\subsection{Thermal diffusion in an Ideal gas}
\label{sec:results_ideal_gas}

As mentioned in Section \ref{sec:method_kempers_hs} there is a distinction between a fluid at infinite dilution, and an ideal gas. Strictly, an ideal gas consists of non-interacting point particles. Although both hard spheres and Mie fluids behave as such when regarding equilibrium properties at infinite dilution, this is not the case regarding non-equilibrium properties. From the derivation in Section \ref{sec:theory_kingas}, one will find that the collision integrals vanish for non-interacting particles. Investigating the limiting case in which the collision integrals vanish, one finds that the solutions to Equation \eqref{eq:kin_d_matrix_form}, that yield the polynomial expansion coefficients from which the thermal diffusion ratio is determined, become ill defined, depending on the manner in which the integrals vanish. Explicitly, the limit is different if one lets the radius of the spheres in a hard sphere mixture approach zero from the case in which one lets the well depth parameters of a Mie fluid approach zero. Even restricting the case to that of hard spheres, the limit will be different depending on the mixing rule employed to obtain the cross-interaction diameter, and the ratio of the hard sphere diameters. It should be mentioned that this was only investigated for the 3rd approximation, i.e. there may exist a well defined limit to some Nth approximation, even though the limit to a lower order approximation is not well defined. This is however of little immediate practical consequence. The argument still stands that for some finite approximation to the Enskog solutions, the transport coefficients in the limit of non-interacting particles depends on how one approaches the limit and is therefore ill-defined.

\begin{table}[htb]
    \centering
    \caption{Mie potential parameters, and the hard sphere diameters producing the same thermal conductivity for the pure fluids at \SI{300}{\kelvin}, \SI{10}{\mol\per\cubic\meter} at the 3rd Enskog approximation.}
    \begin{tabular}{c c c c c c}
    \toprule
        Potential & $\sigma$ [Å] & $m$ [\si{\gram\per\mol}]& $\varepsilon$ [$k_B$] & $\lambda_a$ [-] & $\lambda_r$ [-] \\
    \midrule
        $u_1^{Mie}$ & 3.5 & 10 & 250 & 6 & 12 \\
        $u_2^{Mie}$ & 3 & 5 & 150 & 6 & 12 \\
        HS1 & 4.2747 & 10 & & & \\
        HS2 & 3.2978 & 5 & & & \\ 
    \bottomrule
    \end{tabular}
    \label{tab:eq_hs_diameter}
\end{table}

To demonstrate the consequences of this, the hard sphere diameter of two species, simply termed HS1 and HS2, were fit to the pure component conductivity of two Mie fluids, using the potentials $u_1^{Mie}$ and $u_{2}^{Mie}$ at $T = \SI{300}{\kelvin}$, $\rho = \SI{10}{\mol\per\cubic\meter}$, where the state point was chosen arbitrarily. The Mie potential parameters and the regression results are shown in Table \ref{tab:eq_hs_diameter}. As shown in Figure \ref{fig:eq_hs_cond_x}, using these HS-diameters gave a reasonable match between the HS-conductivity and Mie-conductivity of the mixture at the same state point. However, as shown in Figure \ref{fig:eq_hs_temp_depend}, the temperature dependency is completely different. Furthermore, the diffusion coefficient in the mixtures is completely different. This is by no means surprising, but recognise the fact that because these predictions are made using the non-density corrected Enskog solutions, both these predictions are for an ideal gas.

\begin{figure}
    \centering
    \includegraphics[width=.6\textwidth]{results/figures/eq_hs_cond_x.eps}
    \caption{Conductivity of the mixture described by the parameters shown in Table \ref{tab:eq_hs_diameter} at \SI{300}{\kelvin}, \SI{10}{\mol\per\cubic\meter}. Computed to the 3rd Enskog approximation.}
    \label{fig:eq_hs_cond_x}
\end{figure}

This underlines the fact that speaking of the transport properties of an ideal gas, as Kempers does, is not fruitful. The transport properties of a fluid at infinite dilution depend upon the interaction potentials, even though equilibrium properties do not. This is by no means news,\cite{chapman_cowling,kinetic_review,soret_review_2022} but in the context of the Kempers model it shows that predicting the thermal diffusion coefficient through residual properties that may be determined from equilibrium thermodynamics is bound to fail. This is because different equations of state, based on different underlying potentials, will predict the same ideal gas behaviour in the low density limit, while the transport coefficients behave differently based on the underlying potential.

\begin{figure}
    \centering
    \includegraphics[width=.8\textwidth]{results/figures/eq_hs_D_k_T.eps}
    \caption{Temperature dependency of the conductivity and diffusion coefficient of the hard sphere mixture and Mie mixture described by the parameters in Table \ref{tab:eq_hs_diameter}, at $x_1$ = 0.1, $\rho = \SI{10}{\mol\per\cubic\meter}$.}
    \label{fig:eq_hs_temp_depend}
\end{figure}

These results also tie into the argument presented in Section \ref{sec:enskog_valid}, regarding the possibility that the poor predictions obtained from the Enskog solutions using hard sphere potentials were due to poor estimates of the hard sphere diameters. Even when fitting the hard sphere diameters to pure component conductivities predictions for the mixture deviated from those obtained from a Mie potential. Recall from Figure \ref{fig:noble_gas_conductivity} that the Enskog solutions using a Mie potential predicted the conductivities of mixtures with high accuracy. This demonstrates that using a Mie potential is a significant improvement, even compared to fitting hard sphere diameters to data. Furthermore, the results shown in Figure \ref{fig:eq_hs_temp_depend} illustrate the fact that fitting hard sphere diameters to one property, in this case conductivity, at a given temperature does not give reasonable predictions for another property, diffusion, or even for conductivity at other temperatures.