New Model for the Study of Liquid–Vapor Phase Transitions

A new model is proposed for the study of the liquid–vapor phase transition. The potential energy of a given configuration of N molecules is defined by U(r1, ···, rN) = W(r1, ···, rN) − Nυ0ε/υ0≤0, where W is the volume covered by N interpenetrating spheres each of volume υ0 and each centered on one molecule, and where ε is an arbitrary positive energy. This continuum model proves to have a line of symmetry comparable with those found hitherto only in lattice models. The line is that of the diameter, or mean orthobaric density ρ = 12ρ1 + 12ρg, below the critical point, and continues through the one-phase region to infinite temperature. The existence of this line allows some of the properties to be obtained explicitly, the most unusual of which is that the diameter has a singularity comparable with that in Cυ; hence the law of the rectilinear diameter is not obeyed. An exact solution of the model is obtained in one dimension, in which there is no phase transition, and a mean-field solution in three dimensions. The latter preserves the symmetry. The model is shown to be thermodynamically equivalent to a two-component system in which the pair potential between molecules of like species is zero, while that between molecules of unlike species implies a mutually excluded volume of υ0. In this transcription the symmetry of the model becomes obvious.