Excess Gibbs free energy

Margules expressed the intensive excess Gibbs free energy of a binary liquid mixture as a power series of the mole fractions x_i:

${\displaystyle {\frac {G^{ex}}{RT}}=X_{1}X_{2}(A_{21}X_{1}+A_{12}X_{2})+X_{1}^{2}X_{2}^{2}(B_{21}X_{1}+B_{12}X_{2})+...+X_{1}^{m}X_{2}^{m}(M_{21}X_{1}+M_{12}X_{2})}$

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 ${\displaystyle X_{1}X_{2}}$ 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 x_i. This yields, when applied only to the first term and using the Gibbs–Duhem equation,:^[3]

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

In here A₁₂ and A₂₁ are constants which are equal to the logarithm of the limiting activity coefficients: ${\displaystyle \ln \ (\gamma _{1}^{\infty })}$ and ${\displaystyle \ln \ (\gamma _{2}^{\infty })}$ respectively.

When ${\displaystyle A_{12}=A_{21}=A}$, which implies molecules of same molecular size but different polarity, the equations reduce to the one-parameter Margules activity model:

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

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 is equal to its concentration (mole fraction).

Extrema

Using simple algebraic manipulation, it can be stated that ${\displaystyle d\ln \gamma _{1}/dx_{1}}$ increases or decreases monotonically within all ${\displaystyle x_{1}}$ range, if ${\displaystyle A_{12}<0}$ or ${\displaystyle A_{21}>0}$ with ${\displaystyle 0.5<A_{12}/A_{21}<2}$, respectively. When ${\displaystyle A_{12}<A_{21}/2}$ and ${\displaystyle A_{12}<0}$, the activity coefficient curve of component 1 shows a maximum and compound 2 minimum at:

${\displaystyle x_{1}={\frac {1-2A_{12}/A_{21}}{3(1-A_{12}/A_{21})}}}$

Same expression can be used when ${\displaystyle A_{12}<A_{21}/2}$ and ${\displaystyle A_{12}>0}$, 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 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(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 A₁₂=0.6298 and A₂₁=1.9522. This gives a minimum 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:

${\displaystyle A=\ln \gamma _{1}^{\infty }=\ln \gamma _{2}^{\infty }}$

${\displaystyle \gamma _{1}^{\infty }=\gamma _{2}^{\infty }>\exp(2)\approx 7.38}$

For asymmetric binary systems, A₁₂≠A₂₁, the liquid-liquid separation always occurs for

^[4]

${\displaystyle A_{21}+A_{12}>4}$

Or equivalently:

${\displaystyle \gamma _{1}^{\infty }\gamma _{2}^{\infty }>\exp(4)\approx 54.6}$

The plait point is not located at 50 mol%. It depends on the ratio of the limiting activity coefficients.

Recommended values

An extensive range of recommended values for the Margules parameters can be found in the literature.^[5][6] Selected values are provided in the table below.