Skip to main content
Chemistry LibreTexts

1.12.5: Heat Capacity- Isobaric- Solutions- Excess

  • Page ID
    386473
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    A given solution is prepared using \(1 \mathrm{~kg}\) of solvent (water) and \(\mathrm{m}_{j}\) moles of solute \(j\). If the thermodynamic properties of this solution are ideal, the isobaric heat capacity can be expressed as follows [1].

    \[\mathrm{C}_{\mathrm{p}}\left(\mathrm{aq} ; \mathrm{id} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{C}_{\mathrm{p} 1}^{*}(\ell)+\mathrm{m}_{\mathrm{j}} \, \mathrm{C}_{\mathrm{pj}}^{\infty}(\mathrm{aq})\]

    On the other hand for a real solution the isobaric heat capacity can be expressed in terms of the apparent molar heat capacity of the solute, \(\phi \left(\mathrm{C}_{\mathrm{pj}}\right)\).

    \[\mathrm{C}_{\mathrm{p}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{C}_{\mathrm{pl}}^{*}(\ell)+\mathrm{m}_{\mathrm{j}} \, \phi\left(\mathrm{C}_{\mathrm{pj}}\right)\]

    The difference between \(\mathrm{C}_{\mathrm{p}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)\) and \(\mathrm{C}_{\mathrm{p}}\left(\mathrm{aq} ; \mathrm{id} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)\) defines the relative isobaric heat capacity of the solution \(\mathrm{J}\), an excess property.

    \[\mathrm{J}(\mathrm{aq})=\mathrm{C}_{\mathrm{p}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)-\mathrm{C}_{\mathrm{p}}\left(\mathrm{aq} ; \mathrm{id} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)\]

    Thermodynamics does not define the magnitude or sign of \(\mathrm{J}(\mathrm{aq})\). However, from the definitions of ideal and real partial molar isobaric capacities of solvent and solute, the following condition must hold.

    \[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \mathrm{J}(\mathrm{aq})=0\]

    Relative quantities can also be defined for solute and solvent.

    \[\mathrm{J}_{j}(\mathrm{aq})=\mathrm{C}_{\mathrm{pj}}(\mathrm{aq})-\mathrm{C}_{\mathrm{pj}}^{\infty}(\mathrm{aq})\]

    \[\mathrm{J}_{1}(\mathrm{aq})=\mathrm{C}_{\mathrm{p} 1}(\mathrm{aq})-\mathrm{C}_{\mathrm{p} 1}^{*}(\ell)\]

    Also,

    \[\phi\left(\mathrm{J}_{\mathrm{j}}\right)=\phi\left[\mathrm{C}_{\mathrm{pj}}(\mathrm{aq})\right]-\mathrm{C}_{\mathrm{pj}}^{\infty}(\mathrm{aq})\]

    Hence,

    \[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \mathrm{J}_{\mathrm{j}}(\mathrm{aq})=\mathrm{J}_{1}(\mathrm{aq})=\phi\left(\mathrm{J}_{\mathrm{j}}\right)=0\]

    Equation (c) defines a property \(\mathrm{J}\) which is an excess isobaric heat capacity of a solution prepared using \(1 \mathrm{~kg}\) of water. Thus,

    \[\mathrm{C}_{\mathrm{p}}^{\mathrm{E}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\mathrm{J}(\mathrm{aq})=\mathrm{C}_{\mathrm{p}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)-\mathrm{C}_{\mathrm{p}}\left(\mathrm{aq} ; \mathrm{id} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)\]

    From equations (a) and (b),

    \[\mathrm{C}_{\mathrm{p}}^{\mathrm{E}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\mathrm{m}_{\mathrm{j}} \,\left[\phi\left(\mathrm{C}_{\mathrm{pj}}\right)-\mathrm{C}_{\mathrm{pj}}^{\infty}(\mathrm{aq})\right]\]

    From equation (g),

    \[\mathrm{C}_{\mathrm{p}}^{\mathrm{E}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\mathrm{m}_{\mathrm{j}} \, \phi\left(\mathrm{J}_{\mathrm{j}}\right)\]

    Thus \(\phi \left(\mathrm{J}_{j}\right)\) is the relative apparent molar isobaric heat capacity of the solute in a given real solution. Isobaric heat capacities of solutions and related partial molar isobaric heat capacities reflect in characteristic fashion the impact of added solutes on water water interactions

    Footnote

    [1] \(\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~kg}^{-1}\right]=\left[\mathrm{kg} \mathrm{mol}^{-1}\right]^{-1} \,\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right]+\left[\mathrm{mol} \mathrm{kg}^{-1}\right] \,\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right]\)


    This page titled 1.12.5: Heat Capacity- Isobaric- Solutions- Excess is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.