\subsection{Analysis of the Enskog solutions}
\label{sec:enskog_valid}
The Enskog solutions provide a means of computing both diffusion coefficients and conductivities of gases, in addition to the Soret coefficient. The implemented model was tested with regard to it's ability to reliably reproduce the conductivities, diffusivities and thermal diffusion coefficient of a large number of gases and supercritical fluids. The Mie potential parameters given in Table \ref{tab:mie_params} were used for all computations. In computations using a hard sphere potential, the Mie potential $\sigma$-parameter was used for the hard sphere diameter. 

The average absolute deviation (AAD) between predictions and measurements is reported as
\begin{equation}
    \text{AAD} = \frac{1}{N}\sum_i \frac{\abs{\varphi_i - \hat{\varphi}_i}}{\varphi_i}
\end{equation}
where $\varphi_i$ are the measured values, $\hat{\varphi}_i$ are the predicted values, and $N$ is the number of measurements. Similarly, the average deviation (AD) is reported as 
\begin{equation}
    \text{AD} = \frac{1}{N}\sum_i \frac{\varphi_i - \hat{\varphi}_i}{\varphi_i}.
\end{equation}

Both values are used, as a large AAD may be due to large spread in experimental data that centres around model predictions, in which case the AD should be small. If both the AAD and AD are large, this is indicative of a systematic discrepancy between model predictions and reported data. There may be systematic discrepancies that are not captured by these measures, but they are deemed sufficient to give a good first indication of the models quality. In section \ref{subsub:valid_thermal_diffusion} an additional measure, the Pearson correlation coefficient, $\rho_P$, computed as 
\begin{equation}
    \rho_P = \frac{\text{Cov}[\varphi, \hat{\varphi}]}{\text{Var}[\varphi]^2\text{Var}[\hat{\varphi}]^2}
\end{equation}
where Cov$[\varphi, \hat{\varphi}]$ is the covariance of predicted and reported values, and Var$[\varphi]$ is the variance of $\varphi$, is used to analyse the discrepancy between measured and predicted values.

\subsubsection{Diffusion}
For a large variety of binary noble-gas mixtures, the Enskog solutions for Mie fluids reliably reproduced experimental results, as shown in Figure \ref{fig:D12_noble_gases}. The average absolute deviation over 193 data points was 2.77 \%, with 88 \% of the predictions lying within $\pm 5$ \%, and 96.8 \% of predictions lying within $\pm 7.5$ \% of data reported by Hogervorst, with a reported accuracy of 1-3 \%.\cite{noble_gas_diffusion}

In contrast, using a hard-sphere potential the AAD was 26.4 \%, with a large, systematic under-prediction of the diffusion coefficient, especially at high temperatures. Predictions using hard sphere potentials are shown in Figure \ref{fig:D12_noble_gases_hs}.

\begin{figure}[htb]
    \centering
    \includegraphics[width=.8\textwidth]{kinetic_gas/validation_figures/noble_gases2.eps}
    \caption{Predicted (lines) and measured (marks) diffusion coefficients of noble-gas mixtures at $p = \SI{1}{\atm}$, densities supplied to the Enskog solutions were computed using the SAFT-VR Mie EoS. Predctions made using Mie potentials. Helium mixtures contain 98.5 \% Helium, the remaining mixtures contain 99 \% of the first component. Data reported by Hogervorst.\cite{noble_gas_diffusion}}
    \label{fig:D12_noble_gases}
\end{figure}

Yoshida et al. report the self diffusion coefficient of water along several super-critical isotherms.\cite{water_self_diffusion} As shown in Figure \ref{fig:D12_water} the Enskog solutions for Mie fluids reproduce the diffusion coefficient at the lower reported densities, but becomes unreliable at higher densities and temperatures, with predicted values deviating by 10-20 \% from those reported.

It is evident that the density corrections discussed in Section \ref{subsubsec:density_corr_theory} are unable to capture all the effects of increased density. However, as previously mentioned, the water molecule is notoriously difficult to model, as it is highly polar and prone to forming hydrogen-bonds. In light of this, the discrepancies of 10-20 \% are in fact smaller than what one might expect. Additionally, it is notable that the clear trend of under-predicting the diffusion coefficient is the same as that reported by Lobanova et al. when using these parameters in MD-simulations.\cite{water_mie_params}

Predictions using a hard sphere potential are shown in Figure \ref{fig:D12_water_hs}. Deviations from reported data when using a hard sphere potential is significant, ranging up to 40 \% even at lower densities. 

\begin{figure}[hptb]
    \centering
    \includegraphics[width=.75\textwidth]{kinetic_gas/validation_figures/water.eps}
    \caption{Self-diffusion coefficient of supercritical water at several isotherms. Lines are predictions by the Enskog solutions for Mie fluids, marks are data reported by Yoshida et al.\cite{water_self_diffusion}}
    \label{fig:D12_water}
\end{figure}

Despite the discrepancy between the Enskog solutions and reported data for water at high temperatures and densities, the disagreement was notably $\lesssim 10$ \% at temperatures at or below \SI{573}{\kelvin} when using a Mie potential. Therefore, the high-density regime was investigated further. As shown in Figure \ref{fig:D12_mie}, the deviation from measured data increases rapidly as the density becomes liquid-like.

Afshin et al. report their simulation results in reduced units, and do not report the potential parameters they have used, other than the exponents. The values shown in Figure \ref{fig:D12_mie} are computed using the Mie parameters of Helium but with repulsive and attractive exponents 14 and 7. Using other sets of Mie parameters only resulted in scaling of the results. These results were not compared to predictions using a hard sphere potential, as the results would depend upon the choice of the $\varepsilon$-value one uses to obtain the temperature from the reported dimensionless temperature.

\begin{figure}[hptb]
    \centering
    \includegraphics[width=.75\textwidth]{kinetic_gas/validation_figures/mie_14_7.eps}
    \caption{The self-diffusion coefficient of the Mie-14-7 fluid, as predicted by the Enskog solutions for Mie fluids and measured by MD-simulation, along several isotherms. Measurements by Afshin et al.\cite{mie_14_7_diffusion} Values computed using Mie parameters for helium.}
    \label{fig:D12_mie}
\end{figure}

To compare the different data sets, the reduced phase point of the measurements shown in Figures \ref{fig:D12_noble_gases}, \ref{fig:D12_water} and \ref{fig:D12_mie}, in addition to measurements for a \ce{CO2}/\ce{CH4} mixture reported by Geuvara-Carrion et al,\cite{co2_ch4_diffusion} is shown in Figure \ref{fig:D12_phenv_errors}. The reduced phase point is computed as
\begin{equation}
    \begin{split}
        \rho^* = \rho N_A \sigma^3, \hspace{1cm} \sigma = \sum_i \sum_j x_i x_j \sigma_{ij},\\
        T^* = T \frac{k_B}{\varepsilon}, \hspace{1cm} \varepsilon = \sum_i \sum_j x_i x_j \varepsilon_{ij}
    \end{split}
\end{equation}
where the cross-interaction parameters are obtained via the LB mixing rules. It is clear from Figure \ref{fig:D12_phenv_errors} that the deviation between the predicted and measured diffusion coefficient is stable and on the order of $\lesssim 10-15\%$ until a reduced density of $\approx 0.1$, after which it increases rapidly. There does not appear to be a clear correlation between the temperature and the deviation from measured data. Notably, a reduced density of $0.1$ corresponds to an average inter-particle distance of only $\approx 2.15 \sigma$, a relatively short distance when having in mind the assumptions behind the Enskog solutions.
\begin{figure}[htb]
    \centering
    \includegraphics[width=.8\textwidth]{kinetic_gas/validation_figures/phenv_errors.eps}
    \caption{Relative discrepancy between the Enskog solutions for diffusion using a Mie potential and reported values presented in Section \ref{sec:enskog_valid}, and the reduced phase point of the reported data. The black solid line in the central figure indicates the phase envelope of the LJ-fluid as computed with SAFT-VR Mie. Dotted and dashed lines indicate the 10th and 20th percentiles respectively. $^{(a)}$Hogervorst,\cite{noble_gas_diffusion} $^{(b)}$Yoshida et al.,\cite{water_self_diffusion} $^{(c)}$Guevara-Carrion et al.,\cite{co2_ch4_diffusion} $^{(d)}$Afshin et al.\cite{mie_14_7_diffusion}}
    \label{fig:D12_phenv_errors}
\end{figure}

\newpage
\subsubsection{Conductivity}

Further, the Enskog solutions' ability to reproduce experimental conductivities was investigated. As shown in Figure \ref{fig:noble_gas_conductivity}, the predictive capability of the solutions varies notably between different species, and deteriorates rather quickly with increasing density for some species.

\begin{figure}[htb]
    \centering
    \includegraphics[width=.8\textwidth]{kinetic_gas/validation_figures/noble_gas_conductivities.eps}
    \caption{Conductivity of noble gases and noble gas mixtures at varying density. Predictions by the Enskog solutions with Mie potentials. All measurements at \SI{35}{\celsius}.$^{(a)}x_{\ce{He}} = 0.2172$, $^{(b)}x_{\ce{He}} = 0.4662$, $^{(c)}x_{\ce{He}} = 0.6823$, $^{(d)}x_{\ce{He}} = 0.7156$, $^{(e)}x_{\ce{Ar}} = 0.4088$, $^{(f)}x_{\ce{Ar}} = 0.7059$. Measurements reported by Assael et al.\cite{noble_gas_conductivity}}
    \label{fig:noble_gas_conductivity}
\end{figure}

Using hard sphere potentials the disagreement between predictions and measurements ranged from -40 \% to +30 \%, even at the lowest densities. Very few predictions made with the hard sphere potentials were within $\pm$10 \% of measurements. Conductivity predictions made with hard sphere potentials are shown in Figure \ref{fig:noble_gas_conductivity_hs}. 

\begin{figure}[htb]
    \centering
    \includegraphics[width=.8\textwidth]{kinetic_gas/validation_figures/conductivity_phenv.eps}
    \caption{The reduced phase point of the measurements presented in in Figure \ref{fig:noble_gas_conductivity}. The solid black line in the central figure indicates the vapour-liquid equilibria for the Lennard-Jones fluid, computed using the SAFT-VR Mie EoS. Dotted and dashed lines indicate the 5th and 10th percentiles respectively. $^{(a)}x_{\ce{He}} = 0.2172$, $^{(b)}x_{\ce{He}} = 0.4662$, $^{(c)}x_{\ce{He}} = 0.6823$, $^{(d)}x_{\ce{He}} = 0.7156$, $^{(e)}x_{\ce{Ar}} = 0.4088$, $^{(f)}x_{\ce{Ar}} = 0.7059$. Measurements reported by Assael et al.\cite{noble_gas_conductivity}}
    \label{fig:conductivity_phenv}
\end{figure}

For predictions made with Mie potentials, the reduced state point of the various measurements was investigated. This revealed that the disagreement between predictions and measurements increases far more rapidly at low reduced temperatures, as shown in Figure \ref{fig:conductivity_phenv}. As all the presented conductivity measurements are in the gas phase, it is deemed unlikely that long time-correlations and thereby correlated collisions are the reason for the deviations, even at low temperature. However, it was commented in Section \ref{sec:derivation_eff_density} that the implemented density corrections for conductivity in mixtures are a simplification. These results may be indicative that the assumptions made when simplifying the density corrections are better at higher temperatures.

\newpage
\subsubsection{The thermal diffusion ratio}
\label{subsub:valid_thermal_diffusion}
Vargaftik has compiled an extensive set of thermal diffusion ratios for dilute gas mixtures at varying composition and temperature.\cite{vargaftik} For 970 of these data points, Mie potential parameters have been acquired, as presented in Section \ref{sec:mie_params}. Before comparing the experimental values to model predictions, the data set for each mixture was visually inspected. In total 32 data points were disregarded as clear outliers, these were evenly distributed throughout the data set when regarding mixture components, composition, temperature and reduced temperature.

As shown in Figure \ref{fig:vargaftik_all}, there is a clear trend of the Enskog solutions over-predicting the thermal diffusion ratio when supplied with a Mie potential. However, closer inspection revealed that practically all the bias could be attributed to various mixtures containing hydrogen. Aasen et al. have shown that hydrogen mixtures can exhibit significant non-ideality, and that quantum effects can play a significant role in the behaviour of hydrogen.\cite{saft_mixtures} When predicting equilibrium properties of hydrogen and hydrogen mixtures they find that a Feynmann-Hibbs corrected Mie potential gives significant improvement in model predictions compared to an ordinary Mie potential. Additionally, they report mixing parameters $k_{ij}$ and $l_{ij}$ for mixtures of hydrogen with neon and helium. Using the mixing parameters they report in the computation of the cross interaction parameters, as described in Section \ref{sec:teori_eos} removed the bias in the predictions regarding these mixtures. It is therefore likely that the mixtures of hydrogen with carbon dioxide, nitrogen, and oxygen, indicated in Figure \ref{fig:vargaftik_all_corr} also have a bias due to the mixing rule being skewed, and possibly due to quantum effects that are not captured by the Mie potential.

\begin{figure}[htb]
    \centering
    \subfloat[]{\label{fig:vargaftik_all_corr}\includegraphics[width=.49\textwidth]{kinetic_gas/validation_figures/vargaftik_correlation.eps}}
    \hfill
    \subfloat[]{\label{fig:vargaftik_all_stacked}\includegraphics[width=.49\textwidth]{kinetic_gas/validation_figures/vargaftik_all_stacked.eps}}\\
    \caption{(a) Correlation between predicted and reported thermal diffusion ratio using a Mie potential. Mixtures that showed systematic deviations when using the Mie potential are indicated. (b) Stacked distribution of deviation between predicted and measured thermal diffusion ratios using a Mie potential. Red fields are the specific mixtures indicated in (a), blue fields are the remaining mixtures, indicated with blue dots in (a). AAD and AD are computed using all points.}
    \label{fig:vargaftik_all}
\end{figure}

Separating the hydrogen mixtures for which no mixing parameters are supplied from the remaining data set, as shown in Figure \ref{fig:vargaftik_all_stacked}, clearly demonstrates that the bias for over-predicting the thermal diffusion ratio with the Mie potential is largely confined to these specific mixtures. The remaining data points, indicated as blue dots and a blue field in Figure \ref{fig:vargaftik_all}, are close to normally distributed about the model predictions. It is noted that all the mixtures with large, systematic discrepancies between predictions and measurements contain multi-atomic species, all of which have a non-zero quadrupole moment. For these species, the approximations of a spherical potential and neglecting rotational or vibrational energy are expected to be worse than for monoatomic species.

The AAD of the thermal diffusion coefficients is significant at $16.4$ \% for the data set as a whole, and looking at Table \ref{tab:vargaftik_table} it is clear that very few mixtures have average deviations close to zero. However, the bias does not appear to correlate clearly with anything other than the presence of hydrogen. The correlation between the relative deviation and molar masses, molar mass differences, temperature, reduced temperature, molecular size, molecular density and potential well depth was checked, as shown in Appendix \ref{sec:error_correlation}. The largest correlation coefficient was observed between the deviation and the maximum well depth among the two components, but this appears to arise more from the varying size of the data sets than any actual underlying correlation. Furthermore, the apparent correlation of error with the presence of hydrogen may be due to systematic experimental errors, as hydrogen gas is notorious for being able to diffuse well through most substances, possibly leading to systematically erroneous measurements.

Due to the manner in which the entirety of the data centres around the model predictions, even though few of the individual mixtures' data sets do, it is believed that the large deviations are primarily due to experimental error. This is supported by the large standard deviation in the discrepancy both within the individual data sets, and the the data set as a whole, as well as the lack of a clear correlation between the deviation and any physical property of the mixtures.

\begin{table}[htb]
    \centering
    \caption{Average deviation (AD) and average absolute deviation (AAD) between data reported by Vargaftik and predictions by the Enskog solutions for Mie fluids and hard spheres (HS).\cite{vargaftik} The mixtures for which mixing parameters reported by Aasen et al. were available, these were used in the computations for Mie fluids.\cite{saft_mixtures} }
    \begin{tabular}{c c c c c c | c c c c c c}
    \toprule
        & \multicolumn{2}{c}{AD [\%]} & \multicolumn{2}{c}{AAD [\%]} &  & & \multicolumn{2}{c}{AD [\%]} & \multicolumn{2}{c}{AAD [\%]} & \\
        Mixture & Mie & HS & Mie & HS & NDP & Mixture & Mie & HS & Mie & HS & NDP\\
    \midrule
         \ce{H2}/\ce{CO2} & 15.8 & 49.8 & 16.2 & 49.8 & 194 & \ce{H2}/\ce{D2} & -6.7 & 45.2 & 19.5 & 45.2 & 27\\
        \ce{H2}/\ce{N2} & 12.3 & 55.3 & 14.7 & 55.3 & 165 & \ce{H2}/\ce{O2} & 21.0 & 63.9 & 21.0 & 63.9 & 22\\
        \ce{He}/\ce{Ar} & -3.3 & 33.2 & 10.7 & 33.2 & 85 & \ce{N2}/\ce{Ar} & -29.3 & 58.9 & 29.8 & 58.9 & 22\\
        \ce{He}/\ce{Ne} & -16.5 & 25.1 & 16.7 & 25.1 & 66 & \ce{D2}/\ce{N2} & 6.2 & 55.0 & 7.8 & 55.0 & 15\\
        \ce{H2}/\ce{He} & 17.0 & 50.5 & 19.2 & 50.5 & 58 & \ce{He}/\ce{N2} & -25.8 & 21.7 & 25.9 & 21.7 & 14\\
        \ce{N2}/\ce{CO2} & 32.1 & 66.2 & 32.5 & 66.2 & 58 & \ce{Ne}/\ce{Xe} & 9.6 & 55.6 & 14.2 & 55.6 & 11\\
        \ce{Ne}/\ce{Ar} & 0.0 & 50.9 & 8.2 & 50.9 & 48 & \ce{D2}/\ce{Ne} & 9.1 & 53.3 & 9.1 & 53.3 & 11\\
        \ce{O2}/\ce{CO2} & 7.7 & 54.4 & 14.9 & 54.4 & 43 & \ce{He}/\ce{Xe} & 4.6 & 38.5 & 8.1 & 38.5 & 11\\
        \ce{H2}/\ce{Ne} & -11.0 & 40.4 & 15.3 & 40.4 & 42 & \ce{Ar}/\ce{Xe} & 17.1 & 76.7 & 17.1 & 76.7 & 5\\
        \ce{H2}/\ce{Ar} & 12.6 & 55.3 & 13.6 & 55.3 & 39 & \ce{H2}/\ce{CH4} & 24.9 & 54.3 & 24.9 & 54.3 & 2\\
    \bottomrule
    \end{tabular}

     \label{tab:vargaftik_table}
\end{table}

For comparison, Figure \ref{fig:vargaftik_all_hs} shows the predictions obtained from the Enskog solutions when using a hard sphere (HS) potential. The HS diameters were set to be equal to the $\sigma$-parameters of the Mie potentials used for the predictions displayed in Figure \ref{fig:vargaftik_all}. There is no doubt that there is a large, systematic error in the predictions. Additionally, as can be seen in Figure \ref{fig:vargaftik_all_stacked_hs}, the errors to not appear to follow a normal distribution as one would expect if they were the result of experimental error. 

\begin{figure}
    \centering
    \subfloat[]{\label{fig:vargaftik_all_corr_hs}\includegraphics[width=.49\textwidth]{results/figures/vargaftik_correlation_hs.eps}}
    \hfill
    \subfloat[]{\label{fig:vargaftik_all_stacked_hs}\includegraphics[width=.49\textwidth]{results/figures/vargaftik_all_stacked_hs.eps}}
    \caption{(a) Correlation between predicted and reported thermal diffusion ratio using a HS potential, with HS-diameters equal to the $\sigma$-parameters of the Mie potentials used for the predictions in Figure \ref{fig:vargaftik_all}. Mixtures that showed systematic deviations when using the Mie potential are indicated. (b) Stacked distribution of deviation between predicted and measured thermal diffusion ratios using a HS potential. Red fields are the specific mixtures indicated in (a), blue fields are the remaining mixtures, indicated with blue dots in (a). AAD and AD are computed using all points.}
    \label{fig:vargaftik_all_hs}
\end{figure}

Just as with the Mie-potential predictions displayed in Figure \ref{fig:vargaftik_all}, an investigation was made into whether the errors correlated with any physical parameters. It was found that the error correlated weakly with the inverse temperature, but that this correlation was amplified by the reduced mass of the particles, the combined descriptor 
\begin{equation}
    \hat{\epsilon}_T = \frac{m_1 m_2}{m_1 + m_2}\frac{1}{T}
\end{equation}
visually appeared to capture a correlation in the discrepancy between measurements and predictions, shown in Figure \ref{fig:hs_error_corr_et}. Further, the discrepancy appeared to correlate with the inverse of the mass difference. This correlation was also amplified by the reduced mass of the species, this correlation was largely captured by the descriptor
\begin{equation}
    \hat{\epsilon}_m = \frac{m_1 m_2}{m_1 + m_2}\frac{1}{\abs{\Delta m}},
\end{equation}

as shown in Figure \ref{fig:hs_error_corr_em}. Finally, there appeared to be a positive correlation between the mole fraction of the heavier component and the discrepancy. From these observations, the combined descriptor
\begin{equation}
    \hat{\epsilon} = \frac{m_1 m_2}{m_1 + m_2}\lrp \frac{1}{T} + \frac{0.05}{\abs{\Delta m}} \rrp + 0.05 x_2
\end{equation}
was constructed, where $x_2$ is the mole fraction of the heavier component. The scaling factors of $0.05$ were introduced such that the different terms of the descriptor would be of approximately the same order of magnitude. The correlations for the individual descriptors $\hat{\epsilon}_T$, $\hat{\epsilon}_m$ and $x_2$ are shown in Appendix \ref{sec:error_correlation}.

As shown in Figure \ref{fig:hs_error_correlation} there is a convincing correlation between $\hat{\epsilon}$ and the discrepancy, and the correlation coefficient is 0.59. This is not a very high correlation coefficient, but given the large spread in the data, it is significant. More important is the fact that visually, it is clear that the discrepancy increases with $\hat{\epsilon}$, both within most of the mixtures and between mixtures. 

\begin{figure}
    \centering
    \includegraphics[width=.8\textwidth]{results/figures/hs_error_correlation.eps}
    \caption{Correlation between the descriptor $\hat{\epsilon}$ and the discrepancy between thermal diffusion ratios measured by Vargaftik,\cite{vargaftik} and predicted by the 5th order Enskog solutions with a HS potential. Different mixtures are indicated with different colours and markers.}
    \label{fig:hs_error_correlation}
\end{figure}

The fact that a simple function of the physical parameters of the system correlates significantly with the discrepancy between predictions and measurements indicates that this discrepancy cannot be wholly attributed to measurement error or inaccurate estimates for the hard-sphere diameters. In fact, taking a closer look at the three terms that constitute $\hat{\epsilon}$ one can deduce a physical significance. 

The first term, $\hat{\epsilon}_T = \frac{m_1 m_2}{m_1 + m_2} \frac{1}{T}$, is closely related to the average relative speed as, $\Bar{g}_{12} \propto \hat{\epsilon}_T^{- \frac{1}{2}}$. That is, the discrepancy increases with decreasing average relative speed during collisions. Assuming that the Mie potential is a somewhat realistic model for the molecular interactions, this makes sense. The softness of the potential and the attractive part of the potential have a larger impact on the collision dynamics at low relative speeds, while the particles behave more like hard spheres during high-speed collisions.

The second term, $\hat{\epsilon}_m = \frac{m_1 m_2}{m_1 + m_2}\frac{1}{\abs{\Delta m}}$ is largest for particles of similar mass. In fact, taking $m_2$ to be the larger mass, $\hat{\epsilon}_m$ can be rewritten to only be a function of the mass ratio,
\begin{equation}
    \hat{\epsilon}_m = \frac{m_1 m_2}{m_1 + m_2}\frac{1}{\abs{\Delta m}} = \frac{m_1 / m_2}{1 - (m_1 / m_2)^2}, \hspace{.5cm} m_2 > m_1.
\end{equation}
As mentioned in Section \ref{subsec:intro_models}, several authors discuss that the Soret effect may be split into an isotope effect and configurational or chemical effects.\cite{soret_review_2022,soret_intro} This term captures the fact that for particles of similar mass, the isotope effect will be weak, and the thermal diffusion ratio will to a larger degree be determined by chemical effects, which in turn are determined by the interaction potential. Therefore, it is reasonable that the hard sphere potential systematically gives worse predictions for particles of similar mass.

The final term, $x_2$, the mole fraction of the heavier component, captures the fact that for most mixtures, the discrepancy between predictions and measurements increased with the mole-average molecular mass. This may also have its roots in the fact that molecular masses are connected to the average molecular speed, such that at high mole fractions of the heavy component, the average molecular speed is lower. As argued when interpreting the significance of $\hat{\epsilon}_T$, faster moving molecules behave more like hard spheres. Therefore it is not unreasonable that a hard sphere model will become less accurate as the number of heavy molecules increase, at the expense of lighter molecules. Thus, this term also appears to have a reasonable physical interpretation.

Despite this analysis, one could argue that using the $\sigma$-parameters from a Mie potential to approximate the HS diameters is inaccurate, and that one could achieve more accurate predictions by estimating the HS-diameters in some other way. However, the dependency of the thermal diffusion ratio of hard spheres on temperature and molecular masses, as predicted by the Enskog solutions, is independent of the hard sphere diameters. This means that improving accuracy at some temperature, or for some mixture, even by fitting the hard sphere diameters would likely decrease accuracy at another temperature or for another mixture. This is discussed in more detail in Section \ref{sec:results_ideal_gas}.