The first general multicomponent equations for transport through semipermeable membranes are derived from basic statistical-mechanical principles. The procedure follows that used earlier for open membranes, but semipermeability is modelled mathematically by the introduction of external forces on the impermeant species. Gases are treated first in order to clarify the problems involved, but the final results apply to general nonideal solutions of any concentration. The mixed-solvent effect is treated rigorously, and a mixed-solvent osmotic pressure is defined. A useful specific identification of so-called osmotic flow is given, along with a demonstration that such an identification cannot be unique. Results are obtained both for discontinuous membrane models, and for a continuous model.
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