Are the energy and virial routes to thermodynamics equivalent for hard spheres?

The internal energy of hard spheres (HS) is the same as that of an ideal gas, so that the energy route to thermodynamics becomes useless. This problem can be avoided by taking an interaction potential that reduces to the HS one in certain limits. In this paper the square-shoulder (SS) potential characterized by a hard-core diameter ${\sigma }^{\mathrm{\prime }}$, a soft-core diameter $\sigma >{\sigma }^{\mathrm{\prime }}$ and a shoulder height $ϵ$ is considered. The SS potential becomes the HS one if (i) $ϵ\to 0$, or (ii) $ϵ\to \mathrm{\infty }$, or (iii) ${\sigma }^{\mathrm{\prime }}\to \sigma$ or (iv) ${\sigma }^{\mathrm{\prime }}\to 0$ and $ϵ\to \mathrm{\infty }$. The energy-route equation of state for the HS fluid is obtained in terms of the radial distribution function for the SS fluid by taking the limits (i) and (ii). This equation of state is shown to exhibit, in general, an artificial dependence on the diameter ratio ${\sigma }^{\mathrm{\prime }}\mathrm{/}\sigma$. If furthermore the limit ${\sigma }^{\mathrm{\prime }}\mathrm{/}\sigma \to 1$ is taken, the resulting equation of state for HS coincides with that obtained through the virial route. The necessary and sufficient condition to get thermodynamic consistency between both routes for arbitrary ${\sigma }^{\mathrm{\prime }}\mathrm{/}\sigma$ is derived.