## Liquid-State Physical Chemistry: Fundamentals, Modeling, and Applications (2013)

### 7. Modeling the Structure of Liquids: The Integral Equation Approach

The statistical description using the distribution functions given in __Chapter 6__ led to the pair correlation function *g*(*r*) for characterizing the structure of liquids and options to calculate the thermodynamic properties. In this chapter, we will see that this approach also provides options for calculating *g*(*r*) using methods which are conveniently and collectively described as the “integral equation approach.” Using hard-sphere results as a basis, the effect of a more realistic potential on the thermodynamic properties can be assessed via the perturbation approach. Finally, the necessary steps and associated problems to extend this approach to molecular fluids are indicated briefly.

### 7.1. The Vital Role of the Correlation Function

In __Chapter 6__ we have seen that

__(7.1)__

__(7.2)__

__(7.3)__

where the coupling parameter *ξ* is used in the last equation and all other symbols have their usual meaning. In these expressions, the first term is due to the momenta – that is, the kinetics – while the second term is due to the coordinates – that is, the configuration. Thus, all three equations in principle relate a thermodynamic quantity to the structure as represented by *g*. Now, the thought might occur that, if we could determine the correlation function *g*(*r*) experimentally, we could calculate the thermodynamic properties for a liquid. However, in spite of the significant progress in experimental accuracy for determining the correlation function *g*(*r*), at present these methods are insufficiently accurate to calculate reliable thermodynamic properties, and we thus must seek alternative methods. One option would be to use integral equation methods, as described in this chapter. An extensive review, dealing with several aspects of the integral equation, the associated perturbation approach and several other topics, has been provided by Barker and Henderson [1].

Problem 7.1

Indicate what are the most important reasons why __Eqs (7.1)__, __(7.2)__, and __(7.3)__ yield unreliable thermodynamic results.