The Van Laar equation is a

thermodynamic activity In chemical thermodynamics, activity (symbol ) is a measure of the "effective concentration" of a species in a mixture, in the sense that the species' chemical potential depends on the activity of a real solution in the same way that it would depe ...

model, which was developed by

Johannes van Laar Johannes van Laar (10 July 1860 in The Hague – 9 December 1938 in Montreux) was a Dutch chemist who is best known for the equations regarding chemical activity ( Van Laar equation). Biography Van Laar lost his parents early, his mother in 1862 ...

in 1910-1913, to describe

phase equilibria In thermodynamics, the phase rule is a general principle governing "pVT" systems, whose thermodynamic states are completely described by the variables pressure (), volume () and temperature (), in thermodynamic equilibrium. If is the number of d ...

of liquid mixtures. The equation was derived from the

Van der Waals equation In chemistry and thermodynamics, the Van der Waals equation (or Van der Waals equation of state) is an equation of state which extends the ideal gas law to include the effects of interaction between molecules of a gas, as well as accounting for ...

. The original van der Waals parameters didn't give good description of vapor-liquid equilibria of phases, 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.

Equations

Van Laar derived the excess

enthalpy Enthalpy , a property of a thermodynamic system, is the sum of the system's internal energy and the product of its pressure and volume. It is a state function used in many measurements in chemical, biological, and physical systems at a constant ...

from the van der Waals equation: :

H^= \frac  \left( \frac- \frac \right)^2

In here ''a''_i and ''b''_i are the van der Waals parameters for attraction and excluded volume of component i. He used the conventional quadratic mixing rule for the energy parameter ''a'' and the linear mixing rule for the size parameter ''b.'' Since these parameters didn't lead to good phase equilibrium description the model was reduced to the form: :

\frac= \frac

In here A₁₂ and A₂₁ are the van Laar coefficients, which are obtained by regression of experimental

vapor–liquid equilibrium In thermodynamics and chemical engineering, the vapor–liquid equilibrium (VLE) describes the distribution of a chemical species between the vapor phase and a liquid phase. The concentration of a vapor in contact with its liquid, especially a ...

data. The

activity coefficient In thermodynamics, an activity coefficient is a factor used to account for deviation of a mixture of chemical substances from ideal behaviour. In an ideal mixture, the microscopic interactions between each pair of chemical species are the same ( ...

of component i is derived by differentiation to x_i. This yields: :

\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.

This shows that the van Laar coefficients A₁₂ and A₂₁ are equal to logarithmic limiting activity coefficients

\ln \left( \gamma_1^\infty \right)

and

\ln \left( \gamma_2^\infty \right)

respectively. The model gives increasing (A₁₂ and A₂₁ >0) or only decreasing (A₁₂ and A₂₁ <0) activity coefficients with decreasing concentration. The model can not describe extrema in the activity coefficient along the concentration range. In case

A_{12}=A_{21}=A

, which implies that the molecules are of equal size but different in polarity, then the equations become: :

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

In this case the activity coefficients mirror at x₁=0.5. When A=0, the activity coefficients are unity, thus describing an ideal mixture.

Recommended values

An extensive range of recommended values for the Van Laar coefficients can be found in the literature. Selected values are provided in the table below. {, class="wikitable" !System !''A''₁₂ !''A''₂₁ , - , Acetone(1)-Chloroform(2) , -0.8643 , -0.5899 , - , Acetone(1)-Methanol(2) , 0.6184 , 0.5797 , - , Acetone(1)-Water(2) , 2.1041 , 1.5555 , - , Carbon tetrachloride(1)-Benzene (2) , 0.0951 , 0.0911 , - , Chloroform(1)-Methanol(2) , 0.9356 , 1.8860 , - , Ethanol(1)-Benzene(2) , 1.8570 , 1.4785 , - , Ethanol(1)-Water(2) , 1.6798 , 0.9227

References

{{reflist Thermodynamic equations