  In order to provide a formally correct thermodynamical description of
inhomogeneous fluids valid on all length scales down to the classical limit we
postulate that all extensive quantities have locally extensive analogues. We
derive local thermodynamic relations for the canonical and isothermal- isobaric
ensembles without limitations with respect to the nature of intermolecular
interactions. The inhomogeneity of the system is shown to result in two
distinct entropic contributions. The intrinsic contribution is due the specific
distribution of local environments experienced by a single molecule while the
extrinsic one originates exclusively from the local variation of the density
from its bulk value.
