Chemical-Potential Route: A Hidden Percus-Yevick Equation of State for Hard Spheres

The chemical potential of a hard-sphere fluid can be expressed in terms of the contact value of the radial distribution function of a solute particle with a diameter varying from zero to that of the solvent particles. Exploiting the explicit knowledge of such a contact value within the Percus--Yevick (PY) theory, and using standard thermodynamic relations, a hitherto unknown PY equation of state, $p/\rho k_BT=-(9/\eta)\ln(1-\eta)-(16-31\eta)/2(1-\eta)^2$, is unveiled. This equation of state turns out to be better than the one obtained from the conventional virial route. Interpolations between the chemical-potential and compressibility routes are shown to be more accurate than the widely used Carnahan--Starling equation of state. The extension to polydisperse hard-sphere systems is also presented.