  The basic ingredients of osmotic pressure are a solvent fluid with a soluble
molecular species which is restricted to a chamber by a boundary which is
permeable to the solvent fluid but impermeable to the solute molecules. For
macroscopic systems at equilibrium, the osmotic pressure is given by the
classical van't Hoff Law, which states that the pressure is proportional to the
product of the temperature and the difference of the solute concentrations
inside and outside the chamber. For microscopic systems the diameter of the
chamber may be comparable to the length-scale associated with the solute-wall
interactions or solute molecular interactions. In each of these cases, the
assumptions underlying the classical van't Hoff Law may no longer hold. In this
paper we develop a general theoretical framework which captures corrections to
the classical theory for the osmotic pressure under more general relationships
between the size of the chamber and the interaction length scales. We also show
that notions of osmotic pressure based on the hydrostatic pressure of the fluid
and the mechanical pressure on the bounding walls of the chamber must be
distinguished for microscopic systems. To demonstrate how the theoretical
framework can be applied, numerical results are presented for the osmotic
pressure associated with a polymer of N monomers confined in a spherical
chamber as the bond strength is varied.
