For general mixtures of polyatomic molecules and their constituent atoms, we first rigorously derive an exact statistical mechanical result relating the background pair correlation function y(1,2,...,m) to a certain excess chemical potential difference involving its components, beta Delta mu(e), extending and generalizing our previous results. Second, using only thermodynamic methods, we develop a perturbation theory for the equation of state (EOS) which involves beta Delta mu(e); we then express this EOS in an alternative form involving y(1,2,...,m). The latter form coincides with results recently obtained by Zhou and Stell using a different approach and with the EOS of the Wertheim first-order perturbation theory (TPT1); our approach explicitly exposes the underlying thermodynamic approximations involved. Third, we show for the case of tangent fused-hard sphere (FHS) systems, under the approximation that beta Delta mu(e) is independent of composition, that implementation of the former form of the theory yields results analytically equivalent to these obtained from the Boublik-Nezbeda (BN) EOS; and that the alternative implementation is only slightly less accurate, due to a (numerically small) internal inconsistency in this EOS. This sheds light on the remarkable accuracy obtained for several previous implementations of TPT1 for such systems. We present new computer simulation results for a particular ternary tangent FHS heteronuclear diatomic mixture, which support the approximation that beta Delta mu(e) for mixtures of such molecules is nearly composition independent. Finally, for several FHS mixture model systems, we test the Lewis-Randall rule and several other approximations for calculation of the mixture chemical potentials. The Lewis-Randall rule is generally superior for the individual chemical potentials, and is competitive for beta Delta mu(e).