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).

\[\begin{aligned}
&\mu_{\mathrm{j}}\left(\mathrm{aq} ; \mathrm{m}_{\mathrm{j}} ; \mathrm{T} ; \mathrm{p}\right)= \\
&\mu_{\mathrm{j}}^{0}\left(\mathrm{aq} ; \mathrm{T} ; \mathrm{p}^{0}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)+\int_{\mathrm{p}^{0}}^{\mathrm{p}} \mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq} ; \mathrm{T}) \, \mathrm{dp}
\end{aligned}\]

\[\mathrm{H}_{\mathrm{j}}\left(\mathrm{aq} ; \mathrm{m}_{\mathrm{j}} ; \mathrm{T} ; \mathrm{p}\right)=\mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})-\mathrm{R} \, \mathrm{T}^{2} \,\left[\partial \ln \left(\gamma_{\mathrm{j}}\right) / \partial \mathrm{T}\right]_{\mathrm{p}}\]

where

\[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \mathrm{H}_{\mathrm{j}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})\]

\[\begin{aligned}
&\mathrm{C}_{\mathrm{pj}}\left(\mathrm{aq} ; \mathrm{m}_{\mathrm{j}} ; \mathrm{T} ; \mathrm{p}\right)= \\
&\mathrm{C}_{\mathrm{pj}}^{\infty}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})-2 \, \mathrm{R} \, \mathrm{T} \,\left[\partial \ln \left(\gamma_{\mathrm{j}}\right) / \partial \mathrm{T}\right]_{\mathrm{p}}-\mathrm{R} \, \mathrm{T}^{2} \,\left[\partial^{2} \ln \left(\gamma_{\mathrm{j}}\right) / \partial \mathrm{T}^{2}\right]_{\mathrm{p}}
\end{aligned}\]

Here

\[\mathrm{C}_{\mathrm{pj}}^{\infty}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\left[\partial \mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq}) / \partial \mathrm{T}\right]_{\mathrm{p}}\]

For the solvent, the chemical potential is given by equation (f).

\[\mu_{1}\left(\mathrm{aq} ; \mathrm{m}_{\mathrm{j}} ; \mathrm{T} ; \mathrm{p}\right)=\mu_{1}^{0}\left(\ell ; \mathrm{T} ; \mathrm{p}^{0}\right)-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}+\int_{\mathrm{p}^{0}}^{\mathrm{p}} \mathrm{V}_{\mathrm{l}}^{*}(\ell ; \mathrm{T}) \, \mathrm{dp}\]

\[\mathrm{H}_{1}\left(\mathrm{aq} ; \mathrm{m}_{\mathrm{j}} ; \mathrm{T} ; \mathrm{p}\right)=\mathrm{H}_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p})+\mathrm{R} \, \mathrm{T}^{2} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \,\left(\frac{\partial \phi}{\partial \mathrm{T}}\right)_{\mathrm{p}}\]

Hence,

\[\begin{aligned}
&\mathrm{C}_{\mathrm{pl} 1}\left(\mathrm{aq} ; \mathrm{m}_{\mathrm{j}} ; \mathrm{T} ; \mathrm{p}\right)= \\
&\mathrm{C}_{\mathrm{pl} 1}^{*}(\ell ; \mathrm{T} ; \mathrm{p})+2 \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \,\left(\frac{\partial \phi}{\partial \mathrm{T}}\right)_{\mathrm{p}}+\mathrm{R} \, \mathrm{T}^{2} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \,\left(\frac{\partial^{2} \phi}{\partial \mathrm{T}^{2}}\right)_{\mathrm{p}}
\end{aligned}\]

Where

\[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \mathrm{C}_{\mathrm{p} 1}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mathrm{C}_{\mathrm{pl} 1}^{*}(\ell ; \mathrm{T} ; \mathrm{p})\]

However,

\[\mathrm{C}_{\mathrm{p}}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{C}_{\mathrm{p}_{1}}(\mathrm{aq})+\mathrm{n}_{\mathrm{j}} \, \mathrm{C}_{\mathrm{pj}} \text { (aq) }\]

Hence, from equations (d) and (h),

\[\begin{aligned}
&\mathrm{C}_{\mathrm{p}}\left(\mathrm{aq} ; \mathrm{m}_{\mathrm{j}} ; \mathrm{T} ; \mathrm{p}\right)= \\
&\mathrm{n}_{1} \,\left[\mathrm{C}_{\mathrm{pl}}^{*}(\ell ; \mathrm{T} ; \mathrm{p})+2 \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \,\left(\frac{\partial \phi}{\partial \mathrm{T}}\right)_{\mathrm{p}}\right. \\
&\left.+\mathrm{R} \, \mathrm{T}^{2} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \,\left(\frac{\partial^{2} \phi}{\partial \mathrm{T}^{2}}\right)_{\mathrm{p}}\right] \\
&+\mathrm{n}_{\mathrm{j}} \,\left[\mathrm{C}_{\mathrm{pj}}^{\infty}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})-2 \, \mathrm{R} \, \mathrm{T} \,\left(\frac{\partial \ln \left(\gamma_{\mathrm{j}}\right)}{\partial \mathrm{T}}\right)_{\mathrm{p}}\right. \\
&\left.-\mathrm{R} \, \mathrm{T}^{2} \,\left(\frac{\partial^{2} \ln \left(\gamma_{\mathrm{j}}\right)}{\partial \mathrm{T}^{2}}\right)_{\mathrm{p}}\right]
\end{aligned}\]

We rearrange the latter equation to describe the isobaric heat capacity of a solution prepared using \(1 \mathrm{~kg}\) of water [1].

\[\begin{aligned}
&\mathrm{C}_{\mathrm{p}}\left(\mathrm{aq} ; \mathrm{m}_{\mathrm{j}} ; \mathrm{w}_{1}=1.0 \mathrm{~kg} ; \mathrm{T} ; \mathrm{p}\right)= \\
&\left(1 / \mathrm{M}_{1}\right) \, \mathrm{C}_{\mathrm{pl}}^{*}(\ell ; \mathrm{T} ; \mathrm{p}) \\
&+\mathrm{m}_{\mathrm{j}} \,\left[\mathrm{C}_{\mathrm{pj}}^{\infty}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})-2 \, \mathrm{R} \, \mathrm{T} \,\left(\frac{\partial \ln \left(\gamma_{\mathrm{j}}\right)}{\partial \mathrm{T}}\right)_{\mathrm{p}}-\mathrm{R} \, \mathrm{T}^{2} \,\left(\frac{\partial^{2} \ln \left(\gamma_{\mathrm{j}}\right)}{\partial \mathrm{T}^{2}}\right)_{\mathrm{p}}\right. \\
&\left.\quad+2 \, \mathrm{R} \, \mathrm{T} \,\left(\frac{\partial \phi}{\partial \mathrm{T}}\right)_{\mathrm{p}}+\mathrm{R} \, \mathrm{T}^{2} \,\left(\frac{\partial^{2} \phi}{\partial \mathrm{T}^{2}}\right)_{\mathrm{p}}\right]
\end{aligned}\]

The term inside the [….] brackets is the apparent molar isobaric heat capacity for the solute. Thus,

\[\begin{array}{r}
\phi\left(C_{p j}\right)=C_{p j}^{\infty}(a q ; T ; p)-2 \, R \, T \,\left(\frac{\partial \ln \left(\gamma_{j}\right)}{\partial T}\right)_{p} \\
-R \, T^{2} \,\left(\frac{\partial^{2} \ln \left(\gamma_{j}\right)}{\partial T^{2}}\right)_{p} \\
+2 \, R \, T \,\left(\frac{\partial \phi}{\partial T}\right)_{p}+R \, T^{2} \,\left(\frac{\partial^{2} \phi}{\partial T^{2}}\right)_{p}
\end{array}\]

Hence,

\[\mathrm{C}_{\mathrm{p}}\left(\mathrm{aq} ; \mathrm{m}_{\mathrm{j}} ; \mathrm{w}_{1}=1.0 \mathrm{~kg} ; \mathrm{T} ; \mathrm{p}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{C}_{\mathrm{pl}}^{*}(\ell ; \mathrm{T} ; \mathrm{p})+\mathrm{m}_{\mathrm{j}} \, \phi\left(\mathrm{C}_{\mathrm{pj}}\right)\]

Then,

\[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \phi\left(\mathrm{C}_{\mathrm{pj}}\right)=\phi\left(\mathrm{C}_{\mathrm{pj}}\right)^{\infty}=\mathrm{C}_{\mathrm{pj}}^{\infty}(\mathrm{aq})\]

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 \(j\),

\[\mathrm{J}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{J}_{1}(\mathrm{aq})+\mathrm{n}_{\mathrm{j}} \, \mathrm{J}_{\mathrm{j}}(\mathrm{aq})\]

where

\[J(a q)=C_{p}(a q)-C_{p}(a q ; i d)\]

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

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

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

But

\[\mathrm{C}_{\mathrm{p}}(\mathrm{aq})-\mathrm{C}_{\mathrm{p}}(\mathrm{aq} ; \mathrm{id})=\mathrm{n}_{\mathrm{j}} \,\left[\phi\left(\mathrm{C}_{\mathrm{pj}}\right)-\phi\left(\mathrm{C}_{\mathrm{pj}}\right)^{\infty}\right]\]

Then

\[\mathrm{J}(\mathrm{aq})=\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{J}_{\mathrm{j}}\right)\]

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].

\[\Delta_{s \ln } C_{p j}^{0}=C_{p j}^{\infty}(a q)-C_{p j}^{*}(\ell)\]

Footnotes

[1]

\[\mathrm{C}_{\mathrm{p}}\left(\mathrm{aq} ; \mathrm{m}_{\mathrm{j}} ; \mathrm{w}_{1}=1.0 \mathrm{~kg} ; \mathrm{T} ; \mathrm{p}\right)=\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~kg}^{-1}\right]\]

\[\left(1 / \mathrm{M}_{1}\right) \, \mathrm{C}_{\mathrm{pl}}^{*}(\ell ; \mathrm{T} ; \mathrm{p})=\left[\mathrm{kg} \mathrm{mol}^{-1}\right]^{-1} \,\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right]=\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~kg}^{-1}\right]\]

\[\mathrm{m}_{\mathrm{j}} \, \mathrm{C}_{\mathrm{pj}}^{\infty}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\left[\mathrm{mol} \mathrm{kg}^{-1}\right] \,\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right]=\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~kg}^{-1}\right]\]

\[\mathrm{m}_{\mathrm{j}} \, 2 \, \mathrm{R} \, \mathrm{T} \,\left(\frac{\partial \ln \left(\gamma_{\mathrm{j}}\right)}{\partial \mathrm{T}}\right)_{\mathrm{p}}=\left[\mathrm{mol} \mathrm{kg}^{-1}\right] \,[1] \,\left[\mathrm{J} \mathrm{mol}^{-1} \mathrm{~K}^{-1}\right] \,[\mathrm{K}] \,[\mathrm{K}]^{-1}=\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~kg}^{-1}\right]\]

\[\mathrm{m}_{\mathrm{j}} \, \mathrm{R} \, \mathrm{T}^{2} \,\left(\frac{\partial^{2} \ln \left(\gamma_{\mathrm{j}}\right)}{\partial \mathrm{T}^{2}}\right)_{\mathrm{p}}=\left[\mathrm{mol} \mathrm{kg}^{-1}\right] \,\left[\mathrm{J} \mathrm{mol}^{-1} \mathrm{~K}^{-1}\right] \,[\mathrm{K}]^{2} \,\left[\mathrm{K}^{-2}=\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~kg}^{-1}\right]\right.\]

\[\mathrm{m}_{\mathrm{j}} \, 2 \, \mathrm{R} \, \mathrm{T} \,\left(\frac{\partial \phi}{\partial \mathrm{T}}\right)_{\mathrm{p}}=\left[\mathrm{mol} \mathrm{kg}^{-1}\right] \,[1] \,\left[\mathrm{J} \mathrm{mol}^{-1} \mathrm{~K}^{-1}\right] \,[\mathrm{K}] \,\left[\mathrm{K}^{-1}=\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~kg}^{-1}\right]\right.\]

\[\mathrm{m}_{\mathrm{j}} \, \mathrm{R} \, \mathrm{T}^{2} \,\left(\frac{\partial^{2} \phi}{\partial \mathrm{T}^{2}}\right)_{\mathrm{p}}=\left[\mathrm{mol} \mathrm{kg}{ }^{-1}\right] \,\left[\mathrm{J} \mathrm{mol}^{-1} \mathrm{~K}^{-1}\right] \,[\mathrm{K}]^{2} \,[\mathrm{K}]^{-2}=\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~kg}^{-1}\right]\]

[2] Sucrose(aq); an early determination; F. T. Gucker and F. D. Ayres, J. Am. Chem.Soc.,1937,59,447.

[3] For partial molar isobaric heat capacities of neutral solutes in aqueous solution see,

1. Simple hydrocarbons, G. Olofsson, A. A. Oshodj, E. Qvarnstrom and I. Wadso, J. Chem. Thermodyn., 1984,16,1041.
2. Amides, 2-methyl propan-2-ol, pentanol , R. Skold, J. Suurkuusk and I. Wadso, J. Chem. Thermodyn., 1976,8,1075.
3. Nitrogen and hydrogen, J. Alvarez, R. Crovetto and R. Fernandez-Prini, Ber. Bunsenges. Phys.Chem.,1988,92,935.
4. Amides, ketones, esters and ethers; G. Roux, G. Perron and J. E. Desnoyers, Can. J.Chem.,1978,56,2808.
5. Urea; O. D. Bonner, J. M. Bednarek and R. K. Arisman, J. Am. Chem.Soc.,1977,99,2898.
6. Urea + 2-methylpropan-2-ol(aq); C.de Visser, G.Perron and J.E.Desnoyers, J. Am. Chem. Soc., 1977,99,5894.
7. Alcohols; E. M. Arnett, W. B. Kover and J. V. Carter, J. Am. Chem.Soc.,1969,91,4028.
8. Sugars; O. D. Bonner and P. J. Cerutti, J. Chem. Thermodyn., 1976, 8,105.
9. Amino acids (\(288 \mathrm{~K}\) to \(328 \mathrm{~K}\));
   1. A.W Hakin, M. M. Duke, S. A. Klassen, R. M. McKay and K E. Preuss, Can. J. Chem. 1994,72,362.
   2. M. M. Duke, A.W.Hakin, R.M. McKay and K. M. Preuss, Can. J.Chem.,1994,72,1489.
   3. A. W. Hakin, M. M. Duke, J. L. Marty and K. E. Preuss, J. Chem. Soc. Faraday Trans.,1994,90,14.
   4. A. W. Hakin, M. M. Duke, L. L. Groft, J. L. Marty and M. L.Rushfeldt, Can. J.Chem.,1995,73,725.
   5. A.W.Hakin, C. L. Beswick and M. M. Duke, J. Chem. Soc. Faraday Trans.,1996,92,207.
   6. A. W. Hakin, A. K.Copeland, J. L. Liu, R. A. Marriott and K. E. Preuss, J. Chem. Eng. Data,1997,42,84.
   7. A. W.Hakin and G.R Hedwig, J.Chem. Thermodyn.,2001,33,1709.
10. cyclic dipetides;
    1. A. W. Hakin, B. Cavilla, J. L. Liu and B. Zorzetti, Phys.Chem.Chem.Phys.,2001, 3,3805.
    2. A. W. Hakin, J. L. Liu, M. O’Shea and B. Zorzetti, Phys.Chem.Chem.,Phys.,2003,5,2653.
11. cyclic dipeptides;
    1. C. J. Downes, A. W. Hakin and G. R. Hedwig, J.Chem.Thermodyn.,2001,33,873.
    2. A.W. Hakin, M.G.Kowalchuck, J. L. Liu and R.A. Marriot, J. Solution Chem.2000, 29,131.
12. acetylamides;
    1. J. L. Liu, A. W. Hakin and G. R. Hedwig, J. Solution Chem.,2001,30, 861.
    2. A. W. Hakin and G. R. Hedwig, Phys. Chem. Chem. Phys.,2002,2,1795.
13. alkanolamines; Y. Maham, L. G. Helper, A. E. Mather, A.W. Hakin, and R. A. Marriott, J. Chem. Soc. Faraday Trans.,1997,93,1747.

[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.