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The '''Van Laar equation''' is an activity model, which was developed by [[Johannes van Laar]] in 1910-1913, to describe phase equilibria of liquid mixtures. The equation was derived from the [[Van der Waals equation]]. The original van der Waals parameters didn't give good description of vapor-liquid phase equilibria, which forced the user to fit the parameters to experimental results. Because of this, the model lost the connection to  molecular properties, and therefore it has to be regarded as an empirical model to correlate experimental results.
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==Equations==
Van Laar  derived the excess [[enthalpy]] from the van der Waals equation:<ref>The Open Thermodynamics Journal, 2010, 4, 129-140</ref>
:<math> H^{ex}= \frac {b_1 X_1 b_2 X_2}{b_1 X_1 +b_2 X_2} \left( \frac{\sqrt{a_1}}{b_1}- \frac{\sqrt{a_2}}{b_2} \right)^2</math>
In here  a<sub>i</sub> and b<sub>i</sub> are the van der Waals parameters for attraction and excluded volume of component i.  
Since these parameters didn't lead to good phase equilibrium description the model was reduced to the form:
:<math> \frac{G^{ex}}{RT}= \frac {A_{12} X_1 A_{21}X_2}{A_{12} X_1 +A_{21} X_2} </math>
In here A<sub>12</sub> and A<sub>21</sub> are constants, which are obtained by regression of experimental [[vapor-liquid equilibrium]] data.
 
The activity coefficient of component i is derived by differentiation to  x<sub>i</sub>. This yields:
:<math>
\left\{\begin{matrix} \ln\ \gamma_1=A_{12} \left( \frac{A_{21} X_2}{A_{12} X_1 +A_{21} X_2} \right)^2
\\ \ln\ \gamma_2=A_{21} \left (\frac{A_{12} X_1} { A_{12} X_1 +A_{21} X_2} \right )^2
\end{matrix}\right.</math>
 
This shows that the constants A<sub>12</sub> and A<sub>21</sub> are equal to logarithmic limiting activity coefficients <math> ln(\gamma_1^\infty) </math> and <math> ln(\gamma_2^\infty) </math> respectively. The model gives increasing (A<sub>12</sub> and A<sub>21</sub> >0) or only decreasing  (A<sub>12</sub> and A<sub>21</sub> <0) activity coefficients with decreasing concentration. The model can not describe extrema in the activity coefficient along the concentration range.  
 
In case <math>A_{12}=A_{21}=A</math>, which implies that the molecules are of equal size but different in polarity, then the equations become:
:<math>
\left\{\begin{matrix}  \ln\ \gamma_1=Ax^2_2
\\  \ln\ \gamma_2=Ax^2_1
\end{matrix}\right.</math>
 
In this case the activity coefficients mirror at x<sub>1</sub>=0.5. When A=0 the model the activity coefficients are unity, thus describing an ideal mixture.
 
==References==
{{reflist}}
 
[[Category:Thermodynamic equations]]

Latest revision as of 22:47, 9 May 2014

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