Professor Van der Waals arrived at this relation by relying upon considerations derived from the kinetic theory of gases.  If we keep to the simple idea at the bottom of this theory, we at once demonstrate that the gas ought to obey the laws of Mariotte and of Gay-Lussac, so that the characteristic equation would be obtained by the statement that the product of the number which is the measure of the volume by that which is the measure of the pressure is equal to a constant coefficient multiplied by the degree of the absolute temperature.  But to get at this result we neglect two important factors.

We do not take into account, in fact, the attraction which the molecules must exercise on each other.  Now, this attraction, which is never absolutely non-existent, may become considerable when the molecules are drawn closer together; that is to say, when the compressed gaseous mass occupies a more and more restricted volume.  On the other hand, we assimilate the molecules, as a first approximation, to material points without dimensions; in the evaluation of the path traversed by each molecule no notice is taken of the fact that, at the moment of the shock, their centres of gravity are still separated by a distance equal to twice the radius of the molecule.

M. Van der Waals has sought out the modifications which must be introduced into the simple characteristic equation to bring it nearer to reality.  He extends to the case of gases the considerations by which Laplace, in his famous theory of capillarity, reduced the effect of the molecular attraction to a perpendicular pressure exercised on the surface of a liquid.  This leads him to add to the external pressure, that due to the reciprocal attractions of the gaseous particles.  On the other hand, when we attribute finite dimensions to these particles, we must give a higher value to the number of shocks produced in a given time, since the effect of these dimensions is to diminish the mean path they traverse in the time which elapses between two consecutive shocks.

The calculation thus pursued leads to our adding to the pressure in the simple equation a term which is designated the internal pressure, and which is the quotient of a constant by the square of the volume; also to our deducting from the volume a constant which is the quadruple of the total and invariable volume which the gaseous molecules would occupy did they touch one another.

The experiments fit in fairly well with the formula of Van der Waals, but considerable discrepancies occur when we extend its limits, particularly when the pressures throughout a rather wider interval are considered; so that other and rather more complex formulas, on which there is no advantage in dwelling, have been proposed, and, in certain cases, better represent the facts.