New Model for the Study of Liquid–Vapor Phase Transitions

Abstract

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 $\mathrm{U(}{\mathrm{r}}_1\mathrm{,\; ···,}{\mathrm{r}}_N\mathrm{)\hspace{0.17em}=\hspace{0.17em}[W(}{\mathrm{r}}_1\mathrm{,\; ···,}{\mathrm{r}}_N{\mathrm{)\hspace{0.17em}-\hspace{0.17em}N\upsilon }}_0{\mathrm{]\epsilon /\upsilon }}_0\text{≤0}$ , where $W$ is the volume covered by $N$ interpenetrating spheres each of volume ${\upsilon }_0$ and each centered on one molecule, and where $\epsilon$ 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 $\mathrm{\rho \hspace{0.17em}=\hspace{0.17em}}\frac{1}{2}{\rho }_1\mathrm{\hspace{0.17em}+\hspace{0.17em}}\frac{1}{2}{\rho }_g$ , 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 $C_{\upsilon }$ ; 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 ${\upsilon }_0$ . In this transcription the symmetry of the model becomes obvious.