Stability Analysis in a Two-component System

Consider JV moles of a binary mixture of composition Xi and X2 (mole fractions) at temperature Tand pressure P. Let us compare the stability of the single-phase system with that of the two-phase system at the same temperature and pressure. Similarly to the single-component system in the preceding section, we assume moles of primed phase and N moles of double-primed phase, in the two-phase state. The composition of primed and double-primed phases are Xi -f Ajcj and Xi -f respectively, for component 1 (see Fig. 4.16). 

This time we will work with the Gibbs free energy function. The Gibbs free energy difference between the two states shown in Fig. 4.16 is 

Ag is the molar Gibbs free energy difference between the primed phase and the original single phase, and A g is the molar Gibbs free energy difference between the double-primed phase and the original phase. The constraints (see Fig. 4.16) are constant temperature and pressure and mole numbers of components 1 and 2 Nj = iV- + AT-, i = 1,2 (N = Ni + N2, N = N[ + N2, and N + N ). From the 

The molar Gibbs free energy g = g(T, P, Xj) and since T and P are held constant, then 

The double summation in the above equation can be written as (similarly to Eq, (4.169)) 

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