Abstract

Billiard balls on a table are taken as the model for the adsorption of spherical molecules. In the billiard model, spheres have a finite volume, translational energy and can move freely on the surface. The free-area function is defined as the fraction of area available for a new sphere to adsorb on a surface where other spheres are already present. The free-area function of the billiard model is obtained from simulations. Its adsorption isotherm, which differs from the Langmuir isotherm, as well as the surface pressure are deduced from statistical mechanics. The third virial coefficient of the surface pressure is also calculated for the billiard model. The generalization of the billiard model to three dimensions is the gas of hard spheres. The free-volume function that gives the volume available for a new sphere to be added to the system is obtained from simulations and from the virial coefficients. To deal with real gases, the hardness condition is relaxed, and soft spheres are considered and modeled with the Lennard-Jones potential according to which the exclusion radius of the spheres decreases as the temperature increases. By adding the attractive part of the second virial coefficient and introducing the temperature dependence of the exclusion radius into the equation of state of a hard-sphere gas, universal curves of the compressibility factor as a function of the reduced pressure are obtained, which are in qualitative agreement with the experimental ones.

Keywords: Billiard isotherm; Adsorption; Hard spheres; Soft spheres; Lennard-jones potential; Real gases; Third virial coefficient; Diffuse layer; Double layer

Abbreviations: IHP: Inner Helmholtz Plane; OHP: Outer Helmholtz Plane