Concepts from Chemical Reaction Kinetics - Elements of Environmental Engineering: Thermodynamics and Kinetics - page 212

Note that K † is the pressure units-based equilibrium constant. If molar concen-
trations are used instead of partial pressures, the appropriate conversion has to be
applied. It should also be noted that the partition functions are proportional to T n , and
hence

k ∗ ≈ aT n e − (E 0 /RT) .

(5.71)

Therefore, we have the following equation:

dln k ∗
d T =

E 0 + nRT

RT 2

. (5.72)

The experimental activation energy, E a was defined earlier
dln k ∗
d T =

E a
RT 2 .

(5.73)

Hence, E a = nRT . This gives the relationship between the zero-point activation

energy and the experimental activation energy.
The statistical mechanical expressions of the ACT lead to significant difficulties
since the structure of the activated complex is frequently unknown. This has led to a
more general approach in which the activation process is considered on the basis of
thermodynamic functions. Since K † is the equilibrium constant, we can write

E o +

G † RT ln K †

Δ =−

(5.74)

as the Gibbs free energy of activation . Hence, we have

k B T

h e − ( Δ G † /RT) .

k ∗ = κ (5.75)

G † , namely, the enthalpy of activation We can further obtain the components of Δ

H † and the entropy of activation S † . Hence

Δ Δ

k B T

h e Δ S † /R e − ( Δ H † /RT) .

k ∗ = κ (5.76)

If k ∗ is expressed in L/mol/s (or dm 3 /mol/s), then the standard state for both H † and Δ

S † is 1 mol/L (or 1 mol/dm 3 ) . The experimental activation energy is related to H †

Δ Δ

H † V †

as per the equation E a = Δ − P Δ + RT . For unimolecular gas-phase reactions

V † is zero, and for reactions in solutions V † is negligible. Hence, we have Δ Δ

k B T

h e Δ S † /R e − (E a /RT) .

k ∗ = e κ (5.77)

10 13 e Δ S † /R L/mol/s at 298 K. For a

In terms of the Arrhenius equation, A = 2 ×

V † n † RT H † bimolecular reaction in the gas phase, P Δ = Δ =− RT and E a = Δ +

2 RT , and

k B T

h e Δ S † /R e − (E a /RT) .

k ∗ = e 2 κ (5.78)