Condensation theory for finite, closed systems

Abstract

The applicability of Mayer’s cluster integral formulation of statistical mechanics in the canonical ensemble has been restricted to the gas phase by two difficulties: the enormous labor required for the calculation of cluster integrals from realistic intermolecular potentials and the divergence of the cluster integral expansions of thermodynamic observables and n-particle correlation functions prior to condensation. In this paper the divergence of the cluster expansions is shown to be due to the way in which the limit of a system of infinite size is taken, implicitly, when the partition function, written as a sum over all the cluster size distributions attainable by the closed system, is approximated by its maximum term alone. The use in condensation theory of this ’’maximum term approximation,’’ and of the particular infinite limit it entails, are obviated herein by demonstration that the full theory with use of all the terms in the partition function, can be solved conveniently and exactly by recurrence relation (or formal power series) methods. The cluster size distributions thus found, since they describe finite systems, can have no associated divergences. Because no realistic set of volume dependent cluster integrals is known, we demonstrate the new formalism and the use of the recurrence relations in the case of cluster integrals obtained in the infinite volume limit from the virial expansion determined by the van der Waals equation of state. At low densities these are seen to predict correctly the ideal, imperfect, and metastable states of the homogeneous gas phase. Beyond the spinodal density, they predict bimodal cluster size distributions distinctive of coexisting vapor and liquid; the corresponding pressure isotherms are found to be constant, rather than diverging or displaying a van der Waals loop. Because the sole approximation made is the replacement of the volume dependent cluster integrals by their infinite volume limits, this evidently is the cause of the occurrence of condensation first at the spinodal density rather than at the coexistence density.