B. Widom

It is shown how certain thermodynamic functions, and also the radial distribution function, can be expressed in terms of the potential energy distribution in a fluid. A miscellany of results is derived from this unified point of view. (i) With g(r) the radial distribution function and Φ(r) the pair potential, it is shown that g exp (Φ/kT) may be written as a Fourier integral, or as a power series in r2 the terms of which alternate in sign. (ii) A potential-energy distribution which is independent of the temperature implies an equation of state which is a generalization of a number of well-known approximations. (iii) The grand partition function of the one-dimensional lattice gas is obtained from thermodynamic arguments without evaluating a sum over states. (iv) If in a two-dimensional honeycomb (three-coordinates) lattice gas fr(r=0, 1, 2, 3) is the fraction of all the empty sites which at equilibrium are neighbored by exactly r filled sites, then at the critical density the values of all four of the f's as functions of temperature follow from previously known properties of this system; in particular, at the critical point, f0 = 3/8+5√3/24, f1 = 1/8+√3/24, f2 = 1/8—√3/24, f3 = 3/8–5√3/24.

History of thought on molecular origins of surface phenomena offers a critical and detailed examination and assessment of modern theories, focusing on statistical mechanics and application of results in mean-field approximation to model systems. Emphasis on liquid-gas surface, with a focus on liquid-liquid surfaces in the final chapters. 1989 edition.

A specific form is proposed for the equation of state of a fluid near its critical point. A function Φ(x, y) is introduced, with x a measure of the temperature and y of the density. Fluids obeying an equation of state of van der Waals type (``classical'' fluids) are characterized by Φ being a constant. It is suggested that in a real fluid Φ(x, y) is a homogeneous function of x and y, with a positive degree of homogeneity (Sec. 2). This leads to a nonclassical compressibility, the behavior of which is determined by the degree of homogeneity of Φ (Sec. 3). A previously derived relation connecting the degree of the critical isotherm, the degree of the coexistence curve, and the compressibility index, again follows, this time without the restrictive assumption of effective isochore linearity (Sec. 4). The locus in the temperature—density plane of the points of inflection in the pressure—density isotherms, as determined experimentally by Habgood and Schneider, is accounted for (Sec. 5). It is shown that if a certain combination of the compressibility and coexistence curve indices is an integer, then the constant-volume specific heat on the critical isochore has a logarithmic singularity at the critical temperature with, in general, a superimposed finite discontinuity (Sec. 6).

The van der Waals, Cahn—Hilliard theory of interfacial tension is reformulated for a fluid in the neighborhood of its critical point. The reformulated theory becomes equivalent to the Ornstein—Zernike, Debye theory of molecular correlations when the interface thickness is identified with the correlation length. When account is taken of the nonclassical behavior of the compressibility and coexistence curve, the theory is found to be in good agreement with independently known facts in three-dimensional systems, yet slightly but unambiguously wrong in two dimensions. When one of the hypotheses of the original theory is replaced by an alternative hypothesis, the resulting theory is found to be correct in both two and three dimensions.

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) − Nυ0]ε/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.