Editing Arrhenius equation (section)

== 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.