Osmosis and reverse osmosis. Part 2.—The separation factor of reverse osmosis and its connection with isotonic osmosis

Abstract

Two variational principles control isotonic osmosis and reverse osmosis: the principle of least dissipation of energy referring to a definite time integral and the principle of least constraint referring to the stationary (extremum) value of a volume integral.

An extended form of the Nernst–Planck equation resulting from the time variation has been taken as the basis of our investigations. A theorem of Gyarmati dealing with the minimum conditions of local variation in the presence of constraints yields a lemma concerning the force equilibrium in reverse osmosis. According to Weierstrass's excess function we have obtained linear relations between the potential gradients (forces). Whilst in isotonic osmosis there is an electro-osmotic equilibrium, an electrochemical equilibrium takes place in reverse osmosis. The latter controls the separation effect.

An inversion takes place if c_D=c_F. Experiments have demonstrated that at this point the demineralization effect turns over into a concentration effect.

The mathematical correlations between osmosis and reverse osmosis manifest that reverse osmosis represents the inverse of the transcendental function of osmosis.