Derivation of the basic quasi-chemical formula

Despite the kinship of the present development to these historical works, we don t try to follow upon the historical footsteps directly. Part of our reservation is that these historical works were based upon lattice-gas models that are archaic as contributions to molecular science sometimes useful, but rarely an attempt to come to grips with molecular problems at a molecular resolution. Nevertheless, quasi-chemical and Bethe theories are probably the most effective approximate physical theories available for those lattice gas problems, and an important goal of our present discussion will be to discover corresponding theories of molecular science, as distinct from lattice gases. 

It is interesting that Eq. (4.98), p. 97, offers a discrete-state partition function for the description of the inner-sphere contribution to the thermodynamics. But the discrete coordinate is an occupation number for a precisely defined configurational region, and parameters required for this discrete-state partition function are obtained by molecular-level calculations. Therefore, molecular realism isn t the first casualty of these theories, although strong approximations are typically accumulated after the formulation of quasi-chemical theories. 

Note that the additional factor within the average, the 11 (1 — (7)), would be 

This expression separates solute-solvent interactions into an outer-shell contribution, for which simpler physical approximations might be helpful, and an inner-shell contribution, which we consider from the perspective of quasi-chemical 

Note that the ratio of averages appearing in Eq. (7.2) is the form considered in Section 3.3, p. 41. The averaged quantity is just the indicator function ITy [1 — b (7)]. The ratio evaluates the average with the solute present. Since this function is equal to one (1) for precisely those cases that the defined inner shell is empty, and is equal to zero (0) otherwise, we can recast Eq. (7.2) as 

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