Activation energy

In chemistry and physics, **activation energy** is the energy which must be provided to a chemical or nuclear system with potential reactants to result in: a chemical reaction, nuclear reaction, or various other physical phenomena.

The activation energy (*E*a) of a reaction is measured in joules (J) and or kilojoules per mole (kJ/mol) or kilocalories per mole (kcal/mol).

Activation energy can be thought of as the magnitude of the potential barrier (sometimes called the energy barrier) separating minima of the potential energy surface pertaining to the initial and final thermodynamic state. For a chemical reaction, or division to proceed at a reasonable rate, the temperature of the system should be high enough such that there exists an appreciable number of molecules with translational energy equal to or greater than the activation energy.

The term Activation Energy was introduced in 1889 by the Swedish scientist Svante Arrhenius.

## Temperature dependence and the relation to the Arrhenius equation

Main articles: Arrhenius equation
The Arrhenius equation gives the quantitative basis of the relationship between the activation energy and the rate at which a reaction proceeds. From the equation, the activation energy can be found through the relation

# k
A

e

−

E

a

/

(
R
T
)

{\displaystyle k=Ae^{{-E\_{\textrm {a}}}/{(RT)}}}

where *A* is the pre-exponential factor for the reaction, *R* is the universal gas constant, *T* is the absolute temperature (usually in kelvins), and *k* is the reaction rate coefficient. Even without knowing *A*, *E*a can be evaluated from the variation in reaction rate coefficients as a function of temperature (within the validity of the Arrhenius equation).

At a more advanced level, the net Arrhenius activation energy term from the Arrhenius equation is best regarded as an experimentally determined parameter that indicates the sensitivity of the reaction rate to temperature. There are two objections to associating this activation energy with the threshold barrier for an elementary reaction. First, it is often unclear as to whether or not reaction does proceed in one step; threshold barriers that are averaged out over all elementary steps have little theoretical value. Second, even if the reaction being studied is elementary, a spectrum of individual collisions contributes to rate constants obtained from bulk ('bulb') experiments involving billions of molecules, with many different reactant collision geometries and angles, different translational and (possibly) vibrational energies—all of which may lead to different microscopic reaction rates.

## Negative activation energy

In some cases, rates of reaction *decrease* with increasing temperature. When following an approximately exponential relationship so the rate constant can still be fit to an Arrhenius expression, this results in a negative value of *E*a. Elementary reactions exhibiting these negative activation energies are typically barrierless reactions, in which the reaction proceeding relies on the capture of the molecules in a potential well. Increasing the temperature leads to a reduced probability of the colliding molecules capturing one another (with more glancing collisions not leading to reaction as the higher momentum carries the colliding particles out of the potential well), expressed as a reaction cross section that decreases with increasing temperature. Such a situation no longer leads itself to direct interpretations as the height of a potential barrier.

## Catalysts

Main articles: Catalysis
thumbA substance that modifies the transition state to lower the activation energy is termed a catalyst; a catalyst composed only of protein and (if applicable) small molecule cofactors is termed an enzyme. A catalyst increases the rate of reaction without being consumed in the reaction. In addition, the catalyst lowers the activation energy, but it does not change the energies of the original reactants or products, and so does not change equilibrium. Rather, the reactant energy and the product energy remain the same and only the *activation energy* is altered (lowered).

## Relationship with Gibbs energy of activation

Main articles: Transition state theoryIn the Arrhenius equation, the term activation energy (*E*a) is used to describe the energy required to reach the transition state, and the exponential relationship *k* = *A* exp(*E*a/*RT*) holds. In transition state theory, a more sophisticated model of the relationship between reaction rates and the transition state, a superficially similar mathematical relationship, the Eyring equation, is used to describe the rate of a reaction: *k* = (*k*B*T* / *h*) exp(–Δ*G*‡ / *RT*). However, instead of modeling the temperature dependence of reaction rate phenomenologically, the Eyring equation models individual elementary step of a reaction. Thus, for a multistep process, there is no straightforward relationship between the two models. Nevertheless, the functional forms of the Arrhenius and Eyring equations are similar, and for a one-step process, simple and chemically meaningful correspondences can be drawn between Arrhenius and Eyring parameters.
Instead of also using *E*a, the Eyring equation uses the concept of Gibbs energy and the symbol Δ*G*‡ to denote the Gibbs energy of activation for the transition state. In the equation, *k*B and *h* are the Boltzmann and Planck constants, respectively. Although the equations look similar, it is important to note that the Gibbs energy contains an entropic term in addition to the enthalpic one. In the Arrhenius equation, this entropic term is accounted for by the pre-exponential factor *A*. More specifically, we can write the Gibbs free energy of activation in terms of enthalpy and entropy of activation: Δ*G*‡ = Δ*H*‡ – *T* Δ*S*‡. Then, for a unimolecular, one-step reaction, the *approximate* relationships *E*a = Δ*H*‡ + *RT* and *A* = (*k*B*T*/*h*) exp(1 + Δ*S*‡/*R*) hold. Note, however, that in Arrhenius theory proper, *A* is temperature independent, while here, there is a linear dependence on *T*. For a one-step unimolecular process whose half-life at room temperature is about 2 hours, Δ*G*‡ is approximately 23 kcal/mol. This is also the roughly the magnitude of *E*a for a reaction that proceeds over several hours at room temperature. Due to the relatively small magnitude of *T*Δ*S*‡ and *RT* at ordinary temperatures for most reactions, in sloppy discourse, *E*a, Δ*G*‡, and Δ*H*‡ are often conflated and all referred to as the "activation energy".