Gibbs-Duhem equations membrane

Finally, consider a two-phase, two-component system in which the two phases are separated by an adiabatic membrane that is permeable only to the first component. In this case we know that the temperatures of the two phases are not necessarily the same, and that the chemical potential of the second component is not the same in the two phases. The two Gibbs-Duhem equations for this system are... 

Two cases arise. The simpler case is one in which we imagine that the liquid is confined in a piston-and-cylinder arrangement with a rigid membrane that is permeable to the vapor but not to the liquid, as indicated in Figure 10.1. Pressure may then be exerted on the liquid independently of the pressure of the vapor. The temperatures of the two phases are equal and are held constant. The Gibbs-Duhem equation for the vapor phase is... 

Equation 8.5-3 indicates that the number of natural variables for the system is 6, D = 6. Thus the number D of natural variables is the same for G and G, as expected, since the Legendre transform interchanges conjugate variables. The criterion for equilibrium is dG 0 at constant T,P,ncAoi, ncA(3, /icC, and The Gibbs-Duhem equations are the same as equations 8.4-8 and 8.4-9, and so the number of independent intensive variables is not changed. Equation 8.5-3 yields the same membrane equations (8.4-13 and 8.4-14) derived in the preceding section. 

In the framework of the membrane approach, the film can be treated as a single surface phase, whose Gibbs-Duhem equation reads "... 

Note that the above thermodynamic equations are, in fact, corollaries from the Gibbs-Duhem equation of the membrane approach (Equation 5.130). There is an equivalent and complementary... 

If we add the three equations in equation (8.70), we see that the Gibbs-Duhem equation holds ZQVpi = 0. Let us fix species 3, which might represent the membrane, with velocity = 0. The third equation in (8.70) is superfluous, as Kyy = K. The remaining 2x2 matrix equation can be inverted to determine the velocities and thus the fluxes of species 1 and 2. 

Here, i , represents the chemical potential of water, and a , is the transport coefficient of water. The equation for the membrane is ignored, because it is dependent on the other two of Gibbs-Duhem of equations. From many models in the literatures, these equations were used in a Stefan-Maxwell framework. 

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