THE NATURE OF MOLECULAR COHESION FORCES IN GASEOUS AND LIQUID BODIES

Ya. I. Frenkel.

I. GENERAL CONSIDERATIONS.

§ 1. The Electrical Nature of Molecular Forces.

In contrasting neutral bodies with electrified ones, we usually imagine the former as completely devoid of electric charges. In reality, however, every neutral body contains charges of opposite sign in equivalent quantities; electrification is caused, generally speaking, only by an excess of some of these charges over others. Owing to this circumstance, electrified objects act not only upon one another, but also upon neutral bodies. In the latter case this action reduces to (mutual) attraction, which is the difference between the attractive forces experienced by oppositely charged particles and the repulsive forces experienced by particles with like charges. The preponderance of the attractive forces over the repulsive forces is caused by the approach of the attracted particles and the recession of the repelled ones, in connection with the well-known dependence of electrical forces on distance (Coulomb’s law).

It is not difficult to convince oneself that a completely analogous interaction must also occur between neutral bodies or, more precisely, between the elementary particles of neutral bodies—atoms and molecules. Indeed, atoms and molecules are, as is known, electrical systems consisting of massive positive nuclei and negative electrons rotating around them. Such systems, despite their neutrality, must produce external electrical actions that weaken very rapidly with increasing distance, but which can, nevertheless, at distances not too large in comparison with their own dimensions (i.e., with the dimensions of the electronic orbits), attain very great intensity.

Thus, when two molecules approach one another, the electrons and nuclei composing each of them experience, from the other, oppositely directed forces having the character either

attraction or repulsion. Since the attracted particles must thereby approach one another, and the repelled ones must move apart, the forces of attraction must increase, and the forces of repulsion decrease. The indicated displacement of oppositely charged particles in opposite directions can take place in two ways, namely: 1) by a proper rotation, or orientation, of the molecule as a whole, with its internal structure unchanged, and 2) by a corresponding deformation, or polarization, of the molecule, with its position unchanged. In reality, of course, polarization and orientation must occur simultaneously, and moreover in each of the interacting molecules, producing on the average a more or less considerable attraction between them.

The forces of molecular cohesion in gaseous and liquid bodies are reduced to these electrostatic forces. In solids (crystals), the interparticle cohesion is also conditioned by electrostatic forces. However, in view of the regular arrangement of the atoms and the absence of separate molecules in the strict sense of the word (each crystal represents, as it were, one polymerized molecule), these forces have a character essentially different from that considered above1.

A very vivid illustration of the “orientational” forces of cohesion may be provided by the forces of attraction between permanent magnets, whose poles, in the sense of their interaction, are entirely equivalent to opposite electric charges. Incidentally, magnets can serve as a model only for so-called “dipolar,” or “doublet,” molecules, which, being built not of neutral atoms but of ions (for example, \(HCl\), \(H_2O\), etc.), possess in the normal state (i.e. in the absence of an external field) two opposite electric poles, representing the “centres of gravity” of the positive and negative charges.

In the case when both centres coincide, i.e. with a more symmetrical arrangement of the charges, the molecules may be likened to astatic magnetic pairs, formed by two oppositely directed magnets with equal moments. Such systems are called “quadrupoles,” or “quadruplets,” since they reduce to four pairwise opposite poles which are arranged, alternating, at the vertices of a parallelogram (Fig. 1a); the latter may, moreover, take the form of a rectangle (b) or a straight line (c). Analogously

Fig. 1.

In this way one can construct models of still more symmetrical, or, more precisely, “more neutral,” molecules—octupoles or octuplets (formed by 8 pairwise opposite poles at the vertices of a parallelepiped), and so on. It goes without saying that the cohesive forces between quadrupole molecules should, generally speaking, be of smaller intensity than those between dipole molecules, and greater than those between octupole molecules (if such exist).

As for forces of polarization origin, they may be illustrated by the action of magnets, or of astatic magnetic pairs, on pieces of soft iron, which thereby become “magnetized” and are attracted to them. In this respect the polarization of molecules is completely analogous to magnetization. Let us note that in the case of dipole molecules polarization causes a certain change in the normal electric moment, whereas in the case of quadrupole molecules it causes the appearance of such a moment, absent under normal conditions, i.e. in the absence of an external field.

The approach of molecules must, obviously, have a certain limit, characterizing what is called their “size.” Here, of course, what is meant is not the geometrical dimensions of molecules, i.e. not the distances between the electrons and nuclei that form them, but their dynamic dimensions, corresponding to the equilibrium of the forces acting between them. Above we assumed that these forces reduce to attraction, increasing continuously as the molecules approach one another. This conception obviously needs correction, for at a sufficiently small distance between molecules their mutual attraction must pass into repulsion. The latter circumstance is explained by the fact that all atoms, and consequently molecules as well, possess a shell of one and the same sign, namely a negative one, formed by rapidly rotating electrons. Therefore, when two molecules come too close to one another, the forces of repulsion between these peripheral electrons predominate, while the forces of attraction, due to mutual orientation and polarization, recede into the background. Thus, from the mathematical point of view, the interaction of two molecules may be treated as the sum of two forces—attraction, considered above, and repulsion, which increases with decreasing distance still more rapidly than attraction. For simplicity, however, we shall treat molecules as hard balls possessing a perfectly definite diameter \(d\), i.e. capable of approaching one another to a perfectly definite distance \((a)\), independently of the velocity of their motion. In reality, as this velocity increases, the distance between the centers of colliding molecules may decrease somewhat, so that the quantity \(a\) must be a decreasing function of temperature.

If a heap of magnets is subjected to continuous and vigorous shaking and in this way they are not allowed to preserve that mutual orientation which corresponds to the most intense attraction, then the latter must weaken, and, moreover, the more so the more vigorous the shaking.

The molecules of gaseous and liquid bodies do not remain at rest, but are in rapid motion, translational and rotational. This motion, the energy of which constitutes a measure of the temperature of the body under consideration, continuously disrupts that mutual orientation of the molecules on which, at least in part, the cohesion between them depends. In this respect rotational motion has special significance; its energy, at not too low temperatures, may be regarded as proportional to the energy of translational motion, i.e., to the absolute temperature. Hence it follows that, with an increase in the latter, that part of the forces of molecular cohesion which is due to mutual orientation must tend to zero.

As for the second part of these forces, due to mutual polarization, in a first approximation it must retain a constant magnitude independent of temperature, since, of course, the mean distances between the molecules remain unchanged, i.e., the volume of the body under consideration. It goes without saying that with an increase of volume these forces weaken.

Thus, at high temperatures the predominating influence must belong to the polarization forces. The relative significance of the orientational forces cannot be determined a priori. If, as in the case of dipolar molecules, it is not at too high temperatures that they must play the dominant role. A comparison of the calculations of Keesom1, to whom belongs the orientational theory of the forces of molecular cohesion, with the calculations of Debye2, who put forward for the first time polarization forces, shows that the same thing occurs at ordinary temperatures also in the case of molecules of quadrupolar character.

Taking into account the regular arrangement of atoms in solid bodies, corresponding, evidently, to the regular orientation of those molecules into which they split upon melting, one must suppose that near the melting temperature the forces of molecular cohesion in liquids must have predominantly an orientational character3.

§ 2. Microscopic manifestation of cohesive forces.

The macroscopic characterization of molecular forces is achieved, in the case of gases, by studying the “equation of state,” i.e. the equation relating the volume \((v)\) and pressure \((p)\) to the temperature \((T)\). Thus, for example, in the Van der Waals equation,

\[ \left(p+\frac{a}{v^2}\right)(v-b)=RT \tag{1} \]

the cohesive forces are determined by the constant \(a\), while the volume of the molecules, or, more precisely, the repulsive forces that account for it (see above), are determined by the constant \(b\).

Rewriting formula (1) in the form

\[ p=\frac{RT}{v-b}-\frac{a}{v^2} \]

and assuming, in view of the smallness of \(b\) in comparison with \(v\),

\[ \frac{1}{v-b}=\frac{1}{v}\left(1+\frac{b}{v}\right), \]

we obtain the equation of state in the following form:

\[ \frac{pv}{RT}=1+\frac{B}{v}, \tag{1a} \]

where the coefficient \(B\) is a function of the temperature, defined by the formula

\[ B=b-\frac{a}{RT} \tag{1b} \]

and characterizing simultaneously both the attractive forces and the repulsive forces (volume).

Kamerlingh-Onnes and Keesom, on the basis of extensive experimental material, showed that the equation of state can be represented more exactly by a formula of the form

\[ \frac{pv}{RT}=1+\frac{B}{v}+\frac{C}{v^2}+\frac{D}{v^3}+\cdots, \]

where \(C\), \(D\), etc., just like \(B\), are functions of temperature, called virial coefficients. In the first approximation, which we shall have in mind below, this formula coincides with (1a), although the dependence of \(B\) on temperature is expressed by a series of the form

\[ B=b_0-\frac{b_1}{T}-\frac{b_2}{T^2}-\frac{b_3}{T^3}-\cdots . \tag{1c} \]

Comparison of this formula with (1b) shows that the physical meaning of the coefficients \(b_0, b_1, b_2\) can be interpreted in two ways. Namely, one may set \(b=b_0\) and, consequently,

\[ a=k\left(b_1+\frac{b_2}{T}+\frac{b_3}{T^2}+\cdots\right) \]

or else \(a=kb_1\) and \(b=b_0-\dfrac{b_2}{T^2}-\dfrac{b_3}{T^3}+\cdots\). Although in fact the volume \(b\) cannot be a constant quantity independent of temperature, nevertheless the first interpretation, as we shall see below, is much closer to reality than the second. We have already indicated that the forces of molecular cohesion are composed of a polarization part, which in the first approximation is independent of temperature, and an orientational part, which decreases more or less rapidly with increasing \(T\).

Hence it is clear that in the formula \(a=k\left(b_1+\dfrac{b_2}{T}+\cdots\right)\) the first term characterizes the polarization forces, and the second (also, of course, in the first approximation) the orientational forces.

In the Van der Waals equation the ratio \(\dfrac{a}{v^2}\) represents, as is known, the internal pressure of the gas. It follows from this that the internal work performed by the gas when its volume is increased from \(v\) to \(v'\) is equal to

\[ \int \frac{a}{v^2}\,dv = a\left(\frac{1}{v}-\frac{1}{v'}\right). \]

Denoting the potential energy of the gas by \(U\) and putting \(U=0\) for \(v=\infty\) (i.e. at infinite rarefaction), we may therefore set

\[ U=-\frac{a}{v}. \tag{2} \]

Thus the determination of the Van der Waals “constant” \(a\) reduces to the calculation of the mutual potential energy of the gas under consideration. As for the coefficient \(b\), it can be calculated either from formula (1b) or directly from the virial theorem.

The Van der Waals formula is applicable not only to the gaseous state, but in a rough approximation also to the liquid state. Therefore, taking \(v_1\) to mean the volume of some liquid, and \(v_2\) the volume of the saturated vapor (in the same amount and at the same temperature), we may identify the difference

\[ U_2-U_1=\frac{a}{v_1}-\frac{a}{v_2} \]

with the latent heat of vaporization. However, in view of the smallness of \(v_1\) in comparison with \(v_2\),

the latter can be measured by the quantity \(\dfrac{a}{r_1}\), and represents the mutual electrostatic energy of the molecules of the liquid with the opposite sign \((-U_1)\).

For a macroscopic characterization of the forces of molecular cohesion in liquids one may use, alongside the general (volume) cohesion and the internal pressure, the surface energy or the surface tension. In calculating the potential energy of a liquid it is necessary to take into account the incomplete environment of the outer molecules. Since the potential energy of molecules, becoming zero at a large distance between them, decreases as they approach one another and therefore is a negative quantity, the absence of the normal entourage around surface molecules must be associated with an insufficiently small, i.e. in other words, excessively large value of their energy in relation to all the remaining molecules. The excess energy thus falling upon a unit of surface approximately coincides, in numerical value, with the surface tension of the liquid.

Let us note that for monatomic substances, such as the noble (inert) gases and metallic vapors or liquids, the considerations set forth above are not fully applicable. First of all, in this case the concept of “orientation” acquires a somewhat different meaning than in the case of molecules. In fact, the orientation of a molecule is determined by the arrangement of the atoms composing it or, more precisely, of the corresponding positive nuclei, which may be treated as points; as for the peripheral electrons, in view of their rapid rotation one can speak only of the position of their orbits. Thus the orientation of individual atoms is generally determined by the arrangement of the orbits of the peripheral electrons. If the latter form a system possessing an axis of symmetry, then the orientation of the atom may be determined by the direction of this axis and, alongside the normal rotation of the electrons in their orbits, an additional rotation associated with the turning of this axis may be considered. However, the possibility of such a rotation of atoms (or, more precisely, of their electronic shells around the central nucleus) is excluded by the theory of quanta. The latter shows that the angular momentum of any rotating system can assume only values that are multiples of \(\dfrac{h}{2\pi}\), where \(h\) is Planck’s constant. For a given value of the angular momentum, the kinetic energy, as is not difficult to see, is inversely proportional to the moment of inertia. And since in the case of individual atoms the moment of inertia (owing to the lightness of the electrons) is extremely small, the corresponding “quanta” of energy prove to be extremely large—so large that at ordinary temperatures atoms turn out to be completely incapable of rotating. Under such conditions the relative orientation of interacting

atoms (insofar as one can speak of it at all) remains unchanged and on the average has no influence whatever on the forces of cohesion.

Thus, in monatomic gases the forces of cohesion must have an exclusively polarization character, which is quite consistent with their relatively negligible magnitude, especially in the case of noble gases. The situation is entirely different with metallic elements, which in the liquid state (as opposed to the gaseous state) are characterized by an enormous magnitude of the cohesive forces, manifested, among other things, in exceptionally large values of the surface tension. These properties of metallic liquids are closely connected with their electrical conductivity and depend on the ionization of the atoms, i.e. on the splitting of the latter into ions and “free” electrons1.

Restricting ourselves to these general indications, we shall henceforth not touch at all upon metallic bodies. The theory set forth below, belonging in its essential features to Keesom and Debye, applies mainly to complex substances with diatomic or polyatomic molecules.

II. ELECTROSTATIC THEORY.

§ 3. The electric field of a molecule. Let us imagine an arbitrary system of electrified particles concentrated inside a sphere of a definite radius \((r_0)\). The potential of one of these particles \((Q)\) at some external point \((P')\) is equal to \(\dfrac{E}{r_{QP'}}\), where \(E\) is the charge \(Q\), and \(r_{QP'}\) is the distance.

Let us introduce a rectangular system of coordinates \(X_1, X_2, X_3\), with arbitrary origin and directions of the axes: the coordinates of the center of the sphere \((P)\) we shall denote by \(x_1, x_2, x_3\), those of the point \(P'\) by \(x'_1, x'_2, x'_3\), and, finally, those of the point \(Q\) by \(x_1+\xi_1,\ x_2+\xi_2,\ x_3+\xi_3\). (\(\xi_1, \xi_2, \xi_3\) are the projections of the segment \(PQ\) on the coordinate axes.) Considering the distance

\[ r_{QP'}=\sqrt{(x'_1-x_1-\xi_1)^2+(x'_2-x_2-\xi_2)^2+(x'_3-x_3-\xi_3)^2} \]

as a function of the quantities \(\xi_1, \xi_2, \xi_3\), we can expand \(\dfrac{1}{r_{QP'}}\) in a Taylor series

\[ \frac{1}{r_{QP'}} = \frac{1}{r} +\sum_i \frac{\partial\left(\frac{1}{r}\right)}{\partial x_i}\,\xi_i +\frac{1}{2!}\sum_{i,k}\frac{\partial^2\left(\frac{1}{r}\right)}{\partial x_i\,\partial x_k}\,\xi_i\xi_k +\frac{1}{3!}\sum_{i,k,l}\frac{\partial^3\left(\frac{1}{r}\right)}{\partial x_i\,\partial x_k\,\partial x_l}\,\xi_i\xi_k\xi_l -\cdots \tag{3} \]

where

\[ r=r_{PP'}=\sqrt{(x'_1-x_1)^2+(x'_2-x_2)^2+(x'_3-x_3)^2}. \]

and the indices \(i,k,l\) take the values \(1,2,3\). The derivatives of \(\dfrac{1}{r}\) with respect to the coordinates are expressed, as is not difficult to verify, by the following formulas

\[ \frac{\partial \left(\dfrac{1}{r}\right)}{\partial x_i} = \frac{1}{r^2} f_i = \frac{1}{r^2}\lambda_i; \qquad \frac{\partial^2 \left(\dfrac{1}{r}\right)}{\partial x_k \partial x_i} = \frac{1}{r^3} f_{ik} = \frac{1}{r^3}\left(-\delta_{ik}+3\lambda_i\lambda_k\right); \qquad \frac{\partial^3 \left(\dfrac{1}{r}\right)}{\partial x_l \partial x_k \partial x_i} \]
\[ = \frac{1}{r^4} f_{ikl} = \frac{1}{r^4}\left[-3\left(\delta_{ik}\lambda_l+\delta_{il}\lambda_k+\delta_{kl}\lambda_i\right)+15\lambda_i\lambda_k\lambda_l\right]; \qquad \frac{\partial^4 \left(\dfrac{1}{r}\right)}{\partial x_l \partial x_k \partial x_i \partial x_m} \tag{3a} \]
\[ = \frac{1}{r^5} f_{iklm} = \frac{1}{r^5}\left[3\left(\delta_{ik}\delta_{lm}+\delta_{il}\delta_{km}+\delta_{im}\delta_{kl}\right) -15\left(\delta_{ik}\lambda_l\lambda_m+\ldots\right) +105\lambda_i\lambda_k\lambda_l\lambda_m\right], \]

where

\[ \lambda_i=\frac{x'_i-x_i}{r}\qquad (i=1,2,3) \]

are the cosines of the angles formed by the straight line \(PP'\) with the coordinate axes, and \(\delta_{ik}\) are numbers equal to \(1\) when \(i=k\) and to \(0\) when \(i\ne k\).

For the convergence of the series (3) it is necessary and sufficient that

\[ \frac{\xi_1^2+\xi_2^2+\xi_3^2}{r^2}<1, \]

i.e., that the distance \(P'Q\) be less than \(PP'\). For \(PP'>r\) this condition is satisfied with respect to all particles of the system under consideration.

Thus its potential at the point \(P'\), equal to the sum \(\displaystyle \sum_{(Q)} \frac{\varepsilon_Q}{r_{QP'}}\), extended over all these particles, can be represented in the form of a series arranged according to powers of \(\left(\dfrac{1}{r}\right)\)

\[ \varphi = \frac{\varepsilon_0}{r} + \frac{1}{r^2}\sum_i f_i\mu_i + \frac{1}{2r^3}\sum_{i,k} f_{ik}\mu_{ik} +\cdots . \tag{4} \]

Here \(\varepsilon_0\) is the total charge of the system, and the coefficients \(\mu_i,\ \mu_{ik},\ldots\), which constitute its electrical moments of the 1st, 2nd, etc. orders, are determined by the formulas

\[ \mu_i=\sum_Q \varepsilon \xi_i,\qquad \mu_{ik}=\sum_Q \varepsilon \xi_i\xi_k,\qquad \mu_{ikl}=\sum_Q \varepsilon \xi_i\xi_k\xi_l. \tag{4a} \]

In the case of an electrified system, the potential \(\varphi\) at sufficiently distant points reduces, in the first approximation, to \(\dfrac{\varepsilon_0}{r}\). Exactly likewise

in the case of neutral systems—for example, atoms and molecules—this potential reduces, in the same approximation, to one of the following terms of the series (4), namely to the first term different from zero. Thus the degree of neutrality of different molecules may be unequal, depending on the order of those electric moments with which the series (4) actually begins and by which, consequently, the character of the electric field at sufficiently large distances is determined.

The physical meaning of the quantities \(\mu_i\), i.e. of the components of the moment of the first order, follows from their formal analogy with the corresponding mechanical quantities, in which the role of electric charges is played by the masses of the various particles, and with the aid of which the position of the center of gravity of the bodies formed by them is determined. Dividing the sum \(\sum_Q \varepsilon_i\) into two parts, \(\sum_{(+)} \varepsilon_i\) and \(\sum_{(-)} \varepsilon_i\), belonging respectively to positive and negative charges, we can determine the coordinates of the positive and negative center, or pole, of the neutral system (molecule) under consideration by the formulas

\[ \sum_{(+)} \varepsilon \xi_i=\xi_i^{+}\sum_{(+)} \varepsilon \quad\text{and}\quad \sum_{(-)} \varepsilon \xi_i=\xi_i^{-}\sum_{(-)} \varepsilon . \]

Putting

\[ \sum_{(+)} \varepsilon=e \quad\text{and}\quad \sum_{(-)} \varepsilon=-e, \]

we obtain \(\mu_i=e(\xi_i^{+}-\xi_i^{-})=e\delta_i\), where \(\delta_1,\delta_2,\delta_3\) are the components of the segment drawn from the negative pole to the positive one. Thus the moment of the first order may be treated as a vector whose direction coincides with \(\delta\), and whose magnitude is equal to \(e\delta=\sqrt{\mu_1^2+\mu_2^2+\mu_3^2}=\mu^{(1)}\). Denoting the cosines of the angles formed by the segment \(\delta\) with the coordinate axes by \(a_i\), we can, obviously, represent the corresponding part of the potential \(\varphi\) in the form

\[ \varphi^{(1)}=\frac{\mu^{(1)}}{r^2}\sum_i f_i a_i =\frac{\mu^{(1)}}{r^2}\cos\theta \tag{5} \]

where \(\theta\) is the angle between \(\delta\) and \(r\).

If \(\mu^{(1)}>0\), i.e. if the opposite poles of the molecule do not coincide with one another, then it is called a dipole. The electric field created by it, in the first approximation, is the same as if all like charges were concentrated at the corresponding poles. In this respect, however, all dipoles for which the product \(e\delta\) has one and the same value \((\mu^{(1)})\) are completely equivalent to the given molecule.

The difference between them is determined by the following terms in the expression for the potential \(\varphi\) according to formula (4). In the limiting case of a dipole with finite moment \(\mu^{(1)}=e\delta\) and infinitely small length \((\delta\to 0)\), the moments of higher orders (proportional to \(e\delta^2=\mu^{(1)}\delta\), etc.) vanish, and the potential is exactly expressed by formula (5). Such a dipole is called elementary, and the straight half-line drawn in the direction \(\delta\) is called its axis.

Let us imagine that the negative pole \((-e)\) is at the center of the sphere, while the positive pole \((+e)\) is displaced in the direction of the axis \(A\) by an infinitely small distance \(dA=\delta\). Denoting the component segments \(dA\) by \(dx_1, dx_2, dx_3\), and the cosines of the angles formed by \(A\) with the coordinate axes by \(a_1, a_2, a_3\), we may evidently put

\[ \varphi^{(1)} = e\sum_i \frac{\partial\left(\frac{1}{r}\right)}{\partial x_i}\,dx_i = e\delta\sum_i \frac{\partial\left(\frac{1}{r}\right)}{\partial x_i}\,a_i = \mu^{(1)}\sum_i \frac{\partial\left(\frac{1}{r}\right)}{\partial x_i}\,a_i . \]

Differential expressions of the form \(\sum_i \dfrac{\partial\psi}{\partial x_i}a_i\), or \(\dfrac{\partial\psi}{\partial x'_i}a'_i\), where \(\psi\) is an arbitrary function of the coordinates \(x_i\) and \(x'_i\), are called the derivatives of \(\psi\) along the axis \(A\) at the point \(P\), or along the axis \(A'\) at the point \(P'\), and are denoted by the symbols \(\dfrac{d\psi}{dA}\) or \(\dfrac{d\psi}{dA'}\).

Using the above notation, we can represent the potential of an elementary dipole in the form

\[ \varphi^{(1)} = \mu^{(1)} \frac{d\left(\frac{1}{r}\right)}{dA}. \tag{5a} \]

Let us note that such a dipole is equivalent to an aggregate of three elementary dipoles with axes \(X_i\) and moments \(\mu_i\).

Suppose now that both poles of the molecule coincide, i.e., that the quantities \(\mu_i\) are equal to zero, and that the series (4) begins with the term corresponding to the electric moments of the second order \(\mu_{ik}\). This term,

\[ \varphi^{(2)} = \frac{1}{2}\sum_{i,k} \frac{\partial^2\left(\frac{1}{r}\right)}{\partial r_i\,\partial r_k}\,\mu_{ik}, \]

may be regarded as the potential of nine elementary quadrupoles with axes \((X_i, X_k)\) and moments \(\mu_{ik}\). It is not difficult to see that the aggregate of all these quadrupoles is equivalent to one elementar-

quadrupole with certain axes \(A, B\) and moment \(\mu^{(2)}\), i.e. its potential \(\varphi^{(2)}\) can be represented in the form

\[ \varphi^{(2)} = \frac{\mu^{(2)}}{2}\, \frac{\partial^2\left(\frac{1}{r}\right)}{\partial A\,\partial B} = \frac{1}{2}\frac{\mu^{(2)}}{r^3}\sum_{i,k} f_{ik}\alpha_i\beta_k, \tag{6} \]

where \(\alpha_i=\cos(A,X_i)\) and \(\beta_i=\cos(B,X_i)\). Indeed, owing to the identity

\[ \frac{\partial^2\left(\frac{1}{r}\right)}{\partial x_1^2} + \frac{\partial^2\left(\frac{1}{r}\right)}{\partial x_2^2} + \frac{\partial^2\left(\frac{1}{r}\right)}{\partial x_3^2} =0 \]

or, what is the same,

\[ f_{11}+f_{22}+f_{33}=0, \]

the number of independent parameters \(\mu_{ik}\) characterizing \(\varphi^{(2)}\) can be diminished by 1, i.e. reduced to (5); as for the number of parameters entering into formula (6), in view of the identities \(\sum_i \alpha_i^2=\sum_i \beta_i^2=1\), it is likewise reduced to 5. However, we shall not dwell on the actual reduction of these expressions to one another (i.e. on the determination of \(\mu^{(2)}, \alpha_i, \beta_i\) as functions of \(\mu_{ik}\)).

Recalling the values of the coefficients \(f_{ik}\) [see formula (3a)], we have

\[ \sum_{i,k} f_{ik}\alpha_i\beta_k = -\sum_{i,k}\delta_{ik}\alpha_i\beta_k + 3\sum_{i,k}\lambda_i\alpha_i\lambda_k\beta_k = -\sum_i\alpha_i\beta_i + 3\left(\sum_i\lambda_i\alpha_i\right)\left(\sum_k\lambda_k\beta_k\right). \]

But the sum \(\sum_i \alpha_i\beta_i\) is equal to \(\cos\Theta_{AB}\), i.e. to the cosine of the angle between the axes \(A\) and \(B\), while the sums \(\sum_i \lambda_i\alpha_i\) and \(\sum_k \lambda_k\beta_k\) are the cosines of the angles \(\Theta_A\) and \(\Theta_B\) formed by both axes with the straight line \(PP'\). Substituting these values into (6), we obtain

\[ \varphi^{(2)} = \frac{1}{2}\frac{\mu^{(2)}}{r^3} \left(3\cos\Theta_A\cos\Theta_B-\cos\Theta_{AB}\right). \tag{6a} \]

Let us note that in the particular case of a symmetric quadrupole, characterized by the coincidence of the axes \((A=B,\ \Theta_A=\Theta_B=\Theta)\), formula (6a) reduces to

\[ \varphi^{(2)} = \frac{1}{2}\frac{\mu^{(2)}}{r^3} \left(3\cos^2\Theta-1\right). \tag{6b} \]

Я. И. ФРЕНКЕЛЬ

For \(\mu^{(1)}=\mu^{(2)}=0\) the series (4) begins, generally speaking, with the term \(\varphi^{(3)}\), depending on the electric moments of the 3rd order. This term may be regarded as the potential of 27 elementary octupoles with axes \((x_1, x_2, x_3)\) and moments \(\mu_{abc}\) \((a,b,c=1,2,3)\). It is not difficult to show, just as was done above for the potential of the 2nd order \(\varphi^{(2)}\), that the potential of the 3rd order \(\varphi^{(3)}\) may be represented in the form

\[ \varphi^{(3)}=\frac{\mu^{(3)}}{1\cdot 2\cdot 3}\, \frac{\partial^{3}\left(\frac{1}{r}\right)}{\partial A\,\partial B\,\partial C}, \tag{7} \]

i.e. that the above-mentioned system of octupoles, for arbitrary values of the moments, is equivalent to a single octupole with quite definite axes \(A, B, C\) and moment \(\mu^{(3)}\). Formula (7) is an approximate expression for the potential of an octupole system (with accuracy to terms of the 3rd degree).

In an analogous manner, potentials of higher orders corresponding to the following terms in the expansion (4) can be transformed.

§ 4. Orientational energy of molecules. The preceding results are easily extended to the expression for the potential energy of an arbitrary electric system \((S')\), situated in a given external field, with respect to that system \((S)\) which creates this field. If both systems may be regarded as “rigid,” i.e. if the mutual distances between their constituent particles, and consequently also the “internal” energy, are constant, then this mutual energy determines the forces experienced by each system on the part of the other (see below).

Let us imagine that the center of the second system is placed at the point \(P'\). Let us consider one of its constituent particles, \(Q'\). Denoting the potential of the first system \((S)\) at the point \(P'\) by \(\varphi\), and at the point \(Q'\) by \(\varphi_{Q'}\), we may write, according to Taylor’s formula,

\[ \varphi_{Q'}=\varphi+\sum_i \frac{\partial \varphi}{\partial x_i}\xi_i' +\frac{1}{2!}\sum_{i,k}\frac{\partial^2\varphi}{\partial x_i\,\partial x_k}\xi_i'\xi_k' +\frac{1}{3!}\sum_{i,k,l}\frac{\partial^3\varphi}{\partial x_i\,\partial x_k\,\partial x_l}\xi_i'\xi_k'\xi_l' +\cdots, \]

where \(\xi_i'\) represent the components of the segment \(P'Q'\) along the coordinate axes, and \(x_i'\) are the coordinates of the point \(P'\).

The potential energy of the system under consideration \((S')\) is evidently equal to the sum of these expressions multiplied by the charges of the corresponding particles \((\varepsilon')\). Thus, introducing the electric moments

\[ \mu_i'=\sum_{(Q')}\varepsilon'\xi_i',\qquad \mu_{ik}'=\sum_{(Q')}\varepsilon'\xi_i'\xi_k',\qquad \mu_{ikl}'=\sum_{(Q')}\varepsilon'\xi_i'\xi_k'\xi_l',\ldots \]

and the total charge of the system \(e'_0=\sum_{(p')} e'\), we obtain the formula:

\[ u=e'_0\varphi=\sum_i \frac{\partial \varphi}{\partial x'_i}\,\mu'_i +\frac{1}{2}\sum_{i,k}\frac{\partial^2\varphi}{\partial x'_i\partial x'_k}\,\mu'_{ik}+\cdots . \tag{8} \]

This shows that every neutral system (molecule), with respect both to the derivatives and to the electrical actions experienced by it, is equivalent to a set of elementary dipoles, quadrupoles, octupoles, etc., coinciding at one of its internal points (the “center”).

In what follows we shall treat dipole molecules as elementary dipoles, quadrupole ones as elementary quadrupoles, etc., neglecting higher-order terms in formulas (8) and (8a).

Denoting the axis of a dipole molecule situated at the point \(P'\) by \(A'\), and its moment by \(\mu'\), we may therefore put

\[ u^{(1)}=\mu'\frac{\partial\varphi}{\partial A'} =\mu'\sum_i\frac{\partial\varphi}{\partial x'_i}\,\alpha'_i . \tag{9} \]

The derivatives of the potential with respect to the coordinates are equal in magnitude and opposite in sign to the components of the electric field \(E\) at the point \(P'\). Denoting these components by \(E_i\), and the angle between \(E\) and \(A'\) by \(\Theta_{A'}\), we may rewrite the preceding formula in the form

\[ u^{(1)}=-\mu' E\cos\Theta_{A'} . \tag{9a} \]

Exactly likewise, in the case of a quadrupole molecule with moment \(\mu'^{(2)}=\zeta'\) and axes \((A',B')\), the energy may be represented by the formula

\[ u^{(2)}=\frac{\zeta'}{2}\frac{\partial^2\varphi}{\partial A'\partial B'} =\frac{\zeta'}{2}\sum_{i,k}\frac{\partial^2\varphi}{\partial x'_i\partial x'_k}\,\alpha'_i\beta'_k , \tag{9b} \]

which is easily obtained from the expression

\[ u^{(2)}=\frac{1}{2}\sum_{i,k}\frac{\partial^2\varphi}{\partial x'_i\partial x'_k}\,\mu'_{ik} \]

in connection with Laplace’s equation

\[ \frac{\partial^2\varphi}{\partial x_1'^2} +\frac{\partial^2\varphi}{\partial x_2'^2} +\frac{\partial^2\varphi}{\partial x_3'^2}=0 . \]

Putting in formula (9)

\[ \varphi=\varphi^{(1)}=\mu\,\frac{\partial\left(\frac{1}{r}\right)}{\partial A}, \]

we obtain the mutual potential energy of two dipole molecules, with moments \(\mu\) and \(\mu'\) and axes \(A\) and \(A'\), at a distance \(r\) from one another,

\[ u^{(1,1)} = \mu\mu'\, \frac{\partial^2\left(\frac{1}{r}\right)}{\partial A\,\partial A'} = \mu\mu'\sum_{i i'} \frac{\partial^2\left(\frac{1}{r}\right)}{\partial x_i\,\partial x'_{i'}}\, \alpha_i\alpha'_{i'} . \]

Recalling that

\[ \frac{\partial\left(\frac{1}{r}\right)}{\partial x'_{i'}} = - \frac{\partial\left(\frac{1}{r}\right)}{\partial x_i}, \]

we may rewrite this formula in the form

\[ u^{(1,1)} = - \frac{\mu\mu'}{r^3} \sum_{i i'} f_{i i'}\alpha_i\alpha'_{i'}, \]

or, comparing (9) with (6) and (6a),

\[ u^{(1,1)} = \frac{\mu\mu'}{r^3} \left(\cos\Theta_{AA'}-3\cos\Theta_A\cos\Theta_{A'}\right), \tag{10a} \]

where \(\Theta_A\) and \(\Theta_{A'}\) are the angles of the axes \(A\) and \(A'\) with the straight line \(PP'\), and \(\Theta_{AA'}\) is the angle between them.

In exactly the same way, putting in (9)

\[ \varphi = \frac{\tau}{2} \frac{\partial^2\left(\frac{1}{r}\right)}{\partial A\,\partial B}, \]

we obtain the mutual potential energy of two quadrupole molecules with moments \(\tau\) and \(\tau'\) and axes \(A,B\) and \(A',B'\):

\[ u^{(2,2)} = \frac{\tau\tau'}{4} \frac{\partial^4\left(\frac{1}{r}\right)} {\partial A\,\partial B\,\partial A'\,\partial B'} = \frac{\tau\tau'}{4r^5} \sum f_{i h i' h'}\alpha_i\beta_h\alpha'_{i'}\beta'_{h'} . \]

Without dwelling on the calculation of this expression in the general case [which, incidentally, is not difficult to carry out with the aid of formulas (3a)], we note that in the case of symmetry of both molecules \((A=B\) and \(A'=B')\), it reduces to the form

\[ u^{(2,2)} = \frac{3}{4}\frac{\tau\tau'}{r^5} \left\{ 1 - 5\cos^2\Theta - 5\cos^2\Theta' - 15\cos^2\Theta\cos^2\Theta' + 2\left(5\cos\Theta\cos\Theta' - \cos\Theta_{AA'}\right)^2 \right\}, \tag{10} \]

where \(\Theta=\Theta_A=\Theta_B\) and \(\Theta'=\Theta_{A'}=\Theta_{B'}\).

THE NATURE OF MOLECULAR COHESION

Let us note that the factor before the square bracket \(\left(\dfrac{3}{4}\dfrac{\pi^4}{r^5}\right)\) is equal to the energy of two symmetrical quadrupoles whose axes are perpendicular to one another and to the straight line joining them.

We shall not dwell on the derivation of analogous expressions for the energy of quadrupoles, on the one hand, and of dipoles or octupoles, on the other, since they are of no importance for what follows.

§ 5. The polarization energy of molecules. In the preceding section we treated molecules as “rigid” systems, whose internal structure does not change under the influence of external forces. In reality molecules do not possess such absolute rigidity and, when situated in an electric field of external origin, undergo more or less considerable deformations. These deformations reduce, in the first approximation, to the polarization of the molecule, i.e. to the appearance of an additional electric moment of the first order, proportional to the intensity of the external electric field \((E)\) and parallel to it.^1 Introducing the coefficient of proportionality \(a\), which characterizes the degree of “softness” of the molecule, and denoting the magnitude of the additional moment by \(\Delta\mu\), we may therefore write

\[ \Delta\mu = aE. \tag{11} \]

Denoting the sum of the positive and negative charges in the molecule by \(e\), and the relative displacement of its two poles, caused by the fields under consideration, by \(\delta\), we have \(\Delta\mu=e\delta\), and, consequently,

\[ E=-\frac{e^2}{a}\delta. \]

The latter expression represents the internal force that is due to the displacement of opposite charges in opposite directions and that balances the external forces \(-Ee\) acting on them. Hence it follows that a polarized molecule possesses the internal potential energy

\[ \int_0^\delta \frac{e^2}{a}\delta\,d\delta = \frac{1}{2}\frac{e^2}{a}\delta^2 = \frac{1}{2}eE\delta = \frac{1}{2}\Delta\mu\,E \]

(equal to the work required to overcome the internal forces during polarization). If the polarized molecule were to become rigid, then, according to formula (9a), it would have to possess, at the corresponding point, the (additional) potential energy \(-\Delta\mu E=-aE^2\).

^1 In fact, all molecules possess a more or less pronounced anisotropy, expressed, among other things, in the inequality of the polarization coefficient for different directions of the vector \(E\), and hence in the noncoincidence of the direction of the additional moment with the direction of \(E\).

Thus, the total value of the “polarization energy” of a molecule in an electric field \(E\) is expressed by the formula

\[ u=-\frac{1}{2}\alpha E^{2}. \tag{11a} \]

This result may be explained as follows. If the external force experienced by the pole being attracted is equal to \(-eE\), then the force experienced by the opposite—repelled and, therefore, more distant—pole is, generally speaking, somewhat smaller, namely by

\[ e\,\frac{dE}{dA}\,\delta, \]

where \(A\) is the axis of the electric field, i.e. the straight line determining its direction. This force

\[ -e\delta\,\frac{dE}{dA}=-\alpha\,\frac{dE}{dA}=-\alpha E\,\frac{dE}{dA} \]

is the attraction experienced by the molecule. Noting that

\[ \alpha E\,\frac{dE}{dA}=\frac{d}{dA}\left(\frac{1}{2}\alpha E^{2}\right), \]

we see that it corresponds to the potential energy

\[ -\frac{1}{2}\alpha E^{2}=-\frac{1}{2}\Delta\mu\cdot E, \]

and not to \(-\Delta\mu\cdot E\), as in the case of a rigid dipole \((\delta=\mathrm{const})\). Substituting in (11a)

\[ E^{2}=\sum_k E_k^{2}=\sum_k\left(\frac{\partial \varphi}{\partial x_k}\right)^{2}, \]

we have

\[ u=-\frac{1}{2}\alpha\sum_k\left(\frac{\partial \varphi}{\partial x_k}\right)^{2}. \tag{11b} \]

If the external field is due to a dipole molecule, then

\[ \varphi=\varphi^{(1)}=\mu\,\frac{\partial\left(\frac{1}{r}\right)}{\partial A}, \]

and, consequently,

\[ \frac{\partial\varphi}{\partial x_k} = -\frac{\partial\varphi}{\partial r_k} = -\mu\,\frac{\partial^{2}\left(\frac{1}{r}\right)}{\partial A\,\partial x_k} = -\frac{\mu}{r^{3}}\sum_i f_{ik}a_i. \]

Recalling that

\[ f_{ik}=-\delta_{ik}+3 i_i i_k, \]

we further have

\[ \sum_i f_{ik}a_i=-a_k+3 i_k\cos\vartheta_A, \]

and, consequently,

\[ \sum_k\left(\sum_i f_{ik}a_i\right)^{2} = 1+3\cos^{2}\vartheta_A. \]

i.e.

\[ u^{(1)}=-\frac{1}{2}\frac{\alpha\mu^{2}}{r^{6}}(1+3\cos^{2}\Theta_{1}). \tag{12} \]

To this energy there corresponds, according to the formula \(F_r=-\dfrac{\partial u}{\partial r}\), an attractive force inversely proportional to the 7th power of the distance.

In an analogous manner, if the external field is produced by a symmetric (uniaxial) quadrupole molecule, then expression (11b) takes the following form

\[ u^{(2)}=-\frac{9}{8}\frac{\alpha z^{2}}{r^{8}}(1-2\cos^{2}\Theta+5\cos^{4}\Theta), \tag{12a} \]

and there corresponds to it an attractive force inversely proportional to the 9th power of the distance.

It should be borne in mind that the polarization of molecules is not one-sided, but always has a reciprocal character. Thus, that part of the potential energy of two arbitrary molecules which depends on their mutual polarization is equal to the sum of expressions (12) in the case of dipole molecules, or (12a) in the case of quadrupole molecules.

III. STATISTICAL THEORY.

§ 6. Polarization forces (Debye’s theory). Turning to the calculation of the electrostatic energy of gases and liquids, let us first consider, following Debye, that part of this energy which depends on the mutual polarization of the molecules. We shall denote the number of the latter in a unit of volume by \(n\), the volume occupied by them by \(v\), and, finally, the total number of molecules by \(N=nv\).

Formula (11a) for the potential energy of a polarized molecule makes it possible to treat the potential energy of the entire system (gas or liquid) as a sum of parts corresponding to the individual molecules. In this connection the quantity \(E\) should be understood as the total (resultant) electric-field intensity in which the molecule under consideration is situated, and which is due to the combined action of the surrounding (in particular, of course, neighboring) molecules. Further, taking into account the chaotic character of molecular motion—translational and rotational—we may equate the total energy \(U\) to the product of the total number of molecules \(N\) by the mean value of the energy of any one molecule, corresponding to all possible positions and orientations of all the others. Denoting the components of the electric intensity depending on each of these molecules by \(E_i'\), \(E_i''\), etc., we have \(E_i=E_i'+E_i''+\cdots\), and consequently,

\[ E^{2}=E_{1}^{2}+E_{2}^{2}+E_{3}^{2}=\sum_{i}(E_{i}'+E_{i}''+\cdots)^{2}= \]

\[ \sum_{i}(E_{i}'^{\,2}+E_{i}''^{\,2}+\cdots+2E_{i}'E_{i}''+\cdots). \]

The directions of the various “elementary” stresses, owing to the rotation of the corresponding molecules, change rapidly and irregularly; since all these rotations are not coordinated with one another, the components \(E_i', E_i'', \ldots\) assume equally often, and moreover independently of one another, both positive and negative values. Thus the mean value of products of the form \(E_i' E_i''\) vanishes, and the mean value \(E^2\) reduces to the sum of the mean values \(E'^2=\sum_i E_i'^2\), \(E''^2=\sum_i E_i''^2\), etc. Hence, on the basis of formula (11a), it follows that the mean value of the potential energy of any molecule with respect to all the others is equal to the sum of the mean values of this energy with respect to each of them separately.

The mutual potential energy of two molecules situated at a definite distance \((r)\) from one another depends, generally speaking, on their orientation. We shall denote this energy by \(u_r\), and its mean value for all possible orientations, calculated on the assumption of perfectly equal probability of the latter, by \(\overline{u}_r\). In an analogous way we shall denote the mean value of an arbitrary function of the energy \((F(u_r))\), or of those angular quantities on which it depends.

In reality the different orientations of the molecules do not have the same probability. Each pair of molecules tends to pass into a position corresponding to the greatest possible mutual attraction, i.e. to the least possible potential energy. This tendency is opposed by thermal motion, which continually destroys any more or less ordered arrangement. Thus, in the case of very high temperatures, the indicated circumstance may be disregarded, and \(\overline{u}_r\) may be treated as that mean value of the energy \(u_r\) which is actually observed when all pairs of equidistant molecules are compared with one another (or one and the same pair at different moments of time).

Let us imagine a spherical layer of radius \(r\) and thickness \(dr\), whose center \(P\) coincides with the center of the molecule under consideration. The latter, just like all the rest, we shall treat as a sphere of definite diameter \(d\), in the sense that the distance between the centers of two molecules cannot be less than \(d\). The number of molecules enclosed in the volume of the above-mentioned layer is, if not exactly, then on the average, \(n \cdot 4\pi r^2 dr\). Multiplying this expression by \(\overline{u}_r\) and integrating from \(d\) to \(\infty\), we obtain a quantity which (in view of the extremely rapid decrease of \(\overline{u}_r\) with increasing distance) practically coincides with the potential energy of the molecule under consideration with respect to all the others, independently of the volume \(v\) and of the form of the surface \(S\) bounding it—provided only that the point \(P\) does not lie too close

to the latter. The potential energy of any of the “internal” molecules may therefore be regarded as equal to one and the same quantity

\[ 4\pi \int_d^\infty \bar u_r r^2\,dr. \]

To obtain the total energy of all the molecules, i.e. the potential energy of the system under consideration as a whole \((U)\), it is sufficient to multiply this expression by the total number of molecules \(N\) and divide by 2 [for, by definition, \(U\) is the potential energy corresponding to the mutual polarization of two molecules, i.e. the mean value (12) or (12a), multiplied by 2]. Thus

\[ U=2\pi N^2\int_d^\infty \bar u_r r^2\,dr, \tag{13} \]

i.e. \(U=-\dfrac{a}{v}\), where the coefficient

\[ a=-2\pi N^2\int_d^\infty \bar u_r r^2\,dr \tag{13a} \]

is an essentially positive quantity, in view of the negative value of \(\bar u_r\) [cf. formula (2)].

Formula (13) does not take into account the circumstance that the “external” molecules, owing to their incomplete surroundings—that is, to the absence or deficiency of nearest neighbors on the outside (upon which, chiefly, the energy of each molecule depends)—possess an energy somewhat smaller in absolute value, i.e. greater algebraically, than the internal ones. In the case of rarefied bodies this circumstance may be disregarded. It has, however, substantial importance in the case of liquids, since it determines the surface tension of the latter.

Fig. 2.

Let us imagine the surface of the liquid under consideration in the form of a plane \(AA_1\) (Fig. 2), passing through the centers of the outermost molecules1. In calculating the energy of one of these molecules, for example with its center at the point \(P\), from the integral

\[ 4\pi\int_d^\infty \bar u_r r^2\,dr=u_1 \]

one must, evidently, subtract that part \(u_1'\) which corresponds to the external space.

Let us imagine two cones with axis \(PE\), perpendicular to \(AA_1\), and with generators inclined to it at the angles \(\theta\) and \(\theta+d\theta\). The volume

of a spherical layer of radius \(r\) and thickness \(dr\), enclosed between these cones, is equal to \(2\pi r^2\sin\Theta\,d\Theta\,dr\), and the corresponding part of the integral \(u\):

\[ 2\pi\sin\Theta\,d\Theta\int_{r_0}^{\infty}u_r r^2dr, \]

where the lower limit \(r_0\) depends on the “depth” \(P\), i.e. on the distance \(PC=h\) and on the magnitude of the angle \(\Theta\). Namely, for \(h>d\),

\[ r_0=\frac{h}{\cos\Theta} \]

for all values of \(\Theta\) from 0 to \(\pi\); but if \(h<d\) (as is shown in Fig. 2), then \(r_0=\dfrac{h}{\cos\Theta}\) only for \(\Theta>\Theta_0\), where \(\Theta_0\) is determined by the equality \(d=\dfrac{h}{\cos\Theta_0}\), while for \(\Theta<\Theta_0\), \(r_0=d\), independently of \(\Theta\). Putting \(\cos\Theta=x\) and, consequently, \(\sin\Theta\,d\Theta=-dx\), we obtain in the first case

\[ u'_1=2\pi u\int_{-1}^{1}dx\int_{\frac{h}{x}}^{\infty}u_r r^2dr; \]

and in the second

\[ u'_1=2\pi u\left[\int_{0}^{\frac{h}{d}}dx\int_{\frac{h}{x}}^{\infty}u_r r^2dr+\int_{\frac{h}{d}}^{1}dx\int_{d}^{\infty}u_r r^2dr\right] =2\pi u\int_{0}^{\frac{h}{d}}dx\int_{\frac{h}{x}}^{\infty}u_r r^2dr+\frac{1}{2}u_1\left(1-\frac{h}{d}\right). \]

Multiplying these expressions by half the number of molecules \(\left(\dfrac{1}{2}u\,dh\right)\) contained in a layer of thickness \(dh\) of a vertical column with cross-section \(1\ \mathrm{cm}^2\), and integrating the first of them from \(d\) to \(\infty\), and the second from 0 to \(d\), we obtain the negative value of the excess energy of the external molecules, referred to unit surface area. Thus this excess energy, numerically equal to the surface tension of the liquid, is determined by the formula

\[ \sigma = \pi u^2\left[ \frac{d}{2}\int_{d}^{\infty}u_r r^2dr + \int_{0}^{d}dh\int_{0}^{\frac{h}{d}}dx\int_{\frac{h}{x}}^{\infty}u_r r^2dr + \int_{d}^{\infty}dh\int_{0}^{1}dx\int_{\frac{h}{x}}^{\infty}u_r r^2dr \right]. \tag{13b} \]

As for the calculation of \(u_r\), it is reduced, as formulas (13) and (13a) show, to the determination of the mean values of expressions of the form \(\cos^2\Theta_1\) and \(\cos^4\Theta_1\). The mean value of any function of the angle \(\Theta\) is determined by the following formula

\[ \overline{F(\Theta)} = \frac{1}{4\pi}\int_{0}^{\pi}F(\Theta)\cdot 2\pi\sin\Theta\,d\Theta = \frac{1}{2}\int_{0}^{\pi}F(\Theta)\sin\Theta\,d\Theta, \tag{14} \]

which, for \(F(\Theta)=\cos^{2n}\Theta\), gives

\[ \overline{\cos^{2n}\Theta}=\frac{1}{2n+1}. \tag{14a} \]

i.e.

\[ \overline{\cos^2\Theta}=\frac{1}{3},\qquad \overline{\cos^4\Theta}=\frac{1}{5}, \]

and so on.

According to formula (12), the mean energy \(\bar u_r\) of two mutually polarized molecules of dipole character with identical moments \(\mu\) is equal to \(-\alpha \dfrac{\mu^2}{r^6}(1+3\cos^2\theta)\), i.e., consequently,

\[ \bar u_r=-\frac{2\alpha\mu^2}{r^6}. \tag{15} \]

Substituting this expression into (13a) and (13b), we obtain

\[ a=\frac{4\pi}{3}\frac{\alpha\mu^2 N^2}{d^3} \tag{15a} \]

\[ \varepsilon=\frac{\pi}{2}\frac{\alpha\mu^2 n^2}{d^2}. \tag{15b} \]

In an analogous way, in the case of (symmetric) quadrupole molecules with moment \(z\), we have

\[ \bar u_r=-\frac{3\alpha z^2}{r^8}, \tag{15c} \]

whence it follows that

\[ a=\frac{6\pi}{5}\frac{\alpha z^2 N^2}{d^5} \tag{15a} \]

and

\[ \varepsilon=\frac{3\pi}{8}\frac{\alpha z^2 n^2}{d^4}. \tag{15b} \]

§ 7. Orientational forces (Keesom’s theory). In the preceding paragraph we, following Debye, completely ignored that part of the potential energy of molecules which depends on their orientation. Turning to the consideration of this orientational energy, we shall first, following Keesom, take no account at all of the polarization energy, i.e. we shall assume that the coefficient of polarization of the molecules \(\alpha=0\).

If the orientation of each molecule were completely independent of the orientation of the others, then the mean value of the potential energy of each pair of molecules, and hence also of the whole system formed by them, would be equal to zero. In reality, as was already indicated in § 1, neighboring molecules—and, to a certain extent, more distant ones as well—mutually orient themselves in such a way that the attracting elements in them draw closer together, while the repelling ones move farther apart (respectively turning toward one another or turning away in opposite directions). But such an orientation, ensuring that the forces of attraction possibly predominate over the forces of repulsion, i.e. a possibly more intense attraction between the interacting molecules, corresponds, evidently, to a minimum of their mutual potential energy. To this str-

to a minimum of the potential energy is opposed, however, by thermal motion, translational and rotational, which continuously disturbs any more or less ordered arrangement of the molecules. Nevertheless, the probability of such arrangements and, in particular, of such orientations of interacting molecules as correspond to algebraically small values of the potential energy proves to be increased, while the probability of orientations corresponding to large values of the energy is diminished in comparison with the “geometrical” probability of these orientations, which is determined by the equivalence of all directions in space, quite independently of any dynamical conditions.

If a given arrangement or orientation of molecules is determined not by the forces of interaction, but by some forces of external origin, for example by gravity or by an external electric field, then the ratio of the “physical” probability of different positions or orientations to the “geometrical” probability (in the sense indicated above) is expressed, according to the fundamental theorem of statistical mechanics, by a function of the form

\[ C \cdot F(u)=C e^{-\frac{u}{kT}}, \tag{16} \]

where \(u\) is the potential energy of the molecule under consideration with respect to the source of the external forces acting upon it, \(T\) is the absolute temperature, \(k\) is the constant of molecular energy \(\left(\frac{1}{2}kT\right.\) is the mean value of the kinetic energy corresponding to one degree of freedom), and, finally, \(C\) is a coefficient of proportionality. If \(f\) is some function of the various quantities (coordinates, angles) determining the position or orientation of the molecule, then its mean value, actually observed when different molecules are compared with one another (or one and the same molecule at different moments of time), is expressed by the formula

\[ [f]=C\overline{fF(u)}. \tag{16a} \]

Here the symbol \(\overline{f}\) (without square brackets) denotes the “geometrical” mean, computed on the assumption of equal probability of all possible orientations and positions. Putting \(f=u\) in (16a), we obtain the true, or “physical,” mean value of the energy

\[ [u]=C\overline{uF(u)}. \tag{16b} \]

To determine the coefficient \(C\), let us suppose that \(f=1\). In that case, obviously, \(\overline{f}\) and \([f]\) are also equal to 1. Hence it follows that \(C\overline{F(u)}=1\), i.e., that

\[ C=\frac{1}{\overline{F(u)}}. \tag{16c} \]

In the case where external forces are absent, and the orientation or position of the various molecules is determined by internal forces (i.e., by the forces of their interaction with one another), the determination of the “physical” probability of the various states presents considerable difficulties. From a fundamental point of view the question is resolved by the same formula (16), in which, however, by the quantity \(u\) one understands the potential energy of the molecule under consideration with respect to all the others.

The energy of any “rigid” molecule is determined, as we know (§ 5), by the total value of the electric potential and of its derivatives at the corresponding point. Since this potential is the sum of parts depending on each of the other molecules taken separately, the orientational energy of the molecule under consideration is exactly (and not only on the average, as is the polarization energy) equal to the sum of its energies with respect to each of the others, taken separately, in the corresponding position and orientation. Under such circumstances, in order to determine the total potential energy of any gaseous or liquid body, depending on the mutual orientation of the molecules, one could use formula (13) of the preceding section, replacing in it the geometrical mean \(\overline{u}_r\) by the quantity \(|\overline{u}_r|\), i.e. by the “physically averaged” value of the mutual energy of two (rigid) molecules situated at a given distance \((r)\) from one another (see, however, below).

The exact calculation of the quantity \((|\overline{u}_r|)\) presents great difficulties, since the physical probability of one or another orientation of any two molecules depends, generally speaking, on their energy not only with respect to each other, but also with respect to all the other molecules with which they are in more or less intense interaction. In the case of gases, where this interaction attains appreciable intensity only between individual pairs of molecules (when they collide with one another and, in general, only at exceptionally close approach), one may assume with a sufficient degree of accuracy1

\[ [\overline{u}_r]=C_r \cdot u_r F(u_r)-\frac{\overline{u_r F'(u)}}{F(u_r)} \tag{17} \]

[cf. formulas (16b) and (16c)]. However, in the case of a liquid as well this formula should correctly convey, at least, the order of magnitude of \(|\overline{u}_r|\), and also the general character of its dependence on temperature.

Since the integral \(\displaystyle \int_d^\infty |u_r|r^2\,dr\) cannot be taken in finite form, for its approximate calculation one may expand the function \(F(u_r)=e^{-\,u_r/kT}\) in powers of \(u_r/kT\).

Thus, in the first approximation we obtain

\[ |\overline{u_r}|=-\,\frac{\overline{u}_r^{\,2}}{kT}, \tag{17a} \]

and, consequently, \(U=-\dfrac{a}{v}\), where

\[ a=\frac{2\pi N^2}{kT}\int_d^\infty \overline{u}_r^{\,2}r^2\,dr. \tag{17b} \]

With the same degree of approximation one may also express the surface tension of liquids, if in (13b) \(\overline{u}_r\) is replaced by \(|\overline{u}_r|\). It is clear that in this case the approximation under consideration must be of a very crude character.

In deriving formula (13), which serves, so to speak, as a model for (17a), we assumed that the mean number of molecules surrounding a given one at a distance lying between \(r\) and \(r+dr\) is equal to \(n\,4\pi r^2dr\), where \(n\) is the mean number of molecules in unit volume, i.e. a constant quantity. Meanwhile, owing to the mutual attraction of molecules (since it predominates over repulsion), this number in the immediate vicinity of each of them must be somewhat greater than at more considerable distances. In other words, the physical probability of very small distances must be somewhat greater than the geometrical probability (and of large distances, correspondingly, somewhat smaller).

The ratio of these probabilities is expressed, with the same degree of approximation as in the case of formula (17), by the function \(C\cdot F(u_r)\), where \(C\) is a coefficient of proportionality independent of \(r\). Thus, the mean number of molecules contained in a spherical layer of radius \(r\) and thickness \(dr\), surrounding the given one, is equal to \(C\cdot \overline{F(u_r)}\cdot n\,4\pi r^2dr\). Multiplying this expression by

\[ |u_r|=\frac{\overline{u_rF(u_r)}}{\overline{F(u_r)}} \]

and integrating from \(d\) to \(\infty\), we obtain the energy of one molecule, and, multiplying by \(\dfrac{N}{2}\), the energy of the whole system (gas or liquid)

\[ U=\frac{2\pi N^2C}{v}\int_d^\infty \overline{u_rF(u_r)}\,r^2dr. \]

As for the coefficient \(C\), as is not difficult to see, it must be very close to 1. Indeed, the integral \(\int_d^\infty [C F(u_r)-1]\, n\,4\pi r^2 dr\), representing the total “excess” of molecules around one of them (relative to the norm corresponding to \(n\) molecules per unit volume), must evidently vanish. And since the function \(\overline{F(u_r)}\) differs from 1 only for very small values of \(r\) (whereas the integration is carried out to \(\infty\)), this is possible only if the constant \(C\) is very close to 1. Putting \(U=-\dfrac{a}{v}\) and expanding \(u_r F(u_r)\) in powers of \(u_r\), we obtain the following expression for the van der Waals constant

\[ a=\frac{2\pi N^2}{kT}\int_d^\infty\left\{\overline{u_r^2}-\frac{1}{2}\frac{\overline{u_r^3}}{kT}+\frac{1}{6}\frac{\overline{u_r^4}}{(kT)^2}-\cdots\right\}r^2dr, \tag{18} \]

which, to within quantities of order \(\left(\dfrac{\overline{u_r^2}}{kT}\right)\), agrees with (17b).

The actual calculation of the constant \(a\) according to formula (17b) or (18) reduces to the determination of the quantities \(\overline{u_r}, \overline{u_r^3},\ldots\) for molecules of the type under consideration. In the case of dipolar molecules with moment \(\mu\),

\[ u_r=u^{(1,1)}=\frac{\mu^2}{r^3}(\cos\Theta_{AA'}-3\cos\Theta_A\cos\Theta_{A'}) \]

(see formula 11b). Let us imagine a spherical triangle with sides \(AB=\Theta_A\), \(A'B=\Theta_{A'}\), and \(AA'=\Theta_{AA'}\) (Fig. 3). Denoting the angle \(ARA'\) by \(\psi\), we have

\[ \cos\Theta_{AA'}=\cos\Theta_A\cos\Theta_{A'}+\sin\Theta_A\sin\Theta_{A'}\sin\psi, \]

whence

\[ \cos\Theta_{AA'}-3\cos\Theta_A\cos\Theta_{A'} =\sin\Theta_A\cos\psi-2\cos\Theta_A\cos\Theta_{A'}. \]

Raising \(u_r\) to the square and taking into account the independence of the directions \(A\) and \(A'\), we obtain

\[ \overline{u_r^2}=\frac{\mu^4}{r^6} \left(4\,\overline{\cos^2\Theta_A}\cdot\overline{\cos^2\Theta_{A'}} +\overline{\sin^2\Theta_A}\,\overline{\sin^2\Theta_{A'}}\right) \]

that is, since

\[ \overline{\cos^2\Theta}=\overline{\cos^2\Theta'}=\frac{1}{3},\quad \overline{\sin^2\Theta}=1-\overline{\cos^2\Theta}=\frac{2}{3} \quad\text{and}\quad \overline{\cos^2\psi}=\frac{1}{2}, \]

\[ \overline{u_r^2}=\frac{2}{3}\frac{\mu^4}{r^6}. \]

and, consequently, in the first approximation corresponding to formula (17),

\[ \overline{u_r}=-\frac{2}{3}\frac{\mu^4}{kT r^6}. \tag{19} \]

From this it is clear that the attractive forces caused by the mutual orientation of dipolar molecules are, roughly speaking, inversely proportional

of the 7th power of the distance, in exactly the same way as the polarization forces [see Eq. (15)]; the difference between them consists in the fact that the former decrease rapidly with increasing temperature, whereas the latter are practically independent of it.

In the same approximation, for the constant \(a\), according to Eq. (17c), the following expression is obtained

\[ a=\frac{4\pi}{9}\frac{N^{2}\mu^{4}}{kT\,d^{3}} \tag{19a} \]

and for the surface tension, by replacing in (15b) the coefficient \(a\) by \(\dfrac{\mu^{2}}{3kT}\), as follows directly from comparison of (15) and (19),

\[ \sigma=\frac{\pi}{6}\frac{n^{2}}{kT}\frac{\mu^{4}}{d^{2}}. \tag{19b} \]

In an analogous manner, in the case of symmetric molecules of quadrupolar character, whose potential energy \(u_r\) is determined by formula (10) (with \(\tau'=\tau\)), we obtain \(\overline{u_r^{\,2}}=\dfrac{14}{5}\dfrac{\tau^{4}}{r^{10}}\), or, in the first approximation,

\[ [\overline{u_r}]=-\frac{14}{5}\frac{\tau^{4}}{kT r^{10}}. \tag{20} \]

It follows from this that the orientational forces of attraction between quadrupolar molecules are inversely proportional to the 11th power of the distance and, thus, as the latter increases, decrease more rapidly than the polarization forces (which in this case, as formula (16) shows, are inversely proportional to the 9th power of the distance). Substituting (20) into (17b), we obtain the approximate value of the van der Waals constant

\[ a=\frac{4\pi}{5}\frac{N^{2}\tau^{4}}{kT\,d^{7}}, \tag{20a} \]

and, replacing in (13b) \(\overline{u_r}\) by \([\overline{u_r}]\), the surface tension

\[ \sigma=\frac{7\pi}{30}\frac{n^{2}}{kT}\frac{\tau^{4}}{d^{6}}. \tag{20b} \]

We shall not dwell on a more exact determination of the quantity \(\sigma\), which pertains exclusively to the liquid state, for which the initial formula (21) represents a very crude approximation. As for the quantity \(a\), in the case of gaseous bodies it can be calculated with an incomparably greater degree of accuracy than that corresponding to formulas (19a) and (20a), if, in doing so, one makes use of a sufficient number of terms

in the expansion of the function \(u_rF(u_r)\) in powers of \(u_r\) [see formula (18)]. It is not difficult to see that, in the case of dipolar gases, the mean values \(u_r^n\) vanish for all odd powers of \(u_r\) \((n=1,3,5,\ldots)\), whereas in the case of quadrupolar gases the equality \(u_r^n=0\) holds only for \(n=1\). Thus, in the first case the dependence of the “van der Waals constant” on temperature—since the cohesive forces are due exclusively to the mutual orientation of the molecules—is expressed by a series arranged in odd powers of \(\frac{1}{T}\), or, more precisely, of the ratio \(\frac{T_0}{T}\), where \(T_0=\frac{\mu^2}{kd^3}\) is a quantity playing the role of a kind of characteristic temperature. In the case of quadrupolar gases, the expansion of \(a\) contains both odd and even powers of the ratio \(\frac{T_0}{T}\), apart from the zeroth, i.e. apart from the constant term; here the characteristic temperature is expressed by the formula \(T_0=\frac{\tau^2}{kd^5}\).

Since, however, the cohesive forces are due not only to the mutual orientation of the molecules but also to their mutual polarization, the complete expression for the coefficient \(a\) must, in both cases, contain a constant term determined by formulas (18a) and (19a), as well as additional terms proportional to even powers of the ratio \(\frac{T_0}{T}\). Thus, for example, taking into account both orientational and polarization forces, Falkenhagen\(^{1)}\) obtains for the virial coefficient \(B=b-\frac{a}{RT}\) of dipolar gases the following series

\[ B=b\left[1-\beta_1\left(\frac{T_0}{T}\right)-\beta_2\left(\frac{T_0}{T}\right)^2-\beta_3\left(\frac{T_0}{T}\right)^3-\ldots\right], \tag{21} \]

where the coefficients \(\beta_i\) are polynomials arranged in powers of the ratio \(\frac{a}{d^3}=z\) and determined by the formulas

\[ \beta_1=42,\quad \beta_2=0{,}333+3{,}27z^2,\quad \beta_3=z(0{,}533+4{,}93z^2), \]

\[ \beta_4=0{,}013+0{,}92z^2+9{,}47z^4,\quad \ldots \]

Analogous formulas are given by Rees\(^{2)}\) for quadrupolar gases. Let us note that \(z\) is an abstract number, close in most cases to

\[ \frac{1}{10}. \]

\(^{1)}\) Kohäsion u. Zustandsgleichung d. Dipolgazen, Phys. ZS., 23, 87 (1922).

\(^{2)}\) Die Van-der-Waalsschen Kohäsionskräfte, Phys. ZS., 22, 129 (1921).

If the orientational forces are neglected, and one confines oneself only to the polarization forces, as is done, for example, by Debye’s pupil Zwicky in his investigation of the noble gases1, then in the case of quadrupolar molecules the virial coefficient \(B\) is still given by the series (21), where the characteristic temperature is expressed by the formula \(T_0=\dfrac{3a^{-2}}{kd^8}\), and the coefficients \(\beta_i\) reduce to constant numbers

\[ \left(\beta_1=\frac{3}{5},\ \beta_2=\frac{3}{26},\ \beta_3=-\frac{3}{126},\ldots\right) \]

IV. COMPARISON OF THE THEORY WITH EXPERIMENT.

§ 8. Dipolar substances. A comparison of the theory set forth above with experimental data can be carried out in full measure only in the case of dipolar substances, for which the quantities \(\mu\) and \(a\), i.e. the electric moments of the molecules and their polarization coefficients, are determined directly from the dielectric constant \(\varepsilon\) and the refractive index \(\gamma\). A distinguishing feature of dipolar substances is the extraordinarily large value of the dielectric constant both for static fields and for comparatively slow electrical oscillations. At the same time, their refractive index with respect to rapid (light) oscillations proves to be considerably smaller than the value \((\sqrt{\varepsilon})\) which follows from the electromagnetic theory of light. Thus, for example, for water \(\varepsilon=80\), whereas \(\gamma\) (in the case of visible rays) does not exceed 1.7.

The exceptionally large value of the dielectric constant of such substances is closely connected with the sharp dependence of the latter on temperature, with an increase of which the dielectric constant rapidly decreases. These peculiarities were first noted and explained by Debye2, who reduced them to the dipolar character of the molecules of the corresponding substances.

When placed in an electric field (of external origin), such molecules tend to orient themselves in the direction of the lines of force. This tendency is counteracted by thermal motion. Nevertheless, the “physical” probability of positive orientations proves to be increased, and that of negative ones decreased, in comparison with the geometrical probability, in the ratio already known to us

\[ C.F(u)=C.e^{-\frac{u}{kT}} \]

[cf. formula (16)].

It was shown above that the potential energy of a dipolar molecule with respect to an external field \(E\) is expressed, if its polarization is not taken into account, by the formula

\[ u=-\mu E\cos\vartheta, \]

where \(\theta\) is the angle between the direction of \(E\) and the axis of the molecule, i.e. the vector \(\mu\). Thus the mean value of the component of this vector in the direction of the lines of force \((\mu\cos\theta)\) must be positive.

Denoting it as before by the symbol \(\mu[\overline{\cos\theta}]\) and recalling that \(C=\dfrac{1}{F(u)}\), we have

\[ \mu[\overline{\cos\theta}] =\mu\,\frac{\overline{F(u)\cos\theta}}{F(u)}. \tag{22} \]

Expanding \(F(u)\) in a series in powers of \(\dfrac{u}{kT}\) and using formula (14a) for the mean values of \(\cos^{2n}\theta\), we obtain

\[ \mu[\overline{\cos\theta}] =\frac{1}{3}\rho\, \frac{1+\dfrac{1}{10}\rho^2+\cdots}{1+\dfrac{1}{6}\rho^2+\cdots} =\frac{1}{3}\rho\left(1-\frac{1}{15}\rho^2\right) \tag{22a} \]

where

\[ \rho=\frac{\mu E}{kT}. \tag{22b} \]

Multiplying these expressions by the number of molecules in unit volume \((n)\), we obtain the electric polarization of the body under consideration, \(P\). This polarization, equal to the resultant electric moment of unit volume, is related to the dielectric constant \(\varepsilon\) by the formula

\[ \varepsilon=1+\frac{4\pi P}{E'}, \tag{23} \]

where \(E'\) is the strength of the external field, which is obtained in the void, i.e. when the body under consideration is removed. In the case of gaseous bodies one may, with a sufficient degree of accuracy, put \(E'=E\). In the general case, as Lorentz showed, to this primary field there is added a secondary one, depending on the polarization of the medium and equal approximately to \(\dfrac{4\pi}{3}P\). Thus

\[ E=E'+\frac{4\pi}{3}P. \]

Eliminating \(E'\) from this equality and (23), we have

\[ \frac{P}{E}=\frac{3}{4\pi}\,\frac{\varepsilon-1}{\varepsilon+2}. \tag{23a} \]

The polarization \(P\) is in fact composed of the above-mentioned quantity \(n\mu[\overline{\cos\theta}]\), which depends on the orientation of the molecule, and of an additional,

Ya. I. Frenkel

of moment \(n\Delta\mu = n\alpha E\), acquired by the latter as a result of their polarization, which until now we have not taken into account. Thus we obtain the following relation

\[ a+\frac{\mu}{3kT}\left[1-\frac{1}{15}\left(\frac{\mu E}{kT}\right)^2\right] =\frac{3}{4\pi n}\frac{\varepsilon-1}{\varepsilon+2}. \tag{23b} \]

Knowing the dependence of the dielectric constant on the field strength and, in particular, on the temperature, it is not difficult to determine separately both quantities \(a\) and \(\mu\). The same result can be arrived at still more simply by comparing the dielectric constant in the case of a static field with the refractive index for visible rays, i.e. for very rapid electrical oscillations. In the latter case the molecules do not have time to align themselves in the direction of the electric field; their orientation remains completely disordered, and the dielectric constant, equal to the square of the refractive index \(\gamma\), is determined exclusively by their polarization. Therefore, putting

\[ a=\frac{3}{4\pi n}\frac{\gamma^2-1}{\gamma^2+1}. \tag{23c} \]

one can, with the aid of (23b), directly determine \(\mu\).

The results of such measurements, carried out especially carefully by Jona1 for a whole series of dipolar substances, are given in the following table:

Substance.	\(\mu\cdot 10^{18}\)	\(a\cdot 10^{24}\)	\(d\cdot 10^8\)
\(\mathrm{SO_2}\)	1,76	3,33	3,00
\(\mathrm{NH_3}\)	1,57	2,06	2,74
\(\mathrm{H_2O}\)	1,87	1,41	2,72
\(\mathrm{HCl}\)	2,15	—	—
\(\mathrm{CO_2}\)	0,142	—	4,1
\(\mathrm{CO}\)	0,118	—	3,5

The values of the molecular diameter \(d\), indicated in the last column, are calculated for the corresponding substances from the formulas of the kinetic theory of gases from the internal friction or the van der Waals coefficient \(b\), equal, as is known, to four times the volume of the molecules, i.e. \(\dfrac{2\pi}{3}d^3N\). It should be noted that they cannot lay claim to great accuracy.

Using the data given above, Falkenhagen calculated, by formula (21), the value of the virial coefficient \(B=b-\dfrac{a}{RT}\) for a gram-molecule of various gases and a whole series of temperatures. In the case of gases with a strongly expressed dipole character, i.e. with a large electric moment \((\mathrm{SO}_2,\ \mathrm{NH}_3,\ \mathrm{H}_2\mathrm{O})\), the calculated values are in good agreement with the experimental ones (the latter taken from the measurements of Kamerlingh-Onnes). The degree of this agreement may be judged from the following figures.

Sulfurous anhydride \(\mathrm{SO}_2\)

	331	373	456
Abs. temp. \((T)\)	331	373	456
\(B\) (observed)	0,0138	0,0085	0,0048
\(-B\) (calculated)	0,0151	0,0093	0,0045

Ammonia gas \(\mathrm{NH}_3\)

	273	297	313	373
Abs. temp. \((T)\)	273	297	313	373
\(-B\) (observed)	0,0149	0,0110	0,0095	0,0047
\(-B\) (calculated)	0,0152	0,0095	0,0085	0,0043

The results relating to water vapor are presented graphically in the appended diagram (Fig. 4); here the solid curve corresponds to the observed values, and the dashed curve to the calculated ones.

Fig. 4.

As for substances with a weakly expressed dipole character (for example, carbon monoxide or carbon dioxide), the above-mentioned formulas turn out to be wholly inapplicable to them. This discrepancy is explained by the circumstance that, in view of the smallness of the dipole moments of the corresponding molecules, the predominant importance is acquired by

moments quadrupole. And indeed, if the molecules CO and CO\(_2\) are treated as quadrupoles, then for their moments (\(\tau\)) values of the proper order of magnitude are obtained (see below).

It is curious that in the case of “true” dipolar substances, such as water, alcohol, various kinds of salts (in the molten state), etc., the approximate formulas (19a) and (19b) give, for the van der Waals constant \(a\), as well as for the latent heat of vaporization and for the surface tension \(\sigma\), values fairly close to the experimental ones and, in any case, agreeing with them in order of magnitude.

Thus, for example, for one gram of water \(\left(N=\dfrac{6.06\cdot10^{23}}{18}\right.\), where \(6.06\cdot10^{23}\) is the number of molecules in a gram-molecule), at a temperature of \(17^\circ\)C, i.e. \(T=290\) \(\left(kT=1.39\cdot10^{-16}\cdot290\right)\), we obtain

\[ a=\frac{4\pi N^2\mu^2}{9\,kT\,d^3}=2\cdot10^{10}\ C.G.S \]

(approximately). Recalling that the latent heat of vaporization is equal to \(\dfrac{a}{v}\), and that in the present case \(v=1\), we may identify the preceding number with the latent heat of vaporization of water in ergs. Converting into calories (the small calorie \(=4.2\cdot10^7\) ergs), we obtain \(\dfrac{2\cdot10^{10}}{4.2\cdot10^7}\), or about 500 calories, i.e. a number very close to the experimental one. This agreement shows that the forces of cohesion in water are predominantly of orientational origin. Indeed, formula (15a), corresponding to forces of a polarization character, gives for the quantity \(a\) a value which is to the preceding one as \(\alpha\) is to \(\dfrac{\mu^2}{3kT}\), i.e. approximately 20 times smaller.

Putting, according to formula (19b),

\[ \sigma=\frac{\pi}{6}\frac{n^2}{kT}\frac{\mu^4}{d^2} =\frac{3}{8}\left(\frac{n}{N}\right)^2 ad, \]

we obtain for the surface tension of water about 200 dynes per cm, instead of 75. Bearing in mind the approximate character of formulas (19a) and (19b), especially as applied to liquids, such a discrepancy can by no means be considered excessive.

The scarcity and inaccuracy of the experimental data concerning the electric moments and molecular dimensions of various dipolar substances do not allow the above theory to be tested on more extensive experimental material. The preceding examples, however, are quite sufficient to convince us of the correctness of those ideas concerning the nature of the forces of molecular cohesion on which the formulas given above are based.

§ 9. Quadrupolar substances. In the case of quadrupolar substances, a quantitative test of the theory, in the strict sense of the word, is impossible, since the moments of quadrupolar molecules cannot be determined either from the dielectric constant or from any other

...there were other properties—besides those which are directly connected with the forces of molecular cohesion.

Comparing the theoretical formula (21) for the virial coefficient \(B\) with its experimental values for different temperatures, it is not difficult to determine the characteristic temperature \(T_0\), which is the only unknown parameter of formula (21), if the molecular diameter \(d\) and the coefficient of polarization \(a\) are known; however, in the general case one may also calculate these quantities, determining the two other parameters of formula (21), namely \(b=\dfrac{2\pi}{3}d^3N\) and

\[ z=\frac{a}{d^3}. \]

The first calculations of this kind were made by Keesom1, who, however, took no account at all of the mutual polarization of the molecules, i.e. treated them as rigid quadrupoles (\(a=0\)). In doing so, as a kind of reference points, Keesom used the following two temperatures: the “Boyle point” \((T_B)\), at which the virial coefficient \(B\) becomes zero, passing from negative values to positive ones (i.e., at which the corresponding gas obeys Boyle’s law \(\dfrac{pv}{kT}=1\)), and the “Joule–Thomson inversion point” \((T_i)\), defined by the equality \(B-T\dfrac{dB}{dT}=0\)2. If, for some reason, these temperatures cannot be measured directly, then, knowing the empirical dependence of \(B\) on \(T\), they are not difficult to determine by extrapolation. Without dwelling on the details (which are of no physical interest), we shall give only the results relating to oxygen and nitrogen:

In the first case Keesom obtains \(d=2.65\cdot10^{-8}\ \mathrm{cm}\) and \(\tau=3.55\cdot10^{-26}\) (e.s.u.), and in the second \(d=2.98\cdot10^{-8}\ \mathrm{cm}\) and \(\tau=3.86\cdot10^{-26}\). Recalling that the charge of the electron is equal to \(4.77\cdot10^{-10}\), and taking into account that molecular dimensions are measured in several angstroms \((10^{-8}\ \mathrm{cm})\), we see that the quadrupole moments calculated by Keesom have the proper order of magnitude. As for the dependence of \(B\) on temperature, it also turns out, in general, to be in full agreement with experiment.

In contrast to Keesom, Debye, in the work cited above, calculated the quadrupole moments for a whole series of substances, starting from the assumption that the cohesion forces are due chiefly to the mutual polarization of the molecules, whereas their

mutual orientation (in the absence of a dipole moment) plays a secondary role. Using the simplified theory, which completely neglects the influence of temperature (§ 7), i.e. formulas (16a) and (19b), Debye obtained for the moments of the oxygen and nitrogen molecules figures approximately 3–4 times larger than Keesom (namely, \(11.2 \cdot 10^{-26}\) in the first case and \(13.3 \cdot 10^{-26}\) in the second). This circumstance shows that Keesom’s point of view is closer to the truth than Debye’s point of view. Indeed, if the oxygen and nitrogen molecules possessed those moments which follow from Debye’s theory, then the cohesion forces due to their mutual orientation and proportional to the fourth power of \(\tau\) would be, approximately, 100 times greater than the polarization forces. Conversely, the polarization forces corresponding to those values of \(\tau\) which follow from Keesom’s theory, in view of their proportionality to the square of \(\tau\), must be approximately 10 times weaker than the orientational forces (we note that for oxygen the parameter \(z=\dfrac{a}{d^3}\) is equal to 0.0842, and for nitrogen to 0.0646).

It goes without saying that at sufficiently high temperatures the Keesom forces must, in the end, yield the chief place to the Debye forces; however, these temperatures, even if they are not beyond attainability, are in any case many times higher than the temperatures corresponding to ordinary conditions for observing gaseous and liquid bodies.

An exception in this respect should be made only by monatomic bodies—metallic vapors and noble gases. As was already indicated in § 3, the cohesion forces in such bodies must be due exclusively to the mutual polarization of the atoms, i.e. be determined entirely by Debye’s theory. Taking into account the influence of temperature, Zwicky (see the end of § 9), by means of a method analogous to Keesom’s method, obtained for a gram-atom of argon

\[ T_0=\frac{3\alpha \tau^2}{kd^8}=540 \]

and

\[ b=\frac{2\pi}{3}d^3N=18.2, \]

whence \(d=2.43 \cdot 10^{-8}\) and \(\tau=1.05 \cdot 10^{-26}\). In this case, for the virial coefficient of argon, one obtains a temperature dependence very close to the experimental one.

Assuming that the atoms of argon, in view of their very symmetrical structure, are equivalent in electrical respect not to quadrupoles, but to poles of the 4th order, Zwicky obtained results sharply divergent (as regards the dependence of \(B\) on \(T\)) from the experimental ones. Thus, the atoms of argon must possess a more or less pronounced axial symmetry, which to a certain degree agrees with Bohr’s latest ideas on the structure of atoms.

1. See in particular his last paper, Phys. ZS. 22, p. 121 (1921). ↩↩↩↩↩↩↩↩

2. The Joule–Thomson effect is, as is well known, the change in the temperature of a gas as it passes through a porous plug and undergoes a more or less considerable decrease in pressure. This change in temperature is usually negative; however, it may also be positive at the inversion temperature \(=0\). ↩↩↩

3. Thus, for example, in crystals of common salt the atoms, or, more accurately, the ions, of sodium and chlorine are arranged in a chessboard order, so that, by joining them pairwise, we obtain a set of dipolar molecules NaCl, turned toward one another with their opposite poles. ↩