General Theory of Molecular Forces*

F. London, Paris

Since the time of van der Waals we have been accustomed to picture molecules as centers of force and to regard these so-called molecular forces as the common cause of a variety of phenomena: deviations of the equation of state of a gas from the ideal-gas equation, which, as is known, indicates the identity of molecular forces in liquids and in gases; the phenomena of capillarity and adsorption of the heat of sublimation of molecular crystal lattices; certain effects of broadening of spectral lines, etc. It proved possible to determine these forces in a strictly consistent and quantitative way, using their measurable effects as the basis.

In these semiempirical calculations the molecular forces are represented, for the sake of simplicity, as constant, additively superposed central forces—in the general case attractive forces, similar to the gravitation of masses. Indeed, such an assumption gives a very simple and at the same time unambiguous explanation of the parallelism that is observed in the various manifestations of these forces. Nevertheless, if one attempts to explain molecular forces proceeding only from general premises concerning the electrical structure of molecules, it is apparently impossible to obtain so simple a result.

§ 1. Orientational Effect1

Since usually a molecule as a whole is not charged, the most important constant for intermolecular forces is the dipole moment \(\mu\). The interaction between two dipoles \(\mu_{\mathrm I}\) and \(\mu_{\mathrm{II}}\) depends on their mutual orientation. As is well known, the energy of interaction is given, in the first approximation, by the following equation:

\[ U=\frac{\mu_{\mathrm I}\mu_{\mathrm{II}}}{R^3}\left[2\cos\theta_{\mathrm I}\cos\theta_{\mathrm{II}}-\sin\theta_{\mathrm I}\sin\theta_{\mathrm{II}}\cos(\varphi_{\mathrm I}-\varphi_{\mathrm{II}})\right], \tag{1} \]

where \(\theta_{\mathrm I}, \varphi_{\mathrm I}; \theta_{\mathrm{II}}, \varphi_{\mathrm{II}}\) are the polar coordinates determining the orientation of the dipoles, with the line joining the two centers

and of length equal to \(R\), represents the polar axis. Depending on the direction of orientation, either attraction or repulsion is obtained. If all directions occurred equally often, the mean action \(\mu\) would be equal to zero.

However, according to Boltzmann statistics, orientations with lower energy occur more often, and all the more often the lower the temperature. As a result of this statistical preference, Keesom obtained, by averaging over all positions, the following equation:

\[ \overline{U} = -\frac{2}{3}\frac{\mu_{\mathrm{I}}^{2}\mu_{\mathrm{II}}^{2}}{R^{6}}\frac{1}{kT} \left( \text{for } \frac{\mu_{\mathrm{I}}\mu_{\mathrm{II}}}{R^{3}} \ll kT \right). \tag{2} \]

This expression is not suitable for low temperatures or small distances

\[ \left(kT \sim \frac{\mu_{\mathrm{I}}\mu_{\mathrm{II}}}{R^{3}}\right). \]

It is obvious that the most favorable orientation of the molecules will be obtained when they become parallel to one another along the line joining the two molecules. In this case the following interaction energy would be obtained [see (1)]:

\[ \overline{U} = -\frac{2\mu_{\mathrm{I}}\mu_{\mathrm{II}}}{R^{3}} \left( \text{for } \frac{\mu_{\mathrm{I}}\mu_{\mathrm{II}}}{R^{3}} \gg kT \right), \tag{3} \]

which in any case represents the lower limit for this energy: from equations (2) and (3) it follows that the molecules are attracted as a result of preferential orientation with lower energy. With the help of this so-called orientation effect, Keesom attempted to interpret van der Waals attraction.

§ 2. Induction effect²

Debye noticed that the forces discussed above are not the only ones. According to equation (2), they give an attraction which disappears as the temperature rises. But experiment shows that the empirical van der Waals correction terms do not all disappear equally rapidly with increasing temperature. Debye therefore concluded that there must exist another interaction energy, independent of temperature. A detailed consideration of the actual distribution of charge in the molecule, i.e. the introduction of quadrupoles and multipoles, would not help in this respect. The mean value of these interactions would likewise vanish at high temperatures.

But a molecule is, of course, characterized only very approximately by the distribution of charge alone. In fact, the charge distribution will change under the action of another molecule. This property of a molecule can be described very simply by introducing another constant, the polarizability \(\alpha\). In an external electric field of strength \(F\), a molecule with polarizability \(\alpha\) acquires an induced moment

\[ M=\alpha F . \tag{4} \]

(which is added to the possible permanent dipole moment), and its energy in the field \(F\) is expressed by the equation

\[ U=-\frac{1}{2}\alpha F^{2}. \tag{5} \]

Molecule I can create, near molecule II, an electric field of strength

\[ F=\frac{\mu_{\mathrm{I}}}{R^{3}}\sqrt{1+3\cos^{2}\theta_{\mathrm{I}}}. \tag{6} \]

This field polarizes molecule II and leads to an additional interaction energy, according to equation (5),

\[ U=-\frac{1}{2}\alpha_{\mathrm{II}}F^{2} =-\frac{\alpha_{\mathrm{II}}\mu_{\mathrm{I}}^{2}}{2R^{6}}\left(1+3\cos^{2}\theta_{\mathrm{I}}\right), \tag{7} \]

which is always negative (attraction), and therefore its mean value, even at infinitely high temperatures, is also negative. Since \(\overline{\cos^{2}\theta}=1/3\), one obtains

\[ \overline{U}_{\mathrm{I}\to\mathrm{II}} =-\alpha_{\mathrm{II}}\frac{\mu_{\mathrm{I}}^{2}}{R^{6}}. \]

An analogous expression would also be obtained for \(\overline{U}_{\mathrm{II}\to\mathrm{I}}\), i.e. for the action of \(\mu_{\mathrm{II}}\) on \(\alpha_{\mathrm{I}}\). For the complete interaction of both molecules one obtains:

\[ \overline{U}=-\frac{1}{R^{6}}\left(\alpha_{\mathrm{I}}\mu_{\mathrm{II}}^{2}+\alpha_{\mathrm{II}}\mu_{\mathrm{I}}^{2}\right). \tag{8} \]

If both molecules are identical

\[ (\mu_{\mathrm{II}}=\mu_{\mathrm{I}}=\mu \quad \text{and} \quad \alpha_{\mathrm{II}}=\alpha_{\mathrm{I}}=\alpha), \]

then

\[ \overline{U}=-\frac{2\alpha\mu^{2}}{R^{6}}. \tag{8'} \]

This is the so-called induction effect.

In this way Debye and Falkenhagen consider it possible to explain the van der Waals equation. But many molecules have no permanent dipole moment (rare gases, \(\mathrm{H}_{2}\), \(\mathrm{N}_{2}\), \(\mathrm{CH}_{4}\), etc.). In this case Debye and Falkenhagen assume that the molecules possess a quadrupole moment \(\tau\), which would, of course, cause a similar interaction by mutual induction of dipoles. Instead of (8), one would obtain

\[ \overline{U}=-\frac{2}{3}\frac{\alpha\tau^{2}}{R^{8}}. \tag{9} \]

*

Since there is no other method for measuring these quadrupole moments, the van der Waals correction terms (the second virial coefficient) are used for the inverse determination of \(\tau\), which, after \(\mu\) and \(\alpha\), is considered one of the most fundamental molecular constants.

§ 3. Critique of the Static Model of Molecular Forces

The obvious objection that can be raised against this interpretation of van der Waals forces is that it does not explain the above-mentioned parallelism in the various manifestations of molecular forces. It is unclear why, for example, in the liquid and solid states, between all neighboring molecules there should simultaneously act essentially the very same forces as between the partners of a random pair of molecules in the gaseous state. All these models are very far from making it possible simply to explain, with their aid, the emergence of a general additive attraction.

Let two molecules I and II be oriented by their dipoles in such a way that they are attracted by a third molecule; then, between the first two molecules, in the general case, the most diverse forces act, chiefly repulsive forces. Or else, if the forces are caused by polarization, the acting field will, in the general case, be greatly diminished when many molecules from all sides impose their polarizing fields. Therefore one should expect that, in the liquid and solid states, the forces caused by induced or permanent dipoles or multipoles should, at the very least, be considerably reduced, if not entirely eliminated on grounds of symmetry.

The situation became still more difficult when wave mechanics showed that inert gases are exactly spherically symmetric and that they have neither permanent dipoles, nor quadrupoles, nor any other multipoles. They reveal none of the interactions mentioned above. True, according to wave mechanics, \(\mathrm{H}_2\), \(\mathrm{N}_2\), etc. represent at least quadrupoles. But for \(\mathrm{H}_2\) the magnitude of the quadrupole moment can be calculated with the aid of wave mechanics. One obtains only \(1/100\) of those very van der Waals forces which were previously ascribed to suitably chosen quadrupoles.

On the other hand, from wave mechanics there follows an entirely new conception of the interaction between neutral atomic systems.

§ 4. The Dispersion Effect; a Simplified Model\(^3\)

Let us take two spherically symmetric systems, each with polarizability \(\alpha\), for example, two three-dimensional isotropic harmonic-

... oscillators, having no dipole moment in the position of rest. If the charges \(e\) of these oscillators are artificially displaced from their equilibrium positions by the amounts

\[ \vec r_{\mathrm I}=(x_{\mathrm I},\,y_{\mathrm I},\,z_{\mathrm I})\quad\text{and}\quad r_{\mathrm{II}}=(x_{\mathrm{II}},\,y_{\mathrm{II}},\,z_{\mathrm{II}}) \]

respectively, then for the potential energy one obtains

\[ V=\underbrace{\frac{e^2 r_{\mathrm I}^2}{2\alpha}+\frac{e^2 r_{\mathrm{II}}^2}{2\alpha}}_{\substack{\text{energy of elastic}\\ \text{displacement}\\ \text{of the charges}}} + \underbrace{\frac{e^2}{R^3}(x_{\mathrm I}x_{\mathrm{II}}+y_{\mathrm I}y_{\mathrm{II}}-2z_{\mathrm I}z_{\mathrm{II}})}_{\substack{\text{energy of interaction of dipoles}\\ [\text{see (1)}]}} . \tag{10} \]

Classically, the two systems in the equilibrium position

\[ (x_{\mathrm I}=x_{\mathrm{II}}=\ldots=z_{\mathrm{II}}=0) \]

would not act on one another and, being placed at a finite distance \((R>\sqrt[3]{2\alpha})\), would remain in the equilibrium position. They could not induce a moment in one another.

However, as is well known in quantum mechanics, a particle cannot be at absolute rest at a definite point. This would contradict the uncertainty relation. According to quantum mechanics, our isotropic oscillators, even at the very lowest energy level, perform so-called zero-point motion, which can be described only statistically. This can be done, for example, by means of a certain distribution function determining the probability of how often some configuration occurs. However, the sequence of different configurations cannot be described. For isotropic oscillators these distribution functions give a spherically symmetric distribution of configurations around the equilibrium position (inert gases also possess a spherically symmetric distribution of electrons around the nuclei).

For discussion of this simple model, no deep knowledge of quantum mechanics is required. It is sufficient to know that, according to quantum mechanics, a harmonic oscillator with natural frequency \(\nu\) has, at the lowest level, the energy

\[ E_0=\frac{1}{2}h\nu \tag{11} \]

or the so-called zero-point energy. If one introduces the following “normal” coordinates:

\[ \vec r_{+}\equiv \left\{ \begin{aligned} x_{+}&=\frac{1}{\sqrt{2}}(x_{\mathrm I}+x_{\mathrm{II}})\\ y_{+}&=\frac{1}{\sqrt{2}}(y_{\mathrm I}+y_{\mathrm{II}});\\ z_{+}&=\frac{1}{\sqrt{2}}(z_{\mathrm I}+z_{\mathrm{II}}) \end{aligned} \right. \qquad \vec r_{-}\equiv \left\{ \begin{aligned} x_{-}&=\frac{1}{\sqrt{2}}(x_{\mathrm I}-x_{\mathrm{II}})\\ y_{-}&=\frac{1}{\sqrt{2}}(y_{\mathrm I}-y_{\mathrm{II}})\\ z_{-}&=\frac{1}{\sqrt{2}}(z_{\mathrm I}-z_{\mathrm{II}}) \end{aligned} \right. \]

then the potential energy (10) can be represented as a sum of squares, like the potential energy of six independent oscillators (whereas the kinetic energy does not change its form):

\[ \begin{aligned} V &= \frac{e^{2}}{2\alpha}\left(r_{+}^{2}+r_{-}^{2}\right) +\frac{e^{2}}{2R^{3}}\left(x_{+}^{2}+y_{+}^{2}-2z_{+}^{2}-x_{-}^{2}-y_{-}^{2}+2z_{-}^{2}\right) \\ &= \frac{e^{2}}{2\alpha}\left[ \left(1+\frac{\alpha}{R^{3}}\right)\left(x_{+}^{2}+y_{+}^{2}\right) +\left(1-\frac{\alpha}{R^{3}}\right)\left(x_{-}^{2}+y_{-}^{2}\right)\right. \\ &\qquad\qquad\left. +\left(1-2\frac{\alpha}{R^{3}}\right)z_{+}^{2} +\left(1+2\frac{\alpha}{R^{3}}\right)z_{-}^{2} \right]. \end{aligned} \tag{10′} \]

The frequencies of these six oscillators are determined by the equations:

\[ \left. \begin{aligned} \nu_{x}^{\pm}=\nu_{y}^{\pm} &=\nu_{0}\sqrt{1\pm\frac{\alpha}{R^{3}}} \simeq \nu_{0}\left(1\pm\frac{\alpha}{2R^{3}}-\frac{\alpha^{2}}{8R^{6}}\pm\cdots\right), \\ \nu_{z}^{\pm} &=\nu_{0}\sqrt{1\mp\frac{2\alpha}{R^{3}}} \simeq \nu_{0}\left(1\mp\frac{\alpha}{R^{3}}-\frac{\alpha^{2}}{2R^{6}}\mp\cdots\right). \end{aligned} \right\} \tag{12} \]

Here \(\nu_{0}=\dfrac{e}{\sqrt{m\alpha}}\) represents the natural frequency of our two systems, isolated from one another (for \(R\to\infty\)), and \(m\) is the reduced mass of the systems. Assuming that \(\alpha\ll R\), the square root in (12) may be expanded in powers of \(\dfrac{\alpha}{R^{3}}\).

The zero-point energy of this system of six oscillators will therefore be expressed [according to equation (11)] in the following way:

\[ \begin{aligned} E_{0} &= \frac{h}{2}\left(\nu_{x}^{+}+\nu_{y}^{+}+\nu_{z}^{+}+\nu_{x}^{-}+\nu_{y}^{-}+\nu_{z}^{-}\right) \\ &= \frac{h\nu_{0}}{2} \left[ 6+ \left(\frac12+\frac12-1-\frac12-\frac12+1\right)\frac{\alpha}{R^{3}} -\left(\frac48+\frac22\right)\frac{\alpha^{2}}{R^{6}} +\ldots \right] \\ &= 3h\nu_{0}-\frac34\,\frac{h\nu_{0}\alpha^{2}}{R^{6}}. \end{aligned} \]

The first term \(3h\nu_{0}\) represents simply the internal zero-point energy of two isolated elastic systems. However, the second term

\[ U=-\frac34\,\frac{h\nu_{0}\alpha^{2}}{R^{6}} \tag{13} \]

depends on the distance \(R\). It should be regarded as the interaction energy, and since this energy is negative, the forces turn out to be forces of attraction. We shall assume that this kind of

forces, not due to the existence of a permanent dipole or some higher multipole, is the cause of the van der Waals attraction of inert gases, and also of simple molecules \(H_2\), \(N_2\), etc. For the reasons set forth below, these forces are called the dispersion effect*.

§ 5. Dispersion effect; general formula \(^{5}\)

Although the mechanism of this interaction, of course, cannot be described while remaining within the framework of ordinary classical mechanics, we can nevertheless explain it in a kind of semiclassical language.

If it were possible to take an instantaneous photograph of a molecule at some moment in time, various configurations of nuclei and electrons would be found, in the general case possessing dipole moments. Both in a spherically symmetric molecule of an inert gas and in our isotropic oscillators, averaging over a large number of such instantaneous snapshots would, of course, give no preference to any direction. These very rapidly changing dipoles, created by the zero-point motion in the molecule, excite a field in space and, acting on the polarizability of other molecules, induce dipole moments in them. These latter are in phase and in interaction with the instantaneous dipoles that caused their appearance. The zero-point motion is accompanied, so to speak, by a synchronized electric field, but without radiation. The zero-point energy cannot be dissipated by radiation.

This model can be used in interpreting the generalization of our formula (13) to the case of an arbitrary molecule. An exact derivation of this generalization would, of course, require certain quantum-mechanical calculations.

Let us imagine a molecule in state \(k\) as a set of periodic dipoles \(\mu_{kl}\), whose periods correspond to the frequencies

\[ \nu_{kl}=\frac{E_l-E_k}{h} \]

of the (not forbidden) transitions from state \(k\) to state \(l\). These “oscillator strengths” \(\mu_{kl}\) represent the very same quantities that appear in the “dispersion formula,” which gives the polarizability \(\alpha_k(\nu)\) of a molecule in state \(k\) under the action of an alternating field of frequency \(\nu\)

\[ \alpha_k(\nu)=\frac{2}{3h}\sum_l \frac{\mu_{kl}^{\,2}\nu_{kl}}{\nu_{kl}^{\,2}-\nu^2}. \tag{14} \]

* This type of force first appears in the work of Wang \(^{4}\).

If the acting field of frequency \(\nu_0\) has amplitude \(F_0\), then the induced moment \(M\) is expressed by the formula

\[ M=\chi_k(\nu_0)\cdot F_0=\frac{2}{3h}F_0\sum_l \frac{\mu_{kl}^{\,2}\nu_{kl}}{\nu_{kl}^{\,2}-\nu_0^{\,2}}, \tag{4'} \]

and the interaction energy between the field and the molecule is equal to

\[ U=-\frac{1}{2}\alpha(\nu_0)F_0^{\,2} =-\frac{F_0}{3h}\sum_l \frac{\mu_{kl}^{\,2}\nu_{kl}}{\nu_{kl}^{\,2}-\nu_0^{\,2}}. \tag{5'} \]

This acting field may be produced by another molecule by means of one of its periodic dipoles \(\mu_{\rho\sigma}\), with frequency \(\nu_{\rho\sigma}\), and at an angle \(\theta_{\rho\sigma}\) to the line joining the two molecules. Near the first molecule (let us call it the “Latin” molecule, according to the indices denoting its state, in contrast to the Greek indices of the other molecule) the dipole \(\mu_{\rho\sigma}\) excites an electric field of strength [cf. formula (6)]

\[ F_{\rho\sigma}=\frac{\mu_{\rho\sigma}}{R^3} \sqrt{1+3\cos^2\theta_{\rho\sigma}}. \tag{6'} \]

This field induces in the Latin molecule a periodic dipole of magnitude

\[ M_{\rho\sigma}^{k}=\alpha_k(\nu_{\rho\sigma})F_{\rho\sigma}, \]

and the interaction energy [cf. (5′)] proves to be equal to

\[ -\frac{\alpha_k(\nu_{\rho\sigma})}{2}F_{\rho\sigma}^{2} =-\frac{\mu_{\rho\sigma}^{\,2}}{3hR^6} (1+3\cos^2\theta_{\rho\sigma})\cdot \sum_l\frac{\mu_{kl}^{\,2}\nu_{kl}}{\nu_{kl}^{\,2}-\nu_{\rho\sigma}^{\,2}}. \]

If we now consider the entire ensemble of “Greek” molecules in the state \(\rho\), then we must sum over all states \(\sigma\) and average over all directions \(\theta_{\rho\sigma}\) \(\left(\cos^2\theta=\frac{1}{3}\right)\). This gives us the action of the Greek atom on the polarized Latin atom:

\[ U_{\rho\to k}=-\frac{2}{3hR^6} \sum_{l\sigma} \frac{\mu_{kl}^{\,2}\mu_{\rho\sigma}^{\,2}\nu_{kl}} {\nu_{kl}^{\,2}-\nu_{\rho\sigma}^{\,2}}. \]

If one adds the corresponding expression \(U_{k\to\rho}\) for the action of the Latin molecule on the Greek one, then the complete interaction caused by the “periodic” dipoles of the molecule in a sta-

of \(k\) on another molecule in state \(\rho\):

\[ U_{\rho k}=U_{\rho\to k}+U_{k\to \rho} =-\frac{2}{3hR^6}\sum_{l\sigma}\mu_{kl}^2\mu_{\rho\sigma}^2 \left( \frac{\nu_{kl}}{\nu_{kl}^2-\nu_{\rho\sigma}^2} + \frac{\nu_{\rho\sigma}}{\nu_{\rho\sigma}^2-\nu_{kl}^2} \right) = \]

\[ =-\frac{2}{3hR^6}\sum_{l\sigma}\frac{\mu_{kl}^2\mu_{\rho\sigma}^2}{\nu_{kl}+\nu_{\rho\sigma}}. \tag{15} \]

§ 6. Additivity of the dispersion effect

These considerations do not claim, of course, to serve as a rigorous proof of equation (15), but they may serve as an illustration of the mechanism of action of dispersion forces. It can be shown that formula (15) has the property of additivity. This means that if three molecules act on one another simultaneously, then one need only add the three interaction potentials of the form (15) between the three pairs. The influence of the third molecule on the interaction between the first two is only a small perturbation of a lower order of magnitude than the interaction itself. Therefore these attractive forces may simply be superposed according to the parallelogram rule for forces. Consequently, they can fully explain the fact of universal attraction.

If several molecules interact with one another, this should be conceived as follows: each molecule induces in each of the other molecules a whole group of coordinated periodic dipoles. These dipoles are in a fixed phase relation with the corresponding inducing initial dipoles. Each molecule is thus the carrier of very many independently superposed groups of induced periodic dipoles, caused by the action of various molecules. Each of these induced dipoles always has such a direction that it is attracted to the corresponding producing dipole. At the same time, other dipoles not connected with it by any phase relation lead only to a periodic interaction and, after averaging over all possible phases, contribute nothing to the interaction energy. In this way one may imagine that the simultaneous interaction of many molecules is produced simply as the result of the additive superposition of the attractive forces between individual pairs.

§ 7. Simplified formula; some numerical values

For many simple gas molecules (for example, for the inert gases, \(\mathrm{H_2}\), \(\mathrm{N_2}\), \(\mathrm{O_2}\), \(\mathrm{CH_4}\)) it has turned out that the empirical dispersion curve over a large interval of frequencies can be represented by a dispersion formula of type (14), containing only one term.

This means that, for these molecules, the oscillator strength \(\mu_{kl}\) for a narrow frequency interval so far exceeds the others that the latter may be neglected. For this case, and also for the limiting case \(\nu \to 0\) (polarizability in a static field), formula (14) may be simplified as follows:

\[ \alpha_k = \alpha_k(0) = \frac{2}{3h}\frac{\mu_k^2}{\nu_k} \]

(\(\mu_k\) denotes the dipole strength only of the principal frequency \(\nu_k\)), and formula (15) for the interaction of two systems becomes the following

\[ U_{\rho k}=-\frac{2}{3hR^6}\cdot\frac{\mu_k^2}{\nu_k}\cdot\frac{\mu_\rho^2}{\nu_\rho}\cdot \frac{\nu_k\nu_\rho}{\nu_k+\nu_\rho} = -\frac{3h}{2R^6}\alpha_k\alpha_\rho \frac{\nu_k\nu_\rho}{\nu_k+\nu_\rho}. \tag{13′} \]

This formula coincides with (13) in the case of two molecules of the same kind. It can, of course, be applied only when it is known that the dispersion formula has the special form written above [see (14)]. But in any case, if the dispersion formula for a given kind of molecule has been empirically established, this is sufficient for constructing an expression for the forces of attraction (15). No other details of the molecular structure need be known.

TABLE 1

Dispersion effect between simple molecules

Gas	\(h\nu_J\) in eV	\(h\nu_D\) in eV	\(\alpha\cdot 10^{24}\) in \(\mathrm{cm^3}\)	\(c\cdot 10^{48}=\dfrac{3}{4}\alpha h\nu\cdot 10^{48}\) in \(\mathrm{eV\cdot cm^6}\)
He	24,5	25,5	0,20	0,77
Ne	21,5	25,7	0,39	2,93
Ar	15,4	17,5	1,63	34,7
Kr	13,3	14,7	2,46	69
Xe	11,5	12,2	4,00	146
H\(_2\)	16,4		0,81	8,3
N\(_2\)	17	17,2	1,74	38,6
O\(_2\)	13	14,7	1,57	27,2
CO	14,3		1,99	42,4
CH\(_4\)	14,5		2,58	73
CO\(_2\)		15,45	2,86	94,7
Cl\(_2\)	18,2		4,60	288
HCl	13,7		2,63	71
HBr	13,3		3,58	128
HJ	12,7		5,4	278
Na		2,1	29,7	960

Table 1 gives a series of theoretically calculated values of the attraction constant \(c\) (i.e. the coefficient of \(-\dfrac{1}{R^6}\) in the above-

in the expression given) for inert gases and some other simple gases, whose refractive indices can be well represented by a dispersion formula with a single term. The natural frequency \(\nu_D\), multiplied by \(h\), is in all cases very close to the ionization energy \(h\nu_I\). This may, as a first approximation, justify the use of the ionization energy in those cases where the dispersion formula has not yet been established. As is easy to see, the quantities \(c\) vary in the ratio \(1\) to \(1000\), and this large interval of orders of magnitude makes instructive even the very rough experimental verification that is given in § 11.

Mayer\(^{6}\) showed that, for negative ions having the structure of an inert gas, it would be incorrect to simplify the dispersion formula in the continuous region of the spectrum by taking only one frequency. He uses a simple analytic expression to describe the empirical data on absorption in the continuous region. Replacing the sums in expression (15) by integrals over the continuous region, he obtains the following table of values of \(c\) for 29 possible ion pairs (Table 2).

TABLE 2

Dispersion effect between ions
\((c\cdot 10^{48}\ \text{in eV}\cdot\text{cm}^{6})\)

Ion	\(\mathrm{F}^{-}\)	\(\mathrm{Cl}^{-}\)	\(\mathrm{Br}^{-}\)	\(\mathrm{J}^{-}\)
\(+\)	\(c_{-+}=0.13\)	3.2	4.0	5.4	\(c_{++}=0.11\)
\(\mathrm{Na}^{+}\)	7.14	17.8	22.2	30.3	2.68
\(\mathrm{K}^{+}\)	31.0	76.3	95.3	130	38.6
\(\mathrm{Rb}^{+}\)	49.2	125	157	214	94.3
\(\mathrm{Cs}^{+}\)	82.5	205	259	356	247
	\(c_{--}=23\text{--}30\)	176—206	294—332	600—676

Using another method (variational) and making certain simplifying assumptions concerning the wave functions of atoms (the product of the wave functions of the individual electrons), Slater and Kirkwood\(^{6a}\) also calculated the dispersion forces. They derived the following formula:

\[ U=-\frac{1}{R^{6}}\frac{3eh}{8\pi}\sqrt{\frac{N\alpha^{3}}{m}}. \tag{13''} \]

(\(N\) is the number of electrons in the outer shell.)

This expression usually gives values somewhat larger than (13), but it may be applied in those cases where the natural frequencies in (13) are unknown. At present, however, it is difficult to say to what extent formula (13'') can be relied upon.

§ 8. Classification of forces manifesting themselves at large distances[^7]

Formula (15) is applicable to all freely moving molecules, insofar as the interaction energy is small in comparison with the difference of the energy levels of the molecules under consideration, i.e., under the condition that

\[ \frac{\mu_{kl}\mu_{\rho\sigma}}{R^3} < \left|E_k-E_l+E_\rho-E_\sigma\right|. \tag{16} \]

With this restriction, formula (15) is valid both for freely moving dipolar molecules and for molecules of rare gases. Consequently, there is always some minimum value of the distance \(R\), up to which we may rely on formula (15).

The difference between a molecule possessing a permanent dipole and a molecule of an inert gas consists in the following: an inert-gas molecule has so high an excitation energy (electronic transition) that it may be assumed that at ordinary temperatures all molecules are in the ground state. Therefore here the forces do not depend on the temperature. In the case of dipolar molecules, however, it is necessary at least to consider the Boltzmann distribution of the various rotational states, since the difference in the energies of these states is usually very small in comparison with \(kT\).

Let us first consider an absolutely rigid dipole (in the form of a dumbbell), i.e. a molecule without electronic or vibrational states. Then the probability \(p_{\rho k}\) that the Greek molecule is in the purely rotational state \(\rho\), and the Latin one in the purely rotational state \(k\), is equal to

\[ p_{\rho k}=Ae^{-\frac{1}{kT}(E_k+E_\rho)}, \]

where

\[ A^{-1}=\sum_{k\rho} e^{-\frac{1}{kT}(E_k+E_\rho)}. \]

As a consequence, the mean interaction energy between two such molecules turns out to be equal to

\[ \overline{U} = \sum_{\rho k} p_{\rho k}U_{\rho k} = -\frac{2A}{3R^6} \sum_{\substack{\sigma l\\ \rho k}} \frac{\mu_{kl}^{\,2}\mu_{\rho\sigma}^{\,2}} {E_l-E_k+E_\sigma-E_\rho} e^{-\frac{E_k+E_\rho}{kT}}. \tag{17} \]

If in this expression one replaces the notation of the summation indices \(\rho\) and \(k\) by \(\sigma\) and \(l\), then the value of the sum, of course, does not chan-

is written. Therefore, taking the average of these two equivalent expressions, one may write (since \(\mu_{kl}=\mu_{lk}\)):

\[ \overline{U}=-\frac{A}{3R^6}\sum_{\substack{\sigma l\\ \rho k}}\mu_{kl}^{2}\mu_{\rho\sigma}^{2} \frac{ e^{-\frac{E_k+E_\rho}{kT}}-e^{-\frac{E_l+E_\sigma}{kT}} }{ E_l+E_\sigma-E_k-E_\rho }. \tag{17'} \]

Expanding the exponential terms in a series in powers of \(\frac{1}{kT}\), we note that the constant terms cancel one another (there is no interaction at high temperatures, as in § 1). The first and only significant term of the expansion of expression (17) gives

\[ \overline{U}=-\frac{A}{3R^6kT}\sum_{\substack{kl\\ \rho\sigma}}\mu_{kl}^{2}\mu_{\rho\sigma}^{2}+\ldots =-\frac{2\mu_{\mathrm{I}}^{2}\mu_{\mathrm{II}}^{2}}{3kTR^6}. \tag{18} \]

Here \(\mu_{\mathrm{I}}\) and \(\mu_{\mathrm{II}}\) denote the permanent moments of the dipole molecules, which, in the case of an absolutely rigid molecule, of course do not depend on its state. Thus one obtains exactly the same result that Keesom obtained on the basis of classical mechanics.

Let us note, incidentally, that whereas the applicability of equation (15) is based on condition (16), formula (18) is connected with the weaker condition

\[ \frac{\mu_{\mathrm{I}}\mu_{\mathrm{II}}}{R^3}<kT, \]

which at the same time indicates the limit of applicability of the classical calculations.

Of course, in reality a dipole molecule cannot be treated as a rigid dumbbell. It possesses electronic and vibrational transitions. For the sake of simplification let us assume that \(kT\) is large in comparison with the differences of energy levels for purely rotational transitions, but small in comparison with other differences.

In this general case one should again apply formula (17), but here it is sufficient to extend the Boltzmann sum \(\sum_{\rho k}\) only over those states which involve purely rotational transitions from the ground state, since the thermodynamic probability that other states are occupied may be neglected. Now let us divide the sum over \(\sigma\) and \(l\) in (17) into four parts

\[ \overline{U}=U_{rr}+U_{rg}+U_{gr}+U_{gg} \]

as follows:

1. In \(U_{rr}\) there enter only those \(\sigma\)- and \(l\)-terms which are obtained from the ground state by a purely rotational transition. For this sum (with certain inessential qualifications) the calculation given above for rigid dipoles remains valid. Therefore we obtain formula (18)

\[ U_{rr}=-\frac{2}{3R^6}\cdot\frac{\mu_{\mathrm I}^2\mu_{\mathrm{II}}^2}{kT}, \]

i.e., the orientation effect of Keesom.

1. In \(U_{rg}\) the summation over \(\sigma\), as in the first case, extends only to those terms which are obtained from the ground state by a purely rotational transition; but \(l\) will denote a certain larger (not purely rotational) jump. Then \(E_\sigma-E_\rho\) may be neglected in comparison with \(E_l-E_k\) in the denominator of formula (17), and one may write

\[ U_{rg}=-\frac{2A}{3R^6} \left(\sum_{kl}\mu_{kl}^2 e^{-\frac{E_k}{kT}}\right) \left(\sum_{\rho\sigma}\frac{\mu_{\rho\sigma}^2}{E_\sigma-E_\rho}e^{-\frac{E_\rho}{kT}}\right). \]

Comparison with (14) shows that the terms of the second sum on the right-hand side of the equation may be expressed through the static polarizability \(\alpha_\rho=\alpha_\rho(0)\) of the second molecule, which depends very little on the rotational state \(\rho\) of the molecule, so that it may simply be denoted by \(\alpha_{\mathrm{II}}\). At the same time the first sum again turns out to be the square of the permanent dipole moment of the first molecule, with respect to which an approximate independence of the rotational state may be assumed. We obtain

\[ U_{rg}=-\frac{2}{3R^6}\mu_{\mathrm I}^2\frac{3}{2}\alpha_{\mathrm{II}}(0) =-\frac{\mu_{\mathrm I}^2\alpha_{\mathrm{II}}}{R^6}. \]

1. Analogously,

\[ U_{gr}=-\frac{\mu_{\mathrm{II}}^2\alpha_{\mathrm I}}{R^6}. \]

Points 2 and 3 correspond exactly to the induction effect of Debye.

1. Finally, in \(U_{gg}\) both the \(\sigma\)- and the \(l\)-states are obtained from the ground state by a certain larger (not purely rotational) jump. If it is assumed that the probabilities of such a jump do not depend in any appreciable way on the rotational state, then one may simply take the ground state for \(\rho\) and \(k\). Then one obtains:

\[ U_{gg}=-\frac{2}{3hR^6}\sum_{l\sigma}\frac{\mu_{0l}^2\mu_{0\sigma}^2}{\nu_{0l}+\nu_{0\sigma}}, \]

i.e. the dispersion effect. If the condition for (13′) is satisfied, then all three effects can be combined in the following form

\[ \overline{U}=\frac{1}{R^6}\left(\frac{2}{3}\frac{\mu_{\mathrm{I}}^2\mu_{\mathrm{II}}^2}{kT} +\mu_{\mathrm{I}}^2\alpha_{\mathrm{II}} +\mu_{\mathrm{II}}^2\alpha_{\mathrm{I}} +\frac{3h}{2}\alpha_{\mathrm{I}}\alpha_{\mathrm{II}} \frac{\nu_{\mathrm{I}}\nu_{\mathrm{II}}}{\nu_{\mathrm{I}}+\nu_{\mathrm{II}}}\right). \tag{19} \]

In Table 3 all three effects are briefly listed for several dipolar molecules.

TABLE 3

The three components of van der Waals forces

Gas	\(\mu\cdot 10^{18}\)	\(\alpha\cdot 10^{24}\)	\(h\nu_0\) in V	Orientation effect \(\dfrac{2}{3}\dfrac{\mu^4}{k\cdot 293}\cdot 10^{60}\) in \(\mathrm{erg}\cdot\mathrm{cm}^6\)	Induction effect \(2\mu^2\alpha\cdot 10^{60}\) in \(\mathrm{erg}\cdot\mathrm{cm}^6\)	Dispersion effect \(\dfrac{3}{4}\alpha^2 h\nu\cdot 10^{60}\) in \(\mathrm{erg}\cdot\mathrm{cm}^6\)
CO	0.12	1.99	14.3	0.0034	0.057	67.5
HJ	0.38	5.4	12	0.35	1.68	382
HBr	0.78	3.58	13.3	6.2	4.05	176
HCl	1.03	2.63	13.7	18.6	5.4	105
NH\(_3\)	1.5	2.21	16	84	10	93
H\(_2\)O	1.84	1.48	18	190	10	47

It is evident that in all cases the induction effect can practically be neglected and that even in such strongly dipolar molecules as HCl the permanent dipole moment does not make a particularly noticeable contribution to the van der Waals attraction. The orientation effect becomes comparable with the dispersion effect only for NH\(_3\) and H\(_2\)O. The dispersion effect, apparently, cannot be neglected in any of the cases cited.

§ 9. Limits of Applicability

It remains for us to discuss the physical meaning of condition (16). A characteristic distinction of quantum mechanics from classical mechanics is that, from the point of view of the former, a freely moving polyatomic molecule has a centrally symmetric and (in particular at the very lowest energy level) spherically symmetric structure, i.e. a spherically symmetric distribution function. This means that, even when at the lowest level, a free molecule does not preferentially assume any one direction, but continuously changes its orientation owing to zero-point motion. If another molecule tends to orient the molecule under consideration, then a compromise arises between the zero-point motion and the directing force. However

only in the case when

\[ \frac{\mu_{\mathrm I}\mu_{\mathrm{II}}}{R^3} > |E_0-E_1|, \tag{20a} \]

the orienting forces outweigh the zero-point rotation. Therefore, in this case the motion of the dipoles becomes more similar to oscillations about the equilibrium orientation of the dipoles (parallel to one another along the line connecting the two molecules). Then the interaction acquires the character proper to oriented dipoles, i.e. has the order of magnitude

\[ -\frac{2\mu_{\mathrm I}\mu_{\mathrm{II}}}{R^3}. \]

As is known, according to quantum mechanics the condition

\[ \frac{\mu_{\mathrm I}\mu_{\mathrm{II}}}{R^3} > kT \tag{20b} \]

is insufficient for the orientation of molecules, in contrast to formula (3). The orienting forces must overcome not only thermal motion, but, in addition, zero-point motion as well. Let \(\Theta\) denote the moment of inertia of the molecule. Then the right-hand side of the condition is equal to \(\dfrac{h^2}{4\pi\Theta}\). Using this consideration, one can show that, for example, in HJ molecules at those distances which HJ has in the solid state, the orienting forces of the dipoles still prove too weak to overcome the zero-point rotation. This should be imagined in such a way that these molecules in the solid state rotate even at absolute zero. However, HJ is, undoubtedly, rather an exception.

It is obvious that for the case of large molecules and for small intermolecular distances, as occurs in the solid and liquid states, the magnitude of the dipole interaction is wholly insufficient to explain the orienting forces. To clarify this, one should substitute, in place of the left-hand side of (20), simply the classical expression for the orientational energy, in order to estimate approximately a reasonable value of the limit of free motion.

As long as condition (16) is fulfilled, our arguments in favor of additivity (§ 6) remain completely valid in all cases for all three effects entering into formula (19). Only in the case when, as a result of condition (20), the free motion of the molecule is hindered, must the critical considerations of § 3 be invoked. This applies to the nonadditivity of both the orientational and the induction effect.

However, the fact that the rotation of the molecule as a whole has stopped will not appreciably affect the internal electronic motion of the molecule. Thus it will be legitimate to apply the formula for the dispersion effect also to nonrotating molecules.

General Theory of Molecular Forces

However, it is obvious that only especially compact molecules, such as those listed in Tables 1 and 2, can reasonably be regarded simply as centers of force. In the case of long organic molecules, it is apparently necessary to seek to represent the van der Waals attraction as a sum of the actions of separate parts of the molecule. Since the assignment to separate parts of molecules of the frequencies entering formulas (15) or (13) is rather arbitrary, an attempt was made\(^8\) to eliminate them by using the approximate additivity of atomic refractions and diamagnetic susceptibility.

If there is only one “strong” oscillator \(\mu_k\) [cf. formula (14′)], then the diamagnetic susceptibility has the following simple form:

\[ \chi_k=\frac{\mu_k^2 N_L}{6mc^2}\;(<0)\quad (N_L\text{—Loschmidt number}). \]

Therefore, in consequence of (14′),

\[ \nu_k=\frac{2}{3h}\frac{\mu_k^2}{\alpha_k} =\frac{4mc^2}{hN_L}\frac{\chi_k}{\alpha_k}. \]

Instead of (13′) one may consequently write:

\[ U_{kp}= \frac{3}{2}\frac{h}{R^6}\alpha_k\alpha_p \frac{\dfrac{4mc^2}{hN_L}\dfrac{\chi_k}{\alpha_k}\cdot\dfrac{\chi_p}{\alpha_p}} {\dfrac{\chi_k}{\alpha_k}+\dfrac{\chi_p}{\alpha_p}} = \frac{1}{R^6}\frac{6mc^2}{N_L} \frac{\alpha_k\alpha_p}{\dfrac{\alpha_k}{\chi_k}+\dfrac{\alpha_p}{\chi_p}}. \tag{13'''} \]

In this formula the interaction energy is expressed through approximately additive atomic constants, and it is apparently possible in this way to compose the van der Waals attraction of polyatomic molecules from the interactions of individual atoms. However, comparison of the quantities in Table 4 shows that the accuracy of this method is clearly not great.

TABLE 4

Gas	\(\dfrac{3mc^2}{N_L}\alpha\chi\cdot 10^{48}\)	\(c\cdot 10^{48}\) (from Table 1)
He	0.84	0.77
Ne	4.94	2.93
Ar	69.0	34.7
Kr	180	69
Xe	448	146

For the dispersion effect, condition (16) also indicates a limit characteristic of it. The quantity

\[ \frac{\mu_{kl}^2}{E_k-E_l} \]

is practically identical with the polarizability \(\alpha\), if \(E_k \to E_l\) represents the “principal” electronic transition [cf. (14′)]. Consequently, instead of (16) one may, as a rough approximation, write

\[ \alpha < R^3, \tag{16'} \]

as the condition of applicability of our formulas for the dispersion effect. It is easy to derive what \(\alpha > R^3\) would mean, by consid-

whereas formerly for these forces a law of the form \(\dfrac{b}{R^n}\) was adopted, quantum mechanics now shows that an exponential law of the form

\[ b e^{-\frac{R}{\rho}} \]

is more suitable for repulsion. Usually the expression for the repulsive forces is now taken precisely in this form, and the constants \(b\) and \(\rho\) are determined empirically.

For attraction the expression (15) or (13) is used; it may be supplemented by a term proportional to \(R^{-8}\). The factor \(R^{-8}\) introduces a third constant, which must likewise be determined from experiment. Finally, for ions with a structure similar to that of an inert gas, it remains, of course, to add also the Coulomb expression \(+\dfrac{e_1 e_2}{r}\), where \(e_1\) and \(e_2\) are the charges.

Thus, for the simplest molecules, an expression of the form

\[ U = b e^{-\frac{R}{\rho}} - \frac{c}{R^6} - \frac{d}{R^8} + \frac{e_1 e_2}{R} \tag{21} \]

is now usually employed as a reasonable basis; here \(b\), \(\rho\), and \(d\) are parameters to be determined, whereas \(c\), \(e_1\), and \(e_2\) are regarded as theoretically known. Thus in all applications of van der Waals forces there remains considerable freedom, and this must be borne in mind when one wishes to carry out an experimental verification of the theory.

§ 11. Experimental Verification

It is not our task to describe the various applications which the theory of molecular forces has so far received. We shall confine ourselves here to a very approximate and, as far as possible, simple verification of this theory, without touching on the parameters of equation (21), which still remain to be determined.

1. Recently a direct verification of the asymptotic law \(R^{-6}\) for molecular forces was initiated by means of a very interesting method. This method uses the action of forces, manifested at large distances, on the shape of a spectral line, or the so-called pressure broadening effect. Kuhn \(^{13}\) showed that if the asymptotic law of interaction between atoms has the form

\[ U \simeq \frac{c}{R^p}, \]

then the distribution of intensity in a certain region of the spectral line is given by the formula

\[ I(\nu)=\frac{k}{(\nu_0-\nu)^{\frac{p+3}{p}}}. \]

Thus the slope of the straight line representing \(\lg I\) as a function of \(\lg \nu\) directly gives the exponent \(p\). Minkowski\(^{14}\) interprets his measurements of the broadening of the \(D\)-lines of Na by argon from this same point of view. His paper gives the following diagram of the quantities \(\lg I\) measured by him (Fig. 1).

We have additionally drawn the lines for \(p=5\), \(p=6\), and \(p=7\). As is evident from the figure, the accuracy of the measurements is insufficient for an error-free determination of \(p\). In any case, however, it may be said that \(p=6\) is better than \(p=5\) or \(p=7\), and that \(p=8\) and \(p=4\) can be excluded with certainty.

Fig. 1. Intensity distribution and molecular forces

1. In testing the theory by means of the gas equation of state, we shall confine ourselves to an approximate investigation, considering only the van der Waals quantities \(a\) and \(b\). In the case of a favorable result, one may use the temperature dependence of the second virial coefficient for the inverse determination of the parameters of equation (21) that remain undetermined. However, by fitting the parameters of a formula of type (21), it is always possible to achieve excellent agreement in the second virial coefficients. It is therefore useful to simplify the situation so as to eliminate, if possible, all parameters subject to determination.

For this purpose we replace equation (21) by the following:

\[ U= \begin{cases} -\dfrac{c}{R^6}, & \text{for } R>R_0,\\[6pt] +\infty, & \text{for } R<R_0. \end{cases} \]

This means that 1) the molecules are idealized as though they were impenetrable spheres, and that 2) two terms are neglected: \(be^{-R/\rho}\) and \(-\dfrac{d}{R^8}\) for \(R>R_0\). At large distances the term \(-\dfrac{c}{R^6}\), of course, is the only signif-

term. At intermediate distances \((R \gtrsim R_0)\) the two terms not taken into account can to a considerable extent cancel each other because of their different signs. For \(R < R_0\) the rapid growth of the exponential repulsion is replaced by a sudden increase to \(+\infty\). After such simplifications the minimum value of the energy \(U\) may change by some common factor, but the order of magnitude will remain correct. Instead of the three parameters to be determined in (21), only one is obtained—the distance \(R_0\).

The second virial coefficient \(B_2\) appears in the expansion of the equation of state of a gas in powers of \(\frac{1}{V}\):

\[ \frac{pV}{N_L kT}=1+\frac{B_2(T)}{V}+\frac{B_3(T)}{V^2}+\ldots \]

and is expressed theoretically by the following formula

\[ B_2=2\pi N_L\int_0^\infty \left(1-e^{-\frac{U}{kT}}\right)r^2dr. \tag{23} \]

When \(B_2\) is expanded in powers of \(\frac{1}{T}\), the first two terms are identical with the corresponding terms of the van der Waals equation:

\[ \frac{pV}{N_L kT} = \frac{V}{V-b} - \frac{a}{VN_L kT} \simeq 1+\frac{1}{V}\left(b-\frac{a}{N_L kT}\right) +\frac{1}{V^2}(\cdots)+\ldots \]

Comparison gives

\[ B_2=b-\frac{a}{N_L kT}+\ldots \]

If now (22) is substituted into (23) and it is taken into account that for high temperatures \(U \gg -kT\) for any \(R\), then one obtains

\[ \left\{ \begin{aligned} b&=-\frac{2\pi N_L R_0^3}{3}\\ a&=-2\pi N_L^2\int_{R_0}^{\infty}UR^2\,dR =\frac{2\pi N_L^2 c}{3R_0^3}. \end{aligned} \right. \]

Eliminating \(R_0^3\) from both equations, we obtain:

\[ ab=-\frac{4\pi^2 N_L}{9}\cdot c = 1.51\cdot 10^{54}\cdot c. \tag{24} \]

The numerical coefficient is determined here so that \(a\), as usual, is measured in \(atm\cdot cm^6\cdot g^{-2}\), and \(c\)—in the units indicated in Table 1.

If \(b\) is taken from the empirical equation of state of a gas, then the value of \(a\) can be predicted using equation (24). These values are given in Table 5 under the designation \(a_{\text{theor.}}\). For comparison, the experimental values \(a_{\text{exp.}}\) are also indicated.

TABLE 5

The van der Waals constant \(a\) and the heat of sublimation

Gas	\(b_{\mathrm{expt}}\) \((\mathrm{cm}^3)\)	\(a_{\mathrm{theor}}\cdot 10^{-4}\) \((\mathrm{atm}\cdot\mathrm{cm}^6\cdot\mathrm{g}^{-2})\)	\(a_{\mathrm{expt}}\cdot 10^{-4}\) \((\mathrm{atm}\cdot\mathrm{cm}^6\cdot\mathrm{g}^{-2})\)	\(\rho\)	\(L_{\mathrm{theor}}\cdot 10^{-3}\) \(\left(\dfrac{\mathrm{kg\,cal}}{\mathrm{mol}}\right)\)	\(L_{\mathrm{expt}}\cdot 10^{-3}\) \(\left(\dfrac{\mathrm{kg\,cal}}{\mathrm{mol}}\right)\)
He	24	4.8	3.5
Ne	17	26	21	1.46	0.47	0.59
Ar	32.3	163	135	1.70	1.92	2.03
Kr	39.8	253	240	3.2	3.17	2.80
Xe	51.5	430	410
H\(_2\)	26.5	46	24.5
N\(_2\)	39.6	147	135	1.03	1.64	1.86
O\(_2\)	31.9	135	136	1.43	1.69	2.06
CO	38.6	166	144	1.05	1.86	2.09
CH\(_4\)	42.7	256	224	0.53	2.42	2.70
CO\(_2\)	42.8	334	361
Cl\(_2\)	54.8	680	632	2.00	7.18	7.43
HCl	40.1	283	366	1.56	3.94	5.05
HBr	44.2	510	442	2.73	4.45	5.52
HJ	—	—	—	3.58	6.65	6.21

It is hardly necessary to say how inadequate the use of critical data is for determining the limiting values of the second virial coefficient as \(T\to\infty\). This inadequacy may probably amount to an error of \(30\%\). The simplifications made in deriving formula (22) may likewise introduce an error of the same order of magnitude. However, all these inaccuracies can probably introduce only a general and systematic error in all the examples considered. Although the good agreement of the absolute values in the table may be regarded as accidental, the relative agreement of the theoretical and experimental values of \(a\) over such a wide range of substances is beyond doubt. This fact may justify attempts to improve our ideas about van der Waals forces by using the empirical second virial coefficient to determine the parameters of equation (21). This has hitherto been done\({}^{15}\) only by adding a term of the form

\[ \frac{b}{R^n} \]

for repulsion. But this method inevitably leads to excessively small molecular dimensions, since it attributes to the \(R^{-6}\) forces what is caused by failure to take into account the \(R^{-8}\) forces and the sudden decrease of the exponential repulsion.

1. Table 5 also gives lattice energies \(L\) (heats of sublimation, extrapolated to absolute zero, with the zero-point energy subtracted) for several molecular lattices. These energies were calculated from the same simplified formula (22). In all cases close packing was adopted for the structure, since

this structure is approximately realized in the molecular lattices under consideration. Summation of formula (22) over the entire lattice gives

\[ L=8.36\,N_L^2\,\frac{c}{v^2}\,10^4\ \frac{\mathrm{kg\,cal}}{\mathrm{mol}} =3.04\cdot 10^{52}\,\frac{c\rho^2}{M^2}\ \frac{\mathrm{kg\,cal}}{\mathrm{mol}}. \tag{25} \]

Here \(c\) is taken from Table 1, \(v\) denotes the experimental value of the molar volume, \(\rho\) the density, and \(M\) the molecular weight.

This check is instructive in that it clearly shows the additivity of the forces. In particular, the increase of \(L\) from HCl to HJ with decreasing dipole moment vividly demonstrates the predominance of those forces which are not caused by permanent moments.*

After determining all the coefficients of the complete formula (21), it will be possible, for example, from the empirical second virial coefficient, to calculate all the constants of these molecular lattices (compressibility, elastic moduli, etc.).

Van der Waals attraction also plays a large role in the constitution of ionic lattices. For ions these forces are now known much better than for neutral molecules. Using the expression for the interaction energy of the form (21), Born and Mayer\(^{16}\) calculated the lattice energies of all salts of the alkali metals and hydrohalic acids for a lattice of the NaCl type, and also for the CsCl type. Comparing the stability of both types, they were able to show that the rather strong van der Waals attraction between the heavy ions Cs\(^+\), J\(^-\), Br\(^-\), Cl\(^-\) (cf. Table 2) explains the fact that CsCl, CsBr, CsJ and precisely only these salts possess a lattice structure in which ions of the same sign are at a smaller distance from one another than in lattices of the NaCl type. The share of the van der Waals forces in the total energy of an ionic lattice is, of course, relatively small, varying from 1% to 5. Yet it is precisely this small quantity that is quite sufficient to explain the transition from the NaCl type to the CsCl type.

LITERATURE

1. W. H. Keesom, Leid. Comm. Supp., 24a, 24b, 25, 26, 1912; 39a, 39b, 1915; Proc. Amst., 15, 240, 256, 417, 643, 1913; 18, 636, 1916; 24, 162, 1922. Physik. Z., 22, 129, 643, 1921; 23, 225, 1922.
2. P. Debye, Physik. Z., 21, 178, 1920; 22, 302, 1921. H. Falkenhagen, Physik. Z., 23, 87, 1922.
3. F. London, Z. physik. Chem., B 11, 222, 1931.
4. S. C. Wang, Physik. Z., 26, 663, 1927.
5. R. Eisenschitz u. F. London, Z. Physik, 60, 491, 1930.
6. J. E. Mayer, J. Chem. Phys., 1, 270, 1933.
   6a. J. C. Slater, J. G. Kirkwood, Phys. Rev., 37, 682, 1931.
7. F. London, Z. Physik, 63, 245, 1930.

* In Table 5 the lattice energies of He and H\(_2\) have been omitted, since in these lattices the zero-point energy of nuclear motion contributes such a large share that it cannot be neglected. Therefore H\(_2\) and He cannot be directly compared with other substances\(^{17}\).

1. J. G. Kirkwood, Physik. Z., 33, 57, 1932; A. Müller, Proc. Roy. Soc. A 154, 624, 1936.
2. K. F. Herzfeld, Phys. Rev., 29, 701, 1927.
3. H. Margenau, Phys. Rev., 38, 747, 1931.
4. J. C. Slater, J. Phys. Rev., 32, 349, 1928; see also W. E. Bleand and J. E. Mayer, J. Chem. Phys., 2, 252, 1934.
5. H. Jensen, Z. Physik, 101, 164, 1936; P. Gombas, Z. Physik, 93, 378, 1935.
6. H. Kuhn, Phil. Mag., 18, 987, 1934; Proc. Roy. Soc. A 1936 (in press); see also H. Kuhn, F. London, Phil. Mag. 18, 983, 1934.
7. R. Minkowski, Z. Physik, 93, 731, 1935.
8. K. Wolf, Z. physik. Chem., B 41, 36, 1931; J. E. Lennard-Jones, Proc. Phys. Soc., 43, 461, 1931.
9. M. Born and J. E. Mayer, Z. Physik, 75, 1, 1932; J. E. Mayer, J. Chem. Phys., 1, 270, 1933.
10. F. London, Proc. Roy. Soc., A 153, 576, 1936.

Note added in proof

London’s article contains an error in the calculation of that part of the constant \(a\) which is determined by the dipole attraction of molecules (the orientational effect). The error arises from the fact that London uses equation (23), which is derived (for example, from the virial theorem) under the assumption that the forces of interaction do not depend on the mutual orientation of the molecules. Meanwhile, the dipole interaction depends on orientation, and the term expressing this interaction in equation (19) represents the mean value of the energy, which cannot be introduced into equation (23). The correct transition from the expression for the energy to the equation of state is most simply carried out as follows. Since it is assumed that \(\overline{U} \ll kT\), then, for calculating the potential energy of the gas, the distribution of molecules in space may be regarded as uniform. Since \(1\ \mathrm{cm}^3\) contains \(\dfrac{N_L}{V}\) molecules, the interaction energy of one molecule with its surroundings will be equal to

\[ \int_{R_0}^{\infty} \overline{U}\,\frac{N_L}{V}\,4\pi R^2\,dR, \]

and the energy of all the molecules of a mole of gas is

\[ \frac{N_L}{2}\cdot \int_{R_0}^{\infty} \overline{U}\,\frac{N_L}{V}\,4\pi R^2\,dR \]

(we divide by two so as not to count each pair of molecules twice). Using the equations

\[ \overline{U}=\frac{c}{R^6}\quad \text{for } R \geq R_0 \]

and

\[ b=\frac{2\pi N_L R_0^3}{3}, \]

we obtain for the potential energy of a mole the expression \(-\dfrac{a'}{V}\), where

\[ a'=\frac{4\pi^2 N_L^3}{9b}\cdot c . \]

Thus the internal energy of the gas \(E\) will be determined by the equation

\[ E=-\frac{a'}{V}+E^0, \]

where \(E^0\) depends only on \(T\). We shall further use the well-known thermodynamic relation

\[ P=T\left(\frac{\partial P}{\partial T}\right)_V-\left(\frac{\partial E}{\partial V}\right)_T . \]

Integrating it gives

\[ P=T\left[c(V)+\int \frac{2}{T^2}\left(\frac{\partial E}{\partial V}\right)_T\,dT\right], \]

where \(c(V)\) is some function of \(V\). In our case

\[ \left(\frac{\partial E}{\partial V}\right)_T=\frac{a'}{V^2}. \]

Here \(a'\) consists of terms independent of \(T\), and a term inversely proportional to \(T\) (the orientation effect). Therefore we shall write

\[ a'=\alpha+2\beta T^{-1}. \]

Then we obtain

\[ P=Tc(V)-\frac{\alpha+\beta T^{-1}}{V^2}. \]

It is obvious that

\[ c(V)=\frac{R}{V-b} \]

and

\[ \alpha+\beta T^{-1}=a. \]

We see that the orientation effect gives in \(a\) a term twice as small as in London’s calculation, which is equivalent to identifying \(a'\) and \(a\). Therefore the values in Table 5 that refer to dipole molecules require correction.

I note that in the old works of Keesom the calculation of the second virial coefficient for dipole interaction was carried out correctly.

M. Temkin

1. Trans. Farad. Soc. 33, 8, 1937, translated by V. Vasiliev. ↩