1.15.3: Heat Capacities: Isobaric: Solutions: Unit Volume
A given aqueous solution was prepared using \(\mathrm{n}_{1}\) moles of water(\(\ell\)) and \(\mathrm{n}_{j}\) moles of solute \(j\). Then,
\[\mathrm{C}_{\mathrm{p}}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{C}_{\mathrm{pl}}^{*}(\ell)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{C}_{\mathrm{pj}}\right) \nonumber \]
For the solution, by definition, the isobaric heat capacity per unit volume, (or heat capacitance)
\[\sigma(\mathrm{aq})=\mathrm{C}_{\mathrm{p}}(\mathrm{aq}) / \mathrm{V}(\mathrm{aq}) \nonumber \]
Similarly for the solvent at the same temperature and pressure,
\[\sigma_{1}^{*}(\ell)=\mathrm{C}_{\mathrm{p} 1}^{*}(\ell) / \mathrm{V}_{1}^{*}(\ell) \nonumber \]
With reference to equations (b) and (c), the four experimentally determined quantities are \(\sigma(\mathrm{aq}), \sigma_{1}^{*}(\mathrm{aq}), \rho(\mathrm{aq}) \text { and } \rho_{1}^{*}(\ell)\). The latter two quantities are the densities of the solution and solvent respectively.
Hence \(\sigma(\mathrm{aq})\) is related to the concentration of the solution, \(\mathrm{c}_{j}\) [1,2].
\[\sigma(\mathrm{aq})=\sigma_{1}^{*}(\ell)+\left[\phi\left(\mathrm{C}_{\mathrm{pj}}\right)-\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \sigma_{1}^{*}(\ell)\right] \, \mathrm{c}_{\mathrm{j}} \nonumber \]
The latter equation relates \(\sigma(\mathrm{aq})\) to the property for the pure solvent, \(\sigma_{1}^{*}(\ell)\) and to the concentration of solute, \(\mathrm{c}_{\mathrm{j}}\). Equation (d) relates \(\phi \left(\mathrm{C}_{\mathrm{pj}}\right)\) to the measured quantities \(\sigma(\mathrm{aq})\) and \(\sigma_{1}^{*}(\ell)\) together with the apparent molar volume \(\phi \left(\mathrm{V}_{\mathrm{j}}\right)\). Thus \(\sigma(\mathrm{aq})\) and \(\phi \left(\mathrm{V}_{\mathrm{j}}\right)\) for a given solution yields together with \(\sigma_{1}^{*}(\ell)\), the apparent molar isobaric heat capacity of the solute, \(\phi \left(\mathrm{C}_{\mathrm{pj}}\right)\). In cases where the composition of the solution is expressed using molalities, equation (e) is the equation for \(\phi \left(\mathrm{C}_{\mathrm{pj}}\right)\) [3,4].
\[\begin{aligned}
\phi\left(\mathrm{C}_{\mathrm{pj}}\right)=& {\left[\mathrm{m}_{\mathrm{j}} \, \rho(\mathrm{aq}) \, \rho_{1}^{*}(\ell)\right]^{-1} \,\left[\rho_{1}^{*}(\ell) \, \sigma(\mathrm{aq})-\rho(\mathrm{aq}) \, \sigma_{1}^{*}(\ell)\right] } \\
&+\mathrm{M}_{\mathrm{j}} \, \sigma(\mathrm{aq}) / \rho(\mathrm{aq})
\end{aligned} \nonumber \]
Footnotes
[1] From equations (a) and (b),
\[\sigma(\mathrm{aq})=\left[\mathrm{n}_{1} / \mathrm{V}(\mathrm{aq})\right] \,\left[\mathrm{V}_{1}^{*}(\ell) / \mathrm{V}_{1}^{*}(\ell)\right] \, \mathrm{C}_{\mathrm{p} 1}^{*}(\ell)+\left[\mathrm{n}_{\mathrm{j}} / \mathrm{V}(\mathrm{aq})\right] \, \phi\left(\mathrm{C}_{\mathrm{pj}}\right) \nonumber \]
The term \(\left[\mathrm{V}_{1}^{*}(\ell) / \mathrm{v}_{1}^{*}(\ell)\right]\) has been introduced with the definition of \(\sigma_{1}^{*}(\ell)\) in mind.
But,
\[\mathrm{V}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right) \nonumber \]
or,
\[\mathrm{n}_{\mathrm{l}} \, \mathrm{V}_{1}^{*}(\ell)=\mathrm{V}(\mathrm{aq})-\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right) \nonumber \]
Further concentration, \(\mathrm{c}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} / \mathrm{V}\) Then,
\[\sigma(\mathrm{aq})=\left[\mathrm{V}(\mathrm{aq})-\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\right] \,[\mathrm{V}(\mathrm{aq})]^{-1} \, \sigma_{1}^{*}(\ell)+\mathrm{c}_{\mathrm{j}} \, \phi\left(\mathrm{C}_{\mathrm{pj}}\right) \nonumber \]
Hence,
\[\sigma(\mathrm{aq})=\sigma_{1}^{*}(\ell) \,\left[1-\mathrm{c}_{\mathrm{j}}\left(\mathrm{V}_{\mathrm{j}}\right)\right]+\mathrm{c}_{\mathrm{j}} \, \phi\left(\mathrm{C}_{\mathrm{pj}}\right) \nonumber \]
or
\[\sigma(\mathrm{aq})=\sigma_{1}^{*}(\ell)+\left[\phi\left(\mathrm{C}_{\mathrm{pj}}\right)-\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \sigma_{1}^{*}(\ell)\right] \, \mathrm{c}_{\mathrm{j}} \nonumber \]
[2]
\[\begin{aligned}
&\phi\left(\mathrm{C}_{\mathrm{pj}}\right) \, \mathrm{c}_{\mathrm{j}}=\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right] \,\left[\frac{\mathrm{mol}}{\mathrm{m}^{3}}\right]=\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~m}^{-3}\right] \\
&\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \sigma_{1}^{*}(\ell) \, \mathrm{c}_{\mathrm{j}}=\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right] \,\left[\mathrm{JK}^{-1} \mathrm{~m}^{-3}\right] \,\left[\mathrm{mol} \mathrm{m}^{-3}\right]=\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~m}^{-3}\right]
\end{aligned} \nonumber \]
[3] From [1],
\[\mathrm{V}(\mathrm{aq})=\left[\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell) \, \sigma_{1}^{*}(\ell) / \sigma(\mathrm{aq})\right]+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{C}_{\mathrm{pj}}\right) / \sigma(\mathrm{aq}) \nonumber \]
But,
\[\mathrm{V}(\mathrm{aq})=\left(\mathrm{n}_{1} \, \mathrm{M}_{1}+\mathrm{n}_{\mathrm{j}} \, \mathrm{M}_{\mathrm{j}}\right) / \rho(\mathrm{aq}) \nonumber \]
Then,
\[\frac{\mathrm{n}_{1} \, \mathrm{M}_{1}+\mathrm{n}_{\mathrm{j}} \, \mathrm{M}_{\mathrm{j}}}{\rho(\mathrm{aq})}=\left[\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell) \, \sigma_{1}^{*}(\ell) / \sigma(\mathrm{aq})\right]+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{C}_{\mathrm{pj}}\right) / \sigma(\mathrm{aq}) \nonumber \]
or (dividing by \(\mathrm{n}_{\mathrm{j}}\)),
\[\left[\frac{\mathrm{n}_{1} \, \mathrm{M}_{1}}{\rho(\mathrm{aq}) \, \mathrm{n}_{\mathrm{j}}}\right]+\left[\frac{\mathrm{n}_{\mathrm{j}} \, \mathrm{M}_{\mathrm{j}}}{\mathrm{n}_{\mathrm{j}} \, \rho(\mathrm{aq})}\right]=\left[\frac{\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell) \, \sigma_{1}^{*}(\ell)}{\mathrm{n}_{\mathrm{j}} \, \sigma(\mathrm{aq})}\right]+\frac{\phi\left(\mathrm{C}_{\mathrm{pj}}\right)}{\sigma(\mathrm{aq})} \nonumber \]
But molality \(\mathrm{m}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} / \mathrm{n}_{1} \, \mathrm{M}_{1}=\mathrm{n}_{\mathrm{j}} / \mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell) \, \rho_{1}^{*}(\ell)\) Then,
\[\left[\frac{1}{\rho(\mathrm{aq}) \, \mathrm{m}_{\mathrm{j}}}\right]-\left[\frac{\sigma_{1}^{*}(\ell)}{\rho_{1}^{*}(\ell) \, \mathrm{m}_{\mathrm{j}} \, \sigma(\mathrm{aq})}\right]+\frac{\mathrm{M}_{\mathrm{j}}}{\rho(\mathrm{aq})}=\frac{\phi\left(\mathrm{C}_{\mathrm{pj}}\right)}{\sigma(\mathrm{aq})} \nonumber \]
As an equation for \(\phi \left(\mathrm{C}_{\mathrm{pj}}\right)\);
\[\phi\left(\mathrm{C}_{\mathrm{pj}}\right)=\frac{\sigma(\mathrm{aq})}{\rho(\mathrm{aq}) \, \mathrm{m}_{\mathrm{j}}}-\frac{\sigma_{1}^{*}(\ell)}{\rho_{1}^{*}(\ell) \, \mathrm{m}_{\mathrm{j}}}+\frac{\mathrm{M}_{\mathrm{j}} \, \sigma(\mathrm{aq})}{\rho(\mathrm{aq})} \nonumber \]
Hence
\[\begin{gathered}
\phi\left(\mathrm{C}_{\mathrm{pj}}\right)=\left[\mathrm{m}_{\mathrm{j}} \, \rho(\mathrm{aq}) \, \rho_{1}^{*}(\ell)\right]^{-1} \,\left[\rho_{1}^{*}(\ell) \, \sigma(\mathrm{aq})-\rho(\mathrm{aq}) \, \sigma_{1}^{*}(\ell)\right] \\
+\mathrm{M}_{\mathrm{j}} \, \sigma(\mathrm{aq}) / \rho(\mathrm{aq})
\end{gathered} \nonumber \]
[4]
\[\begin{aligned}
&{\left[\mathrm{m}_{\mathrm{j}} \, \rho(\mathrm{aq}) \, \rho_{1}^{*}(\ell)\right]^{-1} \,\left[\rho_{1}^{*}(\ell) \, \sigma(\mathrm{aq})-\rho(\mathrm{aq}) \, \sigma_{1}^{*}(\ell)\right]=} \\
&{\left[\frac{\mathrm{kg}}{\mathrm{mol}}\right] \,\left[\frac{\mathrm{m}^{3}}{\mathrm{~kg}}\right] \,\left[\frac{\mathrm{m}^{3}}{\mathrm{~kg}}\right] \,\left[\frac{\mathrm{kg}}{\mathrm{m}^{3}}\right] \,\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~m}^{-3}\right]=\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right]}
\end{aligned} \nonumber \]