Editing Arrhenius equation (section)

== Formulation ==
[[File:NO2 Arrhenius k against T.svg|thumb|In almost all practical cases, <math>E_a \gg RT</math> and ''k'' increases rapidly with ''T''.]]
[[File:KineticConstant.png|thumb|Mathematically, at very high temperatures so that <math>E_a \ll RT</math>, ''k'' levels off and approaches ''A'' as a limit, but this case does not occur under practical conditions.]]
The Arrhenius equation describes the [[Exponential function|exponential]] dependence of the [[rate constant]] of a chemical reaction on the [[absolute temperature]] as
<math display="block">k = Ae^\frac{- E_\text{a}}{RT},</math>
where
* {{mvar|k}} is the [[rate constant]] (frequency of collisions resulting in a reaction),
* {{mvar|T}} is the [[absolute temperature]],
* {{mvar|A}} is the [[pre-exponential factor]] or Arrhenius factor or frequency factor. Arrhenius originally considered A to be a temperature-independent constant for each chemical reaction.<ref>[http://goldbook.iupac.org/A00446.html IUPAC Goldbook definition of Arrhenius equation].</ref> However more recent treatments include some temperature dependence – see ''{{slink|#Modified Arrhenius equation}}'' below.
* {{math|''E''<sub>a</sub>}} is the molar [[activation energy]] for the reaction,
* {{mvar|R}} is the [[universal gas constant]].<ref name="Arrhenius96"/><ref name="Arrhenius226"/><ref name="Laidler42"/>

Alternatively, the equation may be expressed as
<math display="block">k = Ae^\frac{-E_\text{a}}{k_\text{B}T},</math>
where
* {{math|''E''<sub>a</sub>}} is the [[activation energy]] for the reaction (in the same unit as ''k''<sub>B</sub>''T''),
* {{math|''k''<sub>B</sub>}} is the [[Boltzmann constant]].

The only difference is the unit of {{math|''E''<sub>a</sub>}}: the former form uses energy per [[mole (unit)|mole]], which is common in chemistry, while the latter form uses energy per [[molecule]] directly, which is common in physics.
The different units are accounted for in using either the [[gas constant]], {{mvar|R}}, or the [[Boltzmann constant]], {{math|''k''<sub>B</sub>}}, as the multiplier of temperature {{mvar|T}}.

The unit of the pre-exponential factor {{mvar|A}} are identical to those of the rate constant and will vary depending on the order of the reaction. If the reaction is first order it has the unit [[second|s]]<sup>−1</sup>, and for that reason it is often called the ''[[frequency]] factor'' or ''attempt frequency'' of the reaction. Most simply, {{mvar|k}} is the number of collisions that result in a reaction per second, {{mvar|A}} is the number of collisions (leading to a reaction or not) per second occurring with the proper orientation to react<ref>{{cite book |last1=Silberberg |first1=Martin S. |title=Chemistry |url=https://archive.org/details/chemistrymolecul00silb_803 |url-access=limited |date=2006 |publisher=McGraw-Hill |location=NY |isbn=0-07-111658-3 |page=[https://archive.org/details/chemistrymolecul00silb_803/page/n728 696] |edition=fourth}}</ref> and <math>e^\frac{- E_\text{a}}{RT}</math> is the probability that any given collision will result in a reaction. It can be seen that either increasing the temperature or decreasing the activation energy (for example through the use of [[catalyst]]s) will result in an increase in rate of reaction.

Given the small temperature range of kinetic studies, it is reasonable to approximate the activation energy as being independent of the temperature. Similarly, under a wide range of practical conditions, the weak temperature dependence of the pre-exponential factor is negligible compared to the temperature dependence of the factor {{tmath|1= e^\frac{- E_\text{a} }{RT} }}; except in the case of "barrierless" [[diffusion]]-limited reactions, in which case the pre-exponential factor is dominant and is directly observable.

With this equation it can be roughly estimated that the rate of reaction increases by a factor of about 2 to 3 for every 10&nbsp;°C rise in temperature, for common values of activation energy and temperature range.<ref>{{cite book |last1=Avery |first1=H. E. |title=Basic Reaction Kinetics and Mechanisms |date=1974 |publisher=Springer |pages=47-58 |url=https://link.springer.com/chapter/10.1007/978-1-349-15520-0_4 |access-date=18 December 2023 |chapter=4. Dependence of Rate on Temperature |quote=However, the rate of reaction varies greatly with temperature, since for a typical process the rate doubles or trebles for a rise in temperature of 10&nbsp;°C.}}</ref>

The <math>e^{\frac{-E_a}{RT}}</math> factor denotes the fraction of molecules with energy greater than or equal to <math>E_a</math>.<ref>{{Cite web |date=2013-10-02 |title=6.2.3.3: The Arrhenius Law – Activation Energies |url=https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Supplemental_Modules_(Physical_and_Theoretical_Chemistry)/Kinetics/06%3A_Modeling_Reaction_Kinetics/6.02%3A_Temperature_Dependence_of_Reaction_Rates/6.2.03%3A_The_Arrhenius_Law/6.2.3.03%3A_The_Arrhenius_Law-_Activation_Energies |access-date= |website=Chemistry LibreTexts |language=en}}</ref>