MODERN THERMODYNAMICS

A. R. Ubbelohde, Oxford1

MAGNETIC PROPERTIES AND THE ATTAINMENT OF VERY LOW TEMPERATURES

Paramagnetism and Ferromagnetism

A statistical treatment of paramagnetism is given in the following paragraph. Paramagnetic solids usually obey the Curie–Weiss law, according to which their susceptibility is

\[ \chi=\frac{C}{T+\Delta}, \]

where \(C\) and \(\Delta\) are constants. At temperatures \(T\) of the same order as the constant \(\Delta\), \(\chi\) is no longer independent of the field strength \(H\), and Curie’s law is no longer applicable.

Near the Curie point, below which a solid becomes ferromagnetic, the heat capacity \(C_v\) and the change in heat content have anomalously large values. Typical Curie points are \(755^\circ\text{C}\) for iron, \(588^\circ\text{C}\) for magnetite, and \(376^\circ\text{C}\) for nickel, although in weakly ferromagnetic substances Curie points are observed down to much lower temperatures. If cooling curves are plotted for paramagnetic substances, then near the Curie point, owing to the change in the heat capacity \(C_v\), a change in the slope of these curves will be observed; however, they contain no portions corresponding to the liberation or absorption of latent heat at a fixed temperature.

In the absence of a strict quantitative theory of ferromagnetism, a formal thermodynamic explanation for the presence of an additional heat capacity in paramagnetic and ferromagnetic substances may be given as follows. If \(H\) is the magnetic field and \(I\) the intensity of magnetization, then the work done on the substance during magnetization is equal to \(\int H dI\), and the heat content may be defined as

\[ H=E-PV+\int HI. \]

If \(E + PV\) does not depend on the state of magnetization of the substance, then, choosing \(H\) and \(T\) as independent variables, we have

\[ \left(\frac{\partial S}{\partial H}\right)_{P,T} = \left(\frac{\partial I}{\partial T}\right)_{P,H}. \]

Hence the change in entropy upon magnetization is

\[ \Delta S = \int_{H_1}^{H_2} \left(\frac{\partial I}{\partial T}\right)_{P,H} \, dH . \]

For paramagnetic substances \(I = XH\), and if the substance obeys Curie’s law \(\left(X=\frac{C}{T}\right)\), so that

\[ \left(\frac{\partial I}{\partial T}\right)_{P,H} = -\frac{CH}{T^2}, \]

where the obtained quantity has an appreciable value only at low temperatures, we have

\[ \Delta S = -\frac{H^2 C}{2T^2}, \]

and the additional specific heat capacity of the paramagnetic substance is equal to

\[ \Delta C_V = T\left(\frac{\partial \Delta S}{\partial T}\right)_V = \frac{H^2 C}{T^2}. \]

Fig. 9.

Ferromagnets. Ferromagnetic substances below the Curie point exhibit spontaneous magnetization, even if \(H=0\).

If \(I_s\) is the saturation magnetization, then, following Weiss, one may write for the internal field

\[ H_m = A I_s, \]

where \(A\) is a constant.

Then the energy of magnetization per unit volume is

\[ E_m = \frac{1}{2} A I_s^2, \]

or, per gram of substance,

\[ E_m = \frac{A}{2\rho} I_s^2. \]

The presence of the additional energy of spontaneous magnetization leads to an additional specific heat capacity

\[ \Delta C_{Vm} = \frac{\partial E_m}{\partial T} = -\frac{A}{\rho} I_s \frac{dI_s}{dT}. \]

Theory and experiment show that \(\frac{dI_s}{dT}\) has its greatest value near the Curie point. Therefore the contribution of the additional heat capacity \(C_{Vm}\) to the total heat capacity of the substance has a maximum near

the Curie point and, generally speaking, is significant where \(I_s\) is still large. Therefore the maximum value of \(\Delta C_{Vm}\) is observed at the Curie point, but \(\Delta C_{Vm}\) falls off much more slowly on the low-temperature side. Fig. 9 shows the course of the addition to the normal specific heat of nickel near the Curie point.

Attainment of Very Low Temperatures

Changes in the energy of atoms in a solid are characterized, generally speaking, by the interaction of atoms with their neighbors. In such phenomena as ferromagnetism, the ordering of alloys, or transitions of the ammonium-chloride type, this interaction makes it difficult to calculate quantitatively the corresponding changes in energy.

In many paramagnetic substances the signs of such interaction are found only at low temperatures. This circumstance gives particular importance to the magnetic method of cooling.

Unattainability of absolute zero. Before proceeding to the consideration of methods used for attaining very low temperatures, it is useful to emphasize the importance of Nernst’s theorem, which establishes the unattainability of absolute zero by means of any reversible process for finite quantities of substance.

This may be illustrated by a certain Carnot cycle. At a temperature \(\Delta T\) above \(0^\circ\mathrm{K}\), a reversible isothermal transition is carried out from state \(A\) to state \(B\), and

\[ \Delta S_{A\to B}=\frac{F-H}{\Delta T}. \]

Then the system is cooled adiabatically to \(0^\circ\mathrm{K}\), so that \(\Delta S=0\). At absolute zero the reverse transition from state \(B\) to state \(A\) is carried out, and on the basis of Nernst’s heat theorem the change of entropy \(\Delta S_{B\to A}=0\). Finally, the system is heated adiabatically to the temperature \(\Delta T\).

If such a process were possible, one could spontaneously increase the entropy of the system at the temperature \(\Delta T\) by the amount \(\Delta S_{A\to B}\) without any other changes, for example, in the working substance. The latter, however, contradicts the second law. Therefore, if the heat theorem is applicable to the system, it is impossible to cool it adiabatically to absolute zero, i.e. absolute zero is unattainable. Consideration of the equation

\[ \Delta S=C_V\frac{\Delta T}{T}=\frac{\Delta Q}{T} \]

shows that this circumstance is essentially trivial. If there are reversible processes that give a noticeable change in entropy down to the very lowest temperatures, then it is always possible to obtain sufficiently large changes of \(\dfrac{\Delta T}{T}\), since the heat capacity \(C_V\) is very small.

From the thermodynamic point of view, a change from \(0.1\) to \(0.001^\circ\mathrm{K}\) is equivalent to a change from 100 to \(1^\circ\mathrm{K}\), since the ratio \(\dfrac{\Delta T}{T}\) remains the same. It is always possible to attain temperatures whose numerical values seem very small, but the theorem on the unattainability of absolute zero automatically comes into force when changes of entropy can no longer be produced, so that the ratio \(\dfrac{\Delta T}{T}\), by which the degree of cooling is measured, can no longer increase.

Magnetic method of cooling. All substances possess a diamagnetic susceptibility that does not depend on temperature. This can be shown from an elementary consideration of electrons in closed orbits as equivalents of electric currents, which are arranged so as to weaken the action of the applied magnetic field. The atoms of a limited number of substances possess additional magnetic moments due to an uncompensated spin or orbital magnetic moment. Such substances are paramagnetic, and their paramagnetic susceptibility depends on temperature. Methods of cooling by liquefying gases, i.e., those based on the change of entropy during evaporation, become insufficient when the pressure of the saturated vapors becomes so small that it is technically impossible to produce sufficiently large changes of entropy within a finite interval of time.

The conditions under which the paramagnetic susceptibility of solids can be used to bring about sufficient changes of entropy down to very low temperatures will become clear from what follows.

The change of entropy upon magnetization is equal to

\[ \Delta S=-\int_{H_1}^{H_2}\left(\frac{\partial I}{\partial T}\right)_{P,H}\,dH, \]

and the corresponding evolution of heat amounts to \(T\Delta S\).

In the magnetic method of cooling, a paramagnetic substance is magnetized, and the heat thereby evolved is absorbed by the surrounding medium; for this purpose the sample being magnetized is placed in a stream of helium, which carries away the evolved heat and transfers it to an absorber made of liquid hydrogen. When thermal equilibrium is established, the helium is pumped out and the magnetic field is removed. In this process the entropy of the working substance must increase, which leads to the absorption of heat and a lowering of the temperature.

In order that large changes of entropy may occur in the working substance at the lowest attainable temperatures, \(\dfrac{\partial I}{\partial T}\) must be large. Since \(I=XH\), where \(X\) is the susceptib—

ity, and since most paramagnetic substances obey the Curie—Weiss law

\[ X(T+\Delta)=C, \]

then

\[ \frac{\partial I}{\partial T}=-\frac{HC}{(T+\Delta)^2};\qquad \Delta S=-\frac{H^2C}{2(T+\Delta)^2}. \]

It is clear that the change of entropy upon magnetization will increase as the temperature is lowered only when \(T\gg\Delta\), so that in a suitable working substance \(\Delta\) must be as small as possible.

Statistical derivation of Curie’s law and the meaning of \(\Delta\). The statistical derivation of Curie’s law for the paramagnetic susceptibility of a solid is based on the assumption that in the solid there are elementary magnets which tend to orient themselves parallel to the applied magnetic field, while the thermal motion of the atoms prevents their complete orientation.

If the elementary magnets have moment \(\mu\), their potential energy, when they are oriented at an angle \(\theta\) to the applied field \(H\), is equal to \(-H\mu\cos\theta\) (it being assumed here that there is no interaction between neighbors). According to Boltzmann’s theorem, the number of molecules in the solid angle

\[ d\Omega=d(\cos\theta)d\varphi \]

at an angle \(\theta\) to the field is proportional to

\[ e^{\frac{\mu H\cos\theta}{kT}}\,d\Omega, \]

and the total susceptibility per atom will be

\[ \mu\cos\theta. \]

The mean magnetic susceptibility per atom will be

\[ \overline{\mu}= \frac{\sum e^{\frac{\mu H\cos\theta}{kT}}\,\mu\cos\theta\,d\Omega} {\sum e^{\frac{\mu H\cos\theta}{kT}}\,d\Omega}. \]

It should be noted that this quantity can be expressed in a more compact form by means of the statistical sum

\[ Z=\sum e^{\frac{\mu H\cos\theta}{kT}}\,d\Omega. \]

The mean potential energy per atom in the magnetic field is equal to \(\bar e=-H\mu\), and, on the other hand,

\[ \bar e=-\frac{\partial\lg Z}{\partial\left(\frac{1}{kT}\right)}=-H\mu. \]

According to quantum theory, the magnetic moment can have only quite definite discrete orientations, so that \(\cos\theta\) can take only a series of discrete values. However, except in the case of very low temperatures or very strong

the fields, the mean susceptibility in this case, calculated according to quantum and classical (i.e. if summation is replaced by integration) statistics, coincides.

If a unit volume contains \(n\) atoms, then the susceptibility \(X\) per unit volume may be written in the form

\[ X= \frac{ n \iint e^{\frac{\mu H \cos\theta}{kT}}\, \mu\cos\theta\, \sin\theta\, d\theta\, d\varphi }{ \iint e^{\frac{\mu H \cos\theta}{kT}}\, \sin\theta\, d\theta\, d\varphi }. \]

Introducing \(x=\dfrac{H\mu}{kT}\), we have

\[ X=n\mu\left[\coth x+\frac{1}{x}\right], \]

which, for small values of \(x\), gives Curie’s law

\[ X=\frac{n\mu}{3kT}. \]

In comparing the formula obtained with experiment, one should subtract the small diamagnetism of the solid.

Weiss’s correction becomes appreciable only when \(T \simeq \Delta\) and is due to the electric and magnetic interaction of the elementary magnets with their neighbors. Convenient working substances for which \(\Delta\) is very small are gadolinium sulfate \(\mathrm{Gd_2(SO_4)_3\cdot 8H_2O}\), iron-ammonium alum \(\mathrm{Fe_2(SO_4)_3(NH_4)_2SO_4\cdot 24H_2O}\), and other salts of the rare earths and transition elements. In these cases the interaction between neighbors is small because the electron orbits responsible for paramagnetism are deeply “immersed” in the molecule and are surrounded by a very symmetrical electric field from the surrounding ions, since the number of attached water molecules is large. From the thermodynamic point of view it is of interest to consider the source of the entropy of magnetization. In gadolinium sulfate the magnetic quantum number of the gadolinium ion is \(j=7/2\), and in a very weak field the electron orbits can be oriented at angles

\[ \cos\theta=\frac{j}{j},\quad \frac{j-1}{j},\quad \frac{j-2}{j},\quad \frac{j-3}{j} \]

with respect to the field.

Taking into account also the difference between spin directions, we have \(4\cdot 2=8\) different orientations. The magnetic moment per gram-atom is equal to \(gj\beta\), where \(\beta\)—the Bohr magneton, equal to \(0.912\cdot 10^{-20}\) CGS units—represents the ratio of the mechanical moment of the electron to the magnetic moment, and \(g=2\). The work required to arrange the orbits parallel to the field is \(gj\beta H(1-\cos\theta)\). If the substance is only in the earth’s field, this work is very small and comparable with \(RT\) only below \(10^{-3}\,{}^\circ\mathrm{K}\). Therefore thermal motion leads to the fact that all states are occupied with equal probability up to this

temperature, if one neglects the orienting action of the electric forces (the electric forces exert a greater influence than the thermal motion, at about \(1^\circ\mathrm{K}\)). If the specific heat could be measured down to this temperature, then an anomaly would be observed, the integral of which

\[ \int C_v d\ln T=-R\ln 8. \]

This anomaly is missed in the usual experimental estimates of the entropy, since the temperature is not kept sufficiently low.

In a strong magnetic field, on the other hand, the gain in energy when the molecules are arranged parallel to the field becomes greater than \(RT\), so that not all positions are occupied to an equal extent. Therefore, in a strong field the entropy defect is less than \(R\ln 8\), and the corresponding difference represents the entropy of magnetization. This value is maximal and cannot be exceeded. The latter is a consequence of the fact that the susceptibility decreases when magnetic saturation sets in in the solid, so that the integral

\[ \Delta S_{\text{magnetiz.}}=\int_{H_1}^{\infty}\left(\frac{\partial I}{\partial T}\right)_{p,H}\,dH \]

has a finite value.

Gaseous State \(^{1}\)

Whereas progress in the thermodynamics of the solid state depends on the correct use of the heat theorem, the thermodynamic functions of gases are calculated exclusively from the distribution of energy levels, derived from band spectra, and the corresponding statistical sums. This is due to the fact that an experimental estimate even of the heat content of gases at high temperatures is not a simple problem.

Energy of translational motion. The distribution of the energy of translational motion among gas molecules is of primary importance in calculating the constant entropy. At ordinary temperatures the theorem of equipartition is fulfilled, and to each degree of freedom there corresponds an energy \(\frac{1}{2}kT\). Accordingly, the heat capacity of an ideal gas due to translational motion is equal to \(C_V=\frac{3}{2}R\) per mole, and the entropy is

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

where \(A\) is the entropy constant.

\(^{1}\) This chapter has been considerably abridged and revised by the translator, since it is intended rather for reference. For a fuller introduction the reader acquainted with quantum statistics is referred to special handbooks: for example, Brillouin, Quantum Statistics.

If gases obey the Nernst heat theorem, i.e., if \(S = 0\) at \(0^\circ\) K, then the heat capacity must fall to zero at very low temperatures. This phenomenon is called the degeneration of gases. Experimental observation of degeneration is difficult because, at very low temperatures and enormous pressures, when degeneration sets in, the van der Waals interaction begins to play a very noticeable role.

Remark on the van der Waals correction. From the experimental point of view, in the case of significant deviations of a gas from the laws of ideal gases, it is convenient to express the results in terms of the so-called fugacity. Fugacity is defined as follows: the change in free energy upon compression of a gas is given by the equation

\[ \Delta F = RT \ln \frac{f_1}{f_2}, \]

where \(f_1\) and \(f_2\) are the fugacities at the pressures \(P_1\) and \(P_2\), respectively.

Then the correction to the value \(\Delta F_1\), which would be obtained in the absence of deviations of the gas from ideality, has the form

\[ A_p = RT \ln \frac{P_1}{P_2} - RT \ln \frac{f_1}{f_2}. \]

In the limit, at very small densities, the gas behaves as an ideal one, and \(f_2 = P_2\).

The correction \(A_p\) at any final pressure is

\[ A_p = RT \ln P - RT \ln f = RT \ln \frac{P}{f}. \]

Writing the equation of state of the gas in the second approximation in the form

\[ V = \frac{RT}{P} - \alpha, \]

we have

\[ RT \ln \frac{f}{f'} = RT \ln \frac{P}{P'} - \alpha(P - P'), \]

and in the limit \((f' = P' \to 0)\)

\[ RT \ln f = RT \ln P - \alpha P, \]

i.e.

\[ A_p = \alpha P. \]

Statistical calculation of degeneration1. Although direct experimental observations of degeneration are unsatisfactory, it is possible to carry out a theoretical calculation of this phenomenon on the basis of statistical theory.

In doing so, it proves necessary to take into account the quantum properties of the particles, i.e., to proceed from quantum statistics. Here we shall confine ourselves only to indicating the basic principles of the latter. As already indicated above (Chapter VI), statistics makes it possible to find

an equilibrium distribution in an ideal gas consisting of \(N\) identical particles. If the particles can be in various states with energies \(\varepsilon_1, \varepsilon_2,\ldots\), then the distribution is specified by indicating the mean number of particles \(N_1, N_2,\ldots,N_n\) in each of these states. In reality the number of particles in a given state is not equal to the mean, but rapidly oscillates (fluctuates) about it.

In the most general case, in a system with a variable number of particles, the distribution function (probability density) has the form

\[ \rho_N=Ae^{\frac{\mu N-\varepsilon_k N}{kT}}=e^{\frac{\Omega+\mu N-\varepsilon_k N}{kT}}, \]

where \(N=\sum N_k\).

It satisfies the normalization condition

\[ \sum_{N,\ldots}\rho_N=1, \]

where \(\sum\) denotes summation over all possible values of the number \(N\) and the numbers \(N_k\).

In classical statistics all identical particles are regarded as fundamentally distinct, or, as is said, they can be renumbered. Two states of a system in which, for example, molecule No. 1 is in the first individual state and molecule No. 2 in the second individual state, and conversely, are regarded as different states.

On the other hand, since all molecules are identical, these two states will be physically indistinguishable from one another.

Therefore, a distribution specified by the numbers \(N_1^{(\varepsilon_1)}N_2^{(\varepsilon_2)}\), where \(\varepsilon_1\) and \(\varepsilon_2\) denote the first and second states, should be regarded as an aggregate of identical states that differ only by the individuality of the particles. The number of such equivalent states is obviously equal to the number of distributions of \(N\) molecules among the \(k\)-states of the system, i.e.

\[ \frac{N!}{\prod_k N_k!}. \]

If this factor is introduced into the sum, then one can easily obtain expressions for the mean number of molecules \(N_k\) in the \(k\)-th state. Namely,

\[ \sum_{N_1,\ldots} e^{\frac{\Omega+(\mu-\varepsilon_k)\sum N_k}{kT}} \frac{N!}{N_1!N_2!\ldots}=1. \]

Whence

\[ e^{-\frac{\Omega}{kT}} = \sum_{N_1\ldots} \frac{N!}{N_1!N_2!\ldots} e^{\frac{(\mu-\varepsilon_k)\sum N_k}{kT}} = \]

\[ = \sum_{N_1\ldots} \frac{N!}{N_1!N_2!\ldots} e^{\frac{(\mu-\varepsilon_1)N_1}{kT}} \cdot e^{\frac{(\mu-\varepsilon_2)N_2}{kT}} = \]

\[ = \sum_{N_1}\sum_{N_2}\cdots \frac{N!}{N_1!N_2!\ldots} e^{\frac{(\mu-\varepsilon_1)N_1}{kT}} e^{\frac{(\mu-\varepsilon_2)N_2}{kT}} \cdots = \]

\[ = \sum_N \left[ e^{\frac{\mu-\varepsilon_1}{kT}} + e^{\frac{\mu-\varepsilon_2}{kT}} +\cdots \right]^N = \sum_N \left( \sum_k e^{\frac{\mu-\varepsilon_k}{kT}} \right)^N \]

by the formula for Newton’s generalized binomial. Summing over \(N\) from \(0\) to \(\infty\), we have

\[ e^{-\frac{\Omega}{kT}} = \frac{1}{1-\sum_k e^{\frac{\mu-\varepsilon_k}{kT}}} \]

\[ \Omega = kT \lg \left( 1-\sum_k e^{\frac{\mu-\varepsilon_k}{kT}} \right). \]

However

\[ \bar N_k = - \frac{\partial\Omega}{\partial(\mu-\varepsilon_k)}. \]

Therefore, in the case of classical statistics, for the distribution of particles over states we have (omitting the sign of the average)

\[ N_k=\mathrm{const}\, e^{-\frac{\varepsilon_k}{kT}}, \]

i.e. the Boltzmann distribution.

It should also be noted that if the individual states \(\varepsilon_1,\varepsilon_2\ldots\) form not a discrete but a continuous sequence, then in all formulas the summation over states is replaced by integration, and they pass into the usual formulas of classical statistics.

This method of obtaining the distribution in an ideal gas can now be applied to a gas consisting of particles obeying quantum mechanics. In this case, the rejection of the individuality of identical particles plays a very essential role.

According to the conclusions of quantum mechanics1, when considering a system consisting of identical particles, it is in principle im-

possible to distinguish some specimens of these particles from others. In other words, in the quantum-mechanical description of a system we are deprived of the possibility of renumbering the particles. Therefore, when specifying the distribution of particles over states, one can indicate only the number of particles found in a certain state, but not their individuality. It follows hence that, in contrast to classical statistics, taking account of the number of permutations of molecules among states loses all meaning. Further, it turns out that a system of identical particles can be described either by symmetric or by antisymmetric wave functions (i.e. those which do not change sign, or which change sign to the opposite, under interchange of two particles). For particles of the latter type (to which, for example, electrons belong) the Pauli principle holds, according to which no more than one particle can be in a quantum state, i.e. \(N_k\) can take only the values 0 or 1.

Particles described by wave functions of the symmetric type obey the so-called “Bose—Einstein statistics.”

Let us obtain an expression for the mean number of particles in the \(k\)-th state in the case of an ideal gas obeying Bose—Einstein statistics.

We have, analogously to the preceding,

\[ e^{-\frac{\Omega}{kT}} = \sum_{N_1 \cdots} e^{\frac{(\mu-\varepsilon_k)\sum N_k}{kT}} = \sum_{N_1} e^{\frac{(\mu-\varepsilon_1)N_1}{kT}} \cdot \sum_{N_2} e^{\frac{(\mu-\varepsilon_2)N_2}{kT}} \cdots = \prod_k \sum_{N_k} e^{\frac{(\mu-\varepsilon_k)N_k}{kT}} . \]

Since no restriction is imposed on the number of particles in the \(k\)-th state, summing over \(N_k\) from 0 to \(\infty\), we have

\[ \Omega = -kT \sum_k \lg \sum_{N_k=0}^{\infty} e^{\frac{(\mu-\varepsilon_k)N_k}{kT}} = -kT \sum_k \lg \frac{1}{1-e^{\frac{\mu-\varepsilon_k}{kT}}}. \]

Whence

\[ N_{\mathrm{B.-E.}} = -\frac{\partial \Omega}{\partial(\mu-\varepsilon_k)} = \frac{1}{e^{\frac{\varepsilon_k-\mu}{kT}}-1}, \]

where the index B.—E. denotes Bose—Einstein statistics.

The statistics of particles described by antisymmetric wave functions is called Fermi—Dirac statistics. In the case of Fermi—Dirac statistics, similarly to the preceding, we have

\[ e^{-\frac{\Omega}{kT}} = \prod_k \sum_{N_k} e^{\frac{(\mu-\varepsilon_k)N_k}{kT}} . \]

Here \(N_k=0,1\), therefore

\[ e^{-\frac{\Omega}{kT}} = \prod_k \sum_{N_k=0}^{1} e^{\frac{(\mu-\varepsilon_k)N_k}{kT}} = \prod_k \left(1+e^{\frac{\mu-\varepsilon_k}{kT}}\right), \]

whence

\[ \Omega=-kT\sum_k \lg\left(1+e^{\frac{\mu-\varepsilon_k}{kT}}\right) \]

\[ N_{\mathrm{F.-D.}}= \frac{1}{e^{\frac{\varepsilon_k-\mu}{kT}}+1}. \]

Using the expressions for \(\Omega\) and \(N_k\), one can calculate all thermodynamic quantities in both new statistics. Of greatest interest here is the case when the distances between neighboring states are sufficiently small. Then summation may be replaced by integration, introducing elements of phase volume corresponding to the individual quantum states.

The number of quantum states contained in the volume \(d\gamma\) is equal to \(\dfrac{g\,d\gamma}{h^3}\), where \(g=2\) for electrons and photons and takes account of the different spin orientations in the first case and of polarizations in the second.

Then, for the total number of particles, in the case of Bose–Einstein statistics we have

\[ N=\sum N_i = A\int_0^\infty \frac{\varepsilon^{\frac12}\,d\varepsilon} {e^{\frac{\varepsilon-\mu}{kT}}-1} \]

\[ E=\sum N_i\varepsilon_i = A\int_0^\infty \frac{\varepsilon^{\frac32}\,d\varepsilon} {e^{\frac{\varepsilon-\mu}{kT}}-1} \]

\[ S=-\frac{\partial\Omega}{\partial T} = \frac{E}{T}+\frac{N\mu}{T} - kA\int_0^\infty \varepsilon^{\frac12} \ln\left(1-e^{\frac{\mu-\varepsilon}{kT}}\right)d\varepsilon . \]

The same quantities for Fermi–Dirac statistics have the form

\[ N= A\int_0^\infty \frac{\varepsilon^{\frac12}\,d\varepsilon} {e^{\frac{\varepsilon-\mu}{kT}}+1} \]

\[ E= A\int_0^\infty \frac{\varepsilon^{\frac32}\,d\varepsilon} {e^{\frac{\varepsilon-\mu}{kT}}+1} \]

\[ S= \frac{N\mu}{T} + \frac{E}{T} + kA\int_0^\infty \varepsilon^{\frac12} \ln\left(1+e^{\frac{\mu-\varepsilon}{kT}}\right)d\varepsilon, \]

where

\[ A = 2\pi v \left(\frac{2m}{h^2}\right)^{\frac{3}{2}} . \]

Degeneracy of the gas becomes significant when the quantum Bose and Fermi distributions differ substantially from the Boltzmann distribution.

This occurs when

\[ e^{\frac{\varepsilon-\mu}{kT}} \sim \pm 1 . \]

Conversely, at a sufficiently high temperature,

\[ e^{\frac{\varepsilon-\mu}{kT}} \gg 1, \]

and degeneracy is absent. In this case both quantum distributions pass over into the Boltzmann distribution.

At the same time one can calculate the entropy constant of an ideal gas, independently of which of the quantum statistics this gas obeys at low temperatures.

Namely, since

\[ N = A \int_0^\infty \frac{\varepsilon^{\frac12}\, d\varepsilon} {e^{\frac{\varepsilon-\mu}{kT}} \pm 1} \simeq \frac{A}{e^{-\frac{\mu}{kT}}} \int_0^\infty \varepsilon^{\frac12} e^{-\frac{\varepsilon}{kT}}\, d\varepsilon, \]

then

\[ e^{\frac{\mu}{kT}} = \frac{v(2\pi mkT)^{\frac32}}{Nh^3} = \frac{M^{\frac32} T^{\frac32}}{n}\cdot 3.074\cdot 10^{-4}, \]

where \(M\) is the atomic weight and \(n\) is the concentration per mole.

Therefore, when

\[ e^{\frac{\varepsilon-\mu}{kT}} \gg 1 . \]

\[ \ln\left(1+e^{\frac{\mu-\varepsilon}{kT}}\right) \simeq e^{\frac{\mu-\varepsilon}{kT}} \]

and for a monatomic gas

\[ S \simeq \frac{N\mu}{T} + \frac{E}{T} + kAe^{\frac{\mu}{kT}} \int_0^\infty \varepsilon^{\frac12} e^{-\frac{\varepsilon}{kT}}\, d\varepsilon = \frac{N\mu}{T} + \frac{E}{T} + Nk = \]

\[ = R\left(\frac52+\frac{\mu}{kT}\right). \]

Substituting \(p=nRT\) and expressing \(\mu\) through \(N\), we obtain

\[ S = R\left[ \frac52 \ln T - \ln p + \ln \frac{(2\pi mk)^{\frac32}k}{h^3} + \frac52 \right]. \]

Whence the entropy constant is

\[ \frac{S_0}{R} = \frac52 + \ln \frac{(2\pi mk)^{\frac32}k}{h^3}. \]

Degenerate gas; heat capacity of metals. Degeneracy of a gas becomes noticeable at such temperatures that

\[ e^{\frac{\mu}{kT}}=\frac{M^{\frac32}T^{\frac32}}{n}\cdot 3.074\cdot 10^{-4}\ll 1. \]

As has already been indicated, for ordinary gases degeneracy cannot be observed, since at these temperatures the van der Waals forces are so large that the application of statistics is impossible. However, the application of Fermi–Dirac statistics to electrons in a metal leads to the consideration of a degenerate gas.

In discussing questions connected with the ionization of gases, the electron gas was regarded as extremely rarefied and as obeying the laws of ideal gases. Inside a metal, however, the situation changes. If one assumes that to each atom there corresponds one conduction electron, then the concentration \(n\) becomes of the order of \(10^{-1}\) mole per cubic centimeter. Owing to the very small atomic weight of the electron gas \((M=5.43\cdot 10^{-4})\), even at ordinary temperatures the condition \(C\ll 1\) is satisfied, and the gas is completely degenerate. The energy of the degenerate electron gas is

\[ E=E_0+aT^2, \]

where \(E_0\) is the energy at absolute zero

\[ E_0=\frac{3}{40}\left(\frac{6N}{\pi v}\right)^{\frac23}\frac{Nh^2}{m}. \]

\(E_0\) is approximately 15 times greater than the classical value \(\frac{3}{2}kT\) at room temperature. Therefore the conduction electrons have considerable mobilities, but their share in the heat capacity is only

\[ C_{\text{el}}=\frac{\partial E}{\partial T}=aT, \]

where

\[ a=\left(\frac{\pi v}{9n}\right)^{\frac23}\frac{m}{h^2}\pi^2\eta k^2. \]

At sufficiently high temperatures the heat capacity of the conduction electrons is comparable with the heat capacity of the lattice. However, at these temperatures a noticeable anharmonicity appears in the lattice vibrations, contributing to the heat capacity a term of the form \(c_{\text{an}}=aT\). Therefore it is not always possible to obtain exact values of the heat capacity due to the conduction electrons at high temperatures. On the other hand, at low temperatures the heat capacity due to lattice vibrations decreases proportionally to \(T^3\), whereas the heat capacity of the conduction electrons decreases as \(T\). This makes it possible to obtain some data concerning the heat capacity

of the electron gas in a metal from investigations at very low temperatures. Although the treatment of conduction electrons as a degenerate electron gas is not entirely rigorous, since the interaction with positive ions is thereby neglected,^1 nevertheless the heat capacity of such metals as Au contains a term of the form \(aT\) at low temperatures.

Thermodynamics of Hydrogen and the Rotational Energy of Gases

The distribution of rotational energy in a gas can be obtained from the distribution of energy levels, derived from band spectra, with the aid of the corresponding statistical sum. However, the thermodynamic functions of hydrogen and deuterium depend so strongly on the correct evaluation of rotational energy that it is preferable to consider these gases in greater detail before proceeding to an evaluation of the special results arising in the case of more complex molecules.

According to classical theory, for each rotational degree of freedom of a molecule there corresponds a specific heat of \(\frac{1}{2}R\) (the equipartition theorem). The rotation of polyatomic molecules may be described as being composed of rotations about three mutually perpendicular axes, which leads to a rotational heat capacity

\[ C_{\mathrm{rot}}=\frac{3}{2}R. \]

According to quantum theory, the energy of rotation proves to be quantized. The energy of the \(r\)-th rotational level, according to the Schrödinger equation, is

\[ E_r=\frac{r(r+1)h^2}{8\pi^2 I}, \]

where \(I\) is the moment of inertia of the molecule with respect to the axis of rotation under consideration. The moments of inertia of comparatively light molecules are

\[ \begin{aligned} \mathrm{F}_2 &\;—\; 25.3\cdot 10^{-40}\ \mathrm{CGS},\\ \mathrm{O}_2 &\;—\; 19.5\cdot 10^{-40}\ \text{''},\\ \mathrm{N}_2 &\;—\; 13.8\cdot 10^{-40}\ \text{''}. \end{aligned} \]

Thus, even in light molecules the moments of inertia are sufficiently large for the quanta of rotational energy to be very small, and the characteristic temperature

\[ \theta_{\mathrm{rot}}=\frac{h^2}{8\pi I k} \]

very low.

Therefore, in the majority of cases, equipartition of the rotational energy of molecules is established. There are two exceptions. First, in diatomic and linear polyatomic molecules it is poss—

^1 And of the electrons among themselves. Translator’s note.

the moment of inertia relative to the axis joining the atoms is very small, and the corresponding quanta of rotational energy are very large, so that this degree of freedom makes no appreciable contribution to the rotational heat capacity, except at very high temperatures. Such molecules have only two rotational degrees of freedom participating in equipartition, and \(C_{\mathrm{rot}}=R\). For simple molecules the effect of rotation adds to the chemical constant the term \(\dfrac{8\pi^{2}Ik}{h^{2}}\dfrac{g}{s}\), provided only that the temperature is sufficiently low for the moment of inertia of the molecule to be regarded as constant. At higher temperatures the vibrations of polyatomic molecules change their linear form and their moment of inertia. Therefore the correction to the rotational energy cannot be expressed in a simple form. The only satisfactory way of calculating, in this case, the change introduced by rotation into the thermodynamic functions is to use the individual energy levels obtained from band spectra.

Thermodynamics of Hydrogen

The moment of inertia of the hydrogen molecule is only \(0.47\cdot 10^{-40}\) CGS units, and the quanta of rotational energy are correspondingly large. This leads to the fact that the departure from equipartition and the decrease of the rotational heat capacity become noticeable at a temperature somewhat below room temperature. At a temperature below \(50^\circ\mathrm{K}\) the rotational heat capacity of hydrogen is zero and it behaves like a monatomic gas. Although the magnitude of the quanta of rotational energy was known, at first it was not possible to construct a statistical sum giving the correct values for the heat capacity. In a diatomic molecule the rotational energy \(\varepsilon_r\) can be distributed in \(2r+1\) different ways with respect to its two axes (see p. 309), so that the statistical weight of each state is \(2r+1\). It therefore seemed that the statistical sum should have the form

\[ Z_{\mathrm{rot}}=\sum_{r=0}^{\infty}(2r+1)e^{-\frac{\varepsilon_r}{kT}}, \qquad \text{where }\varepsilon_r=\frac{r(r+1)h^{2}}{8\pi^{2}I}. \]

Denoting

\[ \theta=\frac{h^{2}}{8\pi^{2}I}; \qquad \frac{\theta}{T}=\sigma, \]

we have

\[ Z=\sum_{0}^{\infty}(2r+1)e^{-r(r+1)\sigma} \]

\[ E_{\mathrm{rot}}=-\frac{Nh^{2}}{8\pi^{2}I}\frac{d\ln Z}{d\sigma} \quad \text{per mole} \]

\[ C_{\mathrm{rot}}=\frac{dE_{\mathrm{rot}}}{dT} \quad \text{per mole} = \frac{dE_{\mathrm{rot}}}{d\sigma}\frac{d\sigma}{dT} = R\sigma^{2}\frac{d^{2}\ln Z}{d\sigma^{2}}, \]

as always when the distribution obeys Boltzmann’s theorem.

From the band spectrum of hydrogen it follows that

\[ I = 0.47 \cdot 10^{-40} \ \text{CGS units} \]

and

\[ \sigma = \frac{85.90}{T}. \]

The expression obtained for \(C_{\mathrm{rot}}\) indicates that, like other thermodynamic quantities (cf., for example, the Schottky effect), \(C_{\mathrm{rot}}\) decreases at low temperatures. At a temperature close to the characteristic temperature \(\theta = 85.90^\circ \mathrm{K}\), \(C_{\mathrm{rot}}\) has a maximum. However, nothing similar is observed on the experimental curves. The discovery of ortho- and para-hydrogen showed that the discrepancy is due to the assumption that transitions between different rotational states are possible under ordinary experimental conditions.

If the nuclei have spin, i.e. an intrinsic moment with a definite direction in space, then one can distinguish symmetric diatomic molecules with parallel spins (ortho) and antiparallel spins (para). From the Schrödinger equation and the Pauli principle it follows that the unexcited molecule \(\mathrm{H}_2\) can exist only in the following states:

Antisymmetric eigenfunction

Nuclear spins parallel
Ortho
Odd rotational states \(1, 3, 5\)

Symmetric eigenfunction

Nuclear spins antiparallel
Para
Even rotational states \(0, 2, 4\)

The statistical weight of the ortho state is three times greater than the weight of the para state. As was first shown by Dennison, if the transitions \(\mathrm{H}_2\) ortho — para occur sufficiently rarely during the time of ordinary heat-capacity measurements, then the correct expression for the mean energy is not

\[ E_{\mathrm{rot}} = \sum (2r + 1)\varepsilon_r e^{-\frac{\varepsilon_r}{kT}}, \]

but

\[ E_{\mathrm{rot}} = \frac{1}{4} \sum_{r\ \text{even}} (2r + 1)\varepsilon_r e^{-\frac{\varepsilon_r}{kT}} + \frac{3}{4} \sum_{r\ \text{odd}} (2r + 1)\varepsilon_r e^{-\frac{\varepsilon_r}{kT}}. \]

In this case the equilibrium ortho/para ratio \(= 3 : 1\) at room and higher temperature is considered “frozen” also at low temperatures. This is possible because ortho-para transitions occur rarely.

Thus,

\[ C_{\mathrm{rot}}=\frac{1}{4}\left(C_{\mathrm{para}}+3C_{\mathrm{ortho}}\right)\quad \text{per mole}, \]

where

\[ C_{\mathrm{para}}=R\sigma^{2}\frac{d^{2}}{d\sigma^{2}}\ln(1+5e^{-6\sigma}+9e^{-20\sigma}+\cdots), \]

\[ C_{\mathrm{ortho}}=R\sigma^{2}\frac{d^{2}}{d\sigma^{2}}\ln(3e^{-2\sigma}+7e^{-12\sigma}+\cdots). \]

The expression obtained for \(C_{\mathrm{rot}}\) is in excellent agreement with experiment and can be subjected to further verification in work with mixtures of ortho- and para-hydrogen of various concentrations—from 100 to 25% para-hydrogen. In all cases the corresponding curve \(C_{\mathrm{rot}}\) coincides with the experimental one. It is interesting to note that, for a mixture enriched in para-hydrogen, the heat capacity \(C_{\mathrm{rot}}\) passes through a maximum, \(C_{\mathrm{rot}}>R\).

Before discussing the influence of such a “frozen” equilibrium on the experimental estimate of the constant entropy of hydrogen, let us indicate the causes responsible for the transitions from the ortho- to the para-state.

At low temperatures a slow transition from the ortho- to the para-state occurs under the influence of high pressure. With a catalyst of activated charcoal at the temperature of liquid air or hydrogen, the transition to the corresponding mixture is completed within half an hour. When an equilibrium mixture at the temperature of liquid hydrogen is pumped off, practically pure para-hydrogen is obtained, the properties of which can be studied.

Catalysts of the ortho \(\rightleftarrows\) para transition are such paramagnetic substances as, for example, oxygen, NO, NO\(_2\), ions of metals of the transition groups, and salts of the rare earths. At high temperatures the activity of a catalyst such as platinum black, which promotes the ortho-para transition, proceeds in parallel with its activity with respect to the hydrogenation process. The hydrogen atoms formed as a result of hydrogenation bring about ortho-, para-transformations, i.e.

\[ \mathrm{H}_{2\,\mathrm{para}}+\mathrm{H}\rightleftarrows \mathrm{H}_{2\,\mathrm{ortho}}+\mathrm{H} \]

\[ \text{spins}\quad \uparrow\downarrow \qquad \downarrow \qquad \uparrow\uparrow \qquad \uparrow \]

The difference in the physical properties of pure para-hydrogen and the ordinary 25% mixture, although small, is nevertheless noticeable. Thus, for example, the pressure at the triple point is, respectively, \(53.0\pm0.1\) mm and \(53.9\pm0.1\) mm for para-hydrogen and the normal mixture.

Constant Entropy of Hydrogen

In estimating the entropy of hydrogen by a purely theoretical or a purely experimental route, certain difficulties arise. The reason for this phenomenon is that, in theoretical calculations in de-

...the expression for the entropy includes terms depending on the nuclear spin. These terms have the same form outside dependence on the phase of hydrogen or hydrogen compounds, but they do not enter into the experimentally determined quantity \(\Delta S=\int CP\,d\ln T\), since the energy of the transitions mentioned above makes no contribution to \(CP\) down to very low temperatures.

In accordance with this, the theoretical formula for the entropy of hydrogen, including the effects of nuclear spin (T. Ya. S.-entropy), gives correct values of the equilibrium constant and of the constant vapor pressure only when it is used in connection with the T. Ya. S.-entropies of other substances. In the opposite case, i.e. when it is used together with entropies of other substances calculated from the heat-capacity integral or with the aid of the heat theorem (T. T.-entropies), the results may be manifestly incorrect. This occurs because the terms introduced into the entropy formula by the presence of nuclear spin mutually cancel when only T. Ya. S.-entropies are used together, but this may not happen when both T. Ya. S.- and T. T.-entropies appear in one equation.

A similar problem, in a somewhat modified form, arises in the case when thermodynamic functions calculated on the basis of thermodynamic laws (T. T.-functions) occur in combination with functions calculated exclusively from the complete statistical sum \(Z\), depending on the energy of all kinds of motion. Since the latter are evaluated from spectroscopic data, thermodynamic functions calculated from the complete statistical sum are T. Ya. S.-functions (theoretical functions that take into account the effects of nuclear spin). As a rule, T. Ya. S.-functions include effects that are absent from T. T.-functions, namely such effects as do not lead to a change of entropy in evaporation, chemical reactions, and similar processes and do not affect the change of heat content (in heat-capacity measurements). Therefore, in each individual case it is necessary to see which parts of the T. Ya. S.-functions must be omitted (i.e. simply how the zeros of the T. Ya. S.-functions must be changed) before using them together with T. T.-functions. Several examples will be considered below.

T. T.-entropy of hydrogen. Since at low temperatures rotation makes no contribution to the total specific heat of hydrogen, it may be regarded as a monatomic gas. At higher temperatures the entropy of hydrogen will include the term \(S_{\mathrm{rot}}=\int C_{\mathrm{rot}}\,d\ln T\), due to the presence of rotational heat capacity. For example, if for the entropy of “monatomic” para-hydrogen one takes the usual expression for the entropy of a monatomic gas at low temperature, then for the change of entropy in the evaporation of solid para-hydrogen (the only means of verification at low temperatures) one obtains values that are...

...in agreement with experiment. At higher temperatures, by means of the expression for the rotational heat capacity of pure para-hydrogen, one can obtain correct values for the part of the thermodynamic entropy due to rotation for any process in which the system remains composed of pure para-hydrogen. The thermodynamic entropy of the ortho–para mixture in the ratio \(3:1\) at low temperatures must be used with caution, since such a mixture is not an equilibrium one. If, during the evaporation process, no changes occur in the mixture, then the experimentally determined change in entropy is in agreement with the value obtained from

\[ S_{3:1}=S_{\text{monatomic}}+\int C_{\mathrm{rot}\,3:1}\,d\ln T. \]

In reality, when calculating the entropy change in sublimation, it is necessary to take into account that in the solid ortho-, para-mixture \(3:1\) the Schottky effect occurs. Thus, if the Schottky effect is neglected, the experimental value of the entropy is only \(29.65\) units at \(298^\circ\mathrm{K}\), instead of \(31.23\).

If, at a very low temperature, an ortho-, para-mixture \(3:1\) takes part in some process in which enrichment of the mixture with para-hydrogen occurs, then the usefulness of applying thermodynamic functions suitable for describing equilibrium phenomena becomes doubtful. Sometimes in this case the entropy is represented in the form

\[ S_{3:1}=\frac{3}{4}S_{\text{ortho}}+\frac{1}{4}S_{\text{para}}+\frac{3}{4}\ln\frac{3}{4}+\frac{1}{4}\ln\frac{1}{4}, \]

i.e. in the form of the entropy of an ordinary mixture. However, this expression has practically nothing in common with the experimental results. In the temperature interval from \(50\) to \(\sim 300^\circ\mathrm{K}\), the equipartition theorem for the rotational degrees of freedom is completely inapplicable; the correct thermodynamic entropy is

\[ S=\frac{1}{4}S_{\text{para}}+\frac{3}{4}S_{\text{ortho}}+\frac{3}{4}\ln\frac{3}{4}+\frac{1}{4}\ln\frac{1}{4}. \]

It is assumed here that the change in entropy is computed only for processes that do not alter the ratio \(3:1\) of the ortho-, para-mixture. Then the last two constant terms in the expression for the entropy cancel and do not enter into the entropy change. At temperatures above \(300^\circ\mathrm{K}\), \(C_{\mathrm{rot}}=R\), and hydrogen may be regarded as a diatomic gas. At \(298^\circ\mathrm{K}\) its entropy is \(31.23\) units per mole. Applying the usual formula for the constant entropy, we have

\[ S=R\left[\ln A+\ln\frac{8\pi^2 Ik}{h^2}+\ln\frac{g}{s}\right]; \]

\[ s=2;\qquad g=2j+1=1. \]

To this must be added the entropy due to the nuclear spin, using the data on p. 311, but it is omitted here, apart from the symmetry factor (see p. 314).

The entropy constant of the ortho-, para-mixture, i.e. of ordinary hydrogen, will then be

\[ S=\frac{1}{4}S_{\text{para}}+\frac{3}{4}S_{\text{ortho}}+\frac{1}{4}\ln \frac{1}{4}+\frac{3}{4}\ln \frac{3}{4}= \]

\[ =R\left[\ln A+\ln \frac{8\pi^{2}Ik}{h^{2}}\right]+\ln \frac{1}{2} +\frac{3}{4}\ln \frac{3}{4}+\frac{1}{4}\ln \frac{1}{4}. \]

The last two terms must be subtracted in order to obtain the T.T. entropy. It should be emphasized that the chief difficulties in estimating the entropy of hydrogen arise at such temperatures when the ortho–para mixture in the ratio \(3:1\) is not strictly an equilibrium one, and that the calculated values can be applied only to processes in which this ratio does not change.

Thermodynamics of deuterium

In connection with the thermodynamics of deuterium, two questions should be considered:

a) Comparison of the rotational heat capacities of \(\mathrm{H}_2\), HD, and \(\mathrm{D}_2\);

b) Comparison of the chemical constants of \(\mathrm{H}_2\), HD, and \(\mathrm{D}_2\) and their influence on the position of equilibrium.

Rotational heat capacities. In order to find the rotational heat capacities, one must construct, for each gas, the parts of the statistical sums depending on rotation. Namely,

\[ Z_{\mathrm{rot}}=\sum q_r e^{-\frac{\varepsilon_r}{kT}}. \]

Then

\[ C_{\mathrm{rot}}=R\zeta^{2}\frac{d^{2}\ln Z_{\mathrm{rot}}}{d\zeta^{2}}. \]

The nuclei of hydrogen have spin \(i=\frac{1}{2}\), the nuclei of deuterium—spin \(i=1\), so that both these gases can exist in ortho- and para-modifications. HD represents an unsymmetrical molecule, so that transitions from even to odd rotational states are possible.

As a result, the following Table 6 can be compiled.

For HD the ordinary expression for the statistical sum proves applicable. Therefore the rotational heat capacity of HD must have a sharp maximum near the characteristic temperature \(\theta_{\mathrm{rot}}\) and fall rapidly after passing through this maximum. Such behavior of the heat capacity has indeed been observed experimentally.

In the case of \(\mathrm{D}_2\), the rotational heat capacity of a “frozen” equilibrium mixture of the ortho- and para-modifications (ratio of ortho to para \(2:1\))

\[ C_{\mathrm{rot}}=\frac{2}{3}C_{\text{ortho}}+\frac{1}{3}C_{\text{para}}. \]

where

\[ C_{\mathrm{ortho}}=R\sigma^2\frac{d^2}{d\beta^2}\ln \sum_{r_{\mathrm{even}}}(2r+1)e^{-\frac{\varepsilon_r}{kT}} \]

\[ C_{\mathrm{para}}=R\sigma^2\frac{d^2}{d\beta^2}\ln \sum_{r_{\mathrm{odd}}}(2r+1)e^{-\frac{\varepsilon_r}{kT}} \]

\[ \frac{\varepsilon_r\;(\text{for }D_2)}{\varepsilon_r\;(\text{for }H_2)} =\frac{I_{H_2}}{I_{D_2}}=\frac{1}{2}. \]

TABLE 6

	H₂	HD	D₂
Nuclear spin . . . .	\(i=1/2\)	. .	\(i=1\)
Rotational states . . . . . .	para \(r=0,2\ldots\) ortho \(r=1,3\ldots\)	all values \(r=0,1,2\ldots\)	ortho \(r=0,2,4\ldots\) para \(r=1,3,5\ldots\)
Quantum weights of the symmetric (antisymmetric) eigenfunctions \(\dfrac{i}{i+1}\) . . . .	\(1:3\)		\(6:3\)
Moment of inertia (assuming a constant distance between atoms) . . . .	\(0.466\cdot10^{-40}\)	\(0.62\cdot10^{-40}\)	\(0.93\cdot10^{-40}\)
Minimum change in quantum number	\(r=0\to2\)	\(r=0\to1\)	\(r=0\to2\)

The resulting curve of the rotational heat capacity turns out to be completely different from the curve for H₂. The course of the theoretical curve has been confirmed experimentally¹).

Chemical constants of the gases H₂, HD, and D₂. The chemical equilibrium constant of the reaction

\[ \mathrm{H_2O}+\mathrm{HD}\rightleftarrows \mathrm{HDO}+\mathrm{H_2} \]

is equal to 3.8 at \(25^\circ\mathrm{C}\) and 2.0 at \(100^\circ\mathrm{C}\).

In the general case, the equilibrium constant in the isotope-exchange reaction

\[ \mathrm{AH}+\mathrm{BD}\rightleftarrows \mathrm{AD}+\mathrm{BH}, \]

equal to

\[ K=\frac{[\mathrm{AD}][\mathrm{BH}]}{[\mathrm{AH}][\mathrm{BD}]}, \]

differs from unity for the following reasons:

¹) Clusius u. Bartolomé, Göttinger Nachr., 1934, I.

1. Difference in energy at absolute zero for isotopic components. Since

\[ RT \ln K = \Delta F = \Delta H - T \Delta S, \]

the difference in energy at absolute zero, for example AD and AH, will change the value of \(\Delta H\) and, consequently, \(\ln K\) when hydrogen is replaced by deuterium.

1. Owing to the different masses of H and D, the corresponding entropy constants will also differ. The significance of this effect can be estimated from the formula on p. 302 and briefly formulated as follows:

\[ S_{\mathrm{D}_2} = \frac{2}{3} S_{\mathrm{ortho}} + \frac{1}{3} S_{\mathrm{para}} + \frac{2}{3}\ln 2 - \ln 3. \]

Terms in the expression for the entropy that depend on the nuclear spin are omitted here, without taking account of the symmetry factor (cf. p. 88),

\[ S_{\mathrm{D}_2} = R\left[\ln A + \frac{3}{2}\ln M_{\mathrm{D}_2} + \ln \frac{8\pi^2 k}{h^2} I_{\mathrm{D}_2}\right] + \ln \frac{1}{2}, \]

where \(A\) is a constant,

\[ S_{\mathrm{D}_2} = S_{\mathrm{H}_2} + R\left[\frac{2}{5}\ln 2\right]. \]

Similarly,

\[ S_{\mathrm{DH}} = S_{\mathrm{H}_2} + R\left[\frac{5}{2}\ln \frac{3}{2} - \ln \frac{1}{2}\right]. \]

From the experimental point of view, the influence of the isotope effect on chemical equilibrium can be readily studied on the example of hydrogen and deuterium, using some catalyst that promotes their exchange, such as platinum. The equilibrium in solution can be studied with the aid of \(\mathrm{H_2O}\) and \(\mathrm{D_2O}\), using acid as the exchange catalyst.

A very important reaction is

\[ \mathrm{H_2 + D_2 \rightleftharpoons 2HD} \qquad K_2 = \frac{[\mathrm{HD}]^2}{[\mathrm{H_2}][\mathrm{D_2}]}. \]

In the temperature region where rotational motion is fully excited, while vibrations are not yet excited,

\[ RT \ln K = -\left(2\varepsilon_{\mathrm{HD}}-\varepsilon_{\mathrm{H}_2}-\varepsilon_{\mathrm{D}_2}\right) + \frac{3}{2}\frac{M_{\mathrm{HD}}^2}{M_{\mathrm{H}_2}M_{\mathrm{D}_2}} + \ln \frac{I_{\mathrm{HD}}^2}{I_{\mathrm{H}_2}I_{\mathrm{D}_2}} + \ln 4. \]

The first term on the right represents the change in energy at absolute zero,

\[ \varepsilon = hc\left(\frac{1}{2}\nu - \frac{1}{4}\nu x\right), \]

where \(\nu\) is the vibrational height and \(x\) is the anharmonicity constant. The term \(\ln 4\) arises because two symmetrical molecules

pass into two antisymmetric ones. It has been determined experimentally that

\[ 2\varepsilon_{\mathrm{DH}}-\varepsilon_{\mathrm{H}_2}-\varepsilon_{\mathrm{D}_2} =\Delta E_0=-155\ \text{cal/mole} \]

and

\[ \ln K_1=-\frac{34}{T}+0.6276. \]

Energy of Vibrations in Gases

One of the most general methods for determining the heat capacities of gases is to determine them from the velocity of sound in the gas. Since, under the conditions corresponding to the experiments, the deviation of gases from ideality is usually small,

\[ C_p-C_V=R, \]

and the ratio \(\dfrac{C_p}{C_V}=\gamma\) is related to the velocity of sound by the relation

\[ u=\sqrt{\frac{\gamma P}{\rho}}, \]

where \(P\) is the pressure and \(\rho\) is the density.

Subtracting from the total heat capacity the parts due to rotation and translational motion, one can estimate the heat capacity arising from vibrations.

In the case of simple diatomic molecules having one vibrational degree of freedom, one may expect that the corresponding part of the statistical sum is simply the Einstein function

\[ Z_{\text{vib.}}=\sum_r e^{-\frac{h\nu_0\left(r+\frac{1}{2}\right)}{kT}}. \]

Here \(\dfrac{1}{2}h\nu_0\) is the energy at absolute zero, and all statistical weights are set equal to unity, in the absence of any conditions. However, this expression agrees with experiment only for the simplest molecules. There are many reasons that considerably complicate the calculation. One such complication is the dependence of the velocity of sound on the frequency in the region of high frequencies. For example, in carbon dioxide at ordinary temperature the velocity of sound begins to increase at a frequency of about \(10^5\) cycles per second, and with a further increase in frequency reaches a new constant value. This increase in velocity occurs for the following reasons.

To establish an equilibrium distribution of vibrational energy a finite interval of time is required, which may become comparable with the time of the experiment in the explosion method or with the period of high-frequency sound waves. Then the vibrations of the molecule no longer follow the change in temperature during an adiabatic change of pressure. Since the velocity of sound is equal to \(a\sqrt{\gamma}\) and

\(\gamma = 1+\dfrac{R}{C_V}\), a decrease in \(C_V\) leads to an increase in the speed of sound. It should be noted that the altered heat capacity can be used in calculations of such a phenomenon as the velocity of blast waves.

The second reason why the simple expression for the statistical sum may become unsuitable is that individual quanta of vibrational energy may be very large (for example, in comparison with quanta of rotational energy), and the vibrations become anharmonic already when the molecule has acquired a small number of quanta. This means that the energy of the \(n\)-th vibrational level is less than \(nh\nu_0\). Therefore, in each term of the statistical sum one must substitute its own altered value of the energy of each vibrational level. Such vibrational levels with energy altered as a result of anharmonicity have been observed in molecular spectra.

Calculation of the complete statistical sum for some simplest atoms and molecules. In calculating the complete statistical sum of even the simplest molecules, besides the complications indicated above, there arises a whole series of further difficulties not connected with thermodynamics, but due to the complex character of band spectra.

Hydrogen atom. The statistical sum for the nucleus of an atom extends over the different orientations of the nuclear spin in an external perturbing field. At all attainable temperatures the energy of orientation is small in comparison with \(kT\). Therefore all exponential terms \(e^{-\varepsilon_r/kT}\) are equal to unity and

\[ Z_{\text{nuc.}} = 2i_s+1, \]

where \(i_s\) is the nuclear spin in units of \(\dfrac{h}{2\pi}\). Apart from nuclear energy, the hydrogen atom has only electronic energy. The quantum weight of each energy level in the hydrogen atom is

\[ q_{r,\text{electr.}} = 2j_s+1 \]

\[ j_s = l \pm s \quad (0 \nless j_s) \]

and the corresponding statistical sum

\[ Z_{\text{electr.}}=\sum (2j_s+1)e^{-\varepsilon_{j_s}/kT}. \]

The summation must be carried out over all electronic levels. In reality, however, \(e^{-\varepsilon_{j_s}/kT}\) makes an appreciable contribution to \(Z\) only when \(T > \dfrac{\varepsilon_{j_s}}{4k}\). Therefore, with the exception of those cases in which the spacings between electronic levels are small, only the normal level plays an essential role. In hydrogen in the normal state the principal quantum number is equal to unity

and \(j_s=\dfrac{1}{2}\). In this case the complete statistical sum (T. S.) for the hydrogen atom will be

\[ Z=Z_{\text{nuc.}}+Z_{\text{electr.}}=2\times 2=4, \]

for chlorine \(i_s=\dfrac{5}{2}\) and \(Z_{\text{nuc.}}=6\). The ground state of chlorine is an inverted doublet \({}^{2}P\); for the red component \(j_s=\dfrac{3}{2}\), for the violet \(j_s=\dfrac{1}{2}\), and the distance between them is \(\Delta \nu=881\ \text{cm}^{-1}\). Thus the statistical weights of both components are, respectively,

\[ 2j_s+1=4\ \text{and}\ 2 \]

and

\[ Z_{\text{electr.}}=4+2e^{-\frac{881hc}{kT}}. \]

The complete statistical sum is

\[ Z=Z_{\text{nuc.}}\cdot Z_{\text{electr.}}=6\left(4+2e^{-\frac{881hc}{kT}}\right). \]

Quantum weight and degeneracy. The origin of the “multiplicity” of energy levels of atoms and molecules can be explained in several ways. From the formal point of view of quantum mechanics, the multiplicity of a \(q\)-state with energy \(E\) arises in the case when the wave equation describing the molecule in this state has \(q\) different solutions. Then the eigenfunction of the molecule represents a linear combination of these solutions with \(q\) arbitrary constants. However, the physical meaning of degeneracy can be clarified in another way.

Quantum weight of electronic levels in an atom. The quantum weight of any electronic energy level is equal to \(2j_s+1\), where \(j_s\) is the so-called internal quantum number, representing the resultant of the orbital and spin mechanical moments. In vector notation

\[ \mathbf{j}=\mathbf{s}+\mathbf{l}. \]

The reason why the quantum weight is introduced into statistical theory can be clearly understood if one considers what happens to an atom placed in a weak magnetic field. According to the rules of spatial quantization, the resultant moment of the atom will be oriented at such angles that its projections onto the direction of the field can take a series of values

\[ m=-j,\ -j+1,\ldots,-1,\ 0,\ 1,\ldots,\ j-1,\ j. \]

If the magnetic moment of the atom is equal to \(\mu m\), then the energy acquired by it in a magnetic field of intensity \(H\) can be calculated from the ordinary statistical sum

\[ Z=\sum e^{\frac{\mu mH}{kT}}, \]

consisting of \(2j+1\) members. In the absence of a magnetic field \(H=0\) and \(Z=2j+1\), which is what should be introduced into statistical theory. Multiplicity may therefore be regarded as the result of the distribution of atoms or molecules among the sublevels of the given energy level and must be taken into account in order to obtain the correct probabilities from the distribution function.

Rotational degeneracy of diatomic molecules.
A rotational level with energy

\[ E_j=\frac{h^2}{8\pi^2 J}j(j+1) \]

has multiplicity \((2j+1)\), since the wave function describing this state contains \(2j+1\) arbitrary constants. This can also be shown in another way, by considering the number of ways in which \(j\) quanta can be distributed between two axes about which the molecule can rotate. A scheme of such a molecule is shown in Fig. 10. If rotation takes place about two mutually perpendicular axes, then two cases must be distinguished: namely, when the rotation about both axes has the same sense of direction, or different senses. Their physical distinction is easily seen in a molecule that is a dipole. Then, depending on the direction of rotation, a magnetic moment of one sign or the other is obtained.

Fig. 10. When \(J_2=0\), \(a\) and \(b\) coincide.

There exist \(j+1\) different distributions of \(j\) quanta between the two axes. Each of them, except \(J_2=0\), must be doubled in order to take account of the sense of the direction of rotation. The total number of sublevels in each rotational level is equal to \(2j+1\).

Complete statistical sum of certain diatomic molecules.
In the complete statistical sum, the summation should be carried out over all electronic, vibrational, and rotational states that make a noticeable contribution to \(Z\). Since usually the normal electronic state is a \({}^1\Sigma\) term, its multiplicity is equal to unity. The most common exceptions are NO, O\(_2\), OH, CN.

It should be noted that such molecules as OH or CN are of special interest, since their thermodynamic functions can hardly be obtained by any means other than spectroscopic data. The thermodynamic functions of radicals are, in particular, important in checking the mechanisms of certain reactions in which the participation of these radicals is assumed. Difficulties in applying the statistical-sum method arise chiefly

thus, with the choice of the correct quantum weights of the various states.

Experimental formula for band spectra. In analyzing molecular band spectra, the results can usually be expressed by the experimental formula

\[ \nu=\nu_0+\omega_e\left(v+\frac{1}{2}\right) -x\omega_e\left(v+\frac{1}{2}\right)^2+\cdots+B_vJ(J+1)+ \]

\[ +D_vJ^2(J+1)^2+E_vJ^3(J+1)^3+\cdots, \]

where \(\nu_0\) is the frequency of the electronic transition, \(\omega_e\) is the natural frequency of vibration in some definite electronic state, \(x\) is the anharmonicity constant, and \(v\) and \(J\) are, respectively, the vibrational and rotational quantum numbers. \(\frac{1}{2}\) is added to \(v\) in order to take account of the energy at absolute zero. \(B_v\) and \(D_v\) depend on \(v\), namely

\[ B_v=B_e-\alpha\left(v+\frac{1}{2}\right)+\gamma\left(v+\frac{1}{2}\right)^2+\cdots \]

\[ D_v=D_e+\beta\left(v+\frac{1}{2}\right)^2+\delta\left(v+\frac{1}{2}\right)^4+\cdots . \]

This formula takes into account the change of the moment of inertia when the rotational level changes (the molecule may be stretched under the action of centrifugal force during rotation) and the interaction of rotational and vibrational quanta (the mean dimensions of the molecule increase together with \(v\)).

For rigid molecules

\[ D_v=E_v=0;\qquad B_v=\frac{h^2}{8\pi^2 I}. \]

Summation for obtaining \(Z\). To obtain the complete statistical sum \(Z\) at a given temperature, one should first sum over all rotational levels, with fixed \(v\), equal to zero. Then one should sum over vibrational levels with \(v=1,2,3\ldots\), if at this temperature they make a noticeable contribution to \(Z\). Finally, one should sum over those electronic levels which lie sufficiently close to the ground level to give noticeable terms in the sum.

Owing to the exponential form of the separate terms of the statistical sum, if the separate parts of the internal energy (electronic, rotational, and vibrational) may be regarded as independent, it decomposes into products

\[ Z=\sum q_{\mathrm{vib.}} e^{-\frac{\varepsilon_{\mathrm{vib.}}}{kT}} \cdot \sum q_{\mathrm{rot.}} e^{-\frac{\varepsilon_{\mathrm{rot.}}}{kT}} \cdot \sum q_{\mathrm{electr.}} e^{-\frac{\varepsilon_{\mathrm{electr.}}}{kT}} = \]

\[ =Z_{\mathrm{vib.}}\cdot Z_{\mathrm{rot.}}\cdot Z_{\mathrm{electr.}}, \]

as was already indicated above.

Sometimes summation can be replaced by integration. Thus, for example, at high temperatures the rotational part of the statistical sum

\[ Z_{\mathrm{rot}}=\sum_{J=0}^{\infty}(2J+1)e^{-\sigma J(J+1)};\qquad \sigma=\frac{h^{2}}{8\pi^{2}IkT} \]

can be represented in the form of an integral, since the intervals between successive terms of the sum are small.

Then

\[ Z_{\mathrm{rot}}=\frac{1}{\sigma}. \]

Such a replacement was made in calculating the entropy constant due to rotation for a simple diatomic molecule.

The statistical sum \(Z\) must be calculated for all temperature ranges, since the dependence of \(Z\) on temperature must be known for complete knowledge of the thermodynamic functions.

Quantum weights in special cases. From the standpoint of the complications introduced by the presence of multiplicity depending on nuclear spin, the quantum weights of the rotational states of a molecule deserve special discussion. Generally speaking, \(q_r\) for a certain state is equal to the product of the rotational degeneracy by an expression due to nuclear spin. The rotational degeneracy of a state with quantum number \(J\) is always equal to \(2J+1\). The degeneracy due to nuclear spin, for a molecule consisting of different atoms, is equal to

\[ Z_{\mathrm{nuc}}=(2i_s+1)(2i_{s'}+1), \]

where \(i_s\) and \(i_{s'}\) are the nuclear spins in units of \(\dfrac{h}{2\pi}\).

As was indicated above, because of the degeneracy due to nuclear spins, terms enter into \(Z\) and, correspondingly, into the thermodynamic functions, which are absent in T. T. functions. This is easily seen from the example of the reaction

\[ AB+CD \rightleftarrows AC+BD \]

\[ Z_{\mathrm{nuc}}\,AB=(2i_A+1)(2i_B+1) \]

\[ Z_{\mathrm{nuc}}\,AC=(2i_A+1)(2i_C+1) \]

\[ Z_{\mathrm{nuc}}\,CD=(2i_C+1)(2i_D+1) \]

\[ Z_{\mathrm{nuc}}\,BD=(2i_B+1)(2i_D+1) \]

\[ \frac{Z_{\mathrm{nuc}}\,AB\,Z_{\mathrm{nuc}}\,CD} {Z_{\mathrm{nuc}}\,BD\,Z_{\mathrm{nuc}}\,AC}=1. \]

Therefore the presence of nuclear spin does not affect \(\Delta F=RT\sum \ln G\), and, in the case of different molecules, the terms entering because of the degeneracy due to nuclear spin cancel in any chemical reactions.

The NO molecule. For diatomic molecules in the ground state \({}^{1}\Sigma\), the rotational part of the statistical sum is simply

\[ Z_{\mathrm{rot}}=Z_{\mathrm{nucl.}}\sum (2J+1)e^{-\sigma J(J+1)} . \]

If in a molecule there is not one electronic level with energy of the order of \(kT\), but several, then two points should be noted in connection with the calculation of the quantum weights of rotational states. For example, the NO molecule has two electronic levels \({}^{2}\Pi_{\frac{1}{2}}\) and \({}^{2}\Pi_{\frac{3}{2}}\), separated only by \(\Delta\nu\sim 120\ \mathrm{cm}^{-1}\). Then, first, in the calculation it should be taken into account that the minimum values of \(J\) may be different. For NO, \(J=\frac{1}{2}\) for the \({}^{2}\Pi_{\frac{1}{2}}\)-level and \(J=\frac{3}{2}\) for the \({}^{2}\Pi_{\frac{3}{2}}\)-level. Secondly, all rotational levels in molecules not in \(\Sigma\)-states undergo the so-called \(\Lambda\)-doubling, i.e. each level splits into two closely lying sublevels. The energies of these sublevels lie so close together that the quantum weight of the level is simply doubled. Since for oxygen \(i_s=0\), and for nitrogen \(i_s=1\), the part of the statistical sum dependent on the nuclear spin is

\[ Z_{\mathrm{nucl.}}=1\cdot 3=3 \]

and the rotational part of the statistical sum, owing to the presence of \(\Lambda\)-doubling, will be equal to

\[ Z_{\mathrm{rot}}=3\cdot 2\sum_{J=\frac{1}{2},\,\frac{3}{2},\,\frac{5}{2},\ldots} (2J+1)e^{-\frac{\varepsilon_J}{kT}} + \]

\[ +3\cdot 2\sum_{J=\frac{3}{2},\,\frac{5}{2},\ldots} (2J+1)e^{-\frac{\varepsilon'_J}{kT}} . \]

Calculation of the part of the statistical sum dependent on rotation, for diatomic molecules consisting of identical atoms. When a molecule is built of two identical atoms, a complication arises connected with the fact that transitions between even and odd rotational states are forbidden, similarly to what takes place for ortho- and para-hydrogen at low temperature. This circumstance was considered above (p. 46).

If the molecule is in the ground \({}^{1}\Sigma\)-state and if the nuclei have an even spin (in units of the nuclear spin \(\frac{1}{2}\frac{h}{2\pi}\)) and obey Bose—Einstein statistics, then the multiplicity of the rotational state with quantum number \(J\), due to nuclear spin, is respectively equal to \((i_s+1)(2i_s+1)\) for even \(J\) and \(i_s(2i_s+1)\) for odd \(J\). An important example is the molecule

deuteron, for which \(i_s=2\) units of nuclear spin, i.e. \(i_s=1\). The rotational part of the statistical sum is

\[ Z_{\mathrm{rot}\,D_2} =6\sum_{J\ \mathrm{even}}(2J+1)e^{-\sigma J(J+1)} +3\sum_{J\ \mathrm{odd}}(2J+1)e^{-\sigma J(J+1)}. \]

Hydrogen, on the other hand, has \(i_s=1\) unit of nuclear spin \(=\frac12\).

The multiplicity of the rotational state due to the nuclear spin is equal to \(i_s(2i_s+1)\) for even \(J\) and to \((i_s+1)(2i_s+1)\) for odd \(J\), i.e. just the reverse.

The rotational part of the statistical sum of hydrogen is

\[ Z_{\mathrm{rot}\,H_2} =3\sum_{J\ \mathrm{odd}}(2J+1)e^{-\sigma J(J+1)} +\sum_{J\ \mathrm{even}}(2J+1)e^{-\sigma J(J+1)}. \]

The molecule \(O^{16}O^{16}\). Since for oxygen \(i_s=0\), its nuclei obey Bose—Einstein statistics. The multiplicity of rotational states with even \(J\) is zero, i.e. they are altogether absent. The ground state of the oxygen molecule is the \({}^{3}\Sigma_g\)-state. The splitting of the level into a triplet is very small at ordinary temperature, but the only possible rotational levels corresponding to the various components of the triplet are

\[ F_1:J=k+1,\qquad F_2:J=k,\qquad F_3:J=k-1, \]

where \(k\) can take only odd values, so that

\[ Z_{\mathrm{rot}} =\sum_{k=1,3,5}(2k+3)e^{-\frac{\varepsilon_{F_1J}}{kT}} +\sum(2k+1)e^{-\frac{\varepsilon_{F_2J}}{kT}}+ \]

\[ +\sum(2k-1)e^{-\frac{\varepsilon_{F_3J}}{kT}}. \]

The excited levels of the \(O^{16}O^{16}\) molecule nearest to the normal one begin to affect the value of the total statistical sum only at temperatures between 1000 and \(2500^\circ\mathrm{K}\).

The rotational part of the statistical sum in the temperature region for which \(\sigma\ll T\). When \(\sigma\ll T\), summations in the statistical sum for symmetric molecules may be replaced by integrations.

Since

\[ \sum_{j=0}^{\infty}(2J+1)e^{-\sigma J(J+1)} =\int_{0}^{\infty}(2J+1)e^{-\sigma J(J+1)}\,dJ= \]

\[ =\int_{0}^{\infty}e^{-\sigma x}\,dx=\frac{1}{\sigma}, \]

where \(x=J(J+1)\), then the rotational part of the statistical sum, including the multiplicity due to nuclear spin, is equal, for symmetric diatomic molecules, to

\[ Z_{\mathrm{rot}}=\frac{2(i_s+1)^2}{2}\cdot \frac{1}{\sigma}. \]

Let us consider reactions in which asymmetric molecules are formed from symmetric molecules, as, for example,

\[ \mathrm{H}_2+\mathrm{D}_2 \rightleftarrows 2\mathrm{HD}, \]

or

\[ A_2+B_2 \rightleftarrows 2AB. \]

Thus, the F. E.—free energies (depending on the volume)

\[ A_{A_2}=-RT\ln \frac{(2i_A+1)^2}{2} \left(\frac{1}{\sigma_{A_2}}Z_{\mathrm{vib.}}\cdot Z_{\mathrm{electr.}}\cdot Z_{\mathrm{transl.}}\right) = \]

\[ =-RT\ln \frac{(2i_A+1)^2}{2}\cdot G'_{A_2}. \]

\[ A_{B_2}=-RT\ln \frac{(2i_B+1)^2}{2}\,G'_{B_2}. \]

But

\[ 2A_{AB}=-RT\ln (2i_A+1)^2(2i_B+1)^2G_{AB}^{\prime 2}. \]

Therefore

\[ \Delta A=-RT\ln [2\cdot 2]\frac{G_{AB}^{\prime 2}}{G'_{A_2}G'_{B_2}}. \]

Thus, in these reactions the multiplicities due to nuclear spin always cancel, except for the symmetry factor \(R\ln 2\). In this case the T. T.-functions are obtained from the T. F.-functions by subtracting the terms that depend on nuclear spin, but retaining the symmetry factor. This was what was done in the theoretical expression for the chemical constant of diatomic molecules indicated on p. 302.

The only important cases where \(\sigma\) is not small in comparison with \(T\) occur for hydrogen at temperatures below \(273^\circ\mathrm{K}\) and for deuterium—below \(200^\circ\mathrm{K}\). Both of them were considered above.

Radical OH. The ground level of the OH radical is an inverted \({}^{2}\Pi\)-term, i.e. with energy \({}^{2}\Pi_{\frac{3}{2}}<{}^{2}\Pi_{\frac{1}{2}}\).

The splitting of each of these terms into \(A\)- and \(B\)-sublevels is so large that, instead of multiplying each term in the rotational part of the statistical sum by 2, as for \(\Lambda\)-doubling in NO, it is better to use four different statistical sums. Mul-

the multiplicity, depending on the nuclear spin, is equal to 2, since \(i_s=\dfrac{1}{2}\) for H and \(i_s=0\) for O,

\[ Z_{\mathrm{rot}}=2\left[\sum_{J=0}^{\infty}(2J+1)e^{-\frac{\varepsilon_J}{kT}}\right] \]

for all values of \(\varepsilon_J\).

One could consider examples of other diatomic molecules, but they are not of special interest. For polyatomic molecules the calculations become very complicated, since their spectra have not been completely investigated. An approximate solution of the problem can be given under the assumption that the molecules are rigid and that the energy levels are independent. Then the statistical sum splits into factors. The vibrational part of the statistical sum is not always easy to estimate, in particular for long molecules. For small molecules it is not of substantial importance, since, for example, for \(\mathrm{N_2O}\) at \(298^\circ\mathrm{K}\), \(Z_{\mathrm{rot}}=496\), \(Z_{\mathrm{vib}}=1.1\).

Linear molecules. \(\mathrm{HCN}\), \(\mathrm{N_2O}\), \(\mathrm{CO_2}\), \(\mathrm{C_2H_2}\). All these molecules may be regarded as diatomic, and

\[ Z_{\mathrm{rot}}=\frac{1}{s\sigma}, \]

where \(s\) is the symmetry factor, and the summation has been replaced by integration, which is permissible since \(T\) and \(I\) are large.

Spherical rotors. \(\mathrm{CH_4}\), \(\mathrm{CCl_4}\). In spherical rotors \(I_A=I_B=I_C\), and if the summation may be replaced by integration, then

\[ Z_{\mathrm{rot}}=\frac{\sqrt{\pi}}{s\sigma^{3/2}}. \]

For both of these molecules \(s=12\) (\(s\) is defined as the number of distinct permutations produced by rotation of the molecule or of its part).

Symmetric tops. \(\mathrm{NH_3}\), \(\mathrm{CHCl_3}\) have \(I_A=I_B\ne I_C\). Upon replacing summation by integration,

\[ Z_{\mathrm{rot}}=\frac{\sqrt{\pi}}{s\sigma_A\sqrt{\sigma_C}}. \]

The molecule \(\mathrm{C_2H_6}\) is a double pyramid. In the same case it has

\[ Z_{\mathrm{rot}}=\frac{\pi}{s\sigma_A\sigma_C}. \]

Asymmetric molecules. \(I_A\ne I_B\ne I_C\). In this case it is convenient to write approximately \(I=\sqrt{I_AI_BI_C}\) and

\[ Z_{\mathrm{rot}}=\frac{1}{s}\sqrt{\frac{\pi}{\sigma_A\sigma_B\sigma_C}}. \]

Typical values of \(s\): for \(\mathrm{H_2O}\), \(s=2\); for \(\mathrm{C_2H_6}\), \(s=12\). For more complex molecules the expression may be proposed

\[ Z_{\mathrm{rot}}=\frac{1}{s\pi}\left[8\pi^3\left(I_A I_B I_C \ldots \frac{1}{h}\right)^{\frac{1}{n}}\right]^{\frac{n}{2}} . \]

However, direct experimental verification of this expression is very difficult.

Information on the thermodynamic functions of various substances, estimated from experimental data, may be found in Ann. Rep. Chem. Soc., 32, 84, 1936.

Among chemical equilibria that are difficult to attain in our measurements, the following may be noted:

\[ \mathrm{H_2O}=\frac{1}{2}\mathrm{H_2}+\mathrm{OH};\quad \mathrm{N_2}=2\mathrm{N};\quad \mathrm{NO}=\mathrm{N}+\mathrm{O}; \]

\[ \mathrm{C}+2\mathrm{H_2}=\mathrm{CH_4};\quad 2\mathrm{C}+2\mathrm{H_2}=\mathrm{C_2H_4};\quad 2\mathrm{C}+\mathrm{H_2}=\mathrm{C_2H_2}; \]

\[ \mathrm{Br_2}=2\mathrm{Br};\quad \mathrm{J_2}=2\mathrm{J};\quad \mathrm{P_2}=2\mathrm{P};\quad \mathrm{S_2}=2\mathrm{S}; \]

\[ \mathrm{SO}=\frac{1}{2}\mathrm{S_2}+\frac{1}{2}\mathrm{O_2}. \]

It should be noted that the thermodynamic functions calculated from spectroscopic data are in excellent agreement with estimates based on other data. The exceptions are one or two substances, such as, for example, \(\mathrm{NO}\) or \(\mathrm{CO}\), already mentioned in connection with the verification of Nernst’s heat theorem, and also substances such as ice or ethane, where the discrepancy may be attributed to an insufficiently fully investigated picture of the energy levels.

Adsorption

The equilibrium of a gas over the surface of a solid body (equilibrium adsorption) deserves discussion both because it represents a very characteristic example of physicochemical equilibrium, the thermodynamic treatment of which is of interest, and because it can be used to illustrate how the formulas of statistical equilibrium may be derived from the concept of dynamic equilibrium.

Langmuir adsorption isotherm. The only case of adsorption that will be considered here consists in the capture of a monomolecular layer of gas by definite adsorption centers on the surface of a solid body. The adsorption centers are assumed to be so far removed from one another that the presence of an adsorbed molecule at one center does not affect the properties of another center. This leads to the independence of the adsorption potential \(\varphi\) from the number of previously adsorbed molecules. Dynamic equilibrium is established when the number of molecules evaporating from the surface per unit time is equal to the number condensing on it. If in all there are \(N\) adsorption cen-

...meters and the \(\alpha\)-th part of them is already occupied, then the number of molecules evaporating from the surface per unit time is equal to

\[ a \alpha N e^{-\frac{E_1}{kT}}, \]

where the introduction of the factor \(e^{-\frac{E_1}{kT}}\) takes into account the fact that a molecule must possess an excess (above the average) energy \(E_1\) in order to be able to leave the surface; \(a\) is a certain constant. The number of condensing molecules will be

\[ bp(1-\alpha)N e^{-\frac{E_2}{kT}}, \]

since the number of molecules reaching the surface per unit time is equal to \(bp\) \([\,b=(2\pi mkT)^{-1},\ \text{where } m \text{ is the mass}\,]\), and the free part of the surface is \((1-\alpha)N\).

In order that the molecules striking the surface of a solid body may be retained on it, they must have a certain activation energy \(E_2\). This fact is taken into account by the factor \(e^{-\frac{E_2}{kT}}\).

Equating the number of evaporating molecules to the number of adsorbed ones, we obtain

\[ N a \alpha = bp(1-\alpha)N e^{-\frac{E_2-E_1}{kT}} = bp(1-\alpha)N e^{-\frac{\varphi}{kT}}. \]

Solving with respect to \(\alpha\), we have for the number of adsorbed molecules

\[ N\alpha = N\frac{bp e^{-\frac{\varphi}{kT}}}{a+bp e^{-\frac{\varphi}{kT}}}. \]

This formula has the characteristic property that, for small \(p\), the number of adsorbed molecules is proportional to

\[ N\lambda = N\left(\frac{b}{a}e^{-\frac{\varphi}{kT}}\right)\cdot p . \]

For large \(p\)

\[ N\alpha=\mathrm{const}, \]

i.e. the surface of a solid body exhibits the phenomenon of saturation. In reality, however, the assumption made about the noninteraction of the adsorbed molecules becomes completely inapplicable long before saturation sets in.

Statistical calculation. The advantage of the statistical calculation over the dynamical one consists in the fact that the values of all constants entering into the final formulas are immediately determined.

\(^{1}\) \(p\) — pressure. Translator’s note.

The distribution of molecules between the sorbent and the gas can be found in the following way. The probability that a certain molecule will be on the surface of the solid is equal to

\[ P_{\text{tv}}=(N_{\max}-N_a)e^{\frac{\varphi}{kT}}, \]

where \(N_{\max}\) is the maximum number of molecules that can be adsorbed, and \(N_a\) is the number of actually adsorbed molecules. Thus \(N_{\max}-N_a\) represents the number of free sites on the surface of the solid. The probability of finding a molecule in the gas is equal to

\[ P_{\text{gas}}=V_{\text{gas}}\frac{(2\pi mkT)^{\frac{3}{2}}}{h^3}, \]

so that

\[ \frac{N_a}{N_g} = \frac{N_{\max}-N_a}{V_g} \frac{h^3}{(2\pi mkT)^{\frac{3}{2}}} e^{-\frac{\varphi}{kT}}. \]

This equation can be given the same form as in the dynamical calculation. Namely, since

\[ \frac{V_g}{N_g}=\frac{kT}{p}, \]

we have

\[ N_a= \frac{bpN_{\max}e^{-\frac{\varphi}{kT}}} {1+bpe^{-\frac{\varphi}{kT}}}, \]

where

\[ b=\frac{h^3}{(2\pi mkT)^{\frac{3}{2}}}\cdot\frac{1}{kT}. \]

A comparison of the statistical and dynamical methods of calculation in this case of equilibrium shows at the same time the general properties of the dynamical method for calculating any equilibria. If the phenomenon under consideration is sufficiently simple, then the dependence of equilibrium on pressure and other quantities can also be found from simpler theoretical premises, and more easily than by the statistical method. However, with the dynamical method of calculation one usually has to leave undetermined constants in the rates.

The statistical method, on the other hand, makes it possible not only to determine the dependence of equilibrium on pressure and other quantities, but also gives the relation of the constants in the rates. This latter property has made it possible to extend thermodynamic theory also to the calculation of reaction rates.

1. See any course in quantum mechanics. ↩↩↩