Statistical Mechanics of Reacting Coulomb Gases

Abstract

A quantum-statistical-mechanical formulation is developed for determining the equation of state of Coulomb gases. The procedure is to develop the activity expansion of the grand partition function into a true perturbation expansion in which the divergences present in the cluster expansion of the Coulomb gas have been eliminated. Ionization and dissociation appear quite naturally in this approach even though only the total number of nuclei and electrons in the system are specified. Considerable insight into the quantum-mechanical perturbation result is obtained from the classical perturbation theory. The form of the static screened Coulomb potential that replaces the classical Debye-Hückel potential is discussed. An application to the equation of state of gaseous hydrogen which takes account of all $e-e$, $p-p$, and $e-p$ interactions is given. The formation of higher clusters such as $\mathrm{H}-e$, $\mathrm{H}-p$, H-H, etc., can also be systematically included. Owing to the extensive numerical calculations required, this has currently only been done in an approximate way. It is shown that the simple one-level Saha equation, including Debye-Hückel corrections for the free charges, is fairly accurate for $\rho <{10}^{-3\mathrm{}}$ g/${\mathrm{cm}}^3$. At greater densities the differences become significant.