UNIFAC

The activity coefficient of the components in a system is a correction factor that accounts for deviations of real systems from that of an Ideal solution, which can either be measured via experiment or estimated from chemical models (such as UNIFAC). By adding a correction factor, known as the activity (${\displaystyle a_{i}}$, the activity of the i^th component) to the liquid phase fraction of a liquid mixture, some of the effects of the real solution can be accounted for. The activity of a real chemical is a function of the thermodynamic state of the system, i.e. temperature and pressure.

Equipped with the activity coefficients and a knowledge of the constituents and their relative amounts, phenomena such as phase separation and vapour-liquid equilibria can be calculated. UNIFAC attempts to be a general model for the successful prediction of activity coefficients.

The UNIFAC model splits up the activity coefficient for each species in the system into two components; a combinatorial ${\displaystyle \gamma ^{c}}$ and a residual component ${\displaystyle \gamma ^{r}}$. For the ${\displaystyle i}$-th molecule, the activity coefficients are broken down as per the following equation:

${\displaystyle \ln \gamma _{i}=\ln \gamma _{i}^{c}+\ln \gamma _{i}^{r}.}$

In the UNIFAC model, there are three main parameters required to determine the activity for each molecule in the system. Firstly there are the group surface area ${\displaystyle Q}$ and volume contributions ${\displaystyle R}$ obtained from the Van der Waals surface area and volumes. These parameters depend purely upon the individual functional groups on the host molecules. Finally there is the binary interaction parameter ${\displaystyle \tau _{ij}}$, which is related to the interaction energy ${\displaystyle U_{i}}$ of molecular pairs (equation in "residual" section). These parameters must be obtained either through experiments, via data fitting or molecular simulation.

The combinatorial component of the activity is contributed to by several terms in its equation (below), and is the same as for the UNIQUAC model.

${\displaystyle \ln \gamma _{i}^{c}=\ln {\frac {\phi _{i}}{x_{i}}}+{\frac {z}{2}}q_{i}\ln {\frac {\theta _{i}}{\phi _{i}}}+L_{i}-{\frac {\phi _{i}}{x_{i}}}\displaystyle \sum _{j=1}^{n}x_{j}L_{j},}$

where ${\displaystyle \theta _{i}}$ and ${\displaystyle \phi _{i}}$ are the molar weighted segment and area fractional components for the ${\displaystyle i}$-th molecule in the total system and are defined by the following equation; ${\displaystyle L_{i}}$ is a compound parameter of ${\displaystyle r}$, ${\displaystyle z}$ and ${\displaystyle q}$. ${\displaystyle z}$ is the coordination number of the system, but the model is found to be relatively insensitive to its value and is frequently quoted as a constant having the value of 10.

${\displaystyle \theta _{i}={\frac {x_{i}q_{i}}{\displaystyle \sum _{j=1}^{n}x_{j}q_{j}}},\quad \phi _{i}={\frac {x_{i}r_{i}}{\displaystyle \sum _{j=1}^{n}x_{j}r_{j}}},\quad L_{i}={\frac {z}{2}}(r_{i}-q_{i})-(r_{i}-1),\quad z=10,}$

${\displaystyle q_{i}}$ and ${\displaystyle r_{i}}$ are calculated from the group surface area and volume contributions ${\displaystyle Q}$ and ${\displaystyle R}$ (Usually obtained via tabulated values) as well as the number of occurrences of the functional group on each molecule ${\displaystyle \nu _{k}}$ such that:

${\displaystyle r_{i}=\displaystyle \sum _{k=1}^{n}\nu _{k}R_{k},\quad q_{i}=\displaystyle \sum _{k=1}^{n}\nu _{k}Q_{k}.}$

The residual component of the activity ${\displaystyle \gamma ^{r}}$ is due to interactions between groups present in the system, with the original paper referring to the concept of a "solution-of-groups". The residual component of the activity for the ${\displaystyle i}$-th molecule containing ${\displaystyle n}$ unique functional groups can be written as follows:

${\displaystyle \ln \gamma _{i}^{r}=\displaystyle \sum _{k}^{n}\nu _{k}^{(i)}\left[\ln \Gamma _{k}-\ln \Gamma _{k}^{(i)}\right],}$

where ${\displaystyle \Gamma _{k}^{(i)}}$ is the activity of an isolated group in a solution consisting only of molecules of type ${\displaystyle i}$. The formulation of the residual activity ensures that the condition for the limiting case of a single molecule in a pure component solution, the activity is equal to 1; as by the definition of ${\displaystyle \Gamma _{k}^{(i)}}$, one finds that ${\displaystyle \ln \Gamma _{k}-\ln \Gamma _{k}^{(i)}}$ will be zero. The following formula is used for both ${\displaystyle \Gamma _{k}}$ and ${\displaystyle \Gamma _{k}^{(i)}}$

${\displaystyle \ln \Gamma _{k}=Q_{k}\left[1-\ln \displaystyle \sum _{m}\Theta _{m}\Psi _{mk}-\displaystyle \sum _{m}{\frac {\Theta _{m}\Psi _{km}}{\displaystyle \sum _{n}\Theta _{n}\Psi _{nm}}}\right].}$

In this formula ${\displaystyle \Theta _{m}}$ is the summation of the area fraction of group ${\displaystyle m}$, over all the different groups and is somewhat similar in form, but not the same as ${\displaystyle \theta _{i}}$. ${\displaystyle \Psi _{mn}}$ is the group interaction parameter and is a measure of the interaction energy between groups. This is calculated using an Arrhenius equation (albeit with a pseudo-constant of value 1). ${\displaystyle X_{n}}$ is the group mole fraction, which is the number of groups ${\displaystyle n}$ in the solution divided by the total number of groups.

${\displaystyle \Theta _{m}={\frac {Q_{m}X_{m}}{\displaystyle \sum _{n}Q_{n}X_{n}}},}$

${\displaystyle \Psi _{mn}=\exp \left[-{\frac {U_{mn}-U_{nm}}{RT}}\right],\quad X_{m}={\frac {\displaystyle \sum _{j}\nu _{m}^{j}x_{j}}{\displaystyle \sum _{j}\displaystyle \sum _{n}\nu _{n}^{j}x_{j}}},}$

${\displaystyle U_{mn}}$ is the energy of interaction between groups m and n, with SI units of joules per mole and R is the ideal gas constant. Note that it is not the case that ${\displaystyle U_{mn}=U_{nm}}$, giving rise to a non-reflexive parameter. The equation for the group interaction parameter can be simplified to the following:

${\displaystyle \Psi _{mn}=\exp {\frac {-a_{mn}}{T}}.}$

Thus ${\displaystyle a_{mn}}$ still represents the net energy of interaction between groups ${\displaystyle m}$ and ${\displaystyle n}$, but has the somewhat unusual units of absolute temperature (SI kelvins). These interaction energy values are obtained from experimental data, and are usually tabulated.