1.10.21: Gibbs Energies- Salt Solutions- DHLL- Empirical Modifications
The success of equations based on the Debye-Huckel equations is often modest and so attempts are made to describe quantitatively the dependences of \(\phi\), \(\ln \left(\gamma_{\pm}\right)\) and \(\mathrm{G}_{\mathrm{m}}^{\mathrm{E}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})\) on \(\mathrm{m}_{j}\) to higher molalities. In most cases attempts are made to moderate the stabilization of the salt with increasing ionic strength. The obvious procedure centres on incorporating a denominator into the DHLL as illustrated by the Guntleberg equation.
\[\ln \left(\gamma_{\pm}\right)=-\left|\mathrm{z}_{+} \, \mathrm{z}_{-}\right| \, \mathrm{S}_{\gamma} \,\left(\mathrm{I} / \mathrm{m}^{0}\right)^{1 / 2} /\left\{1+\left(\mathrm{I} / \mathrm{m}^{0}\right)^{1 / 2}\right\} \nonumber \]
The Guggenheim Equation starts with equation (a) and adds a further term, linear in ionic strength.
\[\ln \left(\gamma_{\pm}\right)=-\left[\left|\mathrm{z}_{+} \, \mathrm{z}_{-}\right| \, \mathrm{S}_{\gamma} \,\left(\mathrm{I} / \mathrm{m}^{0}\right)^{1 / 2} /\left\{1+\left(\mathrm{I} / \mathrm{m}^{0}\right)^{1 / 2}\right\}\right]+\mathrm{b} \,\left(\mathrm{I} / \mathrm{m}^{0}\right) \nonumber \]
The quantity ‘b’ is characteristic of the salt. Another obvious development uses the same approach in the context of the DHLL. An interesting equation takes the following form for the solution containing a salt \(j\).
\[\ln \left(\gamma_{\pm}\right)=-\left[\left|\mathrm{z}_{+} \, \mathrm{z}_{-}\right| \, \mathrm{S}_{\gamma} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{1 / 2}\right]+\mathrm{B} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right) \nonumber \]
Here \(\mathrm{B}\) describes the role of ion size and the impact of cosphere-cosphere interactions specific to a particular salt.
In most approaches, the starting point in an equation for \(\ln \left(\gamma_{\pm}\right)\) as a function of ionic strength, the equation for the dependence of \(\phi\) on ionic strength being obtained using the integral of equation (c). An interesting approach suggested by Bronsted starts out with a virial equation for \(1 - \phi\) in terms of molality \(\mathrm{m}_{j}\).
\[1-\phi=\alpha \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{1 / 2}+\beta \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right) \nonumber \]
Hence [1]
\[\ln \left(\gamma_{\pm}\right)=-3 \, \alpha \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{1 / 2}-2 \, \beta \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right) \nonumber \]
Footnote
[1] From
\[\begin{aligned}
&\ln \left(\gamma_{\pm}\right)=(\phi-1)+\int_{0}^{m_{j}}(\phi-1) \, \mathrm{d} \ln m_{j} \\
&\ln \left(\gamma_{\pm}\right)=(\phi-1)-\int_{0}^{\mathrm{m}_{\mathrm{j}}}\left[\left\{\alpha \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{1 / 2}+\beta \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)\right\} / \mathrm{m}_{\mathrm{j}}\right] \, \mathrm{dm}_{\mathrm{j}} \\
&\left.\ln \left(\gamma_{\pm}\right)=(\phi-1)-\int_{0}^{m_{j}}\left[\left\{\alpha / m_{j} \, m^{0}\right)^{1 / 2}\right\}+\left\{\beta / m^{0}\right\}\right] \, d_{j} \\
&\ln \left(\gamma_{\pm}\right)=(\phi-1)-\left[2 \, \alpha \,\left(m_{j} / m^{0}\right)^{1 / 2}+\beta \,\left(m_{j} / m^{0}\right)\right]_{0}^{m_{j}} \\
&\ln \left(\gamma_{\pm}\right)=-\alpha \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{1 / 2}-\beta \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)-2 \, \alpha \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{1 / 2}-\beta \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right) \\
& \ln \left(\gamma_{\pm}\right)=-3 \, \alpha \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{1 / 2}-2 \, \beta \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)
\end{aligned} \nonumber \]