1.10.27: Gibbs Energies- Binary Liquid Mixtures- General Properties
A given binary liquid mixture ( at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\), which is close to the standard pressure) mole fraction \(x_{1}\left(=1-x_{2}\right)\) can be 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 exists for this mixture granted that the thermodynamic properties are ideal; e.g. \(\Delta_{\text {mix }} G_{m}(\mathrm{id})\). Hence we can 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.
Ideal Mixing Properties
At defined \(\mathrm{T}\) and \(\mathrm{p}\), the molar Gibbs energy for an ideal binary liquid mixture is given by equation (a).
\[\mathrm{G}_{\mathrm{m}}(\mathrm{id})=\mathrm{x}_{1} \,\left[\mu_{1}^{*}(\ell)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1}\right)\right]+\mathrm{x}_{2} \,\left[\mu_{2}^{*}(\ell)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{2}\right)\right] \nonumber \]
Here \(\mu_{1}^{*}(\ell)\) and \(\mu_{2}^{*}(\ell)\) are the chemical potentials of the two pure liquids 1 and 2 at the same \(\mathrm{T}\) and \(\mathrm{p}\). If the same amounts of the two liquid had not been allowed to mix,
\[\mathrm{G}_{\mathrm{m}}(\text { no }-\operatorname{mix})=\mathrm{x}_{1} \, \mu_{1}^{*}(\ell)+\mathrm{x}_{2} \, \mu_{2}^{*}(\ell) \nonumber \]
By definition,
\[\Delta_{\text {mix }} G_{m}(\mathrm{id})=x_{1} \, R \, T \, \ln \left(x_{1}\right)+x_{2} \, R \, T \, \ln \left(x_{2}\right) \nonumber \]
Recalling that \(x_{1}+x_{2}=1\) and \(dx_{1}=-d x_{2}\),
\[\mathrm{d} \Delta_{\text {mix }} \mathrm{G}_{\mathrm{m}}(\mathrm{id}) / \mathrm{dx} \mathrm{x}_{2}=-\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1}\right)-\mathrm{R} \, \mathrm{T}+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{2}\right)+\mathrm{R} \, \mathrm{T} \nonumber \]
We use equation (c) for \(\ln \left(x_{2}\right)\) and substitute in equation (d).
\[\begin{aligned}
&\mathrm{d} \Delta_{\text {mix }} \mathrm{G}_{\mathrm{m}}(\mathrm{id}) / \mathrm{dx} \mathrm{x}_{2}= \\
&-\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1}\right)+\left(1 / \mathrm{x}_{2}\right) \,\left[\Delta_{\text {mix }} \mathrm{G}_{\mathrm{m}}(\mathrm{id})-\mathrm{x}_{1} \, \mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1}\right)\right]
\end{aligned} \nonumber \]
\[\begin{aligned}
&\mathrm{x}_{2} \, \mathrm{d} \Delta_{\operatorname{mix}} \mathrm{G}_{\mathrm{m}}(\mathrm{id}) / \mathrm{dx} \mathrm{x}_{2}= \\
&\left.-\mathrm{x}_{2} \, \mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1}\right)+\Delta_{\text {mix }} \mathrm{G}_{\mathrm{m}}(\mathrm{id})-\mathrm{x}_{1} \, \mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1}\right)\right]
\end{aligned} \nonumber \]
Hence,
\[\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1}\right)=\Delta_{\mathrm{mix}} \mathrm{G}_{\mathrm{m}}(\mathrm{id})-\mathrm{x}_{2} \, \mathrm{d} \Delta_{\mathrm{mix}} \mathrm{G}_{\mathrm{m}}(\mathrm{id}) / \mathrm{dx} \mathrm{x}_{2} \nonumber \]
At all mole fractions, \(\Delta_{\operatorname{mix}} \mathrm{G}_{\mathrm{m}}(\mathrm{id})\) is negative, the plot of \(\Delta_{\operatorname{mix}} \mathrm{G}_{\mathrm{m}}(\mathrm{id})\) against \(x_{1}\) being symmetric about ‘\(x_{1}=0.5\)’. At the extreme, where \(\mathrm{d} \Delta_{\operatorname{mix}} \mathrm{G}_{\mathrm{m}}(\mathrm{id}) / \mathrm{dx} \mathrm{x}_{2}\) is zero; equation (c) shows that \(\Delta_{\operatorname{mix}} \mathrm{G}_{\mathrm{m}}(\mathrm{id}) \text { equals } [\mathrm{R} \, \mathrm{T} \, \ln (0.5)]\). At \(298 \mathrm{~K}\), the latter quantity equals \(– 1.72 \mathrm{~kJ mol}^{-1}\).
Using equation (c), the ratio \(\Delta_{\operatorname{mix}} \mathrm{G}_{\mathrm{m}}(\mathrm{id}) / \mathrm{T}\) is given by equation (h)
\[\Delta_{\operatorname{mix}} \mathrm{G}_{\mathrm{m}}(\mathrm{id}) / \mathrm{T}=\mathrm{R} \,\left[\mathrm{x}_{1} \, \ln \left(\mathrm{x}_{1}\right)+\mathrm{x}_{2} \, \ln \left(\mathrm{x}_{2}\right)\right] \nonumber \]
Hence using the Gibbs-Helmholtz it follows that the molar enthalpy of mixing is zero at all mole fractions. Similarly the molar isobaric heat capacity of mixing is zero at all mole fractions. In terms of entropies,
\[\Delta_{\operatorname{mix}} \mathrm{S}_{\mathrm{m}}(\mathrm{id})=-\mathrm{R} \,\left[\mathrm{x}_{1} \, \ln \left(\mathrm{x}_{1}\right)+\mathrm{x}_{2} \, \ln \left(\mathrm{x}_{2}\right)\right] \nonumber \]
But at all (real) mole fractions (other than \(x_{1} =1 \text { and } x_{2} = 1\)) \(\left[x_{1} \, \ln \left(x_{1}\right)+x_{2} \, \ln \left(x_{2}\right)\right]<0\). Hence \(\Delta_{\operatorname{mix}} \mathrm{S}_{\mathrm{m}}(\mathrm{id})>0\). In the event that mixing of two liquids at temperature \(\mathrm{T}\) to produce a binary liquid mixture having ideal properties, then \(\Delta_{\text {mix }} \mathrm{G}_{\mathrm{m}}(\mathrm{id})\) is negative because \(\Delta_{\operatorname{mix}} \mathrm{S}_{\mathrm{m}}(\mathrm{id})\) is positive, \(\Delta_{\text {mix }} \mathrm{H}_{\mathrm{m}}(\mathrm{id})\) being zero. In other words we have a reference against which to examine the properties of real liquid mixtures.
Excess properties
The molar Gibbs energy of a real binary liquid mixture is related to the mole fraction composition using equation (j).
\[\begin{aligned}
&\mathrm{G}_{\mathrm{m}}= \\
&\mathrm{x}_{1} \,\left[\mu_{1}^{*}(\ell)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1} \, \mathrm{f}_{1}\right)\right]+\mathrm{x}_{2} \,\left[\mu_{2}^{*}(\ell)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{2} \, \mathrm{f}_{2}\right)\right]
\end{aligned} \nonumber \]
For the corresponding mixture having ideal thermodynamic properties,
\[\begin{aligned}
&\mathrm{G}_{\mathrm{m}}(\mathrm{id})= \\
&\mathrm{x}_{1} \,\left[\mu_{1}^{*}(\ell)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1}\right)\right]+\mathrm{x}_{2} \,\left[\mu_{2}^{*}(\ell)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{2}\right)\right]
\end{aligned} \nonumber \]
The excess molar Gibbs energy \({\mathrm{G}_{\mathrm{m}}}^{\mathrm{E}}\) is therefore given by equation (l).
\[\mathrm{G}_{\mathrm{m}}^{\mathrm{E}}=\mathrm{R} \, \mathrm{T} \,\left[\mathrm{x}_{1} \, \ln \left(\mathrm{f}_{1}\right)+\mathrm{x}_{2} \, \ln \left(\mathrm{f}_{2}\right)\right] \nonumber \]
We differentiate equation (l) with respect to mole fraction, \(x_{1}\).
\[\frac{1}{R \, T} \, \frac{\mathrm{dG}_{\mathrm{m}}^{\mathrm{E}}}{\mathrm{dx}_{1}}=\ln \left(\mathrm{f}_{1}\right)+\mathrm{x}_{1} \, \frac{\mathrm{d} \ln \left(\mathrm{f}_{1}\right)}{d \mathrm{x}_{1}}-\ln \left(\mathrm{f}_{2}\right)+\mathrm{x}_{2} \, \frac{\mathrm{d} \ln \left(\mathrm{f}_{2}\right)}{\mathrm{dx}_{1}} \nonumber \]
But from the Gibbs-Duhem equation at fixed \(\mathrm{T}\) and \(\mathrm{p}\),
\[x_{1} \, \frac{d \ln \left(\mu_{1}\right)}{d x_{1}}+x_{2} \, \frac{d \ln \left(\mu_{2}\right)}{d x_{1}}=0 \nonumber \]
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 (k),
\[\ln \left(f_{2}\right)=\ln \left(f_{1}\right)-\frac{1}{R \, T} \, \frac{d G_{m}^{E}}{d x_{1}} \nonumber \]
Hence using equation (j),
\[\frac{\mathrm{G}_{\mathrm{m}}^{\mathrm{E}}}{\mathrm{R} \, \mathrm{T}}=\ln \left(\mathrm{f}_{1}\right)-\frac{\mathrm{X}_{2}}{\mathrm{R} \, \mathrm{T}} \, \frac{\mathrm{dG}_{\mathrm{m}}^{\mathrm{E}}}{\mathrm{dx}} \nonumber \]
Or
\[\ln \left(\mathrm{f}_{1}\right)=\frac{\mathrm{G}_{\mathrm{m}}^{\mathrm{E}}}{\mathrm{R} \, \mathrm{T}}=+\frac{\mathrm{x}_{2}}{\mathrm{R} \, \mathrm{T}} \, \frac{\mathrm{dG}_{\mathrm{m}}^{\mathrm{E}}}{\mathrm{dx}_{1}} \nonumber \]
Equation (o) 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 \(\ln \left(f_{1}\right)\) at that mole fraction.
Perhaps the most direct measure of the extent to which the thermodynamic properties of a given binary liquid mixture differs from that defined as ideal is afforded by the molar enthalpy of mixing which is therefore the excess molar enthalpy of mixing \({\mathrm{H}_{\mathrm{m}}}^{\mathrm{E}\). Thus,
\[\mathrm{G}_{\mathrm{m}}^{\mathrm{E}}=\mathrm{H}_{\mathrm{m}}^{\mathrm{E}}-\mathrm{T} \, \mathrm{S}_{\mathrm{m}}^{\mathrm{E}} \nonumber \]
where
\[\mathrm{C}_{\mathrm{pmn}}^{\mathrm{E}}=\left(\partial \mathrm{H}_{\mathrm{m}}^{\mathrm{E}} / \partial \mathrm{T}\right)_{\mathrm{p}} \nonumber \]
In many reports the properties of a given binary liquid mixture at \(298.15 \mathrm{~K}\) and ambient pressure are summarised in a plot showing the three properties \({\mathrm{G}_{\mathrm{m}}}^{\mathrm{E}\), \({\mathrm{H}_{\mathrm{m}}}^{\mathrm{E}\) and \(\mathrm{T} \, \mathrm{S}_{\mathrm{m}}^{\mathrm{E}}\) as a function of the mole fraction composition. An enormous amount of information can be summarised in such plots. In order to understand the various patterns which emerge two liquid mixtures are often taken as models against which to compare the properties of other liquid mixtures.
-
Trichloromethane + Methanol
For this mixture both \({\mathrm{G}_{\mathrm{m}}}^{\mathrm{E}\) and \({\mathrm{H}_{\mathrm{m}}}^{\mathrm{E}\) are negative, their minima being at approx. mole fractions at 0.5. The pattern in understood in terms of strong inter-component interaction, hydrogen bonding. In these terms the mixing is ‘favourable’ and exothermic. -
Tetrachloromethane + Methanol
For this mixture the mixing is, for the most part endothermic and \({\mathrm{G}_{\mathrm{m}}}^{\mathrm{E}\) is positive with a maximum at mole fractions equal to 0.5. Thus the mixing is unfavourable and endothermic. This pattern points to the impact of added tetrachloromethane disrupting the intermolecular hydrogen bonding between methanol molecules.
These two liquid mixtures provide a basis for the examination of the properties of binary aqueous mixtures for which there is an immense published information. In most cases \({\mathrm{G}_{\mathrm{m}}}^{\mathrm{E}\) is either positive or negative across the mole fraction range for a given liquid mixture although plots of \({\mathrm{G}_{\mathrm{m}}}^{\mathrm{E}\) and \(\mathrm{T} \, \mathrm{S}_{\mathrm{m}}^{\mathrm{E}}\) against mole fraction composition are often S-shaped, nevertheless operating to produce a smooth change in \({\mathrm{G}_{\mathrm{m}}}^{\mathrm{E}\). Plots of \({\mathrm{C}_{\mathrm{pm}}}^{\mathrm{E}}\) and excess molar volume of mixing \({\mathrm{V}_{\mathrm{m}}}^{\mathrm{E}}\) against mole fraction are often quite complicated.
In a few cases the plot of \({\mathrm{G}_{\mathrm{m}}}^{\mathrm{E}\) against composition is S-shaped. One such system is the mixture, water + 1,1,1,3,3,3-hexafluoropropanol [1]. Explanations of such complex patterns are not straightforward. However we might for an alcohol + water mixture envisage a switch from
- at low \(x_{2}\) strong water-water interactions with weak alcohol-water interactions to
- at high \(x_{2}\) strong alcohol-water interactions.
Definition of excess thermodynamic properties is not straightforward in all instances; e.g. isentropic compressibilities.[2]
Footnotes
[1] M. J. Blandamer, J. Burgess, A. Cooney, H. J. Cowles, I. M. Horn, K. J. Martin, K. W. Morcom and P. Warwick, J. Chem. Soc. Faraday Trans.,1990, 86 ,2209`
[2] G. Douheret, C . Moreau and A. Viallard, Fluid Phase Equilibrium, 1985, 22 ,277; 289.