\section{Numeric Minimization}
In order to investigate the foundation of the Kempers model, and \textcolor{red}{something something consistency with Kinetic Gas theory ...}

To compare the results of a numeric minimisation with Kinetic Gas theory, one must relate the Soret coefficient to the mole numbers in two containers (\textcolor{red}{as described ...})

At steady state, from the definition of the Soret coefficient
\begin{equation}
    \DeltaAB x_i = - x_i (1 - x_i) S_{T,i} \DeltaAB T.
\end{equation}
The difference in density between the bulbs is
\begin{equation}
    \begin{split}
        \DeltaAB \rho &\approx \ppder{\rho}{T}_{p, \vx} \DeltaAB T + \sum_i \ppder{\rho}{x_i}_{T, p} \DeltaAB x_i\\
        &= \lsp \ppder{\rho}{T}_{p, \vx} - \sum_i \ppder{\rho}{x_i}_{T, p} x_i (1 - x_i) S_{T,i} \rsp \DeltaAB T
    \end{split}
\end{equation}
Where the differentials are computed at the volume average conditions of the system. With equal bulb volumes, $V^{\alpha} = V^{\beta} = \frac{V}{2}$,
\begin{equation}
    \DeltaAB n = \frac{V}{2} \DeltaAB \rho.
\end{equation}
Finally,
\begin{equation}
    \begin{split}
        \DeltaAB n_i &=  n^{\beta} x_i^{\beta} - n^{\alpha}x_i^{\alpha}\\
        &= (n^{\alpha} + \DeltaAB n) x_i^{\beta} - n^{\alpha} x_i^{\alpha}\\
        &= x_i^{\beta} \DeltaAB n + n^{\alpha} \DeltaAB x_i \\
        &= x_i^{\beta} \DeltaAB n + \frac{n - \DeltaAB n}{2} \DeltaAB x_i\\
        &= \underbrace{x_i^{\beta}}_{\approx x_i} \DeltaAB n + \frac{n}{2} \DeltaAB x_i - \underbrace{\frac{\DeltaAB n \DeltaAB x_i}{2}}_{\approx 0}\\
        &\approx x_i \DeltaAB n + \frac{n}{2} \DeltaAB x_i\\
        &= \lrp \frac{x_i V}{2} \lsp \ppder{\rho}{T}_{p, \vx} - \sum_i \ppder{\rho}{x_i}_{T, p} x_i (1 - x_i) S_{T,i} \rsp - \frac{n}{2} x_i (1 - x_i) S_{T,i}\rrp \DeltaAB T
    \end{split}
\end{equation}