Application of the geometric Gibbs equation: Toward an exact closure condition

The geometric Gibbs equation describes how the available space and correspo nding surface area of a single- component hard particle fluid varies with t he system density. When a closure condition is introduced, i.e., an additio nal equation describing how the surface area depends on the available space , the geometric Gibbs equation reduces to a second-order differential equat ion indicating how the available space varies with the system density. Solu tion of this new equation provides another route to the determination of th e chemical potential and pressure of the hard particle fluid. The simplest proposed closure condition yields the properties of fully penetrable sphere s. A modified closure condition is suggested, and its connection to thermop hysical properties is derived. An extension of the exact form of the closur e condition for the one-dimensional hard rod fluid yields a reasonably good approximation of the properties of the hard sphere fluid at low density, a nd is found to be the required form for densities above the freezing densit y. The simple form of the closure condition and its connection to bulk prop erties may be used to help suggest future closure relations.