# Excess Gibbs Energy models

• siddharth
In summary, there are several empirical excess Gibbs energy models that can be used to calculate activity coefficients for non-ideal liquid mixtures. These models are primarily for binary solutions, but can also be extended to multi-component systems. The Van Laar, Porters, Margules, and Wilsons equations are some examples of these models, each with their own specific calculations for the activity coefficients. The NRTL and UNIQUAC models may also be used for this purpose.
siddharth
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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 $x_i$ refer to the mole fraction of the ith species in the solution.
To calculate the activity coefficient $\gamma_i$ from the excess free energy, we have
$$ln \gamma_i=\left(\frac{\partial\left(nG^{E}/RT\right)}{\partial n_i} \right)_{P,T,n_j}$$

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

1) Van Laar equation

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

So that,

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

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

2) Porters equation

$$\frac{G^{E}}{RT}=Ax_1x_2$$

So that,

$$ln \gamma_1 = Ax_2^2$$

$$ln \gamma_2= Ax_1^2$$

3) Margules equation

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

So that,

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

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

4) Wilsons equation (*local composition model*)

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

So that,

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

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

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

Last edited:
This is a nice summery.

## 1. What is the purpose of excess Gibbs energy models?

Excess Gibbs energy models are used to predict the thermodynamic properties of liquid mixtures, such as vapor-liquid equilibrium, activity coefficients, and excess enthalpy.

## 2. How do excess Gibbs energy models differ from other thermodynamic models?

Excess Gibbs energy models take into account the interactions between different molecules in a mixture, whereas other thermodynamic models assume ideal behavior and no interactions.

## 3. What are the limitations of excess Gibbs energy models?

Excess Gibbs energy models may not accurately predict the properties of mixtures at extreme conditions, such as high pressures or temperatures. They also may not be applicable to mixtures with highly polar or asymmetric molecules.

## 4. How are excess Gibbs energy models validated?

Excess Gibbs energy models are validated by comparing their predictions to experimental data. If the model accurately predicts the properties of a variety of mixtures, it is considered to be a reliable tool for future predictions.

## 5. Are there different types of excess Gibbs energy models?

Yes, there are several types of excess Gibbs energy models, such as Wilson, NRTL, and UNIQUAC. These models use different mathematical expressions to describe the interactions between molecules in a mixture.

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