Van Laar equation

Van Laar derived the excess enthalpy from the van der Waals equation:[1]

${\displaystyle H^{ex}={\frac {b_{1}X_{1}b_{2}X_{2}}{b_{1}X_{1}+b_{2}X_{2}}}\left({\frac {\sqrt {a_{1}}}{b_{1}}}-{\frac {\sqrt {a_{2}}}{b_{2}}}\right)^{2}}$

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:

${\displaystyle {\frac {G^{ex}}{RT}}={\frac {A_{12}X_{1}A_{21}X_{2}}{A_{12}X_{1}+A_{21}X_{2}}}}$

In here A₁₂ and A₂₁ are the van Laar coefficients, which are obtained by regression of experimental vapor–liquid equilibrium data.

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

${\displaystyle \left\{{\begin{matrix}\ln \ \gamma _{1}=A_{12}\left({\frac {A_{21}X_{2}}{A_{12}X_{1}+A_{21}X_{2}}}\right)^{2}\\\ln \ \gamma _{2}=A_{21}\left({\frac {A_{12}X_{1}}{A_{12}X_{1}+A_{21}X_{2}}}\right)^{2}\end{matrix}}\right.}$

This shows that the van Laar coefficients A₁₂ and A₂₁ are equal to logarithmic limiting activity coefficients ${\displaystyle \ln \left(\gamma _{1}^{\infty }\right)}$ and ${\displaystyle \ln \left(\gamma _{2}^{\infty }\right)}$ 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 ${\displaystyle A_{12}=A_{21}=A}$, which implies that the molecules are of equal size but different in polarity, then the equations become:

${\displaystyle \left\{{\begin{matrix}\ln \ \gamma _{1}=Ax_{2}^{2}\\\ln \ \gamma _{2}=Ax_{1}^{2}\end{matrix}}\right.}$

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