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Is it possible for a certain spontaneous reaction to have zero or negative activation energy?

My high school teacher explains that it shouldn't be possible since it breaks the Arrhenius equation, namely the Boltzmann factor

$$\exp\left(\frac{-E_\mathrm{a}}{RT}\right),$$

which gives the fraction of molecules having $E_\mathrm{kin} > E_\mathrm{a}.$ So, instead of putting in $E_\mathrm{a} < 0$, we will just say the Boltzmann factor is equal to 1. The Boltzmann factor just doesn't make sense for a negative $E_\mathrm{a}$ as $E_\mathrm{kin}$ cannot be negative.

Instead of saying it breaks the factor, should we not just say that the factor is undefined for negative energy values and that $E_\mathrm{a}$ can be zero or negative?

Edit-so people are commenting on the question and I am very thankful.but i also want to know that if this was a test question with a simple true or false answer,would you say that $E_a can be<0$ or not?

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  • $\begingroup$ I would rather say that for low enough $E_a$ the Boltzmann energy factor is negligibly different to 1 and the reaction is driven by diffusion. I.e. whenever reactants meet and are properly oriented geometrically, then they react. Typical is neutralization reaction $\ce{H+(aq) + OH-(aq) -> H2O(l)}$. $\endgroup$
    – Poutnik
    May 26 at 10:09
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    $\begingroup$ Bond formation between atoms (e.g. $\ce{H + H -> H2}$) is most often exothermic/exergonic and has zero activation energy. $\endgroup$
    – Nicolau Saker Neto
    May 26 at 10:23
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    $\begingroup$ IMHO, an eventual formal reaction with negative activation energy are 2 reactions closely linked to each other via a common major component, that is thermodynamically preferred to both reagents and products of the former single reaction. $\endgroup$
    – Poutnik
    May 26 at 10:56
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    $\begingroup$ Marcus Theory points to some cases of electron transfer that are in principle barrierless (or very close to barrierless). $\endgroup$
    – Zhe
    May 26 at 13:21
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    $\begingroup$ Negative activation energy is not possible in a simple reaction, the smallest value is zero. Experimentally as @Zhe points out, activationless reactions are seen in solution for electron transfer, but only when the two species are held together, such as intercalation of a dye in DNA, otherwise if free to diffuse the rate constant is limited by how quickly they can diffuse together. $\endgroup$
    – porphyrin
    May 26 at 14:13

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The short answer: yes, but it is not the most common case.

How it can be? You assume that you reaction has a simple reaction profile with a barrier and a single transition state between reactions and products, plus other approximations. When those approximations fail, the model is not helpful and you have to use a deeper model.

A longer, and hopefully, more nuanced explanation.

For many reactions the dependence of the reaction rate with temperature follows the Arrhnius equation $$k(T) = A\exp \left(\frac{-E_a}{R\,T}\right)$$ where A, the preexponential factor, and $E_a$ are independent of temperature.

However, you may have that over an extended temperature range many reactions deviate from this behaviour and their behaviour are described better using more complex expressions that behaves as if $A$ and $E_a$ are functions of temperatura. For example, the Kooij equation that has the form

$$k(T) = A T^m \exp\left(\frac{-E_a}{R\,T}\right) $$

As an actual example you can consider the reactión OH + CH$_4$ in the 220-500 K which behaves as Ahhrenius but in the extended interval 220-2000 deviates from that behaviour as the following image shows. Rate constant temperature dependence for the OH + CH4 reaction

Thus, at present the Arrhenius equation is understood as very useful empirical fitting equation, when it can be applied. But you have to interpret it carefully.

There are some important reactions that have no activation barrier, for example, radical recombinations CH$_{3}\cdot$ + CH$_{3}\cdot$ $\leftarrow$ CH$_3$CH$_3$, or the atmospheric reactions of O($^1$D) -e.g. O($^1$D) + CH$4$-.

Finally, there are some elementary reactions which rate decreases with temperature and fit Arrhenius, thus, its activation energy is negative ($E_a < 0$). For example, the gas phase reaction OH + HNO$_3$ $\leftarrow$ H$_2$O + NO$_3$. Its energy profile does not have a negative barrier, is exhothermic but the reaction profile is complex.

OH + HNO3

Reaction profile OH + HNO3

The reaction profile is adapted from the article Brown S.S. et al., J. Phys. Chem. A, 1999 103, 3031-3037

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    $\begingroup$ Well this was asked in a test and my answer that they can was considered wrong,well thanks anyways $\endgroup$
    – Karan
    May 31 at 5:09

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