1.12.5: Heat Capacity- Isobaric- Solutions- Excess
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].
\[\mathrm{C}_{\mathrm{p}}\left(\mathrm{aq} ; \mathrm{id} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{C}_{\mathrm{p} 1}^{*}(\ell)+\mathrm{m}_{\mathrm{j}} \, \mathrm{C}_{\mathrm{pj}}^{\infty}(\mathrm{aq}) \nonumber \]
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, \(\phi \left(\mathrm{C}_{\mathrm{pj}}\right)\).
\[\mathrm{C}_{\mathrm{p}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{C}_{\mathrm{pl}}^{*}(\ell)+\mathrm{m}_{\mathrm{j}} \, \phi\left(\mathrm{C}_{\mathrm{pj}}\right) \nonumber \]
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.
\[\mathrm{J}(\mathrm{aq})=\mathrm{C}_{\mathrm{p}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)-\mathrm{C}_{\mathrm{p}}\left(\mathrm{aq} ; \mathrm{id} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right) \nonumber \]
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.
\[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \mathrm{J}(\mathrm{aq})=0 \nonumber \]
Relative quantities can also be defined for solute and solvent.
\[\mathrm{J}_{j}(\mathrm{aq})=\mathrm{C}_{\mathrm{pj}}(\mathrm{aq})-\mathrm{C}_{\mathrm{pj}}^{\infty}(\mathrm{aq}) \nonumber \]
\[\mathrm{J}_{1}(\mathrm{aq})=\mathrm{C}_{\mathrm{p} 1}(\mathrm{aq})-\mathrm{C}_{\mathrm{p} 1}^{*}(\ell) \nonumber \]
Also,
\[\phi\left(\mathrm{J}_{\mathrm{j}}\right)=\phi\left[\mathrm{C}_{\mathrm{pj}}(\mathrm{aq})\right]-\mathrm{C}_{\mathrm{pj}}^{\infty}(\mathrm{aq}) \nonumber \]
Hence,
\[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \mathrm{J}_{\mathrm{j}}(\mathrm{aq})=\mathrm{J}_{1}(\mathrm{aq})=\phi\left(\mathrm{J}_{\mathrm{j}}\right)=0 \nonumber \]
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,
\[\mathrm{C}_{\mathrm{p}}^{\mathrm{E}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\mathrm{J}(\mathrm{aq})=\mathrm{C}_{\mathrm{p}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)-\mathrm{C}_{\mathrm{p}}\left(\mathrm{aq} ; \mathrm{id} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right) \nonumber \]
From equations (a) and (b),
\[\mathrm{C}_{\mathrm{p}}^{\mathrm{E}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\mathrm{m}_{\mathrm{j}} \,\left[\phi\left(\mathrm{C}_{\mathrm{pj}}\right)-\mathrm{C}_{\mathrm{pj}}^{\infty}(\mathrm{aq})\right] \nonumber \]
From equation (g),
\[\mathrm{C}_{\mathrm{p}}^{\mathrm{E}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\mathrm{m}_{\mathrm{j}} \, \phi\left(\mathrm{J}_{\mathrm{j}}\right) \nonumber \]
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] \(\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~kg}^{-1}\right]=\left[\mathrm{kg} \mathrm{mol}^{-1}\right]^{-1} \,\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right]+\left[\mathrm{mol} \mathrm{kg}^{-1}\right] \,\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right]\)