1.5.19: Chemical Potentials- Solutions- Salt Hydrates in Aqueous Solution
An aqueous solution is prepared using \(\mathrm{n}_{j}\) moles of salt \(\mathrm{MX}\) and \(\mathrm{n}_{1}\) moles of water. The properties of the system are accounted for using one of two possible Descriptions.
Description I
The solute \(j\) comprises a 1:1 salt MX molality \(\mathrm{m}(\mathrm{MX})\left[=\mathrm{n}_{\mathrm{j}} / \mathrm{w}_{1}\right. \text { where } \mathrm{w}_{1} \text { is the mass of water} \right]\).
The single ion chemical potentials, are defined in the following manner
\[\begin{aligned}
&\mu\left(\mathrm{M}^{+}\right)=\left[\partial \mathrm{G} / \partial \mathrm{n}\left(\mathrm{M}^{+}\right)\right]_{\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}\left(\mathrm{x}^{-}\right)} \\
&\mu\left(\mathrm{X}^{-}\right)=\left[\partial \mathrm{G} / \partial \mathrm{n}\left(\mathrm{X}^{-}\right)\right]_{\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}\left(\mathrm{M}^{+}\right)}
\end{aligned} \nonumber \]
The total Gibbs energy (at fixed \(\mathrm{T}\) and \(\mathrm{p}\) where \(p \approx p^{0}\)) is given by equation (b).
\[\begin{aligned}
&\mathrm{G}(\mathrm{aq} ; \mathrm{I})=\mathrm{n}_{1} \, \mu_{1}^{\mathrm{eq}}(\mathrm{aq}) \\
&\quad+\mathrm{n}_{\mathrm{j}} \,\left\{\mu^{0}\left(\mathrm{M}^{+} ; \mathrm{aq}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{m}\left(\mathrm{M}^{+}\right) \, \gamma_{+}(\mathrm{I}) / \mathrm{m}^{0}\right]\right\} \\
&\quad+\mathrm{n}_{\mathrm{j}} \,\left\{\mu^{0}\left(\mathrm{X}^{-} ; \mathrm{aq}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{m}\left(\mathrm{X}^{-}\right) \, \gamma_{-}(\mathrm{I}) / \mathrm{m}^{0}\right]\right\}
\end{aligned} \nonumber \]
Description II
According to this Description each mole of cations is hydrated by \(\mathrm{h}_{\mathrm{m}\left(\mathrm{H}_{2}\mathrm{O}\right)\) moles of water and each mole of anions is hydrated by \(\mathrm{h}_{\mathrm{x}\left(\mathrm{H}_{2}\mathrm{O}\right)\) moles of water.
The single ion chemical potentials are defined as follows.
\[\mu\left(\mathrm{M}^{+} \, \mathrm{h}_{\mathrm{m}} \mathrm{H}_{2} \mathrm{O}\right)=\left[\partial \mathrm{G} / \partial\left(\mathrm{M}^{+} \, \mathrm{h}_{\mathrm{m}} \mathrm{H}_{2} \mathrm{O}\right)\right] \nonumber \]
at constant \(\mathrm{T}\), \(\mathrm{p}\), \(\mathrm{n}\left(\mathrm{X}^{-} \, \mathrm{h}_{\mathrm{x}} \mathrm{H}_{2} \mathrm{O}\right),\left[\mathrm{n}_{1}-\mathrm{n}_{\mathrm{j}} \,\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{x}}\right)\right]\left(\mathrm{H}_{2} \mathrm{O}\right) \mu\left(\mathrm{X}^{-} \, \mathrm{h}_{\mathrm{x}} \mathrm{H}_{2} \mathrm{O}\right)=\left[\partial \mathrm{G} / \partial \mathrm{n}\left(\mathrm{X}^{-} \, \mathrm{h}_{\mathrm{x}} \mathrm{H}_{2} \mathrm{O}\right)\right]\)
\[\text { at constant } \mathrm{T}, \mathrm{p}, \mathrm{n}\left(\mathrm{M}^{+} \, \mathrm{h}_{\mathrm{m}} \mathrm{H}_{2} \mathrm{O}\right),\left[\mathrm{n}_{1}-\mathrm{n}_{\mathrm{j}} \,\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{x}}\right)\right]\left(\mathrm{H}_{2} \mathrm{O}\right) \nonumber \]
\[\mathrm{m}\left(\mathrm{X}^{-} \, \mathrm{h}_{\mathrm{x}} \mathrm{H}_{2} \mathrm{O}\right)=\mathrm{n}_{\mathrm{j}} / \mathrm{M}_{1} \,\left[\mathrm{n}_{1}-\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{x}}\right) \, \mathrm{n}_{\mathrm{j}}\right] ; \nonumber \]
\[\mathrm{m}\left(\mathrm{M}^{+} \, \mathrm{h}_{\mathrm{m}} \mathrm{H}_{2} \mathrm{O}\right)=\mathrm{n}_{\mathrm{j}} / \mathrm{M}_{1} \,\left[\mathrm{n}_{1}-\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{x}}\right) \, \mathrm{n}_{\mathrm{j}}\right] . \nonumber \]
The (equilibrium) Gibbs energy (at defined \(\mathrm{T}\) and \(\mathrm{p}\)) is given by the following equation.
\[\begin{aligned}
&\mathrm{G}(\mathrm{aq} ; \mathrm{II})=\left[\mathrm{n}_{1}-\mathrm{n}_{\mathrm{j}} \,\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{x}}\right)\right] \, \mu_{1}(\mathrm{aq}) \\
&+\mathrm{n}_{\mathrm{j}} \,\left[\mu^{0} \,\left(\mathrm{M}^{+} \, \mathrm{h}_{\mathrm{m}} \mathrm{H}_{2} \mathrm{O}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left\{\mathrm{m}\left(\mathrm{M}^{+} \, \mathrm{h}_{\mathrm{m}} \mathrm{H}_{2} \mathrm{O}\right) \, \gamma_{+}(\mathrm{II}) / \mathrm{m}^{0}\right\}\right] \\
&+\mathrm{n}_{\mathrm{j}} \,\left[\mu^{0}\left(\mathrm{X}^{-} \, \mathrm{h}_{\mathrm{x}} \mathrm{H}_{2} \mathrm{O}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left\{\mathrm{m}\left(\mathrm{X}^{-} \, \mathrm{h}_{\mathrm{x}} \mathrm{H}_{2} \mathrm{O}\right) \, \gamma_{-}(\mathrm{II}) / \mathrm{m}^{0}\right\}\right]
\end{aligned} \nonumber \]
But the Gibbs energies defined by equations ( b) and (g) are identical (at equilibrium at defined \(\mathrm{T}\) and \(\mathrm{p}\)). After all, it is the same solution. Hence, (dividing by \(\mathrm{n}_{j}\))
\[\begin{aligned}
&{\left[\mu^{0}\left(\mathrm{M}^{+} ; \mathrm{aq}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left\{\mathrm{m}\left(\mathrm{M}^{+} ; \mathrm{I}\right) \, \gamma_{+}(\mathrm{I}) / \mathrm{m}^{0}\right\}\right]} \\
&\quad+\left[\mu^{0}\left(\mathrm{X}^{-} ; \mathrm{aq}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left\{\mathrm{m}\left(\mathrm{X}^{-} ; \mathrm{I}\right) \, \gamma_{-}(\mathrm{I}) / \mathrm{m}^{0}\right\}\right]= \\
&\quad-\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{x}}\right) \, \mu_{1}^{\mathrm{eq}}(\mathrm{aq})+ \\
&\quad\left[\mu^{0}\left(\mathrm{M}^{+} \, \mathrm{h}_{\mathrm{m}} \mathrm{H}_{2} \mathrm{O} ; \mathrm{aq}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left\{\mathrm{m}\left(\mathrm{M}^{+} \, \mathrm{h}_{\mathrm{m}} \mathrm{H}_{2} \mathrm{O}\right) \, \gamma_{+}(\mathrm{II}) / \mathrm{m}^{0}\right\}\right] \\
&+\left[\mu^{0}\left(\mathrm{X}^{-} \, \mathrm{h}_{\mathrm{x}} \mathrm{H}_{2} \mathrm{O} ; \mathrm{aq}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left\{\mathrm{m}\left(\mathrm{X}^{-} \, \mathrm{h}_{\mathrm{x}} \mathrm{H}_{2} \mathrm{O}\right) \, \gamma_{-}(\mathrm{II}) / \mathrm{m}^{0}\right\}\right] \\
&\text { Then, } \mu^{0}\left(\mathrm{M}^{+} ; \mathrm{aq}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left\{\mathrm{m}\left(\mathrm{M}^{+} ; \mathrm{I}\right) \, \gamma_{+}(\mathrm{I}) / \mathrm{m}^{0}\right\} \\
&+\mu^{0}\left(\mathrm{X}^{-} ; \mathrm{aq}^{0}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left\{\mathrm{m}(\mathrm{X} \, ; \mathrm{I}) \, \gamma_{-}(\mathrm{I}) / \mathrm{m}^{0}\right\} \\
&=-\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{x}}\right) \,\left\{\mu_{1}^{*}(\ell)-2 \, \phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\right\} \\
&+\left\{\mu^{0}\left(\mathrm{M}^{+} \, \mathrm{h}_{\mathrm{m}} \mathrm{H}_{2} \mathrm{O} ; \mathrm{aq}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left\{\mathrm{m}\left(\mathrm{M}^{+} \mathrm{h}_{\mathrm{m}} \mathrm{H}_{2} \mathrm{O} ; \mathrm{II}\right) \, \gamma_{+}(\mathrm{II}) / \mathrm{m}^{0}\right\}\right] \\
&+\left[\mu^{0}\left(\mathrm{X}^{-} \, \mathrm{h}_{\mathrm{x}} \mathrm{H}_{2} \mathrm{O} ; \mathrm{aq}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left\{\mathrm{m}\left(\mathrm{X}^{-} \mathrm{h}_{\mathrm{X}} \mathrm{H}_{2} \mathrm{O} ; \mathrm{II}\right) \, \gamma_{-}(\mathrm{II}) / \mathrm{m}^{0}\right\}\right]
\end{aligned} \nonumber \]
We use the latter equation to explore what happens in the limit that \(\mathrm{n}_{j}\) approaches zero. Thus,
\[\begin{aligned}
&\operatorname{limit}\left(\mathrm{n}_{\mathrm{j}} \rightarrow 0\right) \gamma_{+}(\mathrm{I})=1 \quad \gamma_{-}(\mathrm{I})=1 \\
&\gamma_{+}(\mathrm{II})=1 \quad \gamma_{-}(\mathrm{II})=1 \\
&\mathrm{~m}_{\mathrm{j}}=0 \\
&\mathrm{~m}\left(\mathrm{M}^{+} \mathrm{h}_{\mathrm{m}} \mathrm{H}_{2} \mathrm{O} ; \mathrm{II}\right) \, \mathrm{m}\left(\mathrm{X}^{-} \mathrm{h}_{\mathrm{x}} \mathrm{H}_{2} \mathrm{O} ; \mathrm{II}\right) / \mathrm{m}\left(\mathrm{M}^{+} ; \mathrm{I}\right) \, \mathrm{m}\left(\mathrm{X}^{-} ; \mathrm{I}\right)=1.0
\end{aligned} \nonumber \]
\[\begin{gathered}
\text { Hence, } \mu^{0}\left(\mathrm{M}^{+} ; \mathrm{aq}\right)+\mu^{0}\left(\mathrm{X}^{-} ; \mathrm{aq}\right)= \\
\mu^{0}\left(\mathrm{M}^{+} \, \mathrm{h}_{\mathrm{m}} \mathrm{H}_{2} \mathrm{O} ; \mathrm{aq}\right)+\mu^{0}\left(\mathrm{X}^{-} \, \mathrm{h}_{\mathrm{x}} \mathrm{H}_{2} \mathrm{O} ; \mathrm{aq}\right) \\
-\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{x}}\right) \, \mu_{1}^{*}(\ell)
\end{gathered} \nonumber \]
We obtain an equation linking the ionic chemical potentials. Thus,
\[\ln \gamma_{+}(\mathrm{I})+\ln \gamma_{-}(\mathrm{I})=2 \, \phi \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{x}}\right)+\ln \left\{\gamma_{+}(\mathrm{II})\right\}+\ln \left\{\gamma_{-} \text {(II) }\right\} \nonumber \]
\[ \begin{aligned}
&\text { But } \ln \left\{\gamma_{+}(\mathrm{I})\right\}+\ln \left\{\gamma_{-}(\mathrm{I})\right\}=2 \,\left\{\ln \gamma_{\pm}(\mathrm{I})\right\} \\
&\text { Then, } 2 \, \ln \left\{\gamma_{\pm}(\mathrm{I})\right\}=2 \, \phi \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{x}}\right)+2 \, \ln \left\{\gamma_{\pm} \text {(II) }\right\} \nonumber \]
We identify relationships between single ion activity coefficients in an extra-thermodynamic analysis. Thus from equation (l),
\[\ln \left\{\gamma_{+} \text {(II) }\right\}=\ln \left\{\gamma_{+} \text {(I) }\right\}-\phi \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \, \mathrm{h}_{\mathrm{m}} \nonumber \]
\[\ln \left\{\gamma_{-}(\mathrm{II})\right\}=\ln \left\{\gamma_{-}(\mathrm{I})\right\}-\phi \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \, \mathrm{h}_{\mathrm{x}} \nonumber \]
It is noteworthy that in these terms the solution can be ideal using description I where \(\gamma_{\pm} = 1.0\) but non-ideal using description II. Nevertheless, these equations show how the activity coefficient of the hydrated ion (description II) is related to the activity coefficient of the simple ion (description I).