1.12.14: Expansibilities- Isobaric- Binary Liquid Mixtures
A given binary liquid mixture is prepared using liquids 1 and 2 at defined \(\mathrm{T}\) and \(\mathrm{p}\). The molar volume of this mixture is given by equation (a). In the event that thermodynamic properties of the mixture are ideal, the molar volume is given by equation (a).
\[\mathrm{V}_{\mathrm{m}}(\operatorname{mix} ; \mathrm{id})=\mathrm{x}_{1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{x}_{2} \, \mathrm{V}_{2}^{*}(\ell) \nonumber \]
At fixed pressure,
\[\left(\frac{\partial \mathrm{V}_{\mathrm{m}}(\operatorname{mix} ; \mathrm{id})}{\partial \mathrm{T}}\right)_{\mathrm{p}}=\mathrm{x}_{1} \,\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{T}}\right)_{\mathrm{p}}+\mathrm{x}_{2} \,\left(\frac{\partial \mathrm{V}_{2}^{*}(\ell)}{\partial \mathrm{T}}\right)_{\mathrm{p}} \nonumber \]
\[\begin{aligned}
&\frac{\mathrm{V}_{\mathrm{m}}(\operatorname{mix} ; \mathrm{id})}{\mathrm{V}_{\mathrm{m}}(\operatorname{mix} ; \mathrm{id})} \,\left(\frac{\partial \mathrm{V}_{\mathrm{m}}(\operatorname{mix} ; \mathrm{id})}{\partial \mathrm{T}}\right)_{\mathrm{p}}= \\
&\mathrm{x}_{1} \, \frac{\mathrm{V}_{1}^{*}(\ell)}{\mathrm{V}_{1}^{*}(\ell)} \,\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{T}}\right)_{\mathrm{p}}+\mathrm{x}_{2} \, \frac{\mathrm{V}_{2}^{*}(\ell)}{\mathrm{V}_{2}^{*}(\ell)} \,\left(\frac{\partial \mathrm{V}_{2}^{*}(\ell)}{\partial \mathrm{T}}\right)_{\mathrm{p}}
\end{aligned} \nonumber \]
Hence,
\[\mathrm{V}_{\mathrm{m}}(\operatorname{mix} ; \mathrm{id}) \, \alpha_{\mathrm{p}}(\operatorname{mix} ; \mathrm{id})=\mathrm{x}_{1} \, \mathrm{V}_{1}^{*}(\ell) \, \alpha_{\mathrm{p} 1}^{*}(\ell)+\mathrm{x}_{2} \, \mathrm{V}_{2}^{*}(\ell) \, \alpha_{\mathrm{p} 2}^{*}(\ell) \nonumber \]
But
\[\phi_{1}(\operatorname{mix} ; \mathrm{id})=\mathrm{x}_{1} \, \mathrm{V}_{1}^{*}(\ell) / \mathrm{V}_{\mathrm{m}}(\operatorname{mix} ; \mathrm{id}) \nonumber \]
And,
\[\phi_{2}(\operatorname{mix} ; \mathrm{id})=\mathrm{x}_{2} \, \mathrm{V}_{2}^{*}(\ell) / \mathrm{V}_{\mathrm{m}}(\operatorname{mix} ; \mathrm{id}) \nonumber \]
Hence
\[\alpha_{p}(\operatorname{mix} ; \text { id })=\phi_{1}(\operatorname{mix} ; \text { id }) \, \alpha_{p 1}^{*}(\ell)+\phi_{2}(\operatorname{mix} ; \text { id }) \, \alpha_{p 2}^{*}(\ell) \nonumber \]
For a real binary liquid mixture,
\[\mathrm{V}_{\mathrm{m}}(\operatorname{mix})=\mathrm{x}_{1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{x}_{2} \, \mathrm{V}_{2}^{*}(\ell)+\mathrm{V}_{\mathrm{m}}^{\mathrm{E}}(\operatorname{mix}) \nonumber \]
At fixed pressure,
\[\left(\frac{\partial \mathrm{V}_{\mathrm{m}}(\mathrm{mix})}{\partial \mathrm{T}}\right)_{\mathrm{p}}=\mathrm{x}_{1} \,\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{T}}\right)_{\mathrm{p}}+\mathrm{x}_{2} \,\left(\frac{\partial \mathrm{V}_{2}^{*}(\ell)}{\partial \mathrm{T}}\right)_{\mathrm{p}}+\left(\frac{\partial \mathrm{V}_{\mathrm{m}}^{\mathrm{E}}}{\partial \mathrm{T}}\right)_{\mathrm{p}} \nonumber \]
Or,
\[\begin{aligned}
&\frac{\mathrm{V}_{\mathrm{m}}(\mathrm{mix})}{\mathrm{V}_{\mathrm{m}}(\mathrm{mix})} \,\left(\frac{\partial \mathrm{V}_{\mathrm{m}}(\mathrm{mix})}{\partial \mathrm{T}}\right)_{\mathrm{p}}= \\
&\mathrm{x}_{1} \, \frac{\mathrm{V}_{1}^{*}(\ell)}{\mathrm{V}_{1}^{*}(\ell)} \,\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{T}}\right)_{\mathrm{p}}+\mathrm{x}_{2} \, \frac{\mathrm{V}_{2}^{*}(\ell)}{\mathrm{V}_{2}^{*}(\ell)} \,\left(\frac{\partial \mathrm{V}_{2}^{*}(\ell)}{\partial \mathrm{T}}\right)_{\mathrm{p}}+\left(\frac{\partial \mathrm{V}_{\mathrm{m}}^{\mathrm{E}}}{\partial \mathrm{T}}\right)_{\mathrm{p}}
\end{aligned} \nonumber \]
\[\begin{aligned}
&\mathrm{V}_{\mathrm{m}}(\mathrm{mix}) \, \alpha_{\mathrm{p}}(\mathrm{mix})= \\
&\mathrm{x}_{1} \, \mathrm{V}_{1}^{*}(\ell) \, \alpha_{\mathrm{p} 1}^{*}(\ell)+\mathrm{x}_{2} \, \mathrm{V}_{2}^{*}(\ell) \, \alpha_{\mathrm{p} 2}^{*}(\ell)+\left(\frac{\partial \mathrm{V}_{\mathrm{m}}^{\mathrm{E}}}{\partial \mathrm{T}}\right)_{\mathrm{p}}
\end{aligned} \nonumber \]
Or,
\[\begin{aligned}
&\alpha_{p}(\operatorname{mix})= \\
&\frac{1}{V_{\mathrm{m}}(\operatorname{mix})} \,\left[x_{1} \, V_{1}^{*}(\ell) \, \alpha_{p 1}^{*}(\ell)+x_{2} \, V_{2}^{*}(\ell) \, \alpha_{p 2}^{*}(\ell)+\left(\frac{\partial V_{m}^{E}}{\partial T}\right)_{p}\right]
\end{aligned} \nonumber \]
We may also define an excess property using equation (k) but it is important to note that \(\alpha_{\mathrm{p}}^{\mathrm{E}}\) is not a simple second derivative of the excess molar Gibbs energy, \(\mathrm{G}_{\mathrm{m}}^{\mathrm{E}}\).
\[\alpha_{p}^{E}(\operatorname{mix})=\alpha_{p}(\operatorname{mix})-\alpha_{p}(\operatorname{mix} ; \text { id }) \nonumber \]
We start out using an alternative expression for \(\alpha_{p}(\operatorname{mix})\).
\[\alpha_{p}(\operatorname{mix})=\frac{1}{V_{m}(\operatorname{mix})} \,\left[V_{m}(\operatorname{mix} ; \mathrm{id}) \, \alpha_{p}(\operatorname{mix} ; \mathrm{id})+\left(\frac{\partial \mathrm{V}_{\mathrm{m}}^{E}}{\partial T}\right)_{\mathrm{p}}\right] \nonumber \]
\[\begin{aligned}
&\alpha_{p}^{E}(\operatorname{mix})= \\
&\frac{1}{V_{m}(\operatorname{mix})} \,\left[V_{m}(\operatorname{mix} ; \text { id }) \, \alpha_{p}(\operatorname{mix} ; \mathrm{id})+\left(\frac{\partial V_{m}^{E}}{\partial T}\right)_{p}\right]-\alpha_{p}(\operatorname{mix} ; \text { id })
\end{aligned} \nonumber \]
\[\begin{aligned}
&\alpha_{\mathrm{p}}^{\mathrm{E}}(\operatorname{mix})= \\
&\frac{1}{\mathrm{~V}_{\mathrm{m}}(\operatorname{mix})} \,\left[\left(\frac{\partial \mathrm{V}_{\mathrm{m}}^{\mathrm{E}}}{\partial \mathrm{T}}\right)_{\mathrm{p}}+\left[\mathrm{V}_{\mathrm{m}}(\operatorname{mix} ; \mathrm{id})-\mathrm{V}_{\mathrm{m}}(\operatorname{mix})\right] \, \alpha_{\mathrm{p}}(\operatorname{mix} ; \mathrm{id}]\right.
\end{aligned} \nonumber \]
Hence, [1]
\[\alpha_{p}^{E}(\operatorname{mix})=\frac{1}{V_{m}(\operatorname{mix})} \,\left[\left(\frac{\partial V_{m}^{E}}{\partial T}\right)_{p}-V_{m}^{E}(\operatorname{mix}) \, \alpha_{p}(\text { mix } ; \text { id }]\right. \nonumber \]
Footnotes
[1]
\[\begin{aligned}
&{\alpha_{\mathrm{p}}^{\mathrm{E}}(\operatorname{mix})=\left[\mathrm{K}^{-1}\right]} \\
&{\left[\left(\frac{\partial \mathrm{V}_{\mathrm{m}}^{\mathrm{E}}}{\partial \mathrm{T}}\right)_{\mathrm{p}}-\mathrm{V}_{\mathrm{m}}^{\mathrm{E}} \, \alpha_{\mathrm{p}}(\mathrm{mix} ; \mathrm{id})\right]=\frac{\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right]}{[\mathrm{K}]}-\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right] \,\left[\mathrm{K}^{-1}\right]=\left[\mathrm{m}^{3} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}\right]} \\
&\frac{1}{V_{m}(m i x)} \,\left[\left(\frac{\partial V_{m}^{E}}{\partial T}\right)_{p}-V_{m}^{E} \, \alpha_{p}(\text { mix } ; i d)\right] \\
&\quad \quad =\frac{1}{\left[\mathrm{~m}^{3} \mathrm{~mol}^{-1}\right]} \,\left[\mathrm{m}^{3} \mathrm{~mol}^{-1} \quad \mathrm{~K}^{-1}\right]=\left[\mathrm{K}^{-1}\right]
\end{aligned} \nonumber \]