The two-headed arrows in Fig. 8.1 indicate chemical equilibrium of components A and B between phases I and II. According to the discussion in Sect.1.11.14, the criterion of chemical equilibrium at fixed T and p is the minimization of the total free energy of the contents of the cylinder-piston in the figure. Since the total free energy is the sum of those of the two phases, this criterion is:
The free energy of a phase is related to the chemical potentials of its components by Eqs (7.27) Using the latter for components A and B in Eq (8.1) gives:
where nAI nBII are the numbers of moles of each constituent in each phase and pAI.. ,.pBII are their chemical potentials.
The change in the state of the system implied by the differentials in Eq (8.1) is the movement of a small quantity of one of the two components from phase I to phase II without altering the other component. Thus, if dnAI moles of A are transferred from I to II, the change in nAII is dnAII = -dnAI. Because component B is not moved, dnBI = dnBII = 0. Inserting these mole relations into the preceding equation yields the result pAI = pAII. Applying the same argument to component B yields pBI = pBII. In general, for any number of components in the two-phase system, the conditions for chemical equilibrium are:
The chemical potentials are seen to be analogous to the thermal and mechanical potentials which provide the equilibrium conditions Ti = Tn and pI = p n. Equation (8.2) is the multicomponent generalization of the equilibrium condition for two coexisting phases of a pure substance, namely gI = gII, where g is the molar free energy (Eq (5.2)).
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