Phase equilibrium second criterion

The following criterion of phase equilibrium can be developed from the first and second laws of thermodynamics the equilibrium state for a closed multiphase system of constant, uniform temperature and pressure is the state for which the total Gibbs energy is a minimum, whence... 

A plot of 2 vs. -t2 for symmetrical systems (i.e., ii vo) is shown in Fig. 1 for a series of values of the heat lerm, It shows how the partial vapor pressure of a component of a binary solution deviates positively from Raoult s law more and mure as the components become more unlike in their molecular attractive forces. Second, the place of T in die equation shows that tlic deviation is less die higher the temperature. Third, when the heat term becomes sufficiently large, there are three values of U2 for the same value of ay. This is like the three roots of the van der Waals equation, and corresponds to two liquid phases in equilibrium with each other. The criterion is diat at the critical point the first and second partial differentials of a-i and a are all zero. 

We now want to consider the extent to which a solid is soluble in a liquid, a gas. or a supercritical fluid. (This last case is of interest for supercritical extraction, a new separation method.) To analyze these phenomena we again start with the equality of the species fugacities in each phase. However, since the fluid (either liquid, gas, or supercritical fluid) is not present in the solid, two simplifications arise. First, the equilibrium criterion applies only to the solid solute, which we denote by the subscript 1 and second, the solid phase fugacity of the solute is that of the pure solid. Thus we have the single equilibrium relation... 

Consider first point 8. Since this point does not meet the second stability criterion (Ineq. 12.8.2), it cannot represent a stable homogeneous phase. The system must move to points 2 or 10. (Or, it could break down into two phases represented by points 2 and 10. If these two phases, however, are to be in equilibrium, they must meet the criterion that was developed in Section 12.4 the chemical potential of the fluid must be the same in both phases. For this to occur, according to Problem 9.57, the two areas / and II - below and above the constant pressure line - must be equal. This pressure - not equal to Pq of course - represents the vapor pressure of the fluid at the specified temperature according to the EoS used.) This applies to all states described by the line segment from point 5 to point 9 and they are referred to as unstable. 

Our objective in this section is to develop a criterion that tells us when two or more phases can coexist at equilibrium or, conversely, when only a single phase is stable. The second law of thermodynamics can give us some insight into answering this question. We... 

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