1.8.6: Enthalpies- Neutral Solutes
A given aqueous solution at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\) (close to ambient pressure \(\mathrm{p}^{0}\)) is prepared using \(\mathrm{n}_{1}\) moles of water and \(\mathrm{n}_{j}\) moles of a solute, chemical substance-\(j\). The enthalpy of this solution \(\mathrm{H}(\mathrm{aq})\) is given by equation (a) where \(\mathrm{H}_{1}(\mathrm{aq})\) and \(\mathrm{H}_{j}(\mathrm{aq})\) are the (equilibrium) partial molar enthalpies of solvent and solute respectively
\[\mathrm{H}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{H}_{1}(\mathrm{aq})+\mathrm{n}_{\mathrm{j}} \, \mathrm{H}_{\mathrm{j}}(\mathrm{aq}) \nonumber \]
\[\mathrm{H}_{1}(\mathrm{aq})=\left(\frac{\partial \mathrm{H}}{\partial \mathrm{n}_{1}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}(\mathrm{j})} \nonumber \]
\[\mathrm{H}_{\mathrm{j}}(\mathrm{aq})=\left(\frac{\partial \mathrm{H}}{\partial \mathrm{n}_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}(1)} \nonumber \]
Then [1,2],
\[\mathrm{H}_{1}(\mathrm{aq})=\mathrm{H}_{1}^{*}(\lambda)+\mathrm{R} \, \mathrm{T}^{2} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \,\left(\frac{\partial \phi}{\partial \mathrm{T}}\right)_{\mathrm{p}} \nonumber \]
\[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \mathrm{H}_{1}(\mathrm{aq})=\mathrm{H}_{1}^{*}(\lambda) \nonumber \]
Similarly for the solute, chemical substance \(j\) (assuming ambient pressure \(\mathrm{p}\) is close to the standard pressure) [3,4],
\[\mathrm{H}_{\mathrm{j}}(\mathrm{aq})=\mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq})-\mathrm{R} \, \mathrm{T}^{2} \,\left(\frac{\partial \ln \left(\gamma_{\mathrm{j}}\right)}{\partial \mathrm{T}}\right)_{\mathrm{p}} \nonumber \]
\(\mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq})\) is the limiting (infinite dilution) partial molar enthalpy of solute \(j\). The enthalpy of a solution prepared using \(\mathrm{n}_{1}\) moles of water and \(\mathrm{n}_{j}\) moles of solute is given by equation (g).
\[\begin{aligned}
\mathrm{H}(\mathrm{aq})=\mathrm{n}_{1} \,\left[\mathrm{H}_{1}^{*}(\lambda)+\mathrm{R} \, \mathrm{T}^{2} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \,(\partial \phi / \partial \mathrm{T})_{\mathrm{p}}\right] \\
&\left.+\mathrm{n}_{\mathrm{j}} \, \mid \mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq})-\mathrm{R} \, \mathrm{T}^{2} \,\left(\partial \ln \gamma_{\mathrm{j}} / \partial \mathrm{T}\right)_{\mathrm{p}}\right]
\end{aligned} \nonumber \]
For a solution in \(1 \mathrm{~kg}\) of water,
\[\begin{aligned}
\mathrm{H}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=&\left(1 / \mathrm{M}_{1}\right) \,\left[\mathrm{H}_{1}^{*}(\lambda)+\mathrm{R} \, \mathrm{T}^{2} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \,(\partial \phi / \partial \mathrm{T})_{\mathrm{p}}\right] \\
&+\mathrm{m}_{\mathrm{j}} \,\left[\mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq})-\mathrm{R} \, \mathrm{T}^{2} \,\left(\partial \ln \gamma_{\mathrm{j}} / \partial \mathrm{T}\right)_{\mathrm{p}}\right]
\end{aligned} \nonumber \]
We re-arrange equation (h).
\[\begin{aligned}
\mathrm{H}\left(\mathrm{aq} ; \mathrm{w}_{1}=\right.&1 \mathrm{~kg})=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{H}_{1}^{*}(\lambda) \\
&+\mathrm{m}_{\mathrm{j}} \,\left[\mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq})+\mathrm{R} \, \mathrm{T}^{2} \,(\partial \phi / \partial \mathrm{T})_{\mathrm{p}}-\mathrm{R} \, \mathrm{T}^{2} \,\left(\partial \ln \gamma_{\mathrm{j}} / \partial \mathrm{T}\right)_{\mathrm{p}}\right]
\end{aligned} \nonumber \]
Equation (i) is interesting because inside the brackets [….] we have the limiting partial molar enthalpy of the solute and two terms which describe the extent to which the enthalpic properties of the solution differ from those of the corresponding ideal solution. We find it advantageous to describe the property in the brackets […] as the apparent molar enthalpy of the solution, \(\phi\left(\mathrm{H}_{\mathrm{j}}\right)\). By definition,
\[\phi\left(\mathrm{H}_{\mathrm{j}}\right)=\mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq})-\mathrm{R} \, \mathrm{T}^{2} \,\left(\mathrm{d} \ln \gamma_{\mathrm{j}} / \mathrm{dT}\right)_{\mathrm{p}}+\mathrm{R} \, \mathrm{T}^{2} \,(\mathrm{d} \phi / \mathrm{dT})_{\mathrm{p}} \nonumber \]
For a solution prepared using \(1 \mathrm{~kg}\) of solvent water.
\[\mathrm{H}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{H}_{1}^{*}(\lambda)+\mathrm{m}_{\mathrm{j}} \, \phi\left(\mathrm{H}_{\mathrm{j}}\right) \nonumber \]
But at all \(\mathrm{T}\) and \(\mathrm{p}\),
\[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \gamma_{\mathrm{j}}=1.0 ; \ln \left(\gamma_{\mathrm{j}}\right)=0 ; \phi=1.0 \nonumber \]
Hence,
\[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right)\left[\partial \ln \left(\gamma_{\mathrm{j}}\right) / \partial \mathrm{T}\right]_{\mathrm{p}}=[\partial \phi / \partial \mathrm{T}]_{\mathrm{p}}=0 \nonumber \]
\[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \phi\left(\mathrm{H}_{\mathrm{j}}\right)=\phi\left(\mathrm{H}_{\mathrm{j}}\right)^{\infty}=\mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq}) \nonumber \]
We recognize a crucial complication in the treatment of the enthalpies of solutions. Unlike volumetric properties of solutions, we cannot measure the enthalpy of a solution. In other words we need to examine differences. Based on equation (k) we form an equation for the enthalpy of the corresponding solution having thermodynamic properties which are ideal.
\[\mathrm{H}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg} ; \mathrm{id}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{H}_{1}^{*}(\lambda)+\mathrm{m}_{\mathrm{j}} \, \phi\left(\mathrm{H}_{\mathrm{j}}\right)^{\infty} \nonumber \]
The difference between the two enthalpies is given by equation (p)
\[\mathrm{H}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)-\mathrm{H}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg} ; \mathrm{id}\right)=\mathrm{m}_{\mathrm{j}} \,\left[\phi\left(\mathrm{H}_{\mathrm{j}}\right)-\phi\left(\mathrm{H}_{\mathrm{j}}\right)^{\infty}\right] \nonumber \]
Interesting descriptions of the enthalpies of solutions containing simple solutes are based on the concept of excess thermodynamic properties and pairwise solute-solute interaction parameters. Equation (k) describes the enthalpy of a solution prepared using \(1 \mathrm{~kg}\) of water whereas equation (o) describes the enthalpy of the corresponding solution where the thermodynamic properties are ideal. The excess enthalpy \(\mathrm{H}^{\mathrm{E}}\left(\mathrm{aq} ; \mathrm{m}_{\mathrm{j}}\right)\) is given by equation (q).
\[\mathrm{H}^{\mathrm{E}}\left(\mathrm{aq} ; \mathrm{m}_{\mathrm{j}}\right)=\mathrm{h}_{\mathrm{ij}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2}+\mathrm{h}_{\mathrm{iji}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{3} \ldots \ldots \nonumber \]
Footnotes
[1] For the solvent in solutions ( at constant pressure),
\[\mu_{1}(\mathrm{aq})=\mu_{1}^{*}(\lambda)-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \nonumber \]
But
\[-\frac{\mathrm{H}_{1}(\mathrm{aq})}{\mathrm{T}^{2}}=\frac{\partial\left[\mu_{1}(\mathrm{aq}) / \mathrm{T}\right]}{\partial T} \nonumber \]
Then \(-\frac{\mathrm{H}_{1}(\mathrm{aq})}{\mathrm{T}^{2}}=-\frac{\mathrm{H}_{1}^{*}(\lambda)}{\mathrm{T}^{2}}-\mathrm{R} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \,\left(\frac{\partial \phi}{\partial \mathrm{T}}\right)_{\mathrm{p}}\)
[2] \(\mathrm{R} \, \mathrm{T}^{2} \,\left(\frac{\partial \phi}{\partial \mathrm{T}}\right)_{\mathrm{p}}=\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right] \,[\mathrm{K}]^{2} \,[\mathrm{K}]^{-1} =\left[\mathrm{J} \mathrm{mol}^{-1}\right]\)
[3] From \(\mu_{j}(\mathrm{aq})=\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)\) Then, \(-\frac{\mathrm{H}_{\mathrm{j}}(\mathrm{aq})}{\mathrm{T}^{2}}=-\frac{\mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq})}{\mathrm{T}^{2}}+\mathrm{R} \,\left(\frac{\partial \ln \left(\gamma_{\mathrm{j}}\right)}{\partial \mathrm{T}}\right)_{\mathrm{p}}\)
[4] \(\mathrm{R} \, \mathrm{T}^{2} \,\left(\frac{\partial \ln \left(\gamma_{\mathrm{j}}\right)}{\partial \mathrm{T}}\right)_{\mathrm{p}}=\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right] \,[\mathrm{K}]^{2} \,[\mathrm{K}]^{-1}=\left[\mathrm{J} \mathrm{mol}{ }^{-1}\right]\)
[5] See for example,
- amides(aq) and peptides(aq); A. H. Sijpkes, A. A. C. Oudhuis, G. Somsen and T. H. Lilley, J. Chem. Thermodyn.,1989, 21 ,343.
- non-electrolytes in DMSO and H2O; E. M. Arnett and D. R. McKelvey, J. Am. Chem.Soc.,1966, 88 ,2598.
- alkanes(aq); S. Cabani, G. Conti, V. Mollica and L. Bernazzani, J. Chem. Soc. Faraday Trans.,1991, 87 ,2433.
- hydrocarbons in polar solvents; C. V. Krishnan and H. L. Friedman, J. Phys. Chem.,1971, 75 ,3598.
- alkanes in organic solvents; R. Fuchs and W. K. Stephenson, Can. J.Chem.,1985, 63 ,349.
- organic solutes in alkanes and water. W. Riebesehl, E. Tomlinson and H. J. M. Grumbauer, J.Phys.Chem.,1984, 88 ,4775.