1.10.19: Gibbs Energies- Salt Solutions- Debye-Huckel Limiting Law
According to the Debye-Huckel analysis the mean ionic activity coefficient is given by equation (a).
\[\ln \left(\gamma_{\pm}\right)=-\left[\frac{\left|z_{+} \, z_{-}\right| \, e^{2} \, N_{A}}{8 \, \pi \, \varepsilon_{0} \, \varepsilon_{t} \, R \, T} \, \frac{K}{(1+K \, a)}\right] \nonumber \]
In the DHLL the term \((1+\kappa \, a)\) is approximated to unity thereby assuming that \((1>>\kappa \, a)\). By definition
\[\mathrm{S}_{\mathrm{\gamma}}=\left[\frac{2 \, \pi \, \mathrm{N}_{\mathrm{A}} \, \mathrm{M}_{1} \, \mathrm{m}^{0}}{\mathrm{~V}_{1}^{*}(\ell)}\right]^{1 / 2} \,\left[\frac{\mathrm{e}^{2} \, \mathrm{N}_{\mathrm{A}}}{4 \, \pi \, \varepsilon_{0} \, \varepsilon_{\mathrm{r}} \, \mathrm{R} \, \mathrm{T}}\right]^{3 / 2} \nonumber \]
Then[1]
\[\ln \left(\gamma_{\pm}\right)=-\left|z_{+} \, z_{-}\right| \, S_{\gamma} \,\left(m_{j} / m^{0}\right)^{1 / 2} \nonumber \]
The practical osmotic coefficient for quite dilute salt solutions is also a linear function of \(\left(\mathrm{m}_{j}\right)^{1 / 2}. Indeed \(\phi\) and \(\ln \left(\gamma_{\pm}\right)\) are simply related.
\[1-\phi=-(1 / 3) \, \ln \left(\gamma_{\pm}\right) \nonumber \]
Footnotes
[1] From,
\[\mathrm{S}_{\gamma}=\frac{\mathrm{e}^{3} \,\left[2 \, \mathrm{N}_{\mathrm{A}} \, \rho_{1}^{*}(\ell)\right]^{1 / 2}}{8 \, \pi \,\left[\varepsilon_{0} \, \varepsilon_{\mathrm{r}} \, \mathrm{k} \, \mathrm{T}\right]^{3 / 2}} \nonumber \]
Then, \(\frac{1}{\pi}=\frac{\pi^{1 / 2}}{\pi^{3 / 2}}\) and \(\frac{1}{8}=\frac{1}{4^{3 / 2}}\)