1.10.28: Gibbs Energies- Liquid Mixtures- Thermodynamic Patterns
A given binary liquid mixture (at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\), which is close to the standard pressure) mole fraction \(x_{1} \left(= 1 - x_{2}\right)\) is characterised by the molar Gibbs energy of mixing, \(\Delta_{\text{mix}}\mathrm{G}_{\mathrm{m}}\) and related molar enthalpic, volumetric and entropic properties. A corresponding set of properties for this mixture exist granted that the thermodynamic properties are ideal; e.g. \(\Delta_{\text{mix}}\mathrm{G}_{\mathrm{m}}(\text{id})\). As a consequence we define the corresponding excess molar property, \(\mathrm{G}_{\mathrm{m}}^{\mathrm{E}}=\Delta_{\operatorname{mix}} \mathrm{G}_{\mathrm{m}}-\Delta_{\operatorname{mix}} \mathrm{G}_{\mathrm{m}} \text { (id) }\). Interesting patterns emerge relating these properties and the corresponding partial molar properties; e.g. chemical potentials.
For chemical substance 1, (which we will conventionally take as water) the chemical potential in the liquid mixture is related to the chemical potential of the pure liquid at the same \(\mathrm{T}\) and using equation (a) where \(x_{1}\) is the mole fraction and \(\mathrm{f}_{1}\) is the rational activity coefficient.
\[\mu_{1}(\operatorname{mix})=\mu_{1}^{*}(\ell)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1} \, \mathrm{f}_{1}\right) \nonumber \]
where
\[\operatorname{limit}\left(\mathrm{x}_{1} \rightarrow 1\right) \mathrm{f}_{1}=1.0 \text { at all T and } p \nonumber \]
Similarly for component 2,
\[\mu_{2}(\operatorname{mix})=\mu_{2}^{*}(\ell)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{2} \, \mathrm{f}_{2}\right) \nonumber \]
where
\[\operatorname{limit}\left(\mathrm{x}_{2} \rightarrow 1\right) \mathrm{f}_{2}=1.0 \text { at all } \mathrm{T} \text { and } \mathrm{p} \nonumber \]
Here \(\mu_{1}^{*}(\ell)\) and \(\mu_{2}^{*}(\ell)\) are the chemical potentials of the two pure liquids at the same \(\mathrm{T}\) and \(\mathrm{p}\); \(\mathrm{f}_{1}\) and \(\mathrm{f}_{2}\) are rational activity coefficients. These (rational) activity coefficients approach unity at opposite ends of the mixture composition range. For the aqueous component, as \(x_{1}\) approaches 1, so \(\mathrm{f}_{1}\) approaches unity (at the same \(\mathrm{T}\) and \(\mathrm{p}\)). At the other end of the scale, as \(x_{1}\) approaches zero so the chemical potential of water in the binary system approaches ‘minus infinity’. If across the whole composition range (at all \(\mathrm{T}\) and \(\mathrm{p}\)), both \(\mathrm{f}_{1}\) and \(\mathrm{f}_{2}\) are unity, the thermodynamic properties of the liquid mixture are ideal.
A given liquid mixture (at fixed \(\mathrm{T}\) and \(\mathrm{p}\)) is formed by mixing \(\mathrm{n}_{1}\) moles of liquid 1 and \(\mathrm{n}_{2}\) moles of liquid 2. Before mixing the total Gibbs energy of the system, defined as G(no-mix) is given by the following equation where \(\mu_{1}^{*}(\ell)\) and \(\mu_{2}^{*}(\ell)\) are the chemical potentials of the two pure liquids at the same \(\mathrm{T}\) and \(\mathrm{p}\). Then,
\[\mathrm{G}(\mathrm{no}-\operatorname{mix})=\mathrm{n}_{1} \, \mu_{1}^{*}(\ell)+\mathrm{n}_{2} \, \mu_{2}^{*}(\ell) \nonumber \]
After mixing, the Gibbs energy of the mixture is given by equation (f).
\[\begin{aligned}
&\mathrm{G}(\operatorname{mix})= \\
&\quad \mathrm{n}_{1} \,\left[\mu_{1}^{*}(\ell)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1} \, \mathrm{f}_{1}\right)\right]+\mathrm{n}_{2} \,\left[\mu_{2}^{*}(\ell)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{2} \, \mathrm{f}_{2}\right)\right]
\end{aligned} \nonumber \]
By definition,
\[\Delta_{\operatorname{mix}} \mathrm{G}=\mathrm{G}(\operatorname{mix})-\mathrm{G}(\mathrm{no}-\mathrm{mix}) \nonumber \]
Hence the Gibbs energy of mixing ,
\[\Delta_{\text {mix }} \mathrm{G}=\mathrm{R} \, \mathrm{T} \,\left[\mathrm{n}_{1} \, \ln \left(\mathrm{x}_{1} \, \mathrm{f}_{1}\right)\right]+\left[\mathrm{n}_{2} \, \ln \left(\mathrm{x}_{2} \, \mathrm{f}_{2}\right)\right] \nonumber \]
We re-express \(\Delta_{\text{mix}}\mathrm{G}\) in terms of the Gibbs energy of mixing for one mole of liquid mixture. Thus
\[\Delta_{\text {mix }} \mathrm{G}_{\mathrm{m}}=\Delta_{\text {mix }} \mathrm{G} /\left(\mathrm{n}_{1}+\mathrm{n}_{2}\right) \nonumber \]
Hence,
\[\Delta_{\operatorname{mix}} \mathrm{G}_{\mathrm{m}}=\mathrm{R} \, \mathrm{T} \,\left[\mathrm{x}_{1} \, \ln \left(\mathrm{x}_{1} \, \mathrm{f}_{1}\right)+\mathrm{x}_{2} \, \ln \left(\mathrm{x}_{2} \, \mathrm{f}_{2}\right)\right] \nonumber \]
Or,
\[\Delta_{\text {mix }} G_{m}=R \, T \,\left[x_{1} \, \ln \left(x_{1}\right)+x_{1} \, \ln \left(f_{1}\right)+x_{2} \, \ln \left(x_{2}\right)+x_{2} \, \ln \left(f_{2}\right)\right]]
By definition,
\[\Delta_{\text {mix }} \mathrm{G}_{\mathrm{m}}(\mathrm{id})=\mathrm{x}_{1} \, \mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1}\right)+\mathrm{x}_{2} \, \mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{2}\right) \nonumber \]
Recalling that \(x_{1} + x_{2} = 1\) and \(dx_{1} = -dx_{2}\),
\[\mathrm{d} \Delta_{\operatorname{mix}} \mathrm{G}_{\mathrm{m}}(\mathrm{id}) / \mathrm{dx} \mathrm{x}_{2}=-\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1}\right)-\mathrm{R} \, \mathrm{T}+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{2}\right)+\mathrm{R} \, \mathrm{T} \nonumber \]
We use equation (l) for \(\ln \left(\mathrm{x}_{2}\right)\) and substitute in equation (m).
\[\begin{aligned}
&\mathrm{d} \Delta_{\text {mix }} \mathrm{G}_{\mathrm{m}}(\mathrm{id}) / \mathrm{dx} \mathrm{x}_{2}= \\
&-\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1}\right)+\left(1 / \mathrm{x}_{2}\right) \,\left[\Delta_{\text {mix }} \mathrm{G}_{\mathrm{m}}(\mathrm{id})-\mathrm{x}_{1} \, \mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1}\right)\right]
\end{aligned} \nonumber \]
or,
\[\begin{aligned}
&x_{2} \, d \Delta_{\text {mix }} G_{m}(\mathrm{id}) / d x_{2}= \\
&\left.-x_{2} \, R \, T \, \ln \left(x_{1}\right)+\Delta_{\text {mix }} G_{m} \text { (id) }-x_{1} \, R \, T \, \ln \left(x_{1}\right)\right]
\end{aligned} \nonumber \]
Hence,
\[\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1}\right)=\Delta_{\operatorname{mix}} \mathrm{G}_{\mathrm{m}}(\mathrm{id})-\mathrm{x}_{2} \, \mathrm{d} \Delta_{\operatorname{mix}} \mathrm{G}_{\mathrm{m}}(\mathrm{id}) / \mathrm{dx} \mathrm{x}_{2} \nonumber \]
At all mole fractions, \(\Delta_{\operatorname{mix}} G_{m}(\text { id })\) is negative; the plot of \(\Delta_{\text {mix }} G_{m} \text { (id) }\) against \(x_{1}\) is symmetric about ‘\(x_{1} = 0.5\)’. At the extremum, where \(\mathrm{d} \Delta_{\operatorname{mix}} \mathrm{G}_{\mathrm{m}}(\mathrm{id}) / \mathrm{dx} \mathrm{x}_{2}\) is zero, equation (l) shows that \(\Delta_{\operatorname{mix}} \mathrm{G}_{\mathrm{m}}(\mathrm{id})\) equals \(\mathrm{R} \, \mathrm{T} \, \ln (0.5)\).
We define an excess chemical potential for each of the two components of a binary liquid mixture. For liquid component 1,
\[\mu_{1}^{\mathrm{E}}(\mathrm{mix})=\mu_{1}(\mathrm{mix})-\mu_{1}^{*}(\ell)-\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1}\right)=\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{f}_{1}\right) \nonumber \]
Similarly for liquid component 2,
\[\mu_{2}^{\mathrm{E}}(\mathrm{mix})=\mu_{2}(\mathrm{mix})-\mu_{2}^{*}(\ell)-\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{2}\right)=\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{f}_{2}\right) \nonumber \]
Rational activity coefficients \(\mathrm{f}_{1}\) and \(\mathrm{f}_{2}\) depend on mixture composition, \(\mathrm{T}\) and \(\mathrm{p}\).
In summary ‘excess’ means excess over ideal. Excess properties provide a mutually consistent set of perspectives of a given liquid mixture. We define a reference state for binary liquid mixtures so that the thermodynamic properties of a given liquid mixture can be correlated with a common model.
A convenient approach defines an excess molar Gibbs energy of mixing.
\[\mathrm{G}_{\mathrm{m}}^{\mathrm{E}}=\Delta_{\operatorname{mix}} \mathrm{G}_{\mathrm{m}}-\Delta_{\operatorname{mix}} \mathrm{G}_{\mathrm{m}} \text { (id) } \nonumber \]
Then
\[\mathrm{G}_{\mathrm{m}}^{\mathrm{E}}=\mathrm{R} \, \mathrm{T} \,\left[\mathrm{x}_{1} \, \ln \left(\mathrm{f}_{1}\right)+\mathrm{x}_{2} \, \ln \left(\mathrm{f}_{2}\right)\right] \nonumber \]
Where
\[\operatorname{limit}\left(x_{1} \rightarrow 1\right) \mathrm{G}_{\mathrm{m}}^{\mathrm{E}}=0 \nonumber \]
And
\[\operatorname{limit}\left(\mathrm{x}_{2} \rightarrow 1\right) \mathrm{G}_{\mathrm{m}}^{\mathrm{E}}=0 \nonumber \]
Other than the latter two conditions we cannot predict the dependence of \({\mathrm{G}_{\mathrm{m}}}^{\mathrm{E}}\) on a mixture composition for m a given real mixture.[1]
In general terms excess molar properties of binary aqueous mixtures are expressed in terms of the following general equation with respect to the thermodynamic variable \(\mathrm{Q} (= \mathrm{~G}, \mathrm{~V}, \mathrm{~H} \text { and } \mathrm{S})\).
\[\mathrm{Q}_{\mathrm{m}}^{\mathrm{E}}=\mathrm{Q}_{\mathrm{m}}(\text { mix })-\mathrm{Q}_{\mathrm{m}}(\text { mix } ; \text { ideal }) \nonumber \]
Returning to the Gibbs energies, we differentiate equation (t) with respect to mole fraction \(x_{1}\) at fixed \(\mathrm{T}\) and \(\mathrm{p}\).
\[\frac{1}{R \, T} \, \frac{d G_{m}^{E}}{d x_{1}}=\ln \left(f_{1}\right)+x_{1} \, \frac{d \ln \left(f_{1}\right)}{d x_{1}}-\ln \left(f_{2}\right)+x_{2} \, \frac{d \ln \left(f_{2}\right)}{d x_{1}} \nonumber \]
According to the Gibbs-Duhem equation, at fixed \(\mathrm{T}\) and \(\mathrm{p}\),
\[\mathrm{x}_{1} \, \frac{\mathrm{d} \ln \left(\mu_{1}\right)}{\mathrm{dx}_{1}}+\mathrm{x}_{2} \, \frac{\mathrm{d} \ln \left(\mu_{2}\right)}{\mathrm{dx}}=0 \nonumber \]
Hence,
\[x_{1} \, \frac{d \ln \left(f_{1}\right)}{d x_{1}}+x_{2} \, \frac{d \ln \left(f_{2}\right)}{d x_{1}}=0 \nonumber \]
From equation (x),
\[\ln \left(f_{2}\right)=\ln \left(f_{1}\right)-\frac{1}{R \, T} \, \frac{d G_{m}^{E}}{d x_{1}} \nonumber \]
Hence using equation (t),
\[\frac{\mathrm{G}_{\mathrm{m}}^{\mathrm{E}}}{\mathrm{R} \, \mathrm{T}}=\ln \left(\mathrm{f}_{1}\right)-\frac{\mathrm{x}_{2}}{\mathrm{R} \, \mathrm{T}} \, \frac{\mathrm{dG}_{\mathrm{m}}^{\mathrm{E}}}{\mathrm{dx}_{1}} \nonumber \]
Or
\[\ln \left(f_{1}\right)=\frac{G_{m}^{E}}{R \, T}+\frac{x_{2}}{R \, T} \, \frac{d G_{m}^{E}}{d x_{1}} \nonumber \]
Equation (zc) has an interesting feature. At the mole fraction composition where \(\frac{\mathrm{dG}_{\mathrm{m}}^{\mathrm{E}}}{\mathrm{dx}_{1}}\) is zero, \(\frac{\mathrm{G}_{\mathrm{m}}^{\mathrm{E}}}{\mathrm{R} \, \mathrm{T}}\) offers a direct measure of \(\ln \left(\mathrm{f}_{1}\right)\) at that mole fraction. In some systems the plot of \({\mathrm{G}_{\mathrm{m}}}^{\mathrm{E}}\) against composition is S-shaped so we have this information at two mole fractions.
Turning to volumetric properties, the molar volume of an ideal binary liquid mixture is given by equation (zd).
\[\mathrm{V}_{\mathrm{m}}=\mathrm{x}_{1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{x}_{2} \, \mathrm{V}_{2}^{*}(\ell) \nonumber \]
Hence,
\[\mathrm{V}_{\mathrm{m}}^{\mathrm{E}}=\mathrm{x}_{1} \,\left[\mathrm{V}_{1}(\mathrm{mix})-\mathrm{V}_{1}^{*}(\ell)\right]+\mathrm{x}_{2} \,\left[\mathrm{V}_{2}(\mathrm{mix})-\mathrm{V}_{2}^{*}(\ell)\right] \nonumber \]
\[\begin{aligned}
&\frac{\mathrm{dV}_{\mathrm{m}}^{\mathrm{E}}}{\mathrm{dx} \mathrm{x}_{1}}= \\
&{\left[\mathrm{V}_{1}(\mathrm{mix})-\mathrm{V}_{1}^{*}(\ell)\right]+\mathrm{x}_{1} \, \frac{\mathrm{dV}_{1}}{\mathrm{dx}_{1}}-\left[\mathrm{V}_{2}(\mathrm{mix})-\mathrm{V}_{2}^{*}(\ell)\right]+\mathrm{x}_{2} \, \frac{\mathrm{dV}_{2}}{\mathrm{dx}_{1}}}
\end{aligned} \nonumber \]
Using the Gibbs-Duhem equation,
\[\frac{\mathrm{dV}_{\mathrm{m}}^{\mathrm{E}}}{\mathrm{dx}_{1}}=\left[\mathrm{V}_{1}(\operatorname{mix})-\mathrm{V}_{1}^{*}(\ell)\right]-\left[\mathrm{V}_{2}(\operatorname{mix})-\mathrm{V}_{2}^{*}(\ell)\right] \nonumber \]
or,
\[\left[\mathrm{V}_{1}(\mathrm{mix})-\mathrm{V}_{1}^{*}(\ell)\right]=\frac{\mathrm{dV}_{\mathrm{m}}^{\mathrm{E}}}{\mathrm{dx} \mathrm{x}_{1}}+\left[\mathrm{V}_{2}(\mathrm{mix})-\mathrm{V}_{2}^{*}(\ell)\right] \nonumber \]
\[\mathrm{V}_{\mathrm{m}}^{\mathrm{E}}=\mathrm{x}_{1} \, \frac{\mathrm{dV}_{\mathrm{m}}^{\mathrm{E}}}{\mathrm{dx} \mathrm{x}_{1}}+\mathrm{x}_{1} \,\left[\mathrm{V}_{2}(\mathrm{mix})-\mathrm{V}_{2}^{*}(\ell)\right]+\mathrm{x}_{2} \,\left[\mathrm{V}_{2}(\mathrm{mix})-\mathrm{V}_{2}^{*}(\ell)\right] \nonumber \]
Hence,
\[\left[\mathrm{V}_{2}(\mathrm{mix})-\mathrm{V}_{2}^{*}(\ell)\right]=\mathrm{V}_{\mathrm{m}}^{\mathrm{E}}-\mathrm{x}_{1} \, \frac{\mathrm{dV}_{\mathrm{m}}^{\mathrm{E}}}{\mathrm{dx} \mathrm{x}_{1}} \nonumber \]
At the composition where \(\mathrm{dV}_{\mathrm{m}}^{\mathrm{E}} / \mathrm{dx}_{1}\) is zero, \(\left[\mathrm{V}_{2}(\operatorname{mix})-\mathrm{V}_{2}^{*}(\ell)\right]=\mathrm{V}_{\mathrm{m}}^{\mathrm{E}}\).
The analysis set out above is repeated for excess molar enthalpies and excess molar isobaric heat capacities for binary liquid mixtures.
Interesting proposals have been made in which the dependence of excess thermodynamic properties on mixture composition are examined in different composition domains; e.g. the four segment model [2-9].
Footnotes
[1] R. Schumann, Metallurg. Trans.,B,1985, 16B ,807.
[2] M. I. Davis, M. C. Molina and G. Douheret, Thermochim. Acta, 1988, 131 ,153.
[3] M. I. Davis, Thermochim. Acta 1984, 77 ,421; 1985, 90 , 313; 1987, 120 ,299; and references therein.
[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, C. Moreau and A. Viallard, Fluid Phase Equilib.,1986, 26 ,221.
[6] G. Douheret, A. Pal and M. I. Davis, J.Chem.Thermodyn., 1990, 22 ,99.
[7] H. Hoiland, O. Anowi and M. I. Davis, J. Chem. Thermodyn., 1991, 23 ,569.
[8] G. Douheret, A. Pal, H. Hoiland, O. Anowi and M. I. Davis, J. Chem. Thermodyn., 1991, 23 ,569.
[9] G. Douheret, J.C.R. Reis, M. I. Davis, I. J. Fjellanger and H. Hoiland, Phys.Chem. Chem.Phys.,2004, 6 ,784.