\subsection{Kempers assumptions of the steady state}

To investigate the assumption that the steady state for a two-bulb system such as the one described in Section \ref{sec:kempers_model} is a minimum in the target function
\begin{equation}
    \tau = \frac{A^{\alpha}}{T^{\alpha}} + \frac{A^{\beta}}{T^{\beta}}
\end{equation}
the target function was computed as a function of $\Delta n_1$ and $\Delta n_2$ for an ideal gas, using the Sackur-Tetrode equation to compute the pure component entropies. Then, the placement of the curves corresponding to $\Delta p = 0$, $\Delta m = 0$ and the steady state predicted by the Enskog solutions, as derived in Section \ref{subsec:method_steady_state}, were superimposed on this surface.

Note that without density corrections, the Enskog solutions predict that the Soret coefficient is independent of density. The predicted value is that at infinite dilution. Also, without density corrections, the Enskog solutions recover the ideal gas law, regardless of the intermolecular potential that is used. Because the Enskog solutions for Mie fluids are consistent with the ideal gas law, $\tau$ is computed for an ideal gas, rather than using e.g. the SAFT-VR Mie EoS at very low density. It should be commented that SAFT-VR Mie reduces to the ideal gas law at infinite dilution. As a consistency check, the analysis presented in this section was also conducted using the SAFT-VR Mie EoS at very low density. This gave results that, for all practical purposes, were equivalent to those presented here.

The procedure of superimposing the curves corresponding to various constraints on the target function surface allows visualisation of how the target function is permitted to vary with bulb mole numbers when subjected to these constraints. Figure \ref{fig:res_helmholtz_hexe} displays the target function of an ideal He/Xe mixture with $x_{\ce{He}} = 0.5$ (blue grid). Note that the absolute value of the global minimum has been subtracted, such that the global minimum is at \SI{0}{\joule\per\kelvin}, this is simply for clearer visualisation and only changes the absolute value of the target function, not the location of any minima. The red line indicates the path along the surface satisfying $\Delta p = 0$, as determined from Equation \eqref{eq:dp_0_curve}. Likewise, the green line indicates the path along which $\Delta m = 0$. The blue line indicates the steady state predicted by the Enskog solutions, as derived in Section \ref{subsec:method_steady_state}. The marks on each line indicate the constrained minimum in the target function along that line.

To elaborate, the Enskog solutions permit that the steady state lies somewhere on the blue curve, along which the compositional- and pressure difference may vary. The steady state with no pressure difference that satisfies the Enskog solutions is thereby the intersection between the red and blue curves. The steady state with equal container mass is the intersection between the green and blue curves. It is immediately clear neither of these intersections coincide with the constrained minima along the curves, as assumed by Kempers. 
\begin{figure}[htb]
    \centering
    \includegraphics[width=.7\textwidth]{results/figures/he_xe.eps}
    \caption{The objective function in the Kempers approach, normalised to the global minimum $A^{\circ}$, for an ideal He/Xe mixture, with $x_{\ce{He}} = 0.5$ at $T = \SI{300}{\kelvin}$, with a total mole number of $n = \SI{1000}{\mol}$. The red line indicates the curve of no pressure difference, the green line indicates no mass difference, the blue line indicates the separation determined by the Enskog solutions at the bulb average conditions, as derived in Section \ref{subsec:method_steady_state}. Marks indicate the constrained minimum along each curve, the black mark indicates the global minimum.}
    \label{fig:res_helmholtz_hexe}
\end{figure}

While Figure \ref{fig:res_helmholtz_hexe} is an illustrative example, this was checked for a variety of mixtures and at varying conditions. For some mixtures the steady states were closer to the minima, for others further away. There did not appear to be a systematic manner of predicting how far away from the minima the steady states would be for a given mixture. Furthermore, for some mixtures the steady state of equal pressure was closer to the minima than that of equal mass, while for others the opposite was true.

These observations are very similar to what has been observed regarding the predictions of the Kempers model: Sometimes predictions are close to measurements, but determining beforehand whether the model will work for a given mixture beforehand has proven itself to be a significant challenge.\cite{prosjektoppgave} Also, sometimes the Kempers centre-of-volume model yields better predictions than the centre-of-mass model, while other times the opposite is true, with no clear manner of determining which is appropriate for a given system without checking.

This draws the fundamental assumptions behind the Kempers model heavily into question, and two possible explanations for these observations immediately present themselves: Either the Enskog solutions do not correctly predict the Soret coefficient, even for an ideal gas; or the steady state is not at a minimum, constrained or otherwise, in the target function. Both possibilities are in direct contradiction to Kempers' initial assumptions.