1.22.6: Volumes: Apparent Molar and Excess Volumes
For a solution prepared using \(1 \mathrm{~kg}\) of water, the volume is related to the apparent molar volume of the solute \(\phi \left(\mathrm{V}_{j}\right)\) using equation (a).
\[\mathrm{V}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{m}_{\mathrm{j}} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right) \nonumber \]
If the thermodynamic properties of this solution are ideal,
\[\mathrm{V}\left(\mathrm{aq} ; \mathrm{id} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{m}_{\mathrm{j}} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)^{\infty} \nonumber \]
Here \(V_{\mathrm{j}}^{\infty}(\mathrm{aq}) \equiv \phi\left(\mathrm{V}_{\mathrm{j}}\right)^{\infty}\). The difference between \(\mathrm{V}(\mathrm{aq})\) and \(\mathrm{V}(\mathrm{aq}: \mathrm{id})\) defines an excess volume \(\mathrm{V}^{\mathrm{E}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)\). Thus,
\[\mathrm{V}^{\mathrm{E}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\mathrm{V}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)-\mathrm{V}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}: \mathrm{id}\right) \nonumber \]
Hence,
\[\mathrm{V}^{\mathrm{E}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\mathrm{m}_{\mathrm{j}} \,\left[\phi\left(\mathrm{V}_{\mathrm{j}}\right)-\phi\left(\mathrm{V}_{\mathrm{j}}\right)^{\infty}\right] \nonumber \]