The cases of melting of two-component ideal solutions and of phase separation in a regular solution described in the Sect. 8.4 were easily treated by analytical methods. However, as the nonideal behavior of the liquid and solid solutions become more complicated (i.e., do not follow regular solution theory), the analytical methods based on Eq (8.2) as the starting point quickly become sufficiently complex to preclude derivation of simple formulae such Eqs (8.12) and (8.21). The graphical method does not have this restriction. Provided only that the free energy Vs composition curves can be drawn for each phase, construction of the phase diagram is straightforward. Moreover, the graphical method provides a qualitative understanding of the process that would be lost in complex mathematical analysis.
The graphical procedure is based on minimizing the free energy of the system at a fixed temperature (and total pressure). Per mole of solution, the free energy is given by Eq (7.36) with gex approximated by hex for a regular solution :
Where gcomp is the sum of the free energies of the pure (unmixed) components:
where xB represents the overall composition of the two-phase mixture, not the composition of individual phases. smix is the entropy of mixing:
* hex need not be that of a regular solution; see footnote on p. 195
The phase diagram depends on the last two terms in Eq (8.22), which represent the change in the free energy of the system when the pure components are mixed. The composition-dependence of g expressed by Equation (8.22) is called a free energy curve. These curves are parametric in T.
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