Thermodynamic self-consistence of equations of state of gases with first to third laws: Dieterici equation, a peculiar case

As is well-known, equations of state of macroscopic thermodynamic systems can be obtained by using empirical, semi-empirical or theoretical procedures. However, any equation of state must be compatible with the three laws of thermodynamics. In 1973, Tykodi and Hummel (TH) proposed a noteworthy criterion to establish if a given equation of state is consistent or not with the first and second laws of thermodynamics. Nevertheless, in the TH paper, it is shown that the Dieterici equation of state, deduced from empirical and theoretical basis at the end of nineteenth century, did not fulfill their criterion. Hence, it could not be considered as a thermodynamically valid equation of state. In the present article, we generalize the TH criterion; with the generalized criterion, it is shown that the Dieterici equation is, in fact, consistent with the first and second laws of thermodynamics. Furthermore, we extend the TH criterion through the incorporation of the third law to embrace the study of quantum gases which are usually found at temperatures near to absolute zero. We show that the equations of state of quantum gases such as the photon, Fermi and Bose–Einstein gases fulfill such an extended criterion.