# Eyring equation

The Eyring equation also known as Eyring–Polanyi equation in chemical kinetics relates the reaction rate to temperature. It was developed almost simultaneously in 1935 by Henry Eyring, M.G. Evans and Michael Polanyi. This equation follows from the transition state theory (aka, activated-complex theory) and contrary to the empirical Arrhenius equation this model is theoretical and based on statistical thermodynamics.

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:

${\displaystyle k=\left({\frac {k_{\mathrm {B} }T}{h}}\right)\mathrm {exp} \left({\frac {\Delta S^{\ddagger }}{R}}\right)\mathrm {exp} \left(-{\frac {\Delta H^{\ddagger }}{RT}}\right)}$

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.

## References

• Evans, M.G. (1935). Trans. Faraday Soc. 31: 875. Unknown parameter |coauthors= ignored (help); Missing or empty |title= (help)
• Eyring, H. (1935). J. Chem. Phys. 3: 107. Missing or empty |title= (help)
• Eyring, H. (1931). Z. Phys. Chem. Abt. B. 12: 279. Unknown parameter |coauthors= ignored (help); Missing or empty |title= (help)
• Laidler, K.J. (1983). "The development of Transition-State Theory". J. Phys. Chem. 87: 2657–2664. doi:10.1021/j100238a002. Unknown parameter |coauthors= ignored (help)
• Polanyi, J.C. (1987). "Some concepts in reaction dynamics. Science". 236 (4802): 680–690.