Editing Activation energy (section)

== Relationship with Gibbs energy of activation ==
{{Main|Transition state theory}}In the [[Arrhenius equation]], the term activation energy (''E''<sub>a</sub>) is used to describe the energy required [[reaction coordinate|to reach the]] [[transition state]], and the exponential relationship {{math|1=''k'' = ''A'' exp(−''E''<sub>a</sub>/''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 constant of a reaction: {{math|1=''k'' = (''k''<sub>B</sub>''T'' / ''h'') exp(−Δ''G''<sup>‡</sup> / ''RT'')}}.  However, instead of modeling the temperature dependence of reaction rate phenomenologically, the Eyring equation models individual elementary steps 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''<sub>a</sub>, the Eyring equation uses the concept of [[Gibbs energy]] and the symbol Δ''G''<sup>‡</sup> to denote the Gibbs energy of activation to achieve the [[transition state]]. In the equation, ''k''<sub>B</sub> 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 [[entropy|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]]: {{math|1=Δ''G''<sup>‡</sup> = Δ''H''<sup>‡</sup> − ''T'' Δ''S''<sup>‡</sup>}}.  Then, for a unimolecular, one-step reaction, the ''approximate'' relationships {{math|1=''E''<sub>a</sub> = Δ''H''<sup>‡</sup> + ''RT''}} and {{math|1=''A'' = (''k''<sub>B</sub>''T''/''h'') exp(1 + Δ''S''<sup>‡</sup>/''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''<sup>‡</sup> is approximately 23 kcal/mol.  This is also the roughly the magnitude of ''E''<sub>a</sub> for a reaction that proceeds over several hours at room temperature.  Due to the relatively small magnitude of ''T''Δ''S''<sup>‡</sup> and ''RT'' at ordinary temperatures for most reactions, in sloppy discourse,  ''E''<sub>a</sub>, Δ''G''<sup>‡</sup>, and Δ''H''<sup>‡</sup> are often conflated and all referred to as the "activation energy".

The enthalpy, entropy and Gibbs energy of activation are more correctly written as Δ<sup>‡</sup>''H''<sup>o</sup>, Δ<sup>‡</sup>''S''<sup>o</sup> and Δ<sup>‡</sup>''G''<sup>o</sup> respectively, where the o indicates a quantity evaluated between [[standard state]]s.<ref>{{cite journal |title=Enthalpy of activation |url=http://goldbook.iupac.org/terms/view/E02142 |website=IUPAC Gold Book (2nd edition, on-line version) |publisher=IUPAC (International Union of Pure and Applied Chemistry) |access-date=10 May 2020 |date=2019 |doi=10.1351/goldbook.E02142 |archive-date=21 February 2020 |archive-url=https://web.archive.org/web/20200221221256/http://goldbook.iupac.org/terms/view/E02142 |url-status=live |doi-access=free }}</ref><ref>{{cite book |last1=Steinfeld |first1=Jeffrey I. |last2=Francisco |first2=Joseph S.|last3=Hase |first3=William L.|title=Chemical Kinetics and Dynamics |date=1999 |publisher=Prentice Hall |isbn=0-13-737123-3 |page=301 |edition=2nd}}</ref> However, some authors omit the o in order to simplify the notation.<ref>{{cite book |last1=Atkins |first1=Peter |last2=de Paula |first2=Julio |title=Atkins' Physical Chemistry |url=https://archive.org/details/physicalchemistr00atki_572 |url-access=limited |date=2006 |publisher=W.H.Freeman |isbn=0-7167-8759-8 |page=[https://archive.org/details/physicalchemistr00atki_572/page/n914 883] |edition=8th |quote=... but we shall omit the standard state sign to avoid overburdening the notation.}}</ref><ref>{{cite book |last1=Laidler |first1=Keith J. |last2=Meiser |first2=John H.|title=Physical Chemistry |date=1982 |publisher=Benjamin/Cummings |isbn=0-8053-5682-7 |page=381}}</ref>

The total free energy change of a reaction is independent of the activation energy however. Physical and chemical reactions can be either [[Exergonic reaction|exergonic]] or [[Endergonic reaction|endergonic]], but the activation energy is not related to the [[Spontaneous process|spontaneity]] of a reaction. The overall reaction energy change is not altered by the activation energy.