Margules activity model

Introduction

Max Margules introduced in 1895 ^{[1] [2]} 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 γ_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.

Equations

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 X₁X₂ assures that the excess Gibbs energy becomes zero at x₁=0 and x₁=1.

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, ^[3]:

$\left\{\begin{matrix} \ln\ \gamma_1=[A_{12}+2(A_{21}-A_{12})x_1]x^2_2 \\ \ln\ \gamma_2=[A_{21}+2(A_{12}-A_{21})x_2]x^2_1 \end{matrix}\right.$

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 A₁₂ = A₂₁ = A, which implies molecules of same molecular size but different polarity, the equations reduce to the one-parameter Margules activity model:

$\left\{\begin{matrix} \ln\ \gamma_1=Ax^2_2 \\ \ln\ \gamma_2=Ax^2_1 \end{matrix}\right.$

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 A₁₂ > A₂₁ / 2 the activity coefficient curves are monotonic increasing (A₁₂ > 0) or decreasing (A₁₂ < 0), and have the extrema at x₁=0 .

When A₁₂ < A₂₁ / 2 the activity coefficient curve of component 1 shows a maximum and compound 2 minimum at:

$x_1= \frac{1-2A_{12}/A_{21}} {3(1-A_{12}/A_{21})}$ 

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 Chloroform at x₁=0.17.

In general, for the case A=A₁₂=A₂₁, the larger parameter A, the more the binary systems deviates from Raoult's law; i.e. ideal solubility. When A>2 the system starts to demix in two liquids at 50/50 composition; i.e. plait point is at 50 mol%. Since:

$A=ln( \gamma_1^\infty) =ln( \gamma_2^\infty)$

$\gamma_1^\infty = \gamma_2^\infty>exp(2)=7.38$

For assymetric binary systems, A₁₂≠A₂₁, the liquid-liquid separation always occurs for ^[4]:

A21 + A12 > 4

Or equivalent:

$\gamma_1^\infty * \gamma_2^\infty>exp(4)=54.6$

The plait point is not located at 50 mol%. It depends from the ratio in limiting activity coefficients.

See also

• Van Laar equation

Literature

1. ^ Margules, Max (1895). "Über die Zusammensetzung der gesättigten Dämpfe von Misschungen". Sitzungsberichte der Kaiserliche Akadamie der Wissenschaften Wien Mathematisch-Naturwissenschaftliche Klasse II 104: 1243–1278. http://www.archive.org/details/sitzungsbericht10wiengoog
2. ^ Gokcen, N.A. (1996). "Gibbs-Duhem-Margules Laws". Journal of Phase Equilibria 17 (1): 50–51. doi:10.1007/BF02648369. 
3. ^ "Phase Equilibria in Chemical Engineering", Stanley M. Walas, (1985) p180Butterworth Publ. ISBN 0-409-95162-5
4. ^ Wisniak, Jaime (1983). "Liquid—liquid phase splitting—I analytical models for critical mixing and azeotropy". Chem.Eng.Sci. 38 (6): 969–978. doi:10.1016/0009-2509(83)80017-7.