Formation and the Pseudochemical Potential of a Hard Sphere

Cavity Formation and the Pseudochemical Potential of a Hard Sphere 

We derive here a useful relation between the work required to form a cavity and the pseudochemical potential of a hard-sphere solute. We shall still discuss only a system of spherical particles in the T, F, N ensemble. The results are also valid for more general systems and in other ensembles. 

Consider a system of N particles with an effective hard-core diameter da- By effective we mean that two molecules at a distance of da from each other feel almost infinite repulsive forces. Certainly there is no unique method of determining the values of Ja for a real molecule. Nevertheless for, say, Lennard-Jones particles the parameter a is a reasonable choice for an effective hard-core diameter. 

In section 5.9.3, we obtained the pseudochemical potential for a one-component system. We repeat the same process here, but instead of adding the (A + l)th particle, we add a hard-sphere particle of diameter cjhs to a fixed position Ro in a system of particles having an effective hard-core diameter da. The work associated with this process, keeping T, V, N constant, is given by 

Note that since the hard-sphere solute is presumed to have no internal structure, its addition to the system does not introduce an internal partition function. The quantity hs(Ro) stands for the total interaction energy between the iV molecules in the configuration and the hard-sphere particle at Ro. More specifically 

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