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, M.G. 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:

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

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:

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

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

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

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 the transmission coefficient $\ \kappa$ as a 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 $\ B$ ). As discussed by Winzor and Jackson in 2006, this assumption invalidates the description of an equilibrium between the transition state and the reactants and therefore the empirical Arrhenius equation is preferred with a phenomenological interpretation of the prefactor $\ A$ and activation energy $\ E_{a}$ . For more details, see discussion in Winzor and Jackson (2006) pages 399-400 in section "Transition-state theory."

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. doi:10.1039/tf9353100875. Cite uses deprecated parameters (help)

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

Eyring, H.; Polanyi M. (1931). Z. Phys. Chem. Abt. B 12: 279. Cite uses deprecated parameters (help)

Laidler, K.J.; King M.C. (1983). "The development of Transition-State Theory". J. Phys. Chem. 87 (15): 2657–2664. doi:10.1021/j100238a002. Cite uses deprecated parameters (help)

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

Winzor, D.J.; Jackson C.M. (2006). "Interpretation of the temperature dependence of equilibrium and rate constants". J. Mol. Recognit. 19 (5): 389–407. doi:10.1002/jmr.799. PMID 16897812. Cite uses deprecated parameters (help)

Chapman, S. and Cowling, T. G. The Mathematical Theory of Non-uniform Gases: An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases

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