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

  A = (ek_BT/h) exp(ΔS^‡/R),

• while for bimolecular gas reactions

  A = (e²k_BT/h) (RT/p) exp(ΔS^‡/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. The factor is needed because of the pressure dependence of the reaction rate. R′ = .

The value of ΔS^‡ 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 ΔS^‡ 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 $$k=\frac{\kappa k_{\mathrm{B}}T}{h}{e}^{\frac{\mathrm{\Delta }{S}^{‡}}{R}}{e}^{-\frac{\mathrm{\Delta }{H}^{‡}}{RT}}$$ where:

• $k$ = reaction rate constant
• $T$ = absolute temperature
• $\mathrm{\Delta }{H}^{‡}$ = enthalpy of activation
• $R$ = gas constant
• $\kappa$ = transmission coefficient
• $k_{\mathrm{B}}$ = Boltzmann constant = R/N_A, N_A = Avogadro constant
• $h$ = Planck constant
• $\mathrm{\Delta }{S}^{‡}$ = entropy of activation

This equation can be turned into the form $$\ln \frac{k}{T}=\frac{-\mathrm{\Delta }{H}^{‡}}{R}\cdot \frac{1}{T}+\ln \frac{\kappa k_{\mathrm{B}}}{h}+\frac{\mathrm{\Delta }{S}^{‡}}{R}$$ The plot of $\ln (k/T)$ versus $1/T$ gives a straight line with slope $-\mathrm{\Delta }{H}^{‡}/R$ from which the enthalpy of activation can be derived and with intercept $\ln (\kappa k_{\mathrm{B}}/h)+\mathrm{\Delta }{S}^{‡}/R$ from which the entropy of activation is derived.