Editing Arrhenius equation

{{Short description|Formula for temperature dependence of rates of chemical reactions}}
In [[physical chemistry]], the '''Arrhenius equation''' is a formula for the temperature dependence of [[reaction rate]]s. The equation was proposed by [[Svante Arrhenius]] in 1889, based on the work of Dutch chemist [[Jacobus Henricus van 't Hoff]] who had noted in 1884 that the [[Van 't Hoff equation]] for the temperature dependence of [[equilibrium constant]]s suggests such a formula for the rates of both forward and reverse reactions. This equation has a vast and important application in determining the rate of chemical reactions and for calculation of [[Activation energy|energy of activation]]. Arrhenius provided a physical justification and interpretation for the formula.<ref name="Arrhenius96">{{cite journal|first = S. A.|last = Arrhenius|title = Über die Dissociationswärme und den Einfluß der Temperatur auf den Dissociationsgrad der Elektrolyte|journal = [[Z. Phys. Chem.]]|volume = 4|pages = 96–116|year = 1889|doi=10.1515/zpch-1889-0408|s2cid = 202553486|url = https://zenodo.org/record/1448930}}</ref><ref name="Arrhenius226">{{cite journal|first = S. A.|last = Arrhenius|title = Über die Reaktionsgeschwindigkeit bei der Inversion von Rohrzucker durch Säuren|journal = [[Z. Phys. Chem.]]|volume = 4|pages = 226–48|year = 1889|doi = 10.1515/zpch-1889-0416| s2cid=100032801 |url = https://zenodo.org/record/1749766}}</ref><ref name="Laidler1984">{{cite journal|first = K. J.|last = Laidler|title = The development of the Arrhenius equation|journal = [[J. Chem. Educ.]]|volume = 61|pages = 494–498|year = 1984| issue=6 |doi=10.1021/ed061p494| bibcode=1984JChEd..61..494L |url = https://doi.org/10.1021/ed061p494}}</ref><ref name="Laidler42">[[Keith J. Laidler|Laidler, K. J.]] (1987) ''Chemical Kinetics'', Third Edition, Harper & Row, p. 42</ref> Currently, it is best seen as an [[empirical relationship]].<ref name="Connors">Kenneth Connors, Chemical Kinetics, 1990, VCH Publishers {{Google books|nHux3YED1HsC|Chemical Kinetics: The Study of Reaction Rates in Solution}}</ref>{{rp|188}} It can be used to model the temperature variation of [[Mass diffusivity|diffusion coefficients]], population of [[Vacancy defect|crystal vacancies]], [[Creep (deformation)|creep]] rates, and many other thermally induced processes and reactions.{{cn|date=April 2025}} The [[Eyring equation]], developed in 1935, also expresses the relationship between rate and energy.

== 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>

== Derivation ==
Van't Hoff argued that the temperature <math>T</math> of a reaction and the standard equilibrium constant <math> k^0_\text{e}</math> exhibit the relation:
{{NumBlk|:|<math display="block">\frac{d \ln k^0_\text{e} }{dT} = \frac{\Delta U^0}{RT^2} </math>|{{EquationRef|1}}}}
where <math>\Delta U^0</math> denotes the apposite [[Standard temperature and pressure|standard]] [[internal energy]] change value.

Let <math>k_\text{f}</math> and <math>k_\text{b}</math> respectively denote the forward and backward reaction rates of the reaction of interest, then
{{tmath|1= k^0_\text{e} = \frac{k_\text{f} }{k_\text{b} } }},<ref>{{cite web |date=2016-03-11 |title=15.2: The Equilibrium Constant (K) |url=https://chem.libretexts.org/Bookshelves/General_Chemistry/Map%3A_A_Molecular_Approach_(Tro)/15%3A_Chemical_Equilibrium/15.02%3A_The_Equilibrium_Constant_(K) |access-date=2023-06-27 |website=Chemistry LibreTexts |language=en}}</ref> an equation from which <math> \ln k^0_\text{e} = \ln k_\text{f} - \ln k_\text{b}</math> naturally follows.

Substituting the expression for <math> \ln k^0_\text{e} </math> in eq.({{EquationNote|1}}), we obtain {{tmath|1= \frac{d\ln k_\text{f} }{dT} - \frac{d\ln k_\text{b} }{dT} = \frac{\Delta U^0}{RT^2} }}.

The preceding equation can be broken down into the following two equations:
{{NumBlk|:|<math> \frac{d\ln k_\text{f}}{dT} = \text{constant} + \frac{E_\text{f}}{RT^2}</math>|{{EquationNote|2}}}}
and 
{{NumBlk|:|<math> \frac{d\ln k_\text{b}}{dT} = \text{constant} + \frac{E_\text{b}}{RT^2}</math>|{{EquationNote|3}}}}
where <math>E_\text{f}</math> and <math>E_\text{b}</math> are the activation energies associated with the forward and backward reactions respectively, with <math> \Delta U^0 = E_\text{f} - E_\text{b} </math>.

Experimental findings suggest that the constants in eq.({{EquationNote|2}}) and eq.({{EquationNote|3}}) can be treated as being equal to zero, so that {{NumBlk|:|<math> \frac{d\ln k_\text{f}}{dT} = \frac{E_\text{f}}{RT^2}</math>|}} and {{NumBlk|:|<math> \frac{d\ln k_\text{b}}{dT} = \frac{E_\text{b}}{RT^2}</math>|}}

Integrating these equations and taking the exponential yields the results {{tmath|1= k_\text{f} = A_\text{f}e^{-E_\text{f}/RT} }} and {{tmath|1= k_\text{b} = A_\text{b}e^{-E_\text{b}/RT} }}, where each [[pre-exponential factor]] <math>A_\text{f}</math> or <math>A_\text{b}</math> is mathematically the exponential of the constant of integration for the respective [[Antiderivative|indefinite integral]] in question.<ref>{{Cite web |title=Arrhenius Equation |url=https://www.sas.upenn.edu/~moronge/arrhenius.pdf |access-date=27 June 2023 |website=University of Pennsylvania}}</ref>

== Arrhenius plot ==
{{main|Arrhenius plot}}

[[Image:Arrhenius_plot_with_break_in_y-axis_to_show_intercept.svg|thumb|Arrhenius linear plot: ln&nbsp;''k'' against 1/''T''.]]
Taking the [[natural logarithm]] of Arrhenius equation yields:
<math display="block">\ln k= \ln A - \frac{E_\text{a}}{R} \frac{1}{T}.</math>

Rearranging yields:
<math display="block">\ln k = \frac{-E_\text{a}}{R}\left(\frac{1}{T}\right) + \ln A.</math>

This has the same form as an equation for a straight line:
<math display="block">y = a x + b,</math>
where ''x'' is the [[Multiplicative inverse|reciprocal]] of ''T''.

So, when a reaction has a rate constant obeying the Arrhenius equation, a plot of ln&nbsp;''k'' versus ''T''<sup>−1</sup> gives a straight line, whose slope and intercept can be used to determine ''E''<sub>a</sub> and ''A'' respectively. This procedure is common in experimental chemical kinetics. The activation energy is simply obtained by multiplying by (−''R'') the slope of the straight line drawn from a plot of ln&nbsp;''k'' versus (1/''T''):
<math display="block">E_\text{a} \equiv -R \left[ \frac{\partial \ln k}{\partial (1/T)} \right]_P.</math>

== Modified Arrhenius equation ==
The modified Arrhenius equation<ref>[http://goldbook.iupac.org/M03963.html IUPAC Goldbook definition of modified Arrhenius equation].</ref> makes explicit the temperature dependence of the pre-exponential factor. The modified equation is usually of the form
<math display="block">k = A T^n e^\frac{- E_\text{a}}{RT}.</math>

The original Arrhenius expression above corresponds to {{nowrap|1=''n'' = 0}}. Fitted rate constants typically lie in the range {{nobr|−1 < ''n'' < 1}}. Theoretical analyses yield various predictions for ''n''. It has been pointed out that "it is not feasible to establish, on the basis of temperature studies of the rate constant, whether the predicted ''T''<sup>1/2</sup> dependence of the pre-exponential factor is observed experimentally".<ref name="Connors"/>{{rp|190}} However, if additional evidence is available, from theory and/or from experiment (such as density dependence), there is no obstacle to incisive tests of the Arrhenius law.

Another common modification is the [[stretched exponential]] form{{citation needed|date=January 2013}}
<math display="block">k = A \exp \left[-\left(\frac{E_a}{RT}\right)^\beta\right],</math>
where ''β'' is a dimensionless number of order 1. This is typically regarded as a purely empirical correction or ''[[fudge factor]]'' to make the model fit the data, but can have theoretical meaning, for example showing the presence of a range of activation energies or in special cases like the Mott [[variable range hopping]].

== Theoretical interpretation ==

=== Arrhenius's concept of activation energy ===
Arrhenius argued that for reactants to transform into products, they must first acquire a minimum amount of energy, called the [[activation energy]] ''E''<sub>a</sub>. At an absolute temperature ''T'', the fraction of molecules that have a kinetic energy greater than ''E''<sub>a</sub> can be calculated from [[statistical mechanics]]. The concept of ''activation energy'' explains the exponential nature of the relationship, and in one way or another, it is present in all kinetic theories.

The calculations for reaction rate constants involve an energy averaging over a [[Maxwell–Boltzmann distribution]] with <math>E_\text{a}</math> as lower bound and so are often of the type of [[Incomplete gamma function#Special values|incomplete gamma function]]s, which turn out to be proportional to <math>e^{\frac{-E_\text{a}}{RT}}</math>.

=== Collision theory ===
{{main|Collision theory}}
One approach is the [[collision theory]] of chemical reactions, developed by [[Max Trautz]] and [[William Lewis (physical chemist)|William Lewis]] in the years 1916–18. In this theory, molecules are supposed to react if they collide with a relative [[kinetic energy]] along their line of centers that exceeds ''E''<sub>a</sub>. The number of binary collisions between two unlike molecules per second per unit volume is found to be<ref name=LM>{{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 |pages=376–78 |edition=1st}}</ref>
<math display="block"> z_{AB} = N_\text{A} N_\text{B} d_{AB}^2 \sqrt\frac{8 \pi k_\text{B}T}{ \mu_{AB}} ,</math>
where ''N<sub>A</sub>'' and ''N<sub>A</sub>'' are the number densities of ''A'' and ''B'', ''d<sub>AB</sub>'' is the average diameter of ''A'' and ''B'', ''T'' is the temperature which is multiplied by the [[Boltzmann constant]] ''k''<sub>B</sub> to convert to energy, and ''μ<sub>AB</sub>'' is the [[reduced mass]] of ''A'' and ''B''.

The rate constant is then calculated as {{tmath|1= k = z_{AB}e^\frac{-E_\text{a} }{RT} }}, so that the collision theory predicts that the pre-exponential factor is equal to the collision number ''z<sub>AB</sub>''. However for many reactions this agrees poorly with experiment, so the rate constant is written instead as {{tmath|1= k = \rho z_{AB}e^\frac{-E_\text{a} }{RT} }}. Here ''<math>\rho</math>'' is an empirical [[steric factor]], often much less than 1.00, which is interpreted as the fraction of sufficiently energetic collisions in which the two molecules have the correct mutual orientation to react.<ref name=LM/>

=== Transition state theory ===
The [[Eyring equation]], another Arrhenius-like expression, appears in the "[[transition state theory]]" of chemical reactions, formulated by [[Eugene Wigner]], [[Henry Eyring (chemist)|Henry Eyring]], [[Michael Polanyi]] and [[Meredith Gwynne Evans|M. G. Evans]] in the 1930s. The Eyring equation can be written:
<math display="block">k = \frac{k_\text{B}T}{h} e^{-\frac{\Delta G^\ddagger}{RT}} = \frac{k_\text{B}T}{h} e^{\frac{\Delta S^\ddagger}{R}}e^{-\frac{\Delta H^\ddagger}{RT}},</math>
where <math>\Delta G^\ddagger</math> is the [[Gibbs free energy|Gibbs energy]] of activation, <math>\Delta S^\ddagger</math> is the [[entropy of activation]], <math>\Delta H^\ddagger</math> is the [[enthalpy]] of activation, <math>k_\text{B}</math> is the [[Boltzmann constant]], and <math>h</math> is the [[Planck constant]].<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 |pages=378–83 |edition=1st}}</ref>

At first sight this looks like an exponential multiplied by a factor that is ''linear'' in temperature. However, free energy is itself a temperature dependent quantity. The free energy of activation <math>\Delta G^\ddagger = \Delta H^\ddagger - T\Delta S^\ddagger</math> is the difference of an enthalpy term and an entropy term multiplied by the absolute temperature. The pre-exponential factor depends primarily on the entropy of activation. The overall expression again takes the form of an Arrhenius exponential (of enthalpy rather than energy) multiplied by a slowly varying function of ''T''. The precise form of the temperature dependence depends upon the reaction, and can be calculated using formulas from [[statistical mechanics]] involving the [[Partition function (statistical mechanics)|partition functions]] of the reactants and of the activated complex.

=== Limitations of the idea of Arrhenius activation energy ===
Both the Arrhenius activation energy and the rate constant ''k'' are experimentally determined, and represent macroscopic reaction-specific parameters that are not simply related to threshold energies and the success of individual collisions at the molecular level. Consider a particular collision (an elementary reaction) between molecules A and B. The collision angle, the relative translational energy, the internal (particularly vibrational) energy will all determine the chance that the collision will produce a product molecule AB. Macroscopic measurements of ''E'' and ''k'' are the result of many individual collisions with differing collision parameters. To probe reaction rates at molecular level, experiments are conducted under near-collisional conditions and this subject is often called molecular reaction dynamics.<ref>[[Raphael David Levine|Levine, R.D.]] (2005) ''Molecular Reaction Dynamics'', Cambridge University Press</ref>

Another situation where the explanation of the Arrhenius equation parameters falls short is in [[heterogeneous catalysis]], especially for reactions that show [[Langmuir-Hinshelwood kinetics]]. Clearly, molecules on surfaces do not "collide" directly, and a simple molecular cross-section does not apply here. Instead, the pre-exponential factor reflects the travel across the surface towards the active site.<ref>{{Cite journal|last1=Slot|first1=Thierry K.|last2=Riley|first2=Nathan|last3=Shiju|first3=N. Raveendran|last4=Medlin|first4=J. Will|last5=Rothenberg|first5=Gadi|date=2020|title=An experimental approach for controlling confinement effects at catalyst interfaces|journal=Chemical Science|language=en|volume=11|issue=40|pages=11024–11029| doi=10.1039/D0SC04118A|pmid=34123192|pmc=8162257|issn=2041-6520|doi-access=free}}</ref>

There are deviations from the Arrhenius law during the [[glass transition]] in all classes of glass-forming matter.<ref>{{cite journal| last1=Bauer|first1=Th.|last2=Lunkenheimer|first2=P.|last3=Loidl|first3=A.|title=Cooperativity and the Freezing of Molecular Motion at the Glass Transition|journal=Physical Review Letters|date=2013|volume=111|issue=22|page=225702| doi=10.1103/PhysRevLett.111.225702| pmid=24329455|arxiv=1306.4630|bibcode=2013PhRvL.111v5702B|s2cid=13720989}}</ref> The Arrhenius law predicts that the motion of the structural units (atoms, molecules, ions, etc.) should slow down at a slower rate through the glass transition than is experimentally observed. In other words, the structural units slow down at a faster rate than is predicted by the Arrhenius law. This observation is made reasonable assuming that the units must overcome an energy barrier by means of a thermal activation energy. The thermal energy must be high enough to allow for translational motion of the units which leads to [[viscous flow]] of the material.

== See also ==
* [[Accelerated aging]]
* [[Eyring equation]]
* [[Q10 (temperature coefficient)]]
* [[Van 't Hoff equation]]
* [[Clausius–Clapeyron relation]]
* [[Gibbs–Helmholtz equation]]
* [[Cherry blossom front]]{{snd}}predicted using the Arrhenius equation

== References ==
{{reflist|40em}}

== Bibliography ==
* {{cite book |last=Pauling |first=L. C. |date=1988 |title=General Chemistry |publisher=Dover Publications}}
* {{cite book |last=Laidler |first=K. J. |date=1987 |title=Chemical Kinetics |edition=3rd |publisher=Harper & Row}}
* {{cite book |last=Laidler |first=K. J. |date=1993 |title=The World of Physical Chemistry |publisher=Oxford University Press}}

== External links ==
* [https://web.archive.org/web/20100926220628/http://www.composite-agency.com/messages/3945.html Carbon Dioxide solubility in Polyethylene] – Using Arrhenius equation for calculating species solubility in polymers

{{Reaction mechanisms}}

[[Category:Chemical kinetics]]
[[Category:Eponymous equations of physics]]
[[Category:Statistical mechanics]]