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$\left[\vec{d}_{i}\right]^{i=1}_{n}=\textbf{G}_{n\times(n-1)}\left[\vec{d}_{i}\right]^{i=1}_{n-1},$
(B.6)

or contract the mole-fraction gradients in equation (5.1), through a similar
expression where $\vec{\nabla}y_{i}$ replaces $\vec{d}_{i}$. Setting equations
(B.5) and (5.1) equal therefore shows that

$\bm{\Gamma}_{n}\textbf{G}_{n\times(n-1)}=\textbf{G}_{n\times(n-1)}\left(\frac{\overline{\textbf{Q}}_{n-1}}{RT}-\textbf{I}_{n-1}\right),$
(B.7)

where the nonideal correction $\bm{\Gamma}_{n}$ has been used instead of the
thermodynamic matrix $\bm{\chi}_{n}$. This specifies most of
$\bm{\Gamma}_{n}$, but leaves a number of its entries undetermined.

Observe that taking the gradient of the mole-fraction sum and casting the
result in matrix form shows that

$\textbf{1}_{n}^{\top}\left[\vec{\nabla}y_{i}\right]^{i=1}_{n-1}=\vec{0},$
(B.8)

so the domain of the mapping represented by $\bm{\Gamma}_{n}$ lies in a
subspace orthogonal to $\textbf{1}_{n}$. To make $\bm{\Gamma}_{n}$
unambiguous, we stipulate that any quantity outside its domain lies in its
nullspace, that is, that $\bm{\Gamma}_{n}\textbf{1}_{n}=\textbf{o}_{n}$. Since
direct calculation with equation (B.3) shows that $\textbf{1}_{n}$ is also
orthogonal to the column set of $\textbf{G}_{n\times(n-1)}$, one can concisely
represent all of the properties of $\bm{\Gamma}_{n}$ by augmenting equation
(B.7):

$\bm{\Gamma}_{n}\textbf{G}_{n}=\left[\begin{array}[]{cc}\textbf{G}_{n\times(n-1)}\left(\frac{\overline{\textbf{Q}}_{n-1}}{RT}-\textbf{I}_{n-1}\right)&\textbf{o}_{n}\end{array}\right].$
(B.9)

Right multiplication of both sides by $\textbf{G}_{n}^{-1}$ then shows that

$\bm{\Gamma}_{n}=\left[\begin{array}[]{cc}\textbf{G}_{n\times(n-1)}\left(\frac{\overline{\textbf{Q}}_{n-1}}{RT}-\textbf{I}_{n-1}\right)&\textbf{o}_{n}\end{array}\right]\textbf{G}_{n}^{-1},$
(B.10)

the desired explicit equation for $\bm{\Gamma}_{n}$ in terms of
$\overline{\textbf{Q}}_{n-1}$.

Equation (B.10) is useful when expressing the entropy gradients from equation
(5.7) in terms of the natural basis variables. Through a Maxwell relation, the
composition derivatives of $\overline{S}_{i}$ follow from equation (B.2):

$\left(\frac{\partial\overline{S}_{i}}{\partial y_{j}}\right)_{T,p,y_{k\neq
j,n}}=-\frac{1}{y_{i}}\left(\frac{\partial\overline{Q}_{ij}}{\partial
T}\right)_{p,y_{k}}.$ (B.11)

Bearing in mind that $\bm{\Gamma}_{n}=\bm{\chi}_{n}-\textbf{I}_{n}$,
Gibbs–Duhem extension can then be used to find

$\sum_{j=1}^{n-1}\left(\frac{\partial\overline{S}_{i}}{\partial
y_{j}}\right)_{T,p,y_{k\neq
j,n}}\vec{\nabla}y_{j}=-\frac{R}{y_{i}}\sum_{j=1}^{n}\frac{\partial\left(T\chi_{ij}\right)}{\partial
T}\vec{\nabla}y_{j},$ (B.12)

a result that was used when deriving equation (5.8).

##  Appendix C Maxwell relations for activities

It is not immediately obvious, but the thermodynamic matrix $\bm{\chi}_{n}$
from equation (5.1) is also constrained by Maxwell relations. (Here we employ
the conventions for notating matrix size laid out in B.) These constraints are
clarified by relating the thermodynamic factors to entries $\overline{K}_{ij}$
within the isothermal, isobaric Hessian of molar Gibbs energy $\overline{G}$
with respect to the independent mole fractions, defined as

$\overline{K}_{ij}=\left(\frac{\partial^{2}\overline{G}}{\partial
y_{i}\partial y_{j}}\right)_{T,p,y_{k\neq i,j,n}}.$ (C.1)

Thermodynamic stability demands that the Hessian matrix
$\overline{\textbf{K}}_{n-1}$ is positive semidefinite [40], and Maxwell
relations imply its symmetry [64].

Monroe and Newman [25] introduced a composition matrix $\textbf{Y}_{n-1}$ with
entries

$Y_{ij}=\frac{\delta_{ij}}{y_{i}}+\frac{1}{y_{n}},$ (C.2)

and showed that it transforms the free-energy Hessian into the Darken matrix
(cf. B), as

$\overline{\textbf{Q}}_{n-1}=\textbf{Y}_{n-1}\overline{\textbf{K}}_{n-1}.$
(C.3)

It has also been shown that $\textbf{Y}_{n-1}$ is invertible if and only if
all of the mole fractions are nonzero [40], so one can isolate
$\overline{\textbf{K}}_{n-1}$ here in any situation where all the species are
present. In any case, equation (C.3) establishes that the $(n-1)^{2}$ entries
of $\overline{\textbf{Q}}_{n-1}$ depend explicitly on composition and any set
of $n\left(n-1\right)/2$ independent state functions that specifies the
symmetric Hessian $\overline{\textbf{K}}_{n-1}$.

Noting that $\bm{\Gamma}_{n}=\bm{\chi}_{n}-\textbf{I}_{n}$, one can substitute
equation (C.3) into equation (B.10) to get

$\bm{\chi}_{n}=\textbf{I}_{n}+\left[\begin{array}[]{cc}\frac{\textbf{G}_{n\times(n-1)}\textbf{Y}_{n-1}\overline{\textbf{K}}_{n-1}}{RT}-\textbf{G}_{n\times(n-1)}&\textbf{o}_{n}\end{array}\right]\textbf{G}_{n}^{-1}.$
(C.4)

Here the symmetry of $\overline{\textbf{K}}_{n-1}$ justifies the claim that
all $n^{2}$ entries of the thermodynamic matrix in equation (5.1) are
parameterized by $n(n-1)/2$ state functions.

##  Appendix D Construction of Soret diffusivities

For $i,j=1,2,...,n$, take as given a set of Newman–Soret diffusivities
$\tilde{\mathscr{A}}_{ij}$ that satisfy
$\tilde{\mathscr{A}}_{ij}=-\tilde{\mathscr{A}}_{ij}$ and
$\tilde{\mathscr{A}}_{ii}=0$. Practically, these would be taken from
measurements of each binary system, analogous to the construction of Table 3.
Consistency of the modelling framework demands that a proper Newman–Soret
matrix with entries $\mathscr{A}_{ij}$ be based on a set of Soret
diffusivities $\mathscr{D}_{i}$. What constrains us from immediately
identifying $\mathscr{A}_{ij}$ with $\tilde{\mathscr{A}}_{ij}$ is the fact
that it is not clear that this extrapolated $\mathscr{A}_{ij}$ has the form
demanded by equation (4.17) for some set of Soret diffusivities
$\mathscr{D}_{i}^{T}$. For example, in a ternary mixture, equation (4.17)
implies that $\mathscr{A}_{12}+\mathscr{A}_{23}=\mathscr{A}_{13}$, but
inspection of the entries tabulated in Table 3 shows that this does not hold.
Thus the data in $\tilde{\mathscr{A}}_{ij}$ require some regularization to
produce $\mathscr{A}_{ij}$.

To regularize $\mathscr{A}_{ij}$, we choose its entries such that each is as
close to possible to $\tilde{\mathscr{A}}_{ij}$. That is, we impose structure
(4.17) on the coefficients, while minimizing the changes, according to the
minimization problem

$\min_{\mathscr{D}_{i}^{T}\in\mathbb{R}}\sum_{i,j}^{n}\left[\left(\mathscr{D}_{i}^{T}-\mathscr{D}_{j}^{T}\right)-\tilde{\mathscr{A}}_{ij}\right]^{2},$
(D.1)

subject to the constraint given by equation (4.18) with $\bm{\psi}=\bm{1}$. A
direct computation shows that this minimum is obtained by setting

$\mathscr{D}_{i}^{T}=\frac{1}{n}\sum_{j=1}^{n}\tilde{\mathscr{A}}_{ij}.$ (D.2)

Note that the kinematic relation chosen for the minimization problem
implicitly sets a somewhat nonphysical reference velocity for Soret
diffusivities. Since $\mathscr{A}_{ij}$ is independent of the convective
reference velocity, however, this mathematically convenient choice is
immaterial. To enforce a constraint on $\mathscr{D}_{i}^{T}$ with another
$\bm{\psi}$ in (4.18), one could simply perform a shift analogous to that
stated in equation (4.5) post facto.

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