1.12.8: Expansions- Solutions- Isobaric- Partial and Apparent Molar
The volume of a given aqueous solution containing \(\mathrm{n}_{1}\) moles of water and \(\mathrm{n}_{j}\) moles of solute \(j\) is related to the composition by equation (a).
\[\mathrm{V}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{V}_{1}(\mathrm{aq})+\mathrm{n}_{\mathrm{j}} \, \mathrm{V}_{\mathrm{j}}(\mathrm{aq}) \nonumber \]
\(\mathrm{V}_{1}(\mathrm{aq})\) and \(\mathrm{V}_{j}(\mathrm{aq})\) are the partial molar volumes of water and solute \(j\) respectively. The (equilibrium) isobaric thermal expansion of the solution (at fixed pressure) \(\mathrm{E}_{\mathrm{p}}\) characterises the differential dependence of \(\mathrm{V}(\mathrm{aq})\) on temperature.
\[\mathrm{E}_{\mathrm{p}}(\mathrm{aq})=[\partial \mathrm{V}(\mathrm{aq}) / \partial \mathrm{T}]_{\mathrm{p}, \mathrm{A}=0} \nonumber \]
\(\mathrm{E}_{\mathrm{p}}(\mathrm{aq})\) is an extensive property of the solution [1]. Two partial molar isobaric thermal expansions are defined, characteristic of solute and solvent [2].
\[\mathrm{E}_{\mathrm{p} 1}(\mathrm{aq})=\left(\partial \mathrm{V}_{1}(\mathrm{aq}) / \partial \mathrm{T}\right)_{\mathrm{p}} \nonumber \]
\[\mathrm{E}_{\mathrm{pj}}(\mathrm{aq})=\left(\partial \mathrm{V}_{\mathrm{j}}(\mathrm{aq}) / \partial \mathrm{T}\right)_{\mathrm{p}} \nonumber \]
From equation (a),
\[\mathrm{E}_{\mathrm{p}}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{E}_{\mathrm{p} 1}(\mathrm{aq})+\mathrm{n}_{\mathrm{j}} \, \mathrm{E}_{\mathrm{pj}}(\mathrm{aq}) \nonumber \]
In the treatment of volumetric properties of solutions we define an apparent molar volume of the solute, \(\phi\left(\mathrm{V}_{\mathrm{j}}\right)\). By analogy we rewrite equation (e) in a form which defines the apparent molar isobaric expansion of the solute, \(\phi\left(\mathrm{E}_{\mathrm{j}}\right)\). Thus,
\[\mathrm{E}_{\mathrm{p}}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{E}_{\mathrm{p} 1}^{*}(\ell)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{E}_{\mathrm{pj}}\right) \nonumber \]
Here [3],
\[\phi\left(\mathrm{E}_{\mathrm{pj}}\right)=\left(\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{T}}\right)_{\mathrm{p}} \nonumber \]
For the pure solvent,
\[\mathrm{E}_{\mathrm{p} 1}^{*}(\ell)=\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{T}}\right)_{\mathrm{p}} \nonumber \]
Footnotes
[1] \(\mathrm{E}_{\mathrm{p}}\) is an extensive property; the larger the volume \(\mathrm{V}\) the larger the change in volume for a given increase in temperature.
[2] \(E_{p}=\left[m^{3} K^{-1}\right] \quad E_{p 1}=\left[\mathrm{m}^{3} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}\right] \quad E_{p j}=\left[\mathrm{m}^{3} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}\right]\)
[3] \(\phi\left(\mathrm{E}_{\mathrm{j}}\right)=\left[\mathrm{m}^{3} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}\right]\)