  This paper presents a modified grand canonical ensemble which provides a new
simple and efficient scheme to study few-body fluid-like inhomogeneous systems
under confinement. The new formalism is implemented to investigate the exact
thermodynamic properties of a hard sphere (HS) fluid-like system with up to
three particles confined in a spherical cavity. In addition, the partition
function of this system was used to analyze the surface thermodynamic
properties of the many-HS system and to derive the exact curvature dependence
of both the surface tension and adsorption in powers of the density. The
expressions for the surface tension and the adsorption were also obtained for
the many- HS system outside of a fixed hard spherical object. We used these
results to derive the dependence of the fluid-substrate Tolman length up to
first order in density.
