Critical phenomena in gases. II. Vapour pressures and boiling points

TLDR

Dense gas atoms are effectively caged by neighboring atoms, a view that also underlies the van der Waals equation where potential energy scales inversely with volume. The paper aims to develop a simple method for dense gases that yields critical temperatures from atomic force fields and provides a general average potential field for an atom. By treating neighbors at their mean positions determined by gas density, the potential energy of an atom is expressed as a function of volume and relative position, and statistical mechanics yields average potential energy, free volume, and ultimately the equation of state. This framework classifies condensation as a cooperative phenomenon and allows the equation of state to be derived thermodynamically.

Abstract

The object of the first paper on “Critical Phenomena in Gases” (referred to in this paper as Paper I) was to develop a simple method of dealing with dense gases and to calculate critical temperatures in terms of atomic fields of force. Each atom in a dense gas was pictured as caged for most of its time by a cluster of neighbours, equal in number to those which surround it in the solid (and presumably also in the liquid) phase. The model was intended to provide a general average of the potential field in which any one atom moved by replacing its varying environment by an arrangement of neighbours which could be regarded as typical. This arrangement was taken to be the one in which the neighbours were situated at their mean positions as determined by the density of the gas. The potential energy of any one atom could thus be expressed as a function of the volume of the gas—a step which is probably the crucial one in a theory of critical phenomena. This point of view brings the process of condensation within the category of those described by Fowler (1936) as co-operative phenomena. In passing we may observe that the derivation of van der Waals’ equation provides a particular example of this method, for in it the potential energy of each atom is assumed to be inversely proportional to the volume. The present theory goes beyond this simple model, for the potential energy of an atom is considered to be not only a function of volume but also a function of the position of the atom relative to its neighbours. The probability of finding an atom in any assigned position can be calculated by statistical means and its average potential energy and its available free volume easily deduced. The equation of state can then be deduced by thermodynamic methods, as has been pointed out elsewhere (Lennard-Jones 1937).