Combinatorial contribution

The combinatorial contribution accounts for shape differences between molecules and affects the entropy of the mixture and is based on the lattice theory. The Stavermann–Guggenheim equation is used to approximate this term from pure chemical parameters, using the relative Van der Waals volumes r_i and surface areas q_i^{[nb 2]} of the pure chemicals:

${\displaystyle {\frac {G^{E}}{RT}}=\sum _{i}\,x_{i}\ln {V_{i}}+{\frac {1}{2}}z\,q_{i}\,x_{i}\ln {\frac {F_{i}}{V_{i}}}}$

Differentiating yields the excess entropy γ^C,

${\displaystyle \ln \gamma _{i}^{C}=(1-V_{i}+\ln V_{i})-{\frac {z}{2}}q_{i}\left(1-{\frac {V_{i}}{F_{i}}}+\ln {\frac {V_{i}}{F_{i}}}\right)}$

with the volume fraction per mixture mole fraction, V_i, for the i^th component given by:

${\displaystyle V_{i}={\frac {r_{i}}{\sum _{j}x_{j}r_{j}}}}$

The surface area fraction per mixture molar fraction, F_i, for the i^th component is given by:

${\displaystyle F_{i}={\frac {q_{i}}{\sum _{j}x_{j}q_{j}}}}$

The first three terms on the right hand side of the combinatorial term form the Flory–Huggins contribution, while the remaining term, the Guggenhem–Staverman correction, reduce this because connecting segments cannot be placed in all direction in space. This spatial correction shifts the result of the Flory–Huggins term about 5% towards an ideal solution. The coordination number, z, i.e. the number of close interacting molecules around a central molecule, is frequently set to 10. It can be regarded as an average value that lies between cubic (z = 6) and hexagonal packing (z = 12) of molecules that are simplified by spheres.

In the case of infinite dilution for a binary mixture, the equations for the combinatorial contribution reduce to:

${\displaystyle {\begin{cases}\ln \gamma _{1}^{C,\infty }=1-{\dfrac {r_{1}}{r_{2}}}+\ln {\dfrac {r_{1}}{r_{2}}}-{\dfrac {z}{2}}q_{1}\left(1-{\dfrac {r_{1}q_{2}}{r_{2}q_{1}}}+\ln {\dfrac {r_{1}q_{2}}{r_{2}q_{1}}}\right)\\\ln \gamma _{2}^{C,\infty }=1-{\dfrac {r_{2}}{r_{1}}}+\ln {\dfrac {r_{2}}{r_{1}}}-{\dfrac {z}{2}}q_{2}\left(1-{\dfrac {r_{2}q_{1}}{r_{1}q_{2}}}+\ln {\dfrac {r_{2}q_{1}}{r_{1}q_{2}}}\right)\end{cases}}}$

This pair of equations show that molecules of same shape, i.e. same r and q parameters, have ${\displaystyle \gamma _{1}^{C,\infty }=\gamma _{2}^{C,\infty }=1}$.

Residual contribution

The residual, enthalpic term contains an empirical parameter, ${\displaystyle \tau _{ij}}$, which is determined from the binary interaction energy parameters. The expression for the residual activity coefficient for molecule i is:

${\displaystyle \ln \gamma _{i}^{R}=q_{i}\left(1-\ln {\frac {\sum _{j}q_{j}x_{j}\tau _{ji}}{\sum _{j}q_{j}x_{j}}}-\sum _{j}{\frac {q_{j}x_{j}\tau _{ij}}{\sum _{k}q_{k}x_{k}\tau _{kj}}}\right)}$

with

${\displaystyle \tau _{ij}=e^{-\Delta u_{ij}/{RT}}}$

${\displaystyle \Delta u_{ii}}$ [J/mol] is the binary interaction energy parameter. Theory defines ${\displaystyle \Delta u_{ij}=u_{ij}-u_{ii}}$, and ${\displaystyle \Delta u_{ji}=u_{ji}-u_{jj}}$ , where ${\displaystyle u_{ij}}$ is the interaction energy between molecules ${\displaystyle i}$ and ${\displaystyle j}$. The interaction energy parameters are usually determined from activity coefficients, vapor-liquid, liquid-liquid, or liquid-solid equilibrium data.

Usually ${\displaystyle \Delta u_{ij}\neq \Delta u_{ji}}$, because the energies of evaporation (i.e. ${\displaystyle \Delta u_{ii}}$), are in many cases different, while the energy of interaction between molecule i and j is symmetric, and therefore ${\displaystyle u_{ij}=u_{ji}}$. If the interactions between the j molecules and i molecules is the same as between molecules i and j, there is no excess energy of mixing, ${\displaystyle \Delta u_{ij}=\Delta u_{ji}=0}$. And thus ${\displaystyle \gamma _{i}^{R}=1}$.

Alternatively, in some process simulation software ${\displaystyle \tau _{ij}}$ can be expressed as follows :

${\displaystyle \ln \tau _{ij}=A_{ij}+B_{ij}/T+C_{ij}\ln(T)+D_{ij}T+E_{ij}/T^{2}}$ .

The C, D, and E coefficients are primarily used in fitting liquid–liquid equilibria data (with D and E rarely used at that). The C coefficient is useful for vapor-liquid equilibria data as well. The use of such an expression ignores the fact that on a molecular level the energy, ${\displaystyle \Delta u_{ij}}$, is temperature independent. It is a correction to repair the simplifications, which were applied in the derivation of the model.

Newer developments

UNIQUAC has been extended by several research groups. Some selected derivatives are: UNIFAC, a method which permits the volume, surface and in particular, the binary interaction parameters to be estimated. This eliminates the use of experimental data to calculate the UNIQUAC parameters,^[3] extensions for the estimation of activity coefficients for electrolytic mixtures,^[7] extensions for better describing the temperature dependence of activity coefficients,^[8] and solutions for specific molecular arrangements.^[9]

The DISQUAC model advances UNIFAC by replacing UNIFAC's semi-empirical group-contribution model with an extension of the consistent theory of Guggenheim's UNIQUAC. By adding a "dispersive" or "random-mixing physical" term, it better predicts mixtures of molecules with both polar and non-polar groups. However, separate calculation of the dispersive and quasi-chemical terms means the contact surfaces are not uniquely defined. The GEQUAC model advances DISQUAC slightly, by breaking polar groups into individual poles and merging the dispersive and quasi-chemical terms.