Potential distribution theorem view of averages

Now consider the contributions of Eq. (3.19) from intermolecular interactions. Comparing Eq. (3.17) and Eq. (3.18), this is seen to be the logarithm of a ratio of integrals. Simple proportionality factors cancel in forming the ratio. Then the denominator of that ratio is a partition function for the uncoupled N + 1)-molecule system, i.e., without interactions between the A -molecule solution and the distinguished molecule. The numerator is similarly proportional to the partition function for the physical N +1)-molecule system. We thus write 

the doubled brackets with subscript zero in the last term imply no interactions between the A-molecule solution and the distinguished molecule. Thus, one term in that sum will precisely cancel the field contribution of Eq. (3.20). In composing Eq. (3.19), we then write 

Finally, we note that the last combination is the change in the physical field potential energy upon incrementing the molecular number by one, so 

In contrast with the averaging in Eq. (3.20), the averaging here is for the fully coupled system. 

6 For the case considered by Eq. (3.23), explain why the external field makes a contribution p) to the Helmholtz free energy per unit volume, AjV. On this basis, give a thermodynamic derivation of Eq. (3.23) (Landau Lifshitz, vol. 8). 

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