Distribution Functions for Liquid Solutions

The techniques used to describe the properties of pure liquids can be extended in a fairly straightforward fashion to liquid solutions [26, 27]. This treatment is normally restricted to liquids in which the molecules behave as non-interacting hard spheres or as dipolar species interacting via a Lennard-Jones potential. The discussion here is limited to two-component mixtures but it is easily extended to more complex systems. 

Several approaches to estimation of the internal energy have been described. These involve assuming specific relationships between the pair correlation functions gjj and the form of the interaction energy uy. The simplest of these is based on the van der Waals treatment of fluids and its application of the law of corresponding states. Examination of typical radial distribution functions for mixtures such as those shown in fig. 2.16 reveals that the maximum in each distribution function g r) occurs close to the diameter a describing the distance of closest approach for the two molecules involved. Thus, it is better to describe the radial distribution function in terms of the reduced distance r/a instead of the distance r. This conclusion leads to the assumption that 

When the interaction energy is given by the Lennard-Jones potential, one may write 

Changing the variable in equation (2.11.1) from rtoy = r/ y and substituting the expression for uy r) one obtains 

In the case of mixtures of simple molecules, the Gibbs energy for a one-fluid system may be written as 

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