Generalization to any mixture model of water

In this section, we examine the general aspects of the application of the MM formalism to aqueous solutions. We have already seen that the splitting of partial molar quantities into frozen-in and relaxation terms is totally dependent on the choice of the classification procedure. Here, we generalize the discussion of Sec. 3.5.1 and elaborate on the condition under which such a splitting may be useful to the theory of aqueous solutions. 

Let Nuj[a)da be the average number (in the T,P,Nuj,Ns ensemble) of water molecules which are distinguished by some local property having a numerical value between a and a + da. Similarly, Ns p)d is the average number of solute molecules classified according to some other property having a numerical value between p and dfi. [Here, a and are continuous parameters in this section, we will not use the notation P = k T). ] The two normalization conditions are 

The partial molar quantities in (3.5.25) are obtained by functional differentiation of E Nuj,Ns)  

The experimental partial molar energy of the solute s is given by 

This is the most general expression for Es (or any other partial molar quantity) in the MM formalism. We have applied the MM approach here to both the solute and the solvent. The first two terms on the right-hand side of (3.5.27) may be referred to as the frozen-in terms whereas the last two are corresponding relaxation terms for the solvent and solute, respectively. 

---