On the Theory of the Thermal Diffusion Coefficient for Isotopes. II

The temperature dependence of the thermal diffusion constant $\alpha$ of a mixture of isotopes is determined for the 9,5 Lennard-Jones model of intermolecular forces by the use of the Chapman-Enskog theory of transport phenomena in gases. The treatments of the Sutherland model and the special Lennard-Jones model given in the first paper of this series were subject to the drastic restriction that the depth $\epsilon$ of the potential energy minimum must be small compared with $\mathrm{kT}$. The present treatment of the 9,5 model is valid to all orders of $\frac{\epsilon }{\mathrm{kT}}$; this advantage is gained, however, at the expense of substituting laborious numerical methods for the analytical methods that were used in the previous paper. The results indicate that the thermal diffusion constant first increases slightly as the temperature decreases, and then decreases rapidly, passing through zero and becoming negative at a temperature about 1.5 times the critical temperature. The constant becomes strongly negative as the temperature decreases still further, and then approaches zero as the absolute temperature approaches zero. When the theoretical results are modified to account for the fact that the neon molecule is much harder than is indicated by a repulsive force index of 9, a quantitative agreement is obtained with Nier's experimental data on mixtures of the neon isotopes. Fair agreement is obtained with the data of Atkins, Bastick, and Ibbs on mixtures of the noble gases. An approximate method is presented which permits one to obtain from the results for the inverse power model, the first two terms of the series development of $\alpha$ in powers of $\frac{\epsilon }{\mathrm{kT}}$ for the general Lennard-Jones model of intermolecular forces.