MODERN THERMODYNAMICS1

A. R. Ubbelohde, Oxford

Thermodynamic Functions

Thermodynamics deals primarily with systems that are in a state of equilibrium. Its aim is a complete description of the behavior of all possible systems in equilibrium when such quantities as temperature, pressure, and the concentrations of the various components are changed.

From the 92 elements and the innumerable multitude of their various compounds, an enormous number of possible systems can be obtained. If even only those among them that have practical significance had first to be investigated experimentally before anything could be said about them, the situation would be discouraging.

Fortunately, thermodynamics establishes a number of theoretical propositions that greatly simplify the problem. One of these propositions is the phase rule, which states that the number of independent variables, or degrees of freedom, of a thermodynamic system is not arbitrary, but is equal to

\[ F = C - P + 2, \]

where \(C\) is the minimum number of components necessary to describe each phase of the system, and \(P\) is the number of phases.

Another path, which greatly simplifies the study of systems in equilibrium, consists in the use of thermodynamic functions, which are based on the first and second laws of thermodynamics and which can be calculated for each substance separately.

As will be shown below, if the thermodynamic functions of substances \(A\), \(B\), \(C\) are known, then it is possible to find the thermodynamic functions of any systems formed from them, without further experiments, and, in principle, it is possible to calculate the position of equilibrium of all systems obtained as a result of all possible combinations of substances \(A\), \(B\), \(C\), taken two or more at a time, and thus,

describe \(n!\) systems (independently of valence restrictions) with the aid of only \(n\) measurements.

Although the determination of thermodynamic functions encounters theoretical and experimental difficulties, their practical application, once they have been determined, is quite clear. Practically all the progress of modern thermodynamics depends on the development of convenient methods for determining the thermodynamic functions of individual substances.

Thermodynamic Functions Based on the First Law

Heat Content \(H\) and Internal Energy \(E\)

Two of the most important properties of all thermodynamic functions may be illustrated by the simplest of them—the heat content \(H\). If two different states of a simple substance or system can be characterized by different values of temperature, pressure, concentrations, etc., the increase in heat content in the transition of the system from state \(A\) to state \(B\) is defined as the energy received by the system when the transition is carried out at constant pressure. To denote heat content, the symbol \(i\) and the name enthalpy are also used.

The definition of the change in heat content may be written in the form

\[ \Delta H = H_b - H_a, \]

where \(H_a\) and \(H_b\) are the heat contents in states \(A\) and \(B\).

The first important property of the function \(H\) is that the value of \(\Delta H\) depends only on the initial and final states of the simple substance or system under consideration, and not on the process leading to the transition.

This property follows from the first law of thermodynamics, which states that energy can neither be created nor destroyed. If two different processes with different changes in heat content in the transition from \(A\) to \(B\) were possible, then from these processes one could construct a cycle leading to the indefinite creation or destruction of energy without a continuous change in the working substance. But this contradicts the first law. The value of the function \(H\) in any state depends only on the state of the system, and not on its previous history.

In the language of differential calculus this property, which is a common property of all thermodynamic functions, though for different reasons, is expressed by the assertion that \(dH\), \(dS\), etc., are exact differentials; that is, for any infinitesimally small change in the state of a system, which is expressed through changes of the parameters \(x_1, x_2, \ldots x_n, \ldots\), characterizing the system,

\[ dH = \left(\frac{\partial H_1}{\partial x_1}\right) dx_1 + \left(\frac{\partial H_2}{\partial x_2}\right) dx_2 + \left(\frac{\partial H_3}{\partial x_3}\right) dx_3 + \cdots \]

The second important property of heat content and of other thermodynamic functions, such as volume, entropy, etc., is that they are extensive quantities, i.e., that their numerical value is directly proportional to the mass of the working substance. This follows from the fact that the change in heat content in any process is the same whether the entire substance as a whole is transferred from state \(A\) to state \(B\), or whether it is first divided into parts (which requires no work) and the individual parts are then transferred separately from state \(A\) to state \(B\). This property may be characterized by the expression \(\mathrm{H}=n\bar{H}\), where \(\bar{H}\) is the heat content per mole and \(n\) is the number of moles in the mass of the working substance under consideration. The property of extensivity may be contrasted with the property of intensivity of such quantities as temperature or pressure, which must have the same value throughout the entire system in equilibrium and do not depend on the amount of working substance.

Hess’s law. From these properties of heat content it follows that the heat content per mole may be treated like any other quantity in chemical equations.

Thus, from the equations

\[ \mathrm{CaCO}_{3}\to \mathrm{CaO}+\mathrm{CO}_{2}+\Delta H_{1}, \]

\[ \mathrm{C}+\mathrm{CO}_{2}\to 2\mathrm{CO}+\Delta H_{2}, \]

it follows by direct addition that the change in heat content in the reaction

\[ \mathrm{CaCO}_{3}+\mathrm{C}\to \mathrm{CaO}+2\mathrm{CO}+\Delta H_{3} \]

is simply

\[ \Delta H_{3}=\Delta H_{1}+\Delta H_{2}. \]

Hess’s law is very important in thermochemistry for calculating the change in heat content when it cannot be measured directly by calorimetry. Since all the other thermodynamic functions possess the same properties as heat content, this means that the calculation of thermodynamic quantities whose values cannot be obtained directly from experiment must be carried out from two or more equations in which observable quantities appear.

Measurement of heat content; choice of the initial state. The change in heat content can in a large number of cases be measured quite easily in a suitable calorimeter, and also by other methods, for example from spectroscopic data. The heat content itself in any one state, however, is to a certain extent arbitrary, since only differences of heat content can be measured. Therefore any state of a substance or system may be chosen as the initial one, and the heat contents in all other states may be referred to the chosen state as to zero.

When a suitable initial state has been chosen, the change

the heat content upon transition to any state \(A\) can be written in the form

\[ \Delta H_a = H_a + H_{\text{initial}}. \]

Where no misunderstanding can arise, for the heat content in state \(A\), referred to the initial state as to zero, simply the symbol \(H_a\) is used. The choice of the initial state as the zero state does not affect the change in heat content in a subsequent transition from one state \(A\) to another \(B\), since the difference

\[ \Delta H_{a,b} = (H_b - H_{\text{initial}}) - (H_a - H_{\text{initial}}) \]

does not depend on what state has been chosen as the initial one. The derivation of a suitable zero state for thermodynamic functions is, however, important when the latter are connected with statistical theory, and will be considered later. From a practical point of view, the initial state must be chosen so that the changes can be measured accurately, i.e., so that the initial state is experimentally attainable.

Internal energy \(E\). If a system passes from state \(A\) to state \(B\) at constant volume, then the increase in internal energy is defined as

\[ \Delta E = E_b - E_a. \]

Just as in the case of heat content, the values \(E_b\) and \(E_a\) are to a certain extent arbitrary, and the internal energy in a given state must be referred to an appropriate initial state, such that the difference of energies can be measured. From the first law of thermodynamics it follows that the increase in heat content in any process must be equal to the increase in internal energy plus the work performed on the system under external pressure.

The relation between the two functions may therefore be written in the form

\[ \Delta H = \Delta E + P(V_b - V_a) \]

or

\[ H = E + PV. \]

Generally speaking, thermodynamic functions referring to changes at constant pressure are more convenient for experimental measurements, whereas functions referring to processes occurring at constant volume are more convenient for theoretical calculations.

The relations between the various functions which are used for finding one from another will be considered below.

Thermodynamic Functions Based on the Second Law

In calculating equilibrium between chemical substances it is necessary to use thermodynamic functions based not only on the first, but also on the second law. The simplest of the latter is the entropy \(S\). The increase of the entropy of a system under any change of state is defined, for a reversible change, as \(\Delta S=\dfrac{Q}{T}\), where \(Q\) is the heat absorbed by the system, and \(T\) is the absolute temperature.

The requirement of reversibility is imposed in order to preserve an essential property of entropy. Namely, the change in entropy when a system passes from an initial state to some state \(A\) depends only on these states and does not depend on the previous history of the working substance, i.e. entropy is a perfect differential. A consequence of the condition of reversibility is that the quantity \(T\Delta S\), when a system passes from state \(A\) to state \(B\), generally speaking, differs from the calorimetric heat of reaction that would be observed in an irreversible transition. How this property of entropy follows from the second law of thermodynamics can easily be seen from a simple Carnot cycle.

Let \(AB\) and \(CD\) be isotherms, and \(BC\) and \(AD\) adiabats. The change in entropy in passing from \(A\) to \(B\) is

\[ \Delta S = S_B - S_A = \frac{Q_{AB}}{T_1}. \]

In passing from \(B\) to \(C\), the entropy does not change at all. In passing from \(C\) to \(D\),

\[ \Delta S = S_D - S_C = \frac{Q_{CD}}{T_2}. \]

Finally, in passing from \(D\) to \(A\) the change in entropy is equal to zero. From the second law of thermodynamics it follows that if the process occurs reversibly in all parts, then

\[ \frac{Q_{AB}}{T_1} = \frac{Q_{CD}}{T_2}, \]

i.e.

\[ S_B - S_A = S_D - S_C. \]

For any reversible process of heat absorption the integral

\[ \int_A^B \frac{dQ}{T} = S_B - S_A \]

must have the same value. If this were not so, one could construct a certain cycle that would continuously convert heat from the surroundings into work, which would contradict the second law of thermodynamics. Hence it follows that

change in entropy can depend only on the initial and final states, but not on the (reversible) path. This is precisely the property being proved. Like heat content and internal energy, \(S\) has the properties of an extensive quantity; its values are proportional to the active mass of the substance, so that entropy changes can be introduced into the equations of chemical reactions. However, unlike heat content, in the experimental determination of an entropy change one must always exercise caution, taking into account that the process under consideration must be reversible at all stages. All other thermodynamic functions can be derived from these three and possess, correspondingly, the same properties, i.e., they are extensive quantities and depend only on the state of the substance (and, of course, on the initial state).

The most commonly used are the following.

Free energy (for processes occurring at constant pressure). It is defined as \(F = H - TS\) and was also called the Gibbs thermodynamic potential.

Free energy (for processes at constant volume1). It is defined as \(A = E - TS\).

Planck thermodynamic potential \(\varphi\). It is defined simply as \(\dfrac{F}{T}\) and has certain formal advantages in some thermodynamic equations.

Taking into account that these functions are named differently by different authors, it is useful to remember that the quantity \(F\) is measured by the total energy received, \(H\), minus the heat \(T\Delta S\) absorbed in a reversible process occurring at constant pressure. The total energy “liberated” in the process and capable of being converted into mechanical work is \(F\). In the same way, \(A\) represents the total energy that can be used for mechanical work in a reversible process occurring at constant volume.

Thermodynamic Relations2

Before turning to the consideration of problems of physicochemical equilibrium, it is useful to remind the reader of the relations existing between thermodynamic functions and independent variables, such as, for example, pressure, temperature, etc.

For any infinitesimally small reversible change,

\[ dE = T dS - P dV = dQ + dR. \tag{1} \]

The relation (1) is called the thermodynamic identity. Differentiating it, we obtain

\[ T=\left(\frac{\partial E}{\partial S}\right)_V;\qquad P=-\left(\frac{\partial E}{\partial V}\right)_S; \]

\[ \left(\frac{\partial T}{\partial V}\right)_S = \frac{\partial^2 E}{\partial V \partial S} = \frac{\partial}{\partial S}\left(\frac{\partial E}{\partial V}\right)_S = -\left(\frac{\partial P}{\partial S}\right)_V . \]

By definition, the heat capacity at constant volume is

\[ C_V=\left(\frac{\partial Q}{\partial T}\right)_V = T\left(\frac{\partial S}{\partial T}\right)_V = \left(\frac{\partial E}{\partial S}\right)_V \cdot \left(\frac{\partial S}{\partial T}\right)_V = \left(\frac{\partial E}{\partial T}\right)_V, \]

whence

\[ E_{T_1}-E_{T_2}=\int_{T_2}^{T_1} C_V\,dT. \]

The quantity \(E_{T_1}\) is usually found by integrating (most often graphically) the curve of the specific heat capacity as a function of temperature. As the initial energy \(E_{T_2}\), one may choose the energy at absolute zero, which will be considered in more detail below.

Let us now find the identity for the heat content \(H\). By definition we have

\[ dH=d(E+PV)=dE+d(PV)=TdS+VdP, \tag{2} \]

whence

\[ T=\left(\frac{\partial H}{\partial S}\right)_P;\qquad V=\left(\frac{\partial H}{\partial P}\right)_S; \]

\[ \left(\frac{\partial T}{\partial P}\right)_S = \frac{\partial^2 H}{\partial P\partial S} = \frac{\partial}{\partial S}\left(\frac{\partial H}{\partial P}\right)_S = \left(\frac{\partial V}{\partial S}\right)_P . \]

By definition, the heat capacity at constant pressure is

\[ C_P=\left(\frac{\partial Q}{\partial T}\right)_P = T\left(\frac{\partial S}{\partial T}\right)_P = \left(\frac{\partial H}{\partial S}\right)_P \left(\frac{\partial S}{\partial T}\right)_P = \left(\frac{\partial H}{\partial T}\right)_P . \]

From the last relation it follows that

\[ H_{T_1}-H_{T_2}=\int_{T_2}^{T_1} C_P\,dT. \]

Since in practice measurements of specific heat capacity are usually made at constant pressure, it proves much easier to find the value of the heat content \(H\) from experiment. In particular, from experimental data for solids and liquids one most often finds \(H\).

Between \(C_P\) and \(C_V\) there exist several well-known relations (see, for example, any course in thermodynamics). Thus, for example, \(C_P-C_V\) as a function of \(P\) and \(T\) has the form

\[ C_P-C_V=-T\left(\frac{\partial P}{\partial V}\right)_T \left(\frac{\partial V}{\partial T}\right)_P^2 = \frac{9\alpha^2 V}{K}\,T, \]

\[ * \]

where

\[ \alpha=\frac{1}{3V}\left(\frac{\partial V}{\partial T}\right)_P \]

is the linear coefficient of expansion, and

\[ K=-\frac{1}{V}\left(\frac{\partial V}{\partial P}\right)_T \]

is the coefficient of compressibility.

If the coefficient of compressibility \(K\) is unknown, then one often uses the experimental formula

\[ C_P-C_V=AT^{\frac{3}{2}}, \]

where the constant \(A\) is found in each individual case from the known values of \(C_P\) and \(C_V\) at one definite temperature.

Let us now find an identity for the free energy \(F(P,T)\). We have

\[ dF=d(H-TS)=dH-d(TS)=-S\,dT+V\,dP, \tag{3} \]

whence

\[ S=-\left(\frac{\partial F}{\partial T}\right)_P;\qquad V=\left(\frac{\partial F}{\partial P}\right)_T; \]

\[ \left(\frac{\partial S}{\partial P}\right)_T =-\frac{\partial^2 F}{\partial P\partial T} =-\frac{\partial}{\partial T}\left(\frac{\partial F}{\partial P}\right)_T =-\left(\frac{\partial V}{\partial T}\right)_P . \]

With the aid of the last relation one can readily find the expression for the entropy of an ideal monatomic gas.

In the case of an ideal monatomic gas

\[ PV=RT \quad \text{and} \quad C_P=\frac{5R}{2} =T\left(\frac{\partial S}{\partial T}\right)_P . \]

Further,

\[ dS=\left(\frac{\partial S}{\partial P}\right)_T dP +\left(\frac{\partial S}{\partial T}\right)_P dT =-\frac{R}{P}\,dP+\frac{5R}{2}\frac{dT}{T}, \]

whence

\[ S=R\ln\frac{T^{\frac{5}{2}}}{P}+\ln A, \]

where \(A\) is the entropy constant of an ideal monatomic gas. Its value will be obtained below.

From

\[ V=\left(\frac{\partial F}{\partial P}\right)_T \]

one can obtain an expression for the change of \(F\) in an isothermal process, provided only that the equation of state is known. In particular, for an ideal monatomic gas we obtain

\[ (dF)_T=V\,dP=\frac{RT}{P}\,dP, \]

\[ F_1-F_2=RT\lg\frac{P_1}{P_2}=RT\lg\frac{C_1}{C_2}, \]

where \(C_1\) and \(C_2\) are concentrations expressed in the corresponding units.

Finally, we obtain the identity for the free energy \(A(V,T)\)

\[ dA=d(E-TS)=-S\,dT-P\,dV . \]

and

\[ S=-\left(\frac{\partial A}{\partial T}\right)_V;\qquad P=-\left(\frac{\partial A}{\partial V}\right)_T;\qquad \left(\frac{\partial S}{\partial V}\right)_T=-\frac{\partial^2 F}{\partial V\partial T}=\left(\frac{\partial P}{\partial T}\right)_V . \]

It is not difficult to find the expression for the change in \(A\) in an isothermal process for an ideal gas:

\[ (dA)_T=-P\,dV=-\frac{RT}{V}\,dV, \]

whence

\[ A_1-A_2=RT\ln\frac{V_2}{V_1}. \]

Finally, let us also indicate two relations bearing the name of the Gibbs—Helmholtz equations,

\[ S=-\left(\frac{\partial F}{\partial T}\right)_p=-\frac{F-H}{T} \]

by the definition of \(F\), and

\[ S=-\left(\frac{\partial A}{\partial T}\right)_V=-\frac{A-E}{T} \]

by the definition of \(A\).

They prove very useful for finding the dependence of \(F\) and \(A\) on temperature from experimental data.

Thermodynamic Functions and Physicochemical Equilibrium

In principle, each of the functions \(S\), \(F\), \(A\), and \(\varphi\) may be used as a criterion of thermodynamic equilibrium; however, the choice of the most convenient function depends on the physical constraints imposed on the system.

The idea underlying the concept of thermodynamic equilibrium is as follows: in any system, owing to the motion of molecules, small local fluctuations of temperature, density, and energy always arise; however, if the system is in a state of thermodynamic equilibrium, these fluctuations do not lead to a spontaneous change in the state of the system as a whole. The only direction in which a spontaneous change of state could occur is indicated by the second law of thermodynamics.

If entropy is chosen as the function describing a certain system, then it can only either increase or remain constant, depending on whether the change of state of the system under consideration is irreversible or reversible. Formally this may be expressed as follows.

At constant volume and internal energy of the system, small fluctuations of the other variables lead to a change of entropy equal to zero for a system in equilibrium, and to an increase of entropy for a nonequilibrium system, i.e.,

\[ \delta S \geqslant 0. \]

Similarly, at constant heat content, small fluctuations must obey the condition

\[ \delta F \leqslant 0, \]

and at constant internal energy—the condition

\[ \delta A \leqslant 0. \]

If it is possible to express the dependence of the thermodynamic functions of the system through the concentrations of the components, etc., then the concentrations in the state of equilibrium can be obtained by setting equal to zero the first.

From a purely formal point of view, the choice of independent variables that are used in describing the thermodynamic properties of a system is given by Massieu’s rule. It states that from a thermodynamic function whose maximum or minimum, at constant values of the independent variables, expresses the condition of equilibrium, the thermodynamic properties of the system can be obtained by differentiation, i.e. in the most elegant form.

For practical purposes the most important function is the free energy per mole or—for systems consisting of more than one component—the partial free energy per mole. Before discussing how the latter can be applied to the calculation of the equilibrium constant, we shall prove that in each phase of an equilibrium system the free energy per unit mass must be one and the same.

The condition for the existence of two phases of a substance in equilibrium has the form: \(\delta S = 0\), under the additional conditions

\[ \delta m = 0; \qquad \delta E = 0; \qquad \delta V = 0. \]

Let us denote the fraction of the first phase by \(x\). Then, if \(m\) is the total mass of the system, the mass of the first phase is \(mx\), and the mass of the second phase is \(m(1-x)\). If \(S_1\) and \(S_2\) are the entropies per unit mass in the two phases, then the total entropy of the system \(S\) will be equal to

\[ \frac{S}{m}=xS_1+(1-x)S_2 . \]

The total volume \(V\)

\[ \frac{V}{m}=xV_1+(1-x)V_2 . \]

The total energy \(E\)

\[ \frac{E}{m}=xE_1+(1-x)E_2 . \]

Applying the condition \(\delta S=0\), we obtain, since the mass is constant,

\[ x\delta S_1+(1-x)\delta S_2+(S_1-S_2)\delta x=0. \]

Further,

\[ \delta S_1=\frac{\delta E_1+P_1\delta V_1}{T_1}; \qquad \delta S_2=\frac{\delta E_2+P_2\delta V_2}{T_2}. \]

Eliminating \(\delta V_2\) and \(\delta E_2\) with the aid of these equations, we have

\[ x\delta E_1\left(\frac{1}{T_1}-\frac{1}{T_2}\right) +x\delta V_1\left(\frac{P_1}{T_1}-\frac{P_2}{T_2}\right)+ \]

\[ +\left[S_1-S_2-\frac{(E_1-E_2)}{T_2}+P_2\frac{(V_1-V_2)}{T_2}\right]\delta x=0. \]

Since the variations \(\delta E_1\) and \(\delta V_1\) are independent, this equation can, generally speaking, be valid only if the coefficients of the variations vanish. Thus \(T_1=T_2\) and \(P_1=P_2\), and, moreover,

\[ -S_1+\frac{E_1+PV_1}{T}=-S_2+\frac{E_2+PV_2}{T}, \]

i.e.

\[ F_1=F_2. \]

In the case of a system consisting of more than one component, it can similarly be shown that the partial free energy per unit mass must be the same in all phases. Thus this quantity may be called the thermodynamic potential of the components under consideration. If the molecular weights in all phases are the same, then the partial free energy per mole must likewise be the same in all phases.

In conclusion it should be noted that the definition of thermodynamic equilibrium is based on the criterion of the effect of small fluctuations of thermodynamic functions. This leads to a satisfactory consideration of such equilibria as, for example, the vapor pressure of solid nitric oxide, or the equilibrium

\[ 2\mathrm{H}_2+\mathrm{O}_2 \rightleftarrows 2\mathrm{H}_2\mathrm{O}. \]

However, if sufficiently large changes of energy could take place, then over a sufficiently long interval of time nitric oxide at low temperatures would completely decompose into hydrogen and oxygen, and the hydrogen and oxygen nuclei would probably be transformed into helium nuclei. Practically, every thermodynamic equilibrium is a pseudo-equilibrium in the sense that the minimum of free energy is not absolute with respect to all possible changes, but only with respect to those changes which take place during an observable interval of time.

Equilibrium Constant

The most convenient condition of equilibrium for practical purposes is \(\delta F=0\), since most experiments are connected with processes occurring at constant pressure. The application of this condition can be illustrated by calculating the change in free energy in gaseous reactions, such as, for example,

\[ \mathrm{CO}+\mathrm{H}_2\mathrm{O}\rightleftarrows \mathrm{CO}_2+\mathrm{H}_2. \]

Neglecting deviations from the laws of ideal gases that are insignificant for our purposes, one may write the change in free energy upon changing the concentration of one of the components in the form

\[ F_1-F_2=RT\ln \frac{c_1}{c_2}. \]

This expression can be formally simplified if, as the initial state, one chooses the state with concentration equal to unity (or, if partial pressures are used, the state with a pressure of \(1\ atm\)), i.e. \(C_2=1\). Denoting the free energy of this initial state by \(F_0\), we obtain for the free energy in any other state

\[ F_1-F_0=RT\ln c_1 \]

and the total change in free energy in the reaction

\[ \Delta F=\sum F_1-\sum F_0=RT\sum \ln c_1. \]

Up to now all the \(c_1\) have remained arbitrary; they may be chosen so that, for the equilibrium concentrations of the mixture, one has \(\sum F_1=0\), and then

\[ RT\sum c_{\mathrm{eq}}=-\sum F_0 \]

(where \(c_{\mathrm{eq}}\) denotes the concentration in the state of equilibrium. Transl.). This follows from the fact that, in the infinitesimal transformation of \(\delta n\) molecules in the equilibrium mixture,

\[ \delta n(\mathrm{CO}+\mathrm{H_2O})\longrightarrow \delta n(\mathrm{CO_2}+\mathrm{H_2}) \]

the change in free energy (since the system is in equilibrium)

\[ \delta n\sum F_1=\delta F=0, \]

so that

\[ \sum F_{1\mathrm{eq}}=0. \]

Thus, for every state of equilibrium,

\[ -\sum F_0=RT\sum \ln c_{\mathrm{eq}}=RT\ln K, \]

and since the left-hand side is constant, it follows that

\[ K=\mathrm{const}. \]

The equilibrium constant \(K\) is called the constant of the law of mass action. Thus, by this method one directly obtains the constant of the law of mass action referred to the just-specified initial state.

This equation plays an extraordinarily important role in applying thermodynamic functions to the calculation of equilibrium constants. However, if thermodynamic functions were determined experimentally only from the study of equilibrium constants, of the electromotive force of reversible cells, and of limited

of a number of other physicochemical measurements, from which the free energy can be calculated, then this would, in practical terms, add little that is essentially new.

A considerably more general method for calculating thermodynamic functions is to calculate them from thermal data, which in many cases can be obtained very easily directly from calorimetric measurements. The Gibbs—Helmholtz equation

\[ \left(\frac{\partial F}{\partial T}\right)_P = -S = \frac{F-H}{T} \]

after transformation and integration gives

\[ \frac{F}{T}=\varphi=-\int \frac{H}{T^2}\,dT+I, \]

where \(I\) is the constant of integration. Thus the value of \(F\) can be calculated from the value of \(H\), provided that the constant of integration \(I\) is known. Since

\[ F=-RT\ln K, \]

then, introducing the equilibrium constant into the preceding equation, it can be written in another form as

\[ \ln K=\int \frac{H}{RT^2}\,dT-I. \]

With the aid of this formula one can construct interpolation formulas for the values of \(F\) and \(K\) at different temperatures from the following experimental data:

1. The values of \(H\) over the entire temperature range for which the interpolation formula is being constructed. Since for any reaction

\[ \left(\frac{\partial H}{\partial T}\right)_P=\sum C_p, \]

one can always write

\[ H_{T_2}=H_{T_1}+\int_{T_1}^{T_2}\left(\sum C_p\right)dT. \]

Therefore it is sufficient to determine experimentally the value of \(H\) for some single temperature and the specific heat over the entire temperature range under consideration.

1. To determine the constant of integration \(I\), it is also necessary to measure experimentally the value of \(F\) at some one temperature.

Another method can also be given for calculating chemical equilibria from thermal data, which will be used below. For any chemical equilibrium

\[ n_1A+n_2B+\ldots=n_1'A_1'+n_2'B_2'+\ldots \]

\[ F=-RT\ln K=\sum n_r F_r, \]

where the index indicating that the free energy is referred to the initial state with concentration equal to unity has been omitted, since no misunderstanding can arise here.

This expression may be transformed into

\[ F=\sum n_r(H_r-TS_r)=\sum n_rH_r-T\sum n_rS_r=H-T\sum n_rS_r, \]

where \(H\) is the change in heat content of the whole system. Thus the determination of \(F\) is again reduced to the determination of \(H\) and of the entropies of the individual substances \(S_r\) taking part in the reaction. Since the entropy of any state can be expressed through the heat capacities

\[ S_{T_1}=S_{T_2}+\int_{T_1}^{T_2} C_p d\ln T, \]

the construction of an interpolation formula for \(F\) as a function of temperature apparently again requires:

1) the determination of \(H\) at some temperature,
2) knowledge of the quantities \(C_p\) over the entire chosen temperature range,
3) the determination of the entropies of the individual substances at some temperature.

In most cases experimental difficulties arise in determining at least one value of the free energy or of the entropy change in the reaction, since in doing so certain types of reversible processes must be carried out and studied. These difficulties can be avoided with the aid of the third law of thermodynamics.

Determination of thermodynamic functions from thermal data

The practical calculation of the functions \(H\), \(S\), \(F\), etc., from calorimetric measurements is based entirely on two hypotheses, first stated by Nernst and sufficiently well verified and clarified since they were combined into the third law of thermodynamics.

Determination of the quantity \(H\): Nernst’s first hypothesis

The calorimetric determination of the change in heat content in chemical reactions presents no great experimental difficulty, provided that the measurement can be made at a suitable temperature. The value of \(H\) at any other temperature may then be calculated from the equation

\[ H_{T_1}-H_{T_2}=\int_{T_2}^{T_1}\left(\sum C_p\right)dT. \]

The value of \(H\) at absolute zero is of special interest.

To find it, it is necessary to extrapolate the measured values of the specific heat capacity \(C_p\) into the region of temperatures lying below those temperatures at which measurements have still been made. Nernst proposed that for the “condensed” phase (solids and liquids)

\[ \lim_{T \to 0}\left(\frac{\partial H}{\partial T}\right)_P=\lim_{T \to 0} C_p=0. \]

This means that extrapolation from such low temperatures, which can still easily be obtained, to absolute zero may be carried out by means of the integral

\[ \int_0^T C_p\,dT. \]

This is immediately evident from Fig. 1. Usually, for extrapolation at low temperatures one uses the theoretical expression obtained by Debye (see p. 70), \(C_V=\alpha T^3\). Keesom showed that for some metals the expression \(C_V=\alpha' T\) is closer to the experimental results\(^2\), but the use of the more accurate expression gives a very small difference in the area bounded by the curve, since in both cases \(\left(\dfrac{\partial H}{\partial T}\right)_P\) is very small.

Fig. 1.

Determination of the quantity \(S\); Planck’s formulation of the heat theorem.

The calculation of absolute entropy (i.e. entropy referred to the crystalline solid at absolute zero as the initial state) is a considerably more difficult problem than the calculation of heat content, and represents one of the principal tasks of modern thermodynamics. When \(S\) and \(H\) are known, the calculation of \(F\) and of equilibrium constants presents no difficulties. By the definition of absolute entropy,

\[ S_p=\int C_p d(\ln T)+S_0, \]

where \(S_0\) is the entropy at absolute zero.

Planck assumed that the entropy of crystalline bodies at absolute zero is equal to zero, i.e. \(S_0=0\). A narrower assumption consists in the statement that every body possesses some finite positive entropy which, however, for the ideal crystalline state of the substance at absolute zero becomes zero. This Planck formulation of the heat theorem has the further advantage that it is considerably easier to ascertain its statistical meaning. The connection of this formulation with Nernst’s two original hypotheses,

\[ \lim_{T \to 0}\frac{\partial H}{\partial T}=0 \quad \text{and} \quad \lim_{T \to 0}\frac{\partial F}{\partial T}=0 \]

is evident, since

\[ \frac{\partial F}{\partial T} = \frac{\partial H}{\partial T} - S - T \frac{\partial S}{\partial T} = \frac{\partial H}{\partial T} - S - \frac{C_P}{T}, \]

which tends to zero only if

\[ S_0=0 \quad\text{and}\quad \lim_{T\to 0} C_P=0. \]

The equations have been written in the form of limits in order to indicate that the approach to zero is asymptotic, i.e., that \(C_P\) decreases with \(T\) faster than the first power of \(T\).

The application of the third law of thermodynamics reduces the calculation of entropy to the evaluation of the integral

\[ S=\int C_P d(\ln T), \]

since \(S_0=0\). The inaccuracy of the experimental determination of entropy due to heat-capacity anomalies encountered at very low temperatures must always be taken into account.

A more exact study of the question of how widely these anomalies are distributed indicates at the same time the extent to which Nernst’s heat theorem can be applied for practical purposes. It is useful to note that whenever heat-capacity anomalies occur at such low temperatures that measurements become impossible, the entropy values calculated from measurements of specific heat turn out to be too low. In this case agreement with entropy values obtained by other methods can be achieved only by assigning to \(S_0\) a finite positive value. This explains the large number of apparent deviations from the third law of thermodynamics.

Applications and Verification of the Heat Theorem

The methods of practical application of the heat theorem are best clarified by the methods used for its verification.

a) Verification of the heat theorem for condensed systems. If, for a system consisting only of solid and liquid bodies, the equilibrium constant can be measured directly, then there are means suitable for verifying Nernst’s heat theorem. The number of such cases is not very large. An example of such a verification may be the calculation of the transition temperature or of the equilibrium between two solid phases. The method reduces simply to the experimental determination of one value of \(H\) and of heat-capacity values over the entire temperature range. Then the absolute entropies are easily calculated according to Planck’s law, and thereafter, from the entropy and heat content, the free energy and the equilibrium constant are found. The results obtained must agree, within the limits of experimental error, with the directly determined quantities.

Let us consider, for example, gray tin, which is an allotropic modification of white tin and possesses a lower

density. At temperatures below \(18^\circ\text{C}\) white tin tends spontaneously to pass into gray, while above \(18^\circ\text{C}\) gray tin spontaneously passes into white. Thus \(18^\circ\text{C}\) is the transition temperature at ordinary pressure. From measurements of specific heats carried down to very low temperatures, and taking \(S_0=0\), we have for both forms

\[ S_{25^\circ \mathrm{C}}=11.17 \ \text{(white tin)};\qquad S_{25^\circ \mathrm{C}}=9.23 \ \text{(gray tin)}. \]

\[ \text{The difference is } +1.94 \]

(the entropy of the less stable form must inevitably be smaller, since a spontaneous change at constant volume can occur only with an increase of entropy).

This result was compared with the directly determined value \(\Delta S_{\text{gray}\to\text{white}}\). \(\Delta H\) is measured calorimetrically, for example, by dissolving each allotropic modification in acid in a calorimeter. \(\Delta F\) is measured from the electromotive force \(E\) of the cell

\[ \text{gray tin/electrolyte/white tin}, \]

since the change in free energy when a faraday quantity of electricity \(F\) is passed is

\[ -NEF=\Delta F-RT\ln K. \]

From the equation \(\Delta F=\Delta H-T\Delta S\) one obtains an experimental value of \(\Delta S\), equal to \(+1.87\) (gray \(\to\) white). This agrees well with the values obtained directly from thermal data.

Other systems were also investigated, involving reactions only between solids, such as:

\[ \text{rhombic sulfur} \rightleftarrows \text{monoclinic sulfur} \]

\[ \mathrm{Pb}+J_2 \rightleftarrows \mathrm{Pb}J_2, \]

\[ \mathrm{Pb}+2\mathrm{AgCl} \rightleftarrows \mathrm{PbCl}_2+2\mathrm{Ag}, \]

\[ \mathrm{Hg}+\mathrm{AgCl} \rightleftarrows \mathrm{HgCl}+\mathrm{Ag}. \]

Within the limits of experimental error, in all cases agreement was achieved between direct measurements and thermal measurements based on the heat theorem.

b) Verification of the heat theorem by the equilibrium of a condensed system with a gas phase. The heat theorem is directly applicable only to “condensed” systems, and although at very low temperatures and high pressures even gases apparently degenerate, i.e. for them too \(C_p\to 0\), the experimental measurements have not been carried sufficiently far into the region of low temperatures to make it possible to use this fact for calculating entropy. Therefore it is impossible to calculate the absolute entropy of a gas using only thermal data and the laws of thermodynamics.

A statistical calculation (p. 51) shows that the constant

entropy \(A\) for a monatomic ideal gas, occurring in the expression obtained on p. 36,

\[ S=R\left(\ln \frac{T^{\frac{5}{2}}}{P} A\right), \]

has the form

\[ A=\frac{(2\pi m k)^{\frac{3}{2}} k e^{\frac{5}{2}}}{h^3}, \]

where \(m\) is the mass of the molecule, \(k\) is Boltzmann’s constant \(\left(k=\frac{R}{N}\right)\), and \(h\) is Planck’s constant. If the atomic weight is substituted and the pressure is expressed in atmospheres, then the expression takes the form

\[ S=R \ln \frac{M^{\frac{3}{2}}T^{\frac{5}{2}}}{P} - 2.300\ \text{cal/deg. mol.} \]

With the aid of this statistical result one can, in various ways, verify that for crystalline substances \(S_0=0\).

I. Equation for the vapor pressure over a solid

The most direct way in which the third law of thermodynamics can be tested is by comparing theoretical and experimental values for the vapor pressure of ideal gases or monatomic metal vapors, such as, for example, the vapors of Hg, Zn, Cd, over the surface of the corresponding solid specimen.

The increase of entropy per mole upon sublimation is

\[ \Delta S=S_{\text{gas}}-S_{\text{solid}}=\frac{\lambda}{T} =\left[R\ln \left(T^{\frac{5}{2}}M^{\frac{3}{2}}\right)-R\ln P-2.30\right] -\left[\int_0^T C_{p_s}\,d(\ln T)+S_0\right], \]

where \(\lambda\) is the latent heat of sublimation per mole, \(C_{p_{\text{solid}}}\) is the specific heat of the solid, and \(S_0\) is the entropy of the solid at \(0^\circ\mathrm{K}\).

This expression is only another way of writing the usual formula for the vapor pressure over the surface of a solid (giving monatomic vapors), since it can be rewritten in the form

\[ \ln P=-\frac{\lambda}{RT}+\frac{5}{2}\ln T-\frac{1}{R}\int C_{p_{\text{solid}}}\,d(\ln T)+ \]

\[ +\left[\frac{3}{2}\ln M-\frac{2.30}{R}-\frac{S_0}{R}\right]. \]

or, taking into account \(^{1}\) that

\[ \lambda=\lambda_0+\frac{5RT}{2}, \]

then

\[ \ln P=-\frac{\lambda_0}{RT}+\frac{5}{2}\ln T-\frac{1}{R}\int C_{P_{\mathrm{tv}}}\,d(\ln T)+ \]

\[ +\left[\frac{3}{2}\ln M-\left(\frac{S_0+2.30}{R}\right)-\frac{5}{2}\right]. \]

The temperature-independent term in this equation is called the vapor-pressure constant \(i\).

In checking the validity of the heat theorem, the most convenient procedure is usually to compare the value of the entropy of a gas at some temperature, calculated directly from the equation

\[ S_{\mathrm{gas}}=R\ln\left(T^{\frac{5}{2}}\frac{A}{P}\right), \]

with the experimental value determined from the sublimation process

\[ S_{\mathrm{gas}}=\frac{\lambda}{T}+S_{\mathrm{tv}}. \]

In the experimental determination of \(S_{\mathrm{gas}}\), the entropy of the solid is taken, according to the heat theorem, to be equal to zero at absolute zero, and the value of \(\lambda\) is calculated from the slope of the vapor-pressure curve as a function of \(\frac{1}{T}\). Typical results at \(25^\circ\mathrm{C}\) and one atmosphere are given in Table 1.

TABLE 1

Gas	Experimental entropy values	Calculated entropy values
He	29.2	29.8
Ar	36.4	36.7
Cd	40.0	39.8
Hg	41.3	41.5

The good agreement simultaneously gives confirmation of both the statistical theory and the heat theorem.

An older method, which did not use the absolute value of entropy, reduces to comparing the experimental values of the vapor-pressure constant with its theoretical value. According to the Clausius–Clapeyron equation

\[ \frac{dP}{dT}=\frac{\lambda}{(v_{\mathrm{gas}}-v_{\mathrm{tv}})T}. \]

\(^{1}\) The heat of transition solid body—gas is \(\lambda=H_{\mathrm{tv}}-H_{\mathrm{gas}}\). Substituting \(H_{\mathrm{tv}}\) and \(H_{\mathrm{gas}}\), we have

\[ \lambda=T(C_p-C)+N(V_{\mathrm{gas}}-V_{\mathrm{tv}})=\frac{5}{6}RT+\lambda_0, \]

where \(\lambda_0\) does not depend on temperature. Trans. note.

At very low pressures, to which the experimental data refer, \(v_{\text{gas}}=\dfrac{RT}{P}\), where \(v\) is the volume of one mole, and \(v_{\text{tv}}\ll v_{\text{gas}}\). The equation becomes

\[ \frac{1}{P}\frac{dP}{dT}=\frac{\lambda}{RT^{2}} \]

or

\[ \ln P=\int \frac{\lambda\, dT}{RT^{2}}+i, \]

where \(i\) is the vapor-pressure constant. In this expression

\[ \frac{d\lambda}{dT}=C_{p\text{gas}}-C_{p\text{tv}}, \]

i.e.,

\[ \lambda=\lambda_{0}+\int (C_{\text{gas}}-C_{\text{tv}})\,dT, \]

so that

\[ \ln P=\frac{\lambda_{0}}{RT}+\iint \frac{dT}{RT^{2}}(C_{\text{gas}}-C_{\text{tv}})\,dT, \]

or, integrating by parts,

\[ \ln P=-\frac{\lambda_{0}}{RT}+\frac{1}{R}\int (C_{\text{gas}}-C_{\text{tv}})\,d(\ln T)- \]

\[ -\frac{1}{RT}\int (C_{\text{gas}}-C_{\text{tv}})\,dT+i. \]

This may be compared with the direct expression for the entropy of sublimation

\[ S_{\text{gas}}-S_{\text{tv}}=\frac{\lambda}{T}=\frac{\lambda_{0}}{T_{0}}+\frac{1}{T}\int (C_{\text{gas}}-C_{\text{tv}})\,dT, \]

\[ S_{\text{gas}}=\int C_{\text{gas}}\,d(\ln T)-R\ln P, \]

\[ S_{\text{tv}}=\int C_{\text{gas}}\,d(\ln T)+S_{0}. \]

Let us rewrite the equation for the entropy of sublimation:

\[ \ln P=-\frac{\lambda_{0}}{kT}-\frac{1}{RT}\int (C_{\text{gas}}-C_{\text{tv}})\,dT+ \]

\[ +\frac{1}{R}\int (C_{\text{gas}}-C_{\text{tv}})\,d(\ln T)-\frac{S_{0}-A}{R}. \]

We see that if the pressure is expressed in the same units and the heat theorem is applied to the solid phase \((S_{0}=0)\), then

\[ i=\frac{A}{R} \]

(some authors include in the expression for \(i\) the term \(\dfrac{1}{RT}\int C_{\text{gas}}\,dT\), since for an ideal monatomic gas it does not depend on tempe-

temperature and is equal to \(5/2\)). The theoretical value of the vapor-pressure constant \(i\) is simply calculated from the theoretical value of the constant entropy for the corresponding vapors and is compared with the experimentally determined values.

II. Comparison of equilibrium constants with vapor-pressure constants

Although the use of the free-energy and entropy functions leads to the most compact and general method for calculating equilibrium, formerly one often used vapor-pressure constants or “chemical constants.” The relation between these two different methods of calculation may be illustrated by the reaction

\[ aA + bB + cC \rightleftarrows dD + eE \ldots \]

\[ \Delta F = -RT \ln K = \Delta H - T \Delta S = \Delta H_0 + \int_0^T \left( \sum C_p \right) dT - \]

\[ - T \int \left( \sum C_p \right) d(\ln T) - T \sum S_{0\,\mathrm{tv}} - \]

\[ - T \int_0^T \left( \sum C_{p\,\mathrm{gaz}} \right) d(\ln T) - T \sum S_{0\,\mathrm{gaz}} . \]

This expression may be rewritten in the form

\[ -\ln K = \frac{\Delta H_0}{RT} + \frac{1}{RT} \int \sum C_p\, dT - \frac{1}{R} \int \sum (C_{\mathrm{tv}} + C_{\mathrm{gaz}})\, d(\ln T) - \]

\[ - \sum \frac{S^0_{\mathrm{tv}}}{R} - \sum \frac{S^0_{\mathrm{gaz}}}{R}, \]

where \(S_0\) is the entropy, and \(H_0\) is the heat content at \(0^\circ\mathrm{K}\).

This may be compared with the equation for an isochoric reaction

\[ \frac{d \ln K}{dT} = \frac{\Delta H}{RT^2}, \]

which, after integration, gives

\[ \ln K = \int \frac{\Delta H}{RT^2}\, dT + I . \]

Substituting the value of \(\Delta H\), we have

\[ \ln K = - \frac{\Delta H_0}{RT} + \frac{1}{R} \int \frac{dT}{T^2} \int \left( \sum C_p \right) dT + I \]

and, integrating by parts, we obtain

\[ \ln K = - \frac{\Delta H_0}{RT} - \frac{1}{RT} \int \sum C_p\, dT + \frac{1}{R} \sum C_p \frac{dT}{T} + I . \]

From comparison with the preceding formulas it is obvious that the constant of integration is

\[ I=\sum \frac{S_{0\mathrm{tv}}}{R}+\sum \frac{S_{0\mathrm{gaz}}}{R}, \]

so that, if for solids one applies the heat theorem

\[ \sum S_{0\mathrm{tv}}=0, \]

the constant of integration can be expressed through the individual vapor-pressure constants according to the equation

\[ I=\sum \nu\cdot i=\sum \frac{S_{0\mathrm{gaz}}}{R}. \]

(\(\nu\) is the number of gas molecules participating in the reaction).

Thus, by comparing the calculated values of the constant of integration with the sum of the individual vapor-pressure constants, one can test the applicability of Nernst’s heat theorem to solids.

Although, owing to unavoidable experimental errors, heterogeneous equilibrium is not as convenient for exact calculations as the vapor pressure of individual substances, nevertheless this method has the theoretical advantage that it is not connected with statistical theory, since in it—provided that the experimental vapor-pressure curves are known—the theoretical values of the entropy constants for gases do not appear. Therefore this method is directly applicable to the equilibrium of systems including polyatomic gases. Typical results are given in the following Table 2³.

TABLE 2

Reaction	Limits of \(I\) (from equilibrium reactions)	Limits of \(I\) (from equilibrium reactions)	Limits of \(\sum \nu_i\) (from vapor pressures)	Limits of \(\sum \nu_i\) (from vapor pressures)
\(3\mathrm{H}_2+\mathrm{N}_2 \rightleftarrows 2\mathrm{NH}_3\)	\(-7.04\)	\(0.10\)	\(-8.34\)	\(0.09\)
\(\mathrm{N}_2+\mathrm{O}_2 \rightleftarrows 2\mathrm{NO}\)	\(0.93\)	\(0.28\)	\(0.61\)	\(0.15\)
\(2\mathrm{CO}+\mathrm{O}_2 \rightleftarrows 2\mathrm{CO}_2\)	\(-0.8\)	\(0.25\)	\(-1.38\)	\(0.20\)
\(\mathrm{H}_2+\mathrm{J}_2 \rightleftarrows 2\mathrm{HJ}\)	\(-1.5\)	\(0.12\)	\(-2.36\)	\(0.22\)
\(2\mathrm{CO} \rightleftarrows \mathrm{C}+\mathrm{CO}_2\)	\(-0.86\)	\(0.18\)	\(-1.01\)	\(0.18\)
\(2\mathrm{Hg}_{\mathrm{vap}}+\mathrm{O}_2 \rightleftarrows 2\mathrm{HgO}\)	\(4.32\)	\(0.18\)	\(4.44\)	\(0.09\)
\(\mathrm{CaO}+\mathrm{CO}_2 \rightleftarrows \mathrm{CaCO}_3\)	\(0.9\)	\(0.15\)	\(0.91\)	\(0.15\)
\(\mathrm{H}_2+\mathrm{HgO} \rightleftarrows \mathrm{Hg}+\mathrm{H}_2\mathrm{O}\)	\(-3.63\)	\(0.16\)	\(-3.69\)	\(0.03\)

In some cases agreement is observed within the limits of the deviations, which are fairly large in this method of testing. Reactions in which hydrogen participates almost always give poor results. For reactions involving CO and NO, the inaccuracies of this method are too large, but a more precise method (p. 76) shows that the discrepancy really exists and gives some indication for explaining the source of these discrepancies.

Chemical Constants of Polyatomic Gases

The vapor pressures, owing to their role in the integration of the equations of isochoric reactions, have received the name of chemical constants. If their values are known, then in principle it becomes possible to calculate any chemical equilibria exclusively from thermal data.

The chemical constants of gases, and the constant entropies closely connected with them, can be calculated from statistical theory. For monatomic gases the constant entropy \(A\) in the expression

\[ S_p = R \ln \frac{T^{5/2}}{p} A \]

is equal to

\[ A = \frac{(2\pi mk)^{3/2} k e^{5/2}}{h^3}. \]

As will be explained subsequently, if an atom or molecule of a gas can, in addition to the energy of translational motion, also possess energy connected with other kinds of motion (vibrations, rotations, etc.), then the constant entropy must be multiplied by certain factors. This circumstance must always be taken into account, provided that such excitation of non-translational kinds of motion can be observed in the temperature region in which the equilibrium is being considered. For present purposes this rule may be formulated as follows:

1. If atoms have electronic multiplicity (internal quantum number \(j > 0\)), then \(A\) is multiplied by the factor \(g = 2j + 1\), because with the modified statistical counting there can now exist in the gas phase \(2j + 1\) atoms with very close, but nevertheless distinct, energy levels, instead of one at multiplicity equal to unity. This has been experimentally verified for thallium, whose ground state is \(^{2}P_{1/2}\) and \(g = 2\). It indeed has a correspondingly larger entropy of sublimation.^4

2. For polyatomic molecules the term \(A\) is further multiplied by a factor connected with the rotational energy,

\[ \frac{8\pi^2 I k}{h^2} \, \frac{1}{S}. \]

The moment of inertia of the molecule \(I = \sqrt[3]{I_1 I_2 I_3}\) is determined from rotational bands in band spectra. The symmetry factor \(S\) plays a role only for symmetric molecules of the type \(N_2, O_2\), for which it is equal to 2.

Practical Computation of the Position of Equilibrium

Before proceeding to the further discussion of the heat theorem, it is desirable to turn to the question of the practical application of chemical constants. For most purposes, departures from the heat theorem do not exert a great influence on the values of equilibrium constants computed on the assumption of its validity. The only systematic method of computing equilibria consists in determining the entropy and heat-content functions for as large a number of substances as possible, since from them one immediately obtains the value \(\Delta F=-RT\ln K\). The use of constant vapor pressures is formally equivalent to this method, but in practice it leads to tables of data from which the desired results can be obtained only with considerably greater labor.

As an example of the computation of an equilibrium not feasible by other methods, one may cite the case of ionization in stars.

For the process

\[ A \rightleftarrows A^+ + e \]

the change of entropy per mole may be written in the form

\[ \Delta S=S_{A^+}+S_e-S_A. \]

Neglecting the change in the multiplicity of the atom upon ionization, i.e. the change of the factor \(g\), which of course affects the entropy constant, one may put

\[ S_{A^+}=S_A \quad \text{and} \quad \Delta S=S_e. \]

For the equilibria under consideration the pressures are so small, and the temperatures so high, that the electron gas is extremely rarefied and may be regarded, in the first approximation, as an ideal monatomic gas with atomic weight \(M=0.000544\). Then the entropy constant for the electron gas can readily be computed. Using the formula for the entropy of a monatomic gas, we obtain

\[ \Delta S=S_e=\frac{5}{2}R\ln T+\frac{3}{2}R\ln M-2.30=5\ln T-24.70. \]

The change in heat content is approximately equal to the ionization energy per mole. If \(V\) is the ionization potential in volts, then

\[ \Delta H_0=23.3\,V\cdot 10^3\ \text{cal.}, \]

\[ \Delta H=\Delta H_0+\int (C_A+C_e-C_{A^+})\,dT =\Delta H_0+\frac{5RT}{2}, \]

since \(C_A=C_{A^+}\).

Then

\[ \Delta F=-RT\ln\frac{A^+e}{A} =-\frac{23.3\,V\cdot 10^3}{4.57T} +\frac{11.5}{4.57}\lg T-\frac{29.7}{4.57} =-\frac{5.1V\cdot 10^3}{T} \]

\[ +\,2.5\lg T-6.5. \]

Since \(V\) is equal to \(4\text{--}5\) V even for the alkali metals, the right-hand side is positive only at \(T \geq 10^4{}^\circ\mathrm{C}\). In the spectrum of the sun

strong lines of Na are observed, weak lines of K, and the lines of Rb and Cs are entirely absent. This fact can easily be explained with the aid of the formula obtained. Indeed, it follows directly from it that, at a given temperature, the number of un-ionized atoms will be greatest for that of the metals which has the highest ionization potential. At the same time it is obvious that the intensity of the spectral lines of the atomic spectrum is proportional to the number of (non-ionized) atoms of the given substance.

Approximate Nernst Formula

For very rough estimates of the position of equilibrium in various reactions, the approximate Nernst formula often gives quite satisfactory results. It may be written in the form

\[ \lg K_p=-\frac{\Delta H}{4{,}6T}+\sum n(1{,}75\lg T)+\sum C, \]

where all terms of the sum, when summing, are usually taken as positive; \(n\) is the number of molecules of each substance participating in the reaction. For reliable results, \(\sum n\) is no greater than \(\pm 1\); \(C\) is an empirical chemical constant having nothing in common with the vapor-pressure constant and introduced only into this empirical formula. The following Table 3 is given by Nernst\(^5\).

TABLE 3

H₂	1,6	HCl	3,0	H₂S	3,0
N₂	2,6	HBr	3,2	SO₂	3,3
O₂	2,8	HJ	3,4	CO₂	3,2
Cl₂	3,1	NO	3,5	NH₃	3,3
Br₂	3,2	CO	3,5	H₂O	3,6
J₂	3,9	N₂O	3,3

For solids \(C=0\), and for atoms \(C\) is equal to half the value for the corresponding molecules. An example of the extent to which equilibrium can be calculated by this formula is the reaction

\[ \mathrm{H_2+Cl_2 \rightleftarrows 2HCl};\quad \Delta H_{15^\circ \mathrm{C}}=-44\,000\ \text{cal}. \]

Temperature	\(\lg K_p\) Observed	\(\lg K_p\) Calculated
300°	33,2	33,38
1000°	10,43	10,93
2000°	5,77	6,11

However, this equation must be applied with caution, since the values of \(C\) are valid only within certain limits.

A. R. UBBELOHDE

Limits of Applicability of the Heat Theorem and Deviations from It

According to Nernst’s heat theorem in Planck’s formulation, at \(T=0\) the absolute entropy of crystalline solids is \(S_0=0\). In Nernst’s formulation for condensed systems,

\[ \lim_{T \to 0}\frac{\partial H}{\partial T}=0=\lim_{T \to 0}\frac{\partial F}{\partial T}. \]

The principal thermodynamic methods for testing this theorem are always connected with the equilibrium of systems with one or a greater number of components. However, any method of studying the change of thermodynamic functions with temperature on approaching absolute zero can in principle serve to test the theorem. An interesting example is liquid helium.

Liquid Helium

The properties of liquid helium are of great interest, since it is the only substance that remains liquid at the lowest of the attainable temperatures. Its behavior at low temperatures shows that it obeys the heat theorem.

Surface tension. The change in free energy associated with a change of surface is \(\Delta F=\sigma \Delta \Omega\), where \(\sigma\) is the surface tension and \(\Omega\) is the area. The heat theorem requires that

\[ \lim_{T \to 0}\frac{\partial \Delta F}{\partial T}=0=\lim_{T \to 0}\frac{d\sigma}{dT}. \]

This was in fact observed for liquid helium.

Change of the melting point as a function of pressure. Although, at a pressure equal to that of its own saturated vapor, helium remains liquid below the lowest temperatures attained (\(0.8^\circ\ \mathrm{K}\) or less), i.e., it does not give a triple point, it solidifies at a pressure of \(140.5\ \mathrm{atm}\) at \(4.23^\circ\ \mathrm{K}\) and at a pressure of \(25.3\ \mathrm{atm}\) at \(2.15^\circ\ \mathrm{K}\). As the temperature approaches zero, the melting curve tends to become parallel to the temperature axis. This circumstance could have been expected from thermodynamic considerations.

According to the Clausius–Clapeyron equation, the change in entropy on melting, \(\Delta S\), is determined from the formula

\[ \frac{\Delta S}{\Delta V}=\frac{dP}{dT}. \]

Since on melting \(\Delta V \ne 0\), then, if the heat theorem is applicable and \(\Delta S \to 0\) as \(T \to 0\), it follows from the above equation that

\[ \frac{dP}{dT}\to 0. \]

This latter property was indeed observed experimentally.

The fact that helium obeys the heat theorem, despite the fact that at ordinary pressures it remains liquid down to the very lowest temperatures, shows that the entropy of the liquid can in some way decrease at these temperatures. This is apparently connected with the transition into a second liquid phase below \(2^\circ,3\mathrm{K}\). This transition can be traced from measurements of density, surface tension, specific heat, etc. The transformation of the first liquid phase into the second occurs without latent heat of transition. From thermodynamic considerations this could have been expected only in the case of two immiscible liquids in equilibrium at the transition temperature. However, at the transition point there is a sharp maximum of the specific heat, similar to that observed for ammonium chloride (p. 77) and other internal transformations in one and the same phase. Also extremely characteristic are the slow increase from the side of low temperatures and the rapid fall from the side of high temperatures.

Fig. 2.

Some data on the nature of helium II may be obtained from thermodynamic considerations. Between the thermodynamic behavior of liquid helium II and that of solid helium at very low temperatures there exists a well-known analogy. Namely, with decreasing temperature the change of entropy in the transition from the solid phase to liquid helium II gradually decreases and tends to zero.

Further, as the temperature is lowered, solid helium gradually passes into a state of complete ordering, and its entropy decreases and tends to zero at \(T=0\). The heat content of solid helium also decreases. In a similar way, with decreasing temperature, the entropy of liquid helium II decreases. This suggests that liquid helium II is in a state of quasi-crystalline ordering.

The qualitative description reduces to a comparison of liquid helium II with a certain crystalline solid in which the zero-point energy of oscillations of the atoms about their equilibrium positions (Nullpunktsenergie), equal to \(\frac{1}{2}h\nu\), exceeds the work required to displace an atom from one equilibrium position to a neighboring one. Such a solid offers no resistance to shear deformation, i.e. behaves like a liquid. If this solid is subjected to sufficient compression, then the work of displacing an atom from one position to a neighboring one in the lattice increases, and a finite resistance to deformation arises, showing that with increasing pressure liquid helium II passes into solid helium.

Supercooled Liquids (Glasses)

A sharp contrast to the case of helium, where the liquid phase exists in equilibrium down to the very lowest temperatures, is presented by the class of so-called supercooled liquids or glasses. None of them obey the heat theorem, unless it is formulated in the proper way.

It is easier to cool liquids below the normal freezing point the more associative groups there are in the molecule, such as hydroxyl. Typical glasses are provided by glycerin and ethyl alcohol, oxides and acid salts of the oxides \(P_2O_5\), \(As_2O_3\), \(B_2O_3\), \(SiO_2\). Two features promote the transition into the state of a supercooled liquid:

a) The purely crystalline form is too complex, so that the number of crystallization centers newly formed per unit time at temperatures below the melting point is small.

b) The rate of growth of the crystal around the crystallization center is small owing to the high viscosity. In the case of substances possessing these characteristic features, the properties of supercooled liquids can be easily and well studied at temperatures below the melting point. In this case the heat content, volume, viscosity, dielectric constant, and electrical resistance follow a common curve, the form of which is shown in Fig. 3.

Fig. 3. The upper curve is \(\lg c_v\), the lower is \(\dfrac{d^2 c_v}{dT^2}\).

Let us consider one of the properties, very important for glasses—viscosity. The viscosity of a liquid increases rapidly on cooling according to the law

\[ \lg \eta = -A + \frac{B}{T - T_\infty} \]

down to the temperature at which the glass becomes solid.

Over the small interval for which the curve \(\dfrac{d^2 c_v}{dT^2}\) has been drawn, \(\eta\) increases from approximately \(10^3\), through the whole region of practical use of glass \((\eta \sim 10^5—10^{10})\), up to the point \(\eta \approx 10^{13}\). It should be noted that a good glass is one that, without great difficulty, can be annealed in such a way that it has a not too narrow working interval of temperatures. Tammann attributes, for example, the change in the dielectric constant to a gradual rotation of rigid dipoles in the glass with increasing temperature. Thus, the transformation of glass into a supercooled liquid is to some extent analogous to the effect observed in ammonium chloride (p. 77), except that here there is no change in the crystalline structure associated with an increase of rotation, as in a solid body.

The entropy difference between the solid body and the glassy supercooled liquid can be determined by careful measurements of the specific heat from the melting point to

very low temperatures. Fig. 4 shows the results obtained by Lewis and Gibson for glycerine. The initial lowering of the curve for the solid in comparison with the liquid curve is caused by the loss of heat absorbed by atoms and groups of atoms when they are rigidly bound. The subsequent gradual decrease is connected with Debye oscillations of the lattice. The specific heat of the supercooled liquid remains approximately by \(\frac{3}{2}R\) greater than the specific heat of the solid until, at about \(180^\circ\) K, a “glass” is formed; then, over the course of several degrees, the specific heat falls to the corresponding value for the crystal. The equality of the heat capacities of the crystal and the glass at low temperatures is explained by the fact that at these temperatures the difference in the characteristic frequencies of the crystal and the glass is barely noticeable (p. 69).

From these two curves the values \(S_{\text{liquid}}\) and \(S_{\text{solid}}\) at the melting point can be calculated. If for both the solid and the glass \(S_0 = 0\), then the equation must hold

\[ \Delta S_{\text{m}}=\frac{\lambda_{\text{m}}}{T}=S_{\text{liquid}}-S_{\text{solid}}. \]

Fig. 4.

In reality, a discrepancy was found between the different data at the melting temperature. To explain this discrepancy it is necessary to assume that \(S_0\) for the glass is approximately 4.6 entropy units greater than for the crystal. As was explained on p. 44, this positive value of \(S_0\) is sometimes due to transitions occurring at temperatures below the lowest temperature from which measurements still contributing a certain share to the entropy were made. However, such transitions in the case of glass appear very improbable, and one may conclude that glassy glycerine does not obey the heat theorem. For other glasses deviations of the same order of magnitude were found. This result is not very unexpected, since from the thermodynamic point of view the crystal—glass or liquid—glass transitions are not truly reversible. If a molecule first escapes from the glass and then returns to it again, the structure and the gain in energy are, within certain limits, indeterminate, since the molecule has not occupied a definite position in the lattice, as it would have in the case of a crystal.

Entropy of mixing; solid solutions and the heat theorem

The general expression for the increase of entropy in the mixing of two ideal gases by mutual diffusion at constant

at the total pressure can easily be obtained. For each component separately before mixing,

\[ S_1=n_1(C_{p_1}\lg T-R\lg P_1+k_1), \]

\[ S_2=n_2(C_{p_2}\lg T-R\lg P_2+k_2), \]

where the initial pressures—of \(n_1\) moles of the first gas and \(n_2\) moles of the second—are equal, \(P_1=P_2=P\); \(k_1\) and \(k_2\) denote the entropy constants of the first and second gases. For the mixture we have

\[ S=\sum n_1(C_{p_1}\lg T-R\lg P_1' + k_1), \]

where \(P_1'\), etc., are the partial pressures of the corresponding gases.

\[ P_1'=\frac{n_1P}{n_1+n_2}=C_1P, \]

i.e.

\[ S=\sum n_1(C_{p_1}\lg T-R\lg C_1P+k_1). \]

Thus the increase in entropy upon irreversible mixing by diffusion is

\[ \Delta S=S-(S_1+S_2)=-R\sum n_1\lg C_1. \]

After division by \(n_1+n_2\), we obtain the change of entropy per mole

\[ \Delta S=-R\sum C_1\lg C_1. \]

In connection with this result two interesting questions arise. The first of them is known as the “Gibbs paradox.” Gibbs pointed out that if the molecules of two portions of gas being mixed are experimentally indistinguishable, then there should be no increase whatever in entropy upon mixing, since there is no reason to speak of a change in entropy when two identical portions of gas at one and the same pressure come into contact. In this case the molecules still diffuse spontaneously, but no increase in entropy occurs. The question arises: to what extent may the molecules of two gases differ for this result to remain valid?

As will be shown below, an increase in entropy exists even for a mixture of rotational isomers, such as ortho- and parahydrogen; satisfactory experiments have not been carried out for mixtures of isotopes, but they would probably give the same result. It follows from this that, if only the mixing molecules are distinguishable in the statistical sense, the law of mixing remains valid and the entropy increases upon mixing.

Another difficulty arises in the case of solid solutions. For dilute solutions, provided that they obey Raoult’s law, exactly the same expression as for gases applies to the increase of entropy upon mixing. However, according to the heat theorem, the change of entropy in the process pure components \(\to\) mixture is

\[ \Delta S=S_{\text{mixture}}-\sum S_{\text{components}} \]

must tend to zero as the absolute zero is approached, since only condensed phases take part in the reaction. At first glance it might seem that, for mixtures, it is necessary to abandon the heat theorem. However, statistical consideration shows that the entropy of solutions or mixtures may decrease, for various reasons, as the temperature is lowered. This already gives an approximation to the heat theorem, although the processes mentioned in practice proceed so slowly that, for solid solutions, they cannot be detected experimentally. One such process is the formation of ordered alloys from disordered ones, with a decrease in the entropy of the solid solution. It will be shown below that the mixing formula for gases must be modified when degeneracy becomes appreciable, since then the more usual expression for the entropy also becomes inapplicable.

Other deviations from the heat theorem, such as, for example, those of solid hydrogen, carbon monoxide, and nitric oxide, are more conveniently considered from the point of view of statistical theory, as will be done later (p. 76).

Statistical Theory and Thermodynamic Functions¹)

Here we are unable to undertake any very detailed exposition of statistical physics, and refer the reader to special manuals⁵ᵃ, ²). In what follows, however, we shall need only its basic concepts, which we shall assume to be known.

The statistical description of any macroscopic system in a state of equilibrium and constituting a small part of some closed system is carried out by specifying a certain statistical distribution \(dW(p_i, q_i)\), where \(p_i\) and \(q_i\) denote the totality of the momenta and coordinates of the system, \(i = 1, 2, 3 \ldots n\); \(dW(p_i, q_i)\) gives the probability distribution of the various states of the system under consideration.

We shall write this probability in the form

\[ dW = \rho(p_i, q_i)\,d\Gamma, \]

where \(d\Gamma\) denotes the volume element of phase space

\[ d\Gamma = dp_1 dp_2 \ldots dp_n dq_1 dq_2 \ldots dq_n. \]

Since the system under consideration is macroscopic, the number of degrees of freedom \(n\) is, of course, extremely large.

The function \(\rho(p_i, q_i)\) represents the probability density of our system.

¹) The beginning of the paragraph up to the sign ** was written by the translator.
²) In Russian see, for example, Frenkel, Statistical Physics. Translator’s note.

We shall regard the probability distribution as normalized so that

\[ \int dW=\int \rho\,d\Gamma=1. \]

Then the mean value of the physical quantity \(M(p,q)\) will be

\[ \overline{M}=\int M(p,q)\rho(p,q)\,d\Gamma, \]

where the indices \(i\) have been omitted.

It can be shown that, in the most general case of a system with a variable number of particles, the statistical distribution has the form of the Gibbs distribution

\[ dW_N=Ae^{\frac{\mu N-E(p,q)}{kT}}\,d\Gamma_N =e^{\frac{\Omega+\mu N-E(p,q)}{kT}}\,d\Gamma_N, \]

where \(\Omega\) and \(\mu\) are constants.

It can be shown that the quantity \(\mu\), called the chemical potential, is related to the other potentials by the relations

\[ \mu=\left(\frac{\partial A}{\partial N}\right)_{T,V} =\left(\frac{\partial F}{\partial N}\right)_{P,T} =\left(\frac{\partial E}{\partial N}\right)_{V,S} =\left(\frac{\partial H}{\partial N}\right)_{P,S} \]

and

\[ \Omega=A-\mu \overline{N}=A-F=-PV. \]

The probability is regarded as normalized by the condition

\[ \sum_N \int \rho_N\,d\Gamma_N=1, \]

where \(N\) denotes summation over all particles of the system.

In particular, if the number of particles in the system is constant, then

\[ dW=Ce^{\frac{A-E(p,q)}{kT}}\,d\Gamma, \]

and the normalization condition is

\[ C\int e^{\frac{A-E(p,q)}{kT}}\,d\Gamma=1. \]

For the mean energy of the system \(E\) we obtain

\[ \overline{E}= \frac{\int E(p,q)e^{\frac{A-E(p,q)}{kT}}\,d\Gamma} {\int e^{\frac{A-E(p,q)}{kT}}\,d\Gamma}. \]

Since \(A\) does not depend on the coordinates and momenta,

\[ \overline{E}= \frac{\int E(p,q)e^{-\frac{E(p,q)}{kT}}\,d\Gamma} {\int e^{-\frac{E(p,q)}{kT}}\,d\Gamma} = -\frac{\partial}{\partial\left(\frac{1}{kT}\right)} \ln\int e^{-\frac{E(p,q)}{kT}}\,d\Gamma. \]

The quantity

\[ Z=\int e^{-\frac{E(p,q)}{kT}} \]

is called the statistical integral.

Thus, one may write

\[ E=-\frac{\partial}{\partial\left(\frac{1}{kT}\right)}\ln Z . \]

In the particular case of an ideal gas consisting of identical noninteracting molecules with energy \(\varepsilon\),

\[ E(p,q)=N\varepsilon \quad \text{and} \quad d\Gamma=d\gamma_1 d\gamma_2\ldots d\gamma_n, \]

where \(N\) is the total number of gas molecules. Then

\[ Z=\frac{1}{N!}\int e^{-\frac{N\varepsilon}{kT}}\,d\gamma_1\ldots d\gamma_n . \]

The factor \(\frac{1}{N!}\) arises from the fact that all gas molecules are identical, and it is necessary to divide the integral by the number of possible permutations of molecules that lead to the same state of the system. Or

\[ Z=\frac{1}{N!}\left(\int e^{-\frac{\varepsilon}{kT}}\,d\gamma\right)^N . \]

For example, for a monatomic gas

\[ \varepsilon=\frac{m}{2}(v_x^2+v_y^2+v_z^2) \quad \text{and} \]

\[ E=-\frac{\partial}{\partial\left(\frac{1}{kT}\right)} \ln \frac{1}{N!}\left(\int e^{-\frac{\varepsilon}{kT}}\,d\gamma\right)^N =\frac{3}{2}NkT . \]

For the mean number of molecules of an ideal gas in an element of phase volume \(d\gamma\), we shall have

\[ dN=N\rho\,d\gamma=CN e^{-\frac{\varepsilon}{kT}} . \]

Let us now consider the case in which molecules can be only in discrete states with energies \(\varepsilon_1,\varepsilon_2,\ldots,\varepsilon_r,\ldots\). Then integration over all possible states is replaced by summation, and \(Z\) becomes the statistical sum

\[ Z=\sum_r e^{-\frac{\varepsilon_r}{kT}} . \]

Accordingly, the mean number of molecules in the \(r\)-th state will have the form

\[ N_r=NCe^{-\frac{\varepsilon_r}{kT}} . \]

and

$$ E=\frac{\sum N_r\varepsilon_r e^{-\frac{\varepsilon_r}{kT}}}{\sum e^{-\frac{\varepsilon_r}{kT}}} =-N\frac{\partial}{\partial\left(\frac{1}{kT}\right)}\ln\sum e^{-\frac{\varepsilon_r}{kT}} =-N\frac{\partial}{\partial\left(\frac{1}{kT}\right)}\ln Z . $$

Very often the energy states are degenerate, i.e., each state consists of several very close states. The degree of degeneracy of the state \(q_r\) is called the statistical weight. The average number of molecules in a state with statistical weight \(q_r\) is

$$ N_r=NC q_r e^{-\frac{\varepsilon_r}{kT}}. $$

The different statistical weights must also be taken into account when calculating the statistical sum, which gives

$$ Z=\sum q_r e^{-\frac{\varepsilon_r}{kT}}. $$

If a molecule, in addition to translational energy, also possesses vibrational, rotational, and electronic energy, then these can often, in a first approximation, be regarded as independent, i.e.,

$$ \varepsilon=\varepsilon_{\mathrm{tr}}+\varepsilon_{\mathrm{vib}}+\varepsilon_{\mathrm{rot}}+\varepsilon_{\mathrm{el}}. $$

Then the complete statistical sum is determined as

$$ G=Z_{\mathrm{tr}}\cdot Z_{\mathrm{rot}}\cdot Z_{\mathrm{vib}}\cdot Z_{\mathrm{el}}. $$

The reason for this definition is easy to understand if one takes into account that \(Z\) enters the expression for the thermodynamic functions (\(E\), and also see below) under the sign \(\ln\).

Let us find the expression for the remaining thermodynamic quantities in terms of \(Z\). We have

$$ H=E+PV. $$

For an ideal gas \(PV=NkT\),

$$ H=-\frac{\partial}{\partial\left(\frac{1}{kT}\right)}\ln Z+NkT =-N\frac{\partial}{\partial\left(\frac{1}{kT}\right)}\ln ZkT. $$

Directly from the normalization condition we obtain

$$ e^{-\frac{A}{kT}}=\int e^{-\frac{N\varepsilon}{kT}}\,d\gamma_1\ldots d\gamma_n $$

or

$$ A=-NkT\lg Z. $$

The same can be obtained from the Gibbs—Helmholtz equation

$$ A-E=T\frac{dA}{dT}, $$

dividing by \(T^{2}\) and integrating \(\dfrac{d}{dt}\dfrac{A}{T}=-\dfrac{E}{T^{2}}\), and

\[ A=-kT\int E\,d\left(\frac{1}{kT}\right)=-NkT\lg Z. \]

Finally, in a similar manner,

\[ F=NkT\lg ZkT. \]

It follows directly from the condition of equilibrium that \(Z\) must have equal values for all components in all phases.

Thus we see that the values of all thermodynamic potentials can be obtained theoretically by calculating \(Z\).

**. The question of the choice of zero values of the thermodynamic functions, when they are calculated theoretically, requires some caution. The initial values must be chosen so that the calculated thermodynamic functions coincide with those obtained by another route, for example from thermal data and the heat theorem.

Until this has been done, it is convenient to call the thermodynamic functions obtained from thermal data and referred to the initial state of a crystalline solid at absolute zero TT functions (thermal theorem). The functions obtained from the complete statistical sum we shall call TSS functions (theoretical functions including the effects of nuclear spin).

The difference between the TT and TSS functions will become clear if we consider various ways of obtaining the internal energy and entropy of a certain gas. The remaining thermodynamic functions can be obtained from them. The TT-function \(E\) represents the total energy that can be obtained when the gas is cooled from the given temperature to absolute zero at constant volume.

It is not difficult to see that the TSS-function \(E\) differs from the TT-function in that it includes the absolute-zero energy, which cannot be observed under simple cooling,

\[ E_{\mathrm{TT}}=E_{\mathrm{TSS}}-E_{0}. \]

A second complication in comparing TT- and TSS-functions is the presence in the complete statistical sum \(G\) of such factors which remain unchanged under any changes of temperature and concentrations. One of these terms is connected with the presence of nuclear spin (see below). The nuclear spin remains unchanged in all thermal and chemical processes. Therefore this last factor drops out of all thermodynamic calculations.

Another factor is due to the isotope effect.

By definition, the entropy of a mixture of two isotopes mixed in the ratio

\[ \frac{x}{1-x}, \]

would be

\[ S=xS_{1}+(1-x)S_{2}-x\lg x-(1-x)\lg(1-x), \]

however, since ordinary thermal processes do not affect the concentration of isotopes, the terms \(x \lg x\) and \((1-x)\lg(1-x)\), arising from the mixing effect, cancel in the calculations. If only TЯC-functions were used in the calculations, then the corresponding factors in \(G\), not depending on temperature, could be freely left in the expressions for the potentials, since they would in any case cancel in the calculations.

The situation is different, however, with TT-functions. They cannot contain terms that do not depend on temperature, since, for example, the presence of nuclear spin cannot lead to a change in the heat capacity down to the very lowest temperatures.

Therefore, when comparing TЯC- with TT-functions, it is necessary first of all to subtract from the former all terms connected with effects that cannot be observed in ordinary thermodynamic processes.

After subtracting the corresponding terms relating to thermodynamically unobservable processes, the expression for the total change \(\Delta F\) in the reaction \(aA+bB+\ldots \rightleftarrows mM+nN+\ldots\) may be obtained as follows.

For an individual substance

\[ \Delta F_i=-NkT\ln\left[Z_{\mathrm{int}}\left(\frac{2\pi mkT}{h^2}\right)^{\frac{3}{2}}\frac{kT}{a}\right] =-RT\ln\left[G\left(\frac{kT}{a}\right)\right], \]

where

\[ G=Z_{\mathrm{int}}\left(\frac{2\pi mkT}{h^2}\right)^{\frac{3}{2}} \]

represents the product of the statistical sum \(Z_{\mathrm{int}}\) for that part of the internal energy of the molecule whose changes are thermodynamically observable, and the factor

\[ \left(\frac{2\pi mkT}{h^2}\right)^{\frac{3}{2}}, \]

arising from the translational motion of the molecule as a whole.

The constant \(a=1\ \mathrm{atm}=1.0132\cdot10^6\ \mathrm{dyn}/\mathrm{cm}^2\) is introduced into the expression for \(\Delta F_i\) because \(\Delta F_i\) must be referred to a pressure of one atmosphere

\[ \Delta F_{\mathrm{total}}=-RT\ln K_n, \]

where

\[ K_n=\frac{G_M^mG_N^n\ldots}{G_A^aG_B^b\ldots} \left(\frac{kT}{a}\right)^{(\Sigma n-\Sigma b)}. \]

In conclusion, let us remind the reader of the relation between entropy and probability established by Boltzmann,

\[ S=k\ln W_{\mathrm{thermo}}, \]

where \(W_{\mathrm{thermo}}\) refers to the probability of the distribution of molecules among the various states.

This expression makes it possible at once to clarify the statistical meaning of the heat theorem: if all molecules of a crystal pass into one state at absolute zero, then

\[ W_{\mathrm{thermo}}\to 1 \quad\text{and}\quad S\to 0. \]

Statistical Calculation for Substances in the Solid State

The theoretical expressions for the thermodynamic functions of a solid, in view of their importance for the heat theorem, deserve special attention.

The thermal energy of a solid is due mainly to the vibrations of atoms and molecules in the crystal about certain positions of equilibrium (some special effects that contribute to the energy of a solid will be considered below). Therefore the problem consists in finding a sufficiently accurate expression for the energy states in the crystal and for the distribution of atoms among the various energy states.

A. Thermodynamic functions of an individual oscillator. The simplest model of a solid is a set of Planck oscillators vibrating with a constant frequency \(\nu\) and capable of absorbing energy quanta \(h\nu,\ 2h\nu,\ 3h\nu,\ldots,\ rh\nu,\ldots\), etc.

For this case\(^1\)

\[ Z=q_0 e^{-\frac{\varepsilon_0}{kT}} +q_1 e^{-\frac{\varepsilon_0+h\nu}{kT}} +\cdots +q_r e^{-\frac{\varepsilon_0+rh\nu}{kT}} +\cdots \]

All statistical weights are equal to unity, and the sum, which is then obtained as an infinitely decreasing geometric progression, will be

\[ Z=\frac{e^{-\frac{\varepsilon_0}{kT}}}{1-e^{-\frac{h\nu}{kT}}}, \]

which gives

\[ \varepsilon=\frac{h\nu}{e^{\frac{h\nu}{kT}}-1}+\frac{1}{2}h\nu . \]

\[ \frac{1}{2}h\nu \]

represents the energy of the oscillator at absolute zero.

The free energy per oscillator is

\[ \bar a=kT\ln\left(1-e^{-\frac{h\nu}{kT}}\right)+\frac{1}{2}h\nu . \]

If all atoms of a solid vibrate independently with one and the same frequency \(\nu\), then application of the result obtained to a crystalline solid gives, for the energy of \(N\) atoms, simply

\[ E=N\bar\varepsilon = N\frac{h\nu}{e^{\frac{h\nu}{kT}}-1} +\frac{1}{2}Nh\nu . \]

\(^1\) Since there is no interaction and all the natural frequencies of the oscillators are independent, \(Z\) is analogous to the \(Z\) of an ideal gas. Translator’s note.

This elementary theory was first developed by Einstein. From it there indeed follows that the specific heat

$C_V=\dfrac{\partial E}{\partial T}$

falls to zero as one approaches absolute zero, as required by the heat theorem. Nevertheless, Einstein’s elementary theory cannot be regarded as satisfactory, since it is in sharp contradiction with experiment.

Namely, the Einstein distribution function decreases considerably faster than is observed in reality.

The failure of this theory is explained by the fact that, in reality, the vibrations of atoms in a crystal lattice are not independent. On the contrary, when any atom is displaced from its equilibrium position, quasi-elastic forces act on it from all its neighbors, and the frequency of its vibrations will depend substantially on the mutual positions of all the atoms.

Despite this, the thermodynamic functions of an Einstein solid may be applied in known cases, especially in an approximate treatment of the behavior of a three-dimensional solid, when a detailed calculation of the frequencies arising in it is very complicated.

Linear model of a solid. It is often very useful to consider the simplest case—a one-dimensional solid1.

Let us consider a chain of atoms with identical masses and suppose that the restoring forces acting on an atom displaced from its equilibrium position are due exclusively to the interaction of this atom with its nearest neighbors.

Then, denoting by $u_r$ the displacement of the $r$-th atom from its equilibrium position, we obtain the equation of motion

$$ m\ddot u_r+k(u_{r+1}-u_r)+k(u_{r-1}-u_r)=0. $$

In fact, $u_{r+1}-u_r$ and $u_{r-1}-u_r$ represent the displacements of the $r$-th atom relative to its nearest neighbors, and the restoring forces are taken to be proportional to the displacements. The solution of the equation obtained has the form

$$ u_r=X_r e^{i2\pi\nu t}. $$

Substituting, we obtain the equation for

$$ 4\pi^2 mX_r=k(X_{r-1}-x_r)+k(X_{r+1}-x_r). $$

We seek the solution of the equation in the form

$$ X_r=ae^{irf}. $$

Then we have

$$ 4\pi^2 m a e^{irf}=kae^{irf}(e^{-if}+e^{+if}-2)=2kae^{irf}(\cos f-1), $$

i.e.

\[ \cos f - 1 = 2\sin^2\frac{f}{2}=\frac{2\pi^2\nu^2 m}{k};\quad \sin\frac{f}{2}=\pi\nu\sqrt{\frac{m}{k}}. \]

The last expression gives the relation between the natural frequency of the linear chain \(\nu\) and the value \(f\).

Since the end of the chain must remain unconnected, then

\[ X_{n+1}=e^{i(n+1)f}=0. \]

Whence

\[ (n+1)f=2\pi j;\quad f=\frac{2\pi}{n+1}\cdot j^{1}), \]

where \(j=\pm 1,\pm 2,\ldots^{2})\). If \(b\) is the distance between the atoms of the lattice, then the wavelength (i.e. the distance from a given point to a point with the same phase) will be

\[ \lambda=\frac{2\pi}{f}\,b. \]

The velocity of propagation of the disturbance is

\[ v=\nu\cdot\lambda=\frac{2b}{f}\sqrt{\frac{k}{m}}\sin\frac{f}{2}. \]

We see that in the linear model of a solid the velocity changes with the wavelength, i.e. the phenomenon of dispersion exists. For small values of \(f\), \(\lambda\) is very large, and

\[ \frac{\sin f}{f}\approx 1. \]

Whence it follows that

\[ v_\infty=b\sqrt{\frac{k}{m}}\quad \text{for}\quad \lambda \gg b. \]

The smallest value of \(\lambda\) is, evidently, \(2b\). In this case \(f=\pi\) and

\[ v=\frac{2b}{\pi}\sqrt{\frac{k}{m}};\quad \frac{v_\infty}{v_0}=\frac{\pi}{2}, \]

which shows that long waves have a greater velocity.

Three-dimensional lattice. Although such dispersion must also be preserved for a three-dimensional solid, the calculation of the individual natural frequencies for the subsequent calculation of the statistical sum is usually not practiced.

An extremely successful solution of the problem was given by Debye, who considered, in the first approximation, the solid as an elastic continuum, i.e. neglected dispersion effects.

\(^{1}\) It is essential to note that the quantity \(f\) is not determined exactly, but only with accuracy up to \(\bmod\,2\pi\).

\(^{2}\) Zero is excluded, since the equation then loses its meaning.

*

For such a continuum, the number of stationary vibrations or natural frequencies with wavelength between \(\lambda\) and \(\lambda+d\lambda\) is

\[ dZ_\lambda=\frac{V}{\lambda^4}\,4\pi d\lambda, \]

where \(V\) is the volume of the continuum.

Further, if dispersion is neglected, i.e. if the propagation velocity of the vibrations is taken to be the same,

\[ \nu=\frac{v}{\lambda}, \qquad d\nu=-v\frac{d\lambda}{\lambda^2}, \]

then

\[ \frac{3}{v_m^3}=\frac{2}{v_t^3}+\frac{1}{v_c^3}, \]

where \(v_t\) are the propagation velocities of the two transverse acoustic vibrations (equal in an isotropic medium) and \(v_c\) is the velocity of the longitudinal vibration. Introducing the mean velocity \(v_m\), so that

\[ dZ_\nu=4\pi N\nu^2d\nu\cdot \frac{2}{v_t^3}+\frac{1}{v_c^3}, \]

we obtain

\[ dZ_\nu=\frac{12\pi V\nu^2d\nu}{v_m^3}. \]

In order to satisfy the additional physical condition that the total number of degrees of freedom of a solid body consisting of \(N\) atoms is equal to \(3N\), the upper limit of possible frequencies \(\nu_{\max}\) must be such that

\[ 3N=\frac{12\pi V}{v^3}\int_0^{\nu_{\max}}\nu^2d\nu \]

or

\[ 3N=\frac{4\pi V}{v^3}\nu_{\max}^3. \]

This equation makes it possible to eliminate the velocity and to write

\[ dZ_\nu=\frac{9N\nu^2d\nu}{\nu_{\max}^3}. \]

Using this approximation for the number of possible energy states of atoms in the lattice and taking the mean energy \(e_\nu\) of an oscillator of frequency \(\nu\) at temperature \(T\) (p. 65) to be equal to

\[ e_\nu=\frac{h\nu}{e^{\frac{h\nu}{kT}}-1}+\frac{1}{2}h\nu, \]

we obtain for the total energy of a solid body due to vibrations of the lattice

\[ [E_\nu]_T=\int_0^{\nu_{\max}} e_\nu dZ_\nu = \frac{9N}{\nu_{\max}^3}\int_0^{\nu_{\max}}\nu^2d\nu \left( \frac{h\nu}{e^{\frac{h\nu}{kT}}-1} \right) + \frac{9N}{\nu_{\max}^3}\int_0^{\nu_{\max}}\frac{h\nu^3d\nu}{2}. \]

To calculate this expression, we transform it by substituting

\[ \frac{h\nu_{\max}}{k}=\theta;\qquad \frac{h\nu}{kT}=x \]

into the generalized function \(\dfrac{\theta}{T}\), i.e. into

\[ [E_\nu]_T=9RT\left(\frac{T}{\theta}\right)^3 \int_0^{\theta/T}\frac{x^3\,dx}{e^x-1} +\frac{9}{8}R\theta . \]

In this expression the constant \(\theta\) is the characteristic temperature of the substance, whose magnitude depends on its elastic properties. This follows both directly from the preceding, whence

\[ \theta=\frac{h\nu}{k}\sqrt[3]{\frac{3N}{4\pi V}}, \]

and from dimensional considerations, according to which

\[ \theta=A\,\frac{h}{k}\, \frac{N^{1/3}}{m^{1/2}\chi^{1/2}\rho^{1/6}}, \]

where \(m\) is the atomic weight, \(\chi\) the compressibility, \(\rho\) the density, and the proportionality coefficient \(A\) is approximately one and the same for all substances with the same lattice structure. In the absence of experimental data on the specific heat, this expression is sometimes used to determine the thermodynamic functions of solids from calculation of the value of \(\theta\); however, it must be used with caution. Tables of values of this integral for various values of \(\dfrac{\theta}{T}\) are given in the Landolt–Börnstein handbook.^6 The value

\[ C_V=\frac{\partial E}{\partial T} \]

is obtained directly by differentiating the expression for \(E\), which gives (putting \(y=\dfrac{\theta}{T}\))

\[ C_V=9R\left[ \frac{4}{y^3}\int_0^y\frac{x^3\,dx}{e^x-1} -\frac{y^4}{e^y-1} \right]. \]

At such temperatures for which the value of \(\dfrac{\theta}{T}\) is small, \(C_V\) tends to \(3R\) per gram-atom, i.e. to the Dulong–Petit value for the atomic heat capacity of solids. At low temperatures, when \(\dfrac{\theta}{T}\) is large and the share of vibrations with high frequency in the specific heat may be neglected, the integral

\[ \int_0^{\theta/T}\frac{x^3\,dx}{e^x-1} \]

approximately becomes

\[ \int_0^\infty \frac{x^3\,dx}{e^x-1}=\frac{\pi^4}{15}, \]

then

\[ E_\nu=9RT\left(\frac{T}{\theta}\right)^3 \frac{\pi^4}{15} \]

and

\[ C_v=\frac{12}{15}R\frac{T^3}{\theta^3}\pi^4=\alpha T^3, \]

where \(\alpha\) is a constant depending only on the characteristic temperature of the solid.

It is obvious that this expression for the specific heat satisfies the postulate of the heat theorem

\[ \lim_{T\to 0}\frac{\partial H}{\partial T}=0. \]

For a solid with a simple cubic lattice the theory was tested over a large temperature interval by computing \(\theta\) at each given temperature from the observed values of \(C_v\), using the above-mentioned tables. If the theory is strictly correct, then \(\theta\) should not depend on temperature. In experiment, however, deviations of \(\theta\) from a constant value were almost always observed, although in many cases these deviations did not exceed \(5\%\) throughout the entire temperature range in which changes of \(\theta\) with temperature were observed.

At low temperatures it was found that the law \(C_v=aT^3\) is applicable even to complex anisotropic bodies, such as, for example,

\[ \mathrm{CaF_2},\quad \mathrm{NH_4Cl},\quad \mathrm{SiC}, \]

and to crystalline and vitreous quartz and glycerin. This is probably connected with the fact that at low temperatures the wavelengths of the vibrations that are still taken into account in \(C_v\) are large in comparison with the dimensions of the lattice, and the treatment of the solid as a continuous continuum does not introduce large inaccuracies.

Thermodynamic functions and equations of state of a Debye solid

Since Debye’s theory is approximate, the thermodynamic functions of a solid should be found by graphical integration of the specific heats determined experimentally whenever suitable data are available. Sometimes, however, the equation of state of a Debye solid proves useful; it may be obtained in the following way.

Let us suppose that the heat theorem is valid for a Debye solid, and choose as the initial state

the state of the solid at absolute zero. Then the value of the free energy \(A\) at some temperature \(T\) is given by the integral

\[ A=-T\int_0^T \frac{E}{T^2}\,dT, \]

where the value \(E\) is introduced from Debye’s expression

\[ E=9RT\left(\frac{T}{\theta}\right)^3\int_0^{\frac{\theta}{T}}\frac{x^3\,dx}{e^x-1} \]

(omitting the zero-point energy, p. 69). This gives

\[ A=-\frac{E}{3}+3RT\int_\infty^y\frac{dy}{e^y-1}= \]

\[ =3RT\left\{-\left(\frac{T}{\theta}\right)^3\int_0^y\frac{y^3dy}{e^y-1}+\int_\infty^y\frac{dy}{e^y-1}\right\}, \]

where, as before, \(\varphi=\dfrac{\theta}{T}\). This is the free energy of the solid at temperature \(T\), but still at the same constant volume \(V_0\) which the body occupied at absolute zero. In order to calculate the free energy at temperature \(T\), zero external pressure, and volume \(V\), the solid must be compressed at \(0^\circ\mathrm{K}\) to the volume \(V\) and then heated to temperature \(T\) at this same constant volume.

If \(W\) is the work of compression, then

\[ A=W+3RT\left\{-\left(\frac{T}{\theta}\right)^3\int_0^y\frac{y^3dy}{e^y-1}+\int_\infty^y\frac{dy}{e^y-1}\right\}, \]

where the value \(\theta\) (which depends on the volume) now corresponds to the volume \(V\).

To obtain the equation of state we use the equation

\[ P=-\left(\frac{\partial A}{\partial V}\right)_T, \]

which gives

\[ P=\frac{dW}{dV}-9RT\,\frac{1}{T}\frac{d\theta}{dV}\left(\frac{T}{\theta}\right)^4\int_0^{\frac{\theta}{T}}\frac{y^3dy}{e^y-1}, \]

or, multiplying by \(V\) and noting that

\[ \frac{d\theta}{dV}=\frac{\theta}{V}\frac{d\ln\theta}{d\ln V}, \]

after rearrangement we obtain

\[ PV+V\frac{dW}{dV}= \]

\[ =-\frac{d\ln\theta}{d\ln V}\,9RT\left(\frac{T}{\theta}\right)^3 \int_0^{y}\frac{y^3\,dy}{e^y-1} =-\frac{d\ln\theta}{d\ln V}E_v=Y\cdot E_v, \]

where \(E_v\) is the thermal energy associated with lattice vibrations. From this equation of state one readily obtains such properties as, for example, the coefficient of expansion \(\frac{dV}{dt}\).

It may further be noted that the change of \(\theta\) as a function of \(V\) is due to the fact that, upon expansion of a solid, the distance between atoms increases, which leads to a change in the forces of interaction between them and, consequently, to a change in

\[ \nu_{\max}=\frac{k\theta}{h}. \]

Moreover, a Debye solid at low temperatures obeys the heat theorem, since

\[ A=-\beta T^4, \]

\[ C_v=12\beta T^3=-T\frac{\partial^2 A}{\partial T^2}. \]

Other approaches to the question of the heat content of solids associated with vibrations

Let us try to improve the theory of the solid regarded as a continuous continuum by allowing the wave velocity to increase with increasing wavelength. Analysis of equation \((a)\) shows that, among the total number \(3N\) of vibrations, there must be relatively more high-frequency vibrations, whose velocity is smaller, than is allowed for by Debye’s theory. This apparently explains the success of the empirical formula of Nernst and Lindemann \(^8\)

\[ [E_v]=\frac12 E(\nu)+\frac12 E\left(\frac{\nu}{2}\right), \]

where \(E(\nu)\) is the energy per \(1\) g-mole for all oscillators of frequency \(\nu\), i.e. (see Blackman \(^7\)),

\[ E(\nu)=\frac{Nh\nu}{e^{\frac{h\nu}{kT}}-1}+\frac{Nh\nu}{2}. \]

For solids built of complex molecules, such as ice, the vibrations of the lattice as a whole are often practically independent of the internal vibrations of the molecules. In this case Debye’s theory may be applied to the lattice vibrations, and Einstein’s expression to the internal vibrations of the molecule. Then \([E_v]\) is simply the sum of the Debye and Einstein functions

\[ [E_v]=D(\theta_1)+E(\theta_2)+E(\theta_3)+\ldots \]

At temperatures approaching the melting point, the displacements of atoms relative to their equilibrium positions are no longer obeying Hooke’s law; the restoring forces are no longer proportional to the displacement. The resulting anharmonic vibrations of the lattice add to the expression for the specific heat a term depending linearly on temperature,

\[ C_v = 3R + aT. \]

In practice, however, it is rarely possible to separate out this effect from a number of other processes leading to an increase in the specific heat as compared with the Dulong–Petit law.

Other sources of the heat content of a solid

The energy of atoms and molecules in a crystal lattice may increase not only through an increase in vibrational energy, but also as the result of a number of other processes. Such changes in the energy of atoms are very important for the theory of lattice structure and directly give rise to apparent deviations from the third law of thermodynamics.

One such effect, the simplest to calculate, was first considered by Schottky. It may occur in solids whose atoms or molecules can exist in two distinct energy states. Let us denote these states by 1 and 2, and the difference of their energies by \(\varepsilon\). Suppose that the value of \(\varepsilon\) for any atom does not depend on the state of neighboring atoms, i.e. on the number of atoms already in the higher energy state. Although such cases are rarely realized in practice, the calculation reveals certain characteristic features inherent in almost all energy transitions, even when they are considerably more complex.

If the transition energy \(\varepsilon\) does not depend on the number of \(N_2\)-atoms in the higher energy state, then the ratio of \(N_2\) to the number of atoms in the lower energy state \(N_1\) is given by the Boltzmann expression

\[ \frac{N_2}{N_1} = q e^{-\frac{\varepsilon}{kT}}, \]

where \(q\) is the ratio of the statistical weights of the two states. Since some of the atoms may be in the higher energy state, the energy of the solid in this case is somewhat increased. The mean energy per molecule arising from this effect is

\[ \varepsilon = \frac{N_2 \varepsilon}{N_1 + N_2} = \frac{\varepsilon q e^{-\frac{\varepsilon}{kT}}}{1 + q e^{-\frac{\varepsilon}{kT}}}. \]

The additional heat capacity \(C_{VS}\) arising as a result of this will be

\[ C_{VS}=\varepsilon \frac{dN_{2}}{dT} = q\,\frac{\varepsilon^{2}}{kT^{2}}\, \frac{e^{-\frac{\varepsilon}{kT}}} {\left(1+q e^{-\frac{\varepsilon}{kT}}\right)^{2}}. \]

The graph of this additional specific heat is given in Fig. 5.

As for any additional specific heat arising from the energy of transitions in a solid, the maximum value of \(C_{VS}\) is observed near the characteristic temperature

\[ \Theta \simeq \frac{\varepsilon}{k}, \]

and \(C_{VS}\) falls to zero both above and below this temperature.

Fig. 5.

Direct integration or an a priori calculation shows that the maximum addition to the energy \(E\) and to the heat content \(H\), due to this effect, is simply

\[ \frac{Nq\varepsilon}{1+q} \]

per gram-molecule, since for \(T \gg \Theta\)

\[ N_{2}=qN_{1}. \]

If such transitions take place at temperatures lower than the lowest temperatures at which the heat capacities were measured, then the value of the heat content calculated from the specific-heat measurements will be less than the true value by

\[ \frac{Nq\varepsilon}{1+q}. \]

However, this would mean that the characteristic temperature

\[ \Theta=\frac{\varepsilon}{k} \]

is considerably lower than the lowest temperature of the measurements and, consequently, \(\varepsilon\) is very small. Therefore, because of the effect under consideration, no serious errors can arise in calculating the heat content.

The situation is quite different in the calculation of other thermodynamic functions, such as, for example, entropy. The additional entropy associated with the Schottky effect can be calculated from the integral

\[ \Delta S=\int C_{VS}\,d(\ln T), \]

and if the integral is taken from \(0\) to \(\infty\) (practically simply over the whole temperature region where \(C_{VS}\) is noticeably different from zero; p. 44), then finally

\[ \Delta S=R\ln(q-1), \]

However small the value of \(\varepsilon\) may be and, consequently, however low the temperature at which the transitions occur, nevertheless, if the Schottky effect has not been taken into account and the heat capacities have not been measured down to sufficiently low temperatures, the entropy values calculated from thermal data will be less than the true ones by the amounts

\[ R \ln 2,\qquad R \ln 3,\ldots \]

for \(q = 1, 2, 3\ldots\)

These calculations for the simplest case of transitions in a solid may be compared with experiment in two ways.

The additional specific heat is superposed on the normal curve corresponding to lattice vibrations. For a cubic lattice the normal curve corresponds to the Debye function with a constant \(\Theta\). Therefore, from experimental data one can calculate \(C_{VS}\) with sufficient accuracy. The results are given in the following Table 4.

TABLE 4

Substance	\(\varepsilon\), cal/mol	\(\Theta\) (Schottky)	\(\Theta\) (Debye)
Diamond	2,120	1,070	1,840
Quartz	490	246
Gray tin	137	69
Lithium	407	205
Sodium	189	95	159
Potassium	117	59	99.5

An additional test of the theory may be provided by the fact that, since this transition does not lead to a change in volume (a change in the energy state of one of the atoms does not affect its neighbors), the coefficient of expansion obeys Grüneisen’s rule

\[ \frac{dV}{dT}=Y\cdot K_0\cdot C_V, \]

where \(C_V\) is the Debye specific heat obtained by subtracting \(C_{VS}\) from the experimental value of the heat capacity, and \(K_0\) is the compressibility.

The applicability of the theoretical Debye curve for calculating anomalies is doubtful\(^9\), but in some cases, at those temperatures at which the Debye specific heat is small, this apparently does not give rise to serious errors. Thus, for example, crystalline gadolinium sulfate exhibits the Schottky effect; the specific heat of solid gadolinium sulfate obeys the \(T^3\) law down to \(7^\circ\text{K}\), but below this temperature the specific-heat curve begins to rise until, at \(1^\circ,6\text{K}\), the specific heat becomes 500 times greater than the value

... required by the law \(T^3\). In this case the existence of energy transitions finds a satisfactory physical explanation.

The ion \(\mathrm{Ga}^{+++}\) is in the state \({}^{8}S_{\frac{7}{2}}\), i.e. in the absence of external electric and magnetic fields there is an eightfold degeneracy. Interaction with the inhomogeneous electrostatic field of the crystal lattice removes the degeneracy. This means that there are eight energy levels lying close to one another. The difference of their energies is \(\Delta E \simeq 0.52\ \text{cal/mole}\). Consequently, near the temperature for which \(RT = 0.52\ \text{cal}\), the highest levels are gradually emptied.

A similar effect has also been observed for the heat capacity of solid (ortho-, para-) hydrogen; it corresponds to the presence in the solid of three different energy levels \((\Delta E \simeq 7.5\ \text{cal})\). From the thermodynamic point of view the most important cases of the Schottky effect are those in which the transitions occur at temperatures lying below the lowest temperature at which measurements were still made. In this case the presence of the Schottky effect leads to apparent deviations from the third law of thermodynamics.

TABLE 5

Gas	Difference per mole	Difference per mole
\(\mathrm{N_2}\)	\(-0.07\)	\(0.20^{1)}\)
\(\mathrm{CO_2}\)	\(0.32\)	\(0.27^{2)}\)
\(\mathrm{CO}\)	\(1.06\)	\(0.25^{3)}\)
\(\mathrm{CO}\)	\(1.12\)	\(0.10^{3)}\)
\(\mathrm{N_2O}\)	\(0.90\)	\(0.32\)
\(\mathrm{NO}\)	\(0.75\)	\(0.10\)

\(^{1)}\) Within experimental errors.
\(^{2)}\) Doubtful.
\(^{3)}\) Two differences are observed.

The following Table 5\(^{10}\) gives the differences between the entropies of certain gases, calculated from vapor-pressure or chemical-equilibrium curves, and their entropies calculated with the aid of the integral

\[ \int C_p d(\ln T), \]

on the assumption that the solidified gas obeys the heat theorem.

In order to evaluate the significance of Table 5, it should be noted that \(R \ln 2 = 1.4\) and that the positive differences in the second column show that, in calculating the entropy from thermal data, some part of it was omitted (p. 64). This leads to apparent deviations from the heat theorem of the expected order of magnitude. Consequently, this table shows that the molecules \(\mathrm{CO}\), \(\mathrm{N_2O}\), and \(\mathrm{NO}\) in the solid state at temperatures below the lowest temperatures of the measurements undergo a certain transition. Taking into account that the atoms of each of these molecules are very similar, although not identical, one may come to the conclusion that at ordinary temperatures the molecules are arranged in the crystal

completely disordered, for example, as CO, OC, etc. At very low temperatures the energy difference between the arrangement

$$ \begin{array}{llll} \mathrm{C}-\mathrm{O} & \mathrm{C}-\mathrm{O} & \mathrm{C}-\mathrm{O} & \mathrm{C}-\mathrm{O}\\ \mathrm{C}-\mathrm{O} & \mathrm{C}-\mathrm{O} & \mathrm{C}-\mathrm{O} & \mathrm{C}-\mathrm{O}\\ \mathrm{C}-\mathrm{O} & \mathrm{C}-\mathrm{O} & \mathrm{C}-\mathrm{O} & \mathrm{C}-\mathrm{O} \end{array} $$

and an arbitrary arrangement

$$ \begin{array}{llll} \mathrm{C}-\mathrm{O} & \mathrm{O}-\mathrm{C} & \mathrm{C}-\mathrm{O} & \mathrm{C}-\mathrm{O}\\ \mathrm{O}-\mathrm{C} & \mathrm{O}-\mathrm{C} & \mathrm{O}-\mathrm{C} & \mathrm{C}-\mathrm{O}\\ \mathrm{C}-\mathrm{O} & \mathrm{C}-\mathrm{O} & \mathrm{C}-\mathrm{O} & \mathrm{O}-\mathrm{C} \end{array} $$

is comparable with \(kT\) and causes a transition to the lower state with a corresponding decrease in energy and entropy.

The ammonium chloride effect

When the transition to a higher energy level is not independent of neighboring atoms, a complication arises that makes an exact statistical calculation difficult. This occurs because the energy difference between two states depends on the number of molecules already in the higher state. In addition, owing to the changed interaction between neighbors, the transition is accompanied by a change in volume and by changes in the infrared absorption of the crystal.

The graph (Fig. 6) shows the additional specific heat that in this case is superposed on the normal curve. Effects of this kind are found in most ammonium salts near \(230^\circ\mathrm{K}\), and also in a number of other substances, such as: FeO (\(185^\circ\mathrm{K}\)), \(\mathrm{Fe}_3\mathrm{O}_4\) (\(115^\circ\mathrm{K}\)), MnO (\(116^\circ\mathrm{K}\)), \(\mathrm{MnO}_2\) (\(93^\circ\mathrm{K}\)), \(n\)-butyric acid (\(221^\circ\mathrm{K}\)), HBr (\(88^\circ\mathrm{K}\)), HJ (\(70^\circ\mathrm{K}\) and \(120^\circ\mathrm{K}\)), \(\mathrm{CH}_4\) (\(20^\circ\mathrm{K}\)), and also in hydrides, such as: \(\mathrm{SiH}_4\), HF, \(\mathrm{PH}_3\), \(\mathrm{H}_2\mathrm{S}\), but not \(\mathrm{H}_2\mathrm{O}^{11}\).

Fig. 6.

One explanation, proposed by Fowler, appears suitable in the case of hydrogen compounds. It consists in the fact that at temperatures below the transition temperature the asymmetric ions are oriented in the crystal in definite directions.

However, when with increasing temperature the ions acquire a vibrational kinetic energy greater than the maximum energy of mutual orientation, they begin to rotate freely. It is clear that the orienting forces acting on an ion surrounded

already rotating neighbors, than the forces acting when these neighbors are themselves oriented. This explains why the transition energy depends on the number of ions already in the higher state. The anomalous specific heat arising from these interactions increases much more steeply than for the Schottky effect.

Confirmation of this point of view is provided by the increase in the symmetry of the crystal above the transition point. In a similar way, the observed change in volume is caused by a decrease in the interaction between ions. Finally, this hypothesis is confirmed by the fact that different ammonium salts exhibit the effect under consideration at approximately one and the same temperature. For other substances the transition temperature proves to be the lower, the more symmetrical the molecules of the substance are. Thus, for example, in $\mathrm{H_2O}$, which possesses molecules of a very regular structure, the effect is entirely absent, whereas in $\mathrm{H_2S}$, on the contrary, it is observed.

At the same time, any change in the transition energy caused by the action of atoms already occupying states of higher energy leads to specific-heat curves of a similar kind. Examples of this may be transitions from ferromagnetism to paramagnetism, or from ordered to disordered alloys, which will be considered below.

The lattice changes occurring in oxides in transitions of this sort have not yet been explained. In any case it is doubtful that these changes consisted in a transition of mutually oriented ions to free rotation, as in the case of hydrogen-containing compounds.

The influence of such anomalies of the specific heat on thermodynamic functions is similar to the influence of Schottky transitions, but is not so readily accessible to calculation.

General theory of specific heats near the transition point

From the standpoint of formal thermodynamics, the transition point of a substance from one state into another may be represented as the intersection of the free-energy curves of the two states. A well-known case of a phase transition is the melting of ice. The graph (Fig. 7) shows the intersection of the $F$, $P$, $T$ surfaces for ice and water in the $F—T$ plane, corresponding to $P = 1$ atm.

The state with the lower free energy is the more stable; and since the curves intersect at $0^\circ\mathrm{C}$, at this point a spontaneous transition from ice to water takes place.

Although the free energy per unit mass (or per mole, if the same molar units are used for both phases) is the same at this equilibrium point, the entropy and heat content of the two states are different; therefore the transition of ice into water is accompanied by absorption of heat, the so-called latent heat of fusion.

Even in the case of melting, such a formal treatment of the transition point is not entirely adequate, as is shown by experiments on the specific heat of ice near the melting point3. Fig. 8 illustrates the observed sharp rise in the specific heat. In this case the temperature region in which this increase in the specific heat was observed narrowed as the specimen under study was purified of impurities.

This increase in the specific heat is somewhat reminiscent of the effect in ammonium chloride; however, its statistical interpretation is, of course, quite different. In particular, it is necessary to take into account the role of the impurities and contaminants present in the melting specimen.

Fig. 7.

Fig. 8.

Let \(\Delta T_m\) be the total lowering of the melting temperature caused by the presence of an impurity with concentration \(C_0\). Then, if Raoult’s law is satisfied, \(\Delta T_m\) is proportional to the concentration of the impurity.

Let us denote by \(\Delta T\) the lowering of the melting temperature when a fraction \(X\) of the substance has melted.

If no solid solutions are formed, then \(\Delta T\) corresponds to the lowering of the melting temperature of a solution with concentration

\[ \frac{C_0}{X}. \]

Thus we have

\[ \frac{\Delta T}{\Delta T_m}=\frac{1}{X}. \]

If \(L\) is the latent heat of transition, then for the melting of a fraction \(X\) of the substance there will be required

\[ Q=L\cdot X=L\cdot\frac{\Delta T_m}{\Delta T}. \]

Hence, for the anomalous specific heat below the melting point, due to the presence of impurities, we obtain

\[ \Delta C_v=\frac{dQ}{dT}=\frac{\Delta T_m}{(\Delta T)^2}L, \]

since

\[ \frac{d\Delta T}{dT}=1. \]

This simple expression is in agreement with the experimental fact that the curves of specific heat become steeper as the ice is purified. However, the experiments leave unresolved the question of how sharp the transition from a perfectly pure solid to a liquid may be. The rise of the specific heat before the true melting point would entail an anomalously rapid change in the slope of the surfaces \(F\), \(P\), and \(T\) near the transition point. Since, however, latent heat is evolved, these surfaces must intersect at a certain angle. Apart from the phenomenon of melting, a large number of other transitions in a solid are known which occur only within a certain temperature range, such as, for example, transitions from ferromagnetism to paramagnetism and from an ordered to a disordered alloy. Such changes cannot be considered with the aid of surfaces of free energy without introducing an additional degree of freedom, such as the degree of spontaneous magnetization or of spontaneous ordering in the solid; and a statistical treatment is required for the correct interpretation of the phenomena.

Transition from Disordered to Ordered Solid Solutions

It was already pointed out above (p. 59) that mixtures apparently do not obey the heat theorem and that in some solid solutions, owing to the ordering of atoms in the lattice, the entropy may decrease. This effect has been observed for such alloys as brass, \(\mathrm{Fe_3Al}\), \(\mathrm{Cu_3Pd}\), \(\mathrm{Cu_3Pt}\), \(\mathrm{CuAu}\), etc. X-ray investigations show that above a certain temperature pure metals and solid solutions have a face-centered cubic lattice. Upon sufficiently slow cooling of the alloys below the transition temperature, a new “substructure” arises, consisting in the fact that the random distribution of atoms at the nodes of the cubic lattice is replaced by an ordered distribution, in which all atoms of one kind occupy identical positions—for example, all Cu atoms at the vertices of the cube and Au atoms at the centers of the faces. The presence of ordering may also be inferred from other changes in physical properties. For example, the electrical resistance decreases when a “substructure” is formed in the solid.

From the standpoint of thermodynamics and statistics, the ordering process is energetically favorable and therefore becomes spontaneous upon sufficient cooling. It, however, entails a decrease of entropy and, in this sense, is opposed to thermal motion, which tends to introduce disorder and thereby to increase the entropy of the body. Therefore the ordering process can play an essential role only at such temperatures at which the gain in energy due to ordering is large in comparison with the mean thermal energy of the atoms. Only for some alloys are these temperatures sufficiently high that

the corresponding displacements of atoms occurred within a measurable interval of time.

A simple formal treatment was given by Bragg and Williams^13. First of all, it is necessary to determine the degree of ordering in the solid solution. This can be done in several ways; one of them uses the concept of “substitution.” We shall start from a cubic lattice built entirely of atoms of type \(B\). The formation of the solid solution consists in replacing a certain number \(n\) of these atoms by atoms of type \(A\). If the \(r\)-th part of the substituting atoms is located in those positions which are specified by the substructure for atoms of type \(A\), then there will be \(nr\) ordered positions \(\alpha\) and \(n(1-r)\) disordered positions \(\beta\). The degree of order \(S\) is defined as follows.

Let \(p\) be the probability that an ordered position \(\alpha\) will be occupied, by substitution, by an atom of type \(A\), which in the ordered state should be located in it, rather than by an atom of type \(B\).

Then

\[ S=\frac{p-r}{1-r}, \]

i.e., for \(p=1\), \(S=1\) (complete order),

\[ p=r \quad S=0 \]

(complete disorder).

The increase in potential energy \(V=V(S,T)\), when a substituting atom passes from a position \(\beta\) to a position \(\alpha\), will, at constant temperature, be a function of the number of atoms of type \(A\) already situated in positions \(\alpha\). The distribution of atoms, corresponding to \(nr\)-substitutions \((B\to A)\) among \(n\) sites, gives \(nr\) ordered positions \(\alpha\). At any moment the number of available ordered positions \(\alpha\) is equal to

\[ nr-pnr \]

and the number of available positions \(\beta\)

\[ n(1-r)-(1-p)nr. \]

Assuming a Boltzmann distribution of atoms among the states, we obtain

\[ \frac{p}{1-p}=\frac{r(1-p)}{(1-2r+pr)}e^{\frac{V}{kT}}. \]

If \(r=\frac{1}{2}\), for example for \(\mathrm{Fe}_3\mathrm{Al}\), for which only the centers of the cube are ordered positions, and putting

\[ x=-\frac{V}{kT}, \]

then

\[ S=\operatorname{tgh}\frac{x}{4}. \]

In order to obtain a quantitative solution, it is necessary to make assumptions concerning the form of \(V\). Suppose that \(V\) is proportional

degree of order

\[ V=V_0\cdot S, \]

in order to take into account the interaction with neighboring atoms. Let us further assume, for simplicity, that \(V\) does not depend on the temperature, since in the temperature range under consideration there is no large expansion of the lattice.

In the state of equilibrium the degree of order \(S_e\) is determined by the intersection of the curves

\[ V=V_0S \quad \text{and} \quad S=\operatorname{tgh}\frac{V}{kT}, \]

i.e.

\[ S_e=\operatorname{tgh}\frac{S_eV_0}{kT}. \]

This equation gives a rapid increase of \(S_e\) near the critical temperature

\[ T_{\mathrm{cr}}=\frac{V_0}{4k}, \]

above which \(S_e\approx 0\). For \(\mathrm{Fe}_3\mathrm{Al}\)

\[ T_{\mathrm{cr}}=833^\circ\mathrm{K}, \qquad V_0=0.29\,V. \]

The changes in energy associated with changes in \(S_e\) may be estimated as follows:

\[ dp=(1-r)\,dS; \]

\[ dE=Vnrdp=V_0Snr(1-r)\,dS, \]

and, slightly below the critical temperature, for \(S_e\) we have approximately

\[ S_e^2=\frac{3(T_{\mathrm{cr}}-T)}{T_{\mathrm{cr}}}. \]

Differentiating, we obtain

\[ 2S_e\,dS_e=-\frac{3\,dT}{T_{\mathrm{cr}}}. \]

Substituting \(dS\) from the preceding equation, we have

\[ \frac{dE}{dT}=\Delta C_V=6nrk(1-r). \]

For \(r=\frac{1}{2}\), the maximum value of \(r(1-r)\) gives \(\Delta C_{V\max}=\frac{3}{2}nk\). For \(\mathrm{Fe}_3\mathrm{Al}\) there are altogether \(n=\frac{1}{4}N\) displacements per \(N\) atoms of both metals, so that the anomalous specific heat may increase up to \(\frac{1}{8}\) of the Dulong and Petit value for the heat capacity associated with lattice vibrations. A more detailed consideration shows—

It turns out that the disappearance of ordering above \(T_{\mathrm{cr}}\) is less rapid than follows from the simple argument already given, so that above \(T_{\mathrm{cr}}\), \(\Delta C_V\) decreases much more slowly than was shown.

References

1. Nernst, Göttinger Nachrichten, 1906, 1.
2. Keesom, Physik. Z., 35, 939, 1934.
3. Eucken and Fried, Z. Physik, 29, 36, 1924.
4. Coleman and Egerton, Phil. Trans., 234, 177, 1935.
5. Nernst, The New Heat Theorem.
   5a. Fowler, Statistical Mechanics, 2nd ed., Cambridge, 1936.
6. Landolt-Börnstein, Tabellen, 5th ed.
7. Blackman, Proc. Roy. Soc., A 148, 365, 1935.
8. Nernst and Lindemann, Z. Electrochem., 17, 817, 1911.
9. Fowler, Bakerian Lecture, Proc. Roy. Soc., 151 A, 1, 1935.
10. Clusius, Nature, 132, 775, 1932.
11. Landolt-Börnstein, Tabellen, 5th ed.
12. Dickinson and Osborne, Bull. Bur. Stand., 12, 69, 1915.
13. Bragg and Williams, Proc. Roy. Soc., A 145, 699, 1934.

(To be concluded in the next issue)

1. It must be emphasized here that in reality a one-dimensional crystal cannot exist. See Peierls, Ann. d. l’Inst. Poincare. Translator’s note. ↩↩↩

2. The corresponding paragraph of the book has been omitted for purposes of abridgment. The present paragraph was written by the translator and is intended to serve as a reference. ↩

3. visible reference marker in the source. ↩