 Margules activity model

Contents
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}:
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 the excess Gibbs energy becomes zero at x_{1}=0 and x_{1}=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 GibbsDuhem equation, ^{[3]}:
In here A_{12} and A_{21} are constants which are equal to the logarithm of the limiting activity coefficients: and respectively.When A_{12} = A_{21} = A, which implies molecules of same molecular size but different polarity, the equations reduce to the oneparameter Margules activity model:
In that case the activity coefficients cross at x_{1}=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 x_{1}=0 .
When A_{12} < A_{21} / 2 the activity coefficient curve of component 1 shows a maximum and compound 2 minimum at:
It is easily seen that when A_{12}=0 and A_{21}>0 that a maximum in the activity coefficient of compound 1 exists at x_{1}=1/3. Obvious, the activity coefficient of compound 2 goes at this concentration through a minimum as a result of the GibbsDuhem rule.
The binary system ChloroformMethanol 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 A_{12}=0.6298 and A_{21}=1.9522. This gives a maximum in the activity of Chloroform at x_{1}=0.17.
In general, for the case A=A_{12}=A_{21}, 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:
For assymetric binary systems, A_{12}≠A_{21}, the liquidliquid separation always occurs for ^{[4]}:
A_{21} + A_{12} > 4
Or equivalent:
The plait point is not located at 50 mol%. It depends from the ratio in limiting activity coefficients.
See also
 Van Laar equation
Literature
 ^ Margules, Max (1895). "Über die Zusammensetzung der gesättigten Dämpfe von Misschungen". Sitzungsberichte der Kaiserliche Akadamie der Wissenschaften Wien MathematischNaturwissenschaftliche Klasse II 104: 1243–1278.http://www.archive.org/details/sitzungsbericht10wiengoog
 ^ Gokcen, N.A. (1996). "GibbsDuhemMargules Laws". Journal of Phase Equilibria 17 (1): 50–51. doi:10.1007/BF02648369.
 ^ "Phase Equilibria in Chemical Engineering", Stanley M. Walas, (1985) p180Butterworth Publ. ISBN 0409951625
 ^ Wisniak, Jaime (1983). "Liquid—liquid phase splitting—I analytical models for critical mixing and azeotropy". Chem.Eng.Sci. 38 (6): 969–978. doi:10.1016/00092509(83)800177.
Categories: Physical chemistry
 Thermodynamic models
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