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| The '''Eyring equation''' (occasionally also known as '''Eyring–Polanyi equation''') is an equation used in [[chemical kinetics]] to describe the variance of the [[reaction rate|rate of a chemical reaction]] with [[temperature]]. It was developed almost simultaneously in 1935 by [[Henry Eyring]], [[Meredith Gwynne Evans|M.G. Evans]] and [[Michael Polanyi]]. This equation follows from the [[transition state theory]] (''aka'', activated-complex theory) and is trivially equivalent to the [[empirical]] [[Arrhenius equation]] which are both readily derived from [[statistical thermodynamics]] in the [[kinetic theory|kinetic theory of gases]].<ref>Chapman & Enskog 1939</ref>
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| ==General form== | |
| The general form of the Eyring–Polanyi equation somewhat resembles the [[Arrhenius equation]]:
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| <math>\ k = \frac{k_\mathrm{B}T}{h}\mathrm{e}^{-\frac{\Delta G^\Dagger}{RT}}</math>
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| where Δ''G''<sup>‡</sup> is the [[Gibbs free energy|Gibbs energy]] of activation, ''k''<sub>B</sub> is [[Boltzmann's constant]], and ''h'' is [[Planck's constant]].
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| It can be rewritten as: | |
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| <math> k = \left(\frac{k_\mathrm{B}T}{h}\right) \mathrm{exp}\left(\frac{\Delta S^\ddagger}{R}\right) \mathrm{exp}\left(-\frac{\Delta H^\ddagger}{RT}\right)</math>
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| To find the linear form of the Eyring-Polanyi equation:
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| <math> \ln \frac{k}{T} = \frac{-\Delta H^\ddagger}{R} \cdot \frac{1}{T} + \ln \frac{k_\mathrm{B}}{h} + \frac{\Delta S^\ddagger}{R} </math>
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| where:
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| *<math>\ k </math> = [[reaction rate]] constant
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| *<math>\ T </math> = [[absolute temperature]]
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| *<math>\ \Delta H^\ddagger </math> = '''enthalpy of activation'''
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| *<math>\ R </math> = [[gas constant]]
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| *<math>\ k_\mathrm{B} </math> = [[Boltzmann constant]]
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| *<math>\ h </math> = [[Planck's constant]]
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| *<math>\ \Delta S^\ddagger </math> = '''entropy of activation'''
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| A certain chemical reaction is performed at different temperatures and the reaction rate is determined. The plot of <math>\ \ln(k/T) </math> versus <math>\ 1/T </math> gives a straight line with slope <math>\ -\Delta H^\ddagger / R </math> from which the [[enthalpy]] of activation can be derived and with intercept <math>\ \ln(k_\mathrm{B}/h) + \Delta S^\ddagger / R </math> from which the [[entropy]] of activation is derived.
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| ==Accuracy==
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| [[Transition state theory]] requires a value of the [[transmission coefficient]] <math>\ \kappa </math> as a prefactor in the Eyring equation above. This value is usually taken to be unity (i.e., the transition state <math>\ AB^\ddagger </math> always proceeds to products <math>\ AB </math> and never reverts to reactants <math>\ A </math> and <math>\ B </math>). As discussed by Winzor and Jackson in 2006, this assumption invalidates the description of an equilibrium between the transition state and the reactants and therefore the empirical [[Arrhenius equation]] is preferred with a phenomenological interpretation of the prefactor <math>\ A </math> and activation energy <math>\ E_a </math>. For more details, see discussion in Winzor and Jackson (2006) pages 399-400 in section "Transition-state theory."
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| To avoid specifying a value of <math>\ \kappa </math> the ratios of rate constants can be compared to the value of a rate constant at some fixed reference temperature (i.e., <math>\ k(T)/k(T_{Ref}) </math>) which eliminates the <math>\ \kappa </math> term in the resulting expression.
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| ==Notes==
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| {{reflist|2}}
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| ==References==
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| <!-- Copy of 5 References for back-up reasons:
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| * Evans M.G. and Polanyi M. (1935) Trans. Faraday Soc. 31, 875.
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| * Eyring H. (1935) J. Chem. Phys. 3, 107.
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| * Eyring H. and Polanyi M. (1931) Z. Phys. Chem. Abt. B, 12, 279.
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| * Laidler K.J. and King M.C. (1983) The development of Transition-State Theory. J. Phys. Chem. 87, 2657-2664.
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| * Polanyi J.C. (1987) Some concepts in reaction dynamics. Science, 236(4802), 680-690.
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| -->
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| * {{Cite journal
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| | last = Evans
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| | first = M.G.
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| | coauthors = Polanyi M.
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| | year = 1935
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| | title = Some applications of the transition state method to the calculation of reaction velocities, especially in solution
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| | journal = Trans. Faraday Soc.
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| | volume = 31
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| | pages = 875
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| | doi = 10.1039/tf9353100875
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| }}
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| * {{Cite journal
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| | last = Eyring
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| | first = H.
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| | year = 1935
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| | title = The Activated Complex in Chemical Reactions
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| | journal = J. Chem. Phys.
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| | volume = 3
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| | pages = 107
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| | doi = 10.1063/1.1749604
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| |bibcode = 1935JChPh...3..107E
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| | issue = 2 }}
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| * {{Cite journal
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| | last = Eyring
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| | first = H.
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| | coauthors = Polanyi M.
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| | year = 1931
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| | title =
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| | journal = Z. Phys. Chem. Abt. B
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| | volume = 12
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| | pages = 279
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| }}
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| * {{Cite journal
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| | last = Laidler
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| | first = K.J.
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| | coauthors = King M.C.
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| | year = 1983
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| | title = The development of Transition-State Theory
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| | journal = J. Phys. Chem.
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| | volume = 87
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| | pages = 2657–2664
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| | doi = 10.1021/j100238a002
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| | issue = 15
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| }}
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| * {{Cite journal
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| | last = Polanyi
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| | first = J.C.
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| | year = 1987
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| | title = Some concepts in reaction dynamics. Science
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| | volume = 236
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| | issue = 4802
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| | pages = 680–690
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| | doi = 10.1126/science.236.4802.680
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| |bibcode = 1987Sci...236..680P }}
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| * {{Cite journal
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| | last = Winzor
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| | first = D.J.
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| | coauthors = Jackson C.M.
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| | year = 2006
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| | title = Interpretation of the temperature dependence of equilibrium and rate constants
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| | journal = J. Mol. Recognit.
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| | volume = 19
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| | pages = 389–407
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| | doi = 10.1002/jmr.799
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| | pmid = 16897812
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| | issue = 5
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| }}
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| * Chapman, S. and Cowling, T. G. ''The Mathematical Theory of Non-uniform Gases: An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases''
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| == External links ==
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| * [http://www.chemie.uni-regensburg.de/Organische_Chemie/Didaktik/Keusch/eyr-e.htm Eyring equation at the University of Regensburg]
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| * [http://www-jmg.ch.cam.ac.uk/tools/magnus/eyring.html Online-tool to calculate the reaction rate from an energy barrier (in kJ/mol) using the Eyring equation]
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| {{DEFAULTSORT:Eyring Equation}}
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| [[Category:Chemical kinetics]]
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| [[Category:Equations]]
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| [[Category:Physical chemistry]]
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| [[de:Eyring-Theorie]]
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| [[pl:Równanie Eyringa–Polanyiego]]
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