Wilson Equation

**Mainly just playing around right now…
**Still need to work on explanation and stuff…
**Eventually use as template to explain other models…

According to Wilson the excess Gibbs energy is assumed to be

(1)   \begin{equation*} \frac{g^E}{RT}=\sum_i{x_i \ln{\frac{\xi}{x_i}}} \end{equation*}

where \xi_i denotes the so-called local concentration

(2)   \begin{equation*} \xi_i=\frac{x_{ii}v_i}{\sum_j{x_{ji}v_j}} \end{equation*}

with x_{ji} as the local mole fraction of species j in the area around species i. It is primarily a mental concept which can be define but not necessarily measured. Somewhat arbitrarily Wilson related this (mental) local composition to the overall (experimentally accessible) concentrations by using Boltzmann Factors:

(3)   \begin{equation*} \frac{x_{ji}}{x_{ii}}=\frac{x_j}{x_i}\frac{\exp[ -\lambda_{ji}/(RT)]}{\exp[-\lambda_{ii}/(RT)]} \end{equation*}

Inserting Eq. (2) into Eq. (1) gives

(4)   \begin{equation*} \frac{g^E}{RT}=\sum_i{\left[x_i \ln{\left(\frac{\frac{x_{ii}v_i}{\sum_j{x_{ji}v_j}}}{x_i}}\right)\right]} \end{equation*}

Then by combining Eqs. (4) and (3) we obtain

(5)   \begin{equation*} \begin{align} \frac{g^E}{RT}&=\sum_i{x_i \ln{\left(\frac{x_{ii}v_i}{x_i\sum_j{x_{ii}v_j\frac{x_j}{x_i}\frac{\exp[ -\lambda_{ji}/(RT)]}{\exp[-\lambda_{ii}/(RT)]}}}}\right)}\\&=\sum_i{x_i \ln{\left(\frac{v_i}{\sum_j{v_{j}x_{j}\frac{\exp[ -\lambda_{ji}/(RT)]}{\exp[-\lambda_{ii}/(RT)]}}}}\right)}\\&=-\sum_i{x_i\ln{\left(\sum_j{x_j\frac{v_j}{v_i}\exp{\left[-\frac{(\lambda_{ji}-\lambda_{ii})}{RT}\right]}}\right)}} \end{align} \end{equation*}

Setting \lambda_{ij}=\lambda_{ji}, the abbreviation

(6)   \begin{equation*} \Lambda_{ij}=\frac{v_j}{v_i}\exp{\left[-\frac{(\lambda_{ji}-\lambda_{ii})}{RT}\right]}=\frac{v_j}{v_i}\exp{\left[-\frac{\Delta\lambda_{ij}}{RT}\right]} \end{equation*}

can be introduced, where the \Delta\lambda_{ij} are the interaction parameters. The expression for the excess Gibbs energy then becomes

(7)   \begin{equation*} \frac{g^E}{RT}=\sum_i{x_i \ln{\sum_j{x_j\Lambda_{ij}}}} \end{equation*}

The activity coefficient can then be calculated (or derived) via

hup_board

(8)   \begin{equation*} \begin{align} \ln{\gamma_i}&=\left\frac{\partial\left(n_T\frac{g^E}{RT}\right)}{\partial{n_i}}\right\rvert_{T,P,n_{j\neq{i}}}\\&=\frac{\partial}{\partial{n_i}}\left[-\sum_k{n_k\ln{\sum_j{\frac{n_j}{n_T}\Lambda_{kj}}}}\right]_{T,P,n_{j\neq{i}}}\\&=-\ln{\sum_j{x_j\Lambda_{ij}}}-n_i\frac{\partial}{\partial{n_i}}\left(\ln{\sum_j{\frac{n_j}{n_T}\Lambda_{ij}}}\right)_{T,P,n_{j\neq{i}}}+\frac{\partial}{\partial{n_i}}\left[-\sum_{k\neq{i}}{n_k\ln{\sum_j{\frac{n_j}{n_T}\Lambda_{kj}}}}\right]_{T,P,n_{j\neq{i}}}\\&=-\ln{\sum_j{x_j\Lambda_{ij}}}-\frac{n_i}{\sum_j{\frac{n_j}{n_T}\Lambda_{ij}}}\frac{\partial}{\partial{n_i}}\left(\sum_j{\frac{n_j}{n_T}\Lambda_{ij}}\right)_{T,P,n_{j\neq{i}}}-\sum_{k\neq{i}}{\frac{n_k}{\sum_j{\frac{n_j}{n_T}\Lambda_{kj}}}\frac{\partial}{\partial{n_i}}\left(\sum_j{\frac{n_j}{n_T}\Lambda_{kj}}\right)_{T,P,n_{j\neq{i}}}}\\&=-\ln{\sum_j{x_j\Lambda_{ij}}}-\sum_k{\frac{n_k}{\sum_j{\frac{n_j}{n_T}\Lambda_{kj}}}\frac{\partial}{\partial{n_i}}\left(\sum_j{\frac{n_j}{n_T}\Lambda_{kj}}\right)_{T,P,n_{j\neq{i}}}}\\&=-\ln{\sum_j{x_j\Lambda_{ij}}}-\sum_k{\frac{n_k}{\sum_j{\frac{n_j}{n_T}\Lambda_{kj}}}\left(\frac{n_T-n_i}{n^2_T}\Lambda_{ki}-\sum_{j\neq{i}}{\frac{n_j}{n^2_T}\Lambda_{kj}}\right)}\\&=-\ln{\sum_j{x_j\Lambda_{ij}}}-\sum_k{\frac{1}{\sum_j{x_j\Lambda_{kj}}}\left(x_k\Lambda_{ki}-\sum_j{x_jx_k\Lambda_{kj}}\right)}\\&=-\ln{\sum_j{x_j\Lambda_{ij}}}-\sum_k{\frac{x_k\Lambda_{ki}}{\sum_j{x_j\Lambda_{kj}}}}+\sum_k{\frac{\sum_j{x_jx_k\Lambda_{kj}}}{\sum_j{x_j\Lambda_{kj}}}}\\&=-\ln{\sum_j{x_j\Lambda_{ij}}}-\sum_k{\frac{x_k\Lambda_{ki}}{\sum_j{x_j\Lambda_{kj}}}}+\sum_k{\frac{x_k\sum_j{x_j\Lambda_{kj}}}{\sum_j{x_j\Lambda_{kj}}}} \end{align} \end{equation*}

After some tedious steps, an expression describing activity coefficients (via the Wilson equation) is obtained:

(9)   \begin{equation*} \therefore \ln{\gamma_i} =-\ln{\sum_j{x_j\Lambda_{ij}}}-\sum_k{\frac{x_k\Lambda_{ki}}{\sum_j{x_j\Lambda_{kj}}}}+1 \end{equation*}

A Closer Look

The Gibbs excess energy can be written as

(10)   \begin{equation*} \begin{align} g^E&=u^E-Ts^E+Pv^E\\&=h^E-Ts^E \end{align} \end{equation*}

To derive a reliable expression that describes g^E the different excess properties \left(h^E,s^E,v^E\right) should be taken into account, but simplifications can be made by setting either s^E or h^E equal to zero and then developing a relationship that describes the remaining property. Florry and Huggins, for instance, derived a simple expression for g^E for mixtures of molecules which are chemically similar (athermal solutions; h^E=0) and which differ only in size (combinatorial, or s^E effects). So working from the definition of an ideal solution, Flory and Huggins simply replaced the concentration dependence with liquid molar volumes to account for the differences in molecular size:

(11)   \begin{equation*} \frac{g^E}{RT}=\underbrace{\sum^{ncomp}_{i=1}{x_i\ln{x_i}}}_{\text{ideal solution case}}\xrightarrow{\text{is replaced by}}\sum^{ncomp}_{i=1}{x_i\ln{\frac{\Phi_i}{x_i}}} \end{equation*}

Instead of simply replacing x_i with a more convenient measure of composition for asymmetric systems like Flory and Huggins did, Wilson takes nothing more than the combinatorial expression for an ideal solution and introduces (arbitrarily) the local composition instead.

(12)   \begin{equation*} \frac{g^E}{RT}=\underbrace{\sum^{ncomp}_{i=1}{x_i\ln{x_i}}}_{\text{ideal solution case}}\xrightarrow{\text{is instead replaced by}}\sum_i{x_i \ln{\frac{\xi}{x_i}}} \end{equation*}

So there is only exactly one place where Wilson needed the contact energy and that is to calculate the “local composition” (key point). So if contact energies A-A and A-B are very different, you get local compositions very different from the overall mole fraction and very high or low \gamma_i, but only on the sides: huge effect on \gamma_i if x_i = 0.001 and local composition \xi_i = 0.9. But 0.5 and 0.999 is not such a difference… So: only the ratio of local composition to the overall mole fraction determines g^E/RT. The difference in total contact energies between a Raoult’s statistical mixture and the local composition mixture is never included into the calculation of g^E (where, in this case, s^E is taken to be zero).

Temperature Dependence

The excess enthalpy is related to the excess Gibbs energy via the equation:

(13)   \begin{equation*} h^E=\left-T^2\frac{\partial (g^E/T)}{\partial T}\right\rvert_{P,x} \end{equation*}

 Additional Resources…