UNIFAC theory
*************

Below some of the UNIFAC method is explained, but a more complete
description can be found in Ref. [1].

The UNIFAC method is an activity coefficient model derived from the
UNIQUAC model.  Both UNIFAC and UNIQUAC are thermodynamic models based
on local composition theory, which holds that the local environment of
a molecule in solution can be used to calculate the probabilities of
molecular configurations in the bulk solution.  While UNIQUAC requires
parameters for every compound in solution and interaction parameters
for every pair of compounds, UNIFAC estimates these parameters as
functions of the number of occurrences of various molecular
substructures, or groups, in a molecule.  This means that UNIFAC can
be applied to estimate activity coefficients for arbitrary systems, so
long as every group is defined and an interaction parameter exists for
every pair of groups in the composition.

The UNIFAC method calculates the activity coefficient as a function of
two contributions: (1) the residual contribution, meant to account for
the interactions of groups in the mixture; and (2) the combinatorial
contribution, meant to account for entropic effects due to differences
in molecular shape.  Using these two components, the activity
coefficient for each compound *i* is calculated as follows:

   \ln\gamma_i = \ln\gamma_i^R + \ln\gamma_i^C

where \ln\gamma_i^R corresponds to the residual contribution to the
activity coefficient and \ln\gamma_i^C to the combinatorial
contribution.


Residual term
=============

In the UNIFAC method, we first define n_i^k to be the number of times
group *k* occurs in molecule *i*.  Using this with the mole fraction
x_i of each compound *i*, we calculate the group mole fraction, X_k,
or the amount of groups of type *k* as a fraction of the total groups:

   X_k = \frac{\sum\limits_i x_i n_i^k}{\sum\limits_i\sum\limits_j x_i
   n_i^j }

We use the X_k values to then determine the relative surface area each
group represents in the mixture.  This is done by taking an average of
the X_k values weighted with the surface area contributions, Q_k, of
each group *k*.  We define the result as the area fraction of group
*k*:

   \Theta_k = \frac{ X_k Q_k }{\sum\limits_m x_m Q_m }

the surface area and volume of a molecule is estimated as a linear
function of the number and types of groups that are present.  Each
group *k* occurs a n_i^k times in molecule *i*.

Additionally, between every pair of groups, *k* and *m* an interaction
energy is defined as a_{km}.  This energy is given in Kelvin and is
used to calculate the group interaction paramter \Psi_{km}:

   \Psi_{km} = \exp\left(\frac{-a_{km}}{T}\right)

where *T* is the temperature of the system in Kelvin.  Note that
\Psi_{km} \neq \Psi_{mk}.

These parameters are then used to calculate the residual contribution
to the activity coefficient for each group *k*:

   \ln\Gamma_k = Q_k \left( 1 - \ln\sum\limits_m\Theta_m\Psi_{mk} -
   \sum\limits_m\frac{\Theta_m\Psi_{km}}{\sum\limits_n
   \Theta_n\Psi_{nm}} \right)

Additionally, one must also calculate \ln\Gamma_k^{(i)}, which follows
the same procedure as above for each compound assuming it exists in a
pure form, i.e., x_i=1.  Then, the residual contribution to the
activity coefficient is calculated as follows:

   \ln\gamma_i^R = \sum\limits_k n_i^k \left( \ln\Gamma_k -
   \ln\Gamma_k^{(i)} \right)


Combinatorial term
==================

To estimate the combinatorial contribution to the activity
coefficient, first the surface area, q_i, and volume, r_i, of molecule
*i* are estimated as follows:

   q_i = \sum_i Q_k n_i^k \qquad r_i = \sum_i R_k n_i^k

where Q_k is the surface area contribution of group *k* and R_k is the
volume contribution.  Using these parameters we can define the
relative surface area and relative volume (also called fractional
surface area and fractional volume) corresponding to molecule *i* in
solution.  This is simply an average of the surface areas/volumes of
each compound weighted by the mole fraction of that compound in
solution:

   \theta_i = \frac{ x_i q_i }{\sum\limits_j x_j q_j } \qquad \phi_i =
   \frac{ x_i r_i }{\sum\limits_j x_j r_j }

Additionally, we calculate the parameter L_i:

   L_i = \frac{ z }{2} (r_i-q_i)-(r_i-1)

where *z* is the coordination number and is usually taken to be equal
to 10.  All of these parameters are then used to calculate the
combinatorial contribution to the activity coefficient:

   \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} \sum\limits_j
   x_j L_j

[1] A. Fredenslund, R.L. Jones, and J.M. Prausnitz, *Group-
    contribution estimation of activity coefficients in nonideal
    liquid mixtures*, AIChE Journal 21, 1086 (1975)
