# 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 (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

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:

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

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

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

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. 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