The single-phase solid and liquid phase regions in Fig. 8.3 show no structure because the A-B solutions were assumed to be ideal. However, if the components exhibit positive deviations from ideality (i.e., if the A-B molecular interaction is weaker than the average of the A-A and the B-B interactions), the single-phase solutions separate into two distinct phases, either both liquid or both solid. The system in which phase separation has
3 See Sect. 2.6 for application of the lever rule in single-component vapor-liquid systems. A more detailed discussion of the lever rule for binary condensed-phase systems can be found in Sect. 8.8.
occurred is termed partially miscible because the B-rich phase contains some dissolved A and the A-rich phase contains some dissolved B.
Nonideality is assumed (for simplicity) to be represented by the regular solution model (Sect. 7.7), in which the excess free energy is approximated by the excess enthalpy. This latter property is a symmetric function of composition:
where Q is the interaction energy. If this property is negative, the stability of the A-B solution is greater than that of the ideal solution, and no phase separation occurs. If Q > 0, the solution is energetically less stable than an ideal solution. When the destabilizing effect of the excess enthalpy overcomes the stabilizing influence of the entropy of mixing, phase separation occurs.
Labeling the partially-miscible phases as I and II, use of Eq (8.3) in Eq (8.2) with i = A yields the following:
The analogous equation for component B is:
The activity coefficients derived from hex of Eq (8.15) are given by Eq (7.42). Substituting these into Eq (8.16) and replacing xA by 1 - xB for both phases gives:
is a dimensionless (but temperature-dependent) form of the interaction parameter.
Equation (8.17b) can be obtained from Eq (8.17a) by replacing xBI and xBII by 1-xBI and 1-xBII, respectively. This mathematical feature implies that the two equations are mirror-image branches of a function that is symmetric about xB = 0.5. Using the mathematical property xBII = 1 - xBI of such a symmetric function, Eqs (8.17a) and (8.17b) are seen to be identical, and can be represented by the function:
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