\subsection{The Kempers Model}
\label{sec:kempers_model}
In this section, the derivation of the models proposed by Kempers will be presented.\cite{kempers89,kempers01} The model is developed by regarding a closed two-bulb system, as illustrated in Figure  \ref{fig:kemper_two_bulb}, the two containers are kept at uniform temperature, pressure and composition, while the connecting tube is of negligible volume. The underlying assumptions are that the steady state of this system, when the containers are at different temperatures, is the state with a maximum in the canonical partition function ($Q$). The bulbs will be denoted with superscript $\alpha$ and $\beta$, symbols lacking a superscript will be used to refer to the system as a whole. It is further assumed that the Soret coefficient in an ideal gas mixture, denoted with superscript $ig$, is accurately described by kinetic gas theory.

\begin{figure}
    \centering
    \includegraphics[width=.7\textwidth]{theory/figures/kempers_two_bulb.eps}
    \caption{Illustration of the Kempers two-bulb system. The two containers are homogeneous, the connecting tube has negligible volume.}
    \label{fig:kemper_two_bulb}
\end{figure}

\subsubsection{The 1989-approach}
Instead of treating the problem of two bulbs that exchange particles, the bulbs may be treated as independent subsystems, under the constraint of total particle number conservation. In that case, $Q = Q^{\alpha}Q^{\beta}$ and $n_i^{\alpha} + n_i^{\beta} = n_i$ $\forall$ $i$. Additionally, the two subsystems are held at constant and equal volume, such that $\sum_i v_i^{\alpha}n_i^{\alpha} = \sum_i v_i^{\beta}n_i^{\beta} = \frac{V}{2}$, with $V$ as the total system volume and $v_i$ as the partial molar volume of species $i$. Under the current assumptions, the steady state compositions for given temperatures $T^{\alpha}$ and $T^{\beta}$ are then given by the constrained maximisation
\begin{equation}
    \begin{split}
        \max_{n_i^\alpha, n_i^{\beta}}\left\{Q^{\alpha}Q^{\beta}\right\} &= \max_{n_i^\alpha, n_i^{\beta}}\left\{\exp\lrp-\frac{A^{\alpha}}{k_B T^{\alpha}} - \frac{A^{\beta}}{k_B T^{\beta}}\rrp\right\} \\ &=\min_{n_i^\alpha, n_i^{\beta}}\left\{\frac{A^{\alpha}}{T^{\alpha}} + \frac{A^{\beta}}{T^{\beta}}\right\} \\
        n_i^{\alpha} + n_i^{\beta} &= n_i, \hspace{.5cm} \forall \hspace{.5cm} i \\
        \sum_i v_i^{\alpha}n_i^{\alpha} &= \sum_i v_i^{\beta}n_i^{\beta}
    \end{split}
    \label{eq:1989_problem}
\end{equation}

The subscripts on the $\min$ and $\max$ operators indicate the optimisation variables, and are to be understood such that for some function $\psi(\varphi_1, \varphi_2)$,
\begin{equation}
    \begin{split}
        \max_{\varphi_1, \varphi_2}\left\{\psi\right\} = \lrp\varphi_1^{extrema}, \varphi_2^{extrema}\rrp = \min_{\varphi_1, \varphi_2}\left\{-\psi\right\}\\
        \psi\lrp\varphi_1^{extrema}, \varphi_2^{extrema}\rrp = \psi_{max}
    \end{split}
\end{equation}

By the Lagrange multiplier method, the target function corresponding to this optimisation is 
\begin{equation}
    \mathcal{L}\lrp \vn^\alpha, \vn^\beta, \pmb{\phi}, \gamma \rrp = \frac{A^{\alpha}}{T^{\alpha}} + \frac{A^{\beta}}{T^{\beta}} - \sum_i \phi_i \lrp n_i^\alpha + n_i^{\beta} - n_i \rrp - \gamma \sum_i v_i^{\alpha}n_i^{\alpha} - v_i^{\beta}n_i^{\beta}
\end{equation}
with $\phi$ and $\gamma$ as the Lagrange multipliers. Differentiating with respect to the optimisation variables yields the set of equations
\begin{equation}
    \begin{split}
        \frac{\mu_i^{\alpha}}{T^{\alpha}} - \phi_i - \gamma v_i^\alpha &= 0\\
        \frac{\mu_i^{\beta}}{T^{\beta}} - \phi_i + \gamma v_i^\beta &= 0\\
    \end{split}
\end{equation}
Subtracting these two equations eliminates the $\phi_i$, such that
\begin{equation}
    \begin{split}
        \frac{\mu_i^{\alpha}}{T^{\alpha}} - \frac{\mu_i^{\beta}}{T^{\beta}} - \gamma \lrp v_i^\alpha + v_i^\beta \rrp &= 0 \\
        \DeltaAB \pfrac{\mu_i}{T} - \gamma \lrp v_i^\alpha + v_i^\beta \rrp &= 0, \hspace{1cm} i = \{1, ..., N\}
    \end{split}
\end{equation}
where the notation $\DeltaAB\varphi \equiv \varphi^{\alpha} - \varphi^{\beta}$ has been introduced to denote the difference between the bulbs. Solving this equation for $\gamma$ with $i = k$ gives
\begin{equation}
    \gamma = \frac{1}{v_k^{\alpha} + v_k^{\beta}}\DeltaAB \pfrac{\mu_k}{T}
\end{equation}
Inserting this back into the remaining $N - 1$ equations gives
\begin{equation}
    \DeltaAB \pfrac{\mu_i}{T} - \pfrac{v_i^\alpha + v_i^\beta}{v_k^\alpha + v_k^\beta}\DeltaAB \pfrac{\mu_k}{T} = 0, \hspace{1cm} i = \{1, ..., N\} \setminus \{k\}.
\end{equation}
Dividing through by the length of the tube, $\Delta z$ and taking the limit $\Delta z \to 0$, $\DeltaAB T \to 0$ yields the set of equations
\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}
\end{equation}

This set of equations may be solved, in combination with $\sum_i S_{T,i}x_i(1 - x_i) = 0$, following from the definition of the Soret coefficient. However, the observant reader will notice that the reference chemical potentials of the components do not cancel, leaving a result that will depend on the choice of reference state. This is the issue Kempers resolves with the 2001-approach.

\subsubsection{The 2001-approach}
In the 2001-approach,\cite{kempers01} the function subject to maximisation is instead the ratio of the partition function to the partition function in the ideal gas state. That is, the objective is to find
\begin{equation}
    \max_{n_i^{\alpha}, n_i^{\beta}, n_i^{\alpha,ig}, n_i^{\beta,ig}} \lcp \frac{Q}{Q^{ig}}\rcp = \max_{n_i^{\alpha}, n_i^{\beta}, n_i^{\alpha,ig}, n_i^{\beta,ig}} \lcp \frac{Q^{\alpha}Q^{\beta}}{Q^{\alpha,ig} Q^{\beta,ig}}\rcp
\end{equation}
The constraints are expanded to include $n_i^{\alpha,ig} + n_i^{\beta,ig} = n_i$, where the superscript $ig$ indicates the ideal gas state. That is, the particle numbers in the ideal gas state are treated as independent variables, with the constraint that the total number of particles in the ideal gas state must equal that in the real state. Kempers does not use the constraint that $\sum_i v_i^{\alpha,ig}n_i^{\alpha,ig} = \sum_i v_i^{\beta,ig}n_i^{\beta,ig} = \frac{V}{2}$. The resulting optimisation problem is
\begin{equation}
    \begin{split}
        \min_{n_i^{\alpha}, n_i^{\beta}, n_i^{\alpha,ig}, n_i^{\beta,ig}} &\lcp \frac{A^{\alpha} - A^{\alpha,ig}}{T^{\alpha}} + \frac{A^{\beta} - A^{\beta,ig}}{T^{\beta}}\rcp \\
        n_i^{\alpha} + n_i^{\beta} &= n_i\\
        n_i^{\alpha,ig} + n_i^{\beta,ig} &= n_i\\
        \sum_i v_i^{\alpha}n_i^{\alpha} &= \sum_i v_i^{\beta}n_i^{\beta}
    \end{split}
\end{equation}
Because the mole numbers in the ideal gas state and the real state are treated as independent variables, and the Helmholtz energy in each state is only a function of the mole numbers in that state (i.e. $A^{\alpha} = A^{\alpha}(\vn^{\alpha}, T^{\alpha}, V^{\alpha})$, $A^{\alpha, ig} = A^{\alpha,ig}(\vn^{\alpha,ig}, T^{\alpha}, V^{\alpha})$ etc.), this is exactly equivalent to conducting the two independent optimisations
\begin{align}
        \min_{n_i^\alpha, n_i^{\beta}}&\left\{\frac{A^{\alpha}}{T^{\alpha}} + \frac{A^{\beta}}{T^{\beta}}\right\} & \min_{n_i^{\alpha,ig}, n_i^{\beta, ig}}&\left\{- \frac{A^{\alpha, ig}}{T^{\alpha}} - \frac{A^{\beta, ig}}{T^{\beta}}\right\} \\
        n_i^{\alpha} + n_i^{\beta} &= n_i & n_i^{\alpha, ig} + n_i^{\beta, ig} &= n_i \\
        \sum_i v_i^{\alpha}n_i^{\alpha} &= \sum_i v_i^{\beta}n_i^{\beta} & \\
\end{align}
The leftmost minimisation is equivalent to the problem in Equation \eqref{eq:1989_problem}, resulting in the set of Equations \eqref{eq:kempers_89_result}. The rightmost minimisation yields the Lagrange target-function
\begin{equation}
    \mathcal{L}\lrp \vn^\alpha, \vn^\beta, \mathbf{\phi}\rrp = - \frac{A^{\alpha, ig}}{T^{\alpha}} - \frac{A^{\beta, ig}}{T^{\beta}} - \sum_i \phi_i (n_i^{\alpha, ig} + n_i^{\beta, ig} - n_i).
    \label{eq:kempers_ig_target}
\end{equation}
Differentiation with respect to the target variables yields the set of equations
\begin{equation}
    \begin{split}
        - \frac{\mu_i^{\alpha,ig}}{T^{\alpha}} - \phi_i &= 0\\
        - \frac{\mu_i^{\beta,ig}}{T^{\beta}} - \phi_i &= 0
    \end{split}
\end{equation}
Subtracting the first equation from the second yields the set of equations
\begin{equation}
    \DeltaAB\pfrac{\mu_i^{ig}}{T} = 0.
\end{equation}
As before, dividing by the length of the tube $\Delta z$, and taking the limit $\Delta z \to 0$, $\DeltaAB T \to 0$ then gives
\begin{equation}
    \nabla \pfrac{\mu_i^{ig}}{T} = 0.
    \label{eq:kempers_2001_ig_result}
\end{equation}
To arrive at the set of equations proposed by Kempers in 2001, first subtract Equation \eqref{eq:kempers_2001_ig_result} for $i \neq k$ from equation \eqref{eq:kempers_89_result}, arriving at
\begin{equation}
    \begin{split}
        \nabla \pfrac{\mu_i}{T} - \nabla \pfrac{\mu_i^{ig}}{T} - \pfrac{v_i}{v_k}\nabla \pfrac{\mu_k}{T} &= 0, \hspace{1cm} i = \{1, ..., N\} \setminus \{k\}\\
        \DeltaIG \lcp \nabla \pfrac{\mu_i}{T} \rcp - \pfrac{v_i}{v_k}\nabla \pfrac{\mu_k}{T} &= 0
    \end{split}
    \label{eq:kempers_01_intermediate_1}
\end{equation}
where $\DeltaIG \varphi \equiv \varphi - \varphi^{ig}$ denotes the difference between the real state and the ideal gas state. Finally, multiply Equation \eqref{eq:kempers_2001_ig_result} for $i = k$ with $\frac{v_i}{v_k}$ and add the result to Equation \eqref{eq:kempers_01_intermediate_1}, getting
\begin{equation}
    \DeltaIG \lcp \nabla \pfrac{\mu_i}{T} \rcp - \pfrac{v_i}{v_k} \DeltaIG \lcp \nabla \pfrac{\mu_k}{T} \rcp = 0.
    \label{eq:kempers_01_intermediate_2}
\end{equation}
To solve this equation for the Soret coefficient, begin by expanding the gradients
\begin{equation}
    \begin{split}
        \nabla \pfrac{\mu_i}{T} &= \lrp \pder{}{T} \frac{\mu_i}{T} \rrp_{p,x} \nabla T + \sum_j \lrp \pder{}{x_j} \frac{\mu_i}{T} \rrp_{T,p} \nabla x_j \\
        &= -\frac{h_i}{T^2} \nabla T + \frac{1}{T} \sum_j \ppder{\mu_i}{x_j}_{T,p} \nabla x_j,
    \end{split}
    \label{eq:kempers_01_intermediate_3}
\end{equation}
where the relationship
\begin{equation}
    \begin{split}
        \d H &= T \d S + V \d p + \sum_j \mu_j \dn_j\\
        \ppder{H}{n_i}_{T,p} \equiv h_i &= T \ppder{S}{n_i}_{T,p} + \mu_i\\
        &= - T \ppder{\mu_i}{T}_{\vn,p} + \mu_i\\
        &= - T^2 \lrp \pder{\pfrac{\mu_i}{T}}{T} \rrp_{\vn, p}
    \end{split}
\end{equation}
has been used, with $h_i$ denoting the partial molar enthalpy. Insert this expression into Equation \eqref{eq:kempers_01_intermediate_2} and divide through by $\nabla T$ to obtain a set of $N - 1$ equations for the Soret coefficients,
\begin{equation}
    \begin{split}
        &\DeltaIG \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}- \frac{v_i}{v_k} \DeltaIG \lcp -\frac{h_k}{T^2} - \frac{1}{T} \sum_j \ppder{\mu_k}{x_j} x_j (1 - x_j) S_{T,j} \rcp = 0.
    \end{split}
\end{equation}

Given that the optimisation of the target function in Equation \eqref{eq:kempers_ig_target} does in fact yield the bulb compositions in the ideal gas steady state, and that kinetic gas theory can accurately supply the Soret coefficients in the ideal gas state, this final set of equations can be solved for the Soret coefficients in the real state. This set of equations is termed the Kempers CoV model, due to the constraint of no volume translation. It can be shown that this is equivalent to the constraint of no pressure gradient.\cite{kempers89}

If, instead of imposing the constraint of no volume translation, one constrains the system such that the centre-of-mass is stationary i.e. 
\begin{equation}
    \sum_i n_i^{\alpha} m_i = \sum_i n_i^{\beta} m_i,
\end{equation}
where $m_i$ are the molar masses, the only change to the resulting equation will be swapping out the partial molar volumes with the molar masses. This result is termed the Kempers centre-of-mass (CoM) model.