1.12.13: Expansions- Isobaric- Binary Liquid Mixtures
The isobaric (equilibrium) expansion of a liquid, volume \(\mathrm{V}\), is defined by equation (a).
\[\mathrm{E}_{\mathrm{p}}=\left(\frac{\partial V}{\partial T}\right)_{p} \nonumber \]
Both \(\mathrm{E}_{\mathrm{p}}\) and \(\mathrm{V}\) are extensive properties of a mixture. Therefore it is convenient to refer to the molar property, \(\mathrm{E}_{\mathrm{pm}}(\operatorname{mix})\). Thus
\[\mathrm{E}_{\mathrm{pm}}(\mathrm{mix})=\left(\frac{\partial \mathrm{V}_{\mathrm{m}}(\mathrm{mix})}{\partial \mathrm{T}}\right)_{\mathrm{p}} \nonumber \]
At fixed \(\mathrm{T}\) and \(\mathrm{p}\), \(\mathrm{V}_{\mathrm{m}}(\mathrm{mix})\) for a binary liquid mixture is related to the partial molar volumes of the two components.
\[\mathrm{V}_{\mathrm{m}}(\operatorname{mix})=\mathrm{x}_{1} \, \mathrm{V}_{1}(\operatorname{mix})+\mathrm{x}_{2} \, \mathrm{V}_{2}(\mathrm{mix}) \nonumber \]
From equation (b)
\[\mathrm{E}_{\mathrm{pm}}(\mathrm{mix})=\mathrm{x}_{1} \,\left(\frac{\partial \mathrm{V}_{1}(\mathrm{mix})}{\partial \mathrm{T}}\right)_{\mathrm{p}}+\mathrm{x}_{2} \,\left(\frac{\partial \mathrm{V}_{2}(\mathrm{mix})}{\partial \mathrm{T}}\right)_{\mathrm{p}} \nonumber \]
For a binary mixture having molar volume \(\mathrm{V}_{\mathrm{m}}(\mathrm{mix})\) and density \(\rho(\mathrm{mix})\),
\[\rho(\operatorname{mix})=\left(\mathrm{x}_{1} \, \mathrm{M}_{1}+\mathrm{x}_{2} \, \mathrm{M}_{2}\right) / \mathrm{V}_{\mathrm{m}}(\mathrm{mix}) \nonumber \]
Here \(\mathrm{M}_{1}\) and \(\mathrm{M}_{2}\) are the molar masses of liquids 1 and 2 respectively.
\[\mathrm{V}_{\mathrm{m}}(\mathrm{mix})=\left(\mathrm{x}_{1} \, \mathrm{M}_{1}+\mathrm{x}_{2} \, \mathrm{M}_{2}\right) / \rho(\mathrm{mix}) \nonumber \]
Hence,
\[\begin{aligned}
{\left[\partial \mathrm{V}_{\mathrm{m}}(\operatorname{mix}) / \partial \mathrm{T}\right]_{\mathrm{p}} } &=\\
&-\left[\left(\mathrm{x}_{1} \, \mathrm{M}_{1}+\mathrm{x}_{2} \, \mathrm{M}_{2}\right) / \rho(\operatorname{mix})\right] \,[\partial \ln \{\rho(\operatorname{mix})\} / \partial \mathrm{T}]_{\mathrm{p}}
\end{aligned} \nonumber \]
\(\\mathrm{E}_{\mathrm{pm}}(\mathrm{mix})\) is obtained for a given mixture from the isobaric dependence of density on temperature. There is merit in considering equations for \(\mathrm{E}_{\mathrm{pm}}(\operatorname{mix} ; \mathrm{id})\) of a binary mixture having ideal thermodynamic properties and hence for the related excess molar expansion \(\mathrm{E}_{\mathrm{pm}}^{\mathrm{E}}\). With,
\[\mathrm{E}_{\mathrm{pm}}(\operatorname{mix} ; \mathrm{id})=\mathrm{x}_{1} \, \mathrm{E}_{1}^{*}(\ell)+\mathrm{x}_{2} \, \mathrm{E}_{2}^{*}(\ell) \nonumber \]
\[\mathrm{E}_{\mathrm{pm}}^{\mathrm{E}}=\mathrm{E}_{\mathrm{pm}}(\mathrm{mix})-\mathrm{E}_{\mathrm{pm}}(\mathrm{mix} ; \mathrm{id}) \nonumber \]
\(\mathrm{E}_{\mathrm{pm}}(\operatorname{mix} ; \mathrm{id})\) is the mole fraction weighted sum of the isobaric expansions of the pure liquid components at the same \(\mathrm{T}\) and \(\mathrm{p}\). The isobaric expansibility of an ideal binary liquid mixture \(\alpha_{p}(\operatorname{mix} ; \mathrm{id})\) is given by equation (j).
\[\alpha_{\mathrm{p}}(\operatorname{mix} ; \mathrm{id})=\frac{\mathrm{x}_{1} \, \mathrm{E}_{\mathrm{p} 1}^{*}(\ell)+\mathrm{x}_{2} \, \mathrm{E}_{\mathrm{p} 2}^{*}(\ell)}{\mathrm{x}_{1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{x}_{2} \, \mathrm{V}_{2}^{*}(\ell)} \nonumber \]
Or,
\[\alpha_{\mathrm{p}}(\operatorname{mix} ; \text { id })=\frac{\mathrm{x}_{1} \, \mathrm{E}_{\mathrm{p} 1}^{*}(\ell)}{\mathrm{x}_{1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{x}_{2} \, \mathrm{V}_{2}^{*}(\ell)}+\frac{\mathrm{x}_{2} \, \mathrm{E}_{\mathrm{p} 2}^{*}(\ell)}{\mathrm{x}_{1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{x}_{2} \, \mathrm{V}_{2}^{*}(\ell)} \nonumber \]
Hence,
\[\alpha_{\mathrm{p}}(\mathrm{mix} ; \mathrm{id})=\frac{\mathrm{x}_{1} \, \mathrm{V}_{1}^{*}(\ell) \, \alpha_{\mathrm{p} 1}^{*}(\ell)}{\mathrm{x}_{1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{x}_{2} \, \mathrm{V}_{2}^{*}(\ell)}+\frac{\mathrm{x}_{2} \, \mathrm{V}_{2}^{*}(\ell) \, \mathrm{E}_{\mathrm{p} 2}^{*}(\ell)}{\mathrm{x}_{1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{x}_{2} \, \mathrm{V}_{2}^{*}(\ell)} \nonumber \]
Hence, expansibility \(\alpha_{p}(\operatorname{mix} ; 1 \mathrm{~d})\) can be expressed in terms of the volume fractions of the corresponding ideal binary liquid mixture.
\[\alpha_{p}(\operatorname{mix} ; \text { id })=\phi_{1}(\operatorname{mix} ; \text { id }) \, \alpha_{p 1}^{*}(\ell)+\phi_{2}(\operatorname{mix} ; \text { id }) \, \alpha_{p 2}^{*}(\ell) \nonumber \]
The excess (equilibrium) isobaric expansivity \(\alpha_{p}^{E}(\operatorname{mix})\) is given by mix equation (n) [1].
\[\alpha_{\mathrm{p}}^{\mathrm{E}}(\mathrm{mix})=\frac{1}{\mathrm{~V}_{\mathrm{m}}(\mathrm{mix})} \,\left[\left(\frac{\partial \mathrm{V}_{\mathrm{m}}^{\mathrm{E}}(\mathrm{mix})}{\partial \mathrm{T}}\right)_{\mathrm{p}}-\mathrm{V}_{\mathrm{m}}^{\mathrm{E}} \, \alpha_{\mathrm{p}}(\mathrm{mix} ; \mathrm{id})\right] \nonumber \]
From another standpoint the thermal expansion of a binary liquid mixture is analysed in terms of the differential dependence of rational activity coefficients on temperature and pressure. For liquid component 1 at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\),
\[\mu_{1}(\operatorname{mix})=\mu_{1}^{0}(\ell)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1} \, \mathrm{f}_{1}\right)+\int_{\mathrm{p}}^{\mathrm{p}} \mathrm{V}_{1}^{*}(\ell) \, \mathrm{dp} \nonumber \]
Then
\[\mathrm{V}_{1}(\mathrm{mix})=\mathrm{V}_{1}^{*}(\ell)+\mathrm{R} \, \mathrm{T} \,\left[\partial \ln \left(\mathrm{f}_{1}\right) / \partial \mathrm{p}\right]_{\mathrm{T}} \nonumber \]
At temperature \(\mathrm{T}\),
\[\mathrm{E}_{\mathrm{p}_{1}}(\operatorname{mix})=\mathrm{E}_{\mathrm{p} 1}(\operatorname{mix} ; \mathrm{id})+\mathrm{R} \,\left[\partial \ln \left(\mathrm{f}_{1}\right) / \partial \mathrm{p}\right]_{\mathrm{T}}+\mathrm{R} \, \mathrm{T} \,\left[\frac{\partial}{\partial \mathrm{T}}\left(\frac{\partial \ln \left(\mathrm{f}_{1}\right)}{\partial \mathrm{p}}\right)_{\mathrm{T}}\right]_{\mathrm{p}} \nonumber \]
\[\begin{aligned}
&\mathrm{E}_{\mathrm{p} 2}(\operatorname{mix})= \\
&\quad \mathrm{E}_{\mathrm{p} 2}(\mathrm{mix} ; \mathrm{id})+\mathrm{R} \,\left[\partial \ln \left(\mathrm{f}_{2}\right) / \partial \mathrm{p}\right]_{\mathrm{T}}+\mathrm{R} \, \mathrm{T} \,\left[\frac{\partial}{\partial \mathrm{T}}\left(\frac{\partial \ln \left(\mathrm{f}_{2}\right)}{\partial \mathrm{p}}\right)_{\mathrm{T}}\right]_{\mathrm{p}}
\end{aligned} \nonumber \]
Two equations follow for the excess partial molar isobaric expansions of the components of the mixture.
\[\mathrm{E}_{\mathrm{p} 1}^{\mathrm{E}}(\mathrm{mix})=\mathrm{R} \,\left[\partial \ln \left(\mathrm{f}_{1}\right) / \partial \mathrm{p}\right]_{\mathrm{T}}+\mathrm{R} \, \mathrm{T} \,\left[\frac{\partial}{\partial \mathrm{T}}\left(\frac{\partial \ln \left(\mathrm{f}_{1}\right)}{\partial \mathrm{p}}\right)_{\mathrm{T}}\right]_{\mathrm{p}} \nonumber \]
\[E_{\mathrm{p} 2}^{\mathrm{E}}(\mathrm{mix})=\mathrm{R} \,\left[\partial \ln \left(\mathrm{f}_{2}\right) / \partial \mathrm{p}_{\mathrm{T}}+\mathrm{R} \, \mathrm{T} \,\left[\frac{\partial}{\partial \mathrm{T}}\left(\frac{\partial \ln \left(\mathrm{f}_{2}\right)}{\partial \mathrm{p}}\right)_{\mathrm{T}}\right]_{\mathrm{p}}\right. \nonumber \]
Therefore for the mixture,
\[\mathrm{E}_{\mathrm{pm}}^{\mathrm{E}}(\mathrm{mix})=\mathrm{x}_{1} \, \mathrm{E}_{\mathrm{p} 1}^{\mathrm{E}}(\mathrm{mix})+\mathrm{x}_{2} \, \mathrm{E}_{\mathrm{p} 2}^{\mathrm{E}}(\mathrm{mix}) \nonumber \]
Footnotes
[1] For a binary liquid mixture at defined \(\mathrm{T}\) and \(\mathrm{p}\),
\[\mathrm{V}_{\mathrm{m}}(\operatorname{mix})=\mathrm{V}_{\mathrm{m}}(\mathrm{mix} ; \mathrm{id})+\mathrm{V}_{\mathrm{m}}^{\mathrm{E}} \nonumber \]
\[\alpha_{p}(\operatorname{mix})=\frac{1}{V_{m}(\operatorname{mix})} \, \frac{\partial}{\partial T}\left[V_{m}(\text { mix } ; 1 \mathrm{~d})+V_{m}^{E}\right] \nonumber \]
Or, \(\alpha_{p}(\operatorname{mix})=\frac{1}{V_{m}(\operatorname{mix})} \, \frac{\partial \mathrm{V}_{\mathrm{m}}(\mathrm{mix} ; \mathrm{id})}{\partial \mathrm{T}}+\frac{1}{\mathrm{~V}_{\mathrm{m}}(\mathrm{mix})} \, \frac{\partial \mathrm{V}_{\mathrm{m}}^{\mathrm{E}}}{\partial \mathrm{T}}\)
But, \(\alpha_{p}(\operatorname{mix} ; \mathrm{id})=\frac{1}{\mathrm{~V}_{\mathrm{m}}(\mathrm{mix} ; \mathrm{id})} \, \frac{\partial \mathrm{V}_{\mathrm{m}}(\mathrm{mix} ; \mathrm{id})}{\partial \mathrm{T}}\)
By definition,
\[\begin{aligned}
&\alpha_{p}^{E}=\alpha_{p}(\operatorname{mix})-\alpha_{p}(\operatorname{mix} ; \text { id })\\
&\alpha_{p}^{E}(\operatorname{mix})=\left[\frac{1}{V_{m}(\operatorname{mix})}-\frac{1}{V_{m}(\operatorname{mix} ; i \mathrm{~d})}\right] \, \frac{\partial V_{m}(\operatorname{mix} ; \mathrm{id})}{\partial \mathrm{T}}+\frac{1}{\mathrm{~V}_{\mathrm{m}}(\mathrm{mix})} \, \frac{\partial \mathrm{V}_{\mathrm{m}}^{\mathrm{E}}}{\partial \mathrm{T}}\\
&\alpha_{p}^{\mathrm{E}}(\operatorname{mix})=\left[\frac{\mathrm{V}_{\mathrm{m}}(\mathrm{mix} ; \mathrm{id})-\mathrm{V}_{\mathrm{m}}(\mathrm{mix})}{\mathrm{V}_{\mathrm{m}}(\mathrm{mix}) \, \mathrm{V}_{\mathrm{m}}(\mathrm{mix} ; \mathrm{id})}\right] \, \frac{\partial \mathrm{V}_{\mathrm{m}}(\mathrm{mix} ; \mathrm{id})}{\partial \mathrm{T}}+\frac{1}{\mathrm{~V}_{\mathrm{m}}(\operatorname{mix})} \, \frac{\partial \mathrm{V}_{\mathrm{m}}^{\mathrm{E}}}{\partial \mathrm{T}}\\
&\alpha_{\mathrm{p}}^{\mathrm{E}}(\operatorname{mix})=-\left[\frac{\mathrm{V}_{\mathrm{m}}^{\mathrm{E}}}{\mathrm{V}_{\mathrm{m}}(\mathrm{mix})}\right] \, \alpha_{\mathrm{p}}(\mathrm{mix} ; \mathrm{id})+\frac{1}{\mathrm{~V}_{\mathrm{m}}(\mathrm{mix})} \, \frac{\partial \mathrm{V}_{\mathrm{m}}^{\mathrm{E}}}{\partial \mathrm{T}}
\end{aligned} \nonumber \]
Hence,
\[\alpha_{p}^{E}(\operatorname{mix})=\frac{1}{V_{m}(\operatorname{mix})} \,\left[\left(\frac{\partial V_{m}^{E}}{\partial T}\right)_{p}-V_{m}^{E} \, \alpha_{p}(\operatorname{mix} ; \text { id })\right] \nonumber \]