# Entropy of activation

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In [chemical kinetics](/source/Chemical_kinetics), the **entropy of activation** of a reaction is one of the two parameters (along with the enthalpy of activation) that are typically obtained from the temperature dependence of a reaction [rate constant](/source/Rate_constant), when these data are analyzed using the [Eyring equation](/source/Eyring_equation) of the [transition state theory](/source/Transition_state_theory). The standard entropy of activation is symbolized Δ*S*‡ and equals the change in [entropy](/source/Entropy) when the reactants change from their initial state to the [activated complex](/source/Activated_complex) or transition state (Δ = change, *S* = entropy, ‡ = activation).

## Importance

Entropy of activation determines the [preexponential factor](/source/Preexponential_factor) *A* of the [Arrhenius equation](/source/Arrhenius_equation) for temperature dependence of reaction rates. The relationship depends on the [molecularity](/source/Molecularity) of the reaction:

- for reactions in solution and unimolecular gas reactions - *A* = (e*k*B*T*/*h*) exp(Δ*S*‡/*R*),
- while for bimolecular gas reactions - *A* = (e2*k*B*T*/*h*) (*RT*/*p*) exp(Δ*S*‡/*R*).

In these equations e is the base of [natural logarithms](/source/Natural_logarithm), *h* is the [Planck constant](/source/Planck_constant), *k*B is the [Boltzmann constant](/source/Boltzmann_constant) and *T* the [absolute temperature](/source/Absolute_temperature). *R*′ is the ideal gas constant. The factor is needed because of the pressure dependence of the reaction rate. *R*′ = 8.3145×10−2 (bar·L)/(mol·K).[1]

The value of Δ*S*‡ provides clues about the [molecularity](/source/Molecularity) of the [rate determining step](/source/Rate_determining_step) in a reaction, i.e. the number of molecules that enter this step.[2] Positive values suggest that entropy increases upon achieving the transition state, which often indicates a [dissociative mechanism](/source/Dissociative_substitution) in which the activated complex is loosely bound and about to dissociate. Negative values for Δ*S*‡ indicate that entropy decreases on forming the transition state, which often indicates an [associative mechanism](/source/Associative_substitution) in which two reaction partners form a single activated complex.[3]

## Derivation

It is possible to obtain entropy of activation using [Eyring equation](/source/Eyring_equation). This equation is of the form

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

where:

- k = [reaction rate constant](/source/Reaction_rate_constant)
- T = [absolute temperature](/source/Absolute_temperature)
- \Delta H^\ddagger = enthalpy of activation
- R = [gas constant](/source/Gas_constant)
- \kappa = [transmission coefficient](/source/Transmission_coefficient)
- k_\mathrm{B} = [Boltzmann constant](/source/Boltzmann_constant) = *R*/*N*A, *N*A = [Avogadro constant](/source/Avogadro_constant)
- h = [Planck constant](/source/Planck_constant)
- \Delta S^\ddagger = entropy of activation

This equation can be turned into the form

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

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(\kappa k_\mathrm{B} / h) + \Delta S^\ddagger/ R from which the entropy of activation is derived.

## References

1. [Laidler, K.J.](/source/Keith_J._Laidler) and Meiser J.H. *Physical Chemistry* (Benjamin/Cummings 1982) p. 381–382 ISBN 0-8053-5682-7

1. Laidler and Meiser p. 365

1. James H. Espenson *Chemical Kinetics and Reaction Mechanisms* (2nd ed., McGraw-Hill 2002), p. 156–160 ISBN 0-07-288362-6

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Adapted from the Wikipedia article [Entropy of activation](https://en.wikipedia.org/wiki/Entropy_of_activation) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Entropy_of_activation?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.