposition by P. S. Epstein, Commentary on the Scientific Writ* ings of J. W. Gibbs, Article 0, Sections 6, 7. 243 244 TEXTBOOK OF THERMODYNAMICS XV 97 processes from the state A to the state D of a different entropy, and this would be a contradiction with the second law of thermodynamics. On the other hand, this contradiction would not exist if the curve T = were identical with one of the lines S = const, instead of being a stranger in their midst and cutting across their system. There are, however, two reasons why it is not permissible to conclude from this argument that Nernst's postulate is a consequence of the second law. The first of them was realized early, being brought home forcibly by the example of the perfect gas which obeys the second law but contradicts Nernst's theo- rem. In the (p, v) -diagram of the perfect gas the line T = is represented by the axes p = and v = which serve as asymptotes both of the system of isother- mals pv = const and of the system of adiabatics pv y = const. For this reason the line T = cannot be reached by any FIG. 45. State T - in finite and reversible adiabatic process so relation to adiabatics. that the conditions illustrated by Fig. 45 do not apply. It is true that, in the case of the perfect gas, the isothermal T = also belongs, mathematically speaking, to the system of lines of constant entropy, S = const, being its extreme member S = <*> in view of the entropy expression (4.18) 5 = c p log T - R log p + *o, (15.02) (where we suppose p j& 0, v = 0). Nevertheless, this system does not satisfy Nernst's theorem which requires that no change of entropy be possible for T = 0. By changing the pressure from pi to p2 we can change s in formula (15.02) by the finite amount As r-0 = R log (pi/p2), while it retains its singular value oo . The example of the perfect gas is a conclusive proof that the second law of thermodynamics does not contain Nernst's theorem in its en- tirety. But attempts continued to show, at least, that certain conse- quences deduced from it could be also obtained from the second law. In this connection there was advanced by Nernst the principle of the unattainability of the absolute zero of temperature. 1 If one of the states T = could be reached by a reversible adiabatic process, the diagram of the Fig. 45 would be an adequate illustration of the actual conditions. Pointing out that the succession of adiabatics AB, BC, CD could be 1 W. Nernst, Die theoretischen und praktischen Grundlagen des neuen WSrme- eatzes. Halle 1918. XV 98 CRITICAL ANALYSIS OF NERNSTS POSTULATE 245 used to decrease the entropy of the system, Nernst concludes that the assumption of attainability is in contradiction with the second law. This conclusion was challenged by Einstein, 1 who questions that the part BC of the process (corresponding to T = 0) can be carried out. A reversible process is only an ideal, and there is in reality always a certain degree of frictional waste of work, developing heat. However, the slightest amount of heat production would throw the system off the curve T = 0. To bring out the gist of Einstein's objection, we may amplify it as follows. From the point of view of physics, a process can only then be considered as defined and meaningful when an experimen- tal procedure is given by which it can be carried out. In general, the adiabatics and isothermals satisfy this requirement. In the case of a simple system, for instance, the adiabatic process is carried out by compressing (or expanding) the system while it is enclosed in a heat- insulating envelope, the isothermal process by the same procedure while the system is in thermal contact with a heat reservoir. The particular process T = has the peculiarity that, while it is isothermal, no heat is transferred in it to the reservoir. Nernst treats it, therefore, as an adiabatic process. This is equivalent to dropping the heat reservoir altogether and imagining the system adiabatically insulated in all three branches of the Fig. 45. However, the lack of experimental differentiation between the compressions along the lines BA and BC makes the branch BC meaningless. In fact, no experimental direction is given (or, indeed, possible) to insure that a compression, starting from the point B, should cause the system to move in the curve BC and not in BA. This is the second reason why the argument making use of the diagram of Fig. 45 is fallacious. These objections refer, however, only to the attempt to derive the principle of unattainability from the second law. If Nernst's postulate is accepted, unattainability becomes a matter of course since it states (in Planck's formulation) that T = coincides with 5 = 0. The process T = belongs, therefore, to the family S = const and is identical with its extreme member. As the adiabatics S = const do not intersect, it is obvious that 5 = cannot be reached by any reversible process represented by one of the other adiabatics. 98. Solutions, supercooled liquids, and the statistical interpreta- tion of Nernst's postulate. Of great practical and theoretical import- tance is the question whether Nernst's postulate is a general law or admits exceptions for certain condensed systems. Some controversy 1 A. Einstein, Structure de la matiere (Second Solvay Congress of 1913). Pub- lished 1921. 246 TEXTBOOK OF THERMODYNAMICS XV 98 arose with respect to solutions : Planck 1 maintains that his formula- tion of the theorem applies only to chemically pure substances while in the case of a solution or mixture one should add to the entropy the expression so = - R(Ni + . . . + #/)/, log *,, (15.03) *-i where xi, #2, represent the mol fractions of the components of the system and N\, N2> their mol numbers. This term represents the entropy of mixing the components in the case of a mixture of perfect gases (compare section 50). According to Planck, a condensed solu- tion or mixture should have the same value of the entropy constant since it can be converted into the gaseous phase by vaporization. In this country Planck's views were advocated by Lewis and Gibson. 2 These authors went beyond Planck and questioned, on theoretical and experimental grounds, the applicability of Nernst's theorem to supercooled liquids even of a chemically homogeneous constitution. On the other hand, Nernst 3 had always claimed the general validity of his theorem and was supported by other authors. 4 The contradiction was resolved, and partly reconciled, by an investiga- tion of O. Stern's 6 concerning the entropy of mixed crystals. It will be best to say here a few words about the statistical aspect of the problem, as both the nature of the difficulty and its ultimate resolution are reduced in it to its simplest terms. We have seen (in section 30) that in statistical mechanics the entropy of a state is inter- preted as proportional to the logarithm of its probability 5 = k log P f which, in turn, is measured by the number of ways in which the state can be realized. If we consider a chemically homogeneous crystal at T = 0, every one of its Z atoms has a perfectly definite position of rest. We have discussed this case in section 30 and seen that two probability definitions are possible. If we regard permutations of identical atoms as independent realizations of the crystal (specific definition), we obtain P. = Z! and So = k log ZL On the other hand, if (denying the possi- bility of telling whether the atoms were permuted or not) we consider all the permutations as one single realization (generic definition), we find P = 1 and So = 0. Nernsfs -theorem has, therefore, the generic probability definition as its statistical equivalent. 1 M. Planck, Thermodynamik (6th ed), p. 285, 1921. * Lewis and Gibson, J. Am. Chem. Soc. 42, p. 1542, 1920. 1 W. Nernst, Sitzungsberichte Berlin, p. 972, 1913. E. g. W. H. Keesom, Phys. Zs. 14, p. 665, 1913. 1 0. Stern, Ann. Phyaik, 49, p. 813, 1916. XV 98 CRITICAL ANALYSIS OF NERNST'S POSTULATE 247 For a mixed crystal consisting of atoms of two kinds (Zi and Z* in number) the generic definition gives the probability by formula (4.66). By the use of the Stirling approximation (log Z! = Z log Z) and of mol fractions, xi = Z\/(Z\ + Z*), etc., it can be written as Pi 2 - xr Zl x 2 ~ z *, (15.04) so that the zero entropy of the mixed crystal appears to be of Planck's form 5 W ~ R (Ni + N 2 )(xi log xi + x 2 log * 2 ), (15.05) since kZi = RNi, according to (1.14). If we call each of the realiza- tions of the mixed crystal (differing by the arrangement of at least one pair of atoms of the two kinds) a modification of it, the number of modifications is identical with Pi2 and we meet, apparently, with the same difficulty which was encountered by Planck. However, the difficulty is removed by the following remark. The number of possible modifications only then measures the probability in a meaningful way if all these modifications belong to the same thermodynamical state. It was pointed out by Stern that this is not always the case. The rearrangement of atoms changes in a slight degree the internal energy and the other thermodynamical functions of the mixed crystal. Every modification has, therefore, a slightly different thermodynamical potential. Suppose now that we have a very large number Z of copies of the same mixed crystal at the temperature T. Owing to the thermokinetic motions, the atoms will be continually rearranging themselves, chang- ing the modifications to which the crystals belong. We may ask, therefore, how many of these copies are, at any given time, in the modification j having the thermodynamic potential $/, and we shall show in the next section that their number is Z y = const-exp (-*,/W). (15.06) Correspondingly the ratio Z//Z represents the probability of find- ing a given crystal in the modification j. Since the differences in the function $, are very minute, all the numbers Z/ are practically equal at all temperatures which are not too low. This means that all the modifications are equally probable representing statistically equiva- lent realizations of our crystal. It is appropriate to count, as the probability of the crystal, the number of these realizations, as was done in formula (15.04), and this leads to the expression (15.05) for the entropy of mixing. On the other hand, at very low temperatures even very small differences in */ begin to tell because of the factor. 248 TEXTBOOK OF THERMODYNAMICS XV 98 1/r in the exponent. The nurfibers Z, become unequal and, in the extreme case of the immediate vicinity of T = 0, the modification j m with the lowest thermodynamical potential $ m i n dominates to such an extent that all the others are impossible. In the vicinity of the abso- lute zero the mixed crystal is, therefore, in the modification j m . Since every modification can be realized in only one way, the probability of the crystal, instead of (15.04), becomes Pi 2 = 1 and the zero entropy So = 1, in agreement with Nernst's postulate. Offhand it is conceivable that several modifications could have the same thermo- dynamic potential, and its lowest value $ mln might be shared by n of them. In this case, the probability at T = would be Pi 2 = n and the entropy So = k log n. 'The purport of Nernst's postulate is, there- fore (as far as mixed crystals are concerned), that there is one and only one modification of lowest thermodynamic potential. Similar considerations apply to the entropies of supercooled liquids. In a liquid the positions of the atoms and molecules are irregular and the possibilities of arranging them more numerous than in a crystal. This makes their entropy, at high temperatures, larger than that of a crystal. At low temperatures, however, the regular crystalline modification is the most probable and, in the vicinity of r = 0, even the only possible arrangement of atoms. From the theoretical point of view Nernst's postulate is, therefore, completely vindicated since there is no evidence that it admits of any exceptions. However, the differences in $/ on which the above argu- ment depends are so very small that one can expect a perfectly regular arrangement only at extremely low temperatures which are, in many cases, below the range accessible to our experimental technique. Another important consideration is the time element since, under these conditions, all processes are extremely slow. For this reason one can- not be certain that the substances under investigation have reached the state of true thermodynamic equilibrium. In this sense, we may say that the view of Planck and of Lewis and Gibson is justifiable from the practical standpoint of experimental measurements. This is, in fact, borne out by the recent and very accurate observations l of Giauque and his collaborators. The method used by these authors is based on the fact that the absolute entropies of many diatomic and polyatomic gases can be calculated with great accuracy from spec- troscopic data (compare section 119). They take the same sub- stances in the solid state at very low temperatures and measure the 1 Older work pointing in the same direction was due to Gibson, Parks, and Latimer (J. Am. Chem. Soc. 42, p. 1542, 1920) and Gibson and Giauque (ibidem 45, p. 93, 1923). XV 99 CRITICAL ANALYSIS OF NERNST'S POSTULATE 249 entropy difference attending their gradual transformation into gases by heating and subsequent vaporization. In this way it was found that hydrogen, 1 carbon monoxide, 2 nitrous oxide, 3 and ice 4 possess still appreciable entropies as solids at about 15 K. The lattice theory of solid hydrogen was treated by Pauling, 5 who showed that the crystals are built up of para- and ortho-molecules (com- pare section 118) whose distribution is entirely irregular at tem- peratures above 5 K. The interpretation which Giauque and co-workers give with respect to the other substances just mentioned is that their molecules are not uniquely oriented but even at temperatures as low as 15 K have still the choice between several orientations. Pauling 6 investigated the case of ice and finds the entropy R log (3/2) = 0.805 cal deg" 1 mol" 1 , due to the statistical weight of the indeter- minacy of orientation. This is in excellent agreement with the experimental value 0.82 cal deg 1 mol"" 1 found by Giauque and Stout. 99. Equilibrium of modifications of a mixed crystal. There remains to prove the formula (15.06) on which the conclusions of the preceding section rest. We shall do this by an argument which is thermo- dynamical in its main points. Its essential idea (but not its mathe- matical form) is due to Stern. The formula in question refers to the equilibrium of a very large number of copies of the same mixed crystal. In order to deduce it, we must provide an ideal experimental arrangement by which such an equilibrium can be secured. Let all these crystal copies be contained in an enormously large vessel where they are floating in a neutral gas medium removed from the action of gravity. Through their Brownian movements they will set themselves into thermal equilibrium with the medium and with one another. Provided the size of the crystals is small compared with their mean distance, such a suspension can be regarded as a perfect gas with extremely large molecules obeying all the laws valid for gases (compare section 146). In our particular case every modification of the crystal may be considered as a separate gas, so that we have a mixture of as many gases as there are modifications. Any modification (1) can be converted through the internal rearrangement of its atoms into any other modification (2), and we can apply to this process the laws of chemical equilibrium of perfect gases. 1 W. F. Giauque and H. L. Johnston, J. Am. Chem. Soc. 50, p. 3221, 1928. 1 J. O. Clayton and Giauque, ibidem 54, p. 2610, 1932. R. W. Blue and Giauque, ibidem 57, p. 991, 1935. 4 Giauque and Ashley, Phys. Rev. 43, p. 91, 1933; Giauque and Stout, J. Am. Chem. Soc. 58, p. 1144, 1936. L. Pauling, Phys. Rev. 36, p. 430, 1930. L. Pauling, J. Am. Chem. Soc. 57, p. 2680, 1935. 250 TEXTBOOK OF THERMODYNAMICS XV 99 In the law of reaction (6.40) we have to substitute v\ = 1*2 = 1, obtaining Gi - G 2 - 0, (15.07) while the equation of equilibrium (6.43) is reduced to <f>i - <p 2 = 0. (15.08) The difference between the system under consideration and the ordinary perfect gases of Chapter VIII lies in the fact that in the present case the thermodynamic potential consists of two parts 9 = n + v (15.09) <pi is the " internal " potential of the crystals themselves, <p e the " external " part due to the Brownian motions. For <p e we have to use the expression (8.08) of the thermodynamic potential in perfect gases, omitting the terms UQ TSQ which refer to the intrinsic proper- ties of the " molecules " (here crystals) and are, therefore, included in <pi. The total potential is, therefore, V = T(R log p - c p log T + c p ) + Vi. (15.10) At this stage of the reasoning we have to invoke another result of statistical mechanics, the fact that the " external " molal heat c p (due to the movements of the crystals) depends only on the number of degrees of freedom of the " molecules " regarded as rigid bodies. It is, therefore, the same for all modifications: c p \ = C P 2. In substituting (15.10) into the condition (15.08), the terms with c p will cancel out leaving RT log (Pi/fr) = - (<pn - ^2). (15.11) <pn is the molal thermodynamic potential, i.e. it is referred to n A crystals (n A being the Avogadro number). It stands, therefore, in the following relation with the thermodynamic potential <>,- of an individual crystal used in the preceding section <pn = n A $,-. Noting that R = n A k, we obtain log (Pi/fr) --(*i- *2)/*r. (15.12) pi and p2 are the partial pressures in our vessel of the two gases con- sisting of the modifications (1) and (2). These pressures are propor- tional with the numbers Zi and 2 in which these modifications are represented in the vessel, according to the formula (1.15). The last equation becomes, therefore, identical with (15.06), and this is what we set out to prove. XV 99 CRITICAL ANALYSIS OF NERNST'S POSTULATE 251 We have obtained this result by applying to our imaginary sus- pension the laws of the classical perfect gas. It may be objected that these laws have doubtful validity at very low temperatures. How- ever, we shall see in the next chapter that the approximation given by them is the better, the larger the weight of the molecules. Since, in our ideal experiments, we may attribute to the crystals any size, the accuracy may be considered as of any desired degree even in the immediate vicinity of the zero point. CHAPTER XVI DEGENERATE PERFECT GASES 100. Equation of state of the monatomic degenerate gas. Nernst's theorem was enunciated only for condensed systems, and it was realized, from the beginning, that the classical perfect gas fails to fulfill it. In fact, the entropy of the perfect gas has the form (4.18) which would need an infinite entropy constant in order to meet the requirement s r _ The coefficient of expansion of the perfect gas is, according to (1.19), a. = 1/T, while the specific heats c v and c p are constants independent of temperature (section 15), in contradic- tion with the two immediate consequences (13.09) and (13.15) of the theorem which would require lim r . a = and lim r . c = 0. Begin- ning with Sackur and Tetrode (section 95) there were many attempts to subject the perfect gas to quantum conditions and to obtain a cor- rected equation of state. As long as these endeavors were based on the classical statistics of Boltzmann-Planck they failed, but the advent of new types of statistics led to a great success, the discovery of the correct laws of perfect gases which happen to be in agreement with Nernst's theory. A textbook of thermodynamics is not the place to enlarge upon the statistical principles underlying the new theory of the perfect gases. We shall give only a brief reference to questions of statistics in section 104 of this chapter and shall take the equation of state as given, in the same way as we did not inquire into the statistical justification of the classical perfect gas. If we restrict ourselves to monatomic gases, the corrected equation of state can be written in the form (16.01) It differs from the old equation of perfect gases (1.13) only by the factor P(6)/9 appearing on the right side. By P(G) is meant a certain function of the argument ,,, 252 XVI 100 DEGENERATE PERFECT GASES 253 where the letters have the same meaning as before. In particular, /x is the atomic weight, h Planck's quantum of action (6.554 X 10 ~ 27 erg sec), and n A the Avogadro number. The function P(9) is so com- plicated that it cannot be given by an explicit formula but only implicitly. In order to define it we introduce two auxiliary functions F(A) and G(A) of a new argument A by the equations r ^' l- (16.03) Ax"e-*dx 1 d-Ae~* We have to distinguish three types of perfect gases which differ, according to the value assigned to the constant 6 and according to the range of variability of the variable A : (1) The classical or Boltzmann-Planck gas 6 = 0, 0^-4^oo. (2) The Fermi degenerate gas d =- 1, ^ A ^ oo. (3) The Einstein-Bose degenerate gas +!, O^A^l. When 5 =+ 1, A cannot be larger than 1 because otherwise the integrals (16.03) would be divergent. In all three cases the connection between F(A), G(A) t on one hand, and P(0), 0, on the other, is the same, namely: F(A) = 8-*. (16.04) This relation expresses A in terms of 9, so that G(A) can now also be regarded as a function of 6. Therefore, the second relation which we impose, P(0) = 0* G(A), (16.05) amounts to defining P(0) as a function of 0. It should be noted that the functions F(A) and G(A) are not inde- pendent but there exists between them the relation . (16.06) dA A which is readily obtained by multiplying the numerator and denom- 254 TEXTBOOK OF THERMODYNAMICS XVI 100 inator of the integral for G(A) by e*/A, then differentiating the result with respect to A and partially integrating with respect to x. In the range A ^ 1 the functions (16.03) can be represented by the following series (16.07) The opposite case of very large values of A, (A ~2> 1) is of interest only in connection with the Fermi gas, 5 = 1. The approximations for the two functions are, then, as follows 57 (log ^ (16.08) In the case (d = 1) of the Einstein-Bose gas, A cannot be larger than 1 and, for the classical gas (5 = 0), no approximation is needed since F(A) = G(A) = A are, then, the rigorous expressions for the functions. It is clear from the formulas (16.07) and (16.08) that F(A) is parallel with A in the sense of being small or large when A is small or large. On the other hand the equation (16.04) shows that 9 is in an inverse relation to F(A) and, therefore, to A: when A is small, is large, and vice versa. From the four equations (16.07), (16.04), and (16.05), we can eliminate by the method of successive approximations the three quantities A, F(A), G(A) and obtain the following approx- imate relation between P(0) and 6: (0 1) The same procedure carried out with the formulas (16.08), (16.04), and (16.05) applies only to the case d = - 1 of the Fermi gas (0 1) XVI 101 DEGENERATE PERFECT GASES 255 We see from (16.09) that, for very large values of the variable 9, the function P(9)/9 becomes closely equal to 1 so that eq. (16.01) takes the classical form pv = RT. By its definition (16.02) the quantity 6 is proportional to v*T, it is large when the temperature is high or the density low. Under these conditions the gases of Einstein-Bose and Fermi do not deviate in their properties from the classical perfect gas: they are non-degenerate. As 9 decreases the deviation from the classical laws becomes more and more marked : the degree of degener- ation increases. However, we shall postpone the quantitative discus- sion of the degree of degeneration until section 105. 101. The thermodynamic characteristics of the degenerate gas. From the equation of state (16.01) the internal energy can be obtained by means of the relation (4.23) . or integrating with respect to dv, The function f(T) is determined by the condition that the gas is non-degenerate at high temperatures. The internal energy must take, for T >oo, the classical value (3.18) without becoming infinite for T = 0. It is convenient to change the variable of integration from v to 9, given by the relation (16.02) which can be also written f log 9 = f log T + log v + const. (16.13) Since T is to be considered as constant with respect to the inte- gration, dv On the other hand, T(dp/dT) v - 9(3/>/99) t ,. After these substi- tutions the integration can be readily carried out, giving, with a suit- able disposal of the f unction /(r), P(Q) This is, in fact, identical with (3.18) in the limit 9 -oo, since for a classical monatomic gas c v = ^ R. 256 TEXTBOOK OF THERMODYNAMICS XVI 101 Knowing the internal energy u and the pressure p, we can calculate the entropy differential (4.15), ds = (du + pdv)/T. According to the expressions (16.01), (16.13), and (16.15) for p, v, and u, we can choose as variables T and G. The differential dv can be expressed in terms of dT and dQ: it follows from (16.13) 3dQ ^$dT dv 2 9 ~2 r + v 9 This change of variables reduces ds to the form depending only on 9. Nernst's postulate in the form (13.19) can be directly applied only in the case of the Fermi gas. In fact, in the case of the Einstein-Bose gas the variable A cannot be larger than 1, and the relation (16.04) shows that 9 cannot sink below a certain finite limit and is not defined in the vicinity of T = 0. We pointed out in section 1 that the postu- late does not apply to the classical perfect gas. Therefore the validity of the simple entropy expression -i R J. -r (16 - 17) is restricted to the Fermi gas. The integration can be carried out by transforming back to the variable A with the help of the relations (16.04) and (16.05). The first of them can be written as log 9 = f log F(A) t or differentiating d9 2dF 9 ~~3T' In this way the integrand is transformed into dP(Q) __$ G W5dG_2dG 9 ~~ 3 F 2 + 3 F 3 F* The first two terms give $d(G/F) and we can write instead of the last -%dA/A, because of the relation (16.06). Hence The condition 5 * 0, at the limit 9 = 0, or A = oo is taken care of because, for very large values of A, the formulas (16.08) give XVI 102 DEGENERATE PERFECT GASES 257 log A. We shall show by a special investigation in section 103 that the form (16.18) of the entropy expression holds also in the case of the Einstein- Bose gas. Finally, the formulas (16.01), (16.15), and (16.18) give for the molal thermodynamic potential <p = u Ts + pv the simple expression <p = RT log A + u Q , (16.19) which is true for both kinds of degenerate gases. 102. Chemical constant of degenerate gases. Their relation to Nernst's postulate. We have seen in section 1 that for high tempera- tures or low densities (6 <3C 1 and A ^> 1) the gases of Fermi and Einstein-Bose are non -degenerate, approximating in their properties the classical perfect gas. When the variable A is extremely large, the functions (16.07) are reduced to F(A) = G(A) = A, whence eq. (16.04) gives A 9"^. The entropy expression (16.18), therefore, takes the form (16.20) or because of (16.02) and R = n A k, s = R log v + %R log T + R i 1 = %R log T - R log p + Ri, (16.21) The entropy has the classical form (4.18), specialized for the monatomic gas (c v = !?, c p = -j^R), while the constant i exactly coincides, with the Sackur-Tetrode expression (14.12). Putting the question as to the validity of Nernst's postulate we must remember that we so far obtained complete expressions for the energy and equation of state (valid down to the vicinity of T = 0) only in the special case of the Fermi gas.- In deriving them we made explicit use of Nernst's theorem through eq. (16.17) so that there is no question but that it is fulfilled. It will be well, however, to check here the two immediate consequences from the theorem which we have derived in sections 90 and 91, that the specific heat and the coefficient of thermal expansion must vanish for T = 0. For the specific heat we have from (16.15) and (16.10) the value 258 TEXTBOOK OF THERMODYNAMICS XVI 103 which is proportional to T, because of formula (16.02), and vanishes with T. For the coefficient of thermal expansion, eqs. (1.02) and (1.05) give Since p differs from u only by the factor 2/3v, (16.24) (dp\ _ 2* \dT/ v 3 v ' so that a also vanishes for T = in a linear way. To elucidate the conditions in the case of the Einstein-Bose gas we need a separate investigation to which we now turn. 103. Condensation of the Einstein-Bose gas. It was pointed out in section 100 that in the Einstein-Bose case (5 = 1) the variable A cannot be larger than 1. In view of this, 9 has a lower limit 9 m m determined by eq. (16.04) with 4 = 1: F(l) = 6-. (16.25) Now F(l) = 2.6123, and the numerical value of the limit is, there- fore, 6 m i n = 0.528. Since the parameter 9 depends on v and T 9 accord- ing to eq. (16.02), we may ask what will happen if, keeping the temper- ature of the gas constant, we should compress it sufficiently to make 9 fall below the limit 9 mln (or else decrease the temperature keeping the volume constant). To this question Einstein gives the answer that a part of the system will cease to be gaseous and will be precipitated out, forming a condensed phase in equilibrium with the remaining part of the gas. The condition of equilibrium is given by the equation (7.03) 9 - W, (16.26) where <p and <ppr represent the thermodynamic potentials of the gaseous phase and of the precipitate. Since the expressions (16.18) and (16.19), derived for the entropy and thermodynamic potential of the Fermi gas, give the correct value of the chemical constant, it is desirable to conserve them for the Einstein-Bose gas. Under the conditions of condensation (saturated state) 9 = 9 mln and A = 1, so that eq. (16.19) gives <p = UQ. Consequently, <ppr = u pr Ts pr + pVpr must be also equal to UQ, for all temperatures and pressures compatible with 9 9 m in. This requires u^ = o s pr = 0, v pr = and means that in the solid phase the material is much in the same state which prevails at T 0. It does not possess any thermokinetic energy, and XVI 104 DEGENERATE PERFECT GASES 259 it has a perfectly regular arrangement of atoms and a negligible volume. In support of Einstein's explanation we can point to the peculiar shape of the isothermals of the Einstein-Bose gas. 1 By means of (16.02) and (16.05) we obtain from (16.01) the expression for the pressure \ n A ) A 3 G(A), (16.27) and calculate from it the partial derivative /aA _ dp t dA \dv/T~ dA ' dF (^). \dV/T Because of (16.06) and (16.04), this gives dp\ _ RT 1 dA at; A" v 2 Q 3 A dF f V (16.28) It is easy to see from the definition (16.03) of the function F(A) that (in the case 6 = 1) the derivative dF/dA becomes infinite for A = 1. Con- sequently, (dp/dv)T = 0, for A = 1, 6 = 9 m i n . In the state of saturation the gas is, there- fore, characterized by an infinite compress- ibility and its isothermals by horizontal tangents. These conditions are illustrated in Fig. 46, where the solid lines represent the isothermals and the dotted curve cor- responds to 8 = m i n . We have seen in the theory of the Van der Waals equation that the points (dp/d^T = are those where a single phase of the gas begins to be completely unstable. The condensation of the ideal gas of Einstein-Bose is, therefore, closely analo- gous to the conditions in real gases. 104. Statistics underlying the theory of degenerate gases. Rule of alternation. 2 We have introduced and treated eq. (16.01) of degenerate gases in a purely formal manner. It will be well to say now a few words about its origin and about the question of which gases of 1 Compare P. S. Epstein, Commentary on the Scientific Writings of J. W. Gibbs, Article 0, Section 10. 1 For a detailed exposition of quantum statistical principles, see: P. S. Epstein, ibidem, Article V. FIG. 46. Isothermals in Einstein-Bose degenerate gas. 260 TEXTBOOK OF THERMODYNAMICS XVI 104 nature are subject to degeneration of the types of Fermi and of Einstein-Bose. Classical statistics and the statistics of the older quantum theory were dominated by Boltzmann's principle (4.70): If we have a single particle (or other system) which can assume dif- ferent dynamical states, the probability of finding it in a state char- acterized by the energy e, is P i = Cexp(-e,/*r), (16.29) C being a constant. Nothing is changed if we consider Z particles of one and the same kind under joint external conditions (field of forces or vessel), determining a set of energy levels ei, 62, . . . e/, . . . which any of the particles can assume. The mean number Z/ of particles occupying the energy level e/ is again given by Zy/Z = C exp (- e,-/*r), (16.30) because in the classical theory the particles are regarded as statistically independent: the probability of a particle assuming any state is not influenced by the presence of the other particles and remains the same as if it were alone. With the advent of wave mechanics the point of view had to be changed. In the case of a single particle, Boltzmann's principle (16.29) remains valid but Z particles forming a system of the kind mentioned above can no longer be regarded as statistically independent. 1 The primary particles of matter (the proton, neutron, and electron) fulfill the Pauli exclusion principle : No more than one of Z identical particles can occupy any given quantum state. (In the language of wave mechan- 1 It was pointed out in section 30 that the fundamental difference between classical and quantum statistics lies in the question whether identical particles can be told apart. This question has an intimate relation to Heisenberg's principle of indetermination which states that the uncertainties A* (of a coordinate of position) and A/> x (of the conjugate momentum) are restricted by the inequality Ax- &p x <h/4T. In fact, when the uncertainty of position of each particle in a gas is small compared with the mean distance between the particles (v/n A )^, then there exists the con- ceptual possibility of keeping track of any particle and mentally identifying it as distinct from the others. Therefore, classical statistics must give a correct result when the condition (/n 4 )**Ax is satisfied. On the other hand, when the two quantities are of the same order of magnitude, there is no way of telling which particle is which, the concept of permutation loses its sense, and classical statistics can no longer be applied. The uncertainty of the momentum is, obviously, the mean momentum itself A/> x J, since the particlejnay have any velocity. As long as classical statistics (equipartition) is valid p x * M*v x * = MkT. We obtain, therefore the condition (MkT)^(v/n A )^ A/4r, or using the notation (16.02), 6l/8r. This is the deeper reason why degeneration is absent when 6 is large and sets in when this quantity becomes small. XVI 104 DEGENERATE PERFECT GASES 261 ics the same thing is stated by saying that the primary particles have antisymmetric wave functions.) We cannot enter here into the details of the statistical calculations based on the exclusion principle and must be satisfied with their results: they lead to the modified formula (5 a l), which represents the fundamental law of Fermi's statistics. The quantity A gives a measure of the deviation from the classical conditions of the formula (16.29). It is, therefore, obvious that A must be a function both of the particle density in the system and of its temperature. In fact, the sum of all Zj is the total number of particles, Z, leading to the condition Another condition follows from the energy relations. The total energy of the Z, particles in the state j is e,Z, and, therefore, the total internal energy of the system U = 2e,Z, + UQ or U - U = ZC t It is possible to eliminate C from these two equations and to obtain A as a function of U, Z, and T. Let us now turn from primary particles to composite ones, for instance, to atomic nuclei built up of protons and neutrons. If anti- symmetric wave functions are used for the protons and neutrons within the necleus, the translational motions of it, as a whole, are described as follows: (a) by antisymmetric wave functions, if the number of primary particles in the nucleus is odd, (6) by symmetric wave functions, if this number is even. Nuclei of the first kind have the same statistical properties as the primary particles themselves, they obey the exclusion principle and the formula (16.31) of Fermi's statistics (with 5 = 1). On the other hand, nuclei of the second kind possess entirely different qualities: the statistical calculation shows that in their case the number Z, can be expressed by the formula (16.31) with d =+ 1 which characterizes the statistics of Einstein-Bose. The law that the statistics of Fermi applies to composite particles with an odd number of primary elements, and Einstein-Bose's to those with an even number, is known as the rule of alternation. Of course, this rule is theoretically valid also in the case of neutral atoms which 262 TEXTBOOK OF THERMODYNAMICS XVI 104 contain primary particles of three kinds: electrons in addition to protons and neutrons. The experimental test of the rule could not be carried out by observing the degeneration of gases (compare next section) but by study of molecular band spectra whose structure also depends on the statistical properties of the nuclei composing the molecule. In the present state of our knowledge the rule of alternation seems to hold without exception. Doubts exist only with respect to the sulfur nucleus, but the experimental data are insufficient to decide whether they are well founded. There was introduced into the theory of nuclear structure the hypothesis of the existence of another primary particle, the neutrino, which is supposed to have a very minute mass. This assumption proved very useful in the theory /3-ray emission and of nuclear transformations, but it seems that the neutrino could be dispensed with as far as the rule of alternation is concerned. The conditions (16.32) and (16.33) determining the quantity A are closely related to the integrals (16.03) of section 100. If the system is a gas, contained in a vessel of the volume F, the energy levels are determined by the vessel. It is shown in the quantum theory that in the interval of energy values between e and e + Ae there lie 2irF(2/i/w^^ 2 )^e^Ae levels, and, if V is sufficiently large, their dis- tribution is practically continuous. By introducing the variable x = e/kT the sums (16.32) and (16.33) can be thrown into the form of integrals proportional to (16.03). The comparison shows ~ U - Uo = %ZkTG(A}/F(A). (16.34) Thus, the probability definition (16.31) leads directly to an energy expression equivalent with the formula (16.15). The gases here discussed are perfect in the sense that their molecules (or atoms) have no extension and there are no forces of interaction between them. However, the statistical limitations imposed on identi- cal particles by Pauli's exclusion principle or the requirements of Einstein-Bose have an effect upon the thermal and caloric equations of state which resembles the effect of force interactions (compare next section). It must be emphasized that no such limitations exist for non-identical particles. In a mixture of several degenerate perfect gases, there is no mutual influence between them, and each gas behaves as if it were alone and the others absent. This implies, in particular, that the formula P - Pi + p2 + ...+P., (16.35) XVI 105 DEGENERATE PERFECT GASES 263 representing the total pressure as the sum of the partial pressures of the component gases (each calculated as if the rest were not present) remains valid. 105. Chances of observing the degeneration experimentally. Some of the laws of perfect gases have the same mathematical form for all three kinds of them (classical, Fermi, and Einstein-Bose). Such, for instance, is the equation of the adiabatic process: it follows from the expression of the entropy differential (16.16) that s = const, when 9 = const. According to (16.02) this can be written const, (16.36) or multiplying by pv from eq. (16.01) pv* = const. (16.37) The relation between pressure and energy density resulting from (16.01) and (16.15) is also valid for the classical and both degenerate gases. On the other hand, the pressure expressed in terms of the molal volume and temperature depends on the nature of the gas. Written out explicitly the expressions are p = * (i - 5/2*6") or h?n A ** 1 \ 7 ^Ukf^ V + '")' (9 <C 1 occurs only in the Fermi gas) ' Substituting into the expression (16.02) for 9 the numerical values of the constants k,n A , h, and VQ (molal volume at normal conditions), we find 9 = 3.67 M (-) T. (16.41) \VQ/ This gives for helium (/* - 4.00) at C and 1 atm, 9 - 4033, so that the condition 9 ^> 1 is fulfilled for the permanent gases of 264 TEXTBOOK OF THERMODYNAMICS XVI 105 nature. On the other hand, electrons in metals also form a gas (com- pare next chapter) of the atomic weight /z = 1/1821 = 0.000549 and with a remarkably small molal volume, namely of the order V/VQ 1/2000. In such an electron gas the quantity 9 has at room temperature a value of the order of 9 = 0.004, satisfying the opposite condition 9 <C 1. The two extreme cases represented, respectively, by the formulas (16.39) and (16.40) happen, therefore, to be sufficient for the treatment of all the degenerate systems occurring in nature. For the permanent gases, eq. (16.39) must be used, and its form is analogous to the Van der Waals equation of state (1.21) which, in the case of not too small molal volume (v J> b), can be written Perfect gases have, of course, no extension of molecules (6 = 0), but the Einstein-Bose gas (6=1) acts as if its molecules were attract- ing one another, the Fermi gas (5 = 1), as if they exercised repulsive forces. This behavior is in harmony with the statistical meaning of degeneration in the two cases (section 104). The most promising gas for observations is helium: it is the monatomic gas of lowest atomic weight, and, because of this, has the highest degeneration. Moreover, it can be cooled to lower temperatures than any other gas. But even in helium the condi- tions are not favorable, the Van der Waals terms of eq. (16.40) are in it (under normal conditions) b/v = 1.1 X lQ-*,a/RTv = 6.8 X 10~ 5 while the degeneration term of (16.39) has the much smaller value 1/2 H 9** = - 6.9 X 10 ~ 7 . It is true that the ratio becomes more fav- orable at lower temperatures, but barring the immediate vicinity of the critical state the constant 9 remains fairly large even then. The effect is, therefore, completely obscured by the Van der Waals forces. The velocity of sound can be measured with considerable accuracy even at very low temperatures and in a small volume of gas. The de- terminations of Keesom in helium at 4 K and 0.08 atm gave the value to be expected in a non-degenerate gas. It is easy to see the reason for this. Since the equation of the adiabatic (16.37) is the same as for the classical perfect gas, the expression (3.40) for the velocity of sound also remains valid. This gives (with y = ^) - -'RT (1 - 6/2 M 6). (16.43) 3 /* 3/*' XVI 105 DEGENERATE PERFECT GASES 265 Under the conditions of Keesom 9 = 24 and - 1 /2 W 9* = - 1 .5 X H)- 3 . The effect of the degeneration is only 0.15% and entirely within the errors of measurement. Summarizing we can say that the observation of degeneration in atomic gases is hopeless. The only system in which the phenomenon is accessible to the experimental test is the electron gas which is treated in the next chapter. CHAPTER XVII ELECTRON AND ION CLOUDS 106. Thermodynamic properties of charged gases. As long ago as 1888 the pioneers of thermodynamics, Arrhenius and Ostwald, 1 applied the mass law to electrified particles (ions). We are indebted to them and their pupils for extensive experimental data which show conclusively that the thermodynamical theory of chemical equilibrium applies to ions just as well as to neutral atoms. The idea that a cloud of electrons can be also described by the laws of perfect gases proved useful in the theory of metallic conduction and associated phenomena. It was introduced by Drude 2 and Richardson, 3 who dealt with it from the statistical point of view, while the first thermodynamical treatment was due to H. A. Wilson. 4 A part of the internal energy of such a gas must be of electric origin. Let us focus our attention on a portion of the gas so small that we can consider the outer electric potential 12 as constant over its volume r. The potential 2 is produced, partly, by causes extraneous to the gas, partly by charges on the gas particles outside the volume r. The contribution to the total electric energy of the portion within r consists of two items: the mutual energy of this portion and of the rest of the system, and the electrostatic energy of the volume r itself. The first item is proportional to the first power of the volume r ; the second goes mainly with r 2 and only rarely contains a portion linear in r. The latter case may arise when the charges within r are of both signs so that there is a possibility of pairing, i.e. of positive charges hovering, by preference, in the vicinity of negative ones, and vice versa, a phenomenon which we shall discuss in section 115. At present we shall treat the case where pairing is either absent or negligible, so that the inner electric energy of the volume r is quadratic in r. If we select r sufficiently small, the quadratic term becomes negligible com- 1 W. Ostwald, Zs. phys. Chemie 2, pp. 36, 270, 1888. * P. Drude, Ann. Physik 1, p. 566; 3, p. 369, 1901. O. W. Richardson, Proc. Camb. Phil. Soc. 11, p. 286, 1901; Philosophical Transactions (A) 201, p. 497, 1903. H. A. Wilson, Philosophical Transactions (A) 202, p. 258, 1903. 266 XVII 107 ELECTRON AND ION CLOUDS 267 pared with the mutual electric (mentioned above) and the caloric terms which are both proportional to r. Let us denote the charge per mol by / = en A (e being the charge on each particle). 1 The electric energy of the gas in the volume r (due to the outer potential 8) will then be/12 per mol. This is the only part of the electric energy which has to be considered and added to the internal energy of caloric origin. For instance, if we assume that the caloric part has the expression (3.18), valid for a perfect gas, the internal energy becomes u = c v T + uo+fto. (17.01) The additional term /12 is independent of the condition of the gas in the volume r. It is a constant with respect to it and not one of the parameters describing its state. The parameters, subject to change, defined within r, are temperature, pressure, and specific volume, as in every simple system. It has been established by statistical reasoning and by experiment that the conditions concerning the equation of state and the caloric internal energy are in no way different in a charged gas from those in the uncharged. The equation pv = RT, (17.02) together with (17.01) define the classical (or non-degenerate) perfect charged gas and describe the phenomena in clouds of ions and electrons to the same degree of approximation as the equations lacking the electric term describe the ordinary permanent gases. Since /ft must be considered as constant, with respect to changes in the volume r, it may be regarded as forming part of the energy constant UQ: in fact, we have seen in section 98 that UQ itself repre- sents, in part, potential energy. Therefore, the entropy differential ds does not contain the electric potential so that we find for s precisely the same expression (4.18) as in uncharged gases. On the other hand, the thermodynamic potential <p becomes according to (5.41), (5.43) <p = RT log p - CpT log T + (c p - 5 )r + u + /0. (17.03) 107. Equilibrium in charged gases. The problem of equilibrium was treated in section 40 in its generality, but the conditions in charged systems are somewhat peculiar, and it will be necessary to say a word how they fit into the picture there given. The phases in a system consisting of charged gases may be determined by external conditions and separated by surfaces of discontinuity, for instance, when we 1 f becomes identical with the Faraday F when the charge is monovalent and positive. In general, / = <rF, if <r is the valency; for the electron (whose charge ia negative)/ - F. 268 TEXTBOOK OF THERMODYNAMICS XVII 107 consider the free electrons inside a heated metal, as one phase, and the electron cloud outside it, as another. However, when the electric potential Q changes continuously with the position in the gas each small volume r, considered in the preceding section, may be also regarded as a separate phase of the system because the electric condi- tions in it are different from those of the rest of the gas. In either case the potential is one of the essential characteristics of the phase, and this fact has an important bearing on the equilibrium. In section 40, we mentioned two possible cases, represented by eqs. (6.26) and (6.29), arising when there is no transfer of matter from one phase to another. Which of these applies to charged gases? The first equation is valid when the subsidiary condition (6.24) holds: when it is possible to change the volume of the individual phases at the expense of one another, while keeping the total volume of the system constant. Obviously this is not feasible in the system we are now considering. The expansion of any phase would push a part of it into a region of a different electric potential or, in other words, out of the phase. On the other hand, it is clear that the equilibrium will not be disturbed if we imagine each phase surrounded by a rigid heat-conducting envelope, i.e. if we assume the subsidiary conditions underlying the formulas (6.27) and (6.29). We must conclude, therefore, that a system composed of charged gases in an electric field possesses, in its state of equilibrium, a temperature T uni- form for all its parts while the pressure is not uniform but " local," changing with the potential from phase to phase. The remaining condition of equilib- rium (6.31) is derived from the analysis of processes of transfer of matter, from one phase to another and from one component to another, while the tem- perature of the system T and all the local pressures of the phases are kept constant. It applies to a charged system to the extent to which such processes are possible in it. Let us consider, for simplicity, a chemically pure gas (i.e. having molecules of one kind only) in equilibrium at the temper- ature r, regarding the small volumes r and r r (Fig. 47) as two of its phases with respective electric potentials 12 and Q'. The equilibrium will not be disturbed if we imagine the small spherical surfaces L and L' within the phases hardened and transformed into rigid adiabatic FIG. 47. Equilibrium of charg- ed gas in electric field. XVII 107 ELECTRON AND ION CLOUDS 269 shells. If the spheres are taken so small that the electric potential due to the gas within them is negligible compared with the rest, we may imagine this part of the gas removed without prejudice to the equilib- rium. Of course, we could have selected the spherical hole in r larger by an infinitesimal amount and having the surface M (dotted line), and that in r' slightly smaller with the surface M f . Let us adjust the radii in such a way that the spherical layer between L and M contain dN mols of the gas (in its original distribution), and that between L 1 and M 1 an equal number dN' (i.e. dN = dN'), while the tem- perature and the local pressures remain the same. The same effect can be obtained by expanding the sphere in r from L to M and contracting that in r' from M 1 to L'. This last procedure provides the mechanism we were looking for, a process for transferring dN mols of the gas from the phase T' to the phase r without change of temperature or local pressures in any part of the system. We can, therefore, apply the formula (6.31) to it. Let the mol numbers and molal thermodynamic potentials in r and r' be N,<p and N',<p', respectively. Changes of $ occur only in them, and the formula is reduced to <p dN + p'dN' = 0, or because of the relation dN + dN' = 0, <? - j. (17.04) It is well to say here a word about the phase rule. We found in section 43 that only three phases of a chemically pure substance can coexist. However, that applies only to uncharged systems in which temperature and pressure must be uniform throughout. On the other hand, let us envisage a phases of the charged gas we are dis- cussing now. They are described by a + 1 thermodynamical variables r, p (l \ . . . p (a \ and there exist between them a - 1 relations (17.04). Even if the electric potential differences are prescribed, two of the variables can be chosen at random (no matter how large the number a is), for instance, the temperature and the pressure in one of the phases. If the electric potentials are not prescribed, there may exist still further degrees of freedom. The relation (17.04) is valid not only for a chemically pure gas: by suitable use of semipermeable membranes it can be shown to hold for any component of a gas mixture. However, we shall be interested only in mixtures of perfect gases and, then, the following proposition is obvious: if the relation */ = f r , (17.05) holds for the component j, when it is pure, it will continue to hold, when the component forms a part of a mixture. This follows from 270 TEXTBOOK OF THERMODYNAMICS XVII 107 the fact that perfect gases do not exercise forces on one another but behave as if the others were not present. In the particular case of a non-degenerate perfect gas, eqs. (17.03) and (17.05) give RT log W,/pd = ~ /(&'- 0), (17.06) (17.07, This special .form of the relation could have been obtained also in simpler ways, for instance, from Boltzmann's principle (4.69). How- ever, we shall have to fall back upon the general form (17.05) in treating degenerate gases (section 111). The other type of processes included in the equation of equilibrium (6.31) are chemical reactions taking place within the individual phases. In the interior of a phase the electric potential does not vary and merely plays the role of an additional energy constant. The conditions are, therefore, precisely the same in a charged gas as in an uncharged and eq. (6.31) remains valid. In particular, when we consider the case of a reaction of the type (6.46) + . . . + v0G0 = (17.08) between perfect gases, the equation of equilibrium has the form (6.49) of section 42 : + . . + r&t = 0. (17.09) Comparing it with (17.05), we see that the equation is satisfied in all the phases simultaneously so that we have equilibrium in every part of the system. For non-degenerate perfect gases we have to use for <f> the expression (17.03) obtaining the following explicit form of the equilibrium condition log pi = logK p - (Q/RT) },*&. (17.10) K p is here exactly the same equilibrium constant as that defined by eq. (8.15) in the theory of ordinary gas mixtures (section 51). The sum Sv// = nj&vfij (where e$ is the charge on a molecule of the component j) represents the change of electric charge in the reaction. However, there is the law of conservation of electric charge, which cannot be created or destroyed, so that the sum S^,- = 0. The law of equilibrium has, therefore, precisely the same form (8.17) in a mixture XVII 109 ELECTRON AND ION CLOtf>3 of charged as of uncharged gases and can also be expressed in the equivalent formulations (8.18) and (9.60). 108. Remark on heavy gases in a gravitational field. The con- siderations of the preceding section can be transferred in their entirety to a system of heavy gases in a Newtonian gravitational field produced, partly, by external causes, partly, by their own masses. The only necessary change is to interpret 8 as a gravitational potential and to write, instead of /, = n A e it the molecular weights M? = n A Mj, (Mj being the masses of the molecules). The analogue to (17.06) is then fifa - exp [-M/W- Q.)/RT\, (17.11) which is simply a slight generalization of the well-known barometric law of Laplace's. In a gravitational field the equation of chemical equilibrium takes the form log pt = log K p - (0,/RT) T\/M/. ( 17 - 12 > i- In most cases the sum S *>//*/, representing the change of mass in the reaction, may be taken as zero. However, the law of conservation of mass does not hold for matter alone, and there are phenomena for which this sum is significant and may not be neglected (compare section 133). 109. Thermoelectric potential differences in the classical electron theory. As stated in section 106, the theory of electrons in metals achieved some signal successes by assuming that the free electrons form a perfect gas. In ^^^4^3 <2> I the older (classical) theory this gas was re- j garded as non-degenerate, with an osmotic FIG. 48. Thermoelectric pressure obeying the law (17.02) and eq. couple. (17.07) resulting from that law. As an application of these ideas, let us consider two bars (1) and (2), of two different metals (Fig. 48), in contact at the junction J and forming an open chain. Suppose that, in their electrically neutral state at the temperature T, the two metals contain, respectively, z\ and $2 free electrons per unit volume. At the moment the two bars are brought in contact, there exists on both sides of the junction a difference of concentrations and, consequently, of osmotic pressures of the free electrons. As we know from section 65, the osmotic pres- sure tends to equalize the numbers z\ and 212 and drives the electrons from (1) to (2), if z\ > z 2 . How long will this so-called thermoelectric 272 TEXTBOOK OF THERMODYNAMICS XVII 109 action continue? As the electrons are driven toward bar (2), there is set up an electric field: the potential of this bar will fall (because of the negative electronic charge) and that of bar (1) will rise. Equilibrium will be reached when the forces of the electric field just compensate the force of the thermoelectric (osmotic) action. Supposing that the system retains the uniform temperature !T, we can calculate, for the state of equilibrium, the difference of electric poten- tials Oi fa between the two bars. We have only to apply the for- mula (17.07) of section 108, taking into account that pi/pz = zi/Z2i according to (1.15), and R/f = - R/F = - k/e l kT z\ $21 = 81 fo = log (17.13) e 22 The assumption underlying this formula is as follows: the electron density z of each metal is completely determined by its nature and, possibly, by the temperature T, while the loss or gain of electrons necessary for charging up the system is entirely negligible at a small distance from the junction. In the immediate vicinity of the junction, metal (1) carries a positive charge, through loss of electrons, and metal (2) a negative one, through excess of them. The two opposite charges form an electric double layer accounting for the drop of potential. 2 As long as the whole system is kept at the same temperature, the thermoelectric potentials are not accessible to measurement. In fact, their experimental determination would require a pair of wires going from the free ends of bars (1) and (2) to a suitable measuring instru- ment. If the two wires are of the same material (3) we can denote the thermoelectric differences with respect to the two bars by 831 and ^32. We see, then, from the expression (17.13) that 832 = Osi + Oi2, (17.14) so that both wires are at the same potential. It is obvious that taking the wires of different materials will not remove the difficulty. 1 See footnote on p. 267. 1 Compare, however, footnote on p. 274. It was pointed out by some authors that the energy necessary to form the double layer must be taken into account in setting up the conditions of equilibrium (see: P. W. Bridgman, Phys. Rev. 27, p. 173, 1926). Theoretically this is true, but the terms arising from this cause are very small numerically and share with the capillary surface effects the property of decreasing in importance as the size of the system is increased (compare sections 39 and 81). Since capillary forces are neglected in this and the following chapters, consistency demands that the electric surface effects be treated on the same footing. XVII 110 ELECTRON AND ION CLOUDS 273 In the present connection, we are interested in the thermoelectric actions only in so far as they offer an interesting example of potential differences in electron clouds. We shall see in section 139 that it can be indirectly measured from the electromotive force of a thermo- couple with two junctions kept at different temperatures. Without any doubt, the above theory explains the gist of the thermoelectric phenomena, but it is somewhat oversimplified. This appears from the experimental fact that thermoelectricity is structure sensitive: a slight distortion in the lattice structure of a conductor, due to mechan- ical strain, produces an appreciable change of its thermoelectric properties. As the theory stands today, it cannot account for the observations, quantitatively, but does give the correct order of magnitude. It must be borne in mind that the formulas of this section apply (within the approximation of the theory) only to conductors in which the free electrons can be considered as forming a non-degenerate gas. As was shown in section 105, this situation prevails in semi-conductors. Experimentally, it is known only that such materials possess unusually large thermoelectric potential differences (compare Table 56 in section 139). This is unquestionably due to the wide range of the numbers z occurring in them. The thermoelectric conditions in metallic conductors will be considered in section 113. 110. Contact potentials and thermionic emission in the classical theory. Let us consider a conducting plate in vacuo: at its surface the density of free electrons changes abruptly. The forces of osmotic pressure (mentioned in the preceding section) tend, therefore, to drive the electrons out of the conductor into the empty space beyond, producing there a so-called thermionic cloud. We call the potential of the conductor fli, the number of electrons in it (per unit volume) 01, and the same quantities in the thermionic cloud at the surface of the conductor fl'i and z'l, and we apply the formula (17.13) flj- G'i = log^- (17.15) e z i The number z\ is determined by the nature and physical state of the conductor, but the density z'\ of the cloud may assume any value. What condition, then, makes the problem definite and fixes the poten- tial difference Oi 8'i and, through it, the number z'i? The action of the osmotic pressure in producing the outer electron cloud is limited by the fact that a definite amount of work W\ is required in order to liberate one electron from the conductor , which is performed by the electron 274 TEXTBOOK OF THERMODYNAMICS XVII 110 while it passes through the surface. Therefore, the left side of eq. (17.15) is determined by the relation Qi - fl'i = Wi/e, (17.16) whence ^ = ^ ^ (-n A Wi/RT) - zi exp (-bi/T), (17.17) where the quantity bl = n A w 1/R (17 . 18) is called the thermionic work function. 1 It is generally accepted that the force resisting the loss of electrons and pulling them back into the conductor is the image force. As known from the theory of electricity, a perfectly conducting plane acts upon a point charge with a force equivalent to the attraction of the reflected image of the charge. If the distance of the point from the plane is x (its charge being e), the distance to its reflected image becomes 2x and the image force is e?/4x 2 . The work necessary to remove the electron from the position x = a to x = oo is, then, e?/4a. The actual surface of the conductor is, of course, imperfect since it is constituted of atoms with interstices through which the electrons can pass into the outer space. Nevertheless, the above expressions for the image force and the work done against it remain approximately valid as long as x (or a) is not too small. To obtain the total work W\ that must be supplied to an electron in order to get it free, one must attribute to a a value of the order of the atomic distances (10~ 8 cm). The equations Wi = e 2 /4a, fli - fl'i e/4a (17.19) give then, in fact, the correct order of magnitude for the potential difference, namely, a few volts. Sometimes the question is asked how the electron cloud builds up this difference of potentials. This question is illegitimate and based on a misconception resulting from the fact that the term " potential difference " is used in the above formulas in a sense slightly dif- ferent from that given it in most textbooks on electricity. 2 As under- 1 Not to be confused with the "work function" denned in section 34. " n 1 The difference of electric potentials fij Q! between the points (2) and (1) is determined by the work W n which must be supplied to a test body (of the charge e) in order to move it from (1) to (2). Two different definitions are possible: either, 8a - 81 - lim (Wi2/e), when e approaches 0; or fti fli = Wu/e, when e is finite. Most textbooks on electricity do not distinguish between the two definitions. The second, obviously, takes in the work of any field induced by the test body and is, therefore, unsuited for many theoretical purposes. However, it is the definition which must be used in connection with Boltzmann's principle and thermodynamics, since Cpot in eq. (4.69) is the energy imparted to a particle under the actual conditions. In this sense (17.19) represents the actual difference of potential between the con- ductor and the outer space. XVII 110 ELECTRON AND ION CLOUDS 275 stood here, the potential difference (17.19) exists from the start and is not, to any material extent, built up by the electrons leaving the conductor: the potential added by the electron cloud (so called, space charge potential) is usually so small as to be negligible. If a second conducting plate is placed at a short distance from the first facing it in vacuo, it will also set itself in equilibrium with the electron cloud and through it with the first plate. Two potential differences relating to the plates are of interest. On the one hand, the difference fli ^2 between the potentials in the interiors of the two conductors depends (according to the theory of section 107) only on their internal properties. It has, therefore, the same expression as when the plates are in direct contact, and is identical with the thermo- electric potential difference discussed in the preceding section. On the other hand, there is the so-called mutual contact potential 8'i - Q' a of the two substances. It is defined as the difference of potentials between two points in the electronic cloud which are, respectively, close to the surfaces of the two plates. The potential 12' i at the surface of the first plate is given by the eq. (17.16), that at the second plate by the analogous formula $2 ^'2 = W^/e. Subtracting the two equa- tions, we find tti - fl' 2 = Oi - &2 ~ (Wi - W 2 )/e. (17.20) Substituting for &i 82 and W\ W% from the formulas (17.13) and (17.18), we obtain for the contact potential in the classical theory ffi - / 2 = - * [b, - 6 2 - r log (21/22)]. (17.21) e The second term in the brackets is usually very small compared with the first. The work function b can be measured either photoelectrically or by thermionic currents. If we let light of a high frequency v fall upon a metal plate, it emits photoelectrons having a kinetic energy JE^m. Einstein's photoelectric equation hv = Ek* + W (17.22) expresses the hypothesis that the energy of an absorbed photon hv is imparted to the photoelectron. The part W of it is spent in getting through the surface of the metal, the remaining part appears as kinetic energy. As the incident frequency v is decreased, Ewn becomes smaller and smaller and vanishes altogether for a certain limiting value hv<> - W. (17.23) 276 TEXlfcOOtf OP THERMODYNAMICS XVII 110 Light of lower frequency cannot produce any photoelectric effect, and this permits the experimental determination of W. If we identify W'with Wioi eq.(17.16), we find from (17.20) for the contact potential 1 fi'i - o' a - - - (y i - w) + * T log fa/**). (17.24) e e The method of thermionic emission for measuring b is based on the formula from the kinetic theory of gases n = z (RT/2*p)* 9 (17.25) giving the connection between the number z of molecules per unit vol- ume in a perfect gas and the number n of those which pass from one side to the other of any plane per unit time and unit area, n means the molecular weight of the gas, in our case that of the electron Me = 1/1821. This expression holds also for the electrons going in either direction at the surface of the metal. However, of those going outward the number rn are reflected at the surface, if r is the coefficient of reflection. Only the remaining (1 r)n electrons come out of the metal itself. The thermionic current is observed by making the heated metal plate the cathode of a vacuum tube and applying a strong field between the cathode and the anode. The electron cloud in the vacuum is swept away by the field, but the conditions within the metal are not appreciably changed and the electrons leaving it are still the same in number, carrying with them the current J = e(l - r), (17.26) or substituting from (17.25) and (17.17) J = A'T"e-" r , (17.27) where A' - (R/2*n)*zie(l - r). This is Richardson's formula for the thermionic current. It was deduced by its author 2 from kinetic considerations and by H. A. Wilson 3 from thermodynamics. For the reasons pointed out at the end of the preceding section its validity cannot be claimed for metals but is restricted to semi-conductors. 1 It can be made plausible by statistical reasoning that Wi W is, in fact, the cor- rect identification. The data on semi-conductors are not precise enough to test the accuracy of formula (17.24). * See footnote on p. 266. See footnote on p. 266. XVII 111 ELECTRON AND ION CLOUDS 277 111. The degenerate electron gas. We have seen in section 105 that the free electrons in metals must be regarded as a completely degenerate gas. This fact was first pointed out by Pauli, 1 while extensive study of metallic electrons, on the basis of Fermi's statistics, is due to Sommerfeld. 2 The formulas of Chapter XVI, however, need a small correction before they can be applied to electrons. It is caused by the electron spins which obey the following rule: the spins of any two electrons in the cloud are either parallel or anti-parallel. We have, therefore, two types of electrons with respect to the spin which we shall call, for short, the first kind and the second kind. The statistical weight and the intrinsic energy of either kind of electron are the same, so that our gas is a mixture of equal parts of two components. Accord- ing to the remarks made at the end of section 105, the formulas of Chapter XVI apply to each component separately, and it is easy to infer from them the equations for the mixture. The specific volume of each component is 2v, twice the specific volume of the mixture, while the partial pressure of each kind of electrons is p/2, one-half of the total pressure. All we have to do is to substitute in all equations 2v, instead of v, and p/2, instead of p. As this leaves the product pv unchanged, the only correction which the laws of degenerate gases require in the case of electrons is a redefinition of the quantity 9 (replacing v by 2v) instead of (16.02), where /u is the atomic weight of the electron /LI, = 1/1821. Substituting the numerical values of the constants e = 0.403 x io- 5 v*r. (17.29) There is reason to assume that in monovalent metals there are as many free electrons as atoms. The molal volume of the free electrons coincides then with that of the metallic material itself. For instance itisv = 10.26 cm 3 mol^in the case of silver, giving = 1.91 X 10~ 6 r. In other metals the values of 6 are of the same order of magnitude and must be considered as small even at the highest temperatures (about 2300 C) accessible to thermionic measurement. It is, therefore, per- missible to use the expansion (16.10) and to obtain from (16.01) W. Pauli, Zs. Physik 41, p. 81, 1927. * A. Sommerfeld, Zs. Physik 47, pp. 1, 43, 1928. 278 TEXTBOOK OF THERMODYNAMICS XVII 111 When O 2 is negligible, the gas is completely degenerate. Its pressure then becomes independent of temperature, being determined by the specific volume alone which, in turn, is fixed by the nature of the metal. We shall introduce the notation RT where !2< is called the inner potential. Hence its numerical value is !2< = - 25.85 tT H electron-volt. (17.32) For instance, in the case of silver Q- = 5.48 e.-v. With the abbreviation (17.31), eq. (17.30) takes the form while the expression for the energy follows from (16.15) and (17.01): - %pv +/Q + UQ. The entropy is determined by (16.17) and becomes for small values of 9, according to (16.10), -2 / \2 T -'7 5- C1U3) while the molal heat has the expression X Finally, the thermodynamic potential is expressed by IT 2 ^ +0 ~ + *' (17 - 35) The electronic spin has an important bearing also on the entropy expression for high temperatures. In of eq. (16.20), 2v must be sub- stituted for v, and this is equivalent to adding the term R log 2 to the entropy constant, or log 2 to the chemical constant. The entropy (16.21) and the thermodynamic potential (17.03) of the non-degenerate electron gas retain, therefore, the form of their expressions, but the constant JQ has an increased value. The chemical constant (14.15) of the electron gas becomes (with /* = 1/1821): jio, oi =-6.180 (17.36) or jio 0.826 - 1. A similar increase of the chemical constant occurs also in the case of gaseous atoms having an angular momentum and can be calculated XVII 112 ELECTRON AND ION CLOUDS 279 in a quite analogous way. It will be better, however, to postpone this calculation until section 120, where it will be treated together with the chemical constant of diatomic and polyatomic gases. 112. Thermionic emission and thermoelectricity from the modern point of view. We are now going to apply the results of the preceding section to the equilibrium of a metal with the thermionic cloud outside it. While the free electrons within the metal are in a degenerate state, those forming the outer cloud are completely non-degenerate. In fact, after deriving the conditions of equilibrium, under this assump- tion, we shall justify it by showing that the density of the cloud is extremely low and the value of the quantity 9 in it very high. We substitute into the condition (17.04), <p = <p f , the expressions (17.03) for the outer cloud, and (17.35) for the metal. Noticing that c p * | R, we find (with / = - en A ) log P = j + f !og r b/T, (17.360 with the abbreviation - r _2 1.21-21 (17.37) where the potentials 0' and ft refer to the outer cloud and to the metal, respectively. Neglecting second order terms, b - | (Q, + - a')- (17.37') K With the help of (1.15) we throw (17.360 into the form z - 7^ exp (j - b/T), (17.38) and obtain from (17.25) and (17.26) the thermionic current - r)7V" i/p , (17.39) A e (n A /2wpk)* exp j 120.2 amp cm~ 2 deg~ 2 . This formula was first derived by Dushman l from a semi-classical hypothesis of thermionic emission. Its derivation in terms of the degenerate electron gas was due to R. H. Fowler. 2 The work function b receives, according to eq. (17.370, an interpre- tation different from that of section 110. The work which the electron 1 S. Dushman, Phys. Rev. 20, p. 109, 1922; 21, p. 623, 1923. * R. H. Fowler, Proc. Roy. Soc. (A) 122, p. 36, 1929, 280 TEXTBOOK OF THERMODYNAMICS XVII 112 has to do in getting through the metal surface is still determined by the image force w which, in turn, fixes the outer potential difference Q 12' as in eq. (17.17). However, the electrons bring with them the energy eQ* depending on the inner potential (which has the nature of a quantized kinetic energy, as is shown in the statistical version of the theory). Only the difference bk = e(Q 0' + 0<) has to be supplied by thermokinetic or photoelectric forces. The connection of the thermionic work function with the photoelectric threshold fre- quency remains, therefore the same as in the classical theory, namely, hvo - Kb. Our knowledge of the inner potential fi, is not exclusively theoreti- cal: its existence and order of magnitude can be inferred from other phenomena, for instance, from experiments on the diffraction of electron beams falling on the surface of the metal. The values of ft< found by such measurements are not at all accurate, as they are affected by an uncertainty of d= 5 volts, but theoretical results obtained from formula (17.32) do not lie far outside these limits. A few nu- merical data are given in Table 36. 1 TABLE 36 Photoelectric Thermionic Metal kb/e t C kb/e (calc.) o o' (calc.) (obs.) (volt) If V (volt) Ag 4.73 20 4.08 925 5.48 10.2 18 5 4.56 600 Al 2.5-3.6 .... .... (5.57) (8.6) 18 " Au 4.82 20 4.32 1050 5.50 10.3 17 " < 4.73 740 Cu 4.1-4.5 .... 4.38 .... 7.00 11.3 17 " Fe 4.72 Hg 4.53 20 Ni 5.01 (7.35) (12.4) 18 " Pb 3.5-4.1 (3.9) (7.9) 11 " Pd 4.96 .... 4.99 Pt 6.30 .... 6.27 Rh 4.57 20 4.58 41 4.93 240 W 4.58 4.52 1 Taken from Hughes and Du Bridge (Photoelectric Phenomena, pp. 76, 235, New York, 1932) and slightly revised. A more complete list will be found in the article by J. A. Becker, Rev. of Mod. Physics 7, p. 123, 1935. XV11 112 ELECTRON AMD ION CLOUDS 281 The inner potentials Q< in Table 36 are calculated on the assump- tion that there is one free electron per atom. This is sound in the case of metals of the first group (Ag, Au, Cu) but less certain in other cases. From the tabulated values of b one can determine z by means of eq. (17.38) and find the molal volume and the quantity 9 in the ther- mionic cloud at different temperatures. For instance, in the case of tungsten (neglecting the change of b with temperature) these data are as follows : TABLE 37 /C. 20 300* 600 1000 1500 2000 2300 3000 3800 v.... 1.1X10" 4.6X10" 3.4X10* 5.2X10" 1.6X10" 3.4X10" 2.5X10" 4.8X10* 5.1X10 6... 6.0X1078 7.6X10i 1.9X10" 7.8X10" 1.0X10* 2.1X10* 4.3X10* 3.7X10* 1.0X10* We see from this table that 9 has exceedingly high values in the electron gas outside the metal, so that the assumption of its being non-degenerate is entirely justified. In comparing Dushman'* formula (17.39) with experiments, it must be remembered that it has its theoretical limitations. The thermionic work function is not strictly a constant but has a slight temperature dependence. This is due, in the first place, to the neglected terms of the expression (17.35) and to the thermal expansion of the metal which changes the molal volume v in it and also may affect the image force, and through it, the potential difference 8 0'. In the second place, the picture of electrons being completely free is only an approximation; in reality, there are some mutual forces between them and the metal ions which also could account for small deviations from the constancy of b. Finally, the factor (1 r), which is supposed 1 to be of the magnitude 0.9S, is again slightly temperature dependent. On the other hand, the measurement of thermionic cur- rents involves an extremely difficult experimental technique whose accuracy cannot be pushed beyond a certain limit. In view of this situation, the agreement must be considered as quite satisfactory. The results for A, as obtained experimentally, 2 do not agree with the theoretical value 120.2, as shown in Table 38. This is probably due to the work function b depending on T and containing a linear term (b = b' + aT.) Therefore, the formula (17.39) can be rewritten in the same form but with the new constants V = b aT, A' A exp a. From the analysis of the measurements one obtains directly V, A' and not 6, A, but, knowing the theoretical value of A, one can calculate a and the true work function b. L. Nordheim, Proc. Roy. Soc. (A) 117, p. 626, 1928. L. A. DuBridge, Proc. Nat. Acad. 14, p. 788, 1928. 282 TEXTBOOK OF THERMODYNAMICS TABLE 38* XVII 112 Material Authority kb'/e (in volt) kb/e A'/A 0X 10* W Bu Bridge 4.61 5.53 0.99 X 10 4.56 Th-W i < 4.61 2.63 1.05 X 10-' 1.76 Pt 1 1 5.08 6.27 0.57 X 10* 3.41 K 1 1 0.645 2.21 1.37 X 10" 24.70 Cu (liqu) 4.55 3.54 1.65 X 10~ 4 6.91 Zr, Hf Zwikker 3.85 4.84 0.57 X 10 a 4.08 BaO Detels 2.48 1.47 1.89 X 10-* 6.79 * See: S. Dushman, Rev. Modern Physics, 2, p. 381, 1930. A curious and not entirely explained result of DuBridge's is that, in the same metal but for different conditions of its surface, the observed log A' is a linear function of the observed V (i.e. log A' = &b' + const). The coefficients are listed in Table 38. With respect to thermoelectric forces between two metals (1) and (2), the qualitative considerations of section 4 remain valid. The equi- librium of the electrons in the two metals is determined by the same condition <pi = ^2- The only difference is that we have to substitute for the thermodynamic potential in either metal the expression (17.35) since we consider both clouds as degenerate. Thus the potential dif- ference becomes V 2 k 2 I 1 - 2 - (Q - Da) --- T 2 ( - -f- Iz e? \Sm S < 1M > The expression fli $2<2 is independent of the temperature. It represents a potential difference existing between the two metals under all circumstances. It cancels out of every closed circuit and cannot be observed by galvanometric or electrometric methods. Therefore, only the second term need to be taken into account for the thermoelectric difference: 821 - 0.611 x lo-sr 2 ( -f- - a Q<2 volt, (17.41) where Q-i, 12<2 are expressed in volts. This formula was first given by Sommerfeld. 1 For the reasons outlined in section 109, we can hardly expect from it more than the correct order of magnitude. Experi- 1 A. Sommerfeld, note on p. 277. This formula does not represent the e.m.f. in a thermoelectric circuit, which is given at the end of section 139. XVII 113 ELECTRON AND ION CLOUDS 283 mental data and a thermodynamical discussion of thermoelectricity will be given in section 139. As to the contact potential of metals, it can be obtained directly from the formula (17.37), substituting for 12i Ife the expression (17.40), This relation was tested by Millikan, Lukirsky and Prilezaev, and Olpin * and was confirmed in the case of metals with very clean surfaces. 113. lonization in the solar atmosphere. In their systematic study of solar radiation the astronomers succeeded in analyzing spec- troscopically the light coming from the different altitude layers of the chromosphere. They found that the emission of the lower levels mostly consists of arc lines (characteristic of the neutral atoms), while the spark lines (belonging to the ionized atoms) appear higher up and grow in intensity with altitude at the expense of the spark lines. This result seemed puzzling because the lower layers of the sun are hotter and, in the laboratory, the spark lines are enhanced by raising the temperature. The reason for this discrepancy was explained by M. N. Saha, 2 who pointed out the influence of the pressure on the degree of ionization and applied the thermodynamical equations of chemical equilibrium to the problem. There are charged particles of both signs in the solar chromosphere, but at the temperatures and pressures prevailing there the phenomenon of pairing (mentioned in section 106) is quite negligible. Therefore, the form of the equations of equilibrium is not appreciably changed either by the charges on the electrons and ions or by the gravitational field of the sun (section 108). These conditions also exclude degeneration so that we can apply the equation of chemical equilibrium (14.16) for classical perfect gases. The reactions which we have to consider as taking place are those of the single and double ionization of atoms by splitting off one or two electrons. In the symbols of section 42 #++GL-G n = and C++ + 2GL - G n - 0, (17.42) where" 6+ and C++ refer to the positive ions, GL to the electron and G n to the neutral atom. The formula (14.16) needs a slight correction; 1 R. A. Millikan, Phys. Rev. 7, p. 355, 1916; P. Lukirsky and S. Prilezaev, Za. Physik 49, p. 236, 1928; Olpin, Phys. Rev. 36, p. 251, 1930. For a discussion from the point of view of the statistical theory see: Mitchell, Proc. Roy. Soc. (A) 146, p. 442, 1934; 153, p. 513, 1936. * M. N. Saha, Phil. Mag. 40, pp. 472, 809, 1920. 284 TEXTBOOK OF THERMODYNAMICS XVII 113 there must be added log 2 to the chemical constant of the electron (section 111). However, this is not all : it will be shown in section 120 that a similar term log g belongs to the chemical constants of the atoms and ions (because of the statistical weight of the angular momenta of these particles. Since the atomic weights of the atom and the ion are practically the same, we can write f S> A logio MA = f v logio M ~ 4.892v, where p e means the atomic weight of the elec- tron. For monatomic gases c p = %R, and the heat of reaction be- comes, according to (3.28), Q = 2^x* = At/o + fyRT. If we con- tinue to measure the pressure in atmospheres but take as the unit of heat of reaction not the erg but the electron-volt (1 erg/mol = 0.9646 X 10 12 e.-v.), the equations of equilibrium become for the two reactions (17.42) log (P+P-/P*) - ~ 6.180 + logio - f ^ f logio T - 12.360 + ^ + 5 logio T 5037 AC/o 5037 AC/o (17.43) The values of AJ7o and g for the most important ingredients of the solar atmosphere are given in Table 39. The data of this table refer to the normal states of the particles', the statistical weights of the nuclear spins are not included in g because they are the same in the neutral atom and the ion and cancel out of the reaction. TABLE 39 Particle I Al/o (in e.-v.) Particle g ( .^ v } Particle g (in e.-v.) H H + 1} 13.53 Ca 1 Ca + 2 6.09 11.82 Zn Zn + i! 9 36 17.86 He Na Na+ 1} 24.46 5.12 Sr** 1 Sr + 2 Sr ++ 1 5.67 10.98 Cd** Cd + 1 J II 8.96 16.84 Rb Rb+ 1} 4.16 Ba 1 Ba+ 2 5.19 9.96 Hg **g+ ii 10.38 18.67 Cs ?} 3.87 Ba++ 1 rig +4. 1' c... 2! 23.4 XVII 114 ELECTRON AND ION CLOUDS 285 We see from eqs. (17.43) that the ratio P+/p not only is an increas- ing function of the temperature but also depends on the pressure of the free electrons, being inversely proportional to it. This pressure is an appreciable fraction of the total pressure and rapidly decreases with the general attenuation of the chromosphere as the level is raised. The equation, therefore, gives a qualitative explanation of the observations described above. Moreover, it accounts for the observed fact that, the lower the ionization potential, the lower also the level at which the appearance of the spark lines sets in. On the other hand, it is very difficult to calculate p+/p n quantitatively because the pressure of the electrons />_ depends on many factors. All the elements of the mixture which constitutes the solar atmosphere contribute to it, but even if we had atoms of one kind only the degree of ionization would depend on the excited states and its computation would be a difficult problem. 1 However, if our object is only to obtain a check of the thermodynamical formulas (17.43), we can get around this difficulty by choosing one of the elements of the solar chromosphere as a stand- ard. Suppose we have measured, for different levels of the sun, the intensities of the arc lines and the spark line of this standard substance and inferred from these measurements the relative abundance />+<> //><> of its ions and neutral atoms. Writing eq. (17.43) for the standard element and for any other element, we eliminate />- and obtain The theory was tested by similar devices and received a fair con- firmation, but it was pointed out by H. N. Russel 2 that great accuracy in the agreement cannot be expected. The gases of the solar chromo- sphere are not strictly in a condition of thermodynamical equilibrium since they are exposed to the one-sided radiation coming from the lower and hotter regions of the photosphere. 114. Ion concentrations in electrolytes. Galvanic cells. The theory of dilute solutions also offers an important field of applications for the results of sections 106 and 107. A large class of solutes, the electrolytes, undergo a partial or complete dissociation into electrically charged components (ions). This is particularly characteristic of acids and salts in aqueous solutions: when their molecules dissociate, 1 It was treated by Milne, R. H. Fowler, and C. G. Darwin. The most compre- hensive review of the whole field of ionization in the solar atmosphere is due to A. Pannekoek (Handbuch der Astrophysik, Vol. Ill 1, pp. 257-350. Berlin 1930). * H. N. Russel, Astrophys. J. 79, p. 317, 1934. 286 TEXTBOOK OF THERMODYNAMICS XVII 114 the hydrogen or the metal acquires a (single or multiple) positive charge and is called the cathion, or positive ion, the acid radical gets a negative charge and constitutes the anion or negative ion. As an example we take cadmium sulfate which, dissolved in water, dissoci- ates according to the formula Cd++ + SO 4 ' - CdSO 4 = 0, where the superscript ++ indicates a bivalent positive ion (i.e. one carrying two elementary charges). We know from section 68 that non-electrolytic dilute solutions behave like perfect gases. If the conditions of equilibrium are not changed by the electric charges of the ions, we should expect the laws derived in that section, especially the mass law (9.60), to apply also to electrolytic solutions. For instance, the above reaction would give X++X /Xn = K(T), where x++, x , x n are the respective mol fractions of the positive and negative ions and of the neutral (undissociated) salt. In the more general case, when the molecule is ionized into vi, i>2, ... V T positive and negative ions, their total number being v t ~ v\ + . . . + v rt the equation takes the form vid + P2G* + ... + v T G r - G n = (17.45) and the mass law has the expression X) ^ log Xj - log x n = log K. (17.46) i This equation is of the same type as those considered in section 52, and we can draw from it the conclusion that a certain fraction of the salt molecules is ionized (dissociated into ions). In fact, experiments show that the equilibrium constant K is fairly large in these reactions, so that solutions of low concentration are almost completely ionized. The hypothesis that dilute electrolytes are completely dissociated into ions was first advanced by Arrhenius, who adduced in its support their high osmotic pressures and low freezing points. Moreover, he showed that the electric conductivity is in agreement with this assump- tion, being nearly proportional to the molality of the dissolved salt. The work of Arrhenius and his successors had a great influence on the development of thermodynamics, on one hand, and laid the founda- tions of the science of electrochemistry, on the other. The laws of electrolytic conduction, as developed by Kohlrausch, afforded a means XVII 114 ELECTRON AND ION CLOUDS 287 of determining the mol fractions Xj of the individual ions and gave a rough confirmation of the formula (17.46). However, as the technique of these measurements grew more and more accurate, it was found that the law (17.46) is rigorously valid only for a limited class of sub- stances, the so-called weak electrolytes. To this group belong a number of organic acids and their salts. On the other hand, the common inorganic acids and their salts form the class of strong electrolytes which exhibit appreciable deviations from the formula (17.46) even in very low concentrations. At first, it was doubtful whether these discrepancies were due to the inadequacy of the formula itself or to the failure of the conduction measurements to give the correct mol fractions. The investigations based on the freezing method, described in section 78, were under- taken to decide this question. They led to the result that eq. (17.46), actually, is not valid for strong electrolytes, or in other words, the ionic clouds of strong electrolytes do not obey the laws of perfect gases. As was shown in section 77, the form of the mass law can always be retained if the numbers Xj are replaced by the effective mol fractions or activities a,: JjiV log a,- log a n = log K. (17.47) Putting for the neutral salt a n = x n and making use of the notations of section 79, especially of the activity function 7 defined by (11.32), we can write (for dilute solutions) vj log Xi log x n = log K v, log 7. (17.48) The data from conduction measurements turn out to be in excellent agreement with those obtained by the freezing method: when the proper values of 7 are substituted, the formula (17.48) is confirmed with great accuracy. The theoretical reasons for the abnormal behavior of strong electrolytes will be discussed in the next section. The notion of electrolytic ionization affords an insight into the mechanism producing the e.m.f. (electromotive force) of galvanic cells which were treated in section 37 from the more formal point of view of energy relations. In all the constructions mentioned there the chemical reaction is between two metallic salts, MR and M'R (with the same radical R) which ionize in aqueous solutions according to equations similar to MR - M + + -, M'R = M'+ + R-, (17.49) and in which the reaction M'R + M > MR + M' is exothermic. 288 TEXTBOOK OF THERMODYNAMICS XVII 114 The current is, in part, due to the negative radical ion R- traveling through the cell from the vicinity of the positive to the negative plate. 1 We are now able to say what force drives the ions and produces the current: it is the force of osmotic pressure. In fact, the -R_ ions react at the negative plate with the metal M of which it consists. Therefore, their concentration and partial osmotic pressure P_ sinks below the value P'_ which it has at the positive plate. When the resistance of the outer circuit is very large, an electric difference of potentials fl' 12 builds up which almost completely compensates the e.m.f. (electromotive force) of osmotic pressure. According to (17.06) we have then (since/ = F) (17.50) - - f- log (P'/P), provided the cloud of R~ ions in the solution can be regarded as a perfect gas. The explanation for the movement of the M'+ ions to the positive plate is also osmosis (since in equilibrium the total osmotic pressure must be everywhere the same), but the calculation of their concentrations would lead us too far into electrochemistry. It is essential for the operation of the cells described in section 37 that its electrodes consist of two different metals; they are, therefore, called electrode cells. There exist cells of a different kind, known as concentration cells, an example of a modern concentration cell is as follows: 2 Ag AgCl (solid) KG solution (m) K-Hg amalgam KG solution (m f ) AgCl (solid) Ag The first half of this chain (from Ag to the amalgam K-Hg) repre- sents an electrode cell with the reaction AgCl + K = KCl + Ag, which is strongly exothermic. If this half chain were left to itself, the molality m of KCl in the chamber filled with potassium chloride solu- tion would increase producing a current in the direction to the left (i.e. from K-Hg to Ag). In fact, in this case, the K+ ions move to the left and replace some of the silver in the AgCl paste (the reduced silver being deposited on the Ag electrode). At the same time the Cl_ ions move to the right and react with the potassium metal in the amalgam. In both processes KCl is produced and added to the solu- tion. On the other hand, if an electric current is sent through the half 1 In the example of the Weston cell (Fig. 11, p. 95) the radical (R-.) is SO 4 , the negative plate (M) consists of Hg and the positive (M') of Cd. * Maclnnea and Parker. J. Am. Chem. Soc. 37, p. 1445, 1915. XVII 114 ELECTRON AND ION CLOUDS 289 chain in the direction to the right all processes are reversed and the molality of potassium chloride solution must decrease. Now, the second half of the concentration cell is of the same construction and of opposite orientation: the resultant current will be, therefore, the difference of the currents produced in the two half chains and will flow through both of them. If we assume m' > m (see below), it will have the direction from K-Hg to Ag, in the first half, and from Ag to K-Hg, in the second. Therefore, the molality in the first KC1 cham- ber will increase at the expense of that in the second. There is no cur- rent when m r = m : the driving power is due to the difference of osmotic pressures in the two chambers. All this is also characteristic of the general case when the reaction is given by eq. (17.45). The essential feature of every concentration cell are two chambers C and C filled with the same solution in different concentrations. In the beginning, when the chambers are freshly filled, the solutions are electrically neutral (as a whole, containing equal positive and negative charges) so that the osmotic forces are entirely uncompensated by electric potentials. Under the ordinary conditions of construction, there is no tendency for potential differences to build up, and the solutions can be regarded as neutral during the whole length of their operation. Owing to these forces, the ions pass from the chamber of higher osmotic pressure into that of lower in numbers proportional to the coefficients v$ of eq. (17.45) as is implied in the electric neutrality of the solutions. We shall calculate , the e.m.f. of such a cell, only for the case of low concentrations when the electrolyte (in both chambers) can be considered as completely ionized. The change of thermodynamic potential which the ions undergo in being transferred from the chamber C to the chamber C is then (referred to 1 mol of the neutral electrolyte) A* = %v h fo\ ? h ). The left side, according to (5.35), is the (negative) non-mechanical work done by the system APT - W - EJ, where / is the total charge of the ions pertaining to 1 mol. On the right side, we substi- tute the activities from formula (11.17) JE - - RT y>* log (aV**). (17.51) If we denote the valency of the ion h by <r h (taking it positive or negative, according to the sign of the charge) 290 TEXTBOOK OF THERMODYNAMICS XVII 11$ Further, making use of the abbreviation (11.32), we can write E - - (RT/J){v 9 log ( 7 '/7) +Z* log (x'k/tk)}- (17.52) The measurement of electromotive forces of concentration cells offers a method for the determination of relative activity functions of electrolytes. Numerical data for the activity function will be given in the next section. 115. Strong electrolytes. All the results so far obtained in this chapter rest on the following hypothesis made in section 106: the mutual electric energy of the particles in a small volume r is propor- tional to the square of the number of particles in it and, therefore, negligible. However, this argument is sound only as long as the electric forces do not affect the uniformity of distribution of the particles, and it ceases to be strictly applicable in the case of mixtures of positive and negative ions. Owing to the mutual attraction, each ion has the tendency to hover in the neighborhood of other ions of the opposite sign. The mean distance between unlike ions is smaller than that between like ones. This gives rise to energy terms, proportional to the number of particles, which may become significant under certain circumstances. It had been suspected for a long time that these additional electric terms may account for the abnormalities in strong electrolytes mentioned in sections 79 and 114. If this point of view is accepted, the sharp line of division between dissociated and undissociated molecules disappears: the distances between the ions can assume all values and are distributed according to the laws of probability. The transition from association to dissociation is gradual, and all ions can be considered as free, in this sense. The problem was formulated with complete clarity and partially treated by Milner. 1 It was made better amenable to quantitative evaluation by Debye and Hiickel, 2 who gave a simple method for the approximate calculation of the effect at low concentrations. Let us consider a neutral solution containing ions of several kinds, the type (j) being represented by Ni mols. We denote their valency by <r/, which is positive or negative, according to the sign of the charge on this ion. The total charge in the solution is then 0, (17.53) Milner, Phil. Mag. 23, p. 551, 1912; 25, p. 742, 1913. P. Debye and E. HUckel, Phys. Zs. 24, pp. 185, 305, 1923. XVII 115 ELECTRON AND ION CLOUDS 291 where F = en A is the Faraday constant or the charge per 1 mol of monovalent ions. We denote farther by zt-NtnJV, (17.54) the mean density, i.e. the number of ions of the type j per unit volume. Let us now focus our attention on a " preferred ion " of the kind (1) and let us ask how the other ions are distributed around it. The answer is given by the Boltzmann principle (4.69). Suppose that, at a point P, its electric potential is 81 ; in the time average, the density of j-ions at this point will be C exp (-e<rjQi/kT). Let the potential 81 be counted "from infinity", i.e. from a distant point where the field of the preferred ion does not make itself felt (Qi = 0) and the density has the normal value /; then the constant of proportionality has the value C = */. The total density of electric charge at the point P, produced by ions of all kinds, is, therefore, exp ( eat Qi/kT). (17.55) j The potential d is produced by the preferred ion, in part directly, in part indirectly through the action of the " atmosphere " of other ions which hover about it. Assuming the validity of the Coulomb law and, on the average, a spherically symmetrical distribution of the atmosphere, fij. must satisfy Poisson's equation in the form 1 r 2 dr \ dr / D where D is the dielectric constant of the solvent. Eliminating p from (17.55) and (17.56), we obtain a differential equation for the determina- tion of QI. We restrict ourselves to the case when the exponent <TJ e$li/kT is so small that its square can be neglected. Because of (17.53), the term of zero order vanishes, leaving 2 m , H --- = *r 81, (17.57) r dr *-%*%* <"- 58 > The integral of eq. (17.57) is ! = -". (17.59) 292 TEXTBOOK OF THERMODYNAMICS XVII 115 The coefficient A\ must be determined from the "boundary conditions." Let a\ be the mean distance of nearest approach of the other ions to the preferred ion: within the little sphere of the radius a\ the field is not influenced by the ionic atmosphere and the potential in this region may be taken as obeying the simple Coulomb law tf x - ? + ODI, (17.60) while outside this region it follows the law (17.59). At the surface of the sphere r = ai, there must be continuity of potential (&'i = fli) and of dielectric displacement (dQ'i/dr = dtti/dr). This leads to ffl e exp Without the ionic cloud about the preferred ion, its potential (against infinity) would be represented merely by the first term of the expression (17.60). Therefore, QOI gives us the potential difference, built up through mutual interaction, of the ion with respect to the mean potential in the solution. In a similar way the other ions assume the potentials w TT (17 - 62) D 1 + *&/ In arriving at this formula, a dual capacity was attributed to every ion. On the one hand, it acted as a " preferred ion " collecting an atmosphere around it; on the other, it formed part of the atmos- pheres of all the other ions. It is clear that it is not always permissible to separate and superpose these two functions of the ion, as is done in Debye-Huckel's theory. The superposition implies, ultimately, a linearity of the equations controlling the electric field and comes back to the same restriction which was made in eq. (17.57). It seems, therefore, risky trying to improve the theory by taking into account terms of higher order in the exponent of (17.56), as has been done by some authors. 1 The question needs a closer investigation whether the theory is not restricted by its very method to cases in which (ffj-eQi/kT) 2 can be neglected. Another idealization of the theory lies in the schematic character of the " boundary conditions " which lead to the formula (17.62). In view of them, the dependence of the potential Ooy on the ionic radius a,- can hardly be considered as entirely trustworthy. 1 Gronwall, LaMer, and Sandved, Phys. Zs. 29, p. 358, 1928. XVII 115 ELECTRON AND ION CLOUDS 293 We shall show in section 138 that the change of internal energy of the solution produced by the potentials (17.62) is a rather complicated one. The work function SF, however, can be calculated without much trouble. In order to do so, it will be well to bring out the thermo- dynamical significance of our results. An ion of the charge cr,- 6 acquires the potential flo/' let us consider an imaginary, ideal way of producing such a distribution of charges and potentials. Suppose that e is not constant but that the charge of every ion can be changed and built up from to the normal value. 1 The work which must be done in order to increase all the charges from <r,-e to <r,( + de) (e being the same for every ion) is known from the theory of electricity to be DW* = Qo/ffi <fct where the summation is extended over all the ions. Since there are n A N, ions of the kindj, DW 9 = n A / .ffjN&oide. j The total work of building up the distribution at constant temperature and constant volume is, therefore, W. - - n A I y\yQ ,-tf/ de. (17.63) It was shown in section 36 that the non-mechanical work done in a reversible process, at T = const, V = const, is oppositely equal to the increase of the work function. This increase, due to the electric charges and potentials of the ions, we denote by ^, = W 9 . The expression (17.63) is somewhat simplified if we use, instead of the individual ionic radii a/ of eq. (17.62), a mean radius a. In fact, the accuracy of the theory is hardly sufficient to justify such distinctions and, moreover, the term *a/ = *a in the denominator is altogether neglected in our ultimate applications. According to the definition (17.58) of K we can write, then, (17.64) The partial thermodynarnic potential can be obtained directly from the work function by means of eq. (5.39) _ _. kTV "* ( d "\ . T 4ir l + Ka\dN h / VtT 2D 1 + *a " ' This is the increase of the thermodynamic potential due to the electric fields of the ions. If we define the activity coefficient of the ions h as log a h = ?eh/RT (section 78), this means that we compare the 1 R. H. Fowler, Statistical Mechanics, p. 318, Cambridge, 1927. 294 TEXTBOOK OF THERMODYNAMICS XVII 115 activities in strong electrolytes with the theoretical ones in dilute solutions at the same volume, whereas experimentally they are compared at the same pressure. This difference, however, is of slight significance in view of the very small compressibility of water and other condensed solvents. 1 In very dilute solutions K is comparatively small, and Ka in the denominator can be neglected. From (17.58) or in terms of molalities (V = 1000, AT,- = log a* = - ,^* H ( ^- > V- 2 ,- ) (17.66) In water at the temperature of 25 C the dielectric constant is D = 78.8 leading to the expression logio <* h = - 0.356 fft? if we denote by m the molality of the neutral electrolyte originally dis- solved (mj = Vj-m). Hence the activity function, defined by eq. (11.32), is given by log lo 7 . _ while the coefficient becomes 0.345 for the temperature C. This is the so-called limiting law for very dilute electrolytes due to Debye and Hiickel. The agreement of the limiting law with the measured activities is very good for ions of low valency as appears from Table 40. The very accurate measurements by Neumann, 2 on silver chloride and barium sulfate, show, however, that the limiting law is not rigorous even for low-valency salts. In plotting the measured y against m** this author finds curious oscillations of the curves around the theo- retical value (17.67). The deviations, although very small, seem outside the limit of experimental error. High-valency electrolytes 1 The pressure increase due to the effect here considered is only a small fraction of 1 mm Hg. According to (9.42), (11.35), and (17.67), in water of 25 C it is: A - A - 0.006v e m log y - 0.12(2<r,yym)^ mm Hg. E. W. Neumann, J. Am. Chem. Soc. 54, p. 2195, 1932; 55, p. 879, XVII 115 ELECTRON AND ION CLOUDS 295 were investigated by La Mer and co-workers, 1 who found large dis- crepancies with the limiting law. TABLE 40 ACTIVITY FUNCTIONS OF STRONG ELECTROLYTES Type <TI = 1, <r a = 1, n 1, v* 1 0.005 0.932 0.929 0.930 0.926 - 2 0.005 0.753 0.785 0.800 0.734 Type o\ = 2, 0-2 = 2, v\ 1, v\ 1 m .............. 0.001 0.002 0.003 0.005 7 ......... (theor) 0.746 0.661 0.601 0.519 BeSO 4 . . . . (exp) 0.754 0.670 ..... 0.534 CdSOi.... " 0.754 0.671 0.621 0.540 CaS 2 O 8 ... " 0.754 0.674 ..... 0.540 When the solutions are not very dilute (presumably, m > 0.005, for low- valency salts), the term KO, in (17.65) can no longer be neglected. In fact, the measured activity functions usually fall, at the higher concentrations, below the values of formula (17.67). However, in the present state of the theory the dependence on a is not accurately enough known to make its quantitative discussion worth while. 2 i La Mer and Mason, J. Am. Chem. Soc. 49, p. 410, 1927; La Mer and Cook, ibidem 51, p. 2622, 1929; La Mer and Goldman, ibidem 51, p. 2632, 1929. * A review of the work done on the theory of strong electrolytes by statistical methods will be found in an article by O. Halpern (J. Chem. Physics 2, p. 85, 1934). A critical discussion of the foundations of the theory was given by R. H. Fowler, (loc. cU.) m 0.001 0.002 0.003 . . (theor) 0.964 0.949 0.938 HC1 . . . . .. (exp) 0.965 0.953 HNOs 4 0.953 AgNO 8 . 44 0.940 AgC10 3 . KC1 41 41 . . . 0.940 m . Type 01 0.001 = 2, a a = l, 0.002 v\ 1, 0.003 CaCl 2 ..(theor) (exo) 0.881 888 0.836 0.850 0.803 BaBra 44 0.829 H*SO*. . 44 0.876 0.825 CHAPTER XVIII THEORY OF SPECIFIC HEATS 116. General considerations. We had occasion to give the expres- sions for the specific heats of a few ideal systems, such as perfect gases (non-degenerate and degenerate) and the Einstein model of a solid. Purely thermodynamical reasoning does not enable us to say much about the heat capacities of the real substances of nature except that they must vanish at T = 0. The internal energy of a system com- pletely determines its equation of state through eq. (13.22), but the reverse is not true: the knowledge of the equation of state is not sufficient for calculating the energy since eq. (16.12) still contains an unknown function of the temperature which usually gives the major contribution to the specific heat. However, heat capacities are of such fundamental importance as material for the applications of thermodynamics that a textbook of this branch of science would be incomplete if it did not contain some information on the status of our theoretical knowledge about them, even if this knowledge is obtained by non-thermodynamical methods. It seems appropriate, therefore, to include a chapter partially devoted to the way in which the kinetic theory of matter approaches the problem of specific heat. The fundamental tool of the kinetic treatment is eq. (4.70) of the Boltzmann principle. It is true that there are other types of statistics which we have described in section 104. However, as it was stated there, they apply only when many identical particles are distributed over energy states determined largely by joint external conditions. On the other hand, in such systems as an assembly of rotating mole- cules, each particle rotates around its own center of gravity and its quantum states are determined by conditions peculiar to itself as if it were alone and not part of a system. The same thing applies to the other systems occurring in the theory of specific heats, for instance, 296 XVIII 116 THEORY OF SPECIFIC HEATS 297 linear oscillators. There is, therefore, little occasion to use the statistics of Fermi or Einstein-Bose, and the Boltzmann principle reigns supreme as far as these investigations are concerned. If a particle can assume the quantum states 0,1,2, ... with the respective energies eo, ei, 82, ... the principle postulates that the number of particles in the state / be Zi = Cexp ( Bi/kT). The sum of the numbers Zj represents, obvi- ously, the total number Z of particles in the system Z = c exp (-ci/*r) - CY. (18.02) The expression Y = S exp ( i/kT) is usually called the sum of states. Every quantum state of the particle must be represented in it by a term. If the same energy level e< belongs to several (gi) different quantum states, the corresponding term must be repeated g\ times. It is, therefore, more convenient to write it in the form V = Z ft exp (- ei/*r). (18.03) We call the integer gi, as in section 111, the statistical weight of the level /. The energy of all the particles in the state / is X exp ( ei/kT), and the total energy of the system U = C]C gi*i exp (- ei/*r). (18.04) The sum in this expression can be obtained by differentiating Y with respect to -(l/*r,) i.e. U = - CkdY/d(l/T). Eliminating C with the help of (18.02) If we refer our considerations to 1 mol of matter, the number of particles is Z = n A (Avogadro's number n A = R/k), and the molal internal energy becomes u = - R a log F/a(i/r>. (IBM) We may consider this as the fundamental formula of the statistical theory of specific (molal) heats, since they can be derived from u by the formula c = du/$T or R a 2 log Y < 18 - 07 > 298 TEXTBOOK OF THERMODYNAMICS XVIII 117 Comparing (18.06) with the relation (5.S6), we find as the thermo- dynamical interpretation of the sum of states * = - RT log Y, (18.08) where \l/ is the molal work function defined as ^ = u Ts. There- fore, the entropy has the expression s -^A-r + *"* r < IM When the temperature T is very low, all the terms of the sum of states (18.03) become negligible compared with those of the lowest energy eo. The sum is reduced to Y = goesp ( eo/Jfer), whence RT log Y = n A o + RT log go and s = R log go- It may seem, at first sight, that this result is in contradiction with Planck's formula- tion (13.19) of Nernst's postulate. However, these formulas are no longer valid in the vicinity of T = 0. The expression (18.03) refers to the rotational and oscillatory motions of the molecules, and its use implies the assumption that the probabilities of these motions can be evaluated apart from the translational movements. This ceases to be true under conditions of temperature and density in which the statis- tics of Fermi or Einstein- Bose begin to play a role for the translational degrees of freedom. We refer, in this connection, to the remarks made in section 111 with respect to the entropy of the electron gas. In that example the statistical weight is go = 2, as there exist two kinds of spin. In the non-degenerate state the entropy contains, therefore, the term R log 2 (entropy of mixing), but in the degenerate gas this term disappears because the two kinds of electrons are regularly and uniquely arranged over the quantum states. 117. Diatomic gases. The classical kinetic theory inevitably leads to the principle of equipartition of energy (4.71) : each degree of freedom takes up the kinetic energy %RT per mol. It b'assumed that the molecule of a monatomic gas has only 3 (translational) degrees of freedom, the molecule of a diatomic gas 5 (3 translational, 2 rota- tional). The corresponding molal heats are c v = $R and c p = -|J?, respectively. If the gas does not strictly obey the equation pv = RT, the resulting correction is taken care of by eq. (18.01) or by its deriva- tive with respect to T /V^2jA dv+f'Cn, (18.10) while c p can be obtained from (4.27). The main success which the classical theory has to its credit lies in the sphere of monatomic gases, XVIII 117 THEORY OF SPECIFIC HEATS 299 whose specific heats it represents accurately. It is not so satisfactory in application to diatomic substances (compare section 15). For the more perfect permanent gases of nature the agreement is fair in the region of room temperatures, but there are large deviations both for low and very high temperatures. The following treatment is based on the theorem of mechanics that the kinetic energy of any system (molecule) can be represented as a sum of the two terms : the kinetic energy of the center of gravity and the energy of the motion within the system (molecule) relatively to the center of gravity. Correspondingly, the internal energy (as also the entropy and the specific heat) may be divided into two parts. It happens, moreover, that the contributions of the rotations and vibra- tions (though not rigorously additive) are separable with a sufficient degree of approximation 1 u = u t + u r + u vi . (18.11) The first belongs to the translational motions and is calculated by the classical formula wo, (18.12) since the center of gravity is completely determined by the three coordinates of translation, assuming that degeneration of the type of Chapter XVI is negligible (compare preceding section). The second and third parts are due to the rotational motion of the molecule as a whole or to the vibrations of the atoms inside it. They are computed with the help of the quantum eqs. (18.03) and (18.06). As to the rotational heat c r , a good approximation is often afforded by the so-called dumb-bell model. The diatomic molecule is considered as consisting of two material points at an invariable distance from each other rotating around their center of gravity in a joint plane. Quantum dynamics gives for the energy levels of the dumb-bell model j + 1), (18-13) where K is the moment of inertia of the molecule. At the same time the statistical weight of the level e, is g/ = 2j + 1 (provided the nuclei have no spins). 1 Occasionally, more rigorous expressions are used in which the rotational and vibrational energies cannot be separated (compare section 119). This is, however, unnecessary as far as the accuracy of caloric measurements is concerned. In addi- tion to the items of eq. (18.11), sometimes the energy of electronic configurations (excited states) is also of importance. 300 TEXTBOOK OF THERMODYNAMICS XVIII 117 Two cases must be distinguished with respect to the values of the quantum integers j. (A) When the molecule is asymmetrical, con- sisting of two different atoms, it can assume all integral values (j as 0,1,2, . . .). (B) When the molecule is symmetrical, built up of two identical atoms, only even integral values are permissible 1 (j = 0,2,4, . . .) This distinction was recognized even in the old quantum theory and was there interpreted as follows: the asym- metrical molecule must be rotated through the angle 2ir until it returns to the initial position; the cycle of the symmetrical molecule is only half as large, for after being turned by the angle TT it is in a position which cannot be told from the initial. The interpretation of quantum dynamics is somewhat different: when we have two identical atoms (without nuclear spins) they obey the statistics of Einstein-Bose 2 and admit only quantum states with symmetrical wave functions; these happen to be those with even numbers j. No such restriction exists for the asymmetrical molecule. In either case, we can express the sum of states (18.03) by the single formula (j = 1,2,3. . .) (2aj + V exp l- Tfl tffo' + *>]' < 18 - 14 ) r = h 2 /Sir*KkT, (18.15) if we introduce the symmetry number <r, which is, respectively, equal to 1 and 2, for asymmetrical and symmetrical molecules. For high temperatures, r is very small and (18.14) yields the equipar- tition value. In fact, putting n = in eq. (18.22), below, we find in this case Y r - 1/rcr or log Y r = - log (1/r) + const. Hence (18.07) gives c r = R, corresponding to the two degrees of freedom of the dumb-bell model. On the other hand, for very low temperatures, only the first two terms of the sum are significant, log Y r = (2<r + 1) exp [ TO-(<T + 1)], whence c r - R(2a + IJo^cr + 1)V exp [-T<T(<T + 1)]. Therefore, the formula (18.14) represents a gradual rise (with temperature) of the rotational molal heat from zero to the equipartition value, a behavior which is in qualitative agreement with the observed facts. 1 We consider here only nuclei without spins. The case of atoms with nuclear spins will be treated in the next section. 1 Nuclei without spins are, necessarily, built up of an even number of primary particles. XVIII 117 THEORY OF SPECIFIC HEATS 301 When quantitative accuracy is desired, the dumb-bell model often turns out to be oversimplified. For one thing, nuclear spins of the two atoms will introduce changes in the statistical weights, as we shall see in the next section. For another, the molecule may be capable of several electronic configurations (excited states) causing a multiplication of energy levels. The spectroscopy of the band spectra makes available a material of large and rapidly increasing volume relating to the rotational and vibrational energy states. For a great many gases the levels e/ have been accurately measured and tabulated. If a simple formula is not available, the procedure usually followed is to substitute into the sum of states (18.03) the spectroscopically measured levels (or exact theoretical expressions where they are known) and to obtain the specific heats by numerical calculation. In view of this, we shall mention here specifically only a particu- larly simple case, that of the doublet structure of the energy levels. It is not restricted to the rotational energy of molecules but has a quite general application and was first treated by Schottky. 1 Suppose that the energy levels occur in pairs ej and ej + Ae (the increment Ae being constant and independent of /) with the relative statistical weights go : gi. Every term of Y (18.03) will then appear to be multiplied by the same factor Y d = go + gi exp ( Ae/jfer). Writing for short 9 = Ae/fc, 7 = \ log (gi/go), (18.16) we find Y d = go + gi exp (-8/r) = 2go exp (jy - j cosh ^ - T j. Correspondingly, log Y will contain the additional term log Yd = - + log cosh "" 7 + * log ( 4 S<*i),(18.17) whence the doublet heat is calculated from (18.07) (6/2D 2 cosh 2 rz; - This expression is valid, of course, even when no other energy spectrum exists, i.e. when each particle is capable only of the two energy states eo and eo + Ae. In the special case of the rotational doublets, Ae is not strictly constant. Its formula was worked out by 1 W. Schottky, Phys. Zs. 23, p. 448, 1922. 302 TEXTBOOK OF THERMODYNAMICS XVIII 117 Hill and Van Vleck 1 and shows a slight dependence on j. It seems, however, that in most cases the simple formula (18.18) gives an approximation, sufficient within the accuracy of heat measurements (section 119). The presence of the term c d produces a characteristic hump in the specific heat curve as exemplified in Fig. 54. The condi- tions are not very different when each energy level, instead of splitting into two components, splits into several or many closely spaced sub- levels (multiplet structure): the analytical expressions are more complicated but the result is a hump in the curve of specific heat not unlike the doublet hump. It is well known that the energy levels of the linear oscillators are given by the formula e,i = hv Q (v + J), v = 0,1,2, . . . VQ being its characteristic frequency. The vibrational levels of diatomic molecules are not quite so simple as that but in most cases they can be sufficiently well represented by an expression of the form + i) - *(v + ) 2 L (18.19) (anharmonic oscillator) where x is a constant. A theoretical founda- tion for this formula was given by Kratzer 2 and in a different way by Morse. 3 We repeat that, strictly speaking, the vibrational energy is not entirely independent from the rotational but there exist mixed roto-vibrational terms, which are, however, negligible as far as the computation of specific heats is concerned. The fact that e/ and e v are simply additive means that, by taking into account the levels (18.19), each term of the sum F r (18.14) is multiplied by the same expression F,<. Since the statistical weight of each vibrational energy level is 1, we can write F. -53 exp J - || [(v + i) - *(o + *)] J . (18.20) In other words, the rotational and vibrational sums of states are multiplicative, Y = 7 r - F v *, whence log Y = log Y T + log Y vi and c = c r + c vi . The rotational and vibrational heats are additive, within our approximation, and can be evaluated separately. It must be noted that the sum of states with the energy levels * E. L. Hill and J. H. Van Vleck, Phys. Rev. 32, p. 250, 1928. On the basis of thia work a correction to (18.18) was calculated by G. Gregory (Zs. Physik 78, p. 789, 1932). A. Kratzer, Zs. Physik 3, p. 289, 1920. Ph. M. Morse, Phys. Rev. 34, p. 57, 1929. XVIII 118 THEORY OF SPECIFIC HEATS 303 (18.19) would be divergent if v could increase to infinity. This means either that the number of states is finite, being cut short by the disso- ciation of the molecule, or that the expression (18.19) represents an approximation valid only for low order terms. There is no practical difficulty connected with this because the terms decrease rapidly even at high temperatures and no ambiguities arise in the numerical calculations. For a few gases the expression (18.19) does not hold with sufficient precision; recourse is then taken to the empirical energy levels tabulated by spectroscopists. On the other hand, when the coefficient x is small compared with 1 , it is possible to obtain a fair approximation neglecting it altogether, and extending the summation to infinity. 1 In this case it can be readily carried out and gives the result Y vi = 1/2 sinh (0/2r), 1 f (18.21) c vi = *0 2 /4r 2 sinh 2 (e/2r), J where the "characteristic temperature" 9 is an abbreviation for 6 = hvo/k. At any event the law of equipartition of energy does not apply to expressions of the type (18.19) or (18.20). The vibrational specific heats, therefore, rise steadily without reaching a limiting value even at the highest attainable temperatures. The constant hvo in (18.20) is always considerably larger numeri- cally than rkT of (18.15). Therefore, the vibrational heats come into play at much higher temperatures than the rotational. The gases mentioned in the beginning of this section as apparently confirming the classical theory are those for which the rotational degrees of freedom have reached complete equipartition at room temperature while the vibrational degrees are not yet appreciably excited. Exercise 104. Take a sum slightly more general than (18.14) fa + n + $) exp [-rfo* + n)(*j + n +1)] A/. y-o A/ = 1 is the increment which j receives from term to term. Write \/r <rj #/; if T is very small, Ax/ = \fc <rbj is also very small, and the summation can be replaced by an integration with respect to */. Show that the result of the integration is Y = (1/ar) exp [ n(n + l)<r], or when a is extremely small, Y - 1/crr. (18.22) 118. Hydrogen (Protium, E^H 1 )- The decline of rotational specific heats at low temperatures was first observed in hydrogen. 2 Its 1 W. Nernst and K. Wohl, Zs. techn. Physik 10, p. 611, 1929. * A. Eucken, Preuss. Akad. d, Wise., p. 141, 1912. 304 TEXTBOOK OF THERMODYNAMICS XVIII 118 complete theory was developed in slow steps which were highly instructive. The simple dumb-bell model is here inadequate because of the nuclear spin of the hydrogen atom. There exist two modifica- tions of the molecule: para-hydrogen in which the spins of the two nuclei are anti-parallel, and ortho-hydrogen in which they are parallel. The total spin of the para-hydrogen is zero and it is indifferent to a magnetic field, while ortho-hydrogen is capable of assuming in such a field three quantized orientations. Therefore, the factors (2j + 1) in the sum of states (18.14) represent correctly the statistical weights of those energy levels which belong to para-hydrogen while those corre- sponding to ortho-hydrogen must be multiplied by three. It happens that the levels of the two modifications divide neatly, para-hydrogen being capable only of even quantum numbers j, and ortho-hydrogen only of odd. The corrected sum of states as first given by Hund l is, therefore, F= F, + 3F , (18.23) F, - (4j + 1)<T* 2 ' (2 ' +1) , F a - W + 3)<T' (2 ' +1)(2 > +2) . (18.24) This is in complete agreement with the band spectrum data where the lines corresponding to odd levels are three times as strong as those corresponding to even levels. Hund gave also the theoretical explana- tion of the selectivity of the two modifications with respect to even and odd levels. It lies in the requirement of wave mechanics that the wave functions representing the states of the molecule must be anti- symmetric with respect to the two nuclei (compare section 104). These wave functions consist of two factors which characterize, respectively, the spins and the rotational motions. It is clear, there- fore, that the arrangement of the spins will influence the selection of the rotational states. When the spins are anti-parallel, the interchange of the two nuclei produces the same effect as a reversal of the spins: para-hydrogen is, therefore, anti-symmetric in the spins and must have a symmetric rotational function (which happens to be the one with even j). On the other hand, when the spins are parallel an inter- change of the nuclei does not produce any effect: ortho-hydrogen is symmetric with respect to the spins and has an anti-symmetric rota- tional function (the one with odd j). We shall, therefore, drop the prefixes para- and ortho- and speak instead of symmetrical and anti- symmetrical hydrogen (meaning symmetry with respect to the rota- tions). F. Hund, Zs. Physik 42, p. 93, 1927. XVIII 118 THEORY OF SPECIFIC HEATS 305 Yet the specific heats calculated from the sum of states (18.23) do not agree with those obtained from measurements. 1 The reason for this has been cleared up by Dennison 2 and lies in the fact that the two modifications of hydrogen are very slow in reaching equilibrium. The theory of wave mechanics permits one to foresee that transitions between the two modifications of hydrogen are extremely unlikely, and this is borne out by the complete absence of band lines correspond- ing to such transitions. When heat is added to hydrogen, it imme- diately distributes itself over the rotational levels within each modifi- cation separately, but it takes a very long time for the equilibrium ratio of the two kinds of molecules to be established. The number of mole- cules of symmetrical hydrogen is proportional to F t , that of antisym- metrical to 3F ; we give the equilibrium ratio of these numbers for several temperatures using the value <rT = 82.6 deg. which follows from the spectroscopical value 3 K = 4.80 X Temperature F, : 3Y a T = oo 1.00:3.00 293. IK 1.00:2.98 78 1.00:1.07 20.4 1.00:0.0145 While at high temperatures symmetrical hydrogen accounts for only one-quarter of the gas, it is practically pure in equilibrium at the temperature of boiling hydrogen. Under usual conditions, the deter- mination of specific heats takes only a fraction of an hour, and the actual ratio F, : 3 F in the gas is that corresponding to room tempera- ture and not to the temperature of measurement. In other words, symmetrical and anti-symmetrical hydrogen behave like two inde- pendent gases in the permanent ratio of about 1:3. The appropriate procedure is, therefore, to calculate separately, by means of eq. (18.07), the specific heats of the two modifications: c from F,, and c a from F a . The rotational heat of the actual gas is then c r - \c. + \c a . (18.25) The result is indeed in excellent agreement with experiment, as will 1 Experimental determinations were due to: F. A. Giacomini, Phil. Mag. 50, p. 146, 1925; J. H. Brinkworth, Proc. Roy. Soc. (A) 107, p. 510, 1925; Partington and Howe, ibidem, 109, p. 286, 1925; Cornish and Eastman, J. Am. Chem. Soc. 50, p. 627, 1928; Scheel and Reuse, Ann. Physik 40, p. 473, 1913; A. Eucken, Sitzungs- ber. Berlin, p. 141, 1912. * D. M. Dennison, Proc. Roy. Soc. (A) 115, p. 483, 1927. Compare: R. T. Birge, I.C.T. V, p. 409, 1929. 306 TEXTBOOK OF THERMODYNAMICS XVIII 119 be seen from Fig. 49, where the dashed curve is calculated from eq. (18.25). An investigation by Bonhoeffer and Harteck l showed that, under ordinary conditions, the time necessary for reaching equilibrium of para- and ortho-hydrogen is about ten months. However, it can be enormously shortened, in fact, reduced to a few minutes, by letting the gas be absorbed in carbon. In this way it is possible to prepare almost *- 6 100' K 200 300 FIG. 49. Specific heats of hydrogen and deuterium. pure symmetrical hydrogen which keeps for several months when stored in a glass container. 119. Deuterium, N, O2> Ck, NO, Cl. Limitations of space do not permit us to give an exhaustive account of the work on specific heats of gases, and we restrict ourselves to a few typical examples. As to rotational heats, the situation in hydrogen repeats itself in many other gases. Most of them possess a symmetric and an anti- symmetric modification of the molecule, so that the complete sum of states is Y r = g.Y. + gaY a . (18.26) The ratio of statistical weights g : g is, in general, not the same as in hydrogen (where it is 3 : 1). The two modifications interact but little and behave, under the conditions of measurement, like two 1 Bonhoeffer and Harteck, Sitzungsber. Berlin, p. 103, 1929; Zs. phys. Chem. (B) 4, p. 113, 1929. XVIII 119 THEORY OF SPECIFIC HEATS 307 independent gases. The experimental specific heats are, therefore, represented by the formula cr - g ' C t gaC *- (18.27) The vibrational heats are usually calculated from expressions of the sum of states of the form (18.20). (1) Deuterium (H 2 H 2 ). The nucleus of deuterium (deuteron or the heavy isotope of hydro- gen) possesses a double spin (with the quantum number 1, as compared to ^ of the proton). As shown in the quantum theory, the molecule is then capable of 6 symmetrical and 3 antisymmetrical spin configura- tions. Moreover, the deuteron obeys the Einstein-Bose statistics, being built up of an even number (2) of protons. Therefore, the symmetrical spin states combine with the symmetrical rotational states, etc., and we have g = 6, g a = 3. The specific heats of deuterium have been calculated with great accuracy (even taking into account mixed vibro-rotational terms) by Johnston and Long. 1 Their results are represented by the solid curve of Fig. 49. The decline takes place at lower temperatures than in the case of protium, caused by the, roughly, two times larger moment of inertia. The dotted curve of Fig. 49 refers to the mixed proton-deuteron molecule rPH 1 as given by the same authors. The data given below for several gases are mostly taken from Trautz and Ader, 2 in whose paper can be found the references to the older work on the subject. (2) Nitrogen, N 2 (Fig. SO). The rotational heat (determined by K = 39.65 X lO" 40 , <rT = 2.873, g a : g. = 2 : 1) reaches the full equipartition value at 29 abs. The vibrational levels are represented by 2t 3 0,< = fc- 1 ** - 3374.24(0 + J) - 20.66(t> + |) 2 . (18.28) (3) Oxygen, O 2 (Fig. 51). 1 H. L. Johnston and E. A. Long, J. Chem. Phys. 2, p. 389, 1934. 2 M. Trautz and H. Ader, Zs. Physik 89, pp. 1, 12, 15, 1934. The experimental data included in Figs. 50, 51, 52 are taken from the following papers: Eucken and v. Liide, Zs. phys. Chem. 5, p. 413, 1929; Eucken and Miicke, ibidem 18, p. 167, 1932; P. S. Henry, Proc. Roy. Soc. 133, p. 492, 1931; Shillings and Partington, Phil. Mag. 9, p. 1020, 1930. 1 Compare also: Giauque and Clayton, J. Am. Chem. Soc. 55, p. 4875, 1933. 308 TEXTBOOK OF THERMODYNAMICS XVIII 119 The equipartition of the rotational heat is reached at temperatures still lower than in the case of nitrogen. The vibrational levels are 1 2310.7(0 - 16.26(o (18.29) 1.5 1.0 0.5 ( + ( ( "" x i calc.) F< calc.) Fc obiJEu obs.) H< obs.) Sh I 1 jrmula < >r mula ( ckn am nry. illings ai 'artmgto 18.28) 18.21) J Miickt id i / ^ ^ ^^^ / o c x | e vl in Nitrogtn | 200 400 600 800 1000 1200 1400 1600 1800 FIG. 50. Vibrational specific heat of nitrogen. The specific heat of O 2 has been calculated also by Johnston and Walker, 2 who used the rigorous spectroscopic expressions for the energy levels (including the mixed vibro-rotational terms). Their o 15 t" 0.5 ^ Z^Z^ ** f / S* o y / + ( c vl in 2 calc. ) Formula ( 18.29 calc.) Formula (18.21 calc.) Johnston and W obs.) Euckenand Muc obs. > Henry \ ) ilktr ke / / ( ( ( 200 400 600 800 1000 1200 1400 1600 1800 Tabs * FIG. 51. Vibrational specific heat of oxygen. data are represented by the dashed curve in Fig. 51. It will be seen that the difference is appreciable only at very high temperatures. (4) Air (Fig. 52). 1 The constitution of air is 78.06% N 2 , 21.0% O 2 , 0.94% Ar. (5) Chlorine, C1 2 (Fig. S3). 1 1 Trautz and Ader, he. rit. 1 Johnston and Walker, J. Am. Chem. Soc. 55, pp. 172, 187, 1933. XVIII 119 THEORY OF SPECIFIC HEATS 309 The conditions in Cfo are complicated by the existence of the two isotopes Clasi Cls7 which form three kinds of molecules. The band spectra, however, show that the amount of ClsyCls? is small and that chlorine can be taken as consisting of 60% ClasCbs and 40% 1 _ ^ 10.5 200 400 600 800 1000 1200 1400 1600 Tabs >- FIG. 52. Vibrational specific heat of air. -Theor (Exp.) (Exp. c vj in i etical Eucken Henry Mr and Mt ickt / /- ^^ ]/ / / The rotational heat of chlorine is of only academic interest as it reaches equipartition at 1.5 abs. The vibrational levels of are given by 2 = 807.81(t; (18.30) c v , in CI 2 <cak.) Formula (18.30) (obs.) Eucken and Hoffmann + (calc.) Formula (18.21) 500 1000 1500 2000 FIG. 53. Vibrational specific heat of chlorine. In the case of ClasCla? the two coefficients must be multiplied (36/37)* and (36/37), respectively: 2 796.26(0 - 5.57(* (18.31) 1 A. Elliott, Proc. Roy. Soc. (A) 127, p. 638, 1930. * Trautz and Ader, loc. tit. 310 TEXTBOOK OF THERMODYNAMICS XVIII 119 (6) Nitric oxide (NO) does not offer much new with respect to the rotational and vibrational heats. In fact, c r reaches equipartition below 30 abs. and comes into play only above room temperature. In the intermediate range NO is interesting because it offers an example of doublet heat c* as given by the formula (18.18). From spectro- scopic data the doublet difference is Aeo = 354 cal/mol, whence 9 178 deg. As mentioned in section 117, it is not strictly constant. The measurements were carried out by Eucken and d'Or, 1 who also calculated the theoretical values from the approximate formula (18.18), with 7 = 0. Accurate calculations of the specific heat of NO were carried out by Johnston and Chapman, 2 who used the rigorous spectroscopic expressions of the energy levels. We give the values of c d inferred from their results in the last column of Table 41. It will be seen that the approximation given by formula (18.18) is pretty good and certainly sufficient in view of the not very accurate experi- mental data. TABLE 41 SPECIFIC HEATS OF NO T cd/R (obs.) ca/R (calc.) Eucken and d'Or Johnston and Chapman 50. OC 0.34 0.325 75.0 .... 0.44 0.429 100.0 .... 0.39 0.395 127.0 0.27 0.31 0.314 133.3 0.31 0.29 0.295 150.0 0.28 0.25 0.258 173.8 0.22 0.20 0.21 193 0.17 0.17 0.18 218 0.14 0.14 0.15 298.1 0.14 0.09 0.10 (7) Atomic chlorine (Cl) offers another example of doublet heat (18.10). From spectroscopical data Ae = 2 X 580 cal/mol, 9 = 1299, 7 = 2 log. The calculation 3 leads to the curve shown in Fig. 54. There are more gases whose complete curves of specific heats have been accurately calculated, but limitations of space do not 1 A. Eucken and L. d'Or, Nachrichten Gdttingen, p. 107, 1932. * Johnston and Chapman, J. Am. Chem. Soc. 55, p. 159, 1933. * W. Nernst and K. Wohl, Zs. techn. Physik 10, p. 608, 1929. XVIII 120 THEORY OF SPECIFIC HEATS 311 permit us to extend the number of examples, and we can only give here an incomplete list of references: HgO has been treated by Trautz and Ader (loc. cit.) ; CO and N2 by Johnston and Davis (J. Am. Chem. Soc. 56, p. 271, 1934); OH by Johnston and Dawson (ibidem 55, p. 2744, 1933). The work previous to 1930 can be found in I. C. T. and in other Tables of Constants. Figures 50-53 contain also the values calculated by Nernst and Wohl (loc. tit.) with the help of the approximate formula (18.21). 500 1000 1500 2000 2500 3000 3500 T e abs > FIG. 54. Doublet specific heat of atomic chlorine (theoretical). The very good agreement shows that this simple way of computation can be used to great advantage when the coefficient x of the formula (18.19) is small compared with unity (and, presumably, T not too high). The values of the characteristic temperature 0, for the common diatomic gases, as used by them, are given in Table 42. TABLE 42 CHARACTERISTIC TEMPERATURES Gas e Gas e Gas e Gas e Hi 5950 o a 2220 HF 4130 Li 2 495 Fa 1610 S 2 1030 HBr 3660 Na 2 225 Cl* 888 SC2 570 HO 5110 K 2 130 Br 465 Te 2 . . . . 360 NO 2680 NaK 175 Fa 305 Ni 3374 CO 3060 C1J 545 120. The chemical constant of gases. The notion of the chemical constant was introduced in section 95. It is that part of the entropy of a perfect gas which is left undetermined by the first and second laws 312 TEXTBOOK OF THERMODYNAMICS XVIII 120 of thermodynamics. Its determination completes the definition of the thermodynamic functions, especially of the thermodynamic potential <p = u Ts + pv t and so makes definite the equations of equilibrium (6.50). In section 95 we showed that it has the Sackur-Tetrode expression (14.12) in the case of molecules having only translational degrees of freedom. Generally, the thermodynamic potential of a non- degenerate perfect gas can be divided into two parts where <p t refers to the translational and <p* to the inner (rotational, vibrational, etc.) degrees of freedom of the molecules. To <p t applies the expression which was derived for a monatomic gas consisting of mate- rial points, i.e. (5.41), (5.43) with c p = SR/2. On the other hand, the inner motions do not contribute to the term pv and consequently <?,- is identical with the work function ^ whose connection with the sum of states is given by (18.08): ^ = u Qt + RT[lo% p + f log T - log Y - j t ], (18.32) where jt is the expression (14.13). (A) Monatomic gases do not possess rotational or vibrational degrees of freedom. The sum of states Y = Y e refers here only to the energy levels of electronic configurations. The term jRriog Y e contains, however, a contribution to the zero point energy. In fact, factoring off the first exponential in the sum (18.03) we may write Y. = exp (- eo/kT) g, or log Y. = - (eo/kT) + log g e . The quantity gt can be thrown into the form < exp (-GOT, (18.33) i where 6f = (e, eo)/. Consequently, V - uo + RT(lo%p - flog T - j), jio - - 1.589 + f logio M + logio g.> (18.34) (provided the pressure is expressed in atmospheres). o represents the statistical weight of the fundamental state due to the moment of momentum of the atom. The quantities Qf are usually so large that the higher terms of (18.33) are negligible even at fairly high tempera- tures. Only in rare cases are the terms with / = 1,1 2 appreciable. Data relating to a number of atoms are contained in Table 43. 1 ^ Taken from K. Wohl, Landolt-Bornstein, Second Supplement, p. 1254, 1930. XVIII 120 THEORY OF SPECIFIC HEATS TABLE 43 313 Atom Symbol of level * ef Atom Symbol of level 2 ef Noble gases i5 1 'Pa 5 1 H, Cu, Ag, Au, 1 2 5i 2 'Pi 3 (9) 228 alkali metals J M 'Po lj 324 Zn, Cd, Hg, I 'So I F *Pfc 4 earth alkalis f ** 2 Pj4 2 580 C P 9 Cl 2 P& 4 Si 'Po 1] 2 Pn 2 1260 'Pi 3 (9) 110 Br 2 P% 4 'P a 5J 320 2 P^ 2 5270 Sn 'Po 1 J 2 P^ 4 'Pi 3 2420 2 P^ 2 10900 'Pj 5 4900 Fe 6 I>4 9 Pb 'Po 1 5 #, 7 595 'Pi 3 11200 6 Z?2 5 1010 'P* 5 15200 6 Z?i 3 1270 N,P,As,Sb,Bi ^ 4 6Z?0 1 1400 (B) Diatomic gases. The concept of the chemical constant can be extended to diatomic and polyatomic gases only within the approxi- mation with which log F r , pertaining to the rotational states, can be regarded as additive to log Y due to the other degrees of freedom. We have seen in the preceding section that this approximation is, indeed, quite sufficient for thermodynamical purposes. The range of temperatures is restricted to those above equipartition of the rotational energies when Y r = (l/or), according to (18.22) or Y r = - + log K - log <r + log T. In practice, moreover, it is assumed that the molecule is in its fundamental electronic state, so that g e reduces to go and the remainder of log Y becomes log go + log Y vi = log go - t*t/RT. Eq. (18.32), therefore, takes the form * - o< + *.< + RT[logp - |log T - j], where ^* can be calculated either from the approximate equations (18.21), with the data of Table 42, or from the more accurate formula (18.20). Since the numerical value of -logio(h?/&ir*k) is 38.402, we find for the chemical constant the expression jio 36.813 + | logic M + logio K - logic * + logic go- (18.35) 314 TEXTBOOK OF THERMODYNAMICS XVIII 120 TABLE 44 CHEMICAL CONSTANTS OF DIATOMIC GASES Gas 10" K a go j Gas lO 40 ^ <T So 3 H a 0.480 2 1 -3.355 HF 1.35 1 1 -1.10 F a 25.3 2 1 0.285 HC1 2.646 1 1 -0.419 Cla 114 1.57 1 1.45 HBr 3.303 1 1 0.197 Br 8 330 ? 1.42 1 2.49 ? HJ 4.309 1 1 0.609 Ja 742.6 2 1 2.995 HO 1.500 1 2(0 1 e = 173) -0.561 cij 575 ? 1 1 2.89 ? CO 14.9 1 1 0.159 2 19.3 2 3 0.531 NO 16.35 1 2(6^ = 182) 0.862 S 2 67 ? 2 3? 1.08 ? Na 2 179.5 2 1 1.262 Tea 860 ? 2 3 3.09 ? K 2 184 2 1 1.619 N, 13.8 2 1 -0.175 NaK 66 1 1 1.324 The fractional symmetry numbers (for C\2 and Br2 in Table 44) represent the mean values for the isotopes. (C) Polyatomic gases. Without entering into the derivation we only state here the final result for the expression of the thermodynamic potential * 9 = o< + fc< + RT(\ogp - 41og T - j], jio = 56.563 + | log p + | log 'K - log <r + log go, (18.36) where ~R = (^1X2^3)^ is the geometrical mean of the three moments of inertia possessed by the molecule. A few numerical values are given in Table 45. TABLE 45 CHEMICAL CONSTANTS OF POLYATOMIC GASES Gas j Gas j CO, 0.73*0.05 H*O -1.91 N 2 0.68 NH, -1.66 C 3 H 0.008 CH 4 -1.94 In conclusion it must be pointed out that the above expressions are still incomplete in so far as they do not contain the statistical weights g n of the nuclear spins. Strictly speaking, one should add to A. Eucken, Phys. Zs. 30, p. 118, 1929; 31, p. 361, 1930. XVIII 121 THEORY OF SPECIFIC HEATS 315 the expressions (18.34), (18.35), and (18.36) the term log g n . However, at temperatures at which the symmetrical and antisymmetrical modifications of the molecules (sections 118, 119) have practically reached the ratio g a : g a , corresponding to T = oo , this term is the same in the molecules and in an equivalent number of free atoms so that it drops out of the equations of chemical equilibrium. At room temperature this condition is satisfied for all gases except hydrogen, for which it is true from about 70 C on. In Chapter XIX we shall discuss problems of transmutation of matter in which the term log g n may play a role. We mention, therefore, that in the atom of ordinary hydrogen (protium) g n = 2, in the deuteron g n = 3, in the helium atom g n = 0. 121. Heat capacity of a reacting gas mixture. The diagrams of section 119 give the specific heats of several gases up to temperatures of 2000 abs. Under these circumstances dissociation may be already appreciable and must be taken into account. It will, therefore, be well to discuss how chemical reactions, taking place in a gas mixture, affect its heat capacity. We shall restrict ourselves to conditions under which the equation of perfect gases pv = RT gives a sufficient approximation, permitting us to use the equilibrium theory of Chap- ter VIII. Let us consider a mixture composed of the gases 1, 2, ... in the respective mol numbers Ni, N2, . . . N ft , and let us suppose the following reactions to be possible in the mixture v\G\ + . . . + vpGp = 0, 0, etc. (18.37) The condition of equilibrium for the first reaction is represented by the mass law (8.18) X) "i log xi log K(p,T) (18.38) i Substituting the definition of the mol fraction xi = Ni/N (where N = NI + . . . + Np), we can write i i log Ni-vlogN* log K, (18.39) with the abbreviation v v\ + . . . + v ft . According to eqs. (8.19) and (8.21) -~, (18.40) 316 TEXTBOOK OF THERMODYNAMICS XVIII 121 Q being the heat of the reaction I In the same way we find for the other reactions 2>, log Ni - v' log N = log K'> (18.42) 1 m-^ and so on. We are now ready to calculate the heat capacity at constant pres- sure. The total heat function of the gas mixture is , fi Xl, and according to (3.26) The mol numbers change only inasmuch as the molecules of the mixture take part in the reactions (18.37). Each of the variations dNi, therefore, must be expressible in the form dNi = vi da + v'i da' + . . . , (18.45) where the quantities a, a', ... are independent of the coefficients v. Hence Noting (18.46) and the relation (dxi/d^) p = c p i and writing for short 2Nic plt (18.47) 1-1 we have In the case of a mixture of neutral, non-reacting gases this reduces simply to C p =",Nic P i. XVIII 121 THEORY OF SPECIFIC HEATS 317 The process of change of the mol numbers is supposed to take place in such a way that the equations of equilibrium (18.39) and (18.42) are always fulfilled. We can, therefore, obtain (da/dT) p by differ- entiating the first of them and taking into account that the contribu- tion of this reaction to (dNi/QT) p is the first term of (18.46), namely, . With the abbreviations etc., we obtain as the final expression c p = N [r p + F( X ) ^ + TO ~z + The mol fractions x refer here, of course, to the state of chemical equilibrium. They can be obtained either theoretically, by the solution of the simultaneous eqs. (18.38), (18.42), etc., or empirically from measurements of the density and other properties of the gas mixture. For the heat capacity at constant volume, we must start from the internal energy of the mixture U = ^,NiU h (18.51) i whence For dNi we have again to use the expression (18.45). Noting that (dui/dT) v = Cii, 2viUi = AC/, and introducing the abbreviation i^ (18-53) j we obtain In view of the relation (8.19), log K = log K p (T) - vlogp, the equation of equilibrium (18.39) can be transformed by substituting for the pressure its expression from the equation of state, p = NRT/V: Y,n log Ni - log K p - v log T + v log (V/R). (18.54) 2 318 TEXTBOOK OF THERMODYNAMICS XVIII 122 Introducing the abbreviations /(*) = GE'iV*!)" 1 , /'(*) = (XV' 2 /**)' 1 . ( 18 - 55 ) and taking into account the relation xi = i + RT, we obtain by differentiating (18.54) with respect to T ..AJ7 /(*) VaTVV \RT 2 T The expression for the heat capacity (18.52) is, therefore, An interesting application of this theory is due to McCollum, 1 who investigated the heat capacity of nitrogen tetroxide (N2O4). On the one hand, he made calorimetric determinations of C p at temper- tures between 30 and 100 C. On the other hand, he calculated C p from a formula equivalent to (18.50). The equilibrium in N2O4 was treated in section 52: the only reaction in it (8.22) has the coefficients vi 2, V2 = 1 v = 1. As was shown there, the mol fractions can be expressed in terms of the degree of dissociation as follows: *i - 2{/(l + ), * 2 - (1 - *)/(! + *) Hence we find for the (single) functions F(x) and f(x) D, /(*) - 1(1 - 0/(l + Q(2 - Q. McCollum used empirical values of . The excellent correspon- deiice between measurement and calculation which he obtained is shown in Fig. 55. 122. Velocity of sound in gases and gas mixtures. Determina- tions of the velocity of sound in gases have been often used as a simple and, apparently, accurate method of measuring their specific heats. As was mentioned in section 18, the square of the velocity of sound in a pure gas is supposed to be given by the formula > McCollum, J. Am. Chem. Soc. 49, p. 28, 1927. XVIII 122 THEORY OF SPECIFIC HEATS 319 p being the mass density and M the molecular weight. If the gas fol- lows the equation pv = RT, this expression takes the form (3.40) a 2 = yRT/n, 1 = c v (18.58) oexp. theor. FIG. 55. Specific heat of nitrogen tetroxide. It was pointed out that these relations apply without regard to whether c p , c vt y are constants or functions of temperature. However, the specific heats c v measured by the sonic method in oxygen and in air 1 at high temper- atures (from 100 to 1000 C) are considerably below those determined by other methods. In a similar way, the sound velocities in NO give the same specific heat at low tempera- tures as at C, 2 instead of the increase tabulated in Table 41 of section 119. This fact seemed to indicate that the changes of state, produced by the sonic waves, are too rapid to allow the equilibrium of the molecular degrees of freedom to be established. In fact, the disper- sion of sound waves in gases (CC>2) had already been observed a few years earlier 3 and was interpreted as a lag in the molecular adjust- ment. 4 Extensive work on the subject was done by Kneser, 6 who was able to show that it is the vibrational energy which requires periods up to 10~~ 2 sec to assume its equilibrium value, while the rotational degrees of freedom respond much more quickly. At low frequen- cies the ratio 7 of the formula (18.58) was experimentally equal to 1 + R/(c t + c r + c*i), at high frequencies, to 1 + R/(ct + c r ). These investigations were greatly influenced by a paper of Ein- stein's 6 which suggested sonic measurements as a method of deter- 1 W. C. Shillings and J. R. Partington, Phil. Mag. 9, p. 1020, 1930. 1 Partington and Shillings, Phil. Mag. 45, p. 416, 1923. 1 G. W. Pierce, Proc. Am. Acad. 60, p. 271, 1925. 4 K. F. Herzfeld and F. O. Rice, Phys. Rev. 31, p. 691, 1928. *H. O. Kneser, Ann. Physik 11, pp. 761, 778, 1931; 16, p. 337, 1933; H. O. Kneser and V. O. Knudsen, Ann. Physik 21, p. 682, 1935. 6 A. Einstein, Sitzungsber. Berlin, p. 380, 1920. 320 TEXTBOOK OF THERMODYNAMICS XVIII 122 mining the rates of reaction in gas mixtures. While this paper was indirectly responsible for the research on dispersion in pure gases, the results of this research defeated its original purpose. The obser- vations in mixtures are too difficult to interpret when possible abnor- malities of the components themselves must be taken into account. 1 For this reason, we shall calculate here the velocity of sound in mix- tures only for low frequencies when the question of lag does not arise. The velocity of sound in a gas mixture of the total mass M can be written a 2 = (dp/dp) a = ~ V 2 (dp/dV) s /M, (18.59) since the mass density is p = M/V. Let us envisage a mixture of a given initial constitution. If its reacting components are always in equilibrium, the total volume is completely determined by the tempera- ture and pressure. The system is, therefore, simple in the sense of section 2: its entropy differential, dS = (dU + pdV)/T, can be regarded as a function of pressure and volume, and the equation of the adiabatic can be written since OX/9F) P = (dU/dV) p + p, in view of the definition of the heat function X = U + pV. Because of (3.12) and (3.26) faX V 1 = *"V ITI / \ \dp/v \dp/v whence and (dp/dV)a BB y(dp/dT) v /(dV/dT) p = 7(9/9tOrl | (18.60) 7 = Cp/Cv ] This equation holds generally for any simple system. When we apply it to a gas mixture, the heat capacities in the ratio y are, of course, 1 C. E. Teeter, J. Chem. Physics 1, p. 251, 1933. This paper contains a good review of the experimental and theoretical work about measuring rates of dissocia- tion in nitrogen tetroxide by the sonic method. XVIII 123 THEORY OF SPECIFIC HEATS 321 those calculated in the preceding section. From the equation of state, pV = NRT, We transform this with the help of the relations (18.43), (18.46), and (18.54), obtaining 1 + tf ) + //'(*) (18 ' 62) It will be noted that the last factor is much less affected by the heats of reaction than 7: it is linear in Q/RT, etc., while Cv, C p contain the squares of these large numbers. Experimental work concerning the velocity of sound has been published by several authors. 1 While the results are not quite con- sistent in their bearing on the rate of the reaction and the speed with which equilibrium is established, it seems that indications of a lag in reacting gas mixtures are to^be found only in the measurements of absorption of sound. There is no convincing proof of the existence of dispersion, and the formula (18.62) appears to represent the observed velocities fairly well. This would mean that chemical equilibrium is, practically, reached after the lapse of only 10~ 4 sec. 123. Solids. Einstein's model of a solid body was described in section 93. It treats the atoms of the solid as harmonic oscillators, all independent and having the same frequency v. For the energy levels of the three-dimensional harmonic oscillator the quantum theory gives e ni n 2 n, = kv(m + W2 + W3 + f), where n\ t n%, #3 are the quantum numbers of the three degrees of freedom. The sum of states is exp [-*(i +H2 + H3+ f)] = C (18.63) \ l ~- e ) (x = hv/k). The energy and entropy can be derived from it by eqs. 1 Compare the review in the paper of Teeter's (he. cto.) 322 TEXTBOOK OF THERMODYNAMICS XVIII 123 (18.06) and (18.09), and are those given in section 93. In particular, the mean energy of the oscillator is, per degree of freedom, - "* ' e - l ' 2 As was mentioned before, the agreement of Einstein's formulas with the measured specific heats is not very close. The reason for this discrepancy was pointed out in a later paper 1 of his: the model does not represent the actual conditions accurately enough because, in reality, the atoms of a crystal do not oscillate independently of one another with the same frequency but form a coupled system. The forces of interaction influence the characteristic frequencies so that they all become different and are drawn out into a spectrum. More accurate theories were given from this point of view by Debye 2 and independently by Born and Von Karman. 3 Debye's theory is par- ticularly simple: he assumes that the 3Z frequencies of a body con- taining Z atoms can be calculated as if it were a homogeneous elastic continuum. In other words, these frequencies correspond to the fundamental tone and to the first 3Z 1 overtones of its elastic spectrum. The following considerations support this assumption. In the region of high temperatures prevails equipartition and the energy depends only on the number of frequencies (or degrees of freedom) and not on their values. On the other hand, at low tempera- tures, the high order overtones are not appreciably excited ; according to eq. (18.64) their mean energy -becomes very small. All depends then on the low order frequencies, and for these the approximation by a continuum is quite adequate because their wave length is large compared with the atomic distances. Therefore, we should expect accurate results both at low and at high temperatures. The calculations are easily carried through in the case of isotropic elementary materials. If we take the solid in the shape of a cube with the edge /, 4 the theory of elasticity gives the following expressions for the frequencies of the standing longitudinal and transverse waves v - ai(m 2 + 2 2 + w 3 2 ) H /2/, v = a 2 (ni 2 + n 2 2 + w 3 2 ) H /2/, (18.65) where a\ and a* are the velocities of the longitudinal and transverse wave systems. Every triple of integral numbers wi, w 2 , m determines a state of the body (standing wave) in each of the two wave systems, 1 A. Einstein, Ann. Physik 35, p. 679, 1911. P. Dcbye, Ann. Physik 39, p. 789, 1912. *M. Born and Th, Von Karman, Phys. Zs. 13, p. 297, 1912; 14, pp. 15, 65, 1913. 4 R. Ortvay, Ann. Physik 42, p. 745, 1913. XVIII 123 THEORY OF SPECIFIC HEATS 323 the corresponding frequencies being given by eqs. (18.65). It is easy to see that the number of all longitudinal states with frequencies smaller than a given value v is ~ (18.66) . where V = P is the volume of the cube. In the transverse system the number must be doubled: every transverse wave has two degrees of freedom because of the two possible states of polarization (18 ' 67) Exercise 105. If we imagine the numbers ni, n 3 , n 9 plotted in three orthogonal directions, their integral values will represent a cubic lattice. The number Zi(v) is the number of lattice points within the surface given by the first eq. (18.65). Show that the expression (18.66) holds when / is very large. i, n^ n\ are positive. If the solid were really a continuum, there would be no upper limit to its characteristic frequencies. But it has, actually, atomic structure, and the total number of its possible states must be equal to the number of its degrees of freedom, namely 3Z. Debye, therefore, makes the assumption that the elastic spectrum is cut off at an upper limit VQ determined by the condition 3Z. Substituting (18.66) and (18.67) we find The number of characteristic states in the frequency interval from v to v + dv is QZ dZi(v) + dZ*(v) - Mv. v<f Each of these states is a harmonic oscillation whose mean energy is given in the quantum theory by the expression (18.64). The con- tribution of the frequency interval dv to the internal energy is, there- fore, QZpZiPdv/vo 3 * The total internal energy is obtained by integrat- ing this expression from to VQ and is, per mol, " *dv , /40 ^ - + " (18 ' 68) 324 TEXTBOOK OF THERMODYNAMICS XVIII 123 The specific heat results from differentiation with respect to T (contained in x). If we introduce the abbreviation hvo/k = 9, c = (18.69) This formula shares with Einstein's (18.64) the property that the individual characteristics of the substance are all contained in the single constant 9, the characteristic temperature. If we plot c against T/9 (the reduced temperature) we obtain for all bodies one and the same universal curve. In the limiting case of very low tem- peratures (r/9 1) the integral can be regarded as constant and equal to 47r 4 /15. The formula becomes then 127T 4 c = The agreement with experiments can be judged from Fig. 56 l giving the measured c v . The value of 9 for each substance is here 2 -3 -4 -5 -6 -7 -8 -9 2 -1 Pb, Ag, KCI, Zn, NaCI, Cu, Al, CaF 2 C FIG. 56. Specific heat of solids. chosen so as to give the best fit for that substance with the continuous curve representing eq. (18.69). Curve II is in the normal position; the others are displaced because the experimental material was too large to be accommodated in a single curve. The key to the sub- stances is contained in Table 46. 1 Taken from E. Schrodinger, Phys. Zs. 20, pp. 420, 450, 474, 497, 523, 1919. XVIII 123 THEORY OF SPECIFIC HEATS 325 TABLE 46 KEY TO FIG. 56 Substance Chemical symbol Temperature range e Points in Fig. I II III Lead Pb Tl Hg I Cd Na KBr Ag Pt Ca KC1 Zn NaCl Cu Al Fe CaF 2 FeS 2 C 14-573 23-301 31-232 22-298 50-380 50-240* 79-417 35-873 88 96 97 106 129-160 159 177 215 225 230 217.6 235 287 315 392 454 499 645 1860-2230 X O V D A + V X V D A + V D X + A Thallium Mercury Iodine Cadmium Sodium Potassium bromide. . . Silver Platinum Calcium 22-62 23-550 33-673 25-664 14-773* 19-773 32-95 * 17-328 22-57 * 30-1169 Sylvine Zinc Rocksalt Coooer Aluminum Iron Fluorspar Iron pyrites . . Diamond * Rises above the curve after these temperatures. The formula (18.69) was derived on the assumption that the sub- stance is isotropic and elementary. Within this scope, it represents the facts on the whole very well although a few exceptions are known. But Fig. 56 shows that it applies, in some measure, also to substances of a more complicated structure, presumably because they do not deviate greatly from isotropy. The approach of Born and Von Karman permits to treat the more general cases, at least, by approximation. If the substance is an element in an anisotropic system of crystallization, the elastic waves propagate in it with three different velocities and form three sets of overtones. Born l finds that the specific heat is then the arithmetical mean of three terms of the Debye type (18.69), each with a different characteristic temperature. If the substance is a chemical compound containing r different kinds of atoms, there still exist, in addition to the spectrum of overtones, r 1 characteristic frequencies represent- 1 M. Born, Dynamik der Krystallgitter, 1915. 326 TEXTBOOK OF THERMODYNAMICS XVIII 123 ing the oscillations of these atoms with respect to one another. The molal heat is, therefore, expressed by three Debye terms and r 1 terms of the Einstein type (13.25). Unfortunately, these expressions are so unwieldy that their comparison with experimental results is difficult. The third power law at very low temperatures remains valid also in the general case because the Debye terms satisfy it and the Einstein terms become negligible. While a great many substances fulfill the third power law down to the very lowest attainable temperatures, 1 other metallic and non-metallic solids show marked deviations in the region of the temperature of liquid hydrogen and below. The measured specific heats lie, in this case, above those pre- dicted by Debye's formula. It was suggested that in metallic substances the discrepancy may be due to the specific heats of the free conduction electrons (compare sections 102 and 111). In fact, in the case of silver the influence of the free electrons seems well authenticated by recent measurements in the laboratory of Leiden. 2 In Fig. 57 is plot- ted the difference between the .0008 fWtf% .uuuo V0004 / /^ 8 \ \ OrtHO / \ .UOU& c / \ t / \ -T 2 4 6K FIG. 57. Electronic specific heat in silver. measured c and the value predicted by the third power law (18.70). It shows both the linear dependence required by the Sommerfeld formula for free electrons (17.34) and the correct absolute value on the assumption that there is one electron to every atom of silver. The only other (non-supraconductive) metal investigated in the same temperature region is zinc, and here the free electrons do not give a sufficient explanation of the trend of specific heats. Even after correcting for them, there remains a hump-like residual curve with a maximum at 4 abs., not unlike in its shape to that of Fig. 54. Indeed, it was suggested that they are due to the same cause, doubling of the lowest quantum level of the zinc atom. However, the relation 1 Compare in this connection, the work of Simon and collaborators (Ann. Physik 68, p. 241, 1922; Zs. phys. Chemie 123, p. 383, 1926; Zs. Physik 38, p. 227, 1926). W. H. Keesom and J. A. Kok, Physica 1, p. 770, 1934. XVI II 123 THEORY OF SPECIFIC HEATS 327 2.0 1.5 0.5 i 0.4 1.6 2.0 of the position of the hump and its height is such that only a small fraction of the zinc atoms could possess the double level, a circumstance which makes the explanation doubtful. There are some solids (such as gadolinium sulfate, 1 samarium sulfate, 2 orthohydrogen 3 ), in which the doubling or tripling of the atomic and molecular levels gives a complete explanation of the anomalies of specific heats according to (18.18) or to similar formulas. We reproduce here the very instructive curve measured by Giauque and McDougall in gadolinium sulfate (Fig. 58). In other cases (e.g. tin and silicon 4 ) there were observed humps in the curves of specific heats lying at somewhat higher temperatures whose detailed explanation is still lacking. Simon suggested that they may be due to a doubling of the energy levels according to the formula (18.18). It should be men- tioned, however, that electrons which are not completely free and cannot cause electric conduction may nevertheless contribute to the specific heats, sometimes even in a larger measure than the free electrons. At extremely low tem- peratures the specific heat due to them is proportional to r, in non-magnetic materials, and to T*, in ferromagnetic ones. 6 At somewhat higher temperature this cause, being essentially a multiplet action (compare section 117), should produce a hump similar to that of formula (18.18). Recent work by Jaeger and co-workers (on metals at high tempera- tures) drives it home that our theoretical knowledge of specific heats is still incomplete. Measuring very accurately, 6 they found that c p is structure sensitive and varies appreciably with the mechanical treatment of the specimens. Another series of investigations 7 was devoted to testing Neumann's law (section 16) with the result that it was found satisfied, within the experimental accuracy of 1%, in mixed crystals 1 Giauque and McDougall, Phys. Rev. 44, p. 235, 1933. 2 Ahlberg and Freed, Phys. Rev. 39, p. 540, 1932. 1 Mendelssohn, Ruhemann, and Simon, Zs. phys. Chem. (B) 15, p. 121, 1931. 4 F. Simon, Sitzungsber. Berlin, p. 477, 1926. F. Bloch, Zs. Physik 52, p. 555, 1928; Leipziger Vortrage, p. 67, 1930; P. & Epstein, Phys. Rev. 41, p. 91, 1932. 6 Jaeger, Rosenbohm, and Bottema, Proc. Amsterdam 35, pp. 763, 772, 1932* 7 Bottema and Jaeger, ibidem, 35, pp. 352, 916, 929, 1932. 0.8 1.2 abs FIG. 58. Anomaly of specific heat in gadolinium sulfate. 328 TEXTBOOK OF THERMODYNAMICS XVIII 124 (Ag-Au). On the other hand, in homopolar intermetallic compounds (AuSn, PtSn) there were found appreciable deviations from Neumann's law which, moreover, increased with rising temperature (between and 200 C). 124. Liquids. The liquid state of matter is the most complicated and has not yet been made amenable to theoretical treatment. There exists a number of elements and compounds having in the liquid state approximately the same molal heats which are characteristic of solids, according to the laws of Dulong and Petit and of Neumann (section 16), namely, about 6 cal/mol per atom (Table 47). TABLE 47 SPECIFIC HEATS OF LIQUIDS Liquid c n c/n Liquid c n c/n Hg 6.7 1 6.7 H 2 O 18 3 6.0 N, 13.2 2 6.6 S 2 C1 2 22.4 4 5.6 2 12.8 2 6.4 SiCl 4 34.4 5 6.9 AgBr 14.0 2 7.0 TiCU 36.8 5 7.4 T1C1 13.6 2 6.8 20 19 18 17 16 15 C(p) in cal /mol This would indicate that the conditions are similar to those in solids in that one must take into account both the kinetic and the potential energy of the atoms. However, many liquids have atom- ic heats much larger than 6 cal. Because of its practical im- portance, much attention was devoted to the specific heat of water. Of particular interest is the dependence of c p on pressure represented graphically in Fig. 59. These curves were not de- termined by direct measurement but inferred from the thermal expansion at different pressures 1 by means of the relation p (4.59), or Cp = /r(aV3^ 2 )p dp. We shall see that these data have an important bearing on the question of specific heats of aqueous solutions. *Bridgman, Proc. Am, Acad. 48, p. 309, 1913; F. Zwicky, Phys. Zs. 27, p. 271,1926, FIG. 123456789 p-10T 3 inkgXcm 2 59. Specific heat of water as a function of pressure. XVIII 124 THEORY OF SPECIFIC HEATS 329 As was shown by Zwicky, the specific heats of non-electrolytic solvents in water are additive: i.e. the total heat capacity of the solution can be expressed by the formula No c p o + Ni c p i (18.71) where C P Q = 18.01 is the molal heat of water (at 15 C) and c p i of the solute. It is interesting to note that the values of c p \ observed in the solution are pretty close to those observed in the solid state of the solute, as appears from Table 48. TABLE 48 SPECIFIC HEATS OF SOLUTES Substance Dextrose C 6 H 12 6 Lactose CiaH-^Oii Urea OCN 2 H 4 Glycerin C,H 8 0, Tartaric acid C 4 H 6 6 c p i from eq. (18.71) .. Cpi in solid state 55.2 51 7 104.5 98 5 24.8 19 3 56.0 53 67 62 When we turn to aqueous solutions of electrolytes, we meet with the unexpected fact that the heat capacity is not increased but lowered by the solutes. A qualitative explanation of this behavior was given by Tammann, 1 who pointed out that the effect is analogous to a rise of pressure to about 500 atm: the electrolyte must increase, in some way, the pressure under which the particles of water stand. Accurate modern measurements in solutions of alkali chlorides 2 showed that at low molalities (up to m = 0.5) the heat capacity is still well repre- sented by the linear formula (18.71). However, the coefficients c p \ here take negative values of the order 40. Taking up Tammann's suggestion, Zwicky (loc. tit.) developed it into a detailed theory. The field of the ions exercises large ponderomo- tive forces on the electric dipoles of the water molecules which actually produce a strong compression of the aqueous medium in their vicinity. Each ion is, therefore, surrounded by a little sphere within which the specific heat of the water is reduced by pressure. As long as the con- centration is low, the spheres do not overlap and the reduction of the heat capacity is proportional to their number, i.e. to the molality. Hence the validity of the linear expression (18.71). The quantitative calculation of this effect gave the right order of magnitude. 1 G. Tammann, Innere Krafte und Eigenschaften der Losungen, 1907. 1 Richards and Rowe, J. Am. Chem. Soc. 43, p. 770, 1921; 44, p. 684, 1922. CHAPTER XIX EQUILIBRIUM INVOLVING RADIATION 125. Equivalence of mass and energy. We devoted Chapters XIII to XV to questions pertaining to the determination of the abso- lute value of the entropy S. It should not be forgotten, however, that there is another fundamental function, the internal energy U, whose absolute value is also indeterminate. As far as the first and second laws are concerned, S and U are on the same footing since only the differentials of these two functions are defined in thermodynamics (compare sections 13 and 22). Both contain, therefore, additive inte- gration constants. That the indeterminacy of the energy constant was much less emphasized was due to historical and practical reasons. It was taken as a matter of course, because the thermochemical methods of measuring the differences of these constants in a reaction had already been developed before thermodynamics was established as a separate science. It became possible to determine the absolute value of the energy after Einstein l had established the law of proportionality of mass and energy as an important result of his theory of relativity. The fact that the inertial mass of a particle depends on its kinetic energy was not an exclusive feature of relativity but was proved long before its advent. But only the theory of relativity gave a sharp definition of the concepts here involved, and abolished the difference between inertial and gravitational masses. Let M be the mass of a system, E its total energy, and c the velocity of light. Einstein's law is then expressed by the simple relation E Me 2 . (19.01) If we consider 1 mol of a substance at the temperature T = 0, its internal energy will be UQ, while its mass will be equal to its molecular (or atomic) weight ju Einstein's law gives, therefore, *o => 2 . (19.02) 1 A. Einstein, Ann. Physik 18, p. 639, 1905. XIX 126 EQUILIBRIUM INVOLVING RADIATION 331 Strictly speaking, the mass depends on the temperature since the energy changes with it. To be exact we should have specified that M is the molecular weight at T = 0. Chemistry proper deals with dif- ferences Ao (for instance in dissociation of molecules) so small that the attendant change of mass AM = Awo/c 2 cannot be detected by the finest measurements. In this chapter, however, we shall treat proc- esses in which an appreciable part of the energy of the reacting sub- stances is converted into radiation. To the same extent their mass will undergo a loss: conservation of energy and conservation of mass hold only for matter and radiation together and not for the material part of the system separately. Compared with the huge changes of mass involved in these processes the difference between /u at T = and T = 300 abs is irrelevant and we shall understand under the symbol AC the molecular weight as tabulated by international agreement. As an example we write out the explicit expression for the thermo- dynamic potential of a monatomic perfect gas V = RT(log p - | log T - j) + nc*. (19.03) For some applications it will be more convenient to express the pressure in terms of z (the number of molecules or atoms per unit volume) by the relation (1.15), p = zkT, V = RT(log z - | log T - j + log *) + c? (19.04) 126. Energy and entropy of black radiation. The theory of radi- ation is so extensive a subject that it is best treated in separate books, 1 and it is not advisable to discuss it at length in a textbook devoted to thermodynamics. We limit ourselves to a brief review of some of its concepts and laws which shall be needed in the remaining part of this chapter. Let us consider a closed system part of which is free of matter (vacuum) and filled only with radiation, while matter is present in other parts. Left to itself such a system reaches a state of equilibrium in which the temperature T is uniform throughout and the radiation in the evacuated part has quite definite properties. It is called black radiation, and its spectral composition is given by a universal law, independent of the nature of the material part of the system in equilib- rium with it: the energy density u/lv (i.e. energy per unit volume) belonging to the interval of frequencies between v and v + dv is a universal function of v and T only M" - F(v,T)dv. (19.05) 1 For instance, M. Planck, The Theory of Heat Radiation, 1914, 332 TEXTBOOK OF THERMODYNAMICS XIX 126 This implies, of course, that the total energy density /CO /*00 u& - / F(v,T)dv = u(T) (19.06) ^o is a function of the temperature alone. If black radiation is enclosed in an evacuated container, it exerts a pressure p on its walls and through it resists compression. Assuming the walls to be movable, the radiation does the work p dV when the volume is increased by dV (compare section 7). If an element of heat dQ is imparted to the radiation, the first law of thermodynamics requires that it be converted, in part, into an increase of its energy U, in part, into work done by it dQ = dU + pdV. The second law can be also applied to radiation and includes the familiar expression (4.07) for the entropy differential <ZS= (dU + pdV)/T. (19.07) In particular, when the volume of the system remains constant (V = const, dV = 0) the increment of the entropy is dS - dU/T. (19.08) The same results hold, of course, if the vessel is not quite evacuated but contains, in addition to the radiation, gases in equilibrium with it, of so low a density that their refraction is negligible, so that the prop- agation of light is appreciably the same as in vacuo. This is the case we shall consider in the following sections. The restriction to low densities, though not absolutely necessary, greatly simplifies mat- ters and saves lengthy discussions. The general relation (19.08) which contains practically no restrict- ing assumptions about the function U is all we shall need. As far as our applications are concerned, the rest of this section has a purely academic interest. The quantum theory of radiation leads, in its present form, to the result that empty space contains a certain con- stant energy density o even at T = 0. It is usually thought that this is merely a formal defect and that this zero point energy (which is, by the way, infinite) has no physical reality. There were, however, those who argued that a part of it may have physical significance, that it may influence the absolute entropy constant of the radiation and through it the equilibrium with matter. We shall see in the ^t section that the conditions of equilibrium are independent of the XIX 126 EQUILIBRIUM INVOLVING RADIATION 333 absolute entropy of radiation. Nevertheless the question how it is affected by UQ is worth looking into. We shall write, therefore, for the energy density, u + Wo, meaning by u that part which is due to temperature radiation. The total radiation is then U = (u + u$)V\ as to pressure, it is known to be p = w/3, since it is certain from empirical and theoretical considerations that the zero point energy does not contribute to it. The entropy differential (19.08) takes the form V du /4 \ d V dS - -dT + (j u + no ) (19.09) The reciprocity relation (2.10) is easily integrated and gives u = ar 4 - f wo, (19.10) a being a constant. We see that the inclusion of a constant term UQ (not exerting pressure) into the energy density leads to a contradiction. The result gives for p = \u \aT* Jwo, so that UQ does, after all, produce a negative pressure. It follows that the zero point energy, whether it is in other ways real or not, must be left out in computing the entropy. We arrive thus at the usual form of the Stefan-Boltz- mann law u = ar 4 , (19.11) and substituting this into (19.09) 5 = |ar 3 F+5 . So is an integration constant and as such must be independent of V and T. It is obvious that it cannot be anything but So = 0, other- wise we would have still a finite entropy when the volume is reduced to zero, i.e. when we have no system. The entropy density is, therefore, * = | = i r3 - (19tl2) Comparing this with (19.11), we verify ds - du/T. (19.13) This equation is compatible with the following interpretation form- ing an important part of the theory of black radiation. The entropy density 5 is considered as the sum of the densities sjiv belonging to the different frequency intervals /' /o 334 TEXTBOOK OF THERMODYNAMICS XIX 127 where $ is defined by ds v - du,/T. (19.14) In fact, the integration of this relation with respect to dv (at con- stant T) gives eq. (19.13). The relation shows that the frequency intervals of the radiation can be regarded as statistically independent systems in equilibrium with one another at the temperature T (com- pare section 22). In the particular case of constant volume (dV = 0), we can write (1915) ds T 9 (19 * 15) if we denote U v Vu,, S, = Vs . 127. General conditions of equilibrium. Let us envisage a system enclosed by rigid, perfectly reflecting, adiabatic walls and consisting of a sufficiently thin mixture of perfect gases (compare preceding sec- tion) and radiation. Let N\, N%, . . . N ft be the mol numbers of the gases; the total energy and total entropy of the system are then if UR, SR stand for the energy and entropy of the radiation. The total volume of the system is expressed in terms of the molal volume vi of each gas as The conditions of equilibrium (compare section 31) are given by S having its maximum while U and V stay constant, in view of our assumptions about the walls. In variational form they are or JL* + 8U* = 0, (19.16) + 8S* - 0, (19.17) 0, (/ = 1, 2, . . . 0) (19.18) We can regard the first equation as the main condition, the others as subsidiary ones. The difference between this system and that of section 40 lies only in the presence of radiational terms. Apart from XIX nl EQUILIBRIUM INVOLVING RADIATION 33$ this, the procedure is exactly the same as in treating the ordinary gas mixture. We first eliminate the variations of the internal energies by means of (4.15) and (19.13) dU R = T R dS R , whence (19.16) takes the form + uitNi) + T R dS R 0. (19.19) Now we apply the method of Lagrangean multipliers (section 38) : we multiply the first subsidiary condition (19.17) by Xo and the others (19.18) by X/, (/ SB 1, 2, . . . 0), and add them to the main condition (19.19) #i(- Pi X, vi)*Ni] + (T R + Xo)SS* - 0. (19.20) As was explained in section 38, the variations foj, dS R , dvi can now be considered as independent, whence their coefficients must vanish - Xo - Ti = TR - T (i.e. the temperature is uniform) and Xz - pi. The last equation supplies only the physical interpretation of the multipliers Xj. Recalling the definition (5.33) of the molal thermo- dynamic potential ?i = u\ Tsi + pi vi, 1 we can write what remains of (19.20) in the form T] PI#I - 0, which is identical with (6.41) derived without considering radiation. If the reaction taking place in the mixture is expressed by the equation 0, (19.21) the condition of equilibrium is T] n Vi - 0. (19.22) 1 As in (8.08) we use the notation "Qi (and not <PI) to indicate that this function is calculated as if the other gases were not present, i.e. in terms of the partial pressure. 336 TEXTBOOK OF THERMODYNAMICS XIX 128 Though formally identical with (6.49), these conditions have a wider scope of applications. In deriving them we explicitly admitted that the energy and mass of the material part of the system need not remain constant but may be converted, wholly or in part, into radia- tion. We are, therefore, free to apply them also to such reactions (19.21) in which substances disappear altogether under emission of radiation (annihilation of matter) or in which one kind of atoms dis- appears and another is created (permutation of elements). We may also mention that, because of eq. (19.15), we would have obtained the same result writing in (19.16), (19.17), dU v , 5S,, instead of dU Rt &S Rt where v is the frequency of the photons emitted in the reaction. In other words, the reacting substances set themselves in direct equilib- rium with the frequency interval corresponding to their emission and through it, indirectly, with the rest of the radiation. Using the expression (19.04) for the thermodynamic potential of monatomic gases, we obtain from (19.22) "i log zi = log / + I v log r - A-c 2 /RT, (19.23) i where v = v\ + v^ + . . + v$\ the change of mass in the reaction is and log / = Z"tf'i - v log k. (19.24) i According to section 120, the chemical constant of a monatomic gas is jio = 4.417 + logio M + log g, where g must include also the nuclear statistical weight, and logio k = 15.8630. Therefore, ! log Zi = log K n , T- - ' ' - - * (19 2S) logio K. = 20.280? +vi (f logio M< + logio gi) 4.695 X 10 12 A M + f v logio T - 128. Negative and positive electrons. Equilibrium of matter and radiation as a problem of thermodynamics was first treated by O. Stern. 1 He envisaged a state of affairs in which atoms disappear, being converted into radiant energy, and others suddenly appear in the field of radiation, absorbing the amount of energy equivalent to 1 0. Stern, Zs. Elektrochemie 31, p. 448, 1925. XIX 128 EQUILIBRIUM INVOLVING RADIATION 337 their mass. As science progressed processes of this sort were regarded as less and less speculative. With the discovery of the positive elec- tron (positron) by C. D. Anderson 1 the creation and annihilation of electron pairs became an established fact. The tracks of the two electrons (one positive, one negative) produced by a cosmic ray or 7-ray can be observed in the Wilson chamber. There is also direct evidence that they recombine, emitting their whole energy in the form of two photons. If we characterize the electron and positron by the subscripts e and p, eq. (19.21) of the reaction becomes G e + G p - 0, (19.26) whence ? = v p = 1, v = 2, and 2 K. (19.27) For the electron of either sign f log /z + log g = 4.59 (compare section 113), A/* = 2/x = 1.096 X 10"* 5 146 X 10 9 logio K = 31.38 + 3 logic T - - - jr -- (19.28) In the empty space, in the absence of all other types of matter, positive and negative electrons would exist in equal numbers, z p = z et so that the density of positive electrons would be z p = K*. (19.29) This number is exceedingly small at ordinary temperatures and becomes appreciable only when the temperature approaches 10 8 degrees. In fact, for T = 8 X 10 7 deg, we find z p = 2.5 X 10~ 6 cm" 3 , or 1 positron per 40000 cm 3 . From then on the rise is rapid: for T = 2 X 10 8 , we already have z p = 1.9 X 10 15 cm- 3 . Of course, temperatures of this order do not prevail in interstellar space. The only place where they may possibly exist is the interior of stars. Here the conditions are different from those in empty space, in that there is a large excess of free negative electrons, which depress the density of the positrons in the way explained in section 53. Let us denote by ZQ the density of the free electrons which are in the star from the start, being stripped by ionization off the atoms of the stellar material. In addition to this, there are the positive-negative pairs 1 C. D. Anderson, Science 76, p. 238, 1932. * Since there is no clanger of misapprehension, we drop the subscript % of K in the remainder of this chapter. 338 TEXTBOOK OF THERMODYNAMICS XIX 129 created by the reaction (19.26). The strong electric attraction will assure that the stellar matter remain electrically neutral. If the density of positrons is z p , the total density of negatives will be z e = z p + ZQ, the excess negative charge being neutralized by the atomic ions. Equation (19.27) gives, therefore, or Z P (Z P + ZQ) = K, z p = ((K + so 2 /4) H - 20/2]. (19.30) To fix our ideas, let us assume ZQ 5 X 10 21 cm" 3 , which is, pre- sumably, of the right order of magnitude for stars of the G and K types. Table 49 gives the density of positrons z p for a large range of temperatures which may or may not exist in the stars. The last col- umn lists the heat function stored up in the pairs per 1 cm 3 . From about 5 X 10 8 degrees on, z p and x become independent of ZQ and would be the same for all values of this density smaller than 5 X 20 21 . At the same time the kinetic energy of the electrons becomes more important than the intrinsic. TABLE 49 EQUILIBRIUM OF NEGATIVE AND POSITIVE ELECTRONS T K Zp '(-) \cm'/ 8 X 10 7 6.0 X 10- 10 1.2 X 10-" 4.5 X 10~ 45 9 1.2 X 10~ 2 2.4 X 10- 24 9.3 X 10- 38 1 X 108 8.5 X 10 3 1.7 X 10~ 18 6.1 X 10- 32 2 9.1 X 10 29 1.8 X 10 8 7.3 X 10-' 3 5.0 X 10 38 1.0 X 10 17 4.2 X 10 3 5 6.2 X 10" 7.9 X 10 22 3.5 X 10 9 1 X 10" 1 . 7 X 10" 4.1 X 10 26 2.2 X 10 13 1 X 10 10 7.4 X 10' 2.7 X 10 30 5.5 X 10 17 At first sight, one is inclined to doubt if the assumption made in section 127 is fulfilled, namely, if the index of refraction of the radia- tion is approximately 1 in matter as dense as that. However, a check shows that such is indeed the case because the wave length which must be here considered is that emitted in the reaction, and it is extremely short (1.2 X 10 " 10 cm) even at low temperatures. 129. Neutrons and protons. The neutron has a mass slightly larger than the proton. There is, therefore, the possibility of its being XIX 129 EQUILIBRIUM INVOLVING RADIATION 339 converted into a proton under ejection of a negative electron and per- haps of a quantum of radiation. In fact, there is much reason to believe than an intranuclear process of this type plays a role in the emission of radioactive 0-rays. Equation (19.21) takes here the form G n G H + G e = 0, whence v n = 1, J> H + = v e = 1, v = 1 if we characterize the neutron and proton by the subscripts n and H +. The equation of equilibrium (19.23) is then The atomic weights of the neutron and proton are 1 n n = 1.0090, 1.0076, A/z = 0.0009, the (nuclear) statistical weights gn = gH+ = 2, whence 3 4 2 v 10 9 logio K = - 15.688 - - log 10 T - y -- (19.31) The process of dissociation is here exothermic, and, therefore, the conditions are peculiarly inverted. The presence of excess electrons does not depress the degree of dissociation but favors it. The equilib- rium constant does not rise monotonically with temperature but only rises to a maximum (at T = 6.5 X 10 9 ) and then drops again. This maximum value of K is equal to K = 7 X 10- 32 . Of course, the last decimal figure in the atomic weights /-in and /IH+ is not quite certain and the heat of reaction is affected with a con- siderable probable error. But even if Aju were only half as large (0.00045) the maximum value of K would be ttut little increased, namely K = 2 X 10 ~ 30 . As the density of electrons z e in ordinary stars (not white dwarfs) is only of the order of magnitude 10 21 (referred to in the preceding section), we conclude that the ratio of free neutrons to protons in them must be negligible. This conclusion would not be safe with respect to white dwarfs. The density in them is so enormously high that the electrons may form a completely degenerate gas even at temperatures as high as 10 8 de- grees. The thermodynamic potential of these electrons is then (1 7.35), <p e = fQt + jji e c 2 , and the equation of equilibrium, <p n ^>H+ <? = 0, gives simply 1 Bonner and Brubaker, Phys. Rev. 50, p. 308, 1936. 340 TEXTBOOK OF THERMODYNAMICS XIX 130 since /B< is negligible compared with AM- The molal volume of the electrons in the white dwarfs may be as low as v = 3 X 10 ~ 6 cm 3 , whence, by the formula (17.29), 9 = 0.7 X 10~ 8 T. Therefore, the electrons may be regarded as degenerate at temperatures of the order 10 s but not of the order 10 9 . In other words, the temperature within the white dwarfs hardly rises higher than 10 9 degrees. We are, there- fore, justified in concluding that free neutrons do not play any impor- tant role in stars, including the white dwarfs. 130. Hydrogen and deuterium. As early as 1815 the English chemist, William Prout, formulated the hypothesis that the atoms of heavier elements are formed by the association of hydrogen atoms. This view is confirmed by modern science and found its expression in the very name "proton," which is derived from the Greek "TrpSros" meaning " first " or " primary". A scientific theory of nuclear struc- ture became possible after the discovery of the neutron : every nucleus is supposed to consist of neutrons and protons. 1 There is a consider- able body of experimental facts to support this theory. In the so- called experiments on permutation of matter it is observed how fast- moving particles knock protons and neutrons from or into atoms, con- verting them into other elements. Inasmuch as a neutron can be formed by the association of a proton and an electron (preceding sec- tion), we can also say that any nucleus may be produced out of a suitable number of protons and electrons, although the electrons do not retain in it their individuality but merge with the protons, forming neutrons. The simplest examples of the building up of heavier elements is the formation of deuterium and helitim, which we shall discuss in this and the next section. There is direct experimental evidence for these processes. When a fast-moving deuteron hits a target, phenomena are observed which offer evidence of its being split sometimes into a proton and a neutron. On the other hand, we have seen in the pre- ceding section that a neutron dissociates into a proton and an electron. It must be possible, therefore, for two protons and one electron to unite, forming a deuteron 2Gn+ + G e GD+ = 0, PH+ = 2, v, = VD+ =1, v = 2 and * W. Heisenberg, Zs. Phys. 77, p. 1932; 78, p. 156, 1933. XIX 130 EQUILIBRIUM INVOLVING RADIATION 341 The atomic weight of the deuteron 1 is MD+ = 2.0142, the statistical weight 2 is #D+ = 3. Hence A/* = 0.0015, 7 04 v 10 9 lo glo K = 35.63 + 3 logio T - ^^ (19.33) Numerically: TABLE 50 DEUTERON- PROTON EQUILIBRIUM T K K* 10 8 1.6 X 10- 11 4 X 10~ 2 X 10 8 2 X 10 26 4 X 10 12 4 X 10 8 7 X 10 48 8 X 10 21 It is clear from this equation that in the state of equilibrium there should be very little hydrogen below 10 8 degrees. The density of H+ cannot be larger than K^ since zn+/z e is certain to be smaller than 1. At low temperatures the conditions are somewhat different because one cannot treat the gases as completely ionized but must regard them as atomic or even as molecular. For atomic hydrogen and atomic deuterium the equation would be 2G U - CD = 0, J>H ==: 2, VD == 1 v = 1. ZH = CKz D ) M or 2 D = Z H 2 K, 3 7 04 V 10 9 logio K = 20.260 + - lo glo T - -^-~ (19.34) Again we see that in the state of equilibrium there should be an enormous excess of deuterium. There is little doubt that positive and negative electrons, in our universe, are really in equilibrium in the sense of section 127. The recombination of pairs progresses with great rapidity, and under ordinary conditions there are no more of them than calculated from our equations. Also, there is no direct evidence that the free neutrons are more numerous than would be expected theoretically. But when it comes to the formation of deu- terium from hydrogen and to the analogous processes considered in the next two sections, we must conclude that the world is very far from 1 Bonner and Brubaker (footnote on p. 339). 2 At the high temperatures here in question there may exist and be stable another modification of the deuteron (with zero nuclear spin). The joint statistical weight would then be #D+ = 4 and the constant 35.50. 342 TEXTBOOK OF THERMODYNAMICS XIX 131 equilibrium, indeed. It appears that the protons, unless they are in some unusual state of activation, are unable to combine and form heavier nuclei. 131. Hydrogen and helium. The case of formation of helium by the association of four hydrogen atoms is of historical interest because it was the first example of permutation of elements to which the theory of thermodynamical equilibrium was applied. 1 The helium nucleus or a-particle can be formed of four protons and two electrons: 4G H + + 2G. - G a = 0, whence *>H+ =4, v e = 2, v a = 1, v = 5, and z H + 4 Ze 2 /z a = K or SH+ = (Kzjz^z"* . The atomic weight of helium is HH e = 4.0040, and of the a-particle H a = 4.0029, the statistical weight in both cases gn e = g = 1- This gives AM = 0.0284, and i 333 y inn logio K = 92.44 + 7.5 lo glo T - ~ -- (19.35) Numerically: TABLE 51 HELIUM-HYDROGEN EQUILIBRIUM T K K y * 7 X 10 2 X 10- 1.2X10-' 8 X 10 8 4X10- 8 1.4X10-* 10 4 X 10 2 4.6X10' 5 X 10 3 X 10 1 ' 8 4.3 X 10" Seeing that z a < z e and that, in ordinary stars, zj 4 is considerable (up to 10 6 ), hydrogen should be practically absent at temperatures below 10 9 . At low temperature, the gases must be taken as atomic because they are no longer ionized 4G H - G H = 0, or Z H It is hardly necessary to calculate K in this case, because it is imme- diately obvious that the positive terms in its expression are diminished while the negative remain the same as in (19.35) so that logio K has an enormously large negative value. If equilibrium prevailed, with respect to this reaction, no appreciable amount of hydrogen could exist outside the stellar cores. 1 R. C. Tolman, J. Am. Chem. Soc. 44, p. 1902, 1922. XIX 132 EQUILIBRIUM INVOLVING RADIATION 343 The formation of deuterium and helium are merely the beginning of the process of association of protons into heavier elements. The further combination of a-particles, protons, and electrons into stable nuclei is also exothermic, taking place under a reduction of mass (mass defect). The heavier atoms are, thermodynamically, still more probable than helium. Although the nuclei with the largest mass defects are the most abundant, it does not seem that their abundance relatively to helium and to one another is in quantitative agreement with the thermodynamical expectations. It is not certain that the present data on atomic weights are sufficiently accurate to decide this question. Be this as it may, it is quite certain that equilibrium does not exist with respect to hydrogen, as has already been emphasized at the end of the preceding section. 132. Annihilation of hydrogen. The only experimentally estab- lished case of annihilation of matter is the mutual destruction of a posi- tive and a negative electron. The annihilation of protons, hydrogen atoms, or heavier atoms has not as yet been directly observed. How- ever, in the present state of theoretical knowledge no valid reason can be advanced why such a particle could not completely disappear, con- verting its energy into radiation. The equations for the annihilation of a neutron, on one hand, and of a proton-electron system, on the other, are very little different, and we shall consider the latter because of the greater abundance of protons. The reaction is given by G H + + G e = 0, "H+ = " = 1, v = 2, A/x = 1.0081, whence and from (19.25), ZH+ = K/z*> \ i r o*c.7i r 473 X 10 12 (19.36) logio K = 36.275 + 3 logio T Numerically: TABLE 52 T = 5 X 10 10 K - 2 X 10-" 7 X IQio 2 X 10* 8 X 10 10 7 X 10' 10 11 1 X 10" If this process is possible, the universe is extremely far from the state of equilibrium: no matter should exist at temperatures below 7 X 10 10 degrees. The view was expressed that there may exist also negative protons, unobserved, so far, because of their scarcity. If this is true, there is 344 TEXTBOOK OF THERMODYNAMICS XIX 133 the possibility of the creation and mutual annihilation of positive- negative proton pairs. The reaction is the same as in the case of pos- itive-negative electron pairs only the heat is much larger (A/* = 2.015). It follows logio K = 39.948 + 3 logio T - 9.48 X 10 12 (19.37) This means that temperatures of the order of 10 12 degrees are necessary to create such pairs. Expressed in electron-volts, the energy of such a pair is 1.88 X 10 9 e.-v. How exceedingly small the numbers in (19.36) and (19.37) are, appears from the following fact. A few years ago the astronomers calculated the size of the universe as 10 87 cm 3 (in the meantime it has become doubtful whether the universe is finite). The reciprocal of this is still very large compared with K at T = 10 10 or at lower tempera- tures. It follows that there could not be a single proton, either isolated or as a positive-negative pair, in the whole of the universe. The last atom of matter should have been converted into radiation if equilib- rium prevailed. 133. Influence of gravitational fields. The interest of the proc- esses discussed in this chapter lies in their applications to problems of cosmology. However, in treating them we have not taken into con- sideration the gravitational fields existing in and around the stars. It will be well to say a few words of explanation why the effects of the gravitational fields can be neglected on account of their smallness. We have treated the problem of equilibrium of heavy gases in a gravi- tational field in section 108 and have seen there that the only correc- tion which the equations need is a term added to the equilibrium constant log K. This constant, as given by eq. (19.23), already contains the term A/i-c 2 /jRr, which, combined with the correction for gravitation, becomes -(1 + Q g /c 2 )&'C 2 /Rr. In other words, the heat of reaction or the energy available in the process is changed in the proportion (1 + & g /c 2 ) owing to gravitational forces: V = (1 + fl,/c 2 )A/u. (19.38) XIX 133 EQUILIBRIUM INVOLVING RADIATION 345 The gravitational potential Q g vanishes at a great distance from a star and is negative in its vicinity, having its minimum at the surface. The factor is, therefore, smaller than 1, meaning that the heat of reaction is decreased. But even the largest known gravitational potentials, those of the white dwarfs, are materially smaller (in abso- lute value) than 0.002 X c 2 , whereas within the accuracy of the preced- ing chapter a correction of 0.2% is negligible. Equation (19.38) is, of course, quite general and applies to any molecular or atomic process in which energy is converted into radia- tion. According to the quantum theory, the frequency v of the emitted radiation is connected with the available energy per atom or molecule AE = c*&n/n A by the relation AE = hv, whence v' = (1 + tt g /c 2 )v. (19.39) The gravitational field changes the frequency of radiation emitted in an elementary process, decreasing it (gravitational red shift), a phenomenon predicted by Einstein and since confirmed experimentally. We have based these remarks on the Newtonian theory of gravita- tion. On general grounds we are justified in saying that the condi- tions cannot be materially different in the general relativity of Ein- stein as long as tt g /c 2 is small. In fact, chemical equilibrium was investigated from the standpoint of the general relativity by Tolman, 1 who found the same relations as in the absence of any gravitational fields. Einstein's formula of the gravitational red shift / = [(1 + 212,A 2 ) is, in practice (!2 g /c 2 1), identical with (19.39). 1 R. C. Tolman, Proc. Nat. Acad. Sci. 17, p. 159, 1931. CHAPTER XX MAGNETIC AND ELECTRIC PHENOMENA 134. Langevin's theory of magnetization. The systems treated in the preceding chapters were completely described by the thermo- dynamical variables p, V, T, apart from data about their composition. Even in the case of charged gases, discussed in sections 106-115, the electric potential was regarded as only a part of the energy constant and not as an additional parameter of the system. It is instructive to consider also variables of a non-mechanical nature, and we are going to do so in this chapter, beginning with the theory of magnetics or substances capable of magnetization. It will be sufficient to treat the case of isotropic substances because the generalization is obvious and would only encumber our expressions without adding anything essen- tial to them. Suppose that a system of this sort is placed in a homo- geneous magnetic field whose strength is H in the absence of the system (i.e. before the system is brought in). Let M denote the component of the total magnetic moment (or total magnetization) in the direction of the field. We know from the theory of electromagnetism that the work which the field does, in raising the magnetization from M to M + dM, is DW' = HdM. In accordance with the convention of section 7, we count the work done by outer forces against the system as negative. The total element of work (2.02), including the mechanical and mag- netic parts, is then pdV - HdM and eq. (3.04) of the first law has the form DQ - dU + DW = dU + pdV - HdM. (20.01) The state of the magnetized system depends on the two thermal parameters T, V we used all along and, in addition, on the magnetic parameter M. In order to have a complete description of the system, we must know the three functions U = C7(r, F, M), p p(T, F, M), (20.02) H - H(T, V, M) (20.03) 346 XX 134 MAGNETIC AND ELECTRIC PHENOMENA 347 In the sense of section 10, we may call these three relations the caloric, the thermal, and the magnetic equations of state. Of course, it is arbitrary which of the six parameters we regard as independent. For most of the applications it will be convenient to describe the sys- tem in terms of T, p, H, but occasionally we shall use also T, p, M. The first application of the second law of thermodynamics to such a system was due to Langevin. 1 As he pointed out, in the expression (4.07) of the entropy differential, dS = DQ/T, the magnetic work gives rise to the additional term DW II dS' = = dM, (20.04) and he postulated that dS' be an exact differential. This is possible only when the factor H/T is a function of M (compare section 8) and, vice versa, M is a function of H/T, /TT\ (20.05) This is the form of the magnetic equation of state (20.03), accord- ing to Langevin. It is quite obvious that this law is not general, a case in point being the behavior of diamagnetic substances whose mag- netization is known to be independent of temperature: M = - AH, (20.06) where A is a constant. 2 What is, then, the special assumption made in this theory and leading to the formula (20.05)? It lies in the sep- arate treatment of the magnetic part of the entropy (20.04) as an exact differential, independently of the remaining part (dU + pdV)/T. (20.07) This implies, of course, that the expression (20.07) is also an exact differential which cannot depend on the variable M (or H) since its differential is absent in it. Consequently the functions U and p must be also independent of M. Instead of (20.02), U = U(T, V), p = p(T, V). (20.08) We met with a similar case in the theory of perfect gases. In the expression (4.16) of dS the two terms are separately exact differentials, 1 P. Langevin, Ann. Chim. Physique 5, p. 70, 1905. 8 We speak here of the ordinary or atomic diamagnetism. There exist a few substances (as bismuth, antimony, etc.) with so-called crystal diamagnetism which has a temperature coefficient. 348 TEXTBOOK OF THERMODYNAMICS XX 134 and this is due to the internal energy depending only on the thermal variable T. By analogy we shall call a substance whose equations of state follow the simple laws (20.05), (20.08) a perfect magnetic. Langevin initiated also the statistical theory of magnetization and gave an explicit expression of the law (20.05) for paramagnetic mate- rials. As corrected to take into account the restrictions of the quan- tum theory, it can be written M - (20.09) and is sometimes called the Langevin-Brillouin formula. Z represents here the total number of atoms, and the product jgfo the total magnetic moment of one atom. The factors have the following meaning: o the Bohr magneton A) = ^- = (0.9174 db 0.0013) X 10- 20 erg gauss- 1 , (20.10) g the Lande factor 3 * 2+ 2/C7+1) ' ( } j is the quantum number of the total angular momentum of the atom, compounded of the resultant spin momentum s and the resultant angular momentum /. At 20 C the numerical value of fo/kT is 2.28 X 10~ 7 : since jg is rarely larger than 7, the argument of the function F is small (< 0.050) at room temperature, even for fields of the order of 30 000 gauss. Under these circumstances all terms of the expansion of F(a) into a power series are negligible, excepting the first linear term, and the magnetization is well represented by Curie's law *, CH M = , (20.12) where C is a constant. For large values of the argument, the function F asymptotically approaches the value 1 (magnetic saturation), but its indications appear only at very low temperatures. The formula (20.09) was derived for gases and solutions of salts in which the atoms or ions can freely rotate. Even for these sub- stances its validity is not general, being often impaired by doublet or multiplet structure of the energy levels. Only when the width of XX 135 MAGNETIC AND ELECTRIC PHENOMENA 349 these multiplets is very small or very large compared with kT is the formula applicable. It is remarkable, however, that it represents with great accuracy the magnetization of solid paramagnetic salts of rare earths, especially those in which the paramagnetic atoms are not very close together but separated by non-magnetic atoms (as, for instance, in hydrates). The classical example, in this respect, is hydrated gadolinium sulfate investigated in Leiden. 1 The magnetic moment of this substance was measured down to 1?3 K and found to obey strictly the law (20.09) with j = -J and g = 2. At the lowest temperature and highest field the saturation was far advanced, the magnetization being 95% of the possible maximum. Exercise 106. Calculate the entropy term S' from (20.04) for substances whose magnetic moments are expressed by (20.09). Show that S' = kZ\ log sinh ( a ~\ ) /sinh -- aF(a) \ -f- const. I L \ 2j/f 2/J J Exercise 107. In his original formula Langcvin assumed that the orientations of the molecular magnetic moments are not restricted by quantum conditions. This formula can be obtained from (20.09) by the following transition to the limit: let j go to oo, while goes to zero in such a way that jg&o = /3 remains constant where j3 represents the molecular magnetic moment). Show that the result is M = Z/5 f(0H/kT), f(a) coth a - I/a. 135. Magnetothermal and magnetocaloric relations. It will be well to derive a few relations which will elucidate still further the thermodynamical status of perfect magnetics and diamagnetics. We start from the generalized thermodynamic potential * = U - TS + pV - HM, (20.13) whose differential is, according to (20.01) and (4.07), d& = - SdT + Vdp - MAIL (20.14) Hence , V = ( J , M = - P^j (20.15) Substituting this into (20.13) -* - < 20 - 16) \dP/T,H \d 1 H. R. Woltjer and H. Kamerlingh Onnes, Leiden Comm. 167 b, c. 350 TEXTBOOK OF THERMODYNAMICS XX 135 The three partials (20.15) give reciprocity relations when differ* entiated a second time. Thus we obtain from the second and third p.T T.H This is a relation between measurable quantities. The left side is the change of volume due to the field, its relative value (divided by V) is called volume magnetostriction or Barrett effect. The right side represents the change of the total magnetic moment due to pres- sure. Such an effect was foreseen and first measured by Nagaoka and Honda 1 and is named after them. Although accurate modern mea- surements of both effects are available, they were not carried out under comparable conditions and are not suitable for a quantitative test of the relation (20.17). It should be noted that in perfect magnetics the volume does not depend on the magnetic parameters (preceding section). The experimental existence of magnetostriction in ferro- magnetic materials shows, therefore, that these substances do not strictly follow Langevin's law (20.05). However, the effect is very small and the deviation, presumably, not large. Exercise 108. In a particular specimen of iron (at / = 20 C, H = 1 gauss) the partial (dM/dp)T,H was found to be 3 X 10 ~~ 3 gauss per unit volume and per 1 atm. Calculate the volume magnetostriction from formula (20.17). (Use absolute units). The other two reciprocity relations following from (20.15) are no 97 / ptH We shall use them for transforming eq. (20.16) as follows. We differentiate (20.13) partially with respect to p, and with respect to H, and take into account the relations (20.17) and (20.18): -( \ In the case of perfect magnetics we have (dU/QH) TtP 0, and from (20.17) and (20.20) 1 Nagaoka and Honda, Phil. Mag. 46, p. 261, 1898; see also S. R. Williams, Int. Crit. Tables VI, p. 439, 1929. XX 135 MAGNETIC AND ELECTRIC PHENOMENA 351 This is only another proof of Langevin's result because the general integral of this equation is the expression (20.05). On the other hand, the law (20.06) of diamagnetism gives OJ7/9#)r, P = ~~ AH- = -W> so that the internal energy contains a magnetic term whose differential is dU' = (dU/QH) Ttp dH = MdH = HdM. (20.21) A look at the structure of the element of heat (20.01) shows that the additional term in dU just cancels the work of the magnetic field DW' = HdM, so that DQ is independent of magnetic parameters. We see from this that diamagnetic and perfect magnetic substances are complete opposites with respect to the caloric action of putting them into a magnetic field. In diamagnetics, the (positive) work against the field is done at the expense of the internal energy without any contribution from external heat sources. It is a process at the same time adiabatic and isothermal: the loss is sustained, as it were, by the constant of the internal energy and does not affect the temperature of the body or its equilibrium with the environment. On the contrary, the (negative) work of magnetizing perfect magnetics is completely supplied by outer sources, when conducted isothermally. When the process is adiabatic it leads to a rise of temperature in the sub- stance. In the interest of the following section we shall say here a few words about the heat capacity of magnetics. In treating simple sys- tems, we distinguished in section 14 the specific heats at constant vol- ume c v and at constant pressure c p . Similarly, we have to speak here about the following two heat capacities. The heat capacity at constant magnetization (and, say, constant pressure) is the heat that must be imparted to the magnetic when its temperature is raised by 1 degree and, at the same time, the magnetic field is changed in such a way as to keep its magnetization constant: C P M = lim(AQ/Ar) p Af. According to the expression (20.01), no part of this heat is used for magnetic work. On the other hand, the heat capacity at constant field strength C P H = Hm (&Q/&T) P H refers to a process accompanied by a change of M and involving magnetic work. In view of the relation AQ = T&S, we can also write The relation between these quantities analogous to (4.27) is found, in the simplest way, by considering 5 a function of T and M, while M , in its turn, depends on T and H, i.e. S = S[T, M(T, H)]. (We need 352 TEXTBOOK OF THERMODYNAMICS XX 136 not bother about the pressure p, since it is assumed to be always con- stant.) The rules of partial differentiation give us , dT/ P> M \dM/ PtT \d ptH In the partial (dS/dM) PtT the variables p and T are considered as constant so that M is a function of H only: therefore, Substituting this and making use of eq. (20.18), we find the desired relation r c r C vn - C PM = T - 136. Cooling by adiabatic demagnetization. Adiabatic demag- netization of a paramagnetic substance is analogous to the adiabatic expansion of a simple system. In either case, the work against external forces is done at the expense of the internal energy of the system and leads, in general, to a decrease of its temperature. The use of this " magnetocaloric effect " for the production of very low temperatures was proposed independently by Giauque J and Debye. 2 The usual method of lowering the boiling pressure of liquid helium does not per- mit in practice 3 to reach temperatures below 0.7 K. Work with the magnetocaloric method has been under way in recent years, at Berkeley (California) under the direction of Giauque, 4 and at Leiden (Holland), under the direction of deHaas. 5 It has been very suc- cessful in bringing the range of accessible temperatures considerably nearer to the absolute zero point. The equation of the adiabatic dS = dQ/T = is, according to (20.01), dS = (dU + pdV - HdM)/T = 0, (20.24) when S is expressed as a function of T, p t H, also ( a|\ <3 T/ VtH \ OP/ H t T ^O** / T.p i W. F. Giauque, J. Am. Chem. Soc. 49, pp. 1864, 1870, 1927. * P. Debye, Ann. Physik 81, p. 1154, 1926. Keesom, Proc. Amsterdam 35, 136, 1932. W. F. Giauque and C. W. Clarke, J. Am. Chem. Soc. 54, p. 3135, 1932; W. F. Giauque and D. P. McDougall, Phys. Rev. 43, p. 768; 44, p. 235, 1933. W. J. deHaas, E. C. Wiersma, and H. A. Kramers, Physica 13, p. 171; 1, p. 1, 1933; W. J. deHaas and E. C. Wiersma, Physica 1, p. 1107, 1933. XX 136 MAGNETIC AND ELECTRIC PHENOMENA 353 Substituting from (20.22) and (20.18), c f-(f-r) *+(!!?) w -- < 20 - 25 > i \0-l/p,H \0* / Pt ff The procedure consists in precooling the specimen in vacuo, in a strong magnetic field, and then turning off the field. With p = 0, eq. (20.25) gives for the rate of the adiabatic cooling or from (12.23) dH As was pointed out in the preceding section, the heat capacity C P M does not contain any magnetic work. Therefore, it should not be materially different from the ordinary capacity C p as measured in the absence of a magnetic field. Of particular interest are the perfect magnetics: from the kinetic point of view, they are the substances in which every atom is free to adjust itself in the field without being hindered by its neighbors. Other things being equal, they should show the highest magnetization and the highest caloric action. For this reason, both Giauque and Debye suggested the use of gadolinium sulfate as a testing substance because it was known to follow the law (20.09) down to the lowest temperatures. If we denote x = H/T, this law implies dM\ __x_dM /aM\ 1 dM dT/ Pt a~ ~~ T doc ' \9/// Pl r"" T dx ' \dT/ Pi H J. ax \ou/p,r so that (20.27) becomes dT _ x(dM/dx) dH~ C pM +x 2 (dM/dx)' (20.28) Of course, in the case of perfect magnetics there is no need to use a differential formula because the two terms (with p = 0), in the formula (20.24), are separately integrable. At zero pressure there is no differ- ence between c p and c v , and the energy differential is dU CvdT. Hence (20.29) 354 TEXTBOOK OF THERMODYNAMICS XX 137 If the specific heat follows the third power law (18.70), the first integral is equal to 6.45 X 10 9 (r/9) 3 erg/deg mol. Suppose we start the demagnetization at the temperature To and the field HQ. When the field is reduced to H = 0, the temperature has the value s*XQ J To 3 - 1.55X10- 10 8 3 / x- (20.30) where XQ = Ho/To and m is the molal magnetic moment. It was found, however, that the specific heat of gadolinium sulfate does not follow the third power law but exhibits an anomaly due to multiplet structure of the lowest energy level (compare section 117). The curve reproduced on p. 327 was obtained by Giauque and McDou- gall by measuring the magnetocaloric effect dT/dH and by calculating C pM from a formula equivalent to (20.27). Although they reached a temperature of 0?287 K, this substance is not particularly suitable for magnetocaloric cooling. At the time of the writing, the greatest success was obtained by deHaas with a mixture of two alums, K 2 SO 4 Cr 2 (SO4)3 24H 2 O + 14.4K 2 SO4A1 2 (SO4)3 24H 2 O. Demagnet- izing from 24 075 gauss to 1 gauss, he reached 0?0044 K. In all this work the measurements themselves were used to establish the scale of absolute temperatures by the method explained in section 29. However, the last figure (0?0044 K) rests, in part, on an extrapolation and has only approximate validity. Exercise 109. Consider a substance satisfying eq. (20.30) whose magnetization obeys the law (20.09), with j = , g = 2. Start with T 3 K and H = 25 000 gauss and demagnetize to H = 0. What will be the final temperature if = 200? 137. Supraconductivity and thermodynamics. It was discovered by Kamerlingh Onnes that the electric resistance of certain metals suddenly drops to practically nothing when they are cooled below tem- peratures called their transition points. We give the transition points of the principal supraconductive substances in Table 53. The transition point is displaced when the system is put into a magnetic field H. As the magnetic field is increased, the transition point moves continuously to lower temperatures, and at a certain strength of it (threshold value) the supraconductive state ceases to exist altogether. From the point of view of electromagnetism, the immediate effect of switching on a field consists in setting up currents in the supraconductor, in cpnformity with Faraday's law of induction. But the conductivity of these substances is so large that the currents continue to flow, without apparent loss, as long as the state of supra- XX 137 MAGNETIC AND ELECTRIC PHENOMENA 35S TABLE 53 SUPRACONDUCTIVE TRANSITION POINTS Elements Elements Intermetallic compounds Intermetallic compounds MB 70 K V 4 3 K Au a Bi 1 84 K ZrB 2 82 K 7n 78 Nb 9.2 CuS... 1.6 TaSi . . . . 42 CH 0.6 (ca) Ta 4.4 VN.... 1.3 PbS .... 41 HfT 4 22 La 47 WC 2 8 **is Al 1 14. WaC 2 05 Ga . . 1 05 Intermetallic MoC. 7 7 Alloys Tl 9 37 A/fnP 9 1 Ti 1 75 TIN 1 4 Pb-Sn-Bi 8 5 Th 1 5 BiflTlj. 6 5 TiC. 1 1 Pb-As. 8 4 Sn 3.71 Sb 2 Tl 7 .... 5.5 TaC... 9.2 Pb-As-Bi 9.0 Ph 7.2 Na 2 Pb 6 ... 7.2 NbC... 10.1 Pb-Bi-Sb 8.9 In 3.37 Hg 6 Tl 7 ... 3.8 conductivity is maintained: they are therefore called persistent cur- rents. Another result also follows from the theory of electromagnet- ism: when the magnetic field is switched on, after the supraconductive state is established, neither the field nor the currents can penetrate to any appreciable depth of the metal. The persistent currents flow at the surface and produce a magnetic field which exactly compensates within the metal the external field. Experimental work has shown that the same conditions prevail when the metal is cooled below the transition point in an already existing magnetic field. The lines of magnetic induction are pushed out of the metal, as it becomes supra- conductive, until the induction in the interior vanishes. This result was first announced by Meissner and Ochsenfeld 1 and has been con- firmed since by many investigators. Although there is still some doubt whether it holds rigorously, it describes the phenomena with an accuracy sufficient for thermodynamical purposes. In short, a supra- conductor behaves in a magnetic field like a substance of the perme- ability /i = 0. Since /* is connected with the susceptibility K by the relation /* = 1 + 4?r/c, the supraconductors can be formally described as diamagnetics with the susceptibility K = 1/47T. Let us consider a long stretched (needle-shaped) supraconductive body placed with its axis in the direction of the magnetic field. Accord- ing to the electromagnetic theory, the outer magnetic field produced by a body of this shape is negligible . Its j(negative) magnetic energy 1 W. Meiasner and R. Oc'senfeld, Die Naturwiseenachaften 21, p. 787, 1933. 3S6 TEXTBOOK OF THERMODYNAMICS XX 137 is due to the absence of any field in the interior and is equal to per unit volume, or to (20.31) per mol. The magnetization (per unit volume) is an expression of the type (20.06) characteristic of diamagnetics. We can apply, therefore, to supraconductors the results established in the preceding section for diamagnetic substances: the internal energy contains U H as an additive term, while the entropy is independent of the magnetic parameters. Consequently the thermodynamic potential (<p = u Ts + pv) has, in the case of supraconductive materials, the form vH 2 *-*>- , (20.32) O7T where w is the potential in the absence of a field. At the very low temperatures we are here considering, the pressure effects are negligible: there is no observable change of specific volume, so that it is permissible to ignore the pressure as a variable and to regard <po(T) as a function of temperature only, and v as a constant. The normal (non-supraconductive) modification of the metal is, as a rule, non-magnetic, so that its thermodynamic potential does not contain any magnetic term but is, simply, <p n (T). The equation of equilibrium between the two phases becomes, therefore, A3> = ^> n <p = or A* = Vn (T) - ,(D + ~ = 0. (20.33) O7T Of course, it is not a priori certain that the supraconductive state is in true equilibrium and that the second law can be applied to it. The hypothesis that it represents a phase in the thermodynamical sense was first made by Langevin. 1 It was corroborated by the great sharpness of the transition point in good crystalline specimens. The experimental test of the equation (20.33) is also reassuring: its conse- quences hold with good accuracy. This relation gives the dependence of the transition point on the strength of field H and can be tested in several ways. The first and second partials of A$ with respect to T are, according to (5.37) and (7.22), 1 P. Langevin, Rapports du 1* Conseil & vay, p. 301, 1911. XX 137 MAGNETIC AND ELECTRIC PHENOMENA 357 The differentiation of (20.33) gives the equation (l/T)dT (vH/v)dH = or analogous to the Clausius-Clapeyron equation. It was tested by Keesom and Kok l and found to be in agreement with experiments. Of great interest is the observation of these authors that the latent heat decreased as the field diminished and could not be measured at all in the case of field free transitions (i.e. dH/dT 7* oo for H = 0). If this observation is confirmed, it would mean that the equilibrium, in the case H = 0, is of the second order (section 49). Let us consider this case: we denote the transition point in the absence of a field by To and express A$, for T TQ + dT, H = dll, to terms of the second order. From (20.34) l/Ao\ 2\r / (dry - - or AO _ v (dlf^ TQ 4ir\a (20.36) This equation was obtained by Rutgers 2 and in a different way by Gorter and Casimir. 3 All these authors regarded the equilibrium in question as one of the second order. The equation (20.36) was con- firmed with remarkable accuracy by the beautiful measurements of Keesom and Kok on tin and thallium. 4 TABLE 54 ToK v cm l mol" 1 dH/dT gauss deg~ l Aco (calc) cal deg^mol"" 1 Aco (obs) Tin 3.71 14 2 151 2 00229 0024 Thallium 2.36 16 9 137 4 00144 00148 Nevertheless we do not agree with Rutgers, Gorton, and Casimir. 1 W. H. Keesom and J. A. Kok, Physica 1, pp. 503, 595, 1934. 1 Appendix to Ehrenfest's paper, note on p. 128. * C. J. Gorter and H. Casimir, Physica 1, p. 306, 1934. 4 W. H. Keesom and J. A. Kok, Physica 1, p. 175, 1933. 358 TEXTBOOK OF THERMODYNAMICS XX 137 The equilibrium cannot be of the second order since an unsurmount- able difficulty is introduced by the double sign of dH ^ ATT A^oV dT " \v To/ Represented graphically (Fig. 60), the transformation is of the general type of Fig. 18 on p. 132. With any plausible choice of the functions <f>o(T), <p n (T), the analysis shows that the lines representing the equilibrium continue beyond the point TV Therefore, both regions labeled (1) in Fig. 60 correspond to the supraconductive state so that cooling along the line // = does not involve any transition. We maintain, therefore, that the equilibrium in question is, in reality, FIG. 60 FIG. 61 Supraconductive transition point in its dependence on magnetic field. of the first order, there being a small but finite latent heat. According to the general discussion of section 49, this would imply a transition curve of the shape given in Fig. 61. That Rutgers' formula is satisfied with such a remarkable accuracy is due to the following coincidence. Equation (20.35) holds for the whole curve and can be differentiated along it. Since // T = As, its differential is (Ac/T)dT, and we find Ac T (20.37) Where the line of Fig. 61 is straight (d 2 H/dT 2 = 0), the slope is given exactly by Rutgers* expression, whether the equilibrium in TO is of the first or of the second order. 1 There were attempts to make other aspects of supraconductivity amenable to a thermodynamical treatment. However, they failed to 1 In a later paper (Phys. Zs. 35, p. 963, 1934) Gorter and Casimir admit that the supraconductive state would be thermodynamically stable above the transition point "if it existed." They postulate, therefore, that it cannot exist. This seems an adhoc hypothesis, and the difficulty is much better disposed of by the simple assumption made in the text, that the equilibrium is of the first order. XX 138 MAGNETIC AND ELECTRIC PHENOMENA 359 give a complete picture of the phenomena for which they were intended. Limitations of space do not permit us to enter into their discussion. 138. Electrostatic phenomena. Dielectrics or substances capable of electric polarization P form, in many respects, an analogue to the magnetics treated in section 134. The work which must be done on a dielectric body in order to increase its total electric moment (or polarization), in the direction of the field , from P to P+ d P, is -E dP, so that the complete element of work becomes DW = pdV - E dP, (20.38) in analogy with the magnetic expression of section 134. There are two types of dielectric polarization which, from a thermodynamical point of view, correspond closely to the types of magnetization in atomic diamagnetics and in perfect magnetics. (A) The Lorentz type of polarization. If the molecules of a sub- stance have no electric moments in their normal state, the application of an electric field creates such moments in them by displacing the electrons from their normal position. The expression for this sort of polarization (taking account of the field of the electric moments) is P AE', (20.39) where A is a constant and E = E + P is the effective field strength. o The phenomenon is mainly intramolecular, and A is independent of temperature. This has the thermodynamical implications discussed in connection with diamagnetism. (B) Dipole polarization. It was pointed out by Debye l that some dielectric substances may possess permanent molecular dipoles. He suggested, therefore, taking over Langevin's formula (developed for magnetic dipoles) in order to account for empirically observed changes of electric polarization with temperature. The case of polarization is simpler in two respects. In the first place, the electric dipoles have no tendency to orient themselves in discrete quantized directions. The original Langevin formula derived on lines of classical statistics holds for them: this means that one has to take in eq. (20.09) j = oo and jgPo = p e = const. 2 In the second place, the conditions in dielec- 1 P. Debye, Phys. Zs. 13, p. 97, 1912. 2 The transition to the limit is the same as in exercise 107. When the argument a is small, there holds the expansion coth a I/a -f -J-a; therefore, the function of exercise 107 becomes /(a) = ^a, leading to formula (20.40). 360 TEXTBOOK OF THERMODYNAMICS XX 138 tries hardly ever approach saturation so that the term of first order in E! IT represents an excellent approximation . where fa is the electric dipole of a molecule and Z the total number of molecules. Of course, one should expect this expression to hold only in substances where the dipoles can freely rotate, i.e. in gases and liquids. In analogy with the perfect magnetics of section 134 they may be called perfect dielectrics. In general the dipole dielectrics possess also an appreciable polariza- tion of the Lorentz type so that very few of them are, even approxi- mately, perfect. The dielectric constant D (which is accessible to direct measurement) is connected with the polarization by the rela- tion (D 1)/(D + 2) = $irP/VE' and has the general expression F+I" iv + i z kT' (20 - 41) with z = Z/F. By plotting D against \/T one can separate out the part due to dipoles and determine the moment. This proved a very helpful method for investigating the properties of many organic molecules. 1 The electric analogue to magnetostriction is the phenomenon of electrostriction. Because of the close parallelism of the thermody- namical properties of dielectrics and magnetics we can directly take over eq. (20.17), writing T.B We see that electrostriction (represented by the left side) is asso- ciated with another effect called piezoelectricity, which consists in the dependence of the polarization P on pressure. Our formula refers to the volume effect, but piezoelectricity is mostly observed in crystals, where it is particularly strong in certain crystallographic directions. In turmaline the piezoelectric effect (increase of a component of the polarization per unit volume and per unit increase of a strain com- ponent) is 5.78 X 10~ 8 in e.-s. cgs units, in quartz -6.9 X lO" 8 , but it is particularly large in Rochelle salt, where it reaches 8100 X 10"" 8 . An electromagnetic wave, acting upon a plate cut from a piezoelectric crystal, periodically changes its size, because of the electrostriction ' See P. Debye, Polare Molekeln, Leipzig, 1929. XX 139 MAGNETIC AND ELECTRIC PHENOMENA 361 associated with piezoelectricity, and sets up in it elastic oscillations. The amplitude is particularly large when the frequency of the wave is in resonance with the characteristic period of the elastic oscillations in the plate. Such piezoelectric resonators have received important technical applications in the last decades. According to eqs. (20.02), the deeper reason for the existence of piezoelectricity is the dependence of the pressure (or strain) on the polarization P. It is to be expected that the internal energy and entropy should then also depend on P and H. From (20.18) we can, therefore, conclude that piezoelectric substances are at the same time pyroelectric, i.e. their electric moment is influenced by a change of temperature. 1 It is here the place to make a few supplementary remarks about the theory of strong electrolytes (section 115) and to explain why we could not calculate the internal energy U directly but had to take the indirect way over the work function \I>. The reason is that the electric or magnetic energy of a system cannot always be regarded as a part of its internal energy. We have seen that such simple condi- tions obtain only in the case of atomic diamagnetism or of polarization of the Lorentz type, when the electromagnetic work does not depend on the thermal parameters. The opposite extreme are the perfect mag- netics and dielectrics in which U remains completely unaffected by the electromagnetic energy of the system. The strong electrolytes occupy an intermediate position. The addition U to the internal energy due to the ionic interaction should be calculated from the expression (17.64) of the work function V e by means of the relation . = U e - TS e or TJ ty _ T\ - - u * ~~ ^* l \ IT \9* / v The difficulty is, however, that * e depends on T not only explicitly but also implicitly through the dielectric constant D. In a similar way, the thermodynamic potential $ is connected with *, by the formula $ ^$% + P*V, or and here again the dependence of D on the volume is not sufficiently well known for a quantitative evaluation. 139. Thermoelectric phenomena. In section 109 we introduced the concept of the potential difference built up by thermoelectric action 1 For a discussion of these phenomena see: Geiger and Scheel, Handbuch der Physik, Vol. XIII. Berlin 1928. 362 TEXTBOOK OF THERMODYNAMICS XX 139 across a junction of two conductors. The discovery of the e.m.f. (electromotive force) of thermoelectricity was made by Seebeck in 1822. It was supplemented in 1834 by observations of Peltier's on a peculiar heat development due to electric currents flowing through thermoelectric junctions. Let / denote the current flowing from con- ductor (1) to conductor (2): in order to maintain the junction at con- stant temperature, it is necessary to impart to it, in unit time, the heat (20.42) where Hi2 is the so-called Peltier coefficient, depending on the nature of the conductors and on the temperature of the junction. IIi2 is positive (i.e. heat must be imparted to the system) when the current / has the same direction as that produced by the thermoelectric action of the junction. The sign of the Peltier coefficient is reversed when the cur- rent flows in the opposite direction (1X21= IIi2). This fact gives the experimental possibility of separating it from Joule's heat, which does not depend on the direction of the current. From the theoretical point of view, the same fact is taken as an indication that the develop- ment of Peltier heat is a reversible process, amenable to thermodynami- cal treatment. The final step was the discovery (1854) by William Thomson, later Lord Kelvin, of the following phenomenon: when the current / flows through a wire of homogeneous material and cross-section, but of non- uniform temperature, heat must be supplied in order to maintain the temperature gradient. To an element of the wire with the tempera- ture rise dT, there must be imparted in unit time the heat dQ = <rdTJ. (20.43) a is called the Thomson coefficient, and it is positive when the current flows in the direction of rising temperature. The Thomson effect is reversible, in the sense that dQ changes its sign when the direction of the current is reversed. The thermodynamical theory of thermoelectricity is due to Lord Kelvin. Let us consider the circuit of Fig. 62, consisting of two wires (1) and (2) whose junctions are kept at the respective tempera- tures T and T'. When the current J is sent through the (closed) cir- cuit, the total heat which must be imparted to it in unit time to keep the conditions stationary is, according to (20.42) and (20.43), (20.44) XX 139 MAGNETIC AND ELECTRIC PHENOMENA 363 From the point of view of the first law of thermodynamics, Q is the energy which maintains the circulation of the current. In the theory of electricity this energy has the expression Q - /, (20.45) where E is the e.m.f. (electromotive force) of the circuit, J being the electric charge flowing through any cross-section in unit time. Equa- tion (20.44) indicates how this energy is spent. There is a profound analogy between the conduction electrons streaming across a thermo- j unction and a gas forced through a cotton plug in the Joule-Thomson process (section 15): the energy which the electron gas gains or loses is equal to the difference of its heat functions in the two conductors, Q = AX. The same is true for the Thomson effect (20.43): dQ = dX. If the molal heat function is denoted by x and the molal electron charge by/ = F = n A e, eqs. (20.42) and (20.43) can be written _ (X2 - Xl) Idx /IA,HC\ H 12 = - j , *-- (20.46) Combining (20.44) and (20.45), we can write /T' ((72 - ai)dT, (20.47) * as the expression of the first law of thermodynamics. In particular, when the temperature difference of the junctions is infinitesimal, T - r = dT g-f? + -* (20.48, Further relations can be obtained from the second law, assuming that the heat items of eq. (20.44) are imparted to the system in a reversible way and that the thermoelectric effect satisfies the entropy principle separately. Let us follow a portion of the electron gas, responsible for the current, in its motion around the circuit back to the initial position. The process is cyclic in that the final state is identical with the initial and the total entropy change in it must be zero: 2 AS = SAQ/r. The items of heat A() imparted to the electron gas are the same as in eq. (20.44). We find, therefore, (20.49) 364 TEXTBOOK OF THERMODYNAMICS XX 139 or taking again T T' - dT, 02 dT The ratio Uvt/T is called the thermoelectric power of the pair of conductors. There follows from (20.48) dE """ rp 9 (20.50) * 2 ffl T dT\Tj * dT 2 ' Finally, the third law of thermodynamics has also a bearing on thermoelectricity. Since Iliz/T represents the change of entropy which the electrons (or other carriers) undergo in passing across the junction, Nernst's postulate (13.07) gives directly lim r . (H/r) - 0. (20.51) Hence the integrated form of (20.50) becomes / T (7 ~ (T 2 ^ * dT. (20.52) 1 This equation implies, of course, that the integral must be con- vergent, so that lim r . (^2 0*1) = 0. But not only the difference must vanish (in the limit T = 0), but also <n and a* individually. In fact, we see from (20.46) that <r has the physical meaning of the specific heat of the carriers: eq. (13.15) requires, therefore, lim r ..o <r *= 0. (20.53) The empirical representation of the thermoelectric e.m.f. is usually given by the equation E - (at + 10- 2 JW + 10-* $d] X 10- volt, (20.54) supposing that one of the junctions is at C, the other at f C. A few data are given in Table 55. From (20.54) we find (neglecting c) a - (II/TVo, ** - n - bT. It is interesting to note that the coefficients a and b in some cases have opposite signs. Because of the parabolic character of the formula (20.54), as the temperature is increased the e.m.f. rises to a XX 139 MAGNETIC AND ELECTRIC PHENOMENA 365 maximum, then declines again, and finally reverses its sign. For instance, in the case of Cd-Pb the reversal takes place at about 300 C. TABLE 55 THERMOELECTRIC E.M.F. Range in C (1) (2) a b c From To Ag Pb 200 3.34 0.85 Al Pb 200 0.50 0.17 Au Pb 200 2.90 0.93 Bi Pb 100 -74.4 3.2 Cd Pb 200 2.62 -1.79 Cu Pb 100 2.76 1.22 Fe Pb -230 100 16.65 -2.97 -26.75 Ni Pb 200 -19.07 -3.02 Pt Pb 100 - 1.79 -3.46 12.6 Sb Pb 100 35.6 14.5 W Pb 100 1.59 3.4 Zn Pb 250 3.18 0.11 Very large thermoelectric e.m.f.'s are observed in circuits composed of semi-conductors (Table 56). TABLE 56 THERMOELECTRIC E.M.F. IN SEMI-CONDUCTORS Range in C (1) (2) a b From To Bi 2 0, Pb 500 800 1946 -186 Co,0 4 Pb 200 1200 629 - 22.1 Cr 2 0a Pb 950 1285 - 704 43.2 CuO Pb 170 850 - 1029 171 PbO Pb 250 390 -48300 11720 Si Pb -200 350 - 408 - 47 ZnO Pb 355 1350 - 735 25.$ As to the Thomson coefficients, their temperature dependence can be represented by the empirical formula <r - [a + 10- 2 # + 10-V] X 10-* volt deg- 1 . (20.55) 366 TEXTBOOK OF THERMODYNAMICS XX 139 It is apparent from Table 57 that the Thomson effect is positive in some metals and negative in others. TABLE 57 THOMSON EFFECT Range in C Metal a ft 7 From To Ag -123 127 -1.17 -0.50 Al - 13 119 -0.04 +0.475 Au -100 103 -1.49 -0.44 Bi + 25 32 +6.76 +2.8 Cu - 60 127 -1.42 -0.74 Fe - 51 115 4-4.00 +8.4 Pb -153 117 +0.61 +0.221 -0.38 Pt - 72 128 +9.10 -0.475 +4.75 Zn -173 40 -2.74 -1.15 Only the order of magnitude of the constants in Tables 55, 56, and 57 is significant: the thermoelectric power is highly sensitive to the purity and mechanical treatment of the substances and shows con- siderable variation in different samples of the same material. Never- theless, it is possible to test the thermodynamical relation (20.50) by carrying out on the same sample the measurement of the e.m.f. and the calorimetric determinations of II and a. The agreement is very good, as appears from the examples in Table 58. TABLE 58 TEST OF THERMODYNAMICAL RELATIONS n dE (T2 <TI d*E Metals T dT Reference T dT 2 Reference In 10- " fl volt In 10- * volt Cu Pt ... 3 66 3.67 (1) 3 3 6 (2) Cu Fe -10.16 -10.15 (1) 2 2 7 (2) Cu nickeline 18.90 18.88 (i) 6.01 6.28 (2) Cu German silver . 25.25 25.22 (i) 3.92 3.79 (2) Borelius, Ann. Physik 56, p. 388, 1918. * Berg, Ann. Physik 32, p. 477, 1910. XX 139 MAGNETIC AND ELECTRIC PHENOMENA 367 Recent measurements of the Thomson coefficient at very low tem- temperatures l showed that <r vanishes in supraconductors and con- forms to eq. (20.53) in non-supraconductive metals. Let us now find the connection between the Peltier coefficient Hi2 (or the thermoelectric power Ui2/T) and the difference of potentials 12 1 122 which we calculated in sections 109 and 112 from the electron theory of thermoelectricity. We must bear in mind, however, that in its [present form this theory cannot yet account for the detailed phenomena and gives correctly only the order of magnitude of the thermoelectric power. The results of Chapter XVII refer to open chains, but, if we take the case of a closed circuit (Fig. 62) with a very large resistance, the current will be very weak and potentials com- pensating the thermoelectric forces will be FIG. 62. Thermoelectric built up, practically, to the same extent as couple, in the open circuit. Starting from eq. (20.46), we can apply to the heat function the expressions it has in equilibrium. We begin with the case when the electron gas is non- degenerate obeying eqs. (17.01) and (17.02) of the classical perfect gas. The heat function has then the expression x = c p T + Wo + /ft, so that we obtain, taking into consideration (17.13), H 12 = 12 2 -fli = log-- (20.56) e z\ It was mentioned in section 109 that this expression applies only to semi-conductors. As k/e = 86 X 10 ~ 6 volt deg~ ! , this would imply that log (z 2 /zi) for some of the substances of the Table 56 is of the order of magnitude 20, whence z 2 /zi would be about 5 X 10 8 . While it is within reason that a semi-conductor should have an electronic density 10 8 times smaller than that of lead, the signs of a in Table 56 offer a greater difficulty. They seem to indicate that in some of the semi-conductors the carriers of the electric current have a positive charge. As we shall see in the next section this difficulty is not quite so baffling as it was a few years ago. Turning to the case of the degenerate electron gas, we shall make use of the equation ^2 = ^1, from which the results of the section 112 were obtained. In view of the relation v = x Ts, we can write 1 Borelius, Keesom, and Johansson, Comm. Leiden, 196a; Borelius, Keesom, Johansson and Linde, ibidem, 2l7a, b, c. 368 TEXTBOOK OF THERMODYNAMICS XX 140 /Hi2 X2 ~ xi = T($2 si). Hence with the help of eq. (17.33) we obtain the formula due to Sommerfeld _ 1112 " 2 ^ e ) Lft,2 fltiJ ' or ~ = 3.67 X 10~ 8 T\ -^- - -^- ) volt deg- 1 . (20.57) 1 \\li2 A*l/ For the combination lead-silver the formula (with the data of Table 55) gives U 12 /T = 0.86 X 10~ 8 volt deg- 1 , which is of the right order of magnitude. However, the electron theory, in its present form, is incomplete as it does not account for the phenomena quanti- tatively. According to eq. (20.50), the e.m.f. is obtained from the thermo- electric power by the relation E /T ~f ' Intimately connected with thermoelectricity are the so-called galvanomagnetic and thermomagnetic effects describing the influence of a magnetic field on the phenomenon of electric conduction. The scope of this book does not permit us to enter into their discussion, and we refer the reader interested in the subject to its analysis in a paper by Professor Bridgman. 1 140. Semi-conductors. The difference between metallic con- ductors and semi-conductors is not only quantitative but also qualita- tive. The conductivity of metals (in the range about C) has numeri- cal values between 10 15 and 10 17 <abs e-s and is inversely proportional to the absolute temperature Xccl/r, so that it has a negative temperature coefficient. On the other hand, in semi-conductors \ is only of the order 10 4 to 10 13 and its temperature dependence can be best represented by the law 2 X = Xoexp(-a/r) (20.58) with a positive temperature coefficient. Recent work on the quantum theory of electrons in metals 3 has 1 P. W. Bridgman, Phys. Rev. 24, p. 644, 1924. 1 W. Voigt, Krystallphysik, 1910; E. Engelhard, Ann. Physik 17, p. 501, 1933. * A. H. Wilson, Proc. Roy. Soc, 133, p. 458; 134, p. 277, 1931. XX 140 MAGNETIC AND ELECTRIC PHENOMENA 369 greatly contributed to our understanding of the peculiarities of semi- conductors. As a part of the argumentation can be given a thermo- dynamical guise, 1 we shall outline here its main ideas, even though the experimental data for testing the results are extremely meager. While treating electron clouds in metals in Chapter VII, we assumed that the electrons are free, actually implying in these words two different assumptions. In the first place, the free electron does not belong to any particular atom; it is shared by all atoms and can travel from one end of the conductor to the other. In the second place, we assumed that the energy levels of the free electrons can be calculated, as if the atoms were absent and as if the electron gas were contained in an empty box. It is of particular importance that the energy levels in our calculation were distributed practically continu- ously: any small energy increment could raise a few of the electrons to a higher level. We shall show now that electrons, which are free in the first sense, need not necessarily be free in the second, and that in this case they do not contribute to the conductivity, they are not conduction electrons. To fix our ideas let us consider two elements, the one divalent, the other monovalent. The free divalent atom has two valency electrons which, in their normal or 5-states, have the same energy and differ only by the orientation of their spins. There are no other states of the same, or nearly the same, energy available: just as many 5-states as electrons. The next higher electron levels belong to the first excited or -state and have a considerably larger energy. It is differ- ent with the monovalent atom] its single valency electron can assume a new quantum state without change of energy, simply reversing its spin: there are twice as many 5-states as electrons. When Z atoms are brought together to form a crystal lattice, we know from general mechanical principles the following two facts. (1) The number of quantum states remains the same. The Z atoms had in their free condition 2Z 5-states; therefore, the number of 5-states in the crystal is also 2Z. (2) While the 5-states of the free atoms had all the same energy level, this level is split up in the crystal owing to the forces of interaction between the atoms. In general it is replaced by 2Z slightly different levels lying close together. In short, to the 5-states of the free atoms there corresponds in the crystal an 5-band of closely crowded levels (Fig. 63a). The distribution of electrons over the states of the band is very different in the cases of monovalent and divalent elements. In J R. H. Fowler, Pfays. Zs. Sovjetunion 3, p. 507, 1933. 370 TEXTBOOK OF THERMODYNAMICS XX 140 monovalent elements there are only half as many electrons as quantum states. Because of the exclusion principle, there can be only one electron in each level so that, at T = 0, the electrons would fill the lower half of the band. At finite temperatures, they can spread out into the upper half. These are the conditions of freedom, in the second sense mentioned above, which formed the basis of the electron theory of section 111. It is immaterial that the band has an upper limit because the electron gas is under ordinary conditions largely degenerate and the overflow above the lower half is but small. On the other hand, in divalent elements, there are in the s-band exactly as many levels as electrons. Supposing that the excited states (forming (a) FIG. 63. Position of s- and -bands in (a) insulators, (6) metallic conductors. the />-band) lie so high that they are unattainable at ordinary tem- peratures, each state is occupied by an electron: the electrons are restricted to their positions in the band, they are " bound electrons 1 ' in the second sense. 1 What bearing have these facts on the problem of electric conduc- tion? From the point of view of the quantum theory, the mechanism by which an electromotive force produces a current is redistributing the electrons over the quantum states. According to the above analysis, such a redistribution is possible in the case of a monovalent element because it possesses plenty of unoccupied levels of accessible energy. A monovalent element is, therefore, always a conductor (provided its electrons are "free in the first sense"). On the other hand, a divalent crystal may be either an insulator or a conductor, even when its elec- trons are shared by the whole system. When the p-band of excited levels lies so high as to be inaccessible (Fig. 63a) the electrons com- pletely fill the s-band, they are " bound electrons" and their redistri- bution is impossible: the crystal is then an insulator. On the con- 1 Interchanges of electrons need not be considered since they do not lead to a new state of the system. XX 140 MAGNETIC AND ELECTRIC PHENOMENA 371 impurity level trary, when the p-band lies low and partially overlaps with the 5-band (Fig. 63&), it can take the overflow of electrons so that they can be redistributed: the crystal is a metallic conductor. In this way, the quantum theory gives us a simple explanation of why some substances are metals, others dielectrics. It has, more- over, the great advantage that the several kinds of semi-conductors can be easily fitted into the picture. (A) Intrinsic semi-conductors. Suppose that the p-band of excited states lies neither very high nor so low as to overlap with the 5-band, but has an intermediate position (Fig. 64a). We call Ae the energy difference between the lowest />-levels and the highest 5-levels. We consider the case that, at T = 0, the 5-band is completely filled with electrons (as in the preceding ex- ample of divalent elements). At higher temperatures, there is the possibility of an electron spontane- ously leaving the 5-band and rising to one of the ^-levels. We assume that Aeis small enough for the prob- ability of such a rise being appre- ciable at room temperature, so that the number of electrons belonging to the p-band is z_ per unit volume of the crystal. These Z- electrons are free carriers of conductivity because there are plenty of unoccupied states in the vicinity of their energy levels. Because of their removal, there appears in the 5-band an equal number (z+ = 3__) of unoccupied levels or "holes." It was pointed out by Peierls l that the formulas of the quantum theory can be given the following interpretation. The "holes" behave in every way as if they were free carriers similar to the electrons but having the opposite (positive) charge. In this way we arrive at the following model of an intrinsic semi-conductor: it contains two clouds of conductive carriers, the cloud of free negative electrons (G_) and the cloud of "free (positive) holes" (G+). The "bound electrons" (G B ) of the 5-band can be left out of consideration as far as conduction phenomena are concerned. A free electron of the p-band may, of course, drop into a vacancy ("hole") of the 5-band and so become a bound electron. This process can go also in the opposite direction and can be described in the sym- 1 R. Peierls, Zs. Physik S3, p. 255, 1929. Compare also: Bronstein, Phys. Zs. Sovjetunion 2, p. 28, 1932. (a) (b) FIG. 64. Position of 5- and />-bands in (a) intrinsic, (b) impurity semi- conductors. 372 TEXTBOOK OF THERMODYNAMICS XX 140 bols of a chemical reaction as GL + G+ GB = 0. The equation of equilibrium is, therefore, according to (17.09), ^- + (p+ <pB = 0. The concentrations of free electrons and of free holes are very small so that both clouds can be regarded as non-degenerate in the sense of Chapter XVI. We shall assume, as a first approximation, that they obey the equations of classical perfect gases. On the other hand, the bound electrons are completely degenerate, and their thermodynamic potential is reduced to a constant (section 111), namely the mean energy level from which the transitions take place. If we measure all energies from this level we can ignore <f> B altogether and have the same conditions as in the case of equilibrium of positive and negative electrons. Therefore, we can apply eqs. (19.28) and (19.29) of section 128 with the only difference that here the heat of reaction is Ae l *_= 4.9 X 10 15 r % exp (-11 600Ae/2r) (20.59) (if Ae is expressed in electron-volts), a reasonable result as we shall see below. 2 Since the electrons and "holes" are treated as classical perfect gases, the electric conductivity can be computed by means of the Drude-Lorentz formula which was derived precisely under this assumption : X = %e 2 (2irM e kT) " ^lz. The mean free path / is known to be inversely proportional to the temperature, / = feo-293/r (feo being the value at 20 C). Substituting the numerical values of M e and jfe, we can write, therefore, X= 1.01 X10 6 r-*/2os. (20.60) Using the expression (20.59), and seeing that there are two kinds of carriers, we obtain X - 1.0 X 10 21 /2oexp(-5800Ae/r). (20.61) As /2o is of the order 10 ~ 6 cm, the expression is of a reasonable order of magnitude when Ae is of the order 0.1 e-v. (B) Impurity semi-conductors. There exist substances which are insulators in their pure states but semi-conductors when impure. Often as small an admixture of foreign atoms as 1 in 10 6 , or even less, 1 Already J. Koenigsberger (Jahrb. Radioaktiv. 4, p. 158, 1907) suggested an exponential dependence of 2 on 1/3T. 2 A. H. Wilson (loc. cit.) does not treat the electrons as a perfect gas. His formula for * is a little more general, differing from ours by a factor k/ft, where ft measures the tightness of binding of the electrons and is a characteristic constant of the material. XX 140 MAGNETIC AND ELECTRIC PHENOMENA 373 is sufficient to produce an appreciable conductivity. Cuprous oxide (Cu2O), the only semi-conductor investigated with modern accuracy, belongs to this class. The impurity on which the conduction of Cu2O depends is free oxygen: it has been shown 1 that the conductivity of this material is lowered by glowing in vacuo and increased by heating in an oxygen atmosphere. The interpretation in terms of our model is as follows: conduction electrons in the p-band are supplied by the impurity atoms. The distance between the p- and $-bands is so large that in the pure substance the p-band would be empty. On the other hand, each impurity atom possesses an electron which can be more easily separated because its energy level is only little (Ae) below the lowest levels of the p-band (Fig. 646). We denote by 0- the number of conduction electrons in the p-band, by z+ the number of ionized, and by z of neutral, impurity atoms. In analogy with the preceding case, the equation of reaction can be written GL + G<+ d = 0, and the equation of equilibrium ^?- + <?>+ ^< = 0. We regard again the conduction electrons as a classical perfect gas and use for it the expression (19.04). As to (pi = u Ts + pv of the impurity atoms, they must be considered as stuck in the crystal lattice and having a negligible thermokinetic motion: 2 therefore, u = o, po^ 0, while their only entropy is the entropy of mixing given by eqs. (15.03) of section 98 or (1 1 .39) of section 80, so that s t -+ $< = -R log (z+i/zt). Substituting the numerical values of the chemical constant and of Jfe, logio(*-*<+/*<) - 15.69 + f logio r - 5040Ae/r, or *- * 7.00 X lOVSr* exp(- 5800Ae/jT). (20.62) Vogt 3 and Engelhardt measured the coefficient of the exponential (i.e. 7.00 X 10 7 \/ZiT^ in our theory) and obtained values between 1.5 X 10 17 and 9.4 X 10 19 in different samples. This would give for Zi numbers from 1 X 10 16 to 4 X 10 20 , or from 4 X 10~ 7 to 2 X 10~ 2 impurity atoms per molecule of Cu2O. The measured Ae was about 0.65 e-v. The conductivity, as obtained from (20.60), is X - 7.0 X 10 12 Zi"r-*l2Q exp (- SSOOAe/T). (20.63) (C) Light-sensitive semi-conductors. There exist substances whose conductivity increases many times when they are exposed to light, 1 R. Engelhardt, he. cit. * A more rigorous statement is that the impurity atoms have the same thermo- kinetic motion whether they are ionized or not, so that the differences in u and pv drop put of the equation. W. Vogt., Ann. Physik 7, p. 183, 1930. 374 TEXTBOOK OF THERMODYNAMICS XX 140 the best-known example being selenium. This case also fits readily into the model of Fig. 63a. Here the p-band lies so high that the number of its electrons is negligible in the dark. However, when light of sufficiently high frequency is absorbed by the crystal, it raises electrons into the -band by photoelectric action. The return into the s-band requires a radiative transition and may be a very slow process, thus assuring an appreciable density of conduction electrons. One of the most important aspects of the theory outlined above is that it provides a new type of positive carriers of electric conduction : the ''free holes" of the 5-band. We have considered under (B) impuri- ties which supply free electrons. There may exist other impurities which have an affinity for electrons and are able to bind them, thus producing "holes". In this way some conductors may have an excess of negative, others of positive, carriers. More important still, in the more accurate theory the electrons and "holes" are not regarded as entirely free. The binding of the two kinds of carriers is different, and, for this reason, there is an asymmetry in their actions, even if they are present in equal numbers. It was always emphasized by E. H. Hall that the peculiarities of the effect bearing his name could not be explained without an assumption of this sort. The same hypothesis would be helpful in accounting for the great variations in the sign of the thermoelectric e.m.f. and of the Thomson coefficient. CHAPTER XXI THE DIRECTION OF THERMODYNAMICAL PROCESSES 141. General remarks. General rules predicting the direction in which a process is influenced by outer forces are very desirable from a practical point of view. They may offer a quick orientation in the workings of new experimental arrangements and so facilitate their theoretical and practical understanding. This applies to thermo- dynamics even more than to other parts of physical science because the thermodynamical treatment is more abstract and formal and less accessible to visualization. The attempts to establish such rules, with respect to thermodynamical processes, were all influenced, directly or indirectly, by the famous principle of electrodynamics enun- ciated in 1833 by Lenz l : "When a force acting on a primary electric current induces a secondary current, the direction of this secondary current is such that its electrodynamical action opposes the acting force." An analogous principle of thermodynamics was formulated by LeCha teller 2 in 1884 and in an extended form by Braun 3 in 1887. Their procedure was essentially inductive (although Braun included a few formulas in his paper) : they reviewed a large number of examples which they tried to state in a form roughly analogous to Lenz's princi- ple. They claimed that all these examples could be regarded as special cases of a general rule which they proceeded to formulate. However, this formulation was so vague that it was impossible to apply the rule without ambiguity. This fact was first pointed out by Raveau 4 and Ehrenfest 5 and has been generally accepted since. It will serve no purpose to mention here the form they gave to their principle, but it is well to reproduce two of the examples from which it was derived. 1 H. F. Lenz, St. Petersburg Acad. of Sci. 29, XI, 1933; reprinted: Ann. Phys. u. Chem. 31, p, 483, 1934. 2 H. LeChatelier, Comptes Rendus 99, p. 788, 1884. 1 F. Bra'un, Zs. phys. Chem. 1, p. 269, 1887; Ann. Physik 33, p. 337, 1888. * M. C. Raveau, J. Phys. 8, p. 572, 1909. ' P. Ehrenfest, J. Russ. Phys. Soc. 41, p. 347, 1909; Zs. phys. Chemie, 77, p. 227, 1911. 875 376 TEXTBOOK OF THERMODYNAMICS XXI 142 (A) Influence of pressure on solubility. Substances whose solubility increases with pressure dissolve (at constant temperature) with volume contraction; those which have the opposite pressure dependence dis- solve with volume dilatation. This fact (although not new) was rediscovered by Braun a little earlier and was the starting point of his considerations. (B) Influence of temperature on compressibility. Suppose that we compress a gas, increasing the pressure under which it stands by the small amount A. We can do this in two different ways: (1) keeping the temperature constant (T = const) with the help of a heat bath; (2) "leaving the gas to itself" (i.e. adiabatically, 5 = const). The compressibility is smaller in the second case dp dp (21.01) We selected the two examples so as to show the heterogeneity of the cases considered by LeChatelier and Braun. The first involves only three parameters of the system: pressure, volume, and concentra- tion of the saturated solution. In the second we have an interplay of four parameters: pressure, volume, temperature and entropy. The closer analysis will show that the two cases are also quite different in their mathematical characteristics. In fact, Braun himself dis- cussed the two types represented by them separately but claimed, in the end, that they are reducible to the same principle. Without maintaining that all possible examples belong to the types (A) or (B), we assert that the principle of LeChatelier-Braun must be resolved into at least two different and unconnected rules which we shall analyze in the next two sections. 142. Displacement of equilibrium involving transformation of matter. It will be noticed that example (A) of the preceding section is only a special case of the first of the two laws stated at the end of section 59. These laws, in their turn, are consequences of eqs. (6.51) and (6.53) derived from the fundamental conditions of equilibrium, as stated in section 40, namely: dU = 0, (21.02) with the subsidiary conditions 55 =0, 5F = 0. (21.03) The gist of the argument which led in Chapter VI to the explicit equilibrium conditions is as follows. The variables S and V possess XXI 142 DIRECTION OF THERMODYNAMICAL PROCESSES 377 the additive property. In a heterogeneous system the total entropy and the total volume are sums of the contributions of the several phases S = S ( , V = 2] F<. (21 .04) i J As was shown in section 40, this property together with the condi- tions (21.02) and (21.03) immediately leads to the uniformity of the parameters T and />, conjugate to S and V (21.05) The generalization for systems depending on other variables than 5 and V is obvious. For instance, in Chapter XX we discussed mag- netization and electrostatic polarization. We considered systems placed in homogeneous magnetic or electric fields of the strength H or E (i.e. being homogeneous and having this strength when the sys- tem is not there). The element of work was denoted by HdM 9 EdP, where M and P are the magnetic and electric moments of the system in the direction of the fields. When the system consists of several phases, the equations analogous to (21.05) > = E, are a matter of course since all the phases are placed in the same homogeneous field. Moreover, the moments have the additive prop- erty M = Z) Jf ( >\ P =><'>. i J Finally, it is known from the electromagnetic theory that in equi- librium 5M = 0, 6P 0. The set of equations governing the properties of these variables is formally identical with that applying to the parameters 5, V. Passing to the general case we assume a heterogeneous system which has the following properties: (1) Each phase (j) is described by the variables which have the additive property EXi 01 - JT* (21.06) 378 TEXTBOOK OF THERMODYNAMICS XXI 142 In particular let Xo (;) stand for the entropy 5 (j) . (2) The differential of the internal energy has the form y{fldX{*>. (21.07) We call the coefficient yi (j) the generalized force conjugate to the additive variable Xi ( >\ The temperature T (;) = y$ (j) is classed as a generalized force j 1 the pressure p must be counted as a negative force (p = - yi (i)). (3) The equilibrium of the system is determined by 8U = 0, with the subsidiary conditions Mi** 0, (/ = 0, 1,. .. w). (21.08) (4) The chemical composition and mass of each phase are deter- mined by the mol numbers Nk { ' } of its components (k = 1, 2, . . . 0). The variations dN k (i) are given by the reactions possible in the system (section 42): dN k <i} oc ^ (y) , S^C^G* ^ 0. (21.09) The explicit equilibrium conditions can be obtained exactly in the same way as in section 40 where a less general case was treated. The procedure need not be repeated here but only indicated in its bare out- line. The problem is broken up in two partial ones: first, changes of composition and mass are excluded (6JV* (y) = 0). The variation of the internal energy has then the form C 'W>=0, (21.10) j i with the subsidiary conditions (21.08). This gives immediately (as in section 40) yz (y) =yz, (/ = 0,1, . . .n). (21.11) The generalized forces are uniform throughout the system. To treat the second partial problem, we introduce the generalized thermo- dynamic potential * (21-12) 1 As compared with (2.02) the coefficients y are defined with the opposite sign. XXI 142 DIRECTION OF THERMODYNAMICAL PROCESSES 379 whose differential is, according to (21.07), d& = - *,< W>, (21.13) i so that it must be regarded as a function of the variables We denote the partial thermodynamic potentials, as before, When the generalized forces yi are kept constant, the conditions (21.02) and (21.08) can be replaced by a$ = o. for the same reasons which were adduced in section 36. We consider only variations in which yi = const; the condition 6$ = is then valid and can be written in the form (; V ;) =0. (21.15) j t The partials of A< with respect to the generalized forces are, accord- ing to (21.13), (21.16) uyi where is the change of the variable Xi (for the whole system) in the trans- formation or reaction (21.09) Apart from the generalized definition of 3> (/) the condition (21.15) is precisely the same as (6.50) of section 42. The consequences derived in Chapters VII to XI are based on this condition and remain true. In particular, applying the generalized form of the thermodynamic potential to the expressions (9.13) or (11.15) of the equilibrium con- stant, we find from (21.16) 9 log* **' a* " RT (2L17) The increase of the generalized force yi produces a displacement of equilibrium attended by a process (transformation or chemical reaction) in which the change of the conjugate variable AXi is positive. 380 TEXTBOOK OF THERMODYNAMICS XXI 143 In addition to example (A) of Braun's (preceding section), we shall illustrate this principle by considering the influence of a magnetic field on the equilibrium: yi = H, X t = M t The increase of the magnetic strength of field produces a displace- ment of equilibrium in which the magnetic moment of the system (in the direction of the field) is increased. The more paramagnetic phases and components increase their mass at the expense of the less para- magnetic. Since supraconductors are classed as diamagnetics, this includes also the equilibrium between metals in the normal and the supraconductive states: a magnetic field makes the normal phase increase at the expense of the supraconductive. These phenomena are manifestations of the same causes which make an inhomogeneous magnetic field suck in paramagnetic and push out diamagnetic bodies. In a similar way, an electric field favors stronger dielectrics at the expense of weaker ones. 143. The restricted LeChatelier-Braun principle. (Influence of secondary forces.) Example (B) of section 141 is a true analogue of Lenz's principle: when the temperature is allowed to change, it opposes the acting force of compression. This is in keeping with the expres- sion (21.07) in which the temperature is treated as a generalized force. Another example of the same sort is the longitudinal compression of of a solid elastic bar with uniform cross-section. It can be compressed by a longitudinal force pi in two ways: (1) keeping the forces (of pressure) on the lateral surfaces constant (p2 = const) ; (2) keeping the lateral dimensions constant (/2 = const) and preventing a bulging out of the cross-section. If the length of the bar is denoted by /i, it is known from the theory of elasticity that dpi < <2 dpi (21.19) Here, too, the secondary (lateral) force, if permitted to change, opposes the acting (longitudinal) force. As it is sometimes said, the secondary forces help the system to resist the acting force. The analytical connection of the examples of this type with the foundations of thermodynamics was cleared up by Ehrenfest. 1 We shall call the rules pertaining to class (B) the restricted LeChatelier- 1 See footnote on p. 375. XXI 143 DIRECTION OF THERMODYNAMICAL PROCESSES 381 Braun principle. Ehrenfest himself refrained from formulating the principle, 1 but the enunciation given below is based on his analysis. Unlike our treatment in section 142, we may restrict our con- siderations to homogeneous systems described by the parameters Xo, Xi, . . . X n so that the differential of the internal energy has the expression dU = y Q dX + yidXi + . . . + y n dX n , (21.20) defining the generalized forces yo 9 yi, . . . y n - As dU is an exact dif- ferential, the following reciprocity relations hold: In addition to them, we shall assume the validity of the inequalities ~ ^ 0, (21.22) U**- I which are often called conditions of stability. If the variables X are those of section 142, the first of these conditions is QT/QS ^ 0, which expresses the fact that the temperature rises when heat is imparted to the system. As to the condition relating to the volume (X\ = F), it must be borne in mind that y\ = />, so that dp/QV 3* 0. In fact the sign generally given to the hydrostatic pressure is an anomaly among the generalized forces. In hydrodynamics and elasticity the tension, and not the pressure, is taken as positive. The inequality dp/dV ^ was discussed at length in section 5, where it was shown to be a necessary condition of stability. The same applies in all cases where the generalized force yi can be envisaged as different in the system and in the environment. If the condition (21.22) were not fulfilled, the difference would tend to increase and a stable equilibrium between system and environment would be impossible. However, this argument does not apply to cases like magnetization and polarization (sections 134 and 138) where the generalized forces are defined as H and E 9 as they would be in the absence of the system. Here the validity of the conditions (21.22) is simply a coincidence, as far as thermodynamics is concerned. As a matter of fact, diamagnetic bodies form an exception to the rule: for them QH/QM < 0, if M is the mag- netization in the direction of the field. 2 Fortunately, the principle we 1 The reason for this was that he did not segregate class (A) into a separate principle and was looking for a single rule embracing all cases. Planck (Ann. Physik 19, p. 759, 1934) also failed to draw the distinction. * It would not help to define yi * M because this would reverse the sign of i in eq. (21.20). 382 TEXTBOOK OF THERMODYNAMICS XXI 143 are going to formulate need not be amended because of diamagnetics (see below). Sometimes it is said that the inequalities (21.22) hold because the internal energy U has its minimum in equilibrium. But such a conclusion may not be drawn when the minimum is contin- gent upon subsidiary conditions of the type (21.03). In this case, the equations of stability (21.22) are not, in general, invariant with respect to transformations of coordinates. j All we shall need for the proof of the principle are the relations (21.20), (21.21), and (21.22). It is immaterial whether the parameters Xi have the additive property or not; therefore, we drop the requirement of additivity. We are now ready for the enunciation. (I) Suppose that the system is described by the variables Xo, . . . X n and the generalized forces yo, . . . y n for which eqs. (21.20) and (21.21) and the inequalities (21.22) are satisfied. (II) We keep all the variables X constant except two, Xi and Xi, and we give a primary increment dyi to the force yi which acts directly on its conjugate variable Xi. We let this increment change Xi in two different ways: (a) keeping the (secondary) force y^ constant (yi = const) ; (6) keeping the variable Xi constant (Xi = const) and allowing the force yi to change. (III) We assert that the primary increment 8yi is less effective in case (b) (21 . 23) In other words t the secondary force yi, if permitted to act, opposes the primary force yi. The proof is simple: the generalized forces yi and y^ as all the coefficients of the expression (21.20), are functions of the variables X. But since only Xi and Xi are allowed to change, we need to bring in evidence only these two parameters y, - yi(X lt X<), yi - y(jr, f Xi). (21.24) We can resolve the second equation with respect to X < Xi-Xi(Xi, yi ). (21.25) Now dyi/dXi is the reciprocal of QXi/dVit so that the condition (21.23) can be also written XXI 144 DIRECTION OF THERMODYNAMICAL PROCESSES 383 According to the rules of partial differentiation dyi i.e. yi in (21.24) must be differentiated in so far as it contains explicitly and in so far as it contains it through the medium of From the second eq. (21.24) we find, as in section 3, We substitute this, taking into account (21.21) . (2127) V ' ' This finishes the proof of the inequalities (21.26) and (21.23) since all three partials (dyi/dXi) v ., (dyi/dXi) Xi , (dXi/dyi) Xt are positive because of (21.22). It will be seen that diamagnetic bodies do not contradict the principle although they do not satisfy the conditions (21.22). In fact, for them Xi = M, yi = II and (dM/QXi) = 0, since the mag- netic moment depends only on //, as was pointed out in section 134. For diamagnetics the second term on the right side of (21.27) vanishes and they fulfill the limiting case of (21.23) corresponding to the sign of equality. It is conceivable, of course, that applications may arise in which the stability conditions (21.23) are not satisfied any more than in diamagnetics but which are not so innocuous and break through the law (21.23). It is obvious how the principle can be extended to take care of such contingencies: one must write the inequality (21.23) for the absolute values of the partials and must make the sense (< or >) depend on whether (dyi/QXi) and (Qyt/QXi) have the same or opposite signs. 144. Intensive and extensive quantities. The principle of LeChatelier-Braun is sometimes brought in connection with the division of physical variables into " Quantitaeten " and " Inten- sitaeten " proposed by the defunct school of energeticists. 1 This nomenclature is still used by German and Dutch writers and has some relationship to the English terms: extensive and intensive quan- tities. As the reader is likely to come across both designations in his l Main proponents: Mach, Ostwald, Helm. Compare G. Helm, Energetik. Leipzig 1898. 384 TEXTBOOK OF THERMODYNAMICS XXI 144 study of foreign and domestic literature, it will be well to explain here, briefly, their similarities and dissimilarities. As defined by Tolman, 1 extensive quantities are those which have the additive property, for instance, the variables Xo, . . . X n of section 142. Every quantity which is not additive is called by him intensive. The division is intended to be exhaustive so that every variable must fall into one of these two classes. It is possible, and often convenient, to use as parameters describing the system not the variables X but simple functions of them (like the reciprocals 1/X or the squares X 2 ) which are not additive. In this case, the system would be described only by intensive variables. It must be said, however, that Tolman's definitions are not universally accepted and that some authors use the term intensive in the same sense in which we have used the words specific quantity in section 39 (i.e. a homogeneous function of the degree zero in the mol numbers). On the other hand, the classification into Quantitaeten and Inten- sitaeten is not meant to be exhaustive. They are exceptional or preferred variables, and a parameter chosen at random need not belong to either of the two groups. Ostwald and Helm give no sharp definition, saying only that the element of work (section 7) can be represented as a sum of terms, each the product of a " Quantitaet " and an "Intensitaet". It is safe to say that the additive variables X of section 142 (i.e. the extensive quantities) belong to the " Quanti- taeten " and the generalized forces y, conjugate to them, to the " Intensitaeten". But it is vague what other parameters (if any) may be classed in these groups. The only practicable suggestion seems to be that of Mesdames Ehrenfest-Afanassiewa and De Haas-Lorentz, 2 who propose to restrict these terms to such variables and conjugate generalized forces which leave invariant eqs. (21.20) and (21.22) of the preceding section. 1 R. C. Tolman, Phys. Rev. 9, p. 234, 1917. 1 T. Ehrenfest-Afanassiewa and G. L. DeHaas-Lorentz, Physica 2, p. 743, 1935. CHAPTER XXII LIMITATIONS OF THERMODYNAMICS 1 145. The statistical point of view. The statistical interpretation of the entropy concept was treated in section 30. It was stated there that the entropy of a thermodynamical system can be brought in con- nection with its probability P by means of Boltzmann's principle (4.67) 5 = klogP. (22.01) While we emphasized in section 30 the parallelism between the thermodynamical and the statistical points of view, we shall dwell now on the discrepancies between them which have produced in the last decades a profound change in the outlook of science upon the theory of heat and, especially, upon the second law. The identification of entropy and probability involves the following difficulty. According to the second law, the entropy of an adiabatically isolated system can never decrease: AS ^ 0. If we leave such a system to itself, its entropy will either increase monotonically or remain stationary. On the other hand, the probability P shows a different behavior. Owing to the interplay of atomic and molecular movements and forces, the system undergoes, continuously, small but erratic changes. In general they occur in the direction of more probable states. However, this is only an average effect: it lies in the nature of the concept of probability that the less probable states are not totally excluded but only less frequent in their occurrence. Therefore, the probability of a system will, occasionally, decrease. In fact, it was rigorously proved by Poincar 2 that a finite system, subject to the Hamiltonian equations of dynamics, returns again and again to any state through which it once has passed. When Boltzmann's principle as expressed in eq. (22.01) was first advanced, scientific thought divided itself into two schools. One school 1 With the kind permission of Yale University Press parts of this chapter were patterned after the exposition by P. S. Epstein, Commentary on the Scientific Writ- ings of J. W. Gibbs, Article 0, Section 16. * H. Poincare, Acta Mathematica 13, p. 1, 1890. 385 386 TEXTBOOK OF THERMODYNAMICS XXII 145 regarded the lack of conformity between thermodynamics and the kinetic theory as a serious objection against the statistical interpreta- tion. The other contended that the principle of the increase of entropy had itself only a statistical validity and was true only in the time average. As Gibbs put it as early as 1876: "The impossibility of an uncompensated decrease of the entropy seems reduced to an improbability." The views of the two schools were brought to a focus in a very interesting polemic between Zermelo 1 and Boltzmann. 2 The wide attention which this discussion received stimulated new work on the subject, and, before long, the controversy was definitely settled in favor of the statistical point of view through the investigations of Von Smoluchowski 3 and of Einstein. 4 These authors showed that deviations from the entropy principle do, actually, occur and can be observed provided the system is sufficiently small. They accomplished this by turning eq. (22.01) into a heuristic method for the treatment of problems lying outside the scope of classical thermodynamics. Let / be one of the parameters describing the state of the system, and let /o be its value in the normal state of maximum entropy 5. We ask the following question: what is the probability of this parameter assum- ing a value between /o + A/ and /o + A/ + dl? Let the entropy, corresponding to /o + A/, be So + AS, where AS is, necessarily, nega- tive, So being the possible maximum. Inverting the relation (22.01) between entropy and probability, we find that P must be proportional to exp [(So + AS)/]. On the other hand, the probability must be also proportional to the interval dl. Therefore, we can write for it Pidl = Ce* s/ *dl. (22.02) The factor C may depend on the normal values of all the parame- ters of the system. 6 It is determined by the condition Pidl - 1, (22.03) which expresses the fact that the value of / will lie, certainly, in one of the intervals dl. E. Zermelo, Ann. Physik 57, p. 485, 1896; 59, p. 743, 1897. 1 L. Boltzmann, Ann. Physik 57, p. 773, 1896; 60, p. 392, 1897. 1 M. Von Smoluchowski, Boltzmann-Festschrift, p. 626. Leipzig 1904; Ann. Physik 21, p. 756, 1906; 25, p. 205, 1908. 4 A. Einstein, Ann. Physik 17, p. 549, 1905; 19, p. 373, 1906; 33, p. 1275, 1910. * Strictly speaking, the parameter / must be selected in a definite way to make C depend only on the normal values /o and not on the / themselves. We cannot enter here into the rules of selection as they represent a problem of statistical mechanics. XXII 146 LIMITATIONS OF THERMODYNAMICS 387 146. Brownian movements. Classical thermodynamics cannot account for the fact that small particles dispersed in a liquid or gas form a permanent suspension, that is to say, float in the medium and are in a state of irregular agitation called Brownian movement. From the point of view of that theory the particle is merely a part of the boundary of the liquid, and its position is determined not by thermo- dynamical but by dynamical laws. These laws tell us that the position of equilibrium for the particle is its motionless state at the bottom of the vessel. Let us suppose the vessel to be of heat-insulating material so that our thermodynamical system (liquid or gas) is adiabatically isolated. In this case, the dynamical equilibrium of the particle corresponds also to the maximum of entropy of the liquid (or gas). In fact, it would require the energy expenditure e = M'gx to raise the particle from the bottom to the height x, g being the acceleration of the gravity field and M' the effective mass of the particle (its true mass minus the mass of the displaced medium). This energy can be supplied only by the liquid (or gas) which undergoes a corresponding loss of internal energy A 7 = e. In general the entropy is expressed by AS = (AZ7 + &W)/T\ in our particular case, the volume of the liquid remains unchanged when the particle is raised and no work is done by the system (AW = 0). Therefore, AS = e/T: this entropy change is negative, and the strict point of view on the second law would deny the possibility of the particle being permanently afloat. On the other hand, the statistical point of view admits the existence of small entropy fluctuations. The probability of the floating particle's being found at a height between x and x + dx is given directly by the formula (22.02) -. _M*g* P x dx = Ce trdx = Ce *** dx. (22.04) It is equally easy to find the probability for the particle being in motion. Suppose the components of its velocity c lie between the limits c z and c x + dc x , c v and c y + dc v , c g and c g + dc f while its kinetic energy is Mc 2 /2. We conclude, as in the preceding case, that this energy is supplied by the liquid which experiences the changes AJ7 = - Mc 2 /2, AS = - Mc*/2T. The formula (22.02) gives for the probability of this state of the particle Pdctdc^c, - C f exp (-Mc 2 /2kT)dc x dc v dc t . (22.05) If the suspension consists of a large number of identical particles, each having the mass Af, the formulas (22.04) and (22.05) determine their numbers z at the height x and in the velocity interval 388 TEXTBOOK OF THERMODYNAMICS XXII 146 The first equation can be written z = zo exp (-M'gx/kT), (22.06) where z is the number of particles (referred to unit volume) at the height x, and zo at the bottom (x = 0). It is identical with the baro- metric law of Laplace, which was deduced in section 107 in a generalized form starting from the law of perfect gases. We see now that the particles of a suspension also obey the barometric law, and this implies that their osmotic pressure is that of perfect gases: pv = RT or p =5 zkT (section 66). The simplest way of showing this is to invert the usual form of the derivation of Laplace's formula. At the height x the pressure is higher than at x + dx by the amount dp because of the weight of the particles in the layer dx. This weight is dp = zM'gdx per unit area. On the other hand, we find from (20.06) x = kT X log (Z/ZQ)/ M'g and dx = kTdz/zM'g, whence p = zkT. The formula (22.05) represents Maxwell's velocity distribution and is equivalent to the caloric equation of state u = %RT + #o. This follows immediately from calculating the total kinetic energy of all particles by integrating \Mf f J 9 c 2 Pdc ai dCj / dc t and keeping in mind thatffrpdcyidcydcg = 1. Of course, the particles have also rotational kinetic energy which is not included in our calculation of u. It is easy to show that its amount is RT per mol, but the above examples will suffice. Extensive experimental work on suspensions was done by Perrin l and Svedberg. 2 These authors tested the barometric formula (20.06) directly, and Maxwell's distribution law (20.05) indirectly, and found both in excellent agreement with experiments. Brownian movements affect not only particles in a suspension but indeed every movable object. For instance, a delicate tor- sion balance continuously undergoes small deflections from its zero position. If the directing force is a and the angle of deflection tf, the potential energy of the balance becomes e tf = ^ad 2 . As in the case just treated, this energy is taken from the surrounding medium (air) whose internal energy and entropy experience the changes A 7 = - atf 2 , AS = - Ja# 2 /T. Again we find from (20.02) as the probability of a deflection between tf and & + d& Pjto = Cexp (-%atf*/kr) d&. (20.07) 1 J. Perrin, Brownian Movement and Molecular Reality. London 1910. * The Svedberg, Die Existenz der Molekflle. Leipzig 1912. XXII 147 LIMITATIONS OF THERMODYNAMICS 389 The mean potential energy of the balance is, therefore, /+ EtPadd. J> Determining the coefficient from the condition (22.03) we find 2e - ^ kT, (20.08) corresponding to equipartition of energy (section 30). Observation on a torsion balance were carried out by Kappler. 1 He found that the distribution of the deflections is accurately represented by the Gaussian curve (20.07). Substituting into (20.08) the measured mean at? 2 , he obtained an experimental value for k and derived from it (because of n A = R/k) the result n A = (6.06 0.06) X 10 ~ 23 mol" 1 for the Avogadro number. 147, Theory of fluctuations. According to classical thermody- namics, any substance is in equilibrium quite uniform and not subject to spontaneous changes. The statistical view on the second law, on the contrary, permits us to put the question as to local deviations from uniformity. Let us focus our attention on a small part of the system, a group of adjacent molecules, and let us calculate the proba- bility that the parameter / has in it an abnormal value. We denote by MI, M2 the masses of the small part and of the remaining large part of the system (M 2 MI), by so, k the normal specific values of the entropy and of the parameter /. We consider the case, however, when these quantities have in the two parts the slightly abnormal values so + Asi, h + A/i, and SQ + As 2 , / + A/ 2 . The total devia- tion of entropy is, therefore, AS = MiAsi + M 2 As 2 or expanding into a Taylor series with respect to A/i, A/2 (which we assume to be very small quantities) AS - ^ 3A) Since the entropy has its maximum in the normal state, the first term must vanish: AfiA/i + Af 2 A/ 2 =0 or A/2 = (Mi /M 2) A/I. In the second term M 2 (A/ 2 ) 2 = Mi(A/i) 2 Afi/Af 2 becomes then negligible, whence (22.09) The entropy deviation in the small part of the system is expressed 1 E. Kappler, Ann. Physik 11, p. 233, 1931; 15, p. 545, 1932. 390 TEXTBOOK OF THERMODYNAMICS XXII 148 in terms of its own properties. We may, therefore, drop the subscript (1) treating it as if it were alone. Equation (22.02) takes the form Pidl = Cexp The mean quadratic deviation of the parameter / (in the small part of the system) is by definition /^fo (S/) 5 = C I */_00 (A/) 2 P,<tf, while C is to be determined from (22.03). It must be remembered that Qs 2 /Ql 2 is necessarily negative: S being a maximum, the deviation can have only the negative sign. Therefore, the integration is easily carried out giving (22 - 10) 148. Density fluctuations and light scattering. We shall apply this formula to compute the magnitude of the spontaneous density fluctuations in a gas. Let us take as the parameter the molal volume in the small selected portion of the gas: / = v. Considering the fluctuations of one parameter only implies that the others drop out in the averaging process, being on the average constant. The ques- tion arises, therefore, which other parameter we have to regard as constant in taking the partial of 5 with respect to v. The answer is simple in this particular case because we know from section 31 that a subsidiary condition of equilibrium is dU = 0. Unless we want to calculate the fluctuations of the internal energy itself, we must always consider changes at u = const. We have, therefore, for the relative fluctuation /A-,\2 / A A2 I. (22.11) where p oc i/v is the density, whence Ap oc Av/t^. From (4.15), (9s/9i/)t = P/T, the result of the second differentiation at u = const is little different from that at T = 0. We may write with a good approximation (d 2 s/dv*)u = (dp/dv) T /T, or denoting the compressi- bility _^^ ^' (22.12) XXII 148 LIMITATIONS OF THERMODYNAMICS 391 In the case of the ideal gas, /3 =!//>, V = kZT/p, Z being the total number of molecules in the considered small portion of the gas. We obtain, therefore, _ /A.\2 1 (22.13) as found already by Bernoulli for the fluctuations of independent units (molecules). This formula shows us that the mean relative deviation of the density from its normal value is very small for a large mass of gas but becomes appreciable when the mass decreases. Measuring the density of 1 cm of air (Z = 2.7 X 10 19 ) we are not likely to find any deviation from the norm, as it is in the mean only 2 X 10~ 10 . How- ever, the deviation is measurable in an air-filled cube whose edge is of the order of a wave length of light (5 X 10~ 5 cm) because here Z = 3.6 X 10 6 whence 1/Z W = 0.53 X 10~ 3 . There exist, therefore, in the gas numerous condensed and rarefied regions of this size and smaller. This inhomogeneity must produce a scattering action on light passing through the gas, and the amount of scattering to be expected on the basis of the formula (22.12) was computed by Von Smoluchowski l and by Einstein. 2 A simple way of deriving this formula is as follows. 3 The theory of optics gives the following expression for the intensity of light scattered by a small dielectric particle under the angle # from the direction of the incident plane wave (whose intensity is /) and at the distance r from the particle AJ, ^F2(AD)2 1 + cos* tf ~T ~ ~~ (22 ' 14) (Rayleigh scattering). AZ> is the difference between the dielectric constants of the particle and of the surrounding medium. It is sup- posed that AD is small ; under this assumption the scattered intensity depends only on the volume V of the particle and not on its shape. Ao is the wave length of the incident light, as it would be in vacua. The formula applies only to particles whose dimensions are small compared with the wave length. We can regard the spontaneous inhomogeneities of a gas as particles of this sort. Its whole bulk consists of such inhomogeneities lying side by side, but if we focus our attention on one of them, we can say that the average dielectric constant of its environ- 1 See footnote on p. 386. 2 See last paper in footnote on p. 386. 8 P. S. Epstein, EnzyklopSdie der math. Wiss. V 3, p. 519, Leipzig, 1915. 392 TEXTBOOK OF THERMODYNAMICS XXII 148 merit has the normal value D while its own dielectric constant deviates from it by AD. This deviation is connected with the density fluctua- tion (22.12) by the well-known formula (D - !)/(!> + 2)p const, whence ^ = A, P- 1)^ + 2). (22 15) P 3 We now put the question as to the scattering of a mass of gas of the volume V M . The inhomogeneities are distributed in it quite irregu- larly, and there are no phase relations between the amounts of light scattered by them. The total scattering is, therefore, simply the sum of these amounts. According to our previous analysis the inhomo- geneities of small volume greatly dominate, so that there is no objec- tion to applying the formula (22.14): The summation of the expres- sions for all the homogeneities in V M will lead to F 2 (AZ>) 2 being replaced by Substituting from (22.15) and (22.12) Using the abbreviation 8-JT 3 a = ^ kT ^ D ~ " 1 ) S ( Z? + 2 > 2 ' < 22 ' 16 > the scattering by a unit volume of gas becomes 1 - L c2 '\ (22.17) (22.18) "" 16* r 2 or integrated over all space directions - a. a has, therefore, the meaning of the coefficient of extinction through scattering. In the particular case of permanent gases D is very little different from 1, ft is practically 1/p, and kTft = 1/z, where z is the number of molecules per unit volume: Sir 3 - (D - I) 2 . (22.19) XXII 148 LIMITATIONS OF THERMODYNAMICS 393 The last three formulas were first given by Lord Rayleigh 1 in his theory of atmospheric scattering. He explained the blue color of the sky as a consequence of the factor 1/Ao 4 : light of short wave length is scattered more strongly so that the sunlight which indirectly reaches our eyes, after being scattered in the upper layers of the atmosphere, has its intensity maximum in the blue. Numerous measurements confirmed the dependence on the wave length and on the angle # given by these formulas. 2 The direct observations of the coefficient of extinction give Sit*(D - l) 2 /3z == 1.00 X 10 ~ 24 cm- 1 , reduced to 1 atm and C, the calculated value being 1.04 X 10 ~ 24 cm- 1 . The more general formula (22. 16) shows that light scattering must be particularly strong when the gas is in the vicinity of the critical state because the compressibility $ is then very large. This explains the phenomenon of the so-called opalescence in the critical state. It has been investigated quantitatively by Keesom 3 and found in agreement with the theory. In conclusion, we repeat that the entropy principle cannot account for Brownian movements or density fluctuations. The existence of these phenomena is conclusive proof that an uncompensated decrease of the entropy is not impossible but only highly improbable. This fact does not impair the value of thermodynamics as a method of analyzing physical reality: in systems of appreciable size the change of a deviation from the second law is so extremely small as to be entirely negligible. *Lord Rayleigh, Phil. Mag. 41, pp. 107, 274, 447, 1871; 12, p. 81, 1881; 44, p. 28, 1897; 47, p. 375, 1899. 2 H. Dember, Ber. Saxon Acad. 64, p. 289, 1912; F. E. Fowle, Astrophys. J. Obs. Smithsonian Inst. Ill, Washington 1914; E. Kron, Ann. Physik 45, p. 377, 1914. * W. H. Keesom, Ann. Physik 35, p. 591, 1911. APPENDIX I COMPARATIVE TABLE OF NOTATIONS There are in use four major and numerous minor systems of thermodynamical notations. The major systems are as follows: (1) the Gibbs notations; (2) the system developed by the physicists Clausius, Helmholtz, and Planck (C.-H.-P.); (3) the notations adopted by the larger part of the American chemists, especially Lewis and Randall (L.-R.); (4) the system used in technical thermodynamics. We believe that this lack of unity is neither strange nor particularly regrettable. Thermodynamics is primarily a method for deriving relations between measured quantities. Various branches of science make use of it, each applies it to its own distinctive material, and in each the problem of harmonizing the notations and avoiding duplications and ambiguities is entirely different. It would be extremely difficult, if not impossible, to devise a system satisfying all these complicated require- ments. On the other hand, the inconvenience to the reader caused by the lack of unity is slight, as the relation of the different systems of notations can be set down in a simple comparative table. As to the intrinsic merits of the different systems there is not much to choose between them. A systematic principle was first introduced in the C.-H.-P. system in the form of a rule with respect to additive or extensive quantities (sections 39 and 144): the total quantities (i.e. those referred to the whole system or, at least, to a phase) are denoted by capital, the molal and specific quantities by lower-case letters. L.-R. modify this by using for molal quantities a special size of type, an expedient which seems less convenient in classroom teaching. We seriously considered adopting the L.-R. system, which has a large following among the chemists of this country. But their choice of the notations E and H for the internal energy and heat function (enthalpy) is too repugnant to the physicist who has to treat electric and magnetic applications of thermodynamics and is accustomed to denote by E and H the electromagnetic field strengths. Worse still, according to the above rule the molal energy and enthalpy would have to be denoted by e and h, letters which are already badly overtaxed. We have, therefore, chosen to use the C.-H.-P. system with the following slight corrections. (1) The letter F for the work function ( U- TS) seemed undesirable as it is used by L.-R. for the func- tion U TS -f- pV; therefore, we substituted for it *, the choice of Gibbs. (2) The use of UK for the mol number (i.e. number of mols of the component h in a system) and of No for Avogadro's number is inconsistent with the above rule since the first quantity is total and the second molal. We write, therefore Nh and n A . In the following table we list the more important notations, omitting those which are identical in all systems. 395 396 APPENDIX Quantity E. C.-H.-P. Gibbs L.-R. Total Molal Total Molal Internal energy U X s * $ N h u X s t <f> ?A nth n A U X 5 F -r* nn u X s f -TV 7V/i X y t r Mfc nh E H S A F *h n h Wh N Heat function (enthalpy) Entropy Work function (UTS) Thermodynamic potential (U-TS+pV) Partial molal thermodynamic potential Mol number Molality Avogadro's number No .... Mol fraction Xh Xh, Ch N A APPENDIX II CONSTANTS AND CONVERSION FACTORS 1 TABLE OF CONVERSION FACTORS 1 norm, atmosphere 1 013 249 X 10 G dyne cm" 2 logio 6 005 717 1 (15) g-calorie 4.1852 joule 621 720 t < 4 1852 X 10 7 erg 7 621 720 < i 426 78 g m 2 630 199 1 00 095 (20) g-calorie OftOO 4.1? 4 1 99 976 (mean) g-calorie T 999 896 1 e-v. (electron-volt) 2 0017 X 10~ 12 erg 15 30 140 n A e-v 1 2139 X 10 12 erg mol"" 1 1? OS 4.10 < 29 003 cal mol ~ l 4 46 245 1 Partially based on the critical compilations by R. T. Birge, Phys. Rev. Supple- ment 1, p. 1, 1929; Phys. Rev. 40, p. 207, 1932. 398 APPENDIX 9 j* fe c c T ^ *- - . ^-t-(-i QQ c^io OOO'' H -**'^OO*-<*-' iH OOOOOOOOOOOOOOOOOO o T-H X ISIS oooo r- r- o ^ ^ * MM 1 1 1 I. 1 1 5 ooooooooo XXXX XXXXXXXXX & I D d. 2 | i a ! .,< g r i! ' 3J 2 SUBJECT INDEX Absolute entropy, see: entropy constant Absolute temperature, 8, 58, 74 realization of scale, 74-76 Absorption of gases in liquids, 161, 162 coefficient of , 161 Activity, 199, 287 coefficient, 201, 293, 294 function of electrolytes, 203-205, 290, 294, 295 of gas mixtures, 205-208 Additive property of entropy, 59, 60 Additive quantities, 103, 377, 384 Adiabatic, equation of , 48 in gas mixtures, 320 in perfect gases, 49, 263 in simple systems, 320 Adiabatic process, 48, 49 Adsorption in capillary layer, 223 Affinity, chemical , 92, 93, 225 Air, composition of , 308 specific heat of , 309 Annihilation of matter, 336, 343, 344 Atmospheric scattering, 392, 393 Auxiliary constraints, 99-102 Avogadro's number, 6, 7, 398 B Barrett effect, 350 Beattie and Bridgeman, eq. of state of , 16 Binary gas mixtures, 13, 205-208 Binary systems, 179-194 Black radiation, energy and entropy of, 331-333 Boiling point, 116-123 elevation of due to presence of sol- utes, 166, 167, 202 influence of surface tension on , 217, 218 Boiling pressure, 11, 116-123 lowering of by solutes, 163, 166, 202 influence of capillary layer on , 216,217 of water, 1 19 Boiling temperature, see: boiling point Boltzmann's constant, 7, 78, 398 Boltzmann's principle, 78, 79, 260, 296, 385 Born and Stern, formula for surface tension of solids of , 220 Brownian movements, 249, 387-389. Caloric eq. of state (see: eq. of state), 40 Caloric properties of matter, 27 Calorie, 26, 397 Capillarity, 209-224 Capillary layer, 210, 213 Capillary rise, 215 Carbon dioxide, curve of fusion and sublimation of , 127, 128 isothermals of , 17 Carnot's cyclic process, 50-52, 54-56 efficiency of , 52. Centigrade scale of temperatures, 5 Cells, galvanic (see also: galvanic cells), 94, 95, 287, 288 Characteristic functions, 38, 46, 62, 85 construction of , 85 Charged gases, 266-293 equilibrium in , 268-270 Chemical affinity, 92, 93, 225 Chemical compound, 149 Chemical constant (see also: entropy constant of gases and Sackur- Tetrode formula), 236-238 of degenerate perfect gases, 257 of diatomic gases, 313-314 of electron gas, 278 of metallic ions, 284 of monatomic gases, 312, 313 of polyatomic gases, 314 399 400 SUBJECT INDEX Chemical equilibrium, general conditions of , 112, 113 in charged gases, 270 in perfect gases, 137-147, 239 in processes involving radiation, 335, 336 in terms of activities, 200, 201 influence of electric field on , 270 influence of gravitational field on , 271, 344 of solutes in dilute solutions, 176-177 Chlorine, specific heat of , 308, 309, 3 1 1 Clapeyron-Clausius equation, 118, 120 Compressibility, coefficient of , 3, 8 in critical state, 12, 393 Concentration cells, 288 Concentrations of ions, 288 Condensation of the Einstein-Bose de- generate gas, 258, 259 Condensation, process of , 218 Conservation of energy, see: first law of thermodynamics Constant, of entropy, see: entropy of internal energy, see: internal energy Constants, table of fundamental , 398 Contact potential, 275 classical theory of , 273-286 quantum theory of , 283 Continuity of liquid and gaseous states, 9,11 Conversion factors (see also: efficiency), 52, 397 Corresponding states, 12 extended law of , 68, 69, 123-125, 221 law of , 12, 16 Covolume, 9 Critical constants, 12, 14, 15 Critical data, 14, 15 Critical state, 12, 13, 127, 128 opalescence in , 393 Cryohydrates, 188 Crystals, entropy of mixed , 77, 78, 246, 248 D Debye's formula for specific heat of solids, 324 Debye-Huckel, theory of strong electro- lytes, 290-295 limiting law of , 294 Degenerate electron gas, 277, 278, 368 Degenerate perfect gases, 252-265 numerical amount of degeneration in , 263-265 Density fluctuations, 390-392 Deuterium, specific heat of , 306, 307 equilibrium of and hydrogen, 340 Diamagnetism, 347, 351 Dielectrics, 359, 360 Differential, exact , 20 Differential expression, linear , 20, 21 Differential Joule-Thomson effect, 70 Dilute solutions, 152-177 conditions of equilibrium in , 155- 157 Displacement of equilibrium, 156, 376- 380 Dissociation, degree of , 140 depression of by various agents, 144-148, 178 Distillation, 183 Distribution law, 157, 158 Doublet heat, see: specific heat Duhem's equation, 103 Dulong and Petit, law of , 45, 232 Dushman's formula for thermionic cur- rents, 279 Efficiency, of Carnot's process, 52 of galvanic cells, 95 of heat engines, 51, 52, 55, 95 Einstein-Bose gas, see: degenerate per- fect gases condensation of , 258, 259 Einstein's formula for specific heat of solids, 233 Einstein's law of equivalence of mass and energy, 330 Electric field, influence of on equilib- rium in charged gases, 270 influence of on equilibrium of drop- lets, 219 Electrolytes (see also: weak and strong electrolytes, and activity function), 94, 204 specific heat of , 329 SUBJECT INDEX 401 Electromotive force, 94 of galvanic cells, 94, 289, 290 of thermoelectric couples, 364, 365, 368 Electron clouds, 266-285 Electron gas, 266-285 degenerate , 277, 278, 368 Electron pairs, 337, 338 Electronic specific heat in metals, 278, 326 Electrons, positive , see: positrons equilibrium of positive and negative , 337, 338 Electrostriction, 360 Elevation of boiling point by solutes, 166, 167, 202 Endothermic process, 27 Energy, 27 conservation of , see: first law of thermodynamics internal , see: internal energy of black radiation, 333 Enthalpy, see: heat function Entropy, 58, 59, 60 additivity of , 59, 60 measurement of , 66, 229, 230 of black radiation, 333 of degenerate perfect gas, 256, 257 of dilute and perfect solutions, 153, 154 of electron gas, 278 of gas mixtures, 136, 207 of mixed crystals, 77, 78, 246, 248 of perfect gases, 63 of supercooled liquids, 248 of Van der Waals gases, 65 statistical interpretation of , 76-79, 385, 386 Entropy constant, 59, 230 of gases (see also: chemical con- stant), 235, 236, 337 Entropy principle, 60, 61, 82 Eotvos* formula for surface tension, 221 Equation of state (caloric), 40, 68, 347 of degenerate perfect gases, 255 of perfect gases, 43 of Van der Waals gases, 65 Equation of state (magnetic), 347 of Langevin-Brillouin, 348 Equation of state (thermal), 3 of Beattie and Bridgeman, 16 of Kamerlingh Onnes, 16 of degenerate perfect gases, 252, 253 of perfect gases, 6, 7 of Van der Waals, 12 of Van der Waals-Lorentz for binary gas mixtures, 13, 206 Equilibrium, 1 chemical , see: chemical equilibrium displacement of , 156, 376-380 fundamental conditions of , 82, 92, 93, 376-379 influence of capillary forces on , 213-218 influence of electric forces on , 219, 270, 380 influence of gravitational forces on , 271, 344, 345 influence of magnetic forces on , 380 in processes involving radiation, 335-345 role of internal energy in , 82, 376, 378 role of thermodynamic potential in , 93 role of work function in , 92 Equilibrium conditions (explicit) general statement of , 112, 113, 200, 379 in dilute solutions, 155-157 in processes involving radiation, 335-336 in terms of activities, 200, 201 Equilibrium constant, 139, 156, 200, 235, 236, 239, 336 Equipartition of energy, 79 Euler's theorem (about homogeneous functions), 102, 104 Eutectic alloy, 188 Eutectic point, 188 ternary , 196 Exothermic process, 27 Expansion (thermal), 1 coefficient of , 3, 5 in vicinity of T = 0, 228, 258 Extensive quantities, 384 Extinction, atmospheric , 392, 393 402 SUBJECT INDEX Fermi- Dirac gas, see: degenerate perfect gas First law of thermodynamics, 27-51 formulation of , 37-39 history of -, 27-34. Fluctuations, 389, 390 of density, 390-392 Free energy, see: work function and thermodynamic potential Freezing point (see also: fusion), 116 ^"ugacity, 197 of perfect gas, 197 of Van der Waals gas, 198 Fundamental point (see also: triple point), 110, 126-128 Fundamental state, 110, 126-128 Fusion, point of , 116, 119 lowering of due to solutes, 166, 167, 202, 203 Fusion, process of , 219 Gadolinium sulphate, specific heat of , 327 magnetization of , 349 Galvanic cells, 94, 95, 287, 288 electronomotive force of , 94, 289, 290 Galvanomagnetic effects, 368 Gas constant, 6, 7,398 Gas mixtures, 7, 13 activities of binary , 205 chemical equilibrium in , 137-147, 236-239, 270, 284, 336 heat capacity in , 315-318 velocity of sound in , 318-321 Generic probability definition, 78, 246 connection of with principle of indetermination, 260 Gibbs-Helmholtz equation, 93, 96, 225 Gram-molecule, 6 Gravitational forces, influence of on equilibrium, 271, 344 H Heat, 25 latent , see: latent heat of reaction, 27, 67 molal (see also: specific heat), 26 specific , see: specific heat Heat capacity (see also: specific heat). 25,26 at constant magnetic field, 351, 352 at constant magnetization, 351, 352 at constant pressure, 40, 46, 65, 66 at constant volume, 40, 65, 66 of electron pairs, 388 of reacting gas mixtures, 315-318 Heat content, see: heat function Heat function, 46, 85, 86 connection of with latent heat and heat of reaction, 67 of electron pairs, 388 of perfect gases, 47 role of in Joule-Thomson proc- ess, 69 Helium, curve of fusion and sublimation in , 128 equilibrium of and hydrogen, 342 inversion point of , 72 supposed second order transformation in 129 Henry's law, 159-162 Heterogeneous system, 2 Homogeneous system, 2 Hydrogen, annihilation of , 343 equilibrium of and deuterium, 340 equilibrium of and helium, 342 solubility of in molten metals, 162 specific heat of , 44, 303 Hydrogen bromide, dissociation of , 147 Hydrogen iodide, dissociation of , 144, 145 I Image force, 273, 274 Independent component, 107-110 systems with one , 115-133 Independent variables, 98, 99, 107 Integrability, 22, 23 Integrating multiplier, 22, 58, 74 Intensitaet, 383, 384 Intensive quantities, 102, 384 Internal energy, 38, 83 constant of , 330 of degenerate perfect gases, 255 of dilute and perfect solutions, 151, 152 of electron gas, 278 SUBJECT INDEX 403 Internal energy, Continued of perfect gas, 43 of Van der Waals gas, 65 Internal pressure, 9 Inversion temperature in Joule-Thomson effect, 71, 72 Iodine, dissociation of , 148 distribution of between water and carbon tetrachloride, 158 Ion clouds, 266-287 lonization in solar atmosphere, 283-285 Irreversible process, 35, 36 Isobaric process, 46, 47 Isochoric process, 47 Isosmotic solutions, 170 Isothermal process, 47, 48 work in , 87 Isothermals, of Van der Waals gas, 10 of CO 2 , 17 Isotonic, see: isosmotic Joule-Thomson effect, 70-73 differential , 70 integral , 73 Joule-Thomson process, 42 theory of , 70-73 K Kamerlingh-Onnes' empirical reduced equation of state, 16 Kelvin's formula for influence of capil- larity on boiling pressure, 214 Kelvin's scale of temperatures, see: absolute temperature Lagrangean multipliers, 98 Langevin's theory of magnetization, 346-349 Langevin-Brillouin, magnetic equation of state of , 348 Latent heat, 26, 46, 67, 118, 121 temperature dependence of , 121 LeChatelier-Braun, principle of , 375 restricted principle of , 380-383 Legendre transformation, 85 Lenz's principle of electrodynamics, 375 Liquidus curve, 185 Liquidus surface, 195 Lowering, of boiling pressure, 162-167 of point of fusion, 166, 167, 202, 203 M Magnetic equation of state, 347 of Langevin-Brillouin, 348 Magnetics, 346 perfect , 348, 350, 353 Magnetization, 346 influence of on equilibrium, 380 Magnetocaloric cooling, 352-354 Magnetostriction, 350 Mass fraction, 109, 182, 192, 193, 196 Mass law, 138-140, 176-178, 270, 271, 336 Mixture, 149 of gases, see: gas mixtures Mobile equilibrium, principle of , 156 Mol,6 fraction, 13, 102, 136 Molal heat (see also: specific heat), 26 Molal volume, 6 of dilute and perfect solutions, 152 Multiplier, integrating , 22 Lagrangean , 98 N Nagaoka-Honda effect, 350 Nernst's distribution law, 157, 158 Nernst's heat postulate, 225-250 generality of , 246-250 independence of from second law, 244 Nernst's own formulation of , 226, 227, 230 Planck's formulation of , 230, 231 Neumann's law, 45, 327 Neutrons, 260, 339, 343 equilibrium of and protons, 339 Nitric oxide, specific heat of , 44, 310 Nitrogen, solubility of in water, 161 specific heat of , 307, 308 Nitrogen tetroxide, dissociation of , 141, 142 dissociation of in chloroform solu- tion, 177 heat capacity of , 319 Notations, 395 comparative table of , 396 404 SUBJECT INDEX Opalescence in critical state, 393 Osmotic pressure, 167-178 equation for , 169 Van t'Hoff s equation for in dilute solutions, 171 Ostwald-Freundlich, formula for solu- bility of small particles of , 218 Oxygen, specific heat of , 44, 307, 308 Partial molal quantities, 84 Partial molal thermodynamic potential, 90,91 in dilute solutions, 155, 200 in gases, 137 in strong electrolytes, 293 Peltier coefficient, 362 Perfect gases, 4-8 degenerate , 252-265, 277, 278 mixtures of , 7, 134-147 Perfect magnetics, 348, 350, 353 Perfect solutions, 151, 179 Perpetual motion machine of the second kind, 54 Phase, 2 Phase equilibrium, 100, 101, 104-114 displacement of , 156, 376-380 in binary systems, 179-193 in dilute solutions, 154-167 in pure substances, 115-128 number of phases in , 110, 111 of higher order, 128-133 temperature and pressure in , 104 Phase rule, 110, 111 Phosphonium chloride, curve of fusion and sublimation of , 128 Photoelectric effect, 275, 280 Piezoelectricity, 360, 361 Point of fusion, 116-128 influence of capillarity on , 219 lowering of by solutes, 166, 167, 202, 203 Point of inversion, see: inversion point Point of transition, see: transition point Polarization, electric , 359, 360 Positrons (or positive electrons), 337 equilibrium of and negative elec- trons, 337, 338 Principle, Boltzmann's , 78, 79, 240, 260, 296, 385 entropy , 60-62, 82 Lenz's of electrodynamics, 375 of indetermination, 260 of LeChatelier-Braun, 375, 380-383 of mobile equilibrium, 156 of Thomsen-Berthelot, 226 Probability, 76-79, 246, 247, 260 generic definition of 78, 246 relation of generic definition to prin- ciple of indetermination, 260 specific definition of , 78 Protons, annihilation of , 343, 344 equilibrium of and neutrons, 339 Pure substances, phase equilibrium in , 115-128 Pyroelectricity, 361 Q Quadruple point, 179 Quantitaet, 383, 384 R Radiation, black , 331 energy of , 332, 333 entropy of , 332, 333 pressure of , 332 Ramsay-Shields, formula for surface tension of , 221 Raoult's law, 162, 163, 164 Rayleigh's formula for atmospheric light scattering, 392 Reciprocity relations, 21, 64, 70, 89, 350, 360, 381 Reduced variables, 12, 16, 68, 69, 71, 73, 123 Regelation, 120 Reversible process, 35 Richardson's formula for thermionic cur- rent, 276 Rigid envelope, 100, 101 Sackur-Tetrode formula (for absolute entropy of gases), 237-241, 257 Saturated solution, 174, 176 Scale of temperatures, 1 absolute ,8,58, 74-76 centrigrade , 5 SUBJECT INDEX 405 Solutions), Scattering of light, 391, 392 Second law of thermodynamics, 53, 54, 58, 59, 62, 82 Semi-conductors, 368-374 thermoelectric effect in , 276, 365 Semipermeable membrane, 101, 135, 168 Sieverts' law (solubility of gases in molt- en metals), 162 Simon and Glatzel, formula for pressure of fusion of t 122 Simple system, 2 Solidus curve, 185 Solidus surface, 195 Solubility, mutual (complete or par- tial), 149, 150, 184, 186, 189, 190 of gases, see: Henry's law of solids in liquids, 174, 218 Solute, 151 Solution, 149-151 dilute (see also: 152, 177 perfect ,151, T' Solvent, 151 .ula for electronic Sommerf eld's >f metals, 278, 326 specific heanula for inner potential Sommerf eld's f' of met ; ' ula for thermoelectric Sommerf ela 068 power, 2&\ also: heat capacity), Specific heat (s\ 26, 296-329 sure, 40, 46, 65, 66 at constanprefc. 4Q ( 55 ? ^ atconstajvolumct^ 327 doublet , >1, 310, 3 ?2 p in vicinr f T Q t of elect <ytes, 329 of elect" 18 i n metals, 278, 326 of gas*, 44, 298-311 of liqi*s, 328, 329 of pei** gases, 44 of sois, 45, 233, 321-327 of suites, 329 rotaticA 1 ~ 29 ^ 300, 304, 305, 307 transl/onal ,299 vibraanal , 302, 303, 307-309, 311 Specifitf> roDa bih'ty definition, 78 Specifi. quantities and properties, 102, 3/ Stabi# conditions of dynamical , 11, 4 Stability, Continued conditions of thermodynamical , 117,131,381 Statistical average, 76 point of view, 76-79, 246-248, 385, 386 principles, 76-79, 259, 260, 296 Strong electrolytes, 204, 287 activity function of ^ 204 205, 294 theory of , 290-29* 361 Sublimation, 126, lX Sugar solution, '*notic pressure of , 172 s . states, 297, 300-304 ^Jpercooled liquids, 248 Superheated state, 11, 117, 126 Supraconductive transition points, 355 Supraconductivity, 354-358 Surface tension, 209-224 influence of on boiling and melting points, 214, 216-219 temperature dependence of , 224 Symmetry number, 300, 313, 314 System, thermodynamical 2 Table comparative -1 of notations, 396 -- of conv ersion factorSf 397 of f f andamental constants, 398 ratine, 1 a)F ,olute ,8,58, 74-76 centrigrade , 5 Tension, coefficient of , 3 Ternary systems, 194-196 Thermal equation of state (see: equation of state), 3 Thermal equilibrium, see: equilib- rium Thermal expansion (see also: expansion) coefficient of , 3 in vicinity of T 0, 228 Thermal properties of matter, 27 Thermionic emission, 274 in classical theory, 279 in quantum theory, 279 Thermionic work function, 274, 276, 279, 280, 282 406 SUBJECT INDEX Thermodymanic potential, 90 connection of with non-mechanical work, 90 of degenerate perfect gas, 257 of electron gas, 278 of gas mixtures, 137, 207 of perfect gas, 91, 331 of Van der Waals' gas, 91 role of in equilibrium, 91-93, 106, 112, 113,115 Thermodynamics 1 limitations of , 387-393 subject of , 2 Thermoelectric effect, 3oi <^% empirical ,364 in classical theory, 273, 367 in quantum theory, 282, 368 Thermoelectric e.m.f., 368 Thermoelectric power, 364, 366, 368 Thermomagnetic effects, 368 Third law of thermodynamics, see: Nernst's postulate Thomsen-Berthelot, principle of , 226 Thomson coefficient (in thermoelec- tricity), 362, 365, 366 Transformation pressure and tempera- ture, 116, 121, J22 in binary systems. 183-193 01A 910 influence of capillarity 7 on ~~ Z1 *' ' influence of solutes on " 162 " 167 Transition points of supravr onductors ' 355 __ influence of magnetic field J ' 356-358 Trichloroacetic acid, distribution of between water and ethyl ether, 158 Triple point, 125-128 Trouton and Deprez, rule of , 124, 125 U Unattainability of absolute zero, 243-245 Units, conversion factors of , 397 Van der Waals' equation of state, 9, 12 equation of state for binary gas mixtures, 13, 206 formula for vapor pressure, 121 formula for surface tension, 221 Van t'Hoff's equation for osmotic pressure, 171 Vapor pressure, see: boiling pressure in binary systems, 180, 182 'iations, 82 _ 'tv of sound, 49 _ j n Jegenerate gases, 264-265 - Virtual chan ixtures ' 318 ~ 321 Virtual dispfi- 82 t t , 97-99 "nents, method of , Vv Water, activity and f i ^9g of liquid , Water vapor, dissociatit Weak electrolytes, 20*i of , 143 Work, 19 - 287 element of 18, Work function, '**'^ 210 -of perfect^ 7 ' -of stron^ f 8 ' 8 ? \ s electrolytes 293 - - of Van der Waals ga&R9 role of in equilibriumAl, 92 thermionic , see: ther^onic work function