1.10.20: Gibbs Energies- Salt Solutions- DHLL- Derived Parameters
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, \(\mathrm{S}_{\gamma}=\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right]^{1 / 2} \,[\mathrm{K}]^{3 / 2} \,\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right]^{-1 / 2} \,[1]^{-3 / 2} \,[\mathrm{K}]^{-3 / 2}=[1]\)