First-Order Phase Transition in a Gas of Long Thin Rods

Onsager's prediction of a first-order phase transition in a gas of long thin rods is verified by extensive calculations on a simple model. The molecules are rectangular parallelepipeds, of length l and square cross section d×d. The long molecular axes can point in only three mutually perpendicular directions. Virial coefficients, up to the seventh, are calculated exactly as functions of orientation, in the limit l→ ∞, d→0, and l2d=constant. The transition predicted by Onsager's theory, based on a second virial approximation to the potential of mean force in the space of relative orientations, is observed also when all virial coefficients up to the seventh are included. The virial expansion appears to converge well for the isotropic phase; but the properties of the anisotropic phase depend sensitively on the order of approximation.