Chemical potential of hard-sphere fluids by Monte Carlo methods

The ‘potential distribution’ expressions derived by Widom for the pressure and chemical potential of a fluid are developed for the special case of a hard-sphere fluid. The exact equations produced are closely related to those used in Scaled Particle Theory. They have been used to determine the chemical potential of hard-sphere fluids over the reduced density range 0·1 ⩽ σ 3 ρ ⩽ 0·8 by means of Monte Carlo sampling. The chemical potential so obtained is in excellent agreement with that found by integrating over pressure as a function of volume. It is found, however, that the chemical potential is considerably more dependent on the sample size than the pressure. The Monte Carlo results for the nearest-particle distribution around random points in the fluid are compared with the predictions of Scaled Particle Theory. The agreement is close, particularly at low densities. Two simple methods of Monte Carlo sampling of the grand canonical ensemble—an exact method and a more convenient, though inexact, modification—have also been used for the hard-sphere fluid in the same density range. Both give good consistency with the values for pressure and chemical potential obtained by the other methods.