\subsection{A modified Kempers approach}
\label{sec:method_kempers_hs}
When one views the 2001-approach as two independent optimisations, it is not immediately clear why the real state is assumed to obey Equation \eqref{eq:kempers_89_result}, given as 
\begin{equation}
    \nabla \pfrac{\mu_i}{T} - \pfrac{v_i}{v_k}\nabla \pfrac{\mu_k}{T} = 0, \hspace{1cm} i = \{1, ..., N\} \setminus \{k\}.
    \label{eq:kempers_89_result_method}
\end{equation}

while the ideal gas state does not. Additionally, it has previously been noted that the Enskog solutions for hard spheres without density corrections do not represent an ideal gas, but an infinitely dilute mixture of hard spheres.\cite{prosjektoppgave} The difference between the two is discussed in more detail in Section \ref{sec:results_ideal_gas}. It has been proposed that one should therefore not use the ideal gas state as a reference, but rather a hard sphere state, which may be accurately described within the Boublík formalism.\cite{boublik,saft_vr_mie_2}

If, following Kempers, one assumes that all systems obey Equation \eqref{eq:kempers_89_result_method} in the steady state, and that the Soret coefficient in an infinitely dilute mixture of hard spheres is accurately described by kinetic gas theory, one can set up Equation \eqref{eq:kempers_89_result_method} for both the hard sphere state and the real state,
\begin{equation}
    \begin{split}
        \nabla \pfrac{\mu_i}{T} - \frac{v_i}{v_k}\nabla \pfrac{\mu_k}{T} &= 0\\
        \nabla \pfrac{\mu^{HS}_i}{T} - \frac{v^{HS}_i}{v^{HS}_k}\nabla \pfrac{\mu^{HS}_k}{T} &= 0
    \end{split}
\end{equation}
Now, subtracting the two, as in the 2001-approach, and inserting the expansion of $\nabla \pfrac{\mu_i}{T}$ from Equation \eqref{eq:kempers_01_intermediate_3}, one arrives at
\begin{equation}
    \begin{split}
        &\DeltaHS \lcp -\frac{h_i}{T^2} - \frac{1}{T} \sum_j \ppder{\mu_i}{x_j} x_j (1 - x_j) S_{T,j} \rcp \\
        &\hspace{1.5cm}- \DeltaHS \lcp \frac{v_i}{v_k} \lrp -\frac{h_k}{T^2} - \frac{1}{T} \sum_j \ppder{\mu_k}{x_j} x_j (1 - x_j) S_{T,j} \rrp \rcp = 0,
    \end{split}
\end{equation}
where $\DeltaHS \varphi \equiv \varphi - \varphi^{HS}$ indicates the difference between the real value and the hard sphere value for some variable $\varphi$. Together with the condition that $\sum_j x_j(1 - x_j) S_{T,j} = 0$, and with $S_{T,i}^{HS}$ supplied from kinetic gas theory, this set of equations can be solved for the Soret coefficients in the real state. 

All thermodynamic quantities required to solve this set of equations, with the exception of $\ppder{\mu_i}{x_j}_{T,p}$, are directly available in the thermodynamic library ThermoPack.\cite{thermopack} A relationship allowing one to compute $\ppder{\mu_i}{x_j}_{T,p}$ from quantities that are available is derived in Appendix \ref{sec:eos_differentials}. The entirety of the implementation of the Kempers model can be found in a public repository on \href{https://github.com/vegardjervell/Kempers01}{GitHub}.\cite{jervell_git_kempers}