Excess Gibbs Energy models

**siddharth** · Oct24-06, 01:21 PM

These are some empirical excess Gibbs energy models used to calculate activity coefficients for non-ideal liquid mixtures. I didn't find many on the web, so I thought I'll put them here for future reference

All

refer to the mole fraction of the ith species in the solution.
To calculate the activity coefficient $LaTeX Code: \\gamma_i$ from the excess free energy, we have
$LaTeX Code: ln \\gamma_i=\\left(\\frac{\\partial\\left(nG^{E}/RT\\right)}{\\partial n_i} \\right)_{P,T,n_j}$

Most of these are for binary solutions, but some can be extended into multi-component systems.

1) Van Laar equation

$LaTeX Code: \\frac{G^{E}}{RT} = \\frac{ABx_1x_2}{Ax_1+B_x_2}$

So that,

$LaTeX Code: ln \\gamma_1 = A\\left(1+\\frac{Ax_1}{Bx_2}\\right)^{-2}$

$LaTeX Code: ln \\gamma_2 = B\\left(1+\\frac{Bx_2}{Ax_1}\\right)^{-2}$

2) Porters equation

$LaTeX Code: \\frac{G^{E}}{RT}=Ax_1x_2$

So that,

$LaTeX Code: ln \\gamma_1 = Ax_2^2$

$LaTeX Code: ln \\gamma_2= Ax_1^2$

3) Margules equation

$LaTeX Code: \\frac{G^{E}}{RT}=x_1x_2\\left(Ax_1+Bx_2\\right)$

So that,

$LaTeX Code: ln \\gamma_1 = x_2^2\\left(A+2\\left(B-A\\right)x_1\\right)$

$LaTeX Code: ln \\gamma_2 = x_1^2\\left(B+2\\left(A-B\\right)x_2\\right)$

4) Wilsons equation (*local composition model*)

$LaTeX Code: \\frac{G^{E}}{RT}=-x_1ln \\left(x_1+Asingle-quotex_2\\right) - x_2ln \\left(x_2 + Bsingle-quotex_1\\right)$

So that,

$LaTeX Code: ln \\gamma_1=-ln(x1+Asingle-quotex_2) + x2 \\left(\\frac{Asingle-quote}{x_1+Asingle-quotex_2} - \\frac{Bsingle-quote}{x_2+Bsingle-quotex_1}\\right)$

$LaTeX Code: ln \\gamma_2 = -ln(x2+Bsingle-quotex_1) - x1 \\left(\\frac{Asingle-quote}{x_1+Asingle-quotex_2} - \\frac{Bsingle-quote}{x_2+Bsingle-quotex_1}\\right)$

I'll add the NRTL equation and maybe the UNIQUAC model later. Feel free to correct me if you spot any errors.