Kinetic-Energy Corrections to Klein's Theorem for the Thermodynamic Functions of a Plasma

It is shown that the average thermal energy of a system of $N$ free electrons and $N$ free protons with density $n=\frac{N}{V}$ and temperature $T$ has the form $E=2N[\frac{3}{2}kT+Tf(\eta )+f_1(n, T)+{f_2}^{\prime }(n, T\left)\right],$ where $f$, the leading term in the electrostatic energy, is an arbitrary function of $\eta =T{n}^{-\frac{1}{3}}$ in agreement with the original Klein theorem. $f_1$ is a kinetic-energy correction which is related to the electrostatic energy ${f_2}^{\prime }$ by the differential equation $-\frac{n}{T}\frac{\partial }{\partial n}{(f_1+{f_2}^{\prime })}_T=\frac{T}{3}\frac{\partial }{\partial T}{\left(\frac{2f_1+{f_2}^{\prime }}{T}\right)}_n.$ Some consequences of this result are discussed.