Non-equilibrium thermodynamics of electro-osmotic phenomena

We shall first outline the theory of electro-osmosis and streaming potential based on thermodynamics of irreversible processes [9-14], Let us consider two chambers separated by a very thin membrane, so that it merely serves as a dividing surface which we need not consider as a separate phase. 

Electric current. Chambers I and II contain a fluid with n components carrying electrical charge (fe = 1, 2. n) per unit of mass. We assume that both the chambers are at the same temperature T and the concentrations are uniform in each chamber. M and are the masses of component k in chambers I and II, respectively. Similarly 4 and 4 are the potentials in the two chambers. In the system under discussion, we consider the transport of matter and electricity from one chamber to the other. In the process, both mass and change would be conserved. Thus we must have 

Now our object would be to evaluate entropy production dj due to irreversible processes inside the system, which are simply the transport of matter and electricity. The total entropy production d6 due to internal as well as external factors would be given by 

Now dS = dg + di, where d S is the reversible entropy change due to interaction with the surroundings, and hence cr, the rate of entropy production, is given by 

Equation (4.10) enables us to spot out the fluxes and forces in the system by remembering that entropy production is the sum of the product of fluxes 7 and forces X , i.e. 

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