On the equation of state of a classical fluid

Abstract

A rigorous cumulant theory is developed for deriving the Van der Waals equation of state in a direct way in the Van der Waals limit for a system with a hard-sphere potential and a long-range attractive potential; thereby the role played by the Van der Waals limit is cleared out. A linear- potential model is examined in comparison with the exponential potential model. The cases with the Van der Waals attractive potential and other potentials are also investigated. From the rigorous cumulant formula for the free energy, upper and lower free-energy bounds are derived. These bounds agree with what have been derived by the author by a different method. They coincide with each other in the Van der Waals limit to yield the free energy corresponding to the Van der Waals equation of state. This conclusion agrees with Lebowitz and Penrose's, but for certain potentials the radial distribution function of the reference system is needed in the free- energy expression. Considerations are given of the virial type equation of state and the pair distribution function. The uniqueness of the Van der Waals limits in the derivation of a Van der Waals type equation in general and the role played by the limit in the random phase approximation are pointed out.