Combinatorial

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:

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

Residual

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.