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) that 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 $ΔS^{‡}$ 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
 * while for bimolecular gas reactions
 * while for bimolecular gas reactions

In these equations $A = (ek_{B}T/h) exp(ΔS^{‡}/R)$ is the base of natural logarithms, $A = (e^{2}k_{B}T/h) (RT/p) exp(ΔS^{‡}/R)$ is the Planck constant, $e$ is the Boltzmann constant and $h$ the absolute temperature. $k_{B}$ is the ideal gas constant. The factor is needed because of the pressure dependence of the reaction rate. $T$ = $8.315 (bar·L)/(mol·K)$.

The value of $R′$ provides clues about the molecularity of the rate determining step in a reaction, i.e. the number of molecules that enter this step. 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 $R′$ 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.

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/NA, NA = Avogadro constant
 * $$ h $$ = 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.