Definition of activation energy

Question

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 $K \geq U - U_i$.

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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.

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Why does it matter so much?

Activation energy appears in the Arrhenius equation.

$\begin{equation} k = Ae^{-E_a/RT} \end{equation}$

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

(I am currently studying chemical kinetics)

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So what should I answer?

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.

Answer 1

A few basic ideas: A molecule, compound, atom, etc. is stable [possibly only fleetingly so] because it is at a relative energy minimum. For a moiety to react energy must be available to break a bond or attraction. This energy can come from the molecule itself either from the initial KE of the molecules that collide and form a temporary attraction or from being in such a high rotational or vibrational state that the molecular bond is broken [possibly from photoinitiation]. At any temperature above zero Kelvin there is a distribution of energies; the higher the temperature the greater the distribution and the more molecules that either have enough individual energy to become unstable or in collisions with enough total kinetic energy to either break bonds or to cause orbitals to readjust and form orbitals of lower energy [not necessarily of lower energy than the original bonds, transition states can lead to reactive intermediates]. This latter also requires a method to release the excess energy. The activation energy is not added energy it is inherent in the distribution of molecular energies that is a function of the temperature. The molecules with the higher KEs might have it.

The idea is that there is a required energy for a reaction to happen, the activation energy. The activation energy comes from the KINETIC energy in the heat content of the molecules. The reaction energy comes from the different POTENTAL energies from the different bond strengths between the reactants and products in other words how close the electrons can get to the various nuclei. How precise this value is I have no idea, but precisions in kinetics and calculations are usually in the low parts per thousand. The energy or enthalpy change is measured or calculated, and the activation energy is calculated from the kinetics at several temperatures.

The various definitions seem, to me at least, to be fumbled attempts to explain something relatively simple in a more complicated manner.