1.22.9: Volume: Liquid Mixtures
A given binary liquid mixture is prepared using \(\mathrm{n}_{1}\) moles of liquid 1 and \(\mathrm{n}_{2}\) moles of liquid 2 at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\). The term ‘mixture’ usually means that a homogeneous single liquid phase is spontaneously formed on mixing characterized by a minimum in Gibbs energy \(\mathrm{G}\) where the molecular organization is characterized by \(\xi^{\mathrm{eq}}\). \(\mathrm{G}_{\mathrm{m}}^{\mathrm{E}}\) measures the extent to which \(\mathrm{G} /\left(\mathrm{n}_{1}+\mathrm{n}_{2}\right)\) differs from this ratio in the event that the mixing is, in a thermodynamic sense, ideal.
A given binary liquid mixture is displaced to a neighboring state by a change in pressure at constant temperature. The overall composition remains at \(\left(\mathrm{n}_{1} + \mathrm{n}_{2}\right)\) but the organization changes to a new value for \(\xi^{\mathrm{eq}}\) where ‘\(\mathrm{A} = 0\)’. The differential dependence of \(\mathrm{G}_{\mathrm{m}}^{\mathrm{E}}\) on pressure at constant temperature \(\mathrm{T}\) is the excess molar volume \(\mathrm{V}_{\mathrm{m}}^{\mathrm{E}}\).
\[\mathrm{V}(\operatorname{mix})=\left(\frac{\partial \mathrm{G}(\operatorname{mix})}{\partial \mathrm{p}}\right)_{\mathrm{T}, \mathrm{A}=0} \nonumber \]
\[\mathrm{V}_{\mathrm{m}}^{\mathrm{E}}=\left(\frac{\partial \mathrm{G}_{\mathrm{m}}^{\mathrm{E}}}{\partial \mathrm{p}}\right)_{\mathrm{T}, \mathrm{A}=0} \nonumber \]
For the molar volume,
\[\mathrm{V}_{\mathrm{m}}=\left(\frac{\partial \mathrm{G}_{\mathrm{m}}}{\partial \mathrm{p}}\right)_{\mathrm{T}, \mathrm{A}=0} \nonumber \]
The quantities, \(\mathrm{V}(\mathrm{mix})\), \(\mathrm{V}_{\mathrm{m}}\) and \(\mathrm{V}_{\mathrm{m}}^{\mathrm{E}}\) are interesting because they can be determined [1] whereas the same cannot be said for \(\mathrm{G}(\mathrm{mix})\) and \(\mathrm{G}_{\mathrm{m}}\) although \(\mathrm{G}_{\mathrm{m}}^{\mathrm{E}}\) can be obtained m from vapor pressures of mixtures and pure components.
\[\mathrm{V}_{\mathrm{m}}=\mathrm{V}(\mathrm{mix}) /\left(\mathrm{n}_{1}+\mathrm{n}_{2}\right) \nonumber \]
Density,
\[\rho(\operatorname{mix})=\left(\mathrm{n}_{1} \, \mathrm{M}_{1}+\mathrm{n}_{2} \, \mathrm{M}_{2}\right) / \mathrm{V}(\operatorname{mix}) \nonumber \]
\(\mathrm{M}_{1}\) and \(\mathrm{M}_{2} are the molar masses of the two liquid components. By measuring \(\rho(\mathrm{mix})\) as a function of mixture composition, we form a plot of molar volume Vm as a function of mole fraction composition. The plot has two limits;
\[\operatorname{limit}\left(\mathrm{x}_{1} \rightarrow 1\right) \mathrm{V}_{\mathrm{m}}=\mathrm{V}_{1}^{*}(\ell) \nonumber \]
\[\operatorname{limit}\left(\mathrm{x}_{2} \rightarrow 1\right) \mathrm{V}_{\mathrm{m}}=\mathrm{V}_{2}^{*}(\ell) \nonumber \]
If the thermodynamic properties of the binary liquid mixture are ideal (i.e. \(\mathrm{V}_{\mathrm{m}}^{\mathrm{E}}=0\)),
\[\mathrm{V}_{\mathrm{m}}(\mathrm{id})=\mathrm{x}_{1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{x}_{2} \, \mathrm{V}_{2}^{*}(\ell) \nonumber \]
Or,
\[\mathrm{V}_{\mathrm{m}}(\mathrm{id})=\left(1-\mathrm{x}_{2}\right) \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{x}_{2} \, \mathrm{V}_{2}^{*}(\ell) \nonumber \]
Hence,
\[\mathrm{V}_{\mathrm{m}}(\mathrm{id})=\mathrm{V}_{1}^{*}(\ell)+\mathrm{x}_{2} \,\left[\mathrm{V}_{2}^{*}(\ell)-\mathrm{V}_{1}^{*}(\ell)\right] \nonumber \]
The latter is an equation for a straight line. The molar volume of a real binary liquid mixture is usually less than \(\mathrm{V}_{m}(\mathrm{id})\). For a real binary liquid mixture,
\[\mathrm{V}_{\mathrm{m}}(\operatorname{mix})=\mathrm{x}_{1} \, \mathrm{V}_{1}(\operatorname{mix})+\mathrm{x}_{2} \, \mathrm{V}_{2}(\mathrm{mix}) \nonumber \]
The difference between the molar volume of real and ideal binary liquid mixture is the excess molar volume \(\mathrm{V}_{\mathrm{m}}^{\mathrm{E}}\).
\[\mathrm{V}_{\mathrm{m}}^{\mathrm{E}}=\mathrm{x}_{1} \,\left[\mathrm{V}_{1}(\operatorname{mix})-\mathrm{V}_{1}^{*}(\ell)\right]+\mathrm{x}_{2} \,\left[\mathrm{V}_{2}(\operatorname{mix})-\mathrm{V}_{2}^{*}(\ell)\right] \nonumber \]
Or,
\[V_{m}^{E}=x_{1} \, V_{1}^{E}(\operatorname{mix})+x_{2} \, V_{2}^{E}(\operatorname{mix} \nonumber \]
A given mixture, mole fraction \(\mathrm{x}_{2}\) at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\), is perturbed by addition of \(\delta \mathrm{n}_{2}\) moles of chemical substance 2. The system can be perturbed either at constant organization \(\xi\) or constant affinity \(\mathrm{A}\). Here we are concerned with \(\left(\frac{\partial \mathrm{V}}{\partial \mathrm{n}_{2}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}(1), \mathrm{A}=0}\), the (equilibrium) partial molar volume of substance 2 in the mixture, \(\mathrm{V}_{2}(\mathrm{mix})\).The condition '\(\mathrm{A} = 0\)' implies that there is a change in organization \(\xi\) in order to hold the system in the equilibrium state. A similar argument is formulated for the (equilibrium) partial molar volume \(\mathrm{V}_{1}(\mathrm{mix})\). Moreover according to the Gibbs-Duhem equation (at constant temperature and pressure),
\[\mathrm{n}_{1} \, d \mathrm{~V}_{1}(\operatorname{mix})+\mathrm{n}_{2} \, d \mathrm{~V}_{2}(\operatorname{mix})=0 \nonumber \]
Further,
\[\mathrm{V}(\operatorname{mix})=\mathrm{n}_{1} \, \mathrm{V}_{1}(\operatorname{mix})+\mathrm{n}_{2} \, \mathrm{V}_{2}(\operatorname{mix}) \nonumber \]
The property \(\mathrm{V}(\mathrm{mix})\) is directly determined from the density \(\rho(\mathrm{mix})\).
\[\mathrm{V}(\operatorname{mix})=\left(\mathrm{n}_{1} \, \mathrm{M}_{1}+\mathrm{n}_{2} \, \mathrm{M}_{2}\right) / \rho(\operatorname{mix}) \nonumber \]
The important point is that the thermodynamic extensive property \(\mathrm{V}(\mathrm{mix})\) is directly determined by experiment whereas we cannot for example measure the enthalpy \(\mathrm{H}(\mathrm{mix})\). The excess molar volume is given by equation (l).
\[\begin{aligned}
\frac{\mathrm{dV}_{\mathrm{m}}^{\mathrm{E}}}{d \mathrm{x}_{1}}=& {\left[\mathrm{V}_{1}(\mathrm{mix})-\mathrm{V}_{1}^{*}(\ell)\right]+\mathrm{x}_{1} \, \frac{\mathrm{dV} \mathrm{V}_{1}(\mathrm{mix})}{\mathrm{dx}_{1}} } \\
&-\left[\mathrm{V}_{2}(\mathrm{mix})-\mathrm{V}_{2}^{*}(\ell)\right]+\mathrm{x}_{2} \, \frac{\mathrm{dV}_{2}(\mathrm{mix})}{\mathrm{dx}_{1}}
\end{aligned} \nonumber \]
Using the Gibbs -Duhem equation,
\[\frac{\mathrm{dV}_{\mathrm{m}}^{\mathrm{E}}}{\mathrm{dx}_{1}}=\left[\mathrm{V}_{1}(\mathrm{mix})-\mathrm{V}_{1}^{*}(\ell)\right]-\left[\mathrm{V}_{2}(\mathrm{mix})-\mathrm{V}_{2}^{*}(\ell)\right] \nonumber \]
Or,
\[\left[\mathrm{V}_{1}(\mathrm{mix})-\mathrm{V}_{1}^{*}(\ell)\right]=\frac{\mathrm{dV}_{\mathrm{m}}^{\mathrm{E}}}{\mathrm{dx}_{1}}+\left[\mathrm{V}_{2}(\mathrm{mix})-\mathrm{V}_{2}^{*}(\ell)\right] \nonumber \]
From equation (l),
\[\mathrm{V}_{\mathrm{m}}^{\mathrm{E}}=\mathrm{x}_{1} \, \frac{\mathrm{dV}_{\mathrm{m}}^{\mathrm{E}}}{\mathrm{dx}_{1}}+\mathrm{x}_{1} \,\left[\mathrm{V}_{2}(\operatorname{mix})-\mathrm{V}_{2}^{*}(\ell)\right]+\mathrm{x}_{2} \,\left[\mathrm{V}_{2}(\operatorname{mix})-\mathrm{V}_{2}^{*}(\ell)\right] \nonumber \]
Hence,
\[\left[\mathrm{V}_{2}(\mathrm{mix})-\mathrm{V}_{2}^{*}(\ell)\right]=\mathrm{V}_{\mathrm{m}}^{\mathrm{E}}-\mathrm{x}_{1} \, \frac{\mathrm{dV}_{\mathrm{m}}^{\mathrm{E}}}{\mathrm{dx}_{1}} \nonumber \]
The derivation leading up to equation (s) is the 'Method of Tangents'. Moreover at the mole fraction composition where \(\frac{\mathrm{dV}_{\mathrm{m}}^{\mathrm{E}}}{\mathrm{dx}_{1}}\) is zero, \(\left[\mathrm{V}_{2}(\operatorname{mix})-\mathrm{V}_{2}^{*}(\ell)\right]\) equals \(\mathrm{V}_{\mathrm{m}}^{\mathrm{E}}\).
For liquid component 1 the chemical potential in the liquid mixture is related to the mole fraction composition (at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\)).
\[\mu_{1}(\operatorname{mix}, \mathrm{T}, \mathrm{p})=\mu_{1}^{*}(\ell, \mathrm{T})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1} \, \mathrm{f}_{1}\right)+\int_{\mathrm{p}^{0}}^{\mathrm{p}} \mathrm{V}_{1}^{*}(\ell ; \mathrm{T}) \, \mathrm{dp} \nonumber \]
At temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\),
\[\mathrm{V}_{1}(\operatorname{mix})=\mathrm{V}_{1}^{*}(\ell)+\mathrm{R} \, \mathrm{T} \,\left[\partial \ln \left(\mathrm{f}_{1}\right) / \partial \mathrm{p}\right]_{\mathrm{T}} \nonumber \]
But
\[\mathrm{V}_{1}^{\mathrm{E}}(\operatorname{mix})=\mathrm{V}_{1}(\operatorname{mix})-\mathrm{V}_{1}^{*}(\ell) \nonumber \]
Hence,
\[\mathrm{V}_{1}^{\mathrm{E}}(\operatorname{mix})=\mathrm{R} \, \mathrm{T} \,\left[\partial \ln \left(\mathrm{f}_{1}\right) / \partial \mathrm{p}\right]_{\mathrm{T}} \nonumber \]
Then,
\[\mathrm{V}_{\mathrm{m}}^{\mathrm{E}}(\mathrm{mix})=\mathrm{R} \, \mathrm{T} \,\left\{\mathrm{x}_{1} \,\left[\partial \ln \left(\mathrm{f}_{1}\right) / \partial \mathrm{p}\right]_{\mathrm{T}}+\mathrm{x}_{2} \,\left[\partial \ln \left(\mathrm{f}_{2}\right) / \partial \mathrm{p}\right]_{\mathrm{T}}\right\} \nonumber \]
Footnote
[1] R. Battino, Chem.Rev.,1971, 71 ,5.
[2]
- \(\mathrm{C}_{\mathrm{m}}\mathrm{H}_{2\mathrm{m}+2}\) as a solute in \(\mathrm{C}_{\mathrm{n}}\mathrm{H}_{2\mathrm{n}+2}\) solvent; H. Itsuki, S. Terasowa, K. Shinora and H. Ikezawa, J Chem. Thermodyn.,19897, 19 ,555.
- 60[Fullerene} in apolar solvents; P. Ruella, A. Farina-Cuendet and U.W. Kesselring, J. Chem. Soc. Chem. Commun.,1995,1161.