The virial expansion of the grand potential at spherical and planar walls

The graphs of the virial expansion of the grand potential of molecules in a spherical cavity are related to the graphs of the virial coefficients of a homogeneous gas mixture. The second virial coefficient is obtained for hard spheres in a spherical cavity with a square-well wall, and for square-well molecules in a cavity with a hard wall. The planar limit of these results is used to analyse several different proposals for separating the grand potential into a bulk and surface part for arbitrary wall potentials. It is shown that none of these is entirely satisfactory and that, for systems of arbitrary geometry, no unique division is possible.