Eyring equation

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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 rate of a chemical reaction with temperature. It was developed almost simultaneously in 1935 by Henry Eyring, Meredith Gwynne 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 of gases.[1]

General form[edit]

The general form of the Eyring–Polanyi equation somewhat resembles the Arrhenius equation:

\ k = \frac{k_\mathrm{B}T}{h}\mathrm{e}^{-\frac{\Delta G^\Dagger}{RT}}

where ΔG is the Gibbs energy of activation, kB is Boltzmann's constant, and h is Planck's constant.

It can be rewritten as:

 k = \frac{k_\mathrm{B}T}{h} \mathrm{e}^{\frac{\Delta S^\ddagger}{R}} \mathrm{e}^{-\frac{\Delta H^\ddagger}{RT}}

To find the linear form of the Eyring-Polanyi equation:

 \ln \frac{k}{T} = \frac{-\Delta H^\ddagger}{R} \cdot \frac{1}{T} + \ln \frac{k_\mathrm{B}}{h} + \frac{\Delta S^\ddagger}{R}

where:

A certain chemical reaction is performed at different temperatures and the reaction rate is determined. The plot of \ \ln(k/T) versus \ 1/T gives a straight line with slope \  -\Delta H^\ddagger / R  from which the enthalpy of activation can be derived and with intercept \  \ln(k_\mathrm{B}/h) + \Delta S^\ddagger / R from which the entropy of activation is derived.

Accuracy[edit]

Transition state theory requires a value of a certain transmission coefficient, called \ \kappa in that theory, as an additional prefactor in the Eyring equation above. This value is usually taken to be unity (i.e., the transition state \ AB^\ddagger always proceeds to products \ AB and never reverts to reactants \ A and \ B ), and we have followed this convention above. Alternatively, to avoid specifying a value of \ \kappa , the ratios of rate constants can be compared to the value of a rate constant at some fixed reference temperature (i.e., \ k(T)/k(T_{Ref}) ) which eliminates the \ \kappa term in the resulting expression.

Notes[edit]

  1. ^ Chapman & Enskog 1939

References[edit]

  • Evans, M.G.; Polanyi M. (1935). "Some applications of the transition state method to the calculation of reaction velocities, especially in solution". Trans. Faraday Soc. 31: 875–894. doi:10.1039/tf9353100875. 
  • Eyring, H.; Polanyi M. (1931). "Über Einfache Gasreaktionen". Z. Phys. Chem. Abt. B 12: 279–311. 
  • Laidler, K.J.; King M.C. (1983). "The development of Transition-State Theory". J. Phys. Chem. 87 (15): 2657–2664. doi:10.1021/j100238a002. 
  • Chapman, S. and Cowling, T.G. (1991). "The Mathematical Theory of Non-uniform Gases: An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases" (3rd Edition). Cambridge University Press, ISBN 9780521408448

External links[edit]