On the equivalence between the energy and virial routes to the equation of state of hard-sphere fluids

The energy route to the equation of state of hard-sphere fluids is ill defined since the internal energy is just that of an ideal gas, and thus it is independent of density. It is shown that this ambiguity can be avoided by considering a square-shoulder interaction and taking the limit of vanishing shoulder width. The resulting hard-sphere equation of state coincides exactly with the one obtained through the virial route. Therefore, the energy and virial routes to the equation of state of hard-sphere fluids can be considered as equivalent.