The Nonrandom two-liquid segment activity coefficient model (NRTL-SAC) is an activity coefficient model used to calculate the Gibbs free energy of a non-ideal system ^[1] . It is a modification to the NRTL model and it defines the activity coefficient $\gamma_I$ as a function of the molar composition $x_I$. In NRTL-SAC the activity coefficient is defined in a way that:

\[ln \gamma_I = ln \gamma_I^C + ln \gamma_I^R\]

Where $ln \gamma_I^C$ and $ln \gamma_I^R$ are the combinatorial and residual contributions to the activity coefficient of molecule $I$. The residual contribution $ln \gamma_I^R$ is defined as:

\[ln \gamma_I^R = ln \gamma_I^{lc} = \sum_m{r_{m,I} [ln \Gamma_m^{lc} - ln \Gamma_m^{lc, I}]}\]

The residual contribution of the activity coefficient is set equal to the local composition (lc) interaction contribution $ln \gamma_i^{lc}$. The segment activity coefficient $\Gamma_m^{lc}$ and the segment activity coefficient for the molecule $\Gamma_m^{lc, I}$ can be computed as follows:

\[ln \Gamma_m^{lc} = \dfrac{\sum_{j} {\tau_{j m} G_{j m} x_{j}} }{ \sum_{k} {G_{k m} x_{k}} } + \sum_{m'} \dfrac{x_{m'} G_{m m'}}{\sum_{k} { x_k G_{k m'}}}(\tau_{m m'} - \dfrac{\sum_{j} {x_j \tau_{j m'} G_{j m}}}{\sum_{k} {x_k G_{k m'}}})\]

\[ln \Gamma_m^{lc, I} = \dfrac{\sum_{j, I} {\tau_{j m} G_{j m} x_{j}} }{ \sum_{k} {G_{k m} x_{k, I}} } + \sum_{m'} \dfrac{x_{m', I} G_{m m'}}{\sum_{k} { x_{k, I} G_{k m'}}}(\tau_{m m'} - \dfrac{\sum_{j} {x_{j, I} \tau_{j m'} G_{j m}}}{\sum_{k} {x_{k, I} G_{k m'}}})\]

\[x_j = \dfrac{\sum_J {x_J r_{j, J}}}{\sum_I \sum_i x_I r_{i, I}}\]

\[x_{j, I} = \dfrac{r_{j, I}}{\sum_i r_{i, I}}\]

Where i, $j$, $k$, $m$, and $m′$ are the segment-based species indices, $I$ and $J$ are the component indices, $x_j$ is the segment-based mole fraction of segment species $j$, $x_J$ is the mole fraction of component $J$, $r_{m,I}$ is the number of segment species m contained in component $I$. The $G$ and $\tau$ parameters are the same from the standard NRTL model defined by the following:

\[G = e^{-\alpha \tau}\]

The combinatorial contribution of the activity coefficient $ln \gamma_I^C$ is solved from the Flory-Huggins term:

\[ln \gamma_I^C = ln \dfrac{\phi_I}{x_I} + 1 - r_I \sum_J {\dfrac{\phi_J}{x_J}}\]

\[r_I = \sum_i {r_{i, I}}\]

\[\phi_I = \dfrac{r_I x_I}{\sum_J r_J x_J}\]