Stability with respect to diffusion in a binary system

Stability with respect to Diffusion in a Binary System. 

The results of the preceding paragraph may now be applied to a perturbation which consists of the appearance of a heterogeneity in the composition of a binary system which is initially uniform. Stability with respect to perturbations of this kind will be called stability with respect to diffusion. 

Consider a system P which is homogeneous, and characterized by the variables T, p and the mole fraction Xj, and let us direct attention to two elements of volume a and b in the system. The extent of change is denoted by and corresponds to the passage of a molecule of component A from a to 6, together with the passage of an arbitrary number (v) of molecules B from b to a. We then have (c/. chap. I, 7) 

Since the system is in equilibrium Ap = 0, and the chemical potentials have the same values independent of the choice of the volume elements a and b. Furthermore, at constant T and p 

We may also replace the partial derivatives of the chemical potentials in (15.77) by their values in terms of the mean molar Gibbs free energy, g (c/. 6.49 and 6.50). Hence we obtain 

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