Mixing Rules for Hard Spheres and Association

Theory and computer simulation provides information in addition to the virial equation of state that can be used to develop mixing and combining rules. The equation of state for the pure hard-sphere fluid can be represented by the equation of state of Carnahan-Starling  

To match the results of molecular-dynamics computer simulations for square-well molecules, Alder et alP used a double-power-series expansion in reduced inverse temperature and volume for the attractive part of an equation of state. Modifications of their attractive term have been used in a number of equations of state, such as the augmented and perturbed-hard-sphere equations, the perturbed-hard-chain equation, and the BACK equations of state. 

In all the equations that include the double power-series expansion (or variations thereof), such as the family of perturbed-hard-chain equations, the parameters are related to molecular rather than critical properties, and the mixing and combining rules are quadratic in composition for the attractive term and based on hard-sphere theory for the repulsive term. 

Molecules are not hard spheres and other equations of the same general form as eq 5.179 have been proposed for molecules of various geometrical shapes. For example, hard convex bodies Boublik gives 

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