Margules activity model


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 γi 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 xi:


\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 xi. 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. 

Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Margules function — A Margules function is a function added to the Raoult s law description of a liquid solution to account for deviations from ideality. See also Margules activity model. The amended Raoult s law description of the vapor pressure above the solution… …   Wikipedia

  • Max Margules — Born 23 April 1856(1856 04 23) Brody, Galicia, Austrian Empire Died 4 October 1920( …   Wikipedia

  • Non-random two-liquid model — VLE of the mixture of Chloroform and Methanol plus NRTL fit and extrapolation to different pressures The non random two liquid model[ …   Wikipedia

  • weather forecasting — Prediction of the weather through application of the principles of physics and meteorology. Weather forecasting predicts atmospheric phenomena and changes on the Earth s surface caused by atmospheric conditions (snow and ice cover, storm tides,… …   Universalium

  • Dynorphin — protein Name=prodynorphin caption= width= HGNCid=8820 Symbol=PDYN AltSymbols= EntrezGene=5173 OMIM=131340 RefSeq=NM 024411 UniProt=P01213 PDB= ECnumber= Chromosome=20 Arm=p Band=ter LocusSupplementaryData= p12.2Dynorphins are a class of opioid… …   Wikipedia

  • Biodiversity — Some of the biodiversity of a coral reef …   Wikipedia

  • List of Mexican Jews — The Jewish population of Latin America has risen to more than 500,000 more than half of whom live in Argentina, with large communities also present in Brazil and Mexico. In Mexico, Mexico City has the largest Jewish population. The second and… …   Wikipedia