First-order expansion of the coupling work

We end this long section on the chemical potential with one simple and useful expression. We note first that in all of the expressions we had so far, the chemical potential was expressed as integrals over the pair correlation function. It is desirable to have at least one expression of the chemical potential in terms of molecular interactions. This can be obtained for very low densities, for which we know that the pair correlation function takes the form 

Substituting (3.89) into (3.67), we get an immediate integral over , hence 

Using the notation for the second virial coefficient (see section 1.5) 

The last term on the rhs of (3.92) is the first-order term in the expansion of the coupling work in the density. 

The virial expansion for the pressure may be recovered from (3.92) by using the thermodynamic relation 

We wish to obtain the first-order deviation from the ideal-gas expression for the chemical potential. This may be obtained either from (5.9.27) or from (5.9.38). We know from section 5.3.2 that at the limit of low density we have 

The same result can be obtained by expanding the third term on the rhs of (5.9.38) to first order in the density, i.e., 

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