Equation of State of Classical Hard Spheres at High Density

Under certain conditions, an asymptotic expression for the equation of state of a classical mechanical system of N ν-dimensional (ν=1, 2, or 3) hard spheres confined in a volume V is obtained in the form pVNkT=ν(1–1/N)(V/V0)−1+O(1).This expression agrees with the leading term in V/V0—1 of the usual free-volume approximation for N = ∞. The conditions under which this conclusion is established are a restriction to a finite number of molecules N with periodic boundary conditions, and the requirement that as V→V0 the accessible configuration states approach a close-packed configuration whose coordination number c satisfies the requirement c≥2ν−2(ν−1)/N.Such a limiting configuration, from which only an infinitesimal region of configuration space is accessible under an infinitesimal expansion, is called a stable configuration; the above restriction on the coordination number is a necessary condition for stability. The difficulties which appear as N→∞ are indicated.