1.10.14: Gibbs Energies- Salt Hydrates
An aqueous solution is prepared using \(mathrm{n}_{\mathrm{j}}\) 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 \(\mathrm{m}(\mathrm{MX})\left[=\mathrm{n}(\mathrm{MX}) / \mathrm{w}_{1}\right.\) where \(\mathrm{w}_{1}\) is the mass of water]. The single ion chemical potentials, are defined in the following manner
\[\begin{aligned}
&\mu\left(\mathrm{M}^{+}\right)=\left[\partial \mathrm{G} / \partial \mathrm{n}\left(\mathrm{M}^{+}\right)\right]_{\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}\left(\mathrm{x}^{-}\right)} \\
&\mu\left(\mathrm{X}^{-}\right)=\left[\partial \mathrm{G} / \partial \mathrm{n}\left(\mathrm{X}^{-}\right)\right]_{\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}\left(\mathrm{M}^{+}\right)}
\end{aligned} \nonumber \]
Then the total Gibbs energy (at fixed \(\mathrm{T}\) and \(\mathrm{p}\)) is given by equation (b). \(\mathrm{G}(\mathrm{aq} ; \mathrm{I})=\mathrm{n}_{1} \, \mu_{1}(\mathrm{aq})\)
\[\begin{aligned}
&+\mathrm{n}_{\mathrm{j}} \,\left\{\mu^{\#}\left(\mathrm{M}^{+} ; \mathrm{aq}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{m}\left(\mathrm{M}^{+}\right) \, \gamma_{+}(\mathrm{I}) / \mathrm{m}^{0}\right]\right\} \\
&+\mathrm{n}_{\mathrm{j}}\left\{\mu^{\#}\left(\mathrm{X}^{-} ; \mathrm{aq}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{m}\left(\mathrm{X}^{-}\right) \, \gamma_{-}(\mathrm{I}) / \mathrm{m}^{0}\right]\right\}
\end{aligned} \nonumber \]
Description II
According to this Description each mole of cation is hydrated by \(\mathrm{h}_{\mathrm{m}}\left(\mathrm{H}_{2}\mathrm{O}\right)\) moles of water and each mole of anion is hydrated by \(\mathrm{h}_{\mathrm{x}}\left(\mathrm{H}_{2}\mathrm{O}\right)\) moles of water. Hence the single ion chemical potentials are defined as follows.
\[\mu\left(\mathrm{M}^{+} \, \mathrm{h}_{\mathrm{m}} \mathrm{H}_{2} \mathrm{O}\right)=\left\lfloor\partial \mathrm{G} / \partial \mathrm{n}\left(\mathrm{M}^{+} \, \mathrm{h}_{\mathrm{m}} \mathrm{H}_{2} \mathrm{O}\right)\right\rfloor \nonumber \]
at constant \(\mathrm{T}\), \(\mathrm{p}\), \(\mathrm{n}\left(\mathrm{X}^{-} \, \mathrm{h}_{\mathrm{x}} \mathrm{H}_{2} \mathrm{O}\right),\left[\mathrm{n}_{1}-\mathrm{n}_{\mathrm{j}} \,\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{x}}\right)\right]\left(\mathrm{H}_{2} \mathrm{O}\right)\) and,
\[\mathrm{m}\left(\mathrm{X}^{-} \, \mathrm{h}_{\mathrm{x}} \mathrm{H}_{2} \mathrm{O}\right)=\mathrm{n}_{\mathrm{j}} / \mathrm{M}_{1} \,\left[\mathrm{n}_{1}-\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{x}}\right) \, \mathrm{n}_{\mathrm{j}}\right] \nonumber \]
at constant \(\mathrm{T}\), \(\mathrm{p}\), \(\mathrm{n}\left(\mathrm{M}^{+} \, \mathrm{h}_{\mathrm{m}} \mathrm{H}_{2} \mathrm{O}\right),\left[\mathrm{n}_{1}-\mathrm{n}_{\mathrm{j}} \,\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{x}}\right)\right]\left(\mathrm{H}_{2} \mathrm{O}\right)\) Then,
\[\mathrm{m}\left(\mathrm{X}^{-} \, \mathrm{h}_{\mathrm{x}} \mathrm{H}_{2} \mathrm{O}\right)=\mathrm{n}_{\mathrm{j}} / \mathrm{M}_{1} \,\left[\mathrm{n}_{1}-\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{x}}\right) \, \mathrm{n}_{\mathrm{j}}\right] \nonumber \]
\[\mathrm{m}\left(\mathrm{M}^{+} \, \mathrm{h}_{\mathrm{m}} \mathrm{H}_{2} \mathrm{O}\right)=\mathrm{n}_{\mathrm{j}} / \mathrm{M}_{1} \,\left[\mathrm{n}_{1}-\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{x}}\right) \, \mathrm{n}_{\mathrm{j}}\right] \nonumber \]
Hence the (equilibrium) Gibbs energy (at defined \(\mathrm{T}\) and \(\mathrm{p}\)) is given by the following equation.
\[\begin{aligned}
&\mathrm{G}(\mathrm{aq})=\left[\mathrm{n}_{1}-\mathrm{n}_{\mathrm{j}} \,\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{x}}\right)\right] \, \mu_{1}(\mathrm{aq}) \\
&\quad+\mathrm{n}_{\mathrm{j}} \,\left[\mu^{\prime \prime}\left(\mathrm{M}^{+} \, \mathrm{h}_{\mathrm{m}} \mathrm{H}_{2} \mathrm{O}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left\{\mathrm{m}\left(\mathrm{M}^{+} \, \mathrm{h}_{\mathrm{m}} \mathrm{H}_{2} \mathrm{O}\right) \, \gamma_{+}(\mathrm{II}) / \mathrm{m}^{0}\right\}\right] \\
&+\mathrm{n}_{\mathrm{j}} \,\left[\mu^{\mathrm{y}}\left(\mathrm{X}^{-} \, \mathrm{h}_{\mathrm{x}} \mathrm{H}_{2} \mathrm{O}\right)\right. \\
&\left.\quad+\mathrm{R} \, \mathrm{T} \, \ln \left\{\mathrm{m}\left(\mathrm{X}^{-} \, \mathrm{h}_{\mathrm{x}} \mathrm{H}_{2} \mathrm{O}\right) \, \gamma_{-} \text {(II) } / \mathrm{m}^{0}\right\}\right]
\end{aligned} \nonumber \]
But the Gibbs energies defined by equations (b) and (g) are identical (at equilibrium at defined \(\mathrm{T}\) and \(\mathrm{p}\)). Hence [1],
\[\begin{aligned}
&\mu^{\prime \prime}\left(\mathrm{M}^{+} ; \mathrm{aq}\right)+\mu^{\prime \prime}\left(\mathrm{X}^{-} ; \mathrm{aq}\right)+2 \, \mathrm{R} \, \mathrm{T} \, \ln \left[1-\mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{X}}\right)\right]\\
&+\mathrm{R} \, \mathrm{T} \, \ln \left\{\gamma_{+}(\mathrm{I}) \, \gamma_{-}(\mathrm{I})\right\}\\
&=-\left(h_{m}+h_{X}\right) \,\left\{\mu_{1}^{*}(\ell)-2 \, \phi \, R \, T \, M_{1} \, m_{j}\right\}\\
&+\mu^{\prime \prime}\left(\mathrm{M}^{+} \, \mathrm{h}_{\mathrm{m}} \mathrm{H}_{2} \mathrm{O} ; \mathrm{aq}\right)+\mu^{\prime \prime}\left(\mathrm{X}^{-} \, \mathrm{h}_{\mathrm{X}} \mathrm{H}_{2} \mathrm{O} ; \mathrm{aq}\right)\\
&+\mathrm{R} \, \mathrm{T} \, \ln \left\{\gamma_{+} \text {(II) } \, \gamma_{-} \text {(II) }\right\}
\end{aligned} \nonumber \]
We use the latter equation to explore what happens in the limit that \(\mathrm{n}_{j}\) approaches zero. Thus, \(\operatorname{limit}\left(\mathrm{n}_{\mathrm{j}} \rightarrow 0\right) \gamma_{+}(\mathrm{I})=1 ; \gamma_{-}(\mathrm{I})=1 ; \gamma_{+}(\mathrm{II})=1 ; \gamma_{-}(\mathrm{II})=1 ; \mathrm{m}_{\mathrm{j}}=0\) Hence,
\[\begin{aligned}
&\mu^{\#}\left(\mathrm{M}^{+} ; \mathrm{aq}\right)+\mu^{\#}\left(\mathrm{X}^{-} ; \mathrm{aq}\right)= \\
&\mu^{\#}\left(\mathrm{M}^{+} \, \mathrm{h}_{\mathrm{m}} \mathrm{H}_{2} \mathrm{O} ; \mathrm{aq}\right)+\mu^{\#}\left(\mathrm{X}^{-} \, \mathrm{h}_{\mathrm{x}} \mathrm{H}_{2} \mathrm{O} ; \mathrm{aq}\right) \\
&-\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{x}}\right) \, \mu_{1}{ }^{*}(\ell)
\end{aligned} \nonumber \]
We obtain an equation linking the ionic chemical potentials. Thus,
\[\begin{array}{r}
\ln \gamma_{+}(\mathrm{I})+\ln \gamma_{-}(\mathrm{I})=2 \, \phi \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{x}}\right) \\
-2 \, \ln \left[1-\mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{X}}\right)\right] \\
+\ln _{+} \gamma_{+}(\mathrm{II})+\ln \gamma_{-}(\mathrm{II})
\end{array} \nonumber \]
Then in dilute solutions,
\[\begin{array}{r}
\ln \gamma_{+}(\mathrm{I})+\ln \gamma_{-}(\mathrm{I})=2 \,(\phi+1) \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{x}}\right) \\
+\ln \gamma_{+}(\mathrm{II})+\ln \gamma_{-}(\mathrm{II})
\end{array} \nonumber \]
But \(\ln \gamma_{+}(\mathrm{I})+\ln \gamma_{-}(\mathrm{I})=2 \, \ln \gamma_{\pm}(\mathrm{I})\) Then, \(2 \, \ln \gamma_{\pm}(\mathrm{I})=2 \,(\phi+1) \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{x}}\right)+2 \, \ln \gamma_{\pm}(\mathrm{II})\)
We identify relationships between single ion activity coefficients in an extra-thermodynamic analysis. Thus from equation (k),
\[\ln \gamma_{+}(\mathrm{II})=\ln \gamma_{+}(\mathrm{I})-2 \,(\phi+1) \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \, \mathrm{h}_{\mathrm{m}} \nonumber \]
\[\ln \gamma_{-}(\mathrm{II})=\ln \gamma_{-}(\mathrm{I})-2 \,(\phi+1) \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \, \mathrm{h}_{\mathrm{x}} \nonumber \]
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).
Footnote
[1] From equations (b) and (g), (dividing by \(\mathrm{n}_{j}\))
\[\begin{aligned}
&\left[\mu^{n}\left(\mathrm{M}^{+} ; \mathrm{aq}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left\{\mathrm{m}\left(\mathrm{M}^{+} ; \mathrm{I}\right) \, \gamma_{+}(\mathrm{I}) / \mathrm{m}^{0}\right\}\right]\\
&+\left[\mu^{\prime \prime}\left(\mathrm{X}^{-} ; \mathrm{aq}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left\{\mathrm{m}\left(\mathrm{X}^{-} ; \mathrm{I}\right) \, \gamma_{-}(\mathrm{I}) / \mathrm{m}^{0}\right\}\right]=\\
&-\left(h_{m}+h_{x}\right) \, \mu_{1}(a q)+\\
&+\left[\mu^{\prime \prime}\left(\mathrm{M}^{+} \, \mathrm{h}_{\mathrm{m}} \mathrm{H}_{2} \mathrm{O} ; \mathrm{aq}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left\{\mathrm{m}\left(\mathrm{M}^{+} \, \mathrm{h}_{\mathrm{m}} \mathrm{H}_{2} \mathrm{O}\right) \, \gamma_{+} \text {(II) } / \mathrm{m}^{0}\right\}\right]\\
&+\left[\mu^{\prime \prime}\left(\mathrm{X}^{-} \, \mathrm{h}_{\mathrm{x}} \mathrm{H}_{2} \mathrm{O}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left\{\mathrm{m}\left(\mathrm{X}^{-} \, \mathrm{h}_{\mathrm{x}} \mathrm{H}_{2} \mathrm{O}\right) \, \gamma_{-}(\mathrm{II}) / \mathrm{m}^{0}\right\}\right]
\end{aligned} \nonumber \]
Then
\[\begin{aligned}
&\text { en }\left[\mu^{*}\left(\mathrm{M}^{+} ; \mathrm{aq}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left\{\mathrm{m}\left(\mathrm{M}^{+} ; \mathrm{I}\right) \, \gamma_{+}(\mathrm{I}) / \mathrm{m}^{0}\right\}\right]\\
&+\left[\mu^{*}\left(\mathrm{X}^{-} ; \mathrm{aq}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left\{\mathrm{m}\left(\mathrm{X}^{-} ; \mathrm{I}\right) \, \gamma_{-}(\mathrm{I}) / \mathrm{m}^{0}\right\}\right]=\\
&-\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{X}}\right) \,\left\{\mu_{1}^{*}(\ell)-2 \, \phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\right\}\\
&+\left[\mu^{*}\left(\mathrm{M}^{+} \, \mathrm{h}_{\mathrm{m}} \mathrm{H}_{2} \mathrm{O} ; \mathrm{aq}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left\{\mathrm{m}\left(\mathrm{M}^{+} \, \mathrm{h}_{\mathrm{m}} \mathrm{H}_{2} \mathrm{O}\right) \, \gamma_{+}(\mathrm{II}) / \mathrm{m}^{0}\right\}\right]\\
&+\left[\mu^{\prime \prime}\left(\mathrm{X}^{-} \, \mathrm{h}_{\mathrm{x}} \mathrm{H}_{2} \mathrm{O}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left\{\mathrm{m}\left(\mathrm{X}^{-} \, \mathrm{h}_{\mathrm{x}} \mathrm{H}_{2} \mathrm{O}\right) \, \gamma_{-}(\mathrm{II}) / \mathrm{m}^{0}\right\}\right]
\end{aligned} \nonumber \]
Or,
\[\begin{aligned}
&{\left[\mu^{\prime \prime}\left(\mathrm{M}^{+} ; \mathrm{aq}\right)+\mu^{\#}\left(\mathrm{X}^{-} ; \mathrm{aq}\right)\right.} \\
&+\mathrm{R} \, \mathrm{T} \, \ln \left\{\mathrm { m } ( \mathrm { M } ^ { + } ; \mathrm { I } ) \, \mathrm { m } \left(\mathrm{X}^{-} ;(\mathrm{I}) /\left(\mathrm{M}^{+} ; \mathrm{II}\right) \, \mathrm{m}\left(\mathrm{X}^{-} ; \mathrm{II}\right\}\right.\right. \\
&\mathrm{R} \, \mathrm{T} \, \ln \left\{\gamma_{+}(\mathrm{I}) \, \gamma_{-}(\mathrm{I})\right\} \\
&=-\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{X}}\right) \,\left\{\mu_{1}^{*}(\ell)-2 \, \phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\right\} \\
&+\mu^{*}\left(\mathrm{M}^{+} \, \mathrm{h}_{\mathrm{m}}\left(\mathrm{H}_{2} \mathrm{O}\right) ; \mathrm{aq}\right)+\mu^{\#}\left(\mathrm{X}^{-} \, \mathrm{h}_{\mathrm{X}}\left(\mathrm{H}_{2} \mathrm{O}\right) ; \mathrm{aq}\right) \\
&+\mathrm{R} \, \mathrm{T} \, \ln \left\{\gamma_{+}(\mathrm{II}) \, \gamma_{-}(\mathrm{II})\right\}
\end{aligned} \nonumber \]
Using the definition of \(\mu^{\prime \prime}\left(\mathrm{M}^{+} ; \mathrm{I}\right)\) and \(\mu^{\prime \prime}\left(\mathrm{X}^{-} ; \mathrm{I}\right)\) and equations (e) and (f) for description (II),
\[\begin{aligned}
&\frac{\mathrm{m}\left(\mathrm{M}^{+} ; \mathrm{I}\right) \, \mathrm{m}\left(\mathrm{X}^{-} ; \mathrm{I}\right)}{\mathrm{m}\left(\mathrm{M}^{+} ; \mathrm{II}\right) \, \mathrm{m}\left(\mathrm{X}^{-} ; \mathrm{II}\right)}= \\
&\frac{\mathrm{n}_{\mathrm{j}}}{\mathrm{M}_{1} \, \mathrm{n}_{1}} \, \frac{\mathrm{M}_{1} \,\left[\mathrm{n}_{1}-\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{X}}\right) \, \mathrm{n}_{\mathrm{j}}\right]}{\mathrm{n}_{\mathrm{j}}} \, \frac{\mathrm{n}_{\mathrm{j}}}{\mathrm{M}_{1} \, \mathrm{n}_{1}} \, \frac{\mathrm{M}_{1} \,\left[\mathrm{n}_{1}-\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{X}}\right) \, \mathrm{n}_{\mathrm{j}}\right]}{\mathrm{n}_{\mathrm{j}}}
\end{aligned} \nonumber \]
Thus,
\[\frac{\mathrm{m}\left(\mathrm{M}^{+} ; \mathrm{I}\right) \, \mathrm{m}\left(\mathrm{X}^{-} ; \mathrm{I}\right)}{\mathrm{m}\left(\mathrm{M}^{+} ; \mathrm{II}\right) \, \mathrm{m}\left(\mathrm{X}^{-} ; \mathrm{II}\right)}=\left[1-\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{X}}\right) \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\right]^{2} \nonumber \]
Therefore,
\[\begin{aligned}
&\mu^{\# \prime}\left(\mathrm{M}^{+} ; \mathrm{aq}\right)+\mu^{\# \prime}\left(\mathrm{X}^{-} ; \mathrm{aq}\right)+2 \, \phi \, \mathrm{R} \, \mathrm{T} \, \ln \left[1-\mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{X}}\right)\right] \\
&\quad+\mathrm{R} \, \mathrm{T} \, \ln \left\{\gamma_{+}(\mathrm{I}) \, \gamma_{-}(\mathrm{I})\right\} \\
&=-\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{X}}\right) \,\left\{\mu_{1}^{*}(\ell)-2 \, \phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\right\} \\
&\quad+\mu^{\# *}\left(\mathrm{M}^{+} ; \mathrm{h}_{\mathrm{m}} \mathrm{H}_{2} \mathrm{O} ; \mathrm{aq}\right)+\mu^{\# \#}\left(\mathrm{X}^{-} ; \mathrm{h}_{\mathrm{X}} \mathrm{H}_{2} \mathrm{O} ; \mathrm{aq}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left\{\gamma_{+}(\mathrm{II}) \, \gamma_{-}(\mathrm{II})\right\}
\end{aligned} \nonumber \]