Cluster-Variation Exercise Sketched

Here we sketch a heuristic derivation of the quasichemical formula (9.34). Consider a solution of species a and w. The Gibbs free energy is 

If we can identify complexes awm that may form, we might wish to express G as 

Substitution of (9.40) into (9.38) permits a comparison with (9.39). Such a comparison helps, but does not solve the problem that (9.39) seems to have superfluous variables. 

This conceptual hitch is addressed by adjusting the values of the superfluous variables, subject to the constraints of (9.40), to make G stationary - minimal. Accounting for the constraints (9.40) by the standard procedure of Lagrange s undetermined multipliers [49] yields  

In the cases that are the target of the present development, the evaluation of K (T) can be a challenging calculation in its own right. 

---