Critique of primitive model electrolyte theories

Approximate theories for the restricted primitive model electrolyte are compared in the light of Totsuji’s lower bound for the energy (an improvement over Onsager’s), Gillan’s upper bound for the free energy, and thermal stability requirements. Theories based on the Debye-Hückel (DH) approach and the mean spherical approximation (MSA), including extensions due to Bjerrum, Ebeling, Fisher, and Levin, and Stell, Zhou, and Yeh (PMSA1, 2, 3) are tested. In the range $T^{*}{=k}_B{TDa/q}^2\lesssim {10T}_c^{*}\simeq 0.5,$ all DH-based theories satisfy Totsuji’s bound, while the MSA possesses a significant region of violation. Both DH and MSA theories violate Gillan’s bound in the critical region and below unless ion pairing and the consequent free-ion depletion are incorporated. However, the PMSA theories, which recognize pairing but not depletion, fail to meet the bound. The inclusion of excluded-volume terms has only small effects in this respect. Finally, all the pairing theories exhibit negative constant-volume specific heats when $T^{*}\gtrsim {2T}_c^{*}\simeq 0.1;$ this is attributable to the treatment of the association constant.