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The '''Margules activity model''' is  a simple thermodynamic model for the excess [[Gibbs free energy]] of a liquid mixture introduced in 1895 by [[Max Margules]]. <ref>{{Cite journal|first1=Max|last1=Margules|first2=|last2=|title=Über die Zusammensetzung der gesättigten Dämpfe von Misschungen|journal=Sitzungsberichte der Kaiserliche Akadamie der Wissenschaften Wien Mathematisch-Naturwissenschaftliche Klasse II|volume=104|issue=|year=1895|pages=1243–1278|doi=}}http://www.archive.org/details/sitzungsbericht10wiengoog</ref><ref>{{Cite journal|first1=N.A.|last1=Gokcen|first2=|last2=|title=Gibbs-Duhem-Margules Laws|journal=Journal of Phase Equilibria|volume=17|issue=1|year=1996|pages=50–51|doi=10.1007/BF02648369}}</ref> After Lewis had introduced the concept of the activity coefficient, the model could be used to derive an expression for the [[activity coefficient]]s <math>\gamma_i</math>  of a compound i in a liquid, a measure for the deviation from ideal solubility, also known as [[Raoult's law]].
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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, which modern models like [[UNIQUAC]], [[Non-Random Two Liquid model|NRTL]] and [[Wilson]] cannot.
 
==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<sub>i</sub>:
 
<math>
\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)
</math>
 
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 <math>X_1X_2</math> assures that the excess Gibbs energy becomes zero at x<sub>1</sub>=0 and x<sub>1</sub>=1.
 
=== Activity coefficient ===
The activity coefficient of component i is found by differentiation of the excess Gibbs energy towards x<sub>i</sub>.
This yields, when applied only to the first term and using the [[Gibbs-Duhem equation]],:<ref>"Phase Equilibria in Chemical Engineering", Stanley M. Walas, (1985) p180Butterworth Publ. ISBN 0-409-95162-5</ref>
 
<math>
\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.</math>
 
In here A<sub>12</sub> and A<sub>21</sub> are constants which are equal to the logarithm of the limiting activity coefficients: <math>ln( \gamma_1^\infty) </math> and <math> ln(\gamma_2^\infty )</math> respectively.
 
When <math>A_{12}=A_{21}=A</math>, which implies molecules of same molecular size but different polarity, the equations reduce to the one-parameter Margules activity model:
 
<math>
\left\{\begin{matrix} \ln\ \gamma_1=Ax^2_2
\\ \ln\ \gamma_2=Ax^2_1
\end{matrix}\right.</math>
 
In that case the activity coefficients cross at x<sub>1</sub>=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 is equal to its concentration (mole fraction).
 
=== Extrema ===
Using simple algebraic manipulation, can be stated that <math>dln\gamma_1/dx_1</math> increases or decreases monotonically within all <math> x_1</math> range, if <math>A_{12} <0</math> or <math>A_{21} >0 </math> with <math>0.5 < A_{12}/A_{21} < 2 </math>, respectively.
When <math>A_{12} < A_{21}/2 </math> and <math>A_{12} < 0 </math>, the activity coefficient curve of component 1 shows a maximum and compound 2 minimum at:
 
<math>x_1 = \frac{1-2A_{12}/A_{21}} {3(1-A_{12}/A_{21})} </math>
 
Same expression can be used when <math>A_{12} < A_{21}/2 </math> and <math>A_{12} > 0 </math>, but in this situation the activity coefficient curve of component 1 shows a minimum and compound 2 a maximum.  
It is easily seen that when A<sub>12</sub>=0 and A<sub>21</sub>>0  that a maximum in the activity coefficient of compound 1 exists at x<sub>1</sub>=1/3. Obvious, the activity coefficient of compound 2 goes at this concentration through a minimum as a result of the [[Gibbs–Duhem equation|Gibbs-Duhem rule]].
 
The binary system Chloroform(1)-Methanol(2) is an example of a system that shows a maximum in the activity coefficient of Chloroform. The parameters for a description at 20°C are A<sub>12</sub>=0.6298 and A<sub>21</sub>=1.9522. This gives a minimum in the activity of Chloroform at  x<sub>1</sub>=0.17.
 
In general, for the case A=A<sub>12</sub>=A<sub>21</sub>, 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&nbsp;mol%. Since:
 
<math>A = \ln \gamma_1^\infty  = \ln \gamma_2^\infty </math>
 
<math> \gamma_1^\infty = \gamma_2^\infty > exp(2) \approx 7.38</math>
 
For asymmetric binary systems, A<sub>12</sub>≠A<sub>21</sub>, the liquid-liquid separation always occurs for
:<ref>{{Cite journal|first1=Jaime|last1=Wisniak|first2=|last2=|title=Liquid—liquid phase splitting—I analytical models for critical mixing and azeotropy|journal=Chem.Eng.Sci.|volume=38|issue=6|year=1983|pages=969–978|doi=10.1016/0009-2509(83)80017-7}}</ref>
 
<math>A_{21} + A_{12} > 4 </math>
 
Or equivalently:
 
<math>\gamma_1^\infty  \gamma_2^\infty > exp(4) \approx 54.6 </math>
 
The plait point is not located at 50&nbsp;mol%. It depends on the ratio of the limiting activity coefficients.
 
==See also==
* [[Van Laar equation]]
 
==Literature==
<references/>
 
[[Category:Physical chemistry]]
[[Category:Thermodynamic models]]

Latest revision as of 21:21, 19 July 2014

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