1.7.21: Compressions- Isothermal- Binary Aqueous Mixtures
A given binary aqueous mixture is prepared using \(\mathrm{n}_{1}\) moles of water (\(\ell\)) and \(\mathrm{n}_{2}\) moles of liquid 2. The volume of the mixture, \(\mathrm{V}(\mathrm{mix})\) is given by equation (a) (at fixed temperature and pressure).
\[\mathrm{V}(\operatorname{mix})=\mathrm{n}_{1} \, \mathrm{V}_{1}(\operatorname{mix})+\mathrm{n}_{2} \, \mathrm{V}_{2}(\mathrm{mix}) \nonumber \]
The mixture is perturbed by a change in pressure at fixed temperature along an equilibrium pathway where the affinity for spontaneous change remains at zero.
\[\left(\frac{\partial \mathrm{V}(\mathrm{mix})}{\partial \mathrm{p}}\right)_{\mathrm{T}}=\mathrm{n}_{1} \,\left(\frac{\partial \mathrm{V}_{1}(\mathrm{mix})}{\partial \mathrm{p}}\right)_{\mathrm{T}}+\mathrm{n}_{2} \,\left(\frac{\partial \mathrm{V}_{2}(\mathrm{mix})}{\partial \mathrm{p}}\right)_{\mathrm{T}} \nonumber \]
The isothermal compression of the mixture \([\partial \mathrm{V}(\mathrm{mix}) / \partial \mathrm{p}]_{\mathrm{T}}\) is an extensive property. The partial differentials \(\left[\partial V_{1}(\operatorname{mix}) / \partial p\right]_{T}\) and \(\left[\partial V_{2}(\operatorname{mix}) / \partial \mathrm{p}\right]_{\mathrm{T}}\) are intensive properties. There is merit in defining an intensive molar compression using equation (c).
\[\mathrm{K}_{\mathrm{Tm}}=\frac{\mathrm{K}_{\mathrm{T}}(\operatorname{mix})}{\left(\mathrm{n}_{1}+\mathrm{n}_{2}\right)}=-\left(\mathrm{n}_{1}+\mathrm{n}_{2}\right)^{-1} \,[\partial \mathrm{V}(\operatorname{mix}) / \partial \mathrm{p}]_{\mathrm{T}} \nonumber \]
By definition,
\[\mathrm{K}_{\mathrm{T} 1}(\operatorname{mix})=-\left[\partial \mathrm{V}_{1}(\operatorname{mix}) / \partial \mathrm{p}\right]_{\mathrm{T}} \nonumber \]
And
\[\mathrm{K}_{\mathrm{T} 2}(\operatorname{mix})=-\left[\partial \mathrm{V}_{2}(\operatorname{mix}) / \partial \mathrm{p}\right]_{\mathrm{T}} \nonumber \]
Hence
\[\mathrm{K}_{\mathrm{Tm}}(\operatorname{mix})=\mathrm{x}_{1} \, \mathrm{K}_{\mathrm{T} 1}(\operatorname{mix})+\mathrm{x}_{2} \, \mathrm{K}_{\mathrm{T} 2}(\operatorname{mix}) \nonumber \]
For an ideal binary mixture,
\[\mathrm{V}(\operatorname{mix} ; \mathrm{id})=\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{n}_{2} \, \mathrm{V}_{2}^{*}(\ell) \nonumber \]
\(\mathrm{V}_{1}^{*}(\ell)\) and \(\mathrm{V}_{2}^{*}(\ell)\) are the molar volumes of the pure liquids at the same \(\mathrm{T}\) and \(\mathrm{p}\). Therefore, following the argument outlined above,
\[\mathrm{K}_{\mathrm{Tm}}(\mathrm{id})=\mathrm{x}_{1} \, \mathrm{K}_{\mathrm{T} 1}^{*}(\ell)+\mathrm{x}_{2} \, \mathrm{K}_{\mathrm{T} 2}^{*}(\ell) \nonumber \]
By definition,
\[\mathrm{K}_{\mathrm{T}}^{\mathrm{E}}(\operatorname{mix})=\mathrm{K}_{\mathrm{Tm}}(\operatorname{mix})-\mathrm{K}_{\mathrm{Tm}}(\operatorname{mix} ; \mathrm{id}) \nonumber \]
Hence the excess molar compression is given by equation (j).
\[\mathrm{K}_{\mathrm{Tm}}^{\mathrm{E}}(\operatorname{mix})=\mathrm{x}_{1} \,\left[\mathrm{K}_{\mathrm{T} 1}(\operatorname{mix})-\mathrm{K}_{\mathrm{T} 1}^{*}(\ell)\right]+\mathrm{x}_{2} \,\left[\mathrm{K}_{\mathrm{T} 2}(\operatorname{mix})-\mathrm{K}_{\mathrm{T} 2}^{*}(\ell)\right] \nonumber \]