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1.5.19: Chemical Potentials- Solutions- Salt Hydrates in Aqueous Solution

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    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

    \[\begin{aligned}
    &\mu\left(\mathrm{M}^{+}\right)=\left[\partial \mathrm{G} / \partial \mathrm{n}\left(\mathrm{M}^{+}\right)\right]_{\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}\left(\mathrm{x}^{-}\right)} \\
    &\mu\left(\mathrm{X}^{-}\right)=\left[\partial \mathrm{G} / \partial \mathrm{n}\left(\mathrm{X}^{-}\right)\right]_{\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}\left(\mathrm{M}^{+}\right)}
    \end{aligned}\]

    The total Gibbs energy (at fixed \(\mathrm{T}\) and \(\mathrm{p}\) where \(p \approx p^{0}\)) is given by equation (b).

    \[\begin{aligned}
    &\mathrm{G}(\mathrm{aq} ; \mathrm{I})=\mathrm{n}_{1} \, \mu_{1}^{\mathrm{eq}}(\mathrm{aq}) \\
    &\quad+\mathrm{n}_{\mathrm{j}} \,\left\{\mu^{0}\left(\mathrm{M}^{+} ; \mathrm{aq}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{m}\left(\mathrm{M}^{+}\right) \, \gamma_{+}(\mathrm{I}) / \mathrm{m}^{0}\right]\right\} \\
    &\quad+\mathrm{n}_{\mathrm{j}} \,\left\{\mu^{0}\left(\mathrm{X}^{-} ; \mathrm{aq}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{m}\left(\mathrm{X}^{-}\right) \, \gamma_{-}(\mathrm{I}) / \mathrm{m}^{0}\right]\right\}
    \end{aligned}\]

    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.

    \[\mu\left(\mathrm{M}^{+} \, \mathrm{h}_{\mathrm{m}} \mathrm{H}_{2} \mathrm{O}\right)=\left[\partial \mathrm{G} / \partial\left(\mathrm{M}^{+} \, \mathrm{h}_{\mathrm{m}} \mathrm{H}_{2} \mathrm{O}\right)\right]\]

    at constant \(\mathrm{T}\), \(\mathrm{p}\), \(\mathrm{n}\left(\mathrm{X}^{-} \, \mathrm{h}_{\mathrm{x}} \mathrm{H}_{2} \mathrm{O}\right),\left[\mathrm{n}_{1}-\mathrm{n}_{\mathrm{j}} \,\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{x}}\right)\right]\left(\mathrm{H}_{2} \mathrm{O}\right) \mu\left(\mathrm{X}^{-} \, \mathrm{h}_{\mathrm{x}} \mathrm{H}_{2} \mathrm{O}\right)=\left[\partial \mathrm{G} / \partial \mathrm{n}\left(\mathrm{X}^{-} \, \mathrm{h}_{\mathrm{x}} \mathrm{H}_{2} \mathrm{O}\right)\right]\)

    \[\text { at constant } \mathrm{T}, \mathrm{p}, \mathrm{n}\left(\mathrm{M}^{+} \, \mathrm{h}_{\mathrm{m}} \mathrm{H}_{2} \mathrm{O}\right),\left[\mathrm{n}_{1}-\mathrm{n}_{\mathrm{j}} \,\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{x}}\right)\right]\left(\mathrm{H}_{2} \mathrm{O}\right)\]

    \[\mathrm{m}\left(\mathrm{X}^{-} \, \mathrm{h}_{\mathrm{x}} \mathrm{H}_{2} \mathrm{O}\right)=\mathrm{n}_{\mathrm{j}} / \mathrm{M}_{1} \,\left[\mathrm{n}_{1}-\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{x}}\right) \, \mathrm{n}_{\mathrm{j}}\right] ;\]

    \[\mathrm{m}\left(\mathrm{M}^{+} \, \mathrm{h}_{\mathrm{m}} \mathrm{H}_{2} \mathrm{O}\right)=\mathrm{n}_{\mathrm{j}} / \mathrm{M}_{1} \,\left[\mathrm{n}_{1}-\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{x}}\right) \, \mathrm{n}_{\mathrm{j}}\right] .\]

    The (equilibrium) Gibbs energy (at defined \(\mathrm{T}\) and \(\mathrm{p}\)) is given by the following equation.

    \[\begin{aligned}
    &\mathrm{G}(\mathrm{aq} ; \mathrm{II})=\left[\mathrm{n}_{1}-\mathrm{n}_{\mathrm{j}} \,\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{x}}\right)\right] \, \mu_{1}(\mathrm{aq}) \\
    &+\mathrm{n}_{\mathrm{j}} \,\left[\mu^{0} \,\left(\mathrm{M}^{+} \, \mathrm{h}_{\mathrm{m}} \mathrm{H}_{2} \mathrm{O}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left\{\mathrm{m}\left(\mathrm{M}^{+} \, \mathrm{h}_{\mathrm{m}} \mathrm{H}_{2} \mathrm{O}\right) \, \gamma_{+}(\mathrm{II}) / \mathrm{m}^{0}\right\}\right] \\
    &+\mathrm{n}_{\mathrm{j}} \,\left[\mu^{0}\left(\mathrm{X}^{-} \, \mathrm{h}_{\mathrm{x}} \mathrm{H}_{2} \mathrm{O}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left\{\mathrm{m}\left(\mathrm{X}^{-} \, \mathrm{h}_{\mathrm{x}} \mathrm{H}_{2} \mathrm{O}\right) \, \gamma_{-}(\mathrm{II}) / \mathrm{m}^{0}\right\}\right]
    \end{aligned}\]

    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}\]

    We use the latter equation to explore what happens in the limit that \(\mathrm{n}_{j}\) approaches zero. Thus,

    \[\begin{aligned}
    &\operatorname{limit}\left(\mathrm{n}_{\mathrm{j}} \rightarrow 0\right) \gamma_{+}(\mathrm{I})=1 \quad \gamma_{-}(\mathrm{I})=1 \\
    &\gamma_{+}(\mathrm{II})=1 \quad \gamma_{-}(\mathrm{II})=1 \\
    &\mathrm{~m}_{\mathrm{j}}=0 \\
    &\mathrm{~m}\left(\mathrm{M}^{+} \mathrm{h}_{\mathrm{m}} \mathrm{H}_{2} \mathrm{O} ; \mathrm{II}\right) \, \mathrm{m}\left(\mathrm{X}^{-} \mathrm{h}_{\mathrm{x}} \mathrm{H}_{2} \mathrm{O} ; \mathrm{II}\right) / \mathrm{m}\left(\mathrm{M}^{+} ; \mathrm{I}\right) \, \mathrm{m}\left(\mathrm{X}^{-} ; \mathrm{I}\right)=1.0
    \end{aligned}\]

    \[\begin{gathered}
    \text { Hence, } \mu^{0}\left(\mathrm{M}^{+} ; \mathrm{aq}\right)+\mu^{0}\left(\mathrm{X}^{-} ; \mathrm{aq}\right)= \\
    \mu^{0}\left(\mathrm{M}^{+} \, \mathrm{h}_{\mathrm{m}} \mathrm{H}_{2} \mathrm{O} ; \mathrm{aq}\right)+\mu^{0}\left(\mathrm{X}^{-} \, \mathrm{h}_{\mathrm{x}} \mathrm{H}_{2} \mathrm{O} ; \mathrm{aq}\right) \\
    -\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{x}}\right) \, \mu_{1}^{*}(\ell)
    \end{gathered}\]

    We obtain an equation linking the ionic chemical potentials. Thus,

    \[\ln \gamma_{+}(\mathrm{I})+\ln \gamma_{-}(\mathrm{I})=2 \, \phi \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{x}}\right)+\ln \left\{\gamma_{+}(\mathrm{II})\right\}+\ln \left\{\gamma_{-} \text {(II) }\right\}\]

    \[ \begin{aligned}
    &\text { But } \ln \left\{\gamma_{+}(\mathrm{I})\right\}+\ln \left\{\gamma_{-}(\mathrm{I})\right\}=2 \,\left\{\ln \gamma_{\pm}(\mathrm{I})\right\} \\
    &\text { Then, } 2 \, \ln \left\{\gamma_{\pm}(\mathrm{I})\right\}=2 \, \phi \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{x}}\right)+2 \, \ln \left\{\gamma_{\pm} \text {(II) }\right\}\]

    We identify relationships between single ion activity coefficients in an extra-thermodynamic analysis. Thus from equation (l),

    \[\ln \left\{\gamma_{+} \text {(II) }\right\}=\ln \left\{\gamma_{+} \text {(I) }\right\}-\phi \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \, \mathrm{h}_{\mathrm{m}}\]

    \[\ln \left\{\gamma_{-}(\mathrm{II})\right\}=\ln \left\{\gamma_{-}(\mathrm{I})\right\}-\phi \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \, \mathrm{h}_{\mathrm{x}}\]

    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).

    This page titled 1.5.19: Chemical Potentials- Solutions- Salt Hydrates in Aqueous Solution is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.