Hybrid Models¶
Abstract
Equilibria¶
\(pK_a\) (Born-Haber cycle)¶
\(pK_a\) is typically calculated using the Born-Haber cycle with the equation:
Where \(\Delta G^{\circ\prime}=\Delta G^{\circ\prime}_{aq}\)
\[
\begin{align}
pK_a&=-\log\bigg[\exp{\bigg(\frac{-\Delta G^{\circ\prime}}{RT}\bigg)}\bigg]\\
&=\frac{\Delta G^{\circ\prime}}{2.303RT}
\end{align}
\]
Using the free energy cycle:
Process:¶
- For the anion, I need diffuse functions, a big basis set and a good level of theory
- QM packages won’t let you calculate the electronic energy of \(\ce{H+}\), since there’s no electron to calculate
- \(\Delta G^\circ_{(s)}=-264.0\:kcal\cdot mol^{-1}\) (experimentally derived)
- Standard-state concentration-change free energy must be included
- each non-cancelled error of \(1.4\:kcal\cdot mol^{-1}\) in any step will lead to an error of 1 \(pK_a\) unit
- Errors in ionic solvation free energies can be much larger than that
- Function-group systematic errors can be corrected for
The overall equation¶
We can condense this all into the one line equation:
\[
2.303RT\:pK_a=\Delta G_g^\circ(AH)-\Delta G^*_{aq}(AH)+\Delta G^*_{aq}(A^-)+\Delta G_{aq}^*(H^+)
\]
Using this free energy cycle:
Free energy cycles and ion structures¶
Here we’re treating the ion as a cluster, in an attempt to try and reduce the amount error on the cluster:
\[
2.303RT \: pK_a = \Delta G_g^\circ (AH) - \Delta G^*_{aq}(AH) - \Delta G^*_{aq}(\ce{H2O})+\Delta G^*_{aq}(\ce{H2O}\cdot A^-)+\Delta G_{aq}^*(H^+)
\]
Utilising this free energy cycle:
Comparison (Experimental Data \(pK_a=15.5\))¶
Experimental data¶
| Method 1 \(\ce{MeOH/MeO-}\) | Method 2 \(\ce{MeOH/H2O.MeO-}\) |
|---|---|
| \(\Delta G^\circ_g=375.0\:kcal\cdot mol^{-1}\) | \(\Delta G^\circ_g=358.0\:kcal\cdot mol^{-1}\) |
| \(\Delta G_{aq}^*(\ce{H+})=-265.9\:kcal\cdot mol^{-1}\) | \(\Delta G_{aq}^*(\ce{H+})=-265.9\:kcal\cdot mol^{-1}\) |
| \(\Delta G_{aq}^*(\ce{MeOH})=-5.11\:kcal\cdot mol^{-1}\) | \(\Delta G_{aq}^*(\ce{MeOH})=-5.11\:kcal\cdot mol^{-1}\) |
| \(\Delta G_{aq}^*(\ce{H2O})=-6.32\:kcal\cdot mol^{-1}\) |
Computed data (SM6)¶
| Method 1 \(\ce{MeOH/MeO-}\) | Method 2 \(\ce{MeOH/H2O.MeO-}\) |
|---|---|
| \(\Delta G^*_{aq}(\ce{MeO-})=-88.3\:kcal\cdot mol^{-1}\) | \(\Delta G^*_{aq}(\ce{H2O.MeO-})=-81.8\:kcal\cdot mol^{-1}\) |
| \(pK_a=20.4\) | \(pK_a=16.0\) |
Adding more waters¶
This is the experimental \(pK_a\) data
\[
\ce{H2CO3 <=>[pK_{a_1}=6.4] HCO3- <=>[pK_{a_2}=10.3] CO3^{2-}}
\]
Without adding explicit waters of solvation, the results are insanely inaccurate, however when we add water molecules in, the structure becomes much more stabilised and the \(pK_a\) is much more in line with experiment.
| No. \(\ce{H2O}\) | \(pK_{a_1}\) | \(pK_{a_2}\) |
|---|---|---|
| 0 | -0.6 | 1.6 |
| 1 | 1.3 | 5.0 |
| 2 | 2.3 | 7.8 |
| 3 | 4.2 | 9.0 |