Entropy of activation

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

Importance

Entropy of activation determines the preexponential factor $A$ of the Arrhenius equation for temperature dependence of reaction rates. The relationship depends on the 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 = (e 2 k B T / h) (RT / p) exp(Δ S ‡ / R)

.

In these equations $e$ is the base of natural logarithms, $h$ is the Planck constant, $k B$ is the Boltzmann constant and $T$ the absolute temperature. $R$ ' is the ideal gas constant in units of (bar·L)/(mol·K). The factor is needed because of the pressure dependence of the reaction rate. $R' = 8.3145 \times 10 -2$ (bar·L)/(mol·K).[1]

The value of $Δ S ‡$ provides clues about the molecularity of the 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 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 in which two reaction partners form a single activated complex.[3]

Derivation

It is possible to obtain entropy of activation using 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
$T$ = absolute temperature
$\Delta H^{\ddagger }$ = enthalpy of activation
$R$ = gas constant
$\kappa$ = transmission coefficient
$k_{\mathrm {B} }$ = Boltzmann constant = R/N_A, N_A = Avogadro constant
$h$ = Planck's 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

Laidler, K.J. and Meiser J.H. Physical Chemistry (Benjamin/Cummings 1982) p.381-2 ISBN 0-8053-5682-7
Laidler and Meiser p.365
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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[1] Laidler, K.J. and Meiser J.H. Physical Chemistry (Benjamin/Cummings 1982) p.381-2 ISBN 0-8053-5682-7

[2] Laidler and Meiser p.365

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