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There are indeed other assumptions in the derivation you quote. Namely, Chandler considers the classical limit of an ideal quantum mechanical gas with average particle number $<N> = \sum_j <n_j> = \sum [ e^{\beta (e_j - \mu)} ± 1 ]^{-1}$ (plus for Fermi-Dirac, minus for Bose-Einstein statistics). In the classical limit (low density) there ...