Generalized van der Waals theory. I. Basic formulation and application to uniform fluids

Abstract

A generalized van der Waals theory is derived on the basis of simple physical and mathematical arguments. The derivation results in a free- energy functional wherein the independent variable is a coarse-grained particle density. It is assumed that a well defined particle density dominates the free energy and this density is to be obtained by minimizing the free energy functional. The variational theory so obtained can be applied to non-uniform fluids. In the present work the possibility of stable non-uniform structure is neglected and the theory is applied to uniform fluids. It then produces an equation of state identical in form to that proposed originally by van der Waals but the excluded volume is only about half as large in the three-dimensional case. Applications to several two- and three-dimensional systems indicate that the new equation of state is a distinct improvement over the traditional van der Waals theory when the full range of fluid densities is considered. The quantitative accuracy in the case of simple uniform fluids is sufficient to warrant further development and exploitation of the theory.