1.8.4: Enthalpy- Solutions- Partial Molar Enthalpies
The enthalpy of a solution containing n1 moles of water and nj moles of solute, chemical substance j, is defined by the independent variables, \(\mathrm{T}\), \(\mathrm{p}\), \(\mathrm{n}_{1}\) and \(\mathrm{n}_{j}\).
\[\mathrm{H}=\mathrm{H}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{\mathrm{j}}\right] \nonumber \]
where [1],
\[\mathrm{H}=\mathrm{n}_{1} \, \mathrm{H}_{1}(\mathrm{aq})+\mathrm{n}_{\mathrm{j}} \, \mathrm{H}_{\mathrm{j}}(\mathrm{aq}) \nonumber \]
Here \(\mathrm{H}_{1}(\mathrm{aq})\) and \(\mathrm{H}_{j}(\mathrm{aq})\) are the partial molar enthalpies of water and solute \(j\) in the solution.
\[\mathrm{H}_{1}(\mathrm{aq})=\left(\partial \mathrm{H} / \partial \mathrm{n}_{1}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}(\mathrm{j})} \nonumber \]
\[\mathrm{H}_{\mathrm{j}}(\mathrm{aq})=\left(\partial \mathrm{H} / \partial \mathrm{n}_{\mathrm{j}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}(1)} \nonumber \]
For a solution prepared using \(1 \mathrm{~kg}\) of solvent, water and \(\mathrm{m}_{j}\) moles of solute \(j\) [2],
\[\mathrm{H}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{H}_{1}(\mathrm{aq})+\mathrm{m}_{\mathrm{j}} \, \mathrm{H}_{\mathrm{j}}(\mathrm{aq}) \nonumber \]
The chemical potential of the solvent in an aqueous solution is related to the molality of solute \(j\), \(\mathrm{m}_{j}\) using equation (f) where \(\phi\) is the practical osmotic coefficient, a property of the solvent.
\[\mu_{1}(\mathrm{aq})=\mu_{1}^{*}(\lambda)-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \nonumber \]
The chemical potential and partial molar enthalpy are linked using the Gibbs-Helmholtz equation such that at fixed pressure, \(\mathrm{d}\left(\mu_{1}(\mathrm{aq}) / \mathrm{T}\right) / \mathrm{dT}=-\mathrm{H}_{1}(\mathrm{aq}) / \mathrm{T}^{2}\). Hence [3]
\[\mathrm{H}_{1}(\mathrm{aq})=\mathrm{H}_{1}^{*}(\lambda)+\mathrm{R} \, \mathrm{T}^{2} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \,(\mathrm{d} \phi / \mathrm{dT})_{\mathrm{p}} \nonumber \]
By definition the practical osmotic coefficient is unity for ideal solutions at all \(\mathrm{T}\) and \(\mathrm{p}\). Then the partial molar enthalpy of the solvent in an ideal solution,
\[\mathrm{H}_{1}(\mathrm{aq}, \mathrm{id})=\mathrm{H}_{1}^{*}(\lambda) \nonumber \]
The definition of \(\phi\) requires that \(\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \mathrm{H}_{1}(\mathrm{aq})\) equals \(\mathrm{H}_{1}^{*}(\lambda)\). We express the difference between the partial molar enthalpies of the solvent in real and ideal solutions using a relative (partial) molar enthalpy, \(\mathrm{L}_{1}(\mathrm{aq})\).
\[\mathrm{L}_{1}(\mathrm{aq})=\mathrm{H}_{1}(\mathrm{aq})-\mathrm{H}_{1}^{*}(\lambda) \nonumber \]
In equation (i), we encounter another difference in order to take account of the fact that we cannot measure absolute enthalpies of solutions and solvents.
The chemical potential of the solute \(j\) (at fixed \(\mathrm{T}\) and \(\mathrm{p}\), which is close to ambient pressure) is related to the molality \(\mathrm{m}_{j}\) using equation (j).
\[\mu_{\mathrm{j}}(\mathrm{aq})=\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right) \nonumber \]
From the Gibbs-Helmholtz Equation,
\[\mathrm{H}_{\mathrm{j}}(\mathrm{aq})=\mathrm{H}_{\mathrm{j}}^{0}(\mathrm{aq})-\mathrm{R} \, \mathrm{T}^{2} \,\left(\mathrm{d} \ln \gamma_{\mathrm{j}} / \mathrm{dT}\right)_{\mathrm{p}} \nonumber \]
But activity coefficient \(\gamma_{j}\) is defined such that \(\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \gamma_{\mathrm{j}}=1.0\) at all \(\mathrm{T}\) and \(\mathrm{p}\). Moreover for an ideal solution, \(\gamma_{j} = 1.0\). Hence,
\[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \mathrm{H}_{\mathrm{j}}(\mathrm{aq})=\mathrm{H}_{\mathrm{j}}^{0}(\mathrm{aq})=\mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq}) \nonumber \]
In other words, with increasing dilution \(\mathrm{H}_{j}(\mathrm{aq})\) approaches a limiting partial molar enthalpy \(\mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq})\) which equals the partial molar enthalpy of the solute in an ideal solution. We identify a relative (partial) molar enthalpy of solute \(j\), \(\mathrm{L}_{\mathrm{j}}(\mathrm{aq})\).
\[\mathrm{L}_{\mathrm{j}}(\mathrm{aq})=\mathrm{H}_{\mathrm{j}}(\mathrm{aq})-\mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq}) \nonumber \]
Hence, at fixed \(\mathrm{T}\) and \(\mathrm{p}\)
\[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \mathrm{L}_{\mathrm{j}}(\mathrm{aq})=0 \nonumber \]
Therefore for simple solutes in solution in the limit of infinite dilution the relative partial molar enthalpy of solute \(j\) is zero [4].
Footnotes
[1] \([\mathrm{J}]=[\mathrm{mol}] \,\left[\mathrm{J} \mathrm{mol}^{-1}\right]+[\mathrm{mol}] \,\left[\mathrm{J} \mathrm{mol}^{-1}\right]\)
[2] \(\left[\mathrm{J} \mathrm{kg}^{-1}\right]=\left[\mathrm{kg} \mathrm{mol}^{-1}\right]^{-1} \,\left[\mathrm{J} \mathrm{mol}^{-1}\right]+\left[\mathrm{mol} \mathrm{kg}^{-1}\right] \,\left[\mathrm{J} \mathrm{mol}^{-1}\right]\)
[3] Note the advantage of expressing the composition in terms of molalities rather than in concentrations for which we would have to take account of the dependence of volume on temperature.
[4] An interesting comparison is the molar enthalpy of water(\(\lambda\)) and the limiting molar enthalpy of solute water in a solvent such as methanol. We define a transfer quantity, \(\Delta_{\mathrm{tr}} \mathrm{H}^{0}\) [ \(= \mathrm{H}^{\infty}\) (\(\mathrm{H}_{2}\mathrm{O}\) as solute in a defined solvent) \(-\mathrm{H}_{1}^{*}\left(\lambda \mathrm{H}_{2} \mathrm{O}\right)\)], characterizing the difference in molar enthalpy of liquid water and the limiting partial molar enthalpy of solute water at ambient pressure and \(298.15 \mathrm{~K}\)_. \(\Delta_{\mathrm{tr}} \mathrm{H}^{0}\) is \(0.85\), \(4.05\) and \(10.11 \mathrm{~kJ mol}^{-1}\) in \(\mathrm{CH}_{3}\mathrm{OH}(\lambda)\), \(\mathrm{C}_{7}\mathrm{H}_{15}\mathrm{OH}(\lambda)\) and \(\mathrm{C}_{2}\mathrm{H}_{4}(\mathrm{O.CO.C}_{3}\mathrm{H}_{7})_{2} (\lambda)\) respectively [5].
[5] S.-O. Nilsson, J. Chem. Thermodyn., 1986, 18 , 1115.