The MM and KS expansions

The Mayer-Montroll (Mayer and Montroll, 1941) and Kirkwood-Salsburg (Kirkwood and Salsburg, 1953) expansions are storied parts of basic statistical thermodynamics (Stell, 1985), but have been neglected for practical purposes because of a lack of recognition of how simple and simplifying they can be. 

We introduce results with the specific example of a hard-core solute that was previously considered in Section 4.3. The hard-core results give perspective for a direct generalization to more realistic interactions. 

Consider again (0. S ) = exp [—(. )] for hard-core solutes as in Section 4.3. The most immediate guiding theory is the inclusion-exclusion 

Here the random variable m is the number of solvent centers within the observation volume. As examples m 0l )Q is the expected number of centers within the observation volume, and — 1). i )o = ((2) l )o number of pairs of 

These results are obtained straightforwardly from the potential distribution theorem. We write (Kirkwood and Salsburg, 1953) 

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The MM and KS expansions For the case that three molecules are involved we write... 

Consider the KS expansion. Fig. 6.3, applied to a one-component fluid and truncated at the second term displayed this will be satisfactory for low density because the subsequent terms have higher-power initial density multipliers. What is the corresponding MM approximate theory for the excess chemical potential Show that this KS approximate theory, expressed for is... 

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