A Monte Carlo method for the evaluation of equilibrium thermodynamic properties of a system of interacting particles is described. The classical configurational partition function at a given density is calculated from the estimated value of the weight function γ(Φ), up to a factor that is independent of temperature; γ(Φ)dΦ is the fraction of all accessible configurations of N particles confined to a fixed volume for which the total potential energy has a value between Φ and Φ+dΦ. The weight function is found to vary approximately as (Φ—Φ0)n, where Φ0 and n are constants; n increases roughly linearly with density. The method is applied to the calculation of thermodynamic properties of gaseous argon at four densities in the temperature range between −100° and 150°C. A Lennard‐Jones potential function is used and though the agreement with experiment is good there is evidence of inadequacy in the Leonnard‐Jones model. All calculations are made for N=32.

Machine Calculation of Thermodynamic Properties of a Simple Fluid at Supercritical Temperatures

The equation of state of a mixture of hard spheres of diameter σ11 and σ22 is evaluated by expanding the free energy of the mixture to second order in powers of the differences σαβ − σ, where σαβ = 12(σαα + σββ) and σ is the diameter of the unperturbed hard spheres The parameter σ is chosen to make the sum of the first‐order terms zero The second‐order terms are evaluated using the superposition approximation, and the equation of state is determined by analytic differentiation The agreement of this perturbation calculation with quasiexperimental Monte Carlo and molecular dynamics results is excellent, even for quite large values of R = σ22 / σ11, but becomes less satisfactory as R becomes very large

Perturbation Theory and the Equation of State of Mixtures of Hard Spheres

Chapter 7 – Liquid Mixtures

John Adair Barker 1925-1995

John Adair Barker 24 March 1925–27 October 1995

The pseudo pair potential theory of mixtures developed by Salsburg1 has been tested by carrying out Monte Carlo calculations for hard sphere systems of radius ratios 5 : 3 and 3 : 1 at low and moderate densities. Comparison between the results obtained by use of the pseudo-potential model with those obtained earlier by direct Monte Carlo simulation studies2, 3 for such mixtures shows that the model is not a good approximation. An explanation of the observed discrepancies is proposed.

M. Rigby

Papers

Pseudo-potential model for hard sphere mixtures