1.15.1: Heat Capacities: Isobaric: Neutral Solutes
An aqueous solution molality \(\mathrm{m}_{j}\), at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\), contains a simple neutral solute, \(j\). The chemical potential of the solute is given by equation (a).
\[\begin{aligned}
&\mu_{\mathrm{j}}\left(\mathrm{aq} ; \mathrm{m}_{\mathrm{j}} ; \mathrm{T} ; \mathrm{p}\right)= \\
&\mu_{\mathrm{j}}^{0}\left(\mathrm{aq} ; \mathrm{T} ; \mathrm{p}^{0}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)+\int_{\mathrm{p}^{0}}^{\mathrm{p}} \mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq} ; \mathrm{T}) \, \mathrm{dp}
\end{aligned} \nonumber \]
\[\mathrm{H}_{\mathrm{j}}\left(\mathrm{aq} ; \mathrm{m}_{\mathrm{j}} ; \mathrm{T} ; \mathrm{p}\right)=\mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})-\mathrm{R} \, \mathrm{T}^{2} \,\left[\partial \ln \left(\gamma_{\mathrm{j}}\right) / \partial \mathrm{T}\right]_{\mathrm{p}} \nonumber \]
where
\[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \mathrm{H}_{\mathrm{j}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p}) \nonumber \]
\[\begin{aligned}
&\mathrm{C}_{\mathrm{pj}}\left(\mathrm{aq} ; \mathrm{m}_{\mathrm{j}} ; \mathrm{T} ; \mathrm{p}\right)= \\
&\mathrm{C}_{\mathrm{pj}}^{\infty}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})-2 \, \mathrm{R} \, \mathrm{T} \,\left[\partial \ln \left(\gamma_{\mathrm{j}}\right) / \partial \mathrm{T}\right]_{\mathrm{p}}-\mathrm{R} \, \mathrm{T}^{2} \,\left[\partial^{2} \ln \left(\gamma_{\mathrm{j}}\right) / \partial \mathrm{T}^{2}\right]_{\mathrm{p}}
\end{aligned} \nonumber \]
Here
\[\mathrm{C}_{\mathrm{pj}}^{\infty}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\left[\partial \mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq}) / \partial \mathrm{T}\right]_{\mathrm{p}} \nonumber \]
For the solvent, the chemical potential is given by equation (f).
\[\mu_{1}\left(\mathrm{aq} ; \mathrm{m}_{\mathrm{j}} ; \mathrm{T} ; \mathrm{p}\right)=\mu_{1}^{0}\left(\ell ; \mathrm{T} ; \mathrm{p}^{0}\right)-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}+\int_{\mathrm{p}^{0}}^{\mathrm{p}} \mathrm{V}_{\mathrm{l}}^{*}(\ell ; \mathrm{T}) \, \mathrm{dp} \nonumber \]
\[\mathrm{H}_{1}\left(\mathrm{aq} ; \mathrm{m}_{\mathrm{j}} ; \mathrm{T} ; \mathrm{p}\right)=\mathrm{H}_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p})+\mathrm{R} \, \mathrm{T}^{2} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \,\left(\frac{\partial \phi}{\partial \mathrm{T}}\right)_{\mathrm{p}} \nonumber \]
Hence,
\[\begin{aligned}
&\mathrm{C}_{\mathrm{pl} 1}\left(\mathrm{aq} ; \mathrm{m}_{\mathrm{j}} ; \mathrm{T} ; \mathrm{p}\right)= \\
&\mathrm{C}_{\mathrm{pl} 1}^{*}(\ell ; \mathrm{T} ; \mathrm{p})+2 \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \,\left(\frac{\partial \phi}{\partial \mathrm{T}}\right)_{\mathrm{p}}+\mathrm{R} \, \mathrm{T}^{2} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \,\left(\frac{\partial^{2} \phi}{\partial \mathrm{T}^{2}}\right)_{\mathrm{p}}
\end{aligned} \nonumber \]
Where
\[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \mathrm{C}_{\mathrm{p} 1}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mathrm{C}_{\mathrm{pl} 1}^{*}(\ell ; \mathrm{T} ; \mathrm{p}) \nonumber \]
However,
\[\mathrm{C}_{\mathrm{p}}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{C}_{\mathrm{p}_{1}}(\mathrm{aq})+\mathrm{n}_{\mathrm{j}} \, \mathrm{C}_{\mathrm{pj}} \text { (aq) } \nonumber \]
Hence, from equations (d) and (h),
\[\begin{aligned}
&\mathrm{C}_{\mathrm{p}}\left(\mathrm{aq} ; \mathrm{m}_{\mathrm{j}} ; \mathrm{T} ; \mathrm{p}\right)= \\
&\mathrm{n}_{1} \,\left[\mathrm{C}_{\mathrm{pl}}^{*}(\ell ; \mathrm{T} ; \mathrm{p})+2 \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \,\left(\frac{\partial \phi}{\partial \mathrm{T}}\right)_{\mathrm{p}}\right. \\
&\left.+\mathrm{R} \, \mathrm{T}^{2} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \,\left(\frac{\partial^{2} \phi}{\partial \mathrm{T}^{2}}\right)_{\mathrm{p}}\right] \\
&+\mathrm{n}_{\mathrm{j}} \,\left[\mathrm{C}_{\mathrm{pj}}^{\infty}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})-2 \, \mathrm{R} \, \mathrm{T} \,\left(\frac{\partial \ln \left(\gamma_{\mathrm{j}}\right)}{\partial \mathrm{T}}\right)_{\mathrm{p}}\right. \\
&\left.-\mathrm{R} \, \mathrm{T}^{2} \,\left(\frac{\partial^{2} \ln \left(\gamma_{\mathrm{j}}\right)}{\partial \mathrm{T}^{2}}\right)_{\mathrm{p}}\right]
\end{aligned} \nonumber \]
We rearrange the latter equation to describe the isobaric heat capacity of a solution prepared using \(1 \mathrm{~kg}\) of water [1].
\[\begin{aligned}
&\mathrm{C}_{\mathrm{p}}\left(\mathrm{aq} ; \mathrm{m}_{\mathrm{j}} ; \mathrm{w}_{1}=1.0 \mathrm{~kg} ; \mathrm{T} ; \mathrm{p}\right)= \\
&\left(1 / \mathrm{M}_{1}\right) \, \mathrm{C}_{\mathrm{pl}}^{*}(\ell ; \mathrm{T} ; \mathrm{p}) \\
&+\mathrm{m}_{\mathrm{j}} \,\left[\mathrm{C}_{\mathrm{pj}}^{\infty}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})-2 \, \mathrm{R} \, \mathrm{T} \,\left(\frac{\partial \ln \left(\gamma_{\mathrm{j}}\right)}{\partial \mathrm{T}}\right)_{\mathrm{p}}-\mathrm{R} \, \mathrm{T}^{2} \,\left(\frac{\partial^{2} \ln \left(\gamma_{\mathrm{j}}\right)}{\partial \mathrm{T}^{2}}\right)_{\mathrm{p}}\right. \\
&\left.\quad+2 \, \mathrm{R} \, \mathrm{T} \,\left(\frac{\partial \phi}{\partial \mathrm{T}}\right)_{\mathrm{p}}+\mathrm{R} \, \mathrm{T}^{2} \,\left(\frac{\partial^{2} \phi}{\partial \mathrm{T}^{2}}\right)_{\mathrm{p}}\right]
\end{aligned} \nonumber \]
The term inside the [….] brackets is the apparent molar isobaric heat capacity for the solute. Thus,
\[\begin{array}{r}
\phi\left(C_{p j}\right)=C_{p j}^{\infty}(a q ; T ; p)-2 \, R \, T \,\left(\frac{\partial \ln \left(\gamma_{j}\right)}{\partial T}\right)_{p} \\
-R \, T^{2} \,\left(\frac{\partial^{2} \ln \left(\gamma_{j}\right)}{\partial T^{2}}\right)_{p} \\
+2 \, R \, T \,\left(\frac{\partial \phi}{\partial T}\right)_{p}+R \, T^{2} \,\left(\frac{\partial^{2} \phi}{\partial T^{2}}\right)_{p}
\end{array} \nonumber \]
Hence,
\[\mathrm{C}_{\mathrm{p}}\left(\mathrm{aq} ; \mathrm{m}_{\mathrm{j}} ; \mathrm{w}_{1}=1.0 \mathrm{~kg} ; \mathrm{T} ; \mathrm{p}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{C}_{\mathrm{pl}}^{*}(\ell ; \mathrm{T} ; \mathrm{p})+\mathrm{m}_{\mathrm{j}} \, \phi\left(\mathrm{C}_{\mathrm{pj}}\right) \nonumber \]
Then,
\[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \phi\left(\mathrm{C}_{\mathrm{pj}}\right)=\phi\left(\mathrm{C}_{\mathrm{pj}}\right)^{\infty}=\mathrm{C}_{\mathrm{pj}}^{\infty}(\mathrm{aq}) \nonumber \]
Equation (m) shows that \(\phi \left(\mathrm{C}_{\mathrm{pj}}\right)\) is a complicated property of a solution and that ‘the devil is in the detail’ [1]. A simplification in the algebra emerges if we define a set of J-properties which are excess properties [2,3]. Thus for a given solution prepared using \(\mathrm{n}_{1}\) moles of water and \(\mathrm{n}_{j}\) mole of solute \(j\),
\[\mathrm{J}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{J}_{1}(\mathrm{aq})+\mathrm{n}_{\mathrm{j}} \, \mathrm{J}_{\mathrm{j}}(\mathrm{aq}) \nonumber \]
where
\[J(a q)=C_{p}(a q)-C_{p}(a q ; i d) \nonumber \]
\[\mathrm{J}_{1}(\mathrm{aq})=\mathrm{C}_{\mathrm{p} 1}(\mathrm{aq})-\mathrm{C}_{\mathrm{pl} 1}^{*}(\ell) \nonumber \]
\[\mathrm{J}_{\mathrm{j}}(\mathrm{aq})=\mathrm{C}_{\mathrm{pj}}(\mathrm{aq})-\mathrm{C}_{\mathrm{pj}}^{\infty}(\mathrm{aq}) \nonumber \]
\[\phi\left(\mathrm{J}_{\mathrm{j}}\right)=\phi\left(\mathrm{C}_{\mathrm{p}_{\mathrm{j}}}\right)-\phi\left(\mathrm{C}_{\mathrm{pj}}\right)^{\infty} \nonumber \]
But
\[\mathrm{C}_{\mathrm{p}}(\mathrm{aq})-\mathrm{C}_{\mathrm{p}}(\mathrm{aq} ; \mathrm{id})=\mathrm{n}_{\mathrm{j}} \,\left[\phi\left(\mathrm{C}_{\mathrm{pj}}\right)-\phi\left(\mathrm{C}_{\mathrm{pj}}\right)^{\infty}\right] \nonumber \]
Then
\[\mathrm{J}(\mathrm{aq})=\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{J}_{\mathrm{j}}\right) \nonumber \]
An extensive literature reports the partial molar heat capacities of solutes in aqueous solution [2,3].
Further \({\mathrm{C}_{\mathrm{pj}}}^{\infty}(\mathrm{aq})\) for a range of related solutes can be analysed to yield group contributions [4-6]; e.g. at \(298.15 \mathrm{~K}\) the contribution of a methyl group, \(\mathrm{CH}_{3}\) to \({\mathrm{C}_{\mathrm{pj}}}^{\infty}\) for an aliphatic solute is \(178 \mathrm{~J K}^{-1} \mathrm{~mol}^{-1}\). Granted that \({\mathrm{C}_{\mathrm{pj}}}^{\infty}(\mathrm{aq})\) has been obtained for solute \(j\) and that the molar heat capacity of pure liquid \(j\), \(\mathrm{C}_{\mathrm{pj}}^{*}(\ell)\) is known , the isobaric heat capacity of solution \(\Delta_{s \ln } \mathrm{C}_{\mathrm{pj}}^{0}\) is obtained [7].
\[\Delta_{s \ln } C_{p j}^{0}=C_{p j}^{\infty}(a q)-C_{p j}^{*}(\ell) \nonumber \]
Footnotes
[1]
\[\mathrm{C}_{\mathrm{p}}\left(\mathrm{aq} ; \mathrm{m}_{\mathrm{j}} ; \mathrm{w}_{1}=1.0 \mathrm{~kg} ; \mathrm{T} ; \mathrm{p}\right)=\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~kg}^{-1}\right] \nonumber \]
\[\left(1 / \mathrm{M}_{1}\right) \, \mathrm{C}_{\mathrm{pl}}^{*}(\ell ; \mathrm{T} ; \mathrm{p})=\left[\mathrm{kg} \mathrm{mol}^{-1}\right]^{-1} \,\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right]=\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~kg}^{-1}\right] \nonumber \]
\[\mathrm{m}_{\mathrm{j}} \, \mathrm{C}_{\mathrm{pj}}^{\infty}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\left[\mathrm{mol} \mathrm{kg}^{-1}\right] \,\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right]=\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~kg}^{-1}\right] \nonumber \]
\[\mathrm{m}_{\mathrm{j}} \, 2 \, \mathrm{R} \, \mathrm{T} \,\left(\frac{\partial \ln \left(\gamma_{\mathrm{j}}\right)}{\partial \mathrm{T}}\right)_{\mathrm{p}}=\left[\mathrm{mol} \mathrm{kg}^{-1}\right] \,[1] \,\left[\mathrm{J} \mathrm{mol}^{-1} \mathrm{~K}^{-1}\right] \,[\mathrm{K}] \,[\mathrm{K}]^{-1}=\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~kg}^{-1}\right] \nonumber \]
\[\mathrm{m}_{\mathrm{j}} \, \mathrm{R} \, \mathrm{T}^{2} \,\left(\frac{\partial^{2} \ln \left(\gamma_{\mathrm{j}}\right)}{\partial \mathrm{T}^{2}}\right)_{\mathrm{p}}=\left[\mathrm{mol} \mathrm{kg}^{-1}\right] \,\left[\mathrm{J} \mathrm{mol}^{-1} \mathrm{~K}^{-1}\right] \,[\mathrm{K}]^{2} \,\left[\mathrm{K}^{-2}=\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~kg}^{-1}\right]\right. \nonumber \]
\[\mathrm{m}_{\mathrm{j}} \, 2 \, \mathrm{R} \, \mathrm{T} \,\left(\frac{\partial \phi}{\partial \mathrm{T}}\right)_{\mathrm{p}}=\left[\mathrm{mol} \mathrm{kg}^{-1}\right] \,[1] \,\left[\mathrm{J} \mathrm{mol}^{-1} \mathrm{~K}^{-1}\right] \,[\mathrm{K}] \,\left[\mathrm{K}^{-1}=\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~kg}^{-1}\right]\right. \nonumber \]
\[\mathrm{m}_{\mathrm{j}} \, \mathrm{R} \, \mathrm{T}^{2} \,\left(\frac{\partial^{2} \phi}{\partial \mathrm{T}^{2}}\right)_{\mathrm{p}}=\left[\mathrm{mol} \mathrm{kg}{ }^{-1}\right] \,\left[\mathrm{J} \mathrm{mol}^{-1} \mathrm{~K}^{-1}\right] \,[\mathrm{K}]^{2} \,[\mathrm{K}]^{-2}=\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~kg}^{-1}\right] \nonumber \]
[2] Sucrose(aq); an early determination; F. T. Gucker and F. D. Ayres, J. Am. Chem.Soc.,1937, 59 ,447.
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Amino acids (\(288 \mathrm{~K}\) to \(328 \mathrm{~K}\));
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cyclic dipeptides;
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[4] ROH(aq); D. Mirejovsky and E. M. Arnett, J. Am. Chem.Soc.,1983, 105 ,112.
[5] Ph-X; group additivity; G. Perron and J. E. Desnoyers, Fluid Phase Equilib, 1979, 2 ,239.
[6] Amides(aq); R. Skold, J. Suurkuus and I. Wadso, J.Chem. Thermodyn., 1976, 8 ,1075.
[7] \(\Delta_{s \ln } \mathrm{C}_{\mathrm{pj}}^{0}\) for gases(aq); G. Olofsson, A. A. Oshodj, E. pj Qvarstrom and I. Wadso, J. Chem. Thermodyn., 1984, 16 ,1041.