1.8.12: Enthalpies- Born-Bjerrum Equation- Salt Solutions
- Page ID
- 375300
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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.
\[\begin{aligned}
&C_{p j}\left(\operatorname{sln} ; c_{j}=1 \mathrm{~mol} \mathrm{dm}{ }^{-3} ; \mathrm{id} ; \mathrm{T} ; \mathrm{p}\right) \\
&=-\left[\mathrm{N}_{\mathrm{A}} \,\left(\mathrm{z}_{\mathrm{j}} \, \mathrm{e}\right)^{2} / 8 \, \pi \, \mathrm{r}_{\mathrm{j}} \, \varepsilon_{0}\right] \,\left[\partial\left\{\left(1 / \varepsilon_{\mathrm{r}}\right)+\left(\mathrm{T} / \varepsilon_{\mathrm{r}}\right) \, \partial \ln \varepsilon_{\mathrm{r}} / \partial \mathrm{T}\right\} / \partial \mathrm{T}\right]
\end{aligned} \nonumber \]
Footnotes
[1] F. A. Askew, E. Bullock, H. T. Smith, R. K. Tinkler, O. Gatty and J. H. Wolfenden, J. Chem. Soc., 1934, 1368.
[2] For estimation of single in enthalpies see M. Booij and G. Somsen, Electrochim Acta,1983,28,1883.