Margules activity model

Introduction

Max Margules introduced in 1895 a simple thermodynamic model for the excess Gibbs free energy of a liquid mixture. After Lewis had introduced the concept of the activity coefficient, the model could be used to derive an expression for the activity coefficients $\gamma _{i}$ of a compound i in a liquid. The activity coefficient is a measure for the deviation from ideal solubility, also known as Raoult's law. In Chemical Engineering the Margules' Gibbs free energy model for liquid mixtures is better known as the Margules activity or activity coefficient model. Although the model is old it has the characteristic feature to describe extrema in the activity coefficient, while modern models like UNIQUAC, NRTL and Wilson can not.

Excess Gibbs free energy

Margules expressed the excess Gibbs free energy of a binary liquid mixture as a power series of the mole fractions x_i:

${\frac {G^{ex}}{RT}}=X_{1}X_{2}(A_{21}X_{1}+A_{12}X_{2})+X_{1}^{2}X_{2}^{2}(B_{21}X_{1}+B_{12}X_{2})+...+X_{1}^{m}X_{2}^{m}(M_{21}X_{1}+M_{12}X_{2})$

In here the A, B are constants, which are derived from regressing experimental phase equilibria data. Frequently the B and higher order parameters are set to zero.

The leading term $X_{1}X_{2}$ assures that at x₁=0 and 1 the excess Gibbs energy becomes zero.

Activity coefficient

The activity coefficient of component i is found by differentiation of the excess Gibbs energy towards x_i. This yields, when applied only to the first term and using the Gibbs-Duhem equation, :

$\left\{{\begin{matrix}\ln \ \gamma _{1}=[A_{12}+2(A_{21}-A_{12})x_{1}]x_{2}^{2}\\\ln \ \gamma _{2}=[A_{21}+2(A_{12}-A_{21})x_{2}]x_{1}^{2}\end{matrix}}\right.$

In here A₁₂ and A₂₁ are constants which are equal to the logarithm of the limiting activity coefficients: $ln(\gamma _{1}^{\infty })$ and $ln(\gamma _{2}^{\infty })$ respectively.

When $A_{12}=A_{21}=A$, which implies molecules of same molecular size but different polarity, the equations reduce to the one-parameter Margules activity model:

$\left\{{\begin{matrix}\ln \ \gamma _{1}=Ax_{2}^{2}\\\ln \ \gamma _{2}=Ax_{1}^{2}\end{matrix}}\right.$

In that case the activity coefficients cross at x₁=0.5 and the limiting activity coefficients are equal. When A=0 the model reduces to the ideal solution, i.e. the activity of a compound equals to its concentration (mole fraction).

Extrema

When $A_{12}>A_{21}/2$ the activity coefficient curves are monotonic increasing $(A_{12}>0)$ or decreasing $(A_{12}<0)$, and have the extrema at x₁=0 .

When $A_{12}<A_{21}/2$ the activity coefficient curve of component 1 shows a maximum and compound 2 minimum at mole fraction:

It is easily seen that when A₁₂=0 and A₂₁>0 that a maximum in the activity coefficient of compound 1 exists at x₁=1/3. Obvious, the activity coefficient of compound 2 goes at this concentration through a minimum as a result of the Gibbs-Duhem rule.

The binary system Chloroform-Methanol is an example of a system that shows a maximum in the activity coefficient, i.c. Chloroform. The parameters for a description at 20°C are A₁₂=0.6298 and A₂₁=1.9522. This gives a maximum in the activity of compound 1, i.e. Chloroform at x₁=0.17.

In general the larger A, the more the binary systems deviates from Raoult's law; i.e. ideal solubility. For the one-parameter Margules equation it can be derived that when A>2 the system starts to demix in two liquids. So when:

$\gamma _{1}^{\infty }=\gamma _{2}^{\infty }>exp(2)=7.38$

For assymetric binary systems one can expect that liquid-liquid demixing occurs when:

$\gamma _{1}^{\infty }*\gamma _{2}^{\infty }>exp(4)=54.6$