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.