The Carnahan-Starling Approximation

An accurate reduced equation of state for the hard-sphere approximation using the virial expression was determined by Ree and Hoover [5]. The first six virial coefficients are given by 

(275) has been written here in the form of two terms. The first term on the right-hand side is the same as the ideal gas. One may think of the second term as a correction to the ideal gas. In Fig. 118 is a comparison of Eq. (275) with the virial equation derived by Ree and Hoover. It is apparent that this is a good approximation above a value of V/VJl) 2. Comparison to experimental data is difficult since, firstly, there is no such gas represented by hard spheres and, secondly, experimental virial coefficients even for gases such as argon are not readily available to the fifth term. This, however, seems to be a reasonable staring point for modeling. 

Notice that this does not include any attractive potential as one would add in, for example, the van der Waal equation. Some authors have added in 

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Many generalizations of van der Waals ideas have been proposed for improving the prediction of the fluid phase behavior in a wide range of thermodynamic states and to extend the description to molecular fluids and mixtures [79]. Longuet-Higgins and Widom [80] suggested that the repulsive term Po be replaced with the accurate expressions for the pressure of hard spheres elaborated in the theory of liquids [81], such as the Carnahan-Starling approximation,[82]... 

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