C The Relationship Between \(\Delta H^\ddagger\) and the \(E_\mathrm{a}\)
In Section 6.4, we identified a correspondence between the transition state theory rate equation and the Arrhenius equation. The enthalpy of activation \(\Delta H^\ddagger\) and the Arrhenius activation energy \(E_\mathrm{a}\) play similar roles in these equations, but they are not identical quantities. Here we examine their precise relationship.
C.1 Temperature Dependence of Rate Constants
The Arrhenius equation defines the temperature dependence of rate constants empirically:
\[\begin{equation} k = A\mathrm{e}^{-E_\mathrm{a}/RT} \tag{5.5} \end{equation}\]
Taking logarithms:
\[\begin{equation} \ln k = \ln A - \frac{E_\mathrm{a}}{RT} \tag{C.1} \end{equation}\]
Differentiating with respect to temperature10
\[\begin{equation} \frac{\mathrm{d}\ln k}{\mathrm{d}T} = \frac{E_\mathrm{a}}{RT^2} \tag{C.2} \end{equation}\]
This equation can be considered a formal definition of the Arrhenius activation energy.
For a bimolecular reaction, the thermodynamic form of the transition state theory rate equation is:
\[\begin{equation} k = \frac{k_\mathrm{B}T}{hc^\circ}\mathrm{e}^{\Delta S^\ddagger/R}\mathrm{e}^{-\Delta H^\ddagger/RT} \tag{C.3} \end{equation}\]
Taking logarithms:
\[\begin{equation} \ln k = \ln\left(\frac{k_\mathrm{B}T}{hc^\circ}\right) + \frac{\Delta S^\ddagger}{R} - \frac{\Delta H^\ddagger}{RT} \tag{C.4} \end{equation}\]
For an ideal gas, \(\frac{1}{c^\circ} = \frac{V}{N} = \frac{RT}{p}\), and we can rewrite our equation as
\[\begin{equation} \ln k = \ln\left(\frac{k_\mathrm{B}T}{h}\right) + \ln\left(\frac{RT}{p}\right) + \frac{\Delta S^\ddagger}{R} - \frac{\Delta H^\ddagger}{RT} \tag{C.5} \end{equation}\]
Differentiating with respect to temperature gives,11
\[\begin{equation} \frac{\mathrm{d}\ln k}{\mathrm{d}T} = \frac{1}{T} + \frac{1}{T} + \frac{\Delta H^\ddagger}{RT^2} \tag{C.6} \end{equation}\]
Comparing equations (C.2) and (C.6) we find:
\[\begin{equation} E_\mathrm{a} = \Delta H^\ddagger + 2RT \tag{C.7} \end{equation}\]
C.2 Effect of Reaction Molecularity
The derivation above is for a bimolecular reaction. For a unimolecular reaction, the same procedure gives a different relationship between \(E_\mathrm{a}\) and \(\Delta H^\ddagger\):
\[\begin{equation} E_\mathrm{a} = \Delta H^\ddagger + RT \tag{C.8} \end{equation}\]
The difference arises because the unimolecular TST rate equation is
\[\begin{equation} k = \frac{k_\mathrm{B}T}{h}\mathrm{e}^{\Delta S^\ddagger/R}\mathrm{e}^{-\Delta H^\ddagger/RT} \tag{C.9} \end{equation}\]
with no \(1/c^\circ\) term in the prefactor, and we lose the corresponding factor of \(RT\) when we evaluate \(\mathrm{d} \ln k / \mathrm{T}\).
Similarly, the TST rate equation for a termolecular reaction is
\[\begin{equation} k = \frac{k_\mathrm{B}T}{h{c^\circ}^2}\mathrm{e}^{\Delta S^\ddagger/R}\mathrm{e}^{-\Delta H^\ddagger/RT} \tag{C.10} \end{equation}\]
and
\[\begin{equation} E_\mathrm{a} = \Delta H^\ddagger + 3RT \tag{C.11} \end{equation}\]
where \(2RT\) comes from the squared concentration factor in the rate equation.
At room temperature, \(RT \approx 2.5\) kJ mol\(^{-1}\). The difference between \(E_\mathrm{a}\) and \(\Delta H^\ddagger\) therefore ranges from about 2.5 kJ mol\(^{-1}\) for unimolecular reactions to 7.5 kJ mol\(^{-1}\) for termolecular processes. While these differences are relatively small compared to typical activation energies, they become important when comparing activation parameters determined using different methods or analysing temperature-dependent kinetic data over wide temperature ranges.