Generic van der Waals equation of state and statistical mechanical representations of the van der Waals parameters

In this paper, we show that in the case of the potential made up of a repulsive and an attractive part the virial equation of state can be put into a form similar to the van der Waals equation of state and thus the form of the van der Waals equation of state is generic to such a class of potentials. The derivation provides exact statistical mechanical representations for the van der Waals parameters. The generic van der Waals parameters are evaluated as functions of density and temperature by using the Percus-Yevick integral equation for the pair correlation function in the case of a square-well potential. They become disjointed functions of density, which are not defined in a density interval in the subcritical regime, if the temperature is less than the critical temperature. On the basis of the numerical solution results for the parameters we conjecture that they are irrational functions of density and thus nonanalytic.