Kinetic model of osmosis through semipermeable and solute-permeable membranes

The gas analogy of the van't Hoff equation for osmotic pressure deltapi = RT/V, where R is gas constant, T absolute temperature and V mole volume of water, remained unexplained for a century because of a few misconceptions: (1) Use of supported membranes prevented the recognition that osmotic forces exert no effect on the solid membrane. During osmotic flow frictional force of solvent within membrane channels equals osmotic kinetic force pi at the interface against the solution containing impermeant solute. (2) Retrograde diffusion of water is much less than osmotic flow even when dx in the gradient dc/dx approaches zero. (3) The gas analogy was thought to be accidental. Actually, the internal kinetic pressure is P = RT/V, because intermolecular forces cancel out at the liquid interface, just as within a gas. The kinetic osmotic pressure is the difference in solvent pressure across the interface: pi = RT/V-(RT/V)X1 = (RT/V)X2, where X1 and X2 are the mole fractions of water and impermeant solute, respectively. Integration gives pi = -(RT/V)lnX1, identical to the thermodynamic equation. This equation is correct up to 25 atmospheres, and up to 180 atmospheres by assuming that a sucrose molecule binds 4 and a glycerol molecule 2.5 water molecules. For solute-permeable membranes, the reflection coefficient sigma can be calculated by formulas proposed for ultrafiltration. Because the fraction (1-sigma) of solute concentration behaves as solvent, osmosis may well proceed against the chemical potential gradient for water. The analogy to an ideal gas applies because pi = -(RT/V)lnX1 is the small difference between enormous internal solvent pressures.