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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 [itex]x_i[/itex] refer to the mole fraction of the ith species in the solution.

To calculate the activity coefficient [itex]\gamma_i[/itex] from the excess free energy, we have

[tex]ln \gamma_i=\left(\frac{\partial\left(nG^{E}/RT\right)}{\partial n_i} \right)_{P,T,n_j}[/tex]

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

[tex]\frac{G^{E}}{RT} = \frac{ABx_1x_2}{Ax_1+B_x_2}[/tex]

So that,

[tex] ln \gamma_1 = A\left(1+\frac{Ax_1}{Bx_2}\right)^{-2} [/tex]

[tex] ln \gamma_2 = B\left(1+\frac{Bx_2}{Ax_1}\right)^{-2} [/tex]

[tex]\frac{G^{E}}{RT}=Ax_1x_2[/tex]

So that,

[tex]ln \gamma_1 = Ax_2^2[/tex]

[tex]ln \gamma_2= Ax_1^2[/tex]

[tex]\frac{G^{E}}{RT}=x_1x_2\left(Ax_1+Bx_2\right)[/tex]

So that,

[tex] ln \gamma_1 = x_2^2\left(A+2\left(B-A\right)x_1\right)[/tex]

[tex]ln \gamma_2 = x_1^2\left(B+2\left(A-B\right)x_2\right)[/tex]

[tex]\frac{G^{E}}{RT}=-x_1ln \left(x_1+A'x_2\right) - x_2ln \left(x_2 + B'x_1\right) [/tex]

So that,

[tex] ln \gamma_1=-ln(x1+A'x_2) + x2 \left(\frac{A'}{x_1+A'x_2} - \frac{B'}{x_2+B'x_1}\right)[/tex]

[tex]ln \gamma_2 = -ln(x2+B'x_1) - x1 \left(\frac{A'}{x_1+A'x_2} - \frac{B'}{x_2+B'x_1}\right)[/tex]

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

All [itex]x_i[/itex] refer to the mole fraction of the ith species in the solution.

To calculate the activity coefficient [itex]\gamma_i[/itex] from the excess free energy, we have

[tex]ln \gamma_i=\left(\frac{\partial\left(nG^{E}/RT\right)}{\partial n_i} \right)_{P,T,n_j}[/tex]

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

**1) Van Laar equation**[tex]\frac{G^{E}}{RT} = \frac{ABx_1x_2}{Ax_1+B_x_2}[/tex]

So that,

[tex] ln \gamma_1 = A\left(1+\frac{Ax_1}{Bx_2}\right)^{-2} [/tex]

[tex] ln \gamma_2 = B\left(1+\frac{Bx_2}{Ax_1}\right)^{-2} [/tex]

**2) Porters equation**[tex]\frac{G^{E}}{RT}=Ax_1x_2[/tex]

So that,

[tex]ln \gamma_1 = Ax_2^2[/tex]

[tex]ln \gamma_2= Ax_1^2[/tex]

**3) Margules equation**[tex]\frac{G^{E}}{RT}=x_1x_2\left(Ax_1+Bx_2\right)[/tex]

So that,

[tex] ln \gamma_1 = x_2^2\left(A+2\left(B-A\right)x_1\right)[/tex]

[tex]ln \gamma_2 = x_1^2\left(B+2\left(A-B\right)x_2\right)[/tex]

**4) Wilsons equation**(*local composition model*)[tex]\frac{G^{E}}{RT}=-x_1ln \left(x_1+A'x_2\right) - x_2ln \left(x_2 + B'x_1\right) [/tex]

So that,

[tex] ln \gamma_1=-ln(x1+A'x_2) + x2 \left(\frac{A'}{x_1+A'x_2} - \frac{B'}{x_2+B'x_1}\right)[/tex]

[tex]ln \gamma_2 = -ln(x2+B'x_1) - x1 \left(\frac{A'}{x_1+A'x_2} - \frac{B'}{x_2+B'x_1}\right)[/tex]

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

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