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

Abstract

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 ${\sigma }^{\prime }$, a soft-core diameter $\sigma >{\sigma }^{\prime }$ and a shoulder height $\epsilon$ is considered. The SS potential becomes the HS one if (i) $\epsilon \to 0$, or (ii) $\epsilon \to \infty$, or (iii) ${\sigma }^{\prime }\to \sigma$ or (iv) ${\sigma }^{\prime }\to 0$ and $\epsilon \to \infty$. 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 ${\sigma }^{\prime }/\sigma$. If furthermore the limit ${\sigma }^{\prime }/\sigma \to 1$ 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 ${\sigma }^{\prime }/\sigma$ is derived.

10 pages, 4 figures; v2: minor changes; to be published in the special issue of Molecular Physics dedicated to the Seventh Liblice Conference on the Statistical Mechanics of Liquids (Lednice, Czech Republic, June 11-16, 2006)

References in corpus (1)

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

Cited by in corpus (8)

• Penetrable Square-Well fluids: Exact results in one dimension
• Structural and Thermodynamic Properties of Hard-Sphere Fluids
• Structure of the square-shoulder fluid
• Low-temperature and high-temperature approximations for penetrable-sphere fluids. Comparison with Monte Carlo simulations and integral equation theories
• Chemical-Potential Route: A Hidden Percus-Yevick Equation of State for Hard Spheres
• Penetrable-Square-Well fluids: Analytical study and Monte Carlo simulations
• Janus fluid with fixed patch orientations: theory and simulations
• Thermodynamic consistency between the energy and virial routes in the mean spherical approximation for soft potentials