THEORIES OF THE LIQUID STATE

J. de Boer *)

§ 1. INTRODUCTION

The purpose of the present review of theories of the liquid state is a critical comparison of theories proposed by various authors **) and, in particular, consideration of the question of which of them are capable, in terms of intermolecular forces, of giving a quantitative explanation of the well-known thermodynamic properties of liquids. Therefore many interesting attempts to establish relations between various properties of liquids, which, however, bypass an explanation of these properties in terms of intermolecular forces, such as, for example, certain investigations based on the “hole theory” of liquids (see ²), will not be considered in the present article.

Our consideration will be restricted to the simplest case of monatomic liquids of the nonmetallic type, for which the assumption of a central character of the intermolecular forces may serve as a good approximation. In addition, in this article we shall not touch upon quantum effects, since their influence on the properties of liquids has been considered elsewhere³, ⁴. Therefore the experimental material that can be used for comparing theory with experiment is limited to the heavy noble gases Ne, A, Kr, Xe, and Rn, among which Ne already exhibits weak quantum effects.

Rigorous theories of the liquid state are based on the evaluation of the configuration integral

\[ Z=\lambda^{-3N}\frac{1}{N!}\int e^{-\beta\Phi} d\mathbf r_1\ldots d\mathbf r_N, \tag{1} \]

where \(\beta=1/kT\). The first factor \(\lambda^{-3N}\), with \(\lambda=h/(2\pi m kT)^{1/2}\), appears

) J. de Boer, Proceedings of the Royal Society*, A 1120, 4 (1952). Translated by I. Z. Fisher.

**) See, in particular, the excellent review of lattice theories of the liquid state¹.

as a result of integration over the kinetic energy in classical theory. The factor \(1/N!\) is introduced because, in integration over all possible configurations with potential energy \(\Phi(\mathbf r_1,\ldots,\mathbf r_N)\), configurations differing only by permutations of molecules are regarded as different “states,” whereas in the state integral they must be regarded as identical.

The theoretical treatment of the liquid state is usually based on the following assumptions concerning the function \(\Phi(\mathbf r_1,\ldots,\mathbf r_N)\):

1. Between any two molecules separated by a distance \(r\) from one another there acts the same central force, determined by the intermolecular potential \(\varphi(r)\).

2. The total potential energy \(\Phi(\mathbf r_1,\ldots,\mathbf r_N)\) is additively composed of the interaction energies of all pairs of molecules, so that \(\Phi=\sum_{i>k}\varphi(r_{ik})\).

It is further assumed that the potential \(\varphi(r)\) has the form

\[ \varphi(r)=4\varepsilon\left[\left(\frac{\sigma}{r}\right)^{12}-\left(\frac{\sigma}{r}\right)^6\right], \tag{2} \]

where \(\varepsilon\) and \(\sigma\) are two quantities (of the dimensions of energy and length, respectively) that determine the magnitude of the intermolecular force and vary from substance to substance.

These quantities \(\varepsilon\) and \(\sigma\) can be used to introduce “molecular units” for the molar volume \((N\sigma^3)\), energy \((N\varepsilon)\), temperature \((N/R=\varepsilon/k)\), and pressure \((N\varepsilon/N\sigma^3=\varepsilon/\sigma^3)\), where \(N\) is Avogadro’s number. From the circumstance that in classical theory there can occur no constants that would be dimensionally independent of the molecular mass \(m\) and of the quantities \(\varepsilon\) and \(\sigma\), it follows that such dimensionless quantities as \(p^*(=p\sigma^3/\varepsilon)\) and \(U^*(=U/N\varepsilon)\) can be functions only of the two other dimensionless quantities \(V^*(=V/N\sigma^3)\) and \(T^*(=Tk/\varepsilon)\). To construct a dimensionless quantity from the constants \(m\), \(\varepsilon\), and \(\sigma\) alone is impossible. Therefore, in classical theory

\[ p^*=p^*(V^*,T^*),\quad U^*=U^*(V^*,T^*). \tag{3} \]

Although we do not intend here to consider quantum effects, it may nevertheless be noted that the stated proposition no longer holds in quantum theory, where from the new constant \(h\), together with \(m\), \(\varepsilon\), and \(\sigma\), one can form the dimensionless quantity \(\lambda^*=h/\sigma(m\varepsilon)^{1/2}\). Consequently, in quantum theory equations (3) must be corrected: \(p^*=p^*(V^*,T^*,\lambda^*)\) and \(U^*=U^*(V^*,T^*,\lambda^*)\). For heavy substances, however, the quantity \(\lambda^*\) is very small in comparison with unity, and the dependence of \(p^*\) and \(U^*\) on \(\lambda^*\) may be neglected.

Thus, in classical theory there exists a principle of corresponding states between the “reduced” thermodynamic

quantities, i.e., thermodynamic quantities expressed in molecular units. Such a state of affairs greatly facilitates comparison of theoretical results with experimental data, since theoretical results expressed in the form of relations of type (3) can be directly compared with experimental data for the heavy noble gases, if these data are expressed in the form (3) using known values of the molecular units. Since the mean observed values of the various reduced thermodynamic quantities of liquids are apparently not widely known, a summary of the experimental data will be given in the following section.

The comparison of experimental data with theoretical data will be made in §§ 3—8. There are two types of theories which in principle permit an estimate of the properties of liquids in terms of intermolecular forces: the so-called “free-volume theory” and theories constructed on the study of the radial distribution function. These theories will be discussed and compared with experimental data in §§ 3—6 and § 8, respectively.

§ 2. EXPERIMENTAL DATA EXPRESSED IN MOLECULAR UNITS

The values of the molecular units used below are obtained either from the behavior of the second virial coefficient, or from data for the critical point3, 5, 6. These data are collected in Table I. For completeness, data are also given here for He, H₂, D₂, and N₂, although these substances will not be considered in our article.

Table I

Molecular units of the noble gases

Substance	\(N\sigma^{3}\) \((\text{cm}^{3}/\text{mol})\)	\(N\varepsilon\) \((\text{cal}/\text{mol})\)	\(\varepsilon/k\) \((^\circ\text{K})\)	\(\varepsilon/\sigma^{3}\) \((\text{atm})\)
He	10,06	20,31	10,22	83,40
Ne	12,51	70,74	35,60	233,6
A	23,70	233,0	119,8	413,2
Kr	29,35	330,1	166,1	464,4
Xe	35,74	456,6	229,8	513,2
H₂, D₂	15,12	73,52	37,00	200,8
N₂	31,27	191,9	96,57	253,5

The liquid state extends from the triple point with reduced temperature \(T_{\mathrm{pl}}^{*} \approx T_{\mathrm{tr}}^{*} = 0,7\) to the critical point with temperature \(T_{\mathrm{cr}}^{*} = 1,26\). The experimental data given for

Table II

Triple and critical points

	$T^*_{\mathrm{tr}}$	$p^*_{\mathrm{tr}}\times 10^3$	$V^*_{\mathrm{tr}}$	$T^*_{\mathrm{cr}}$	$p^*_{\mathrm{cr}}$	$V^*_{\mathrm{cr}}$
Ne	0,690	1,882	1,292	1,247	0,1150	3,333
A	0,699	1,646	1,182	1,258	0,1162	3,164
Kr	0,698	1,556	1,161	(1,26)	0,1167	3,163
Xe	0,702	1,570	1,162	(1,26)	0,1134	3,095
Average	0,70	1,56	1,16	1,26	0,117	3,16

of the triple and critical points3, 5 are given in Table II, where the average values are also indicated.

In Fig. 1 are shown the reduced molar volumes for the liquid between the triple and critical points as a function of the reduced temperature. In Fig. 2 are shown the “densities” $1/V^*_{\mathrm{l}}$ and $1/V^*_{\mathrm{g}}$ of the liquid and gas and the “rectilinear diameter.” In Fig. 3 are shown

Fig. 1. Reduced experimental values of the liquid volume $V^*_{\mathrm{l}}$.
$\times$ — A; $\bullet$ — Kr; $\circ$ — Xe.

the melting curves for Ne and A, and, finally, in Fig. 4—the logarithms of the reduced pressure of saturated vapors as a function of $1/T^*$.

The heat and entropy of vaporization are thermodynamically related to the difference between the volumes $V^*_{\mathrm{g}}-V^*_{\mathrm{l}}$ of the gas and liquid and to the vapor pressure $p^*=f(T^*)$ at the condensation point. Therefore these caloric data cannot be regarded as independent experimental quantities subject to comparison with theory.

Fig. 2. Reduced experimental values of the densities of gas and liquid \(1/V_g^*,\ 1/V_l^*\), expressed in molecular units. \(\times\) — A; \(\bullet\) — Kr; \(\circ\) — Xe.

Fig. 3. Reduced experimental melting curve in molecular units and the theoretical curve (§ 7). ○ — Ne; ● — A.

Fig. 4. Reduced experimental vapor pressure in molecular units. The theoretical curves were calculated according to the original theory of Lennard-Jones and Devonshire \((z = 12)\) and with the aid of the reduced coordination number \(z' = 10\) (§ 4). The value of the dashed line is given in § 5. × — A; ● — Kr; ○ — Xe.

§ 3. THE FREE-VOLUME THEORY OF LIQUIDS
OF LENNARD-JONES AND DEVONSHIRE

The first attempt to give a description of liquids on the basis of the concept of “free volume” was made by Eyring^7 and by Eyring and Hirschfelder^8. However, they used the free-volume theory to establish the connection between various observed properties of liquids. On the other hand, Lennard-Jones and Devonshire used the concept of free volume in order to express the properties of liquids in terms of intermolecular forces^9, ^10.

Each molecule is constrained to move in a cell near a node of a virtual face-centered cubic lattice with a total number of nodes equal to the number of molecules \(N\). The volume of a cell is equal to \(v = V/N\). The field \(\psi^*=\psi/\varepsilon\), in which a molecule moves in its cell, is the molecular field of the surrounding molecules (2), averaged over directions:

\[ z\psi^*(r^*)=z\left\{\frac{l(r^*/a^*)}{v^{*4}}-\frac{2m(r^*/a^*)}{v^{*2}}\right\}, \tag{4} \]

where \(z(=12)\) is the coordination number of the lattice, \(r^*=r/\sigma\), \(v^*=v/\sigma^3\), \(a^*=a/\sigma\) is the distance to the nearest neighboring molecule \((a^3=v\sqrt{2})\), and the functions \(l\) and \(m\) are given by the expressions:

\[ \left. \begin{aligned} l(\rho)&=(1+12\rho^2+25.2\rho^4+12\rho^6+\rho^8)(1-\rho^2)^{-10}-1,\\ m(\rho)&=(1+\rho^2)(1-\rho^2)^{-4}-1. \end{aligned} \right\} \tag{5} \]

The potential energy of the liquid, when all molecules are situated exactly at the centers of their cells, is equal to

\[ \frac{z}{2}N\varphi_0^*=\frac{z}{2}N\varphi^*(a^*)=\frac{z}{2}N(v^{*-4}-2v^{*-2}), \tag{6} \]

where the coefficients 1 and 2 must be replaced by 1.011 and 2.409 if the interaction not only of nearest neighbors, but also of more distant molecules, is taken into account.

Since the total energy of the liquid may be written as

\[ E=\frac{z}{2}N\varphi_0+\sum_{i=1}^{N}z\psi(r_i), \tag{7} \]

the configurational integral is equal to

\[ Z=\lambda^{-3N}\exp\left(-\frac{1}{2}z\beta N\varphi_0\right)v_f^N, \tag{8} \]

where \(\beta=1/kT\), \(\lambda=h/(2\pi mkT)^{1/2}\), and where the “free volume” \(v_f\) is equal to

\[ v_f(v,T)=\int_{0}^{r_{\max}} e^{-\beta z\psi(r)}\,4\pi r^2dr. \tag{9} \]

The maximum value of \(r\) is determined by the condition \((4\pi/3)r_{\max}^{3}v=a^{3}/\sqrt{2}\), or \(\rho_{\max}=r_{\max}/a=(3/4\pi\sqrt{2})^{1/3}\). If we denote \(v_f=2\pi a^{3}g\), then the integral for \(g\)

\[ g=\int_{0}^{\rho_{\max}} e^{-\beta z\psi(\rho)}\,2\rho^{2}\,d\rho \tag{10} \]

contains no constants and depends only on \(v\) and \(T\). Tables of this integral are contained in the original paper\(^9\). Extensive tables have recently been published in Ref. 11.

The limiting case of low densities

A typical difficulty arises if one considers the limiting value of the configurational integral for low densities \((V\to\infty)\). Then

\[ Z\to \lambda^{-3N}v^{N}=\lambda^{-3N}V^{N}/N^{N}, \tag{11} \]

whereas for an ideal gas it should be

\[ Z_{\text{ideal gas}}=\lambda^{-3N}V^{N}/N! \tag{12} \]

The factor \(N!\) appears because in the configurational integral all states obtained from one another by permutation of identical particles must themselves be regarded as identical (the counting must be carried out by “phases of kind,” and not by “phases of appearance”*). Although in deriving expression (11) the identity of the particles was taken into account, the correct expression (12) is not obtained. Since \(1/N!\sim e^{N}/N^{N}\), Lennard-Jones and Devonshire were forced additionally to introduce into the configurational integral of their theory a factor \(e^{N}\), which in the limit \(V\to\infty\) makes expression (11) equal to (12). But since there is no sufficient basis for such a procedure—except, perhaps, as we shall see below, the empirical fact that the entropy of a liquid calculated from (8) is too small—we shall omit this factor, so as not to complicate the theory.

Equation of state

The equation of state can be obtained from the configurational integral by using the relation for the free energy \(F=U-TS=-kT\ln Z\):

\[ p=-\left(\frac{\partial F}{\partial V}\right)_{T} = kT\left(\frac{\partial\ln Z}{\partial V}\right)_{T}, \tag{13} \]

* See J. W. Gibbs, Elementary Principles in Statistical Mechanics, translated by K. V. Nikolsky, Gostekhizdat, Moscow—Leningrad, 1946. (Translator’s note.)

which gives us

\[ \frac{pV}{RT}=-\frac{z}{2kT}v\frac{\partial \varphi_0}{\partial v} +v\left(\frac{\partial \ln v_f}{\partial v}\right)_T . \tag{14} \]

The first term here is proportional to the “potential part of the pressure,” corresponding to the potential energy of a system of molecules arranged strictly at the nodes of a lattice, while the second part is the “thermal pressure,” arising from the motion of molecules in their cells.

A direct check of the theory is provided by calculating the volume of a liquid under the pressure of its saturated vapor; this can be obtained approximately from (14) if one sets \(p=0\), since the very small vapor pressure may be neglected. The isotherm (14) between \(T^*=0.7\)

Fig. 5. Isotherms of the liquid state according to the original theory of Lennard-Jones and Devonshire (\(z=12\)) and for a reduced coordination number (\(z'=10\)). \(V^*_{\mathrm{тв}.0}\), \(V^*_{\mathrm{тв.пл}}\), and \(V^*_{\mathrm{ж.пл}}\) denote the volume of the solid at \(T=0\), the volume of the solid at the triple point, and the volume of the liquid at the triple point, respectively.

(\(\sim\) melting point) and \(T^*=1.0\) is shown in Fig. 5, from which it is seen that the volume of the liquid found in this way is underestimated by \(10 \div 20\%\).

For large volumes the \(p\)–\(V\) isotherm exhibits an S-shaped bend, analogous to the bend of the van der Waals isotherms, which represents the transition from the liquid state to the gaseous state. Data for the critical point are given in Table III. Here too the critical volume is too small, although the critical temperature proves to be approximately correct.

Table III

Critical data according to Lennard-Jones and Devonshire

	$T_{\mathrm{cr}}^{*}$	$V_{\mathrm{cr}}^{*}$	$p_{\mathrm{cr}}^{*}$	$p_{\mathrm{cr}}^{*}V_{\mathrm{cr}}^{*}/T_{\mathrm{cr}}^{*}$
Theory	1.30	1.77	0.434	0.591
Experiment	1.26	3.16	0.117	0.294

§ 4. REDUCTION OF THE COORDINATION NUMBER

The situation can be noticeably improved if it is assumed that the coordination number in the liquid state $z'=\xi z$ is smaller than in the solid. But here a difficulty arises owing to the circumstance that a reduction of the coordination number will affect the relation between $a$ and $v$, which for $z=12$ (a face-centered cubic lattice) reduced to $a^3=v\sqrt{2}$. Now $a^3=\gamma v$, and one of the possibilities for determining the value of $\gamma$ for smaller values of the coordination number consists in interpolation between the values for the face-centered cubic, body-centered cubic, and simple cubic lattices (see Table IV). The values of $\gamma$ obtained by interpolation will be only an upper limit, since the disorder inherent in the system, for example for $8<z'<12$, will undoubtedly lead to still smaller values of $\gamma$.

The values $\varphi_0$ and $v_f$ for the reduced coordination number $z'$ can be determined directly from the values for $z=12$ by means of the obvious relations

\[ \varphi_0(v; z')=\varphi_0(v'; z)\equiv \varphi_0', \tag{15} \]

\[ v_f(v; T; z')=v_f(v', T'; z)\equiv v_f', \tag{16} \]

where $\varphi_0'$ and $v_f'$ are the values of these functions for $z=12$, but calculated for $v'$ and $T'$ determined by the expressions

\[ v'=(\gamma_{z'}/\gamma_z)v,\qquad T'=T/\xi. \tag{17} \]

This leads to the equation of state

\[ \frac{pV}{RT} = -\frac{z}{2kT'}\,v'\frac{\partial \varphi_0'}{\partial v'} + v'\left(\frac{\partial \ln v_f'}{\partial v'}\right)_{T'} . \tag{18} \]

It follows from this that for smaller coordination numbers $z'=\xi z$ with $\xi<1$ and given $V$ and $T$ the ratio $pV/RT$ is the same as for $z'=z$ ($\xi=1$), but for the values $V'=(\gamma_{z'}/\gamma_z)V$ and $T'=T/\xi$.

Using the “interpolation” definition of $\gamma$, we obtain the curves for $z' = 10$ ($\xi = 0.834$), shown in Fig. 5. It is clear from the figure that the volumes of the liquid calculated in this way are in better agreement with experiment than for $z' = 12$. But it must be noted that the coordination number $z'$ is a function of temperature and density, which makes the task of determining it near the critical point very difficult.

Table IV

Determination of $\gamma = a^3/v$

$\xi$	$z' = \xi z$	$\gamma z'$ (interpolation)
1	12	$\sqrt{2} = 1.414$
0.834	10	$\approx 1.35$
0.666	8	$\dfrac{3}{4}\sqrt{3} = 1.299$
0.500	6	$1.000$

It must also be noted that the idea of a coordination number decreasing from 12 to 10 or 9 has been introduced into the theory arbitrarily, for the purpose of obtaining better agreement with experiment. The only conclusion that can be drawn is that a decreasing coordination number improves this agreement, but the magnitude of the decrease must, of course, follow from the theory. Below we shall see that a new interpretation of the decrease of the coordination number is provided in the generalized theory of free volume, which will be considered in § 6.

§ 5. VAPOR PRESSURE. TROUTON’S RULE

The saturated vapor pressure can be obtained by equating the chemical potentials of the liquid and the vapor. Neglecting, in the case of the liquid, the quantity $pV$, which changes the chemical potential only slightly, we have:

\[ \mu_{\text{liq}} = -\frac{kT}{N}\ln Z = \frac{3}{2}kT\ln\lambda + \frac{z}{2}\varphi_0 - kT\ln v_f, \tag{19} \]

whereas for the vapor, treating it as an ideal gas (which is correct for temperatures not too close to the critical one),—

\[ \mu_{\text{g}} = -\frac{kT}{N}\ln Z + kT = \frac{3}{2}kT\ln\lambda + kT\ln p - kT\ln kT. \tag{20} \]

The equation $\mu_{\text{liq}} = \mu_{\text{g}}$ then gives:

\[ \ln p = \ln kT - \ln v_f + \frac{z\varphi_0}{2kT}. \tag{21} \]

The results of the calculations are shown in Fig. 4. The theoretical curve \((z=12)\) is too high (by approximately 0.40 units along the \(\ln p\) axis). This discrepancy increases still further (to 1.6) if, in accordance with the preceding discussion, the smaller value of the coordination number \(z'=10\) is adopted for the calculations, which is essentially necessary in order to explain the observed volume of the liquid. This contradiction may be somewhat smoothed out if a larger statistical weight is assigned to the liquid, i.e., by increasing the entropy of the liquid and correspondingly decreasing its chemical potential.

The additional factor in the state integral required for this is approximately \(5^N\), which gives an additional entropy \(S_{\mathrm{neup}}\sim R\ln 5\sim 1.6R\). This shows that the adopted model insufficiently accounts for the disorder of the liquid, and chiefly for this reason, as already mentioned, Lennard-Jones and Devonshire were compelled to introduce into the state integral the factor \(e^N=(2.718)^N\). The result of including the factor \(e^N\) is shown by the dotted curve in Fig. 4.

However, our consideration shows that for a satisfactory description of the liquid state it would be necessary to modify the state integral by a factor greater than \(e\;(=2.718)\).

This additional statistical weight which must be introduced into the theory can undoubtedly be attributed to the disorder of the liquid state, consisting in displacements of the centers of the cells, near which the molecules oscillate, from the nodes of the crystalline lattice. The notion of a crystalline lattice was introduced into the theory of the liquid state intuitively, as an arbitrary assumption in view of the great similarity between the liquid and solid states. The fact that this is only a rough approximation evidently does not noticeably affect the equation of state. But it is not surprising that it has a strong effect on the magnitude of the vapor pressure, since the latter is very sensitive to the degree of disorder contained in the model, owing to the fact that in the corresponding calculations the entropy plays the principal role. The fact that from time to time more than one molecule can occupy one of the cells might seem capable of explaining the increase of entropy. However, recent calculations\({}^{12}\) show that these states make a negligibly small contribution to the state integral of the liquid.

Trauton’s rule consists in the assertion that the molar heat of vaporization \(Q_{\mathrm{исп}}\) at the boiling temperature \(T_{\mathrm{кип}}\), divided by this temperature, i.e., the entropy of vaporization \((\Delta S)_{\mathrm{исп}}\), is approximately the same for all liquids.

The entropy of vaporization can easily be represented in the form

\[ \frac{(\Delta S)_{\mathrm{исп}}}{R}\simeq \frac{d\ln p}{d\ln T} = -\left(\frac{\partial \ln v_f}{\partial \ln T}\right)_v -\frac{z\varphi_0}{2kT}, \tag{22} \]

or in reduced form—

\[ (\Delta S^*)_{\mathrm{vap}} \simeq \frac{d \ln p^*}{d \ln T^*} = -\left(\frac{\partial \ln v_f^*}{\partial \ln T^*}\right)_{v^*} -\frac{z\varphi^*}{2T^*}. \tag{23} \]

At the triple point (or approximately at the melting point) \(T^*_{\mathrm{tr}}=0.7\), which at this point gives \((\Delta S^*)_{\mathrm{tr}}=9.35\). But the boiling point is not a characteristic temperature of a liquid and, correspondingly, its reduced value is not a constant, as it is for the critical or triple points. However, because of the small difference in the molecular unit of pressure for heavy substances, the reduced value of the pressure at the boiling point (at \(1\) atm) also changes little, and as a consequence the difference between \((\Delta S^*)_{\mathrm{tr}}\) and \((\Delta S^*)_{\mathrm{vap}}\) is very small. The smallness of the aforementioned variation of the molecular unit of pressure \(\varepsilon/\sigma^3\) follows from the fact that, in the case of large molecules, as their diameters increase the energy of attraction of the molecules \(\varepsilon\) also increases, being approximately proportional to the volume (polarizability) of the molecule. Table V shows that \((\Delta S^*)_{\mathrm{vap}}(=(\Delta S^*)_{\mathrm{vap}}/R)\)

Table V

Validity of Trauton’s rule

Substance	\(\varepsilon/\sigma^3\) (atm)	\(p^*\)	\(T^*\)	\((\Delta S^*)_{\mathrm{vap}}\)
A	413	0.00242	0.726	9.20
Kr	465	0.00215	0.722	9.22
Xe	513	0.00195	0.716	9.26
Triple point	—	0.00160	0.700	9.35

indeed changes little, being somewhat larger for heavy substances such as Xe. It is therefore necessary to conclude that, although deviations from Trauton’s rule must exist, they will necessarily be small.

§ 6. GENERALIZATION OF THE THEORY OF FREE VOLUME

Following an idea first put forward by Bernal and Eyring \(^{13}\), the Lennard-Jones theory has in recent years been generalized so as to include the possibility of the appearance of empty lattice sites in the liquid state \(^{14,15,1}\). In paper \(^{1}\), in particular, there is a complete and systematic discussion of this generalization of the Lennard-Jones theory.

A volume \(V\) of liquid, consisting of \(N\) molecules, is divided into \(L\) cells (with \(L>N\)), forming a virtual face-centered cubic structure. The sizes of the cells are assumed to be so large that the interaction of molecules in nonadjacent cells may be neglected. The fraction \(x=N/L\) of occupied cells is a function of temperature and density.

The volume per molecule, \(v=V/N\), is now larger than the cell volume \(\omega=V/L=xv\). Since some of the cells surrounding a given \(i\)-th molecule may be unoccupied, the coordination number \(z_i\) of this molecule will, generally speaking, be smaller than the coordination number of a face-centered cubic structure with \(z=12\): \(z_i=\xi_i z\) (\(\xi_i\leq 1\)). We define \(\xi=\sum_i \xi_i/N\). Since the distance \(a\) between the centers of neighboring cells is completely determined by the volume per lattice cell \(\omega\) by means of the relation \(a^3=\omega\sqrt{2}\), the interaction between nearest-neighbor molecules in their equilibrium positions depends on \(\omega\), and the potential energy of a molecule in a cell at a distance \(r\) from its center \(\psi(r)\) depends on \(\omega\) and \(r/a\). Consequently, the energy of the system is now equal to

\[ E=\frac{\xi z}{2}N\varphi_0+\sum_{i=1}^{N}\xi_i z\psi(r_i), \tag{24} \]

where \(\varphi_0\) and \(\psi(r)\) are related to \(\omega\) (the cell volume) by relations analogous to (4), (5), (6):

\[ \varphi_0^*=\varphi_0^*(a)=-\frac{1}{\omega^{*4}}-\frac{2}{\omega^{*2}}, \tag{25} \]

\[ \psi^*(r^*)=\frac{l(r^*/a^*)}{\omega^{*4}}-\frac{2m(r^*/a^*)}{\omega^{*2}}. \tag{26} \]

The free volume, i.e. the potential part of the configuration integral per molecule, now depends on the number of neighboring molecules:

\[ v_f(v,T;\,x,\xi)=\int_{0}^{r_{\max}} e^{-\beta \xi z\psi(r)}\,d\mathbf r, \tag{27} \]

where \(r_{\max}\) is determined from the relation \((4\pi/3)r_{\max}^3=\omega=xv\).

From comparison of equation (27) for \(v_f\) with (9), it is obvious that

\[ v_f(v,T;\,x,\xi)=v_f(\omega,T'), \tag{28} \]

where \(v_f(\omega,T')\) is the value of the free volume determined by (9), but taken for the “volume” \(\omega=xv\) and “temperature” \(T'=T/\xi\).

For the subsequent application of the methods of statistical mechanics it is necessary to have an analytic expression for the dependence of \(v_f\) on

$\xi$ at constant $\omega\;(=xv)$ and $T$. We shall use for this the linear expression

\[ \ln v_f(\omega,T;\xi)=\ln v_f^0+\gamma(1-\xi) \tag{29} \]

or

\[ v_f(\omega,T;\xi)=v_f^0 e^{\gamma(1-\xi)} . \tag{30} \]

The various approximations introduced hitherto by different authors for the dependence of $v_f$ on $\xi$ reduce to a choice of values for $v_f^0$ and $\gamma$. The following have been proposed:

Cernuschi and Eyring$^{13}$

\[ v_f^0=v_f(\xi=0)=v_f(\omega,T), \qquad \gamma=0; \]

Ono$^{14}$

\[ v_f^0=v_f(\omega,T), \qquad \gamma=\ln [v_f(\xi=0)/v_f(\xi=1)]; \]

Peek and Hill$^{15}$

\[ v_f^0=v_f(\xi=\bar{\xi})=v_f(\omega,T'), \qquad \gamma=0; \]

Rowlinson and Curtiss$^{1}$

\[ v_f^0=v_f(\omega,T')\exp(1-\bar{\xi})\gamma, \qquad \gamma=\left(\frac{\partial\ln v_f}{\partial \xi}\right)_{\xi=\bar{\xi}} . \]

The first two cases are too simple, since in them neither $v_f^0$ nor $\gamma$ depends on $\xi$. Since $\ln v_f(\xi)$, represented as a function of $\xi$, turns out to be approximately constant between $\xi=1$ and $\xi=0.5$, it is convenient to replace the assumption of Rowlinson and Curtiss by the following:

\[ v_f^0=v_f(\xi=1)=v_f(\omega,T); \qquad \gamma=\left(\frac{\partial\ln v_f}{\partial \xi}\right)_{\xi=1}, \tag{31} \]

the latter has the advantage that neither $v_f^0$ nor $\gamma$ again depends on $\xi$. For a liquid volume $\gamma\simeq 1$.

Consequently, the state integral corresponding to a definite value $x=\omega/v$ is equal to

\[ Z(x)=\lambda^{-3N}e^{-\frac{1}{2}\beta zN\varphi_0}v_f^{0N} \sum_{\xi} g(x,\xi)e^{\frac{1}{2}\beta zN\gamma(1-\xi)}, \tag{32} \]

where $g(x,\xi)$ is the number of distributions corresponding to the assigned values of $x$ and $\xi$, and

\[ \xi=\varphi_0+\frac{2\gamma}{z}kT. \tag{33} \]

The sum appearing in (32) can be transformed by the Bethe method, or, as was done in papers \(^{1,15}\), with the aid of Guggenheim’s quasi-chemical method, which ultimately leads to the Bethe method. First of all, the mean value \(\bar{\xi}\) is determined from the equation

\[ \frac{2\bar N_{aa}\cdot 2\bar N_{hh}}{(N_{ah})^2} = \frac{\bar{\xi}(1-2x+x\bar{\xi})}{x(1-\bar{\xi})^2} = e^{-\beta\zeta}, \tag{34} \]

where \(N_{aa}\) and \(N_{hh}\) are, respectively, the numbers of pairs of neighboring cells that are simultaneously occupied or simultaneously empty, and \(N_{ah}\) is the number of pairs of neighboring cells of which only one is occupied. This equation is the condition of chemical equilibrium between empty and occupied sites; \(\zeta\) is the energy required to create a hole—molecule pair. The solution of this equation is

\[ 1-\bar{\xi} = \frac{\left[1+4x(1-x)(e^{-\beta\zeta}-1)\right]^{1/2}-1} {2x(e^{-\beta\zeta}-1)}, \tag{35} \]

which determines \(\bar{\xi}\) as a function of \(x\) and \(\zeta/kT\). Guggenheim showed that the state integral (32) can be approximated by the expression

\[ Z(x)=\lambda^{-3N}e^{-\frac12\beta zN\varphi_0}v_f^{0N}g(x)e^{\frac12\beta zN\zeta(1-\bar{\xi})}, \tag{36} \]

or

\[ Z(x)=\lambda^{-3N}e^{-\frac12\beta z\bar{\xi}N\varphi_0}g(x)v_f(\omega,T')^N, \tag{37} \]

where

\[ T'=T/\bar{\xi},\qquad g(x)=\sum_{\xi} g(x;\xi)=L!/N!(L-N)! \]

and \(\bar{\xi}\) is determined by the equation

\[ 1-\bar{\xi} = \frac{kT}{\zeta x} \left\{ x\ln\frac{x}{\alpha-1+2x} + (1-x)\ln\frac{1-x}{\alpha+1-2x} + \ln(1+\alpha) \right\}, \tag{38} \]

where \(\alpha\) is an abbreviated notation for the square root,

\[ \alpha=\{1+4x(1-x)(e^{-\beta\zeta}-1)\}^{1/2}. \tag{39} \]

Equation (38) determines the effective value \(\bar{\xi}\) as a function of \(\zeta/kT\). Assumption (30) makes \(\bar{\xi}\) independent of \(\xi\), which makes it possible to find the solution of (38) easily.

Limiting Case of Low Densities

First of all it is necessary to note that, in the limiting case of low densities \((cN \ll L,\ x \to 0,\ \alpha \to 1,\ \upsilon_f \to \omega)\), we have:

\[ Z \to \lambda^{-3N}\omega^N(\langle n\rangle) = \lambda^{-3N}\frac{L^N\bar{\omega}^N}{N!} = \lambda^{-3N}\frac{V^N}{N!}, \tag{40} \]

which coincides exactly with the partition integral of an ideal gas. Thus, the generalized theory of free volume is in principle capable of giving a partition integral equally suitable for both the liquid and the gaseous states.

The second improvement in the case of low densities consists in the fact that, in this theory, for the equation of state and the energy one obtains terms tending to zero as the first powers of the density, and not as the squares of the density, as was the case in the original Lennard-Jones theory. The second virial coefficient calculated from the present model is equal to

\[ B=\frac{N\omega}{2}\{1-2[\exp(-\varphi_0/kT)-1]\}. \tag{41} \]

For values \(\omega^* = 1\) to \(2\), acceptable values are obtained, in agreement with the rigorous values of the second virial coefficient \(^{1}\).

Equation of State

The partition integral \(Z\) is a function of \(T\), \(x\), and \(\omega\), with \(\upsilon=\omega/x\). The equation of state can be obtained by differentiating the free energy with respect to \(\upsilon\).

In the first approximation we shall regard \(\omega\) as a constant independent of \(\upsilon^{13}\). A change in \(\upsilon\) then means a change in \(1/x\). Consequently,

\[ pV=-\upsilon\left(\frac{\partial F}{\partial \upsilon}\right)_T =x\left(\frac{\partial F}{\partial x}\right)_{T,\omega}, \tag{42} \]

which gives us

\[ \frac{pV}{RT} =\frac{z-2}{2x}\ln(1-x) -\frac{z}{2x}\ln\frac{\alpha+1-2x}{\alpha+1}, \tag{43} \]

where it must be borne in mind that \(x=\omega/\upsilon\) is a measure of density. The isotherms obtained from this are S-shaped for values of \(\alpha\) smaller than \(\alpha_{\mathrm{cr}}=z/z-2\), which for \(z=12\) and \(\gamma\approx 1\) gives

\[ -\zeta/kT_{\mathrm{cr}} =2\ln\frac{z}{z-2}=0.364; \qquad kT_{\mathrm{cr}}=-0.530/\varphi_0. \tag{44} \]

These equations determine the critical temperature.

For temperatures below the critical temperature, the condensation point can be obtained from the two simultaneous equations \(p_{\ell}=p_{\mathrm{g}}\) and \(\mu_{\ell}=\mu_{\mathrm{g}}\),

which leads to the equations

\[ x_{\mathrm{ж}}=1-x_{\mathrm{г}}, \tag{45} \]

\[ \ln \frac{1-x_{\mathrm{ж}}}{x_{\mathrm{ж}}} = \frac{z-2}{z} \ln \frac{\alpha+1-2x_{\mathrm{ж}}}{\alpha-1+2x_{\mathrm{ж}}}. \tag{46} \]

Hence the “densities” \(1/V^*\) of the saturated vapor and of the liquid at the point of coexistence, plotted as functions of \(T\), are two curves symmetric with respect to the line \(x=1/2\) (Fig. 6).

Fig. 6. Experimental and theoretical curves of the densities of coexisting liquid and vapor. The theoretical curves are drawn according to the Lennard-Jones theory for \(z=12\) and \(z'=10\), and according to the generalized free-volume theory for \(\gamma=1,\ \omega^*=1\); \(\gamma=1,\ \omega^*=1.6\), and for \(v_f=\mathrm{const},\ \gamma=0\).

The critical density is \(x=0.5\). The numerical values of the densities depend on the choice of the value for \(\omega\), as is seen from Table VI.

Table VI

Critical data for \(\omega=\mathrm{const}\)

	\(T^*_{\mathrm{cr}}\)	\(V^*_{\mathrm{cr}}\)	\(p^*_{\mathrm{cr}}\)	\(p^*_{\mathrm{cr}}V^*_{\mathrm{cr}}/T^*_{\mathrm{cr}}\)
\(\omega^*=1\)	1.89	2.00	0.322	0.342
\(=1.6\)	1.18	3.20	0.126	0.342
\(=2\)	0.82	4.00	0.070	0.342
Experiment	1.26	3.16	0.117	0.294

Table VII

Critical data for \(\gamma = 0\) (according to \(^{13}\))

	\(T^*_{\mathrm{cr}}\)	\(V^*_{\mathrm{cr}}\)	\(p^*_{\mathrm{cr}}\)	\(p^*_{\mathrm{cr}} V^*_{\mathrm{cr}} / T^*_{\mathrm{cr}}\)
\(\omega^* = 1\) . . . . . .	2,75	2,00	0,470	0,342
\(= 1,6\) . . . . . .	1,72	3,20	0,184	0,342
\(= 2\) . . . . . .	1,20	4,00	0,103	0,342
Experiment . . . .	1,26	3,16	0,117	0,294

Consequently, satisfactory agreement with experiment is obtained when \(\omega^*\) is chosen to be approximately \(1.6\). In Fig. 7 the critical isotherms calculated for \(\omega^* = 1\) and \(\omega^* = 1.6\) are shown. The experimental critical isotherm is also given there, and from their comparison it is evident that there is rather satisfactory agreement of the theory (for \(\omega^* = 1.6\)) with experiment.

Fig. 7. Experimental and theoretical critical isotherms according to equation (43) with \(\omega^* = 1\) and \(\omega^* = 1.6\), according to the original Lennard-Jones theory and according to work \(^{15}\).

The calculations presented here differ from the original calculations in work \(^{13}\) in that we took into account the dependence of the free volume \(v_f\) on \(\xi\). The introduction into the calculations of the change of \(v_f\) with \(\xi\), in accordance with (31), is an improvement that makes it possible to obtain agreement with the experimental data, whereas according to the theory of Cernuschi and Eyring, as is seen from Table VII, the choice of \(\omega^*\) could not have been made.

In the second approximation, which is more satisfactory from the theoretical point of view, the determination of the required values of \(\omega\) (or \(x\)) for given \(v\) and \(T\) is not carried out by trial, as above, but from the condition of a minimum of the free energy. This leads to the equation

\[ x\left(\frac{\partial F}{\partial x}\right)_{v,T} = x\left(\frac{\partial F}{\partial x}\right)_{v,T,\omega} + \omega\left(\frac{\partial F}{\partial \omega}\right)_{v,T,x} =0 \tag{47} \]

and then the equation of state will be

\[ pV=-v\left(\frac{\partial F}{\partial v}\right)_T = -\omega\left(\frac{\partial F}{\partial \omega}\right)_{T,x} = x\left(\frac{\partial F}{\partial x}\right)_{T,\omega}, \tag{48} \]

which gives us

\[ \frac{pV}{RT} = \frac{z-2}{2x}\ln(1-x) - \frac{z}{2x}\ln\frac{\alpha+1-2x}{\alpha+1} = \]

\[ = \omega\left(\frac{\partial \ln v_f'}{\partial \omega}\right)_{T'} - \frac{z}{2kT'}\,\omega\frac{\partial \varphi_0}{\partial \omega}. \tag{49} \]

Here \(v_f'\) is the free volume at \(\omega\) and \(T'\), with

\[ \omega=vx,\qquad T'=T/\xi. \tag{50} \]

In the first form of writing, the equation of state coincides with the equation of state obtained for \(\omega=\mathrm{const}\). In the second form it coincides with the Lennard-Jones equation of state for the smaller coordination number \(z'=\overline{\xi}z\) and the volume \(\omega=vx\). Here \(\overline{\xi}\) is determined as a function of \(x\) by equations (38), (39), and the value of \(x\) is determined from the condition of a minimum of the free energy by equation (47).

This shows that the concept of a smaller coordination number follows from the theory of the generalized free volume of a liquid.

The determination of \(x\) as a function of \(V\) and \(T\) turns out to be rather difficult. But Table VIII shows that for the liquid volume values \(\xi=0.8\text{--}0.9\) are obtained, which corresponds to the concept of a smaller coordination number introduced ad hoc in § 4.

Table VIII

Equilibrium values of \(x\) and \(\xi\).

\(V^*\)	\(\xi\) (\(T^*=0.7\))	\(x\) (\(T^*=0.7\))	\(\omega^*\) (\(T^*=0.7\))	\(\xi\) (\(T^*=1.0\))	\(x\) (\(T^*=1.0\))	\(\omega^*\) (\(T^*=1.0\))
1.0	0.998	0.998	0.998	0.999	0.999	0.999
1.2	0.94	0.93	1.12	0.97	0.97	1.17
1.4	0.90	0.88	1.23	0.96	0.95	1.34
1.6	0.86	0.83	1.33	0.94	0.94	1.50
1.8	0.82	0.78	1.41	0.92	0.92	1.65

In the end it may be stated that, in the case of a liquid at not very high temperature, the generalized theory of free volume gives a satisfactory extension of the original theory of Lennard-Jones and Devonshire, explaining the decrease in the coordination number as a feature necessarily inherent in the liquid state.

However, along this path it is impossible to explain the additional entropy appearing in a liquid as compared with a solid; it could be explained only by the presence of a much greater degree of disorder than is contained in the “lattice” theory of the liquid state.

§ 7. THEORY OF MELTING *)

In the theory of the liquid state based on the lattice model, it is of interest to discuss in greater detail the process of transition of a solid into a liquid, i.e., melting.

The first attempt to interpret the process of melting in terms of intermolecular forces was made by Herzfeld and Goeppert-Mayer^16 and by Born^17. Both of these theories are based on the investigation of the properties only of the solid. In the work of Herzfeld and Goeppert-Mayer, the region of stability of a solid with respect to uniform compression or tension was investigated. The stability limit, where \((\partial p/\partial V)_T=0\), is reached for a crystal with a cubic lattice at values of the elastic constants satisfying the relation \(c_{11}+2c_{12}=0\). On the other hand, Born investigated stability with respect to shear stresses and found that the stability limit corresponds to the condition \(c_{44}=0\) **).

The values of the coefficients \(c_{11}, c_{12}, c_{44}\) can be expressed in terms of intermolecular forces, and both stability principles lead to a melting temperature very close to the experimental value \(T^*=0.7\).

However, these investigations of the stability limits of the crystalline lattice can give only an upper limit for the melting temperature and volume. An exact theory of the melting process must be based on calculation of the state integral for both the solid and the liquid states. Therefore, in the present article these two theories will not be considered.

More promising from the point of view that interests us is the approximation proposed by Lennard-Jones and Devon-

) See also Ya. I. Frenkel, Kinetic Theory of Liquids, published by the USSR Academy of Sciences, M.–L., 1946. (Translator’s note.*)

) One of the first works in this direction was the paper by Ya. I. Frenkel, Acta Physicochim., 3, 633 (1935). See also the works of V. A. Zhdanov and V. F. Konusov, ZhETF 17, 977 (1947); 20, 3 (1950). (Translator’s note.)

by Hirshfelder, Eyring, et al.18, 19, who attempted to interpret melting as a disordering process. The simplest type of disorder that can be introduced in a solid crystalline state is the transition of molecules from their equilibrium positions into interstitial positions. Taking, for the ordered crystal, the model of a cubic face-centered lattice, we obtain, for the totality of all interstitial positions, a second lattice, also cubic face-centered. These two lattices are related to one another as the Na- and Cl-lattices in an NaCl crystal. At low temperatures only one of these two lattices is completely filled, but at higher temperatures the molecules are distributed over both lattices.

The degree of long-range order \(s\) is introduced in the following way. Let \(N_0\) denote the number of molecules at the sites of the crystalline lattice, and \(N_1\) the number of molecules in interstitial sites. Then

\[ N_0 = \frac{1}{2} N(1+s), \qquad N_1 = \frac{1}{2} N(1-s). \tag{51} \]

For \(s = 1\) only the sites of the crystalline lattice are occupied; for \(s = 0\) both lattices are filled with equal probabilities.

Each site of the crystalline lattice is surrounded by \(z_0(=12)\) sites at a distance \(a_0 = v^{1/3}2^{1/6}\) and by \(z_1(=6)\) interstitial sites at a distance \(a_1 = a_0/\sqrt{2}\). Conversely, each interstitial site is surrounded by \(z_1\) crystalline sites at a distance \(a_1\) and by \(z_0\) interstitial sites at a distance \(a_0\).

The total energy of the system is now equal to

\[ E = \frac{1}{2} z_0 N\varphi_0 + \frac{1}{4}N(1-s^2)\chi + \]

\[ + \sum_i \{ z_{0i}\psi_0(r_i) + z_{1i}\psi_1(r_i)\}, \tag{52} \]

where \(\chi = z_1\varphi_1 - z_0\varphi_0\), and \(\psi_0(r_i)\) and \(\psi_1(r_i)\) are the mean potentials in the neighborhood of a site of the crystalline lattice, due to the action of a single molecule at a distance \(a_0\) or \(a_1\) from the site, respectively.

Lennard-Jones and Devonshire did not take into account the difference between the free volumes of a molecule near a lattice site and in an interstitial site. But the theory can easily be generalized so that in \(v_f\) the contribution of the surrounding molecules both near the site and in the interstitial site is taken into account. Since it is assumed that, for a given \(s\), the distribution of neighboring molecules over the possible positions is completely random, for the free volume of a molecule near a lattice site we write

\[ v_{f_0}(s) = \int e^{-\frac{1}{2}\beta z_0(1+s)\psi_0(r) - \frac{1}{2}\beta z_1(1-s)\psi_1(r)}\,dr \simeq \]

\[ \simeq v_{f_0}^{\frac{1}{2}(1+s)}\, v_{f_1}^{\frac{1}{2}(1-s)}. \tag{53} \]

and for a molecule in an internode

\[ v_{f_1}(s)=\int e^{-\frac{1}{2}\beta z_0(1-s)\psi_0(r)-\frac{1}{2}\beta z_1(1+s)\psi_1(r)}\,dr \simeq \]

\[ \simeq v_{f_0}^{\frac{1}{2}(1-s)}\,v_{f_1}^{\frac{1}{2}(1+s)}, \tag{54} \]

where

\[ v_{f_0}=\int e^{-\beta z_0\psi_0(r)}\,dr,\quad v_{f_1}=\int e^{-\beta z_1\psi_1(r)}\,dr. \tag{55} \]

The configuration integral is now equal to

\[ Z=\lambda^{-3N}e^{-\frac{1}{2}\beta N z_0\varphi_0}\,v_{f_0}^{N}g(s)\, e^{-\frac{1}{4}\beta N\zeta(1-s^2)}, \tag{56} \]

where

\[ \zeta=z_1\varphi_1-z_0\varphi_0+2kT\ln(v_{f_0}/v_{f_1}) \tag{57} \]

and

\[ g(s)=\left(\frac{N!}{N_0!N_1!}\right)^2. \tag{58} \]

The equation of state can now be obtained from \(Z\), using \(F=-kT\ln Z\) and \(p=-(\partial F/\partial V)_T\), which gives:

\[ p=-\frac{1}{2}z_0N\frac{\partial\varphi_0}{\partial v} +kT\frac{\partial\ln v_f}{\partial v} +(1-s^2)\frac{N}{4}\frac{\partial\zeta}{\partial v}. \tag{59} \]

The first and second terms in this expression represent the potential and thermal contributions to the total pressure, respectively, while the third term gives the pressure due to disorder. This pressure is greatest at \(s=0\), i.e., in the state of complete disorder, and vanishes in the completely ordered state at \(T=0\).

The value of the disorder parameter \(s\) is determined from the condition of a minimum of the free energy \(F=-kT\ln Z\), which gives

\[ (\partial F/\partial s)=0,\quad \ln\frac{1+s}{1-s}=\frac{1}{2}\beta\zeta s. \tag{60} \]

For values of \(\zeta/2kT\) greater than 2, the value of \(s\) is different from zero, being equal to \(s=1\) at \(T=0\), since \(\zeta\) is positive. When the temperature or volume increases, \(\zeta/2kT\) decreases (\(\zeta\) is a function of \(T\)) and, consequently, \(s\) also decreases and reaches the value \(s=0\) at \(\zeta/2kT=2\). At still smaller values of \(\zeta/2kT\) we have a completely disordered system.

Before discussing the crystal–liquid transition in more detail, it is important to consider the configuration integral above the transition point in the state of complete disorder, which we identify-

with the liquid state. Here the configurational integral is equal to

\[ Z=\lambda^{-3N}4^N \exp\left\{-\beta\frac{N}{4}(z_0\varphi_0+z_1\varphi_1)\right\} v_{f_0}^{\frac12 N} v_{f_1}^{\frac12 N}. \tag{61} \]

The molecules are distributed over both lattices with equal probabilities. But now \((v_{f_0}v_{f_1})^{1/2}\) is the free volume \(v_{f_0}(s=0)\) or \(v_{f_0}(s=0)\) of a molecule near a site of a simple cubic lattice with only half-occupied sites. Consequently, the configurational integral of the liquid in this model is identical with the configurational integral of a liquid possessing a virtual simple cubic lattice with the number of sites (occupied only halfway) equal to twice the number of sites of the original lattice. This is the decisive step, consisting in the assumption that upon disordering the type of lattice changes with a change in the volume per cell by a factor \(2\sqrt{2}\), although the distance between nearest-neighbor molecules, roughly speaking, does not change (the factor \(\sqrt{2}\) is due to the loose packing of the simple cubic lattice, the factor 2 to the fact that only half the sites are occupied).

To apply this model to the real state of affairs, Lennard-Jones and Devonshire assumed that local disturbances of order substantially change the picture of the division into cells. One may suppose, for example, that the local parameters \(a_0\) and \(a_1\) of the volume of cells near the sites and in the interstices are related in such a way that the local pressures are identical in cells of both types. For zero pressure one must put \(a_0 \simeq a_1 \simeq a\). For large pressures \(a_1\) will be somewhat smaller than \(a_0\). However, in this case \(\varphi_0=\varphi_1\) and \(v_{f_0}=v_f(a,z=12)\), \(v_{f_1}=v_f(a,z=6)\).

Using the fact that

\[ (v_{f_0}\cdot v_{f_1})^{1/2}\simeq v_{f_0}(s=0)\simeq v_{f_1}(s=0), \tag{62} \]

for the configurational integral at complete disorder we obtain:

\[ Z=\lambda^{-3N}4^N e^{-\frac12\beta N\bar{z}\varphi(a)}\,v_f(a,\bar{z})^N, \tag{63} \]

where \(\bar{z}=\frac12(z_0+z_1)=9\), and where the volume per molecule \(v\) is related to \(a\) by the condition that it is the geometric mean of the volume per molecule in the cubic face-centered structure \(a^3/\gamma_{12}\) \((\gamma_{12}=\sqrt{2})\) and the volume per molecule in the simple cubic structure \(a^3/\gamma_6\) \((\gamma_6=1)\). This gives \(\gamma=(\gamma_{12}\gamma_6)^{1/2}\simeq1.19\).

The equation of state then takes the form

\[ p=-\frac12\bar{z}N\frac{\partial\varphi(a)}{\partial v}+kT\frac{\partial\ln v_f(a,\bar{z})}{\partial v}, \tag{64} \]

and it again has the simple form of the Lennard-Jones equation of state for a liquid.

But this is a configurational integral and an equation of state for a liquid with coordination number \(z = 9\) and with volume \(v\), related to \(a\) by the relation \(v=\gamma a^3\). In § 2 it was shown that reducing the coordination number from \(z=12\) to \(z=10\) or 9 improves the numerical values of the liquid volume at zero pressure. It should also be noted that the configurational integral (63) contains the factor \(4^N\), which is larger than the originally introduced factor \(e^N\) and which is closer to the required value \(5^N\), as we saw in § 5.

The consideration of a completely disordered two-lattice structure as a “liquid state” shows that this model should not be taken too literally. In a real liquid, most likely, local disturbances of order entirely destroy the lattice laid at the basis of our consideration. Therefore there is no point in dwelling in greater detail on the consequences that follow from the dependence of \(\zeta\) on temperature and volume required according to (57).

It is more appropriate to follow Lennard-Jones and Devonshire, who at this stage of the investigation introduce the approximation \(\zeta^* = 5.54\cdot v^{*-4}\), leading to the correct value of the triple-point temperature \(T_{\mathrm{tr}}^* = 0.70\). Figure 8 shows \(p\)–\(V\) isotherms obtained using such an approximation for \(\zeta^*\). By the method of equating areas, as is well known, one can calculate the transition pressure for each temperature. It is interesting that in this case, as was first mentioned in \({}^{18}\), for temperatures above \(T^* \approx 1.1\) there is no discontinuity of volume at the transition point. The transition changes from a first-order phase transition into a second-order phase transition. This means that there must exist a critical point on the melting curve, at which melting as a first-order transition changes into a second-order transition. It is very difficult to su—

Fig. 8. Isotherms of the solid and liquid states and the transition corresponding to melting, according to the theory of Lennard-Jones and Devonshire.

decide whether this effect is real or whether it appears as a result of the rough approximations introduced for calculating the melting curve. But it is important to emphasize that such a critical point does not mean that at a certain temperature the transition disappears, as is the case with the liquid—gas transition, but means only a point of change in the thermodynamic type of transition.

In Fig. 3 the pressure at the melting point is shown as a function of temperature. The figure shows that here there is very good agreement between theory and experiment.

Finally, Table IX gives a comparison with experiment for the remaining data for the melting point (i.e., the jumps in volume and entropy). This comparison shows that, although in determining \(\zeta(v)\) the value of the triple-point temperature was used, the satisfactory agreement of the melting curve and the values \(\Delta V\) and \(\Delta S\) with experiment makes it very probable that this theory gives a correct interpretation of the melting process.

Table IX

Data for the triple point

	Theory	Experiment
\(V^*_{\mathrm{тв.\ тр}}\)	1.08	1.01
\(V^*_{\mathrm{ж.\ тр}}\)	1.21	1.16
\(\Delta V^*_{\mathrm{тр}}\)	0.13	0.15
\(\Delta S^*_{\mathrm{тр}}\)	1.7	1.7

It is interesting to note that the theory presented is capable of giving an explanation of Simon’s formula for the melting curve*):

\[ p=a+bT^c . \tag{65} \]

This can be done if, as was proposed by Domb\(^{20}\), the melting point is identified with the point of discontinuity, where \(s\) becomes equal to zero, and all equations are written in dimensionless form. Accordingly, equation of state (59) in reduced form is

\[ p^*=p^*_{\mathrm{пот}}+p^*_{\mathrm{терм}}+p^*_{\mathrm{неуп}} . \tag{66} \]

The first term \(p^*_{\mathrm{пот}}=-\dfrac{1}{2}z_0(\partial \varphi^*/\partial v^*)\) is approximately constant for those volumes at which melting occurs. The second term \(p^*_{\mathrm{терм}}\), equal to

\[ p^*_{\mathrm{терм}}= \left(\frac{\partial \ln \nu_f}{\partial \ln v^*}\right)_{T^*} \frac{T^*}{v^*}, \]

is of the order \(aT^*/v^*\), where \(a\), defined as \((\partial \ln \nu_f^*/\partial \ln v^*)_{T^*}\), is approximately constant \((a \simeq 8)\).

) See, for example, P. W. Bridgman, Recent work in the field of high pressures, translated by L. F. Vereshchagin, IL Publishing House, Moscow, 1948. (Translator’s note.*)

The disorder pressure \(p^*_{\text{neord}}\) at the transition point \(s=0\) is equal to \(\zeta^* / v^* = \beta v^{*-5}\) (where \(\beta=5.56\)). The volume of the liquid \(v_{\text{trans}}\) at the transition point is specified by the condition \(\chi^*/T^*=4\), which leads to \(\beta=4v_{\text{trans}}^{*4}T^*\). Consequently, combining the second and third terms, at the discontinuity point one may write:

\[ p^*_{\text{trans}}=p^*_{\text{pot}}+a\,\frac{T^*}{v^*}+\frac{\beta}{v^{*5}} =p^*_{\text{pot}}+\frac{(a+4)\sqrt{2}}{\beta^{1/4}}\,T^{*5/4}. \tag{67} \]

Substitution of the constants gives the formula \(p^*_{\text{trans}}\simeq -7+11T^{*5/4}\), representing the melting curve. Passing from reduced quantities to ordinary ones, for the constants of Simon’s formula we find:

\[ a=-p_{\text{pot}};\qquad b=\frac{(a+4)\sqrt{2}\,k^{5/4}}{\sigma^3(\beta\varepsilon)^{1/4}};\qquad c=1.25. \tag{68} \]

The interpretation of the quantity \(a\) as the pressure corresponding to the potential energy (often incorrectly called the “internal pressure”) was already proposed by Simon.

§ 8. RADIAL DISTRIBUTION FUNCTION

A completely different approach to the equation of state of a liquid is based on the study of the so-called molecular distribution functions. These molecular functions are defined according to the equations (see \(^{21}\))*:

\[ n_h(\mathbf r_1,\ldots,\mathbf r_h)= \frac{N!}{(N-h)!}\, \frac{\displaystyle \int e^{-\beta\Phi}\,d\mathbf r_{h+1}\cdots d\mathbf r_N} {\displaystyle \int e^{-\beta\Phi}\,d\mathbf r_1 d\mathbf r_2\cdots d\mathbf r_N}, \tag{69} \]

Consequently, \(n_h(\mathbf r_1,\ldots,\mathbf r_h)\) is the probability density of finding an arbitrarily chosen set of \(h\) molecules in the configuration \(\mathbf r_1,\mathbf r_2,\ldots,\mathbf r_h\).

The importance of these molecular distribution functions, in particular \(n_2(\mathbf r,\mathbf r_2)=n_2(r_{12})\), which for a gas or for a liquid is a function only of the distance between molecules, consists in the fact that the equation of state and the internal energy are completely determined by the binary distribution function \(n_2(r_{12})\), respectively by the equations \(^{21}\)

\[ U=\frac{3}{2}RT+\frac{1}{2}\iint n_2(r_{12})\varphi(r_{12})\,d\mathbf r_1\,d\mathbf r_2, \tag{70} \]

\[ pV=RT-\frac{1}{6}\iint n_2(r_{12})\,r_{12}\,\frac{d\varphi}{dr_{12}}\,d\mathbf r_1\,d\mathbf r_2, \tag{71} \]

* For more detail see N. N. Bogolyubov, Problems of Dynamical Theory in Statistical Physics, Gostekhizdat, Moscow–Leningrad, 1946.

Thus knowledge of \(n_2(r_{12})\) leads directly to knowledge of the thermal and caloric equations of state.

The function \(n_2(r_{12})\) can be determined by experimental studies of the scattering of X-rays by liquids. The “radial distribution function” \(g(r)\) is equal to \(g(r)=n_2(r)/n^2\). However, the experimental determination of \(g(r)\) does not provide the accuracy required for establishing the equation of state.

The molecular distribution functions \(n_h(\mathbf r_1,\ldots,\mathbf r_h)\) are directly connected with the so-called potential of mean force \(\psi(\mathbf r_1,\ldots,\mathbf r_h)\):

\[ n_2(\mathbf r_1,\mathbf r_2)=n^2 e^{-\beta \psi(\mathbf r_1,\mathbf r_2)} =\frac{N^2}{Q_N}\int e^{-\beta\Phi}\,d\mathbf r_3\ldots d\mathbf r_N, \tag{72} \]

\[ n_3(\mathbf r_1,\mathbf r_2,\mathbf r_3)=n^3 e^{-\beta \psi(\mathbf r_1,\mathbf r_2,\mathbf r_3)} =\frac{N^3}{Q_N}\int e^{-\beta\Phi}\,d\mathbf r_4\ldots d\mathbf r_N \tag{73} \]

and so on, where

\[ Q_N=\int e^{-\beta\Phi}\,d\mathbf r_1\,d\mathbf r_2\ldots d\mathbf r_N. \tag{74} \]

\(\psi(\mathbf r_1,\ldots,\mathbf r_h)\) is the potential of mean force acting between molecules \(1,2,\ldots,h\), averaged over all possible positions of the remaining \(N-h\) molecules. Obviously, the direct calculation of \(n_2(r_{12})\) encounters the same difficulties as the calculation of the configuration integral itself. Therefore Kirkwood\(^{22,23}\), as well as Born and Green\(^{24}\), took an indirect route in the hope of obtaining a usable approximate expression for \(n_2(r)^*\).

The Born–Green equation can be obtained if one takes into account that

\[ \Phi(\mathbf r_1,\ldots,\mathbf r_N)=\sum_{i>k}\varphi(r_{ik}), \tag{75} \]

and differentiates the quantity \(\ln n_2(r_{12})\) (proportional to \(\psi(r_{12})\)) with respect to \(\mathbf r_1\). This gives

\[ -kT\,\frac{\partial \ln n_2(r_{12})}{\partial \mathbf r_1} = \frac{\partial \varphi_{12}}{\partial \mathbf r_1} + \int \frac{\partial \varphi_{13}}{\partial \mathbf r_1}\, \frac{n_3(\mathbf r_1,\mathbf r_2,\mathbf r_3)}{n_1(\mathbf r_1,\mathbf r_3)} \,d\mathbf r_3. \tag{76} \]

Kirkwood considered a system of \(N\) molecules in which molecule 1 is connected with the others in a distinguished way, namely:

\[ \Phi(\mathbf r_1,\ldots,\mathbf r_N) = \xi\sum_{i=2}^{N}\varphi(\mathbf r_1,\mathbf r_i) + \sum_{i>k\ge 2}^{N}\sum \varphi(r_{jk}), \]

\[ \text{*) See in particular the already mentioned work of N. N. Bogoliubov, where these same equations are obtained independently by more rigorous methods. (Translator’s note.)} \]

provided that $\xi \leqslant 1$. Then one may write:

\[ -kT \frac{\partial \ln n_2(r_1, r_2; \xi)}{\partial \xi} = \varphi_{12}+ \int \varphi_{13}\left\{ \frac{n_3(r_1, r_2, r_3; \xi)}{n_2(r_1, r_3; \xi)} -\frac{n_2(r_1,r_2;\xi)}{n} \right\}dr_3 . \tag{77} \]

In both equations the derivatives of $n_2(r_{1c})$ are expressed in terms of $n_3$. In a similar way one can express $n_3$ in terms of $n_4$, and so on. Consequently, an integral equation for determining $n_2$ can be obtained only in the case where it is possible to express $n_3$ in terms of $n_2$. This is the aim of the so-called superposition approximation, which can be obtained as follows. The quantities $\psi(r_{12})$, $\psi(r_1,r_2,r_3)$, etc., may be represented in the form of expansions in powers of the density:

\[ \psi(r_1,r_2)=\varphi(r_{12})+O(n). \tag{78} \]

\[ \psi(r_1,r_2,r_3)=\varphi(r_{12})+\varphi(r_{23})+\varphi(r_{31})+O(n), \tag{79} \]

and so forth. Consequently, neglecting terms proportional to the density, one may write:

\[ \psi(r_1,r_2,r_3)=\psi(r_{12})+\psi(r_{23})+\psi(r_{31}), \tag{80} \]

whence it follows directly that

\[ n_3(r_1,r_2,r_3)=n_2(r_{12})\,n_2(r_{23})\,n_2(r_{31})/n^3 . \tag{81} \]

Kirkwood assumes that equation (81) is valid not only at small densities, but that it is also a good approximation for all densities. This so-called superposition approximation makes it possible to eliminate $n_3$ from the Kirkwood or Born–Green equations, and as a result we arrive at an integral equation for a single function only, $n_2$.

Before discussing the results obtained with the aid of such an integral equation, it is important to discuss the validity of the superposition approximation. Nijboer and van Hove$^{25}$ computed $\psi(r_{12})$ for the case of hard spheres with accuracy up to terms quadratic in the density, strictly and, on the other hand, using the Born–Green integral equation. Comparison shows that the terms in $\psi(r_{12})$ proportional to the density $n$ are equal in both methods of calculation, but the terms proportional to $n^2$ already differ. This indicates that it is impossible to judge how suitable the superposition approximation may prove at densities as large as in a liquid. A similar result was obtained in work$^{26}$, where it was shown that for the case of hard spheres the Born–Green equations give correct values for the second and third virial coefficients, but the fourth virial coefficient is already incorrect.

Calculations based on these equations were carried out by Green, and also by Rodriguez$^{27}$, and they show the necessity of introducing

into the theory various approximations whose validity is not easy to ascertain. Therefore it seems to us that the method of radial distribution functions at present encounters many more difficulties in obtaining practical results than does the free-volume theory.

References

1. J. S. Rowlinson, C. F. Curtiss, J. Chem. Phys. 19, 1519 (1951).
2. R. Fürth, Proc. Cambr. Phil. Soc. 37, 252, 276, 281 (1941).
3. J. de Boer, Physica 14, 139 (1948).
4. J. de Boer, R. J. Lunbeck, Physica 14, 510, 520 (1948).
5. J. de Boer, A. Michels, Physica 5, 945 (1938).
6. R. J. Lunbeck, Thesis, Amsterdam, 1950.
7. H. Eyring, J. chem. Phys. 4, 283 (1936).
8. H. Eyring, J. Hirschfelder, J. chem. Phys. 5, 249 (1937).
9. J. E. Lennard-Jones, A. F. Devonshire, Proc. Roy. Soc. A-163, 63 (1937).
10. J. E. Lennard-Jones, A. F. Devonshire, Proc. Roy. Soc. A-164, 1 (1938).
11. R. H. Wentorf, R. J. Buehler, J. O. Hirschfelder, C. F. Curtiss, J. chem. Phys. 18, 1484 (1950).
12. J. A. Pople, Phil. Mag. 42, 459 (1951).
13. F. Cernuschi, H. Eyring, J. chem. Phys. 7, 547 (1939).
14. S. Ono, Mem. Fac. Eng. Kyusha 10, 190 (1947).
15. H. M. Peek, F. L. Hill, J. chem. Phys. 18, 1252 (1950).
16. J. F. Herzfeld, M. Goeppert-Mayer, Phys. Rev. 46, 995 (1934).
17. M. Born, J. chem. Phys. 7, 591 (1939).
18. J. E. Lennard-Jones, A. F. Devonshire, Proc. Roy. Soc. A-169, 317 (1939).
19. J. E. Lennard-Jones, A. F. Devonshire, Proc. Roy. Soc. A-170, 464 (1939).
20. C. Domb, Phil. Mag. 42, 1316 (1951).
21. J. de Boer, Rep. Progr. Phys. 12, 305 (1949).
22. J. G. Kirkwood, E. M. Boggs, J. chem. Phys. 10, 394 (1942).
23. J. G. Kirkwood, E. K. Mann, B. J. Alder, J. chem. Phys. 18, 1040 (1950).
24. M. Born, H. S. Green, Proc. Roy. Soc. A-188, 10 (1946).
25. B. R. Nijboer, L. van Hove, Proc. Acad. Sci. Amst. B-54, 256 (1951).
26. G. S. Rushbrooke, H. J. Scoins, Phil. Mag. 42, 582 (1951).
27. A. E. Rodriguez, Proc. Roy. Soc. A-196, 73 (1949).