# Definition of activation energy

There seems to be contradictions in the definitions of activation energy ($$E_a$$) stated by different sources. I found two such popular definitions. Before that, I state my understanding of how an elementary reaction generally occurs, as it will be used to ask the main question.

The reactants have a certain potential energy $$U_i$$ per molecule (which is constant for each molecule unless the molecules come too close). The reactants must pass through the transition state in order to form products, and the configuration in the transition state has a potential energy $$U$$ (per molecule) associated with it. Thus the goal is to increase the potential energy of the reactants from $$U_i$$ to $$U$$. In order to do this, the configuration of the molecules of the reactants must change, which can either be achieved by physical distortion of the molecule, or changing their location in space (because electrostatic interactions are present). The former can be achieved by collisions, and the latter by the molecules just approaching each other and coming too close. In any case, the increasing potential energy must be compensated by a decrease in the molecules' kinetic energies. If the molecules have kinetic energies $$K_1$$ and $$K_2$$, then those two molecules would be able to convert into products if $$K_1 + K_2 \geq 2(U - U_i)$$. If I assume $$K_1 = K_2 = K$$ in order to simplify things for the following discussion, then the condition to form products would become $$K \geq U - U_i$$.

Definition 1:

Activation energy is the minimum amount of energy that must be provided to the reactants for them to form products. (source: Wikipedia) (not stated exactly)

This implies that $$E_a$$ is the additional energy that needs to be supplied to the reactants in order to form products. According to this definition, $$E_a = U - U_i$$. I can rephrase that into this (using the relation $$K \geq U - U_i$$): $$E_a$$ is the minimum kinetic energy a molecule should possess in order to form products.

Definition 2:

The activation energy is the minimum energy reactants must have in order to form products. (source: Physical Chemistry by Peter Atkins)

This definition suggests that $$E_a = U$$, which is not in agreement to the first one. Someone could argue that energy in this definition means kinetic energy, and then the definition is equivalent to the first one. But in the first definition, energy was interpreted as potential energy, and that should be the case with this definition too. If I instead interpret it as kinetic energy, then again these two definitions contradict each other. Also, in the text preceding this definition, kinetic energy was not mentioned anywhere, so it is natural to think they are talking about potential energy.

Why does it matter so much?

Activation energy appears in the Arrhenius equation.

$$$$k = Ae^{-E_a/RT}$$$$

The two definitions would give different values of $$k$$, so obviously one of the definitions is wrong.

(I am currently studying chemical kinetics)

I would like you to clarify all this and state a final definition along with interpretation (and verifying my understanding of a reaction will be much appreciated). Also, I have come across the term threshold energy while going through a few sources, and they define it to be $$U$$ (i.e., potential energy of the activated complex). This leads me into thinking that $$E_a = U - U_i$$, and so the second definition might be wrong.

• Distortion may happen and may be necessary for some reactions. But it is not necessary generally and may not be even possible. Sep 3, 2022 at 15:14
• It may depend on what you mean by distortion. Consider 2 O -> O2. PE increase may be based on initial electrostatic repulsion, without distortion in sense of change of atomic nuclei relative placement. Sep 3, 2022 at 15:32
• Chemistry is the part of physics about electrostatic interactions of atoms. The Coulomb law is the basis of quantum chemistry. Sep 3, 2022 at 16:38
• Is the reason for all this you didn't got that if you "provide" energy to reactants then they "have" energy? Because it sounds like that... Sep 3, 2022 at 17:00
• It seems as if it is a matter of where the zero of energy is taken to be. Definition 1 assumes that the energy is initially zero, and definition 2 assumes that it is the current thermal energy. Both are correct although neither states things clearly. Sep 3, 2022 at 17:35