1.8.12: Enthalpies- Born-Bjerrum Equation- Salt Solutions

It is generally assumed that the Born Equation yields a difference in Gibbs energies rather than Helmholtz energies and so one can use the Gibbs-Helmholtz Equation for the dependence on temperature at fixed pressure to yield the Born-Bjerrum Equation, assuming that (\(\mathrm{dr}_{\mathrm{j}} / \mathrm{dT}\)) is zero.

\[\begin{aligned}
&\Delta(\mathrm{pfg} \rightarrow \mathrm{s} \ln ) \mathrm{H}_{\mathrm{j}}\left(\mathrm{c}_{\mathrm{j}}=1 \mathrm{moldm} \mathrm{dm}^{-3} ; \mathrm{id} ; \mathrm{T} ; \mathrm{p}\right)= \\
&\quad-\left[\mathrm{N}_{\mathrm{A}} \,\left(\mathrm{z}_{\mathrm{j}} \, \mathrm{e}\right)^{2} / 8 \, \pi \, \mathrm{r}_{\mathrm{j}} \, \varepsilon_{0}\right] \,\left[1-\left(1 / \varepsilon_{\mathrm{r}}\right)-\left(\mathrm{T} / \varepsilon_{\mathrm{r}}\right) \,\left(\partial \ln \varepsilon_{\mathrm{r}} / \partial \mathrm{T}\right)_{\mathrm{p}}\right]
\end{aligned} \nonumber \]

In fact an early calorimetric study showed that in terms of predicting the enthalpies of solution for salts, the Born equation is inadequate, often predicting the wrong sign. [1,2]

Differentiation of equation (a) with respect to temperature yields an equation for the partial molar isobaric heat capacity of ion \(j\) in a solution having ideal thermodynamic properties.

\[\begin{aligned}
&C_{p j}\left(\operatorname{sln} ; c_{j}=1 \mathrm{~mol} \mathrm{dm}{ }^{-3} ; \mathrm{id} ; \mathrm{T} ; \mathrm{p}\right) \\
&=-\left[\mathrm{N}_{\mathrm{A}} \,\left(\mathrm{z}_{\mathrm{j}} \, \mathrm{e}\right)^{2} / 8 \, \pi \, \mathrm{r}_{\mathrm{j}} \, \varepsilon_{0}\right] \,\left[\partial\left\{\left(1 / \varepsilon_{\mathrm{r}}\right)+\left(\mathrm{T} / \varepsilon_{\mathrm{r}}\right) \, \partial \ln \varepsilon_{\mathrm{r}} / \partial \mathrm{T}\right\} / \partial \mathrm{T}\right]
\end{aligned} \nonumber \]

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