1.22.8: Volume: Salt Solutions: Born-Drude-Nernst Equation
Differentiation of the Born Equation with respect to pressure (at fixed temperature) yields the Born-Drude-Nernst Equation which describes the difference in partial molar volumes of ion \(j\) in the gas phase and in solution. The simplest model assumes that the radius \(\mathrm{r}_{j}\) is independent of pressure [1].
\[\begin{aligned}
\Delta(\mathrm{pfg}&\rightarrow \mathrm{s} \ln ) \mathrm{V}_{\mathrm{j}}\left(\mathrm{c}_{\mathrm{j}}=1 \mathrm{~mol} \mathrm{dm} \mathrm{m}^{-3} ; \mathrm{id} ; \mathrm{p}, \mathrm{T}\right)=\\
&-\mathrm{N}_{\mathrm{A}} \,\left(\mathrm{z}_{\mathrm{j}} \, \mathrm{e}\right)^{2} \,\left[\frac{1}{\varepsilon_{\mathrm{r}}} \,\left(\frac{\partial \varepsilon_{\mathrm{r}}}{\partial \mathrm{p}}\right)_{\mathrm{T}}\right] \, \frac{1}{8 \, \pi \, \mathrm{r}_{\mathrm{j}} \, \varepsilon_{0}}
\end{aligned} \nonumber \]
A more complicated equation emerges if radius \(\mathrm{r}_{j}\) is assumed to depend on pressure, but there seems little merit in taking account of such a dependence.
Footnote
[1]
\[\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right]=\left[\mathrm{mol}^{-1}\right] \,\left[\mathrm{A}^{2} \mathrm{~s}^{2}\right] \,\left[\mathrm{N} \mathrm{m}^{-2}\right]^{-1} \,[1]^{-1} \,[1]^{-1} \,\left[\mathrm{m}^{-1}\right] \,\left[\mathrm{F} \mathrm{m}^{-1}\right]^{-} \nonumber \]
where, \(\left[\mathrm{F} \mathrm{m}^{-1}\right]=\left[\mathrm{A}^{2} \mathrm{~s}^{4} \mathrm{~kg}^{-1} \mathrm{~m}^{-3}\right]\)