The Ideal Gas and Real Gas Heat Capacity of Sodium Atoms

Abstract

The ideal gas heat capacity of sodium atoms in the vapor phase is calculated to high temperatures using statistical mechanics. Since there are, in principle, an infinite number of atomic energy levels, the partition function and the heat capacity will grow very large unless the summation over energy levels is constrained as temperature increases. At higher temperatures, the increasing size of the atoms, which is a consequence of the increased population of highly excited energy levels, is used as a mechanism for limiting the summation over energy levels. The "IP − kT " and "Bethe" procedures for cutting off the summation over energy levels will be discussed, and the results obtained using the two methods will be compared. In addition, although experimental information is available about lower atomic energy levels and some theoretical calculations are available for excited energy levels, information is lacking for most individual atomic states associated with highly excited energy levels. A "fill" procedure for approximating the energy of the unknown states will be discussed. Sodium vapor will also be considered to be a real gas that obeys the virial equation of state. The first non-ideal term in the power series expansion of the heat capacity in terms of virial coefficients involves the second virial coefficient, B(T ). This depends on the interaction potential energy between two sodium atoms, i.e., the potential energy curves for the sodium dimer. Accurate interaction potential energies can be obtained from either experimental or theoretical information for the lowest ten electronic states of the sodium dimer. These are used to calculate B(T ) for each state, and the averaged value of B(T ) for all ten states is used to calculate the non-ideal contribution to the heat capacity of sodium atoms as a function of temperature.