1.8.9: Enthalpies- Salt Solutions- Apparent Molar- Partial Molar and Relative Enthalpies
Description of the enthalpies of salt solutions is similar to that given for neutral solutes except that account is taken of the fact that one mole of a given salt can with complete dissociation produce \(v\) moles of ions. The chemical potential of the solvent in an aqueous salt solution (at constant temperature and ambient pressure) is given by equation (a).
\[\mu_{1}(\mathrm{aq})=\mu_{1}^{\star}(\lambda)-\mathrm{v} \, \phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \nonumber \]
Here \(\phi\) is the practical osmotic coefficient where \(\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \phi=1.0\) at all \(\mathrm{T}\) and \(\mathrm{p}\). Using the Gibbs-Helmholtz Equation,
\[\mathrm{H}_{1}(\mathrm{aq})=\mathrm{H}_{1}^{*}(\lambda)+\mathrm{v} \, \mathrm{R} \, \mathrm{T}^{2} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \,(\partial \phi / \partial \mathrm{T})_{\mathrm{p}} \nonumber \]
Also
\[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \mathrm{H}_{1}(\mathrm{aq})=\mathrm{H}_{1}^{*}(\lambda) \nonumber \]
By definition,
\[\mathrm{L}_{1}(\mathrm{aq})=\mathrm{H}_{1}(\mathrm{aq})-\mathrm{H}_{1}^{*}(\lambda) \nonumber \]
The chemical potential of a salt \(j\) in aqueous solution is given by equation (e).
\[\mu_{j}(a q)=\mu_{j}^{0}(a q)+v \, R \, T \, \ln \left(Q \, m_{j} \, \gamma_{\pm} / m^{0}\right) \nonumber \]
where, at all \(\mathrm{T}\) and \(\mathrm{p}\),
\[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \gamma_{\pm}=1.0 \nonumber \]
Using the Gibbs-Helmholtz Equation,
\[\mathrm{H}_{\mathrm{j}}(\mathrm{aq})=\mathrm{H}_{\mathrm{j}}^{0}(\mathrm{aq})-\mathrm{v} \, \mathrm{R} \, \mathrm{T}^{2} \,\left[\partial \ln \left(\gamma_{\pm}\right) / \partial \mathrm{T}\right]_{\mathrm{p}} \nonumber \]
For a salt solution having ideal thermodynamic properties,
\[\mathrm{H}_{\mathrm{j}}(\mathrm{aq} ; \mathrm{id})=\mathrm{H}_{\mathrm{j}}^{0}(\mathrm{aq})=\mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq}) \nonumber \]
By definition, the relative partial molar enthalpy of the salt,
\[\mathrm{L}_{\mathrm{j}}(\mathrm{aq})=\mathrm{H}_{\mathrm{j}}(\mathrm{aq})-\mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq}) \nonumber \]
In the limit of infinite dilution the relative partial molar enthalpy of a salt is zero. Thus
\[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \mathrm{L}_{\mathrm{j}}(\mathrm{aq})=0 \nonumber \]
For a solution prepared using \(\mathrm{w}_{1} \mathrm{~kg}\) of water(\(\lambda\)),
\[\begin{aligned}
\mathrm{H}\left(\mathrm{aq} ; \mathrm{w}_{1} \mathrm{~kg}\right) &=\mathrm{n}_{1} \, \mathrm{H}_{1}^{*}(\lambda)+\mathrm{n}_{1} \, \mathrm{v} \, \mathrm{R} \, \mathrm{T}^{2} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \,(\partial \phi / \partial \mathrm{T})_{\mathrm{p}} \\
+\mathrm{n}_{\mathrm{j}} \, \mathrm{H}_{\mathrm{j}}^{0}(\mathrm{aq})-\mathrm{n}_{\mathrm{j}} \, \mathrm{v} \, \mathrm{R} \, \mathrm{T}^{2} \,\left[\partial \ln \left(\gamma_{\pm}\right) / \partial \mathrm{T}\right]_{\mathrm{p}}
\end{aligned} \nonumber \]
But
\[\mathrm{n}_{1} \, \mathrm{m}_{\mathrm{j}}=\mathrm{n}_{1} \, \mathrm{n}_{\mathrm{j}} / \mathrm{w}_{1}=\mathrm{n}_{\mathrm{j}} / \mathrm{M}_{1} \nonumber \]
\[\begin{aligned}
\mathrm{H}\left(\mathrm{aq} ; \mathrm{w}_{1} \mathrm{~kg}\right)=\mathrm{n}_{1} \, \mathrm{H}_{1}^{*}(\lambda)+\mathrm{n}_{\mathrm{j}} \, \mathrm{v} \, \mathrm{R} \, \mathrm{T}^{2} \,(\partial \phi / \partial \mathrm{T})_{\mathrm{p}} \\
&+\mathrm{n}_{\mathrm{j}} \, \mathrm{H}_{\mathrm{j}}^{0}(\mathrm{aq})-\mathrm{n}_{\mathrm{j}} \, \mathrm{v} \, \mathrm{R} \, \mathrm{T}^{2} \,\left[\partial \ln \left(\gamma_{\pm}\right) / \partial \mathrm{T}\right]_{\mathrm{p}}
\end{aligned} \nonumber \]
\[\begin{aligned}
\mathrm{H}\left(\mathrm{aq} ; \mathrm{w}_{1} \mathrm{~kg}\right)=& \mathrm{n}_{1} \, \mathrm{H}_{1}^{*}(\lambda)+\mathrm{n}_{\mathrm{j}} \,\left\{\mathrm{v} \, \mathrm{R} \, \mathrm{T}^{2} \,(\partial \phi / \partial \mathrm{T})_{\mathrm{p}}\right.\\
&\left.+\mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq})-\mathrm{v} \, \mathrm{R} \, \mathrm{T}^{2} \,\left[\partial \ln \left(\gamma_{\pm}\right) / \partial \mathrm{T}\right]_{\mathrm{p}}\right\}
\end{aligned} \nonumber \]
The term in the brackets {….} defines the apparent molar enthalpy of salt \(j\), \(\phi\left(\mathrm{H}_{\mathrm{j}}\right)\).
\[\phi\left(\mathrm{H}_{\mathrm{j}}\right)=\mathrm{v} \, \mathrm{R} \, \mathrm{T}^{2} \,(\partial \phi / \partial \mathrm{T})_{\mathrm{p}}+\mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq})-\mathrm{v} \, \mathrm{R} \, \mathrm{T}^{2} \,\left[\partial \ln \left(\gamma_{\pm}\right) / \partial \mathrm{T}\right]_{\mathrm{p}} \nonumber \]
Using equation (o),
\[\mathrm{H}\left(\mathrm{aq} ; \mathrm{w}_{1} \mathrm{~kg}\right)=\mathrm{n}_{1} \, \mathrm{H}_{1}^{*}(\lambda)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{H}_{\mathrm{j}}\right) \nonumber \]
In other words we have grouped all the parameters describing the properties of the salt in a real solution under a single term, \(\phi(\mathrm{H}_{j})\). For a solution prepared using \(1 \mathrm{~kg}\) of water,
\[\mathrm{H}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{H}_{1}^{*}(\lambda)+\mathrm{m}_{\mathrm{j}} \, \phi\left(\mathrm{H}_{\mathrm{j}}\right) \nonumber \]
At all \(\mathrm{T}\) and \(\mathrm{p}\),
\[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \gamma_{\pm}=1 ; \ln \left(\gamma_{\pm}\right)=0 ; \phi=1 \nonumber \]
Hence,
\[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right)\left[\partial \ln \left(\gamma_{\pm}\right) / \partial \mathrm{T}\right]_{\mathrm{p}}=[\partial \phi / \partial \mathrm{T}]_{\mathrm{p}}=0 \nonumber \]
\[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \phi\left(\mathrm{H}_{\mathrm{j}}\right)=\mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq}) \nonumber \]