An explicit formula for the chemical potential (&mgr;) of simple fluids is derived for a whole class of integral equation theories, including the Percus–Yevick approximation and some other, more recently proposed closures. This formula only requires the pair correlation functions for one single state of the system, and applies to both homogeneous and inhomogeneous fluids. The coupling parameter integration method to calculate the chemical potential—where one particle is gradually coupled to the rest—is also investigated. It is shown that the &mgr; value obtained in the approximate theories is not unique, but depends on the integration path. This behavior is due to inconsistencies of the approximation, which are discussed in some detail. For a certain choice of integration path the &mgr; values obtained using the latter method agree with those from the explicit formula. Numerical results for the different closures are presented for a hard sphere fluid. The accuracy of &mgr; depends strongly on the quality of the bridge function inside the hard core diameter.
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