On the van der Waals Theory of the Vapor-Liquid Equilibrium. III. Discussion of the Critical Region

Abstract

The discussion of the properties of the Kac one‐dimensional fluid model presented in Parts I and II of this series of papers breaks down near the critical point. In Sec. II of the present paper we develop a new successive‐approximation method for the eigenvalues and eigenfunctions of the Kac integral equation which is valid in the critical region and which connects smoothly with the developments in the one‐ and two‐phase regions given in Part I. The perturbation parameter is (γδ)^½ where γδ is the ratio of the ranges of the repulsive and attractive forces. The main physical consequence is that in the critical region the long‐range behavior of the two‐point distribution function is represented by an infinite series of decreasing exponentials with ranges all of order 1/γ(γδ)^½, and with amplitudes of order (γδ)^⅔. This leads to deviations from the Ornstein‐Zernike theory and to a specific heat anomaly which are discussed in Sec. V. We conclude with some comments on the possible relevance of our results for the three‐dimensional problem.