## Calculation of Virial Coefficients and Their Uses

##### Abstract

This work is centers on the classical statistical derivation of thermodynamic properties of a system from its pair potential through the calculation of virial coefficients and also develops a useful method for modeling systems with the square-well potential. It is organized into 5 chapters.Chapter 1 lays the statistical mechanics background for the use of virial coefficients and the square-well model. It is intended for the reader unfamiliar with these concepts and with the exception of a derivation of C(T) for the square- well potential using overlapping spheres, does not contain any novel discoveries.Chapter 2 introduces a Monte Carlo hit-and-miss technique for calculating virial coefficients for the square-well named ratio integration. Unlike traditional Monte Carlo hit-and-miss integration, only the percentage of hits are measured and not the volume of the integration. The integration is then scaled by the literature values for virial coefficients of the hard sphere model. This has a significant increase in computation speed.In chapter 3 the virial coefficients calculated in chapter 2 are used to produce phase diagrams using the Maxwell construction. A technique is developed to fit the square-well parameters to second virial coefficient data. The method is examined for argon and then generalized for mixtures.Mayer sampling is a technique for calculating virial coefficients at specific temperatures for any pair potential. In chapter 4 it is used to examine two models for the C60 fullerene, the widespread Girifalco potential and its suggested replacement, the Smith-Thakkar potential. These are both also modeled by the square-well potential using the techniques developed in chapters 2 and 3.work.Chapter 5 presents possible future extensions and improvements of this