Activity Coefficient - Composition Models

  • Activity coefficient of a component \(\gamma_i\) is defined as is \[\gamma_i = \exp\left(\frac{\bar{G_i}^{\text{E}}}{RT}\right)\] The above equation can be written as \[\ln \gamma_i = \frac{\bar{G_i}^{\text{E}}}{RT}\] \(\ln\gamma_i\) is a partial molar property.

  • Since \(M=\sum x_i \bar{M_i}\), for the property \(G^{\text{E}}/RT\), we can write, \[\frac{G^{\text{E}}}{RT} = x_1\frac{\bar{G_1}^{\text{E}}}{RT} + x_2\frac{\bar{G_2}^{\text{E}}}{RT} = x_1\ln\gamma_1 + x_2\ln\gamma_2\]

  • Using the definition of partial molar property, we can write \(\ln\gamma_i\) as, \[\ln\gamma_i = \left[\frac{\partial(nG^E/RT)}{\partial n_i}\right]_{T,P,n_j} \tag*{(60)}\]

  • A very simple model of \(G^E\) as given by Porter is \[\frac{G^E}{RT} = Ax_1x_2 \tag*{(61)}\] where \(A\) is a constant for a given system at a temperature.

    From this Porter’s equation, equations for activity coefficients are obtained from Eqn.(60). They are given as: \[\begin{align*} \ln \gamma_1 &= Ax_2^2 \tag*{(62a)}\\ \ln \gamma_2 & = Ax_1^2 \tag*{(62b)}\end{align*}\] Eqn.(62) is known as two-suffix Margules equations (or one parameter Margules, or Porter equations). This model provides a good representation for many simple mixtures. Typically, this requires that the molecules are of similar size, shape, and chemical nature.

  • Model for \(G^E\) as given by Redlich-Kister expansion form (given below) is very generic and can be used for most of the systems with as many fitting parameters needed. \[\frac{G^E}{RT} = x_1x_2\left[A+B(x_1-x_2)+C(x_1-x_2)^2+\cdots \right] \tag*{(63)}\] With only one parameter (except \(A\), others are zero), the above form reduces to that of one parameter Margules equations.

  • By considering two parameters (\(A\) and \(B\)) of Redlich-Kister expansion form, the two-parameter Margules equations are obtained. \[\begin{align*} \ln \gamma_1 &= x_2^2\left[A_{12}+2x_1(A_{21}-A_{12})\right] \tag*{(64a)} \\ \ln \gamma_2 & = x_1^2\left[A_{21}+2x_2(A_{12}-A_{21})\right] \tag*{(64b)}\end{align*}\] Eqn.(64) is known as three-suffix Margules equations or two-parameter Margules equations (or simply called as Margules equations). \(A_{12}\) and \(A_{21}\) are called as Margules parameters. They are constant for a given system, and depends on temperature. For the limiting conditions of infinite dilution, i.e., when \(x_1\rightarrow0\),   \(\ln\gamma_1^\infty=A_{12}\); and, when \(x_2\rightarrow0\),  \(\ln\gamma_2^\infty=A_{21}\).

  • van Laar equations: From the modified form of Redlich-Kister equations, the following equations, known as van Laar equations are obtained: \[\begin{align*} \ln\gamma_1 &= A_{12}'\left(1+\frac{A_{12}'x_1}{A_{21}'x_2}\right)^{-2} \tag*{(65a)} \\ \ln\gamma_2 &= A_{21}'\left(1+\frac{A_{21}'x_2}{A_{12}'x_1}\right)^{-2} \tag*{(65b)}\end{align*}\] where \(A_{12}'\) and \(A_{21}'\) are van Laar constants. When \(x_1\rightarrow0\), \(\ln\gamma_1^\infty=A_{12}'\); when \(x_2\rightarrow0\), \(\ln\gamma_2^\infty=A_{21}'\).

    From van Laar equations, it can be shown that, \[\frac{x_1\ln\gamma_1}{x_2\ln\gamma_2} = \frac{A_{21}'x_2}{A_{12}'x_1}\] Using the above in Eqn.(65), we get \[\begin{align*} A_{12}' &= \left(1 + \frac{x_2\ln\gamma_2}{x_1\ln\gamma_1}\right)^2 \ln\gamma_1 \tag*{(66a)} \\ A_{21}' &= \left(1 + \frac{x_1\ln\gamma_1}{x_2\ln\gamma_2}\right)^2\ln\gamma_2 \tag*{(66b)}\end{align*}\] These equations are very convenient to evaluate the van Laar constants \(A_{12}'\) and \(A_{21}'\), given the value of \(x_i\) and \(\gamma_i\).

  • The Wilson equation, like Margules and van Laar equations, contains just two parameters for a binary system (\(\Lambda_{12}\)) and (\(\Lambda_{21}\)), and is given as: \[\begin{align*} \ln\gamma_1 &= -\ln(x_1+x_2\Lambda_{12}) + x_2\left(\frac{\Lambda_{12}}{x_1+x_2\Lambda_{12}} - \frac{\Lambda_{21}}{x_2+x_1\Lambda_{21}}\right) \tag*{(67a)}\\ \ln\gamma_2 &= -\ln(x_2+x_1\Lambda_{21}) - x_1\left(\frac{\Lambda_{12}}{x_1+x_2\Lambda_{12}} - \frac{\Lambda_{21}}{x_2+x_1\Lambda_{21}}\right) \tag*{(67b)}\end{align*}\] For infinite dilution, these equations become \[\begin{align*} \ln\gamma_1^\infty &= -\ln\Lambda_{12} + 1 -\Lambda_{21} \tag*{(68a)}\\ \ln\gamma_2^\infty &= -\ln\Lambda_{21} + 1 -\Lambda_{12} \tag*{(68b)}\end{align*}\] The parameters \(\Lambda_{12}\) and \(\Lambda_{21}\) are positive numbers.