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1.22.3: Volume: Partial and Apparent Molar

  • Page ID
    397781
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    In descriptions of the volumetric properties of solutions, two terms are extensively used. We refer to the partial molar volume of solute \(j\) in, for example, an aqueous solution \(\mathrm{V}_{j}(\mathrm{aq})\) and the corresponding apparent molar volume \(\phi\left(\mathrm{V}_{j}\right)\). Here we explore how these terms are related. We consider an aqueous solution prepared using water, \(1 \mathrm{~kg}\), and \(\mathrm{m}_{j}\) moles of solute \(j\). The volume of this solution at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\) is given by equation (a).

    \[\mathrm{V}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{V}_{1}(\mathrm{aq})+\mathrm{m}_{\mathrm{j}} \, \mathrm{V}_{\mathrm{j}}(\mathrm{aq}) \nonumber \]

    The chemical potential of solvent, water, in the aqueous solution \(\mu_{1}(\mathrm{aq})\) is related to the molality \(\mathrm{m}_{j}\) using equation (b) where \(\mu_{1}^{*}(\ell)\) is the chemical potential of pure water(\(\ell\)), molar mass \(\mathrm{M}_{1}\), at the same \(\mathrm{T}\) and \(\mathrm{p}\).

    \[\mu_{1}(\mathrm{aq})=\mu_{1}^{*}(\ell)-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \nonumber \]

    Here practical osmotic coefficient \(\phi\) is defined by equation (c).

    \[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \phi=1 \quad \text { at all T and } \mathrm{p} \nonumber \]

    But

    \[\mathrm{V}_{1}(\mathrm{aq})=\left[\partial \mu_{1}(\mathrm{aq}) / \partial \mathrm{p}\right]_{\mathrm{T}} \nonumber \]

    Then[1]

    \[\mathrm{V}_{1}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\mathrm{V}_{1}^{*}(\ell)-\mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \,(\partial \phi / \partial \mathrm{p})_{\mathrm{T}} \nonumber \]

    For the solute, the chemical potential \(\mu_{j}(\mathrm{aq})\) is related to the molality of solute \(\mathrm{m}_{j}\) using equation (f) where pressure \(\mathrm{p}\) is close to the standard pressure.

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

    where

    \[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \gamma_{j}=1 \quad \text { at all } \mathrm{T} \text { and } \mathrm{p} \nonumber \]

    Then[2]

    \[\mathrm{V}_{\mathrm{j}}(\mathrm{aq})=\mathrm{V}_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \,\left[\partial \ln \left(\gamma_{\mathrm{j}}\right) / \partial \mathrm{p}\right]_{\mathrm{T}} \nonumber \]

    \[\operatorname{Limit}\left(m_{j} \rightarrow 0\right) V_{j}(a q)=V_{j}^{0}(a q)=V_{j}^{\infty}(a q) \nonumber \]

    Combination of equations (a), (e) and (h) yields equation (j).

    \[\begin{gathered}
    \mathrm{V}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \,\left[\mathrm{V}_{1}^{*}(\ell)-\mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \,(\partial \phi / \partial \mathrm{p})_{\mathrm{T}}\right] \\
    \quad+\mathrm{m}_{\mathrm{j}} \,\left\{\mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \,\left[\partial \ln \left(\gamma_{\mathrm{j}}\right) / \partial \mathrm{p}\right]_{\mathrm{T}}\right\}
    \end{gathered} \nonumber \]

    An important point emerges if we re-arrange equation (j).

    \[\begin{aligned}
    &\mathrm{V}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{V}_{1}^{*}(\ell) \\
    &\quad+\mathrm{m}_{\mathrm{j}} \,\left\{\mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \,\left[\partial \ln \left(\gamma_{\mathrm{j}}\right) / \partial \mathrm{p}\right]_{\mathrm{T}}-\mathrm{R} \, \mathrm{T} \,(\partial \phi / \partial \mathrm{p})_{\mathrm{T}}\right\}
    \end{aligned} \nonumber \]

    Equation (k) has an interesting form in that the brackets {….} contain \(\mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq})\) and terms describing the extent to which the volumetric properties of the solution are not ideal in a thermodynamic sense. It is therefore convenient to define an apparent molar volume of solute \(j\), \(\phi\left(\mathrm{V}_{\mathrm{j}}\right)\) using equation (l).

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

    Then

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

    Therefore we obtain equation (n).

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

    Interest in equation (n) arises from the fact that the for a given solution \(\mathrm{V}(\mathrm{aq})\) can be measured {using the density \(\rho(\mathrm{aq})\)} and hence knowing \(\mathrm{V}_{1}^{*}(\ell)\), \(\phi\left(\mathrm{V}_{\mathrm{j}}\right)\) is obtained. If we measure \(\phi\left(\mathrm{V}_{j}\right)\) as a function of \(\mathrm{m}_{j}\), equation (m) indicates how one obtains \(\mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq})\). Moreover the difference \(\left[\phi\left(\mathrm{V}_{\mathrm{j}}\right)-\mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq})\right]\) signals the role of solute - solute interactions.

    Footnotes

    [1]

    \[\begin{array}{r}
    \mathrm{R} \, \mathrm{T} \,\left[\frac{\partial \phi}{\partial \mathrm{p}}\right]=\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right] \,[\mathrm{K}] \,\left[\frac{1}{\left[\mathrm{~N} \mathrm{~m}^{-2}\right]}\right] \\
    =\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right]
    \end{array} \nonumber \]

    [2]

    \[\begin{aligned}
    \mathrm{R} \, \mathrm{T} \,\left[\frac{\partial \ln \left(\gamma_{\mathrm{j}}\right)}{\partial \mathrm{p}}\right]=\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right] \,[\mathrm{K}] \,\left[\frac{1}{\left[\mathrm{~N} \mathrm{~m}^{-2}\right]}\right] \\
    =& {[\mathrm{N} \mathrm{m} \mathrm{mol}] \,\left[\frac{1}{\left[\mathrm{~N} \mathrm{~m}^{-2}\right.}\right]=\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right] }
    \end{aligned} \nonumber \]


    This page titled 1.22.3: Volume: Partial and Apparent Molar 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.