The objective here is to use Eq (8.2) to relate the concentrations of species A and B (mole fractions xA and xB) in a condensed phase containing only these two species to their equilibrium partial pressures in the vapor, pA and pB. The condensed phase is assumed to contain only A and B (although this is not required). The gas phase may contain an inert diluent such as helium that does not enter the liquid phase. As long as the total pressure is not very large, its value does not affect the distributions of A and B between the two phases. Only the distribution laws for component A are derived; those for component B are determined in the same manner.
From Eq (8.2) the equilibrium condition for component A is |xA(g) = ^A(s or L), where g denotes the gas phase and s and L denote solid and liquid, respectively. The relationships between the chemical potential and the partial pressure in the gas phase and between the chemical potential and the mole fraction in the condensed phase were derived in Sects 7.10 and 7.5, respectively. For the solid or liquid, combination of Eqs (7.29) and (7.30) yields:
gA is the molar free energy of pure A and yA is the activity coefficient of A in the A-B solution. The chemical potential of A in the gas phase is given by Eq (7.44):
where gA(g) is the molar free energy of pure gaseous A at one atm pressure (indicated by the superscript o). The corresponding quantity for the condensed phase, gA, does not need an indication of 1 atm because it is essentially pressure-insensitive. Both molar free energies are at the same temperature T.
Equating the right hand sides of Eqs (8.3) and (8.4) as required by the equilibrium criterion yields:
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