1.10.30: Gibbs Energies- General Equations

For binary liquid mixtures at fixed \(\mathrm{T}\) and \(\mathrm{p}\), an important task is to fit the dependence of \({\mathrm{G}_{\mathrm{m}}}^{\mathrm{E}}\) on \(x_{2}\) to an equation in order to calculate the derivative \(\mathrm{dG}_{\mathrm{m}}^{\mathrm{E}} / \mathrm{dx}_{2}\) at required mole fractions. The Guggenheim - Scatchard [1,2] (commonly called the Redlich - Kister [3] ) equation is one such equation. This equation has the following general form.

\[\mathrm{X}_{\mathrm{m}}^{\mathrm{E}}=\mathrm{x}_{2} \,\left(1-\mathrm{x}_{2}\right) \, \sum_{\mathrm{i}=1}^{\mathrm{i}=\mathrm{k}} \mathrm{A}_{\mathrm{i}} \,\left(1-2 \, \mathrm{x}_{2}\right)^{\mathrm{i}-1}\]

\(\mathrm{A}_{\mathrm{i}}\) are coefficients obtained from a least squares analysis of the dependence of \({\mathrm{G}_{\mathrm{m}}}^{\mathrm{E}}\) on \(x_{2}\).The equation clearly satisfies the condition that \({\mathrm{G}_{\mathrm{m}}}^{\mathrm{E}}\) is zero at \(x_{2} = 0\) and at \(x_{2} = 1\). In fact the first term in the \(\mathrm{G} - \mathrm{S}\) equation has the following form.

\[X_{m}^{E}=X_{2} \,\left(1-X_{2}\right) \, A_{1}\]

According to equation (b) \({\mathrm{X}_{\mathrm{m}}}^{\mathrm{E}}\) is an extremum at \(x_{2} = 0.5\), the plot being symmetric about the line from \({\mathrm{X}_{\mathrm{m}}}^{\mathrm{E}}\) to ‘\(x_{2} = 0.5\)’. In fact for most systems the \(\mathrm{A}_{1}\) term is dominant. For the derivative \(\mathrm{dG}_{\mathrm{m}}^{\mathrm{E}} / \mathrm{dx} \mathrm{x}_{2}\), we write equation (a) in the following general form.

\[\mathrm{X}_{\mathrm{m}}^{\mathrm{E}}=\left(\mathrm{x}_{2}-\mathrm{x}_{2}^{2}\right) \, \mathrm{Q}\]

Then

\[\mathrm{dX}_{\mathrm{m}}^{\mathrm{E}} / \mathrm{dx} \mathrm{x}_{2}=\mathrm{x}_{2} \,\left(1-\mathrm{x}_{2}\right) \, \mathrm{dQ} / \mathrm{dx} \mathrm{x}_{2}+\left(1-2 \, \mathrm{x}_{2}\right) \, \mathrm{Q}\]

where

\[\mathrm{dQ} / \mathrm{dx}_{2}=-2 \, \sum_{\mathrm{i}=2}^{\mathrm{i}=\mathrm{k}}(\mathrm{i}-1) \, \mathrm{A}_{\mathrm{i}} \,\left(1-2 \, \mathrm{x}_{2}\right)^{\mathrm{i}-2}\]

Equation (a) fits the dependence with a set of contributing curves which all pass through points, \({\mathrm{X}_{\mathrm{m}}}^{\mathrm{E}}=0\) at \(x_{1} = 0\) and \(x_{1} =1\). The usual procedure involves fitting the recorded dependence using increasing number of terms in the series, testing the statistical significance of including a further term. Although equation (a) has been applied to many systems and although the equation is easy to incorporate into computer programs using packaged least square and graphical routines, the equation suffers from the following disadvantage. As one incorporates a further term in the series, (e.g. \(\mathrm{A}_{j}\)) estimates of all the previously calculated parameters (i.e. \(\mathrm{A}_{2}, \mathrm{~A}_{3 \ldots} \ldots \mathrm{A}_{\mathrm{j}-1}\)) change. For this reason orthogonal polynomials have been increasingly favoured especially where the appropriate computer software is available. The only slight reservation is that derivation of explicit equations for the required derivative \(\mathrm{d}\mathrm{X}_{\mathrm{m}}{ }^{\mathrm{E}}\) is not straightforward. The problem becomes rather more formidable when the second and higher derivatives are required. The derivative \(\mathrm{d}^{2} \mathrm{X}_{\mathrm{m}}{ }^{\mathrm{E}}\) is sometimes required by calculations concerning the properties of binary liquid mixtures.

The derivative \(\mathrm{dG}_{\mathrm{m}}^{\mathrm{E}} / \mathrm{dx} \mathrm{x}_{1}\) and \({\mathrm{G}_{\mathrm{m}}}^{\mathrm{E}}\) are combined to yield an equation for \(\ln\left(\mathrm{f}_{1}\right)\).

\[\ln \left(f_{1}\right)=\frac{G_{m}^{E}}{R \, T}+\frac{\left(1-x_{1}\right)}{R \, T} \, \frac{d G_{m}^{E}}{d x_{1}}\]

A similar equation leads to estimates of \(\ln\left(\mathrm{f}_{2}\right)\). Hence the dependences are obtained of both \(\ln\left(\mathrm{f}_{1}\right)\) and \(\ln\left(\mathrm{f}_{2}\right)\) on mixture composition. It is of interest to explore the case where the coefficients \(\mathrm{A}_{2}, \mathrm{~A}_{3} \ldots\) in equation (a) are zero. Then

\[\mathrm{X}_{\mathrm{m}}^{\mathrm{E}}=\mathrm{x}_{2} \,\left(1-\mathrm{x}_{2}\right) \, \mathrm{A}_{1}\]

and

\[\mathrm{dX}_{\mathrm{m}}^{\mathrm{E}} / \mathrm{dx_{2 }}=\left(1-2 \, \mathrm{X}_{2}\right) \, \mathrm{A}_{1}\]

With reference to the Gibbs energies,

\[\ln \left(\mathrm{f}_{2}\right)=(1 / \mathrm{R} \, \mathrm{T}) \,\left[\mathrm{x}_{2} \,\left(1-\mathrm{x}_{2}\right)+\left(1-\mathrm{x}_{2}\right) \,\left(1-2 \, \mathrm{x}_{2}\right)\right] \, \mathrm{A}_{1}^{\mathrm{G}}\]

\[\ln \left(\mathrm{f}_{2}\right)=\left(\mathrm{A}_{1}^{\mathrm{G}} / \mathrm{R} \, \mathrm{T}\right) \,\left[1-2 \, \mathrm{x}_{2}+\mathrm{x}_{2}^{2}\right]\]

or,

\[\ln \left(\mathrm{f}_{2}\right)=\left(\mathrm{A}_{1}^{\mathrm{G}} / \mathrm{R} \, \mathrm{T}\right) \,\left[1-\mathrm{x}_{2}\right]^{2}\]

In fact the equation reported by Jost et al. [4] has this form.

Rather than using the Redlich-Kister equation, recently attention has been directed to the Wilson equation [5] written in equation (l) for a two-component liquid [6].

\[\mathrm{G}_{\mathrm{m}}^{\mathrm{E}} / \mathrm{R} \, \mathrm{T}=-\mathrm{x}_{1} \, \ln \left(\mathrm{x}_{1}+\Lambda_{12} \, \mathrm{x}_{2}\right)-\mathrm{x}_{2} \, \ln \left(\mathrm{x}_{2}+\Lambda_{21} \, \mathrm{x}_{1}\right)\]

Then , for example [7],

\[\ln \left(\mathrm{f}_{1}\right)=-\ln \left(\mathrm{x}_{1}+\Lambda_{12} \, \mathrm{x}_{2}\right)+\mathrm{x}_{2} \,\left(\frac{\Lambda_{12}}{\mathrm{x}_{1}+\Lambda_{12} \, \mathrm{x}_{2}}-\frac{\Lambda_{21}}{\Lambda_{21} \, \mathrm{x}_{1}+\mathrm{x}_{2}}\right)\]

The Wilson equation forms the basis for two further developments, described as the NRTL (non-random, two-liquid) equation [8-10] and the UNIQUAC equation [9-10]. Nevertheless Douheret et al. [11] show how an excess property must be carefully defined. Davis et al. have explored how excess molar properties for liquid mixtures can be analysed in terms of different mole fraction domains [12].

Footnotes

[1] E. A. Guggenheim, Trans. Faraday Soc.,1937,33,151; equation 4.1.

[2] G. Scatchard, Chem. Rev.,1949,44,7;see page 9.

[3] O. Redlich and A. Kister, Ind. Eng. Chem.,1948,40,345; equation 8.

[4] F. Jost, H. Leiter and M. J. Schwuger, Colloid Polymer Sci., 1988, 266, 554.

[5] G. M. Wilson, J. Am. Chem. Soc.,1964,86,127.

[6] See also

1. R. V. Orye and J. M. Prausnitz, Ind. Eng. Chem.,1965,57,19.
2. S. Ohe, Vapour-Liquid Equilibrium Data, Elsevier, Amsterdam, 1989.
3. C. W. Bale and A. D. Pelton, Metallurg. Trans.,1974,5,2323.
4. R. Schuhmann, Metallurg. Trans.,B,1985,16,807.
5. M. Prehal, V. Dohnal and F. Versely, Collect. Czech. Chem.Commun.,1982,47,3171.

[7] From equation (l),

\[\begin{aligned}
\frac{1}{\mathrm{R} \, \mathrm{T}} \, \frac{\mathrm{dG}_{\mathrm{m}}^{\mathrm{E}}}{\mathrm{dx}_{1}}=&-\ln \left(\mathrm{x}_{1}+\Lambda_{12} \, \mathrm{x}_{2}\right)-\frac{\mathrm{x}_{1} \,\left(1-\Lambda_{12}\right)}{\mathrm{x}_{1}+\Lambda_{12} \, \mathrm{x}_{2}} \\
&+\ln \left(\Lambda_{21} \, \mathrm{x}_{1}+\mathrm{x}_{2}\right)-\frac{\mathrm{x}_{2} \,\left(\Lambda_{21}-1\right)}{\Lambda_{21} \, \mathrm{x}_{1}+\mathrm{x}_{2}}
\end{aligned}\]

Then using equation (f) with \(1 − x_{1} = x_{2}\),

\[\begin{aligned}
\ln \left(\mathrm{f}_{1}\right)=&-\mathrm{x}_{1} \, \ln \left(\mathrm{x}_{1}+\Lambda_{12} \, \mathrm{x}_{2}\right)-\mathrm{x}_{2} \, \ln \left(\Lambda_{21} \, \mathrm{x}_{1}+\mathrm{x}_{2}\right) \\
&-\mathrm{x}_{2} \, \ln \left(\mathrm{x}_{1}+\Lambda_{12} \, \mathrm{x}_{2}\right)-\frac{\mathrm{x}_{1} \, \mathrm{x}_{2} \,\left(1-\Lambda_{12}\right)}{\mathrm{x}_{1}+\Lambda_{12} \, \mathrm{x}_{2}} \\
&+\mathrm{x}_{2} \, \ln \left(\Lambda_{21} \, \mathrm{x}_{1}+\mathrm{x}_{2}\right)+\frac{\left(\mathrm{x}_{2}\right)^{2} \,\left(1-\Lambda_{21}\right)}{\Lambda_{21} \, \mathrm{x}_{1}+\mathrm{x}_{2}}
\end{aligned}\]

Or,

\[\begin{aligned}
\ln \left(f_{1}\right) &=-\left(x_{1}+x_{2}\right) \, \ln \left(x_{1}+\Lambda_{12} \, x_{2}\right) \\
+x_{2} \,\left[\frac{\Lambda_{12} \, x_{1}-x_{1}}{x_{1}+\Lambda_{12} \, x_{2}}-\frac{\Lambda_{21} \, x_{2}-x_{2}}{\Lambda_{21} \, x_{1}+x_{2}}\right]
\end{aligned}\]

But

\[\Lambda_{12} \, \mathrm{x}_{1}-\mathrm{x}_{1}=\Lambda_{12} \,\left(1-\mathrm{x}_{2}\right)-\mathrm{x}_{1}=\Lambda_{12}-\left(\mathrm{x}_{1}+\Lambda_{12} \, \mathrm{x}_{2}\right)\]

Hence,

\[\begin{aligned}
\ln \left(\mathrm{f}_{1}\right) &=-\ln \left(\mathrm{x}_{1}+\Lambda_{12} \, \mathrm{x}_{2}\right) \\
&+\mathrm{x}_{2} \,\left[\frac{\Lambda_{12}-\left(\mathrm{x}_{1}+\Lambda_{12} \, \mathrm{x}_{2}\right)}{\mathrm{x}_{1}+\Lambda_{12} \, \mathrm{x}_{2}}-\frac{\Lambda_{21}-\left(\Lambda_{21} \, \mathrm{x}_{1}+\mathrm{x}_{2}\right)}{\Lambda_{21} \, \mathrm{x}_{1}+\mathrm{x}_{2}}\right]
\end{aligned}\]

Or,

\[\ln \left(f_{1}\right)=-\ln \left(x_{1}+\Lambda_{12} \, x_{2}\right)+x_{2} \,\left[\frac{\Lambda_{12}}{x_{1}+\Lambda_{12} \, x_{2}}-\frac{\Lambda_{21}}{\Lambda_{21} \, x_{1}+x_{2}}\right]\]

[8] D. Abrams and J. M. Prausnitz, AIChE J.,1975,21,116.

[9] R. C. Reid, J. M. Prausnitz and E. B. Poling, The Properties of Gases and Liquids, McGraw-Hill, New York, 4th edn.,1987, chapter 8.

[10] J. M. Prausnitz, R. N. Lichtenthaler and E. G. de Azevedo, Molecular Thermodynamics of Fluid Phase Equilibria, Prentice –Hall, Upper Saddle River, N.J., 3rd edn.,1999,chapter 6.

[11] G. Douheret, C. Moreau and A. Viallard, Fluid Phase Equilib.,1985,22,277,287.

[12]

1. M. I. Davis, Thermochim. Acta, 1984,77,421; 1985,90,313;1987,120,299; 1990,157,295.
2. M. I. Davis, M. C. Molina and G. Douheret, Thermochim Acta, 1988,131,153.
3. G. Douheret, A. Pal and M. I. Davis, J.Chem.Thermodyn.,1990,22,99.
4. G. Douheret, A. H. Roux, M. I. Davis, M. E .Hernandez, H. Hoiland and E. Hogseth, J. Solution Chem.,1993,22,1041.
5. G. Douheret, M. I. Davis., J. C. R. Reis , I. J. Fjellanger, M. B. Vaage and H. Hoiland, Phys.Chem.Chem.Phys.,2002,4,6034.

[13] Finally we note that the Redlich-Kister equation can be expressed in the following form.

\[\mathrm{X}_{\mathrm{m}}^{\mathrm{E}}=\mathrm{x}_{1} \,\left(1-\mathrm{x}_{1}\right) \, \sum_{\mathrm{i}=1}^{\mathrm{i}=\mathrm{k}} \mathrm{B}_{\mathrm{i}} \,\left(1-2 \, \mathrm{x}_{1}\right)^{i-1}\]

Then \(\mathrm{A}_{\mathrm{i}}=\mathrm{B}_{\mathrm{i}} \,(-1)^{i}\)

J. D. G. de Oliveira and J. C. R.Reis, Thermochim. Acta 2008,468, 119