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1.7.17: Compressions- Isothermal- Solutes- Partial Molar Compressions

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    A given aqueous solution at temperature \(\mathrm{T}\) and near ambient pressure \(\mathrm{p}\) contains a solute \(j\) at molality \(\mathrm{m}_{j}\). The chemical potential \(\mu_{j}(\mathrm{aq})\) is related to the molality \(\mathrm{m}_{j}\) using equation (a).

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

    Then

    \[\mathrm{V}_{\mathrm{j}}(\mathrm{aq})=\mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \,\left[\partial \ln \left(\gamma_{\mathrm{j}}\right) / \partial \mathrm{p}\right]_{\mathrm{T}}\]

    By definition, the partial molar isothermal compression of solute \(j\),[1]

    \[K_{T_{j}}(a q)=-\left(\frac{\partial V_{j}(a q)}{\partial p}\right)_{T}\]

    Then

    \[\mathrm{K}_{\mathrm{Tj}}(\mathrm{aq})=\mathrm{K}_{\mathrm{TJ}}^{\infty}(\mathrm{aq})-\mathrm{R} \, \mathrm{T} \,\left[\partial^{2} \ln \left(\gamma_{\mathrm{j}}\right) / \partial \mathrm{p}^{2}\right]_{\mathrm{T}}\]

    Thus by definition,

    \[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \mathrm{K}_{\mathrm{T}_{\mathrm{j}}}(\mathrm{aq})=\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}^{\infty}(\mathrm{aq})\]

    Hence the difference between \(\mathrm{K}_{\mathrm{Tj}_{\mathrm{j}}}(\mathrm{aq})\) and \(\mathrm{K}_{\mathrm{T}_{j}}^{\infty}(\mathrm{aq})\) depends on the second differential of \(\ln \left(\gamma_{j}\right)\) with respect to pressure.

    Footnotes

    [1] The formal definition of \(\mathrm{K}_{\mathrm{Tj}}(\mathrm{aq})\) is given by equation (a).

    \[\mathrm{K}_{\mathrm{Tj}}(\mathrm{aq})=\left(\frac{\partial \mathrm{K}_{\mathrm{T}}}{\partial \mathrm{n}_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}(\mathrm{i} \neq \mathrm{j})}\]

    However,

    \[\mathrm{K}_{\mathrm{T}}=-(\partial \mathrm{V} / \partial \mathrm{p})_{\mathrm{T}, \mathrm{n}(\mathrm{j})}\]

    Then,

    \[\mathrm{K}_{\mathrm{Tj}}(\mathrm{aq})=-\left(\frac{\partial\left(\partial \mathrm{V} / \partial \mathrm{n}_{\mathrm{j}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}(\mathrm{i} \neq \mathrm{j})}}{\partial \mathrm{p}}\right)_{\mathrm{T}, \mathrm{n}(\mathrm{j})}\]

    Or,

    \[\mathrm{K}_{\mathrm{Tj}}(\mathrm{aq})=-\left(\frac{\partial \mathrm{V}_{\mathrm{j}}(\mathrm{aq})}{\partial \mathrm{p}}\right)_{\mathrm{T}}\]

    In other words, equation (c) shows that \(\mathrm{K}_{\mathrm{Tj}}(\mathrm{aq})\) is a Lewisian partial molar property

    This page titled 1.7.17: Compressions- Isothermal- Solutes- Partial Molar Compressions is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.