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1.8.8: Enthalpies- Solutions- Simple Solutes- Interaction Parameters

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    The excess Gibbs energy \(\mathrm{G}^{\mathrm{E}}\) for a dilute aqueous solution containing a simple solute \(j\) prepared using \(1 \mathrm{~kg}\) of solvent, water is given by equation (a).

    \[\mathrm{G}^{\mathrm{E}}=\mathrm{R} \, \mathrm{T} \, \mathrm{m}_{\mathrm{j}} \,\left[1-\phi+\ln \left(\gamma_{\mathrm{j}}\right)\right]\]

    In terms of Gibbs energies pairwise solute-solute interaction parameters,

    \[\mathrm{G}^{\mathrm{E}}=\mathrm{g}_{\mathrm{jj}} \,\left[\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right]^{2}\]

    The excess enthalpy[1]

    \[\mathrm{H}^{\mathrm{E}}=\mathrm{h}_{\mathrm{ij}} \,\left[\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right]^{2}\]

    where [cf. Gibbs –Helmholtz Equation],

    \[\mathrm{h}_{\mathrm{ij}}=-\mathrm{T}^{2} \,\left\{\partial\left[\mathrm{g}_{\mathrm{jj}} / \mathrm{T}\right] / \partial \mathrm{T}\right\}_{\mathrm{p}}\]

    Here \(\mathrm{h}_{\mathrm{jj}}\) is the pairwise solute-solute enthalpic interaction parameter. For the solvent,

    \[\mu_{1}(\mathrm{aq})=\mu_{1}^{*}(\lambda)-\mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}-\mathrm{M}_{1} \, \mathrm{g}_{\mathrm{j}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2}\]

    Using the Gibbs-Helmholtz Equation,

    \[\mathrm{H}_{1}(\mathrm{aq})=\mathrm{H}_{1}^{*}(\lambda)-\mathrm{M}_{1} \, \mathrm{h}_{\mathrm{ij}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2}\]

    For the solute,

    \[\mu_{\mathrm{j}}(\mathrm{aq})=\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)+2 \, \mathrm{g}_{\mathrm{ij}} \,\left(\mathrm{m}^{0}\right)^{-2} \, \mathrm{m}_{\mathrm{j}}\]

    Then using the Gibbs-Helmholtz Equation [2]

    \[\mathrm{H}_{\mathrm{j}}(\mathrm{aq})=\mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq})+2 \, \mathrm{h}_{\mathrm{ij}} \,\left(\mathrm{m}^{0}\right)^{-2} \, \mathrm{m}_{\mathrm{j}}\]

    Alternatively we may express the enthalpy of the solution in terms of the apparent molar enthalpy of the solute, \(\phi\left(\mathrm{H}_{\mathrm{j}}\right)\).

    \[\mathrm{H}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{H}_{1}^{*}(\lambda)+\mathrm{m}_{\mathrm{j}} \, \phi\left(\mathrm{H}_{\mathrm{j}}\right)\]

    For the ideal solution,

    \[\mathrm{H}\left(\mathrm{aq} ; \mathrm{id} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{H}_{1}^{*}(\lambda)+\mathrm{m}_{\mathrm{j}} \, \phi\left(\mathrm{H}_{\mathrm{j}}\right)^{\infty}\]

    where \(\phi\left(\mathrm{H}_{\mathrm{j}}\right)^{\infty}=\mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq})\). Then

    \[\mathrm{H}^{\mathrm{E}}=\mathrm{m}_{\mathrm{j}} \,\left[\phi\left(\mathrm{H}_{\mathrm{j}}\right)-\phi\left(\mathrm{H}_{\mathrm{j}}\right)^{\infty}\right]\]

    Hence using equation (c),

    \[\phi\left(\mathrm{H}_{\mathrm{j}}\right)=\phi\left(\mathrm{H}_{\mathrm{j}}\right)^{\infty}+\mathrm{h}_{\mathrm{ji}} \,\left(\mathrm{m}^{0}\right)^{-2} \, \mathrm{m}_{\mathrm{j}}\]

    We use these equations in the analysis of a calorimetric data where a given solution is diluted. The solution is prepared using \(\mathrm{n}_{1}\) moles of solvent (water) and \(\mathrm{n}_{j}\) moles of a simple solute \(j\). Then

    \[\mathrm{H}(\mathrm{I} ; \mathrm{aq})=\mathrm{n}_{1} \, \mathrm{H}_{1}^{*}(\lambda)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{H}_{\mathrm{j}} ; \mathrm{I} ; \mathrm{aq}\right)\]

    A new solution is prepared by adding (in the calorimeter) \(\Delta \mathrm{n}_{1}\) moles of solvent, Then

    \[\mathrm{H}(\mathrm{II} ; \mathrm{aq})=\left(\mathrm{n}_{1}+\Delta \mathrm{n}_{1}\right) \, \mathrm{H}_{1}^{*}(\lambda)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{H}_{\mathrm{j}} ; \mathrm{II} ; \mathrm{aq}\right)\]

    Thus the molality of solute \(j\) changes from \(\mathrm{m}_{j}\)(I) \(\left[=\mathrm{n}_{\mathrm{j}} / \mathrm{n}_{1} \, \mathrm{M}_{1}\right]\) to \(\mathrm{m}_{j}\)(II) \(\left[=\mathrm{n}_{\mathrm{j}} /\left(\mathrm{n}_{1}+\Delta \mathrm{n}_{1}\right) \, \mathrm{M}_{1}\right]\). Therefore,

    \[\phi\left(\mathrm{H}_{\mathrm{j}} ; \mathrm{I} ; \mathrm{aq}\right)=\phi\left(\mathrm{H}_{\mathrm{j}}\right)^{\infty}+\left[\mathrm{h}_{\mathrm{ij}} \,\left(\mathrm{m}^{0}\right)^{-2} \, \mathrm{n}_{\mathrm{j}} / \mathrm{n}_{1} \, \mathrm{M}_{1}\right]\]

    And

    \[\phi\left(\mathrm{H}_{\mathrm{j}} ; \mathrm{II} ; \mathrm{aq}\right)=\phi\left(\mathrm{H}_{\mathrm{j}}\right)^{\infty}+\left[\mathrm{h}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-2} \, \mathrm{n}_{\mathrm{j}} /\left(\mathrm{n}_{1}+\Delta \mathrm{n}_{1}\right) \, \mathrm{M}_{1}\right]\]

    In fact we record the heat \(\mathrm{q}\) (at constant pressure) when \(\Delta \mathrm{n}_{1}\) moles of solvent are added to solution I to form solution II. Thus,

    \[\mathrm{q}=\mathrm{H}(\mathrm{II} ; \mathrm{aq})-\mathrm{H}(\mathrm{I} ; \mathrm{aq})-\Delta \mathrm{n} \, \mathrm{H}_{1}^{*}(\lambda)\]

    Footnotes

    [1] From equation (a) and (b) \(\mathrm{G}^{\mathrm{E}}=\left[\mathrm{J} \mathrm{kg}^{-1}\right]\) From equation (c) \(\mathrm{H}^{\mathrm{E}}=\left[\mathrm{J} \mathrm{kg}^{-1}\right]\)

    [2] A check on the equations with reference to solution prepared using \(1 \mathrm{~kg}\) of solvent.

    \[\begin{aligned}
    \mathrm{H}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \,\left[\mathrm{H}_{1}^{*}(\lambda)-\mathrm{M}_{1} \, \mathrm{h}_{\mathrm{ji}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2}\right] \\
    &+\mathrm{m}_{\mathrm{j}} \,\left[\mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq})+2 \, \mathrm{h}_{\mathrm{jj}} \,\left(\mathrm{m}^{0}\right)^{-2} \, \mathrm{m}_{\mathrm{j}}\right]
    \end{aligned}\]

    Or,

    \[\begin{aligned}
    \mathrm{H}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{H}_{1}^{*} &(\lambda)-\mathrm{h}_{\mathrm{jj}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2} \\
    &+\mathrm{m}_{\mathrm{j}} \, \mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq})+2 \, \mathrm{h}_{\mathrm{ij}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2}
    \end{aligned}\]

    Then

    \[\mathrm{H}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{H}_{1}^{*}(\lambda)+\mathrm{m}_{\mathrm{i}} \, \mathrm{H}_{\mathrm{i}}^{\infty}(\mathrm{aq})+\mathrm{H}^{\mathrm{E}}\]

    This page titled 1.8.8: Enthalpies- Solutions- Simple Solutes- Interaction Parameters is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.