1.10.20: Gibbs Energies- Salt Solutions- DHLL- Derived Parameters
- Page ID
- 384734
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Granted that the DHLL forms a starting point for understanding the role of ion-ion interactions in aqueous solutions, we use the DHLL to explain the impact of these interactions on related properties such as volumes and enthalpies. We write the Debye-Huckel coefficient as a function of three variables:
- the molar volume of the solvent \(\mathrm{V}_{1}^{*}(\ell)\),
- the relative permittivity of the solvent, \(\varepsilon_{r}\) and
- the temperature.
Here we take account of the fact that \(\mathrm{V}_{1}^{*}(\ell)\) and \(\varepsilon_{r}\) depend on both temperature and pressure.
\[S_{\gamma}=\left[\frac{2 \, \pi \, \mathrm{N}_{\mathrm{A}} \, \mathrm{M}_{1} \, \mathrm{m}^{0}}{\mathrm{~V}_{1}^{*}(\ell)}\right]^{1 / 2} \,\left[\frac{\mathrm{e}^{2} \, \mathrm{N}_{\mathrm{A}}}{4 \, \pi \, \varepsilon^{0} \, \varepsilon_{\mathrm{r}} \, \mathrm{R} \, \mathrm{T}}\right]^{3 / 2} \nonumber \]
\(\mathrm{S}_{\gamma}\) is written in the following form.
\[\mathrm{S}_{\gamma}=\mathrm{E} \,\left[\mathrm{V}_{1}^{*}(\ell)\right]^{-1 / 2} \,\left(\varepsilon_{\mathrm{r}}\right)^{-3 / 2} \,(\mathrm{T})^{-3 / 2} \nonumber \]
where
\[\mathrm{E}=\left[2 \, \pi \, \mathrm{N}_{\mathrm{A}} \, \mathrm{M}_{1} \, \mathrm{m}^{0}\right]^{1 / 2} \,\left[\frac{\mathrm{e}^{2} \, \mathrm{N}_{\mathrm{A}}}{4 \, \pi \, \varepsilon^{0} \, \mathrm{R}}\right]^{3 / 2} \nonumber \]
Hence[1]
\[\mathrm{S}_{\gamma}=\mathrm{E} \, \mathrm{F} \nonumber \]
where
\[\mathrm{F}=\left[\mathrm{V}_{1}^{*}(\ell)\right]^{-1 / 2} \,\left(\varepsilon_{\mathrm{r}}\right)^{-3 / 2} \,(\mathrm{T})^{-3 / 2} \nonumber \]
In terms of our interest in the dependence of \(\mu_{\mathrm{j}}(\mathrm{aq})\) for salt \(j\) on temperature leading to partial molar enthalpies we require \(\left[\partial \mathrm{S}_{\gamma} / \partial \mathrm{T}\right]_{\mathrm{p}}\) which is calculated using the dependences of both \(\mathrm{V}_{1}^{*}(\ell)\) and \(\varepsilon_{r}\) on temperature yielding \((\partial \mathrm{F} / \partial \mathrm{T})_{\mathrm{p}}\). For partial molar isobaric heat capacities we require the second differential \(\left(\partial^{2} \mathrm{~F} / \partial \mathrm{T}^{2}\right)_{\mathrm{P}}\). The predicted dependence by DHLL of the partial molar volume \(\mathrm{V}_{\mathrm{j}}(\mathrm{aq})\) on salt molality involves the derivative \((\partial \mathrm{F} / \partial \mathrm{p})_{\mathrm{T}}\).
Calculations are considerably helped using a PC in conjunction with equations describing the \(\mathrm{T} - \mathrm{p}\) dependences of \(\mathrm{V}_{1}^{*}(\ell)\) and \(\varepsilon_{r}\).
Footnotes
[1] \(\mathrm{E}=\left[[1] \,[1] \,\left[\mathrm{mol}^{-1}\right] \,\left[\mathrm{kg} \mathrm{mol}^{-1}\right] \,\left[\mathrm{mol} \mathrm{kg}^{-1}\right]\right]^{1 / 2} \,\left[\frac{[\mathrm{C}]^{2} \,\left[\mathrm{mol}^{-1}\right]}{[1] \,[1] \,\left[\mathrm{C}^{2} \mathrm{~J}^{-1} \mathrm{~m}^{-1}\right] \,\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right.}\right]^{3 / 2}\)
or, \(\mathrm{E}=[\mathrm{mol}]^{-1 / 2} \,[\mathrm{m}]^{3 / 2} \,[\mathrm{K}]^{3 / 2}=\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right]^{1 / 2} \,[\mathrm{K}]^{3 / 2}\)
and \(\mathrm{F}=\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right]^{-1 / 2} \,[1]^{-3 / 2} \,[\mathrm{K}]^{-3 / 2}\)
Hence, \(\mathrm{S}_{\gamma}=\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right]^{1 / 2} \,[\mathrm{K}]^{3 / 2} \,\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right]^{-1 / 2} \,[1]^{-3 / 2} \,[\mathrm{K}]^{-3 / 2}=[1]\)