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

The geometric Gibbs equation describes how the available space and corresponding surface area of a single- component hard particle fluid varies with the system density. When a closure condition is introduced, i.e., an additional equation describing how the surface area depends on the available space, the geometric Gibbs equation reduces to a second-order differential equation indicating how the available space varies with the system density. Solution of this new equation provides another route to the determination of the chemical potential and pressure of the hard particle fluid. The simplest proposed closure condition yields the properties of fully penetrable spheres. A modified closure condition is suggested, and its connection to thermophysical properties is derived. An extension of the exact form of the closure condition for the one-dimensional hard rod fluid yields a reasonably good approximation of the properties of the hard sphere fluid at low density, and is found to be the required form for densities above the freezing density. The simple form of the closure condition and its connection to bulk properties may be used to help suggest future closure relations.