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:

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

where ΔG^‡ is the Gibbs energy of activation, k_B 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

↑ 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. (1935). "The Activated Complex in Chemical Reactions". J. Chem. Phys. 3 (2): 107–115. Bibcode:1935JChPh...3..107E. doi:10.1063/1.1749604.

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.

Polanyi, J.C. (1987). "Some concepts in reaction dynamics". Science. 236 (4802): 680–690. Bibcode:1987Sci...236..680P. doi:10.1126/science.236.4802.680.

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

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