Physikalische Zeitschrift, Vol. 21, pages 178-187, 1920.
As is well known the great success of van der Waals' equation of state is based essentially on the assumption of attractive forces between the molecules. These forces cause, in addition to the external pressure, an internal pressure which is proportional to the square of the density. According to van der Waals, these forces of attraction exist between molecules of any kind, and constitute a general property of matter. It appears, therefore, of particular interest to consider the origin of this universal attraction.
Today we know with absolute certainty that the molecule is a system of electric charges, and we are led to an attempt to find an electric origin for van der Waals' forces. It will certainly be unnecessary to consider details of molecular structure. A property of matter as general as the van der Waals' attraction can require for its explanation nothing but structural features common to all molecules. We shall show in the following that it is in fact sufficient only to know that the molecules are electrical systems in which the charges are not rigidly bound to their rest positions. A relation between van der Waals' constant of attraction on the one hand and the index of refraction and the widening of spectral lines on the other hand will be derived on the basis of this hypothesis.
I. Van der Waals' Equation
We start by presenting a few known relations which will be used subsequently. . . . [N.B. Here he goes through a discussion which winds up in the common form of van der Waals' equation.]
II. Cause of the Cohesion
If we imagine the molecules to be rigid electrical systems, then there will, of course, always be a force acting between two such systems which will change sign and magnitude with their mutual orientation. Since all possible orientations occur in a gas, the average over these orientations must be taken in order to compute the attraction term appearing in the equation of state.
In general, in carrying through this averaging process, the probability of an arbitrary orientation would have to be determined on the basis of the Boltzmann-Maxwell principle. The higher the temperature, however, the less important is the dependence on the mutual energy. In the limit for high temperatures, all orientations will have to be considered equally probable. Obviously, van der Waals' assumption requires that the characteristic cohesion introduced into the equation persist in this limiting case.
It can be easily shown that two rigid electrical systems do not, in the mean, exert a force on one another. The potential which is generated at a distant point by a molecule can be considered as originating from a series of concentric spheres covered with a layer of electric charges of constant surface density. If the molecule assumes all possible orientations in space, each charge occupies, on the average, all points of a sphere with equal frequency. Since it is known that a sphere with a charge of constant surface density affects points outside of the sphere as if its total charge were concentrated at its center, and since the molecule carries no total charge, the average of the potential at the point considered will be zero. Thus, no force is effective on the average, between two rigid molecules.
The situation is immediately and essentially changed if we consider molecules that are not completely rigid. The fact that each gas has a refractive index different from unity is proof of the mobility of the separate charges in the molecule. Taking this into consideration, it will be clear that a given molecule assumes an electric moment in the field of another molecule, which moment is proportional to the field . Thereby a mutual energy arises between the two molecules which is proportional to the product of field strength times moment, i.e., proportional to the square of . Thus the average of the corresponding force cannot vanish. Further it will be readily seen that the force is always one of attraction. Hence we may conclude that we have found in this force the origin of van der Waals' universal attraction.*
*The situation can be illustrated by the following example. Two dipoles are situated opposite to each other.
(a) In a position 1. Here the main effect is repulsive. As a consequence of the action, the fleld on the elastically bound charges, the latter are displaced in such a way that the electric moments are reduced. Thereby the repulsive force decreases; in other words, an attractive force appears as a secondary effect.
(b) In position II. Here the main effect is attractive. The field strength now displaces the charges so that the moments increase. The main effect is now increased, or expressed differently: again an attractive force has been added as secondary effect.
The main effect vanishes when the average over all possible orientations is taken; the secondary effect is always positive and thus can never vanish.
We add the comment, that In terms of our concept van der Waals' force has the same origin as the forces effective In the basic experiment of electrostatics. There a charged body attracts other bodies irrespective of whether they are conductors or insulators i.e., forces appear which drive the mobile particles from places of lower to places of higher density of lines of force. Similarly the molecules are subject to forces which tend to bring the particles nearer together and thus bring them into as strong a field as possible.