\subsection{The Carnahan-Starling EoS}
The Carnahan-Starling (CS) EoS is given as en equation for the compressibility factor in terms of the packing fraction, $\eta$.
\begin{equation}
    Z = \frac{1 + \eta + \eta^2 - \eta^3}{(1 - \eta)^3}, \hspace{.5cm} \eta = \frac{\pi N_A n \sigma^3}{6V}
\end{equation}
For a mixture, the applied mixing rules are 
\begin{equation}
    \sigma = \sum_i \sum_j x_i x_j \sigma_{ij}, \hspace{.5cm} \sigma_{ij} = \frac{1}{2}(\sigma_{ii} + \sigma_{jj}).
\end{equation}
The reduced Helmholtz energy may also be given as a function of the packing fraction,
\begin{equation}
    a = \frac{A^r}{nRT} = \frac{4\eta - 3 \eta^2 }{(1 - \eta)^2}.
\end{equation}
Thereby, the residual chemical potential may be obtained directly by differentiation as
\begin{equation}
    \begin{split}
        \ppder{a}{n_i}_{T,V} &= \frac{\ppder{A^r}{n_i}_{T,V}}{nRT} - \frac{A^r}{n^2RT}\\
            &= \frac{\mu_i^r}{nRT} - \frac{a}{n}\\
        \mu_i^r &= RT\left(a +  n\ppder{a}{n_i}_{T,V}\right).
    \end{split}
\end{equation}
The total chemical potential is then given as
\begin{equation}
    \begin{split}
        \mu_i(T,V,n) &= \mu_i^{ig}(T,V,n) + \mu_i^r(T,V,n)\\
        \mu_i^{ig}(T,V,n) &= \underbrace{\mu_i^{\circ}(T,p^{\circ})}_{\text{T.P.?}} + RT\ln\pfrac{n_iRT}{p^{\circ}V}.
    \end{split}
\end{equation}
The differential of the reduced Helmholtz energy is expanded,
\begin{equation}
    \begin{split}
        \ppder{a}{n_i}_{T,V} &= \ppder{a}{\eta}_{T,V}\ppder{\eta}{n_i}_{T,V}\\
        &= \ppder{a}{\eta}_{T,V} \frac{\pi N_A }{6V}\left(\sigma^3 + 3n\sigma^2 \ppder{\sigma}{n_i}\right).
    \end{split}
\end{equation}
Finally, the differentiation of the mixing rule gives
\begin{equation}
    \begin{split}
        \ppder{\sigma}{n_k}_{n_{\ell \neq k}} &= 2x_k\pfrac{1 - x_k}{n}\sigma_{kk} + \sum_{i \neq k} \sum_{j \neq k} \left[\left(-\frac{x_i}{n}\right)x_j + x_i\left(-\frac{x_j}{n}\right)\right]\sigma_{ij}\\
        \ppder{\sigma}{n_k}_{n_{\ell \neq k}} &= 2x_k\pfrac{1 - x_k}{n}\sigma_{kk} - \frac{2}{n} \sum_{i \neq k} \sum_{j \neq k} x_i x_j \sigma_{ij}.
    \end{split}
\end{equation}
