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1.15.3: Heat Capacities: Isobaric: Solutions: Unit Volume

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    A given aqueous solution was prepared using \(\mathrm{n}_{1}\) moles of water(\(\ell\)) and \(\mathrm{n}_{j}\) moles of solute \(j\). Then,

    \[\mathrm{C}_{\mathrm{p}}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{C}_{\mathrm{pl}}^{*}(\ell)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{C}_{\mathrm{pj}}\right)\]

    For the solution, by definition, the isobaric heat capacity per unit volume, (or heat capacitance)

    \[\sigma(\mathrm{aq})=\mathrm{C}_{\mathrm{p}}(\mathrm{aq}) / \mathrm{V}(\mathrm{aq})\]

    Similarly for the solvent at the same temperature and pressure,

    \[\sigma_{1}^{*}(\ell)=\mathrm{C}_{\mathrm{p} 1}^{*}(\ell) / \mathrm{V}_{1}^{*}(\ell)\]

    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].

    \[\sigma(\mathrm{aq})=\sigma_{1}^{*}(\ell)+\left[\phi\left(\mathrm{C}_{\mathrm{pj}}\right)-\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \sigma_{1}^{*}(\ell)\right] \, \mathrm{c}_{\mathrm{j}}\]

    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].

    \[\begin{aligned}
    \phi\left(\mathrm{C}_{\mathrm{pj}}\right)=& {\left[\mathrm{m}_{\mathrm{j}} \, \rho(\mathrm{aq}) \, \rho_{1}^{*}(\ell)\right]^{-1} \,\left[\rho_{1}^{*}(\ell) \, \sigma(\mathrm{aq})-\rho(\mathrm{aq}) \, \sigma_{1}^{*}(\ell)\right] } \\
    &+\mathrm{M}_{\mathrm{j}} \, \sigma(\mathrm{aq}) / \rho(\mathrm{aq})
    \end{aligned}\]

    Footnotes

    [1] From equations (a) and (b),

    \[\sigma(\mathrm{aq})=\left[\mathrm{n}_{1} / \mathrm{V}(\mathrm{aq})\right] \,\left[\mathrm{V}_{1}^{*}(\ell) / \mathrm{V}_{1}^{*}(\ell)\right] \, \mathrm{C}_{\mathrm{p} 1}^{*}(\ell)+\left[\mathrm{n}_{\mathrm{j}} / \mathrm{V}(\mathrm{aq})\right] \, \phi\left(\mathrm{C}_{\mathrm{pj}}\right)\]

    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,

    \[\mathrm{V}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\]

    or,

    \[\mathrm{n}_{\mathrm{l}} \, \mathrm{V}_{1}^{*}(\ell)=\mathrm{V}(\mathrm{aq})-\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\]

    Further concentration, \(\mathrm{c}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} / \mathrm{V}\) Then,

    \[\sigma(\mathrm{aq})=\left[\mathrm{V}(\mathrm{aq})-\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\right] \,[\mathrm{V}(\mathrm{aq})]^{-1} \, \sigma_{1}^{*}(\ell)+\mathrm{c}_{\mathrm{j}} \, \phi\left(\mathrm{C}_{\mathrm{pj}}\right)\]

    Hence,

    \[\sigma(\mathrm{aq})=\sigma_{1}^{*}(\ell) \,\left[1-\mathrm{c}_{\mathrm{j}}\left(\mathrm{V}_{\mathrm{j}}\right)\right]+\mathrm{c}_{\mathrm{j}} \, \phi\left(\mathrm{C}_{\mathrm{pj}}\right)\]

    or

    \[\sigma(\mathrm{aq})=\sigma_{1}^{*}(\ell)+\left[\phi\left(\mathrm{C}_{\mathrm{pj}}\right)-\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \sigma_{1}^{*}(\ell)\right] \, \mathrm{c}_{\mathrm{j}}\]

    [2]

    \[\begin{aligned}
    &\phi\left(\mathrm{C}_{\mathrm{pj}}\right) \, \mathrm{c}_{\mathrm{j}}=\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right] \,\left[\frac{\mathrm{mol}}{\mathrm{m}^{3}}\right]=\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~m}^{-3}\right] \\
    &\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \sigma_{1}^{*}(\ell) \, \mathrm{c}_{\mathrm{j}}=\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right] \,\left[\mathrm{JK}^{-1} \mathrm{~m}^{-3}\right] \,\left[\mathrm{mol} \mathrm{m}^{-3}\right]=\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~m}^{-3}\right]
    \end{aligned}\]

    [3] From [1],

    \[\mathrm{V}(\mathrm{aq})=\left[\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell) \, \sigma_{1}^{*}(\ell) / \sigma(\mathrm{aq})\right]+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{C}_{\mathrm{pj}}\right) / \sigma(\mathrm{aq})\]

    But,

    \[\mathrm{V}(\mathrm{aq})=\left(\mathrm{n}_{1} \, \mathrm{M}_{1}+\mathrm{n}_{\mathrm{j}} \, \mathrm{M}_{\mathrm{j}}\right) / \rho(\mathrm{aq})\]

    Then,

    \[\frac{\mathrm{n}_{1} \, \mathrm{M}_{1}+\mathrm{n}_{\mathrm{j}} \, \mathrm{M}_{\mathrm{j}}}{\rho(\mathrm{aq})}=\left[\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell) \, \sigma_{1}^{*}(\ell) / \sigma(\mathrm{aq})\right]+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{C}_{\mathrm{pj}}\right) / \sigma(\mathrm{aq})\]

    or (dividing by \(\mathrm{n}_{\mathrm{j}}\)),

    \[\left[\frac{\mathrm{n}_{1} \, \mathrm{M}_{1}}{\rho(\mathrm{aq}) \, \mathrm{n}_{\mathrm{j}}}\right]+\left[\frac{\mathrm{n}_{\mathrm{j}} \, \mathrm{M}_{\mathrm{j}}}{\mathrm{n}_{\mathrm{j}} \, \rho(\mathrm{aq})}\right]=\left[\frac{\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell) \, \sigma_{1}^{*}(\ell)}{\mathrm{n}_{\mathrm{j}} \, \sigma(\mathrm{aq})}\right]+\frac{\phi\left(\mathrm{C}_{\mathrm{pj}}\right)}{\sigma(\mathrm{aq})}\]

    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,

    \[\left[\frac{1}{\rho(\mathrm{aq}) \, \mathrm{m}_{\mathrm{j}}}\right]-\left[\frac{\sigma_{1}^{*}(\ell)}{\rho_{1}^{*}(\ell) \, \mathrm{m}_{\mathrm{j}} \, \sigma(\mathrm{aq})}\right]+\frac{\mathrm{M}_{\mathrm{j}}}{\rho(\mathrm{aq})}=\frac{\phi\left(\mathrm{C}_{\mathrm{pj}}\right)}{\sigma(\mathrm{aq})}\]

    As an equation for \(\phi \left(\mathrm{C}_{\mathrm{pj}}\right)\);

    \[\phi\left(\mathrm{C}_{\mathrm{pj}}\right)=\frac{\sigma(\mathrm{aq})}{\rho(\mathrm{aq}) \, \mathrm{m}_{\mathrm{j}}}-\frac{\sigma_{1}^{*}(\ell)}{\rho_{1}^{*}(\ell) \, \mathrm{m}_{\mathrm{j}}}+\frac{\mathrm{M}_{\mathrm{j}} \, \sigma(\mathrm{aq})}{\rho(\mathrm{aq})}\]

    Hence

    \[\begin{gathered}
    \phi\left(\mathrm{C}_{\mathrm{pj}}\right)=\left[\mathrm{m}_{\mathrm{j}} \, \rho(\mathrm{aq}) \, \rho_{1}^{*}(\ell)\right]^{-1} \,\left[\rho_{1}^{*}(\ell) \, \sigma(\mathrm{aq})-\rho(\mathrm{aq}) \, \sigma_{1}^{*}(\ell)\right] \\
    +\mathrm{M}_{\mathrm{j}} \, \sigma(\mathrm{aq}) / \rho(\mathrm{aq})
    \end{gathered}\]

    [4]

    \[\begin{aligned}
    &{\left[\mathrm{m}_{\mathrm{j}} \, \rho(\mathrm{aq}) \, \rho_{1}^{*}(\ell)\right]^{-1} \,\left[\rho_{1}^{*}(\ell) \, \sigma(\mathrm{aq})-\rho(\mathrm{aq}) \, \sigma_{1}^{*}(\ell)\right]=} \\
    &{\left[\frac{\mathrm{kg}}{\mathrm{mol}}\right] \,\left[\frac{\mathrm{m}^{3}}{\mathrm{~kg}}\right] \,\left[\frac{\mathrm{m}^{3}}{\mathrm{~kg}}\right] \,\left[\frac{\mathrm{kg}}{\mathrm{m}^{3}}\right] \,\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~m}^{-3}\right]=\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right]}
    \end{aligned}\]

    This page titled 1.15.3: Heat Capacities: Isobaric: Solutions: Unit Volume is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.