The Ideal and Real Gas Heat Capacity of Potassium Atoms at High Temperatures

Abstract

The ideal gas heat capacity, $C_{p}$, of potassium atoms is calculated to high temperatures using statistical mechanics. Since there are a large number of electronic energy levels in the partition function (Boltzmann sum) below the first ionization potential, the partition function and $C_{p}$ will become very large as the temperature increases unless the number of energy levels contributing to the partition function is constrained. Two primary categories of arguments are used to do this. First, at high temperatures, the increased size of the atoms constrains the sum (Bethe method). Second, an argument based on the existence of interacting charged species at higher temperatures is used to constrain the sum (ionization potential lowering method). When potassium atoms are assumed to constitute a real gas that obeys the virial equation of state, the lowest non-ideal contribution to $C_{p}$ depends on the second derivative of the second virial coefficient, B(T), which depends on the interaction potential energy curves between two potassium atoms. When two ground-state ($^{2}\hbox {S}$) atoms interact, they can follow either of the two potential energy curves. When a $^{2}\hbox {S}$ atom interacts with an atom in the first electronically excited ($^{2}\hbox {P}$) state, they can follow any of the eight potential energy curves. The values of B(T) for the ten states are determined, then averaged, and used to calculate the nonideal contribution to $C_{p}$.