Rankin and, subsequently, Recknagel, and then Hirn, formerly proposed formulas of that kind; but the most famous, the one which first appeared to contain in a satisfactory manner all the facts which experiments brought to light and led to the production of many others, was the celebrated equation of Van der Waals.
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.