Relations between thermodynamic quantities and generalized molecular distribution functions

6 Relations between thermodynamic quantities and generalized molecular distribution functions 

In section 2.7, we introduced the generalized molecular distribution functions GMDFs. Of particular importance are the singlet GMDF, which may be re-interpreted as the quasi-component distribution function (QCDF). These functions were deemed very useful in the study of liquid water. They provided a firm basis for the so-called mixture model approach to liquids in general, and for liquid water in particular (see Ben-Naim 1972a, 1973a, 1974). 

In this section we shall derive some new relationships between thermodynamic quantities and GMDF. In previous sections we have derived a few relationships 

Let E be any extensive thermodynamic quantity expressed as a function of the variables T, P, and N (where N is the total number of molecules in the system). Viewing the same system as a mixture of quasi-components, we can express E as a function of the new set of variables T, P, and N. For correctness, consider a QCDF based on the concept of CN. The two possible functions mentioned above are then 

For the sake of simplicity, we henceforth use N(K) in place of N (K), so that the treatment will be valid for any discrete QCDF. Since E is an extensive quantity, it has the property 

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