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{{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 [[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 rate of chemical reactions and for calculation of 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|url = https://zenodo.org/record/1448930/files/article.pdf}}</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 = 226–248|year = 1889|doi = 10.1515/zpch-1889-0116|url = https://zenodo.org/record/1749766}}</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 diffusion coefficients, population of crystal vacancies, creep rates, and many other thermally-induced processes/reactions. The [[Eyring equation]], developed in 1935, also expresses the relationship between rate and energy.

==Equation==
[[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.]]
Arrhenius equation gives the dependence of the [[rate constant]] of a chemical reaction on the [[absolute temperature]], a [[pre-exponential factor]] and other constants of the reaction.
:<math>k = Ae^\frac{-E_{\rm a}}{RT},</math>
where
: {{mvar|k}} is the [[rate constant]],
: {{mvar|T}} is the [[absolute temperature]] (in [[Kelvin]]),
: {{mvar|A}} is the [[pre-exponential factor]], a constant for each chemical reaction. According to [[collision theory]], ''A'' is the frequency of collisions in the correct orientation,
:{{Math|''E''<sub>a</sub>}} is the [[activation energy]] for the reaction (in the same units as ''RT''),
: {{mvar|R}} is the [[universal gas constant]].<ref name="Arrhenius96"/><ref name="Arrhenius226"/><ref name="Laidler42"/>

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

The only difference is the energy units 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 units 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 units: [[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 |date=2006 |publisher=McGraw-Hill |location=NY |isbn=0--07-111658-3 |page=696 |edition=fourth}}</ref> and <math>e^{-E_{\rm 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 <math>\exp(-E_{\rm a}/(RT))</math> factor; except in the case of "barrierless" [[diffusion]]-limited reactions, in which case the pre-exponential factor is dominant and is directly observable.

==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>\ln k= \ln A - \frac{E_{\rm a}}{R} \frac{1}{T}.</math>

Rearranging yields:

:<math>\ln k = \frac{-E_{\rm a}}{R}\left(\frac{1}{T}\right) + \ln A.</math>

This has the same form as an equation for a straight line:

: <math>y = m x + b,</math>

where ''x'' is the [[Multiplicative inverse|reciprocal]] of ''T''.

So, when a reaction has a rate constant that obeys Arrhenius equation, a plot of ln&nbsp;''k'' versus ''T''<sup>−1</sup> gives a straight line, whose gradient and intercept can be used to determine ''E''<sub>a</sub> and ''A'' . This procedure has become so common in experimental chemical kinetics that practitioners have taken to using it to ''define'' the activation energy for a reaction. That is the activation energy is defined to be (−''R'') times the slope of a plot of ln&nbsp;''k'' vs. (1/''T''):

:<math>E_{\rm 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>k = A T^n e^{-E_{\rm a}/(RT)}.</math>

The original Arrhenius expression above corresponds to ''n''&nbsp;= 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>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 of the equation==

===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_{\rm 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_{\rm 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=376-8 |edition=1st}}</ref>

: <math> 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>B</sub> are the number of molecules of A and B per unit volume, d<sub>AB</sub> is the average diameter of A and B, k<sub>B</sub> is the [[Boltzmann constant]], and μ is the [[reduced mass]].

The rate constant is then calculated as <math>k = Z_{AB}e^\frac{-E_{\rm a}}{RT},</math> 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 <math>k = PZ_{AB}e^\frac{-E_{\rm a}}{RT},</math>. Here P is an empirical [[steric factor]], often much less than 1, 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|Wigner]], [[Henry Eyring (chemist)|Eyring]], [[Michael Polanyi|Polanyi]] and [[Meredith Gwynne Evans|Evans]] in the 1930s. The Eyring equation can be written:

:<math>k = \frac{k_{\rm B}T}{h} e^{-\frac{\Delta G^\ddagger}{RT}} = \frac{k_{\rm B}T}{h} e^{\frac{\Delta S^\ddagger}{R}}e^{-\frac{\Delta H^\ddagger}{RT}},</math>

where <math>\Delta G^\ddagger</math> = 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_{\rm B}</math> is [[Boltzmann's constant]], and <math>h</math> is [[Planck's 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=378-383 |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>

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}}</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==
* [http://www.composite-agency.com/messages/3945.html Carbon Dioxide solubility in Polyethylene] – Using Arrhenius equation for calculating species solubility in polymers

[[Category:Chemical kinetics]]
[[Category:Equations]]
[[Category:Statistical mechanics]]