\subsection{Equations of state}
\label{sec:teori_eos}
An equation of state (EoS) is an equation that relates various thermodynamic state variables.\cite{properties_liq_and_gas} Often, these are expressed on the form $p = p(n, V, T; \varphi_1, \varphi_2, ... \varphi_n)$ where $p, V, n, T$ are the pressure, mole number, volume and temperature and the $\varphi_i$ are parameters determined either empirically, from first principles or a combination of the two. Common manners of expressing an EoS are by the use of the compressibility factor $Z \equiv \frac{pV}{nRT}$, or by expressing pressure as a polynomial in $\frac{1}{v}$, where $v$ denotes the molar volume, known as a virial expansion. 

An EoS worth mentioning is the Carnahan-Starling (CS) EoS for hard spheres.\cite{carnahan_starling} This equation is based on approximating the virial expansion in volume as an infinite geometric series. Manipulation of this series yields an explicit expression for the compressibility factor for hard spheres as a function of the packing fraction, $\eta$, given as
\begin{equation}
    \eta = \frac{\pi N_A n \left(\sigma^{HS}\right)^3}{6V}
\end{equation}
where $\sigma^{HS}$ is the hard sphere diameter and $N_A$ is Avogadros number. Once an expression for the compressibility factor is known, other properties of interest may be related by various derivatives of this. This EoS has been proven to be highly accurate when measured against MD-simulations of hard-sphere particles.\cite{carnahan_starling} Nevertheless, several propositions of extensions and modifications have been made to the theory from its introduction until today, in attempts to further improve its performance.\cite{carnahan_starling_extend,carnahan_starling_pertub} An important area of usage for the CS EoS is the ability to accurately represent a theoretical hard sphere state, which can be a convenient reference state.

A frequent starting point in the development of an EoS is describing the Helmholtz energy ($A$) of the system, this can be done by regarding the intermolecular potential. Observable properties, and other properties of interest, such as the entropy, chemical potential and so on can be calculated in terms of the derivatives of the Helmholtz energy. This approach opens several doors: The possibility to estimate molecular parameters from macroscopic observations;\cite{saft_regression} firm control of the microscopic description of the system; the option to explicitly include quantum effects,\cite{saft_mixtures} and the possibility of creating equations of state for theoretical fluids, such as the Lennard-Jones fluid. Apart from accurately describing real fluids, this means that one can construct an EoS that, in principle, should reproduce the results of molecular dynamics (MD) simulations. This can be useful in the development of thermodynamic models.

Many equations of state have been developed using the approach outlined above. Of particular interest in the context of this work is the SAFT-VR Mie EoS. This EoS aims to describe a fluid molecule as a chain of monomers interacting through a Mie-potential, given as

\begin{equation}
    u_{ij}^{Mie} = C\varepsilon_{ij} \lsp \pfrac{\sigma_{ij}}{r_{ij}}^{\lambda_{r,ij}} - \pfrac{\sigma_{ij}}{r_{ij}}^{\lambda_{a,ij}} \rsp, \hspace{.5cm} C = \frac{\lambda_{r_{ij}}}{\lambda_{r,ij} - \lambda_{a,ij}}\pfrac{\lambda_{r,ij}}{\lambda_{a,ij}}^{\frac{\lambda_{a,ij}}{\lambda_{r,ij} - \lambda_{a,ij}}},
    \label{eq:def_mie_potential}
\end{equation}

where $r_{ij}$ is the inter-particle distance, $\varepsilon_{ij}$ is the potential well depth, $\sigma_{ij}$ is the point at which $u_{ij}^{Mie} = 0$, related to the particle sizes, and $\lambda_{r,ij}$ and $\lambda_{a,ij}$ are adjustable exponents. Further, the Helmholtz energy of the fluid is related to the radial distribution function (rdf.) of these monomers. Using the framework of Barker-Henderson (BH) perturbation theory it can then be shown that the residual Helmholtz energy of the fluid may be written as the sum of a hard sphere term, which may be determined by the CS EoS,\cite{carnahan_starling} and a series of perturbation terms. This EoS has been shown to give highly accurate predictions of fluid-phase equilibria, pressure-volume-temperature (PVT) behaviour and second-derivative properties for both pure fluids and mixtures. 

When applying an EoS to a mixture, composition must be taken into account. This is accomplished by the use of mixing rules.\cite{properties_liq_and_gas} A mixing rule is an equation that in some way combines or averages the parameters of the components in a mixture such that an EoS can be applied. These equations are to a large degree empirical in nature, and are often expressed on the form
\begin{equation}
    \varphi_m = \sum_i \sum_j x_i x_j \varphi_{ij}
\end{equation}
where $\varphi_m$ is some property of the mixture, and $\varphi_{ii}$, $\varphi_{jj}$ are the properties of the pure components. Various combining rules can be chosen to determine the interaction terms $\varphi_{ij}$. An example of such combining rules are the Lorentz-Berthelot (LB) rules for combining the parameters of a Mie-potential
\begin{equation}
    \sigma_{ij} = \frac{1}{2}\left(\sigma_{ii} + \sigma_{jj}\right), \hspace{1cm} \varepsilon_{ij} = \sqrt{\varepsilon_{ii}\varepsilon_{jj}}.
\end{equation}
These are of special significance in this work, and may be modified to create the more general mixing rules hereafter termed the modified Lorentz-Berthelot rules,\cite{thermopack,LJ_binary_mixtures} given as
\begin{equation}
    \begin{split}
       \sigma_{ij} = \frac{1}{2}\left(\sigma_{ii} + \sigma_{jj}\right),& \hspace{.5cm} \varepsilon_{ij} = (1 - k_{ij})\frac{\left(\sigma_{ii}\sigma_{jj}\right)^{\frac{3}{2}}}{\sigma_{ij}^3}\sqrt{\varepsilon_{ii}\varepsilon_{jj}}\\
       \lambda_{r,ij} - 3 &= (1 - \ell_{ij}) \sqrt{(\lambda_{r,ii} - 3)(\lambda_{r,jj} - 3)} 
    \end{split}
    \label{eq:modified_LB}
\end{equation}
where $k_{ij}$ and $\ell_{ij}$ are parameters that may be adjusted represent the behaviour of fluids for which particle interaction is energetically favourable ($k_{ij} > 0$), stabilising the liquid phase; or fluids for which interaction is less favourable ($k_{ij} < 0$), destabilising the liquid phase and lowering the boiling point.