Density Virial Series for Nonpolar Fluids

For gases at low density, the cluster series in Eqs. (16) and (17) for si and g provide almost directly a virial series in powers of the density, provided the interaction potential is short ranged. By short ranged we mean that there must exist some distance R, some energy A, and some number p greater than 3, such that 

For short-ranged potentials, each of the cluster integrals for the graphs (16) and (17) is finite. If we liked, we could introduce a length I characteristic of the range of the interaction and then introduce dimensionless lengths defined as r,//. Eadi integral in Eq. (16) for Vsd would then be of the form 

If pl is a small number, we can collect all terms with a given number of field points and the sum represents a coefficient of jz/ or g in powers of pl. This is an example of the first strategy mentioned in Section 5. Thus we find 

Equation (32) can be used to derive the familiar virial series for the pressure  

Thus the problem of calculating the thermodynamic properties and pair correlation function of a low-density gas has been reduced to that of evaluating the integrals corresponding to the graphs with certain specified numbers of field points. There is an extensive literature that makes use of this virial expansion. 

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