Physical Chemistry , 1st ed.

sition state, and rewrite the Eyring equation in terms of the Gibbs free energy.
We g e t

k

c

k

°

T

h

eG*/RT (20.80)

where kon the left is the rate constant, whereas kon the right is Boltzmann’s
constant. Since Gcan be written in terms ofHand S, we can rewrite equa-
tion 20.80 as

k

c

k

°

T

h

eH/RTeS*/R (20.81)

where now we are referring to an enthalpy change for the formation of the
transition state,H, and an entropy change for the formation of the transi-
tion state,S. Equation 20.81 is similar to the Arrhenius equation, but not
exactly. However, if we write the logarithm of the Arrhenius equation as

ln kln A

R

E

T

A

we differentiate with respect to temperature:







ln

T

k







ln

T

A







T



R

E

T

A



The term (ln A)/Tis zero (Ais a constant, so the derivative is zero), and we
can evaluate the derivative of the activation energy. We get







ln

T

k



R

E

T

A

 (^2) 
We can rearrange this to
EART^2 





ln

T

k

RT^2 

1

k







T

k

 (20.82)

We can substitute the expression for kfrom equation 20.81 into equation 20.82
to find that

EAH*  2 RT or H* EA 2 RT (20.83)

In terms of the activation energy,kcan be written as

k

c

k

°

T

h

e(EA^2 RT)/RTeS*/R



c

k

°

T

h

eEA/RTe(^2 RT)/RTeS*/R

20.10 Transition-State Theory 723

Table 20.2 Experimental and calculated pre-exponential factors

A[cm^3 /(mols)]

Reaction Experimental Calculated

H H 2 →H 2 H 5.4 1013 7.4 1013

H 2 Br →HBr H 3 1013 1 1014

H CH 4 →H 2 CH 3 1 1013 2 1013

CH 3 H 2 →CH 4 H 2 1012 1 1012

ClO ClO →Cl 2 O 2 6 1010 1 1011

Source:J. Nicholas,Chemical Kinetics: A Modern Survey of Gas Reactions,Wiley, New York, 1976.