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 $$\frac{\mathrm{pV}}{\mathrm{NkT}}=\frac{\text{ν(1–1/N)}}{{\mathrm{(V/V}}_0\text{)−1}}\text{+O(1).}$$ 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 $$\text{c≥2ν−2(ν−1)/N.}$$ 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.