Entropy of activation

Entropy of activation determines the preexponential factor {{math|1=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
• : {{math|1=A = (ek_BT/h) exp(ΔS^‡/R)}},
• while for bimolecular gas reactions
• : {{math|1=A = (e²k_BT/h) (RT/p) exp(ΔS^‡/R)}}.

In these equations {{math|1=e}} is the base of natural logarithms, {{math|1=h}} is the Planck constant, {{math|1=k_B}} is the Boltzmann constant and {{math|1=T}} the absolute temperature. {{math|1=R′}} is the ideal gas constant. The factor is needed because of the pressure dependence of the reaction rate. {{math|1=R′}} = {{val|8.3145|e=−2|u=(bar·L)/(mol·K)}}.Laidler, K.J. and Meiser J.H. Physical Chemistry (Benjamin/Cummings 1982) p. 381–382 {{ISBN|0-8053-5682-7}}

The value of {{math|1=ΔS^‡}} provides clues about the molecularity of the rate determining step in a reaction, i.e. the number of molecules that enter this step.Laidler and Meiser p. 365 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 {{math|1=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.James H. Espenson Chemical Kinetics and Reaction Mechanisms (2nd ed., McGraw-Hill 2002), p. 156–160 {{ISBN|0-07-288362-6}}

It is possible to obtain entropy of activation using Eyring equation. This equation is of the form

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

where:

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

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

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Category:Chemical kinetics