# Eyring equation

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 (a.k.a. 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

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

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

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

It can be rewritten as:

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

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

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

where:

• ${\displaystyle \ k}$ = reaction rate constant
• ${\displaystyle \ T}$ = absolute temperature
• ${\displaystyle \ \Delta H^{\ddagger }}$ = enthalpy of activation
• ${\displaystyle \ R}$ = gas constant
• ${\displaystyle \ k_{\mathrm {B} }}$ = Boltzmann constant
• ${\displaystyle \ h}$ = Planck's constant
• ${\displaystyle \ \Delta S^{\ddagger }}$ = entropy of activation

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

## Accuracy

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

## Notes

1. ^ Chapman & Enskog 1939

## References

• 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. 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