Physical Chemistry , 1st ed.

Thus, if we can determine Kc and k, we can calculate the rate of the elemen-
tary process.
The rate constant k for the second step of our hypothetical two-step process
is relatively easy to express. Consider how a transition state rearranges to a set
of product species: typically, one chemical bond of the transition state will
lengthen as two parts of the transition state separate into the ultimate prod-
ucts. The lengthening of a chemical bond is part of a molecular motion known
as a vibration,just like the vibrations considered for stable molecules in
Chapters 12 and 14. Therefore, we assume that the transition state has some
vibrational frequency  that “connects” the transition state and the ultimate
products. (Like any polyatomic species, a transition state has other vibrations,
but only one particular vibration represents the movement from the transition
state to the products.) Understanding that a unimolecular rate constant like k
has units of s^1 (that is, seconds in the denominator), we submit that the rate
constant k should be proportional to the transition state’s vibrational fre-
quency  (also units of s^1 ) that promotes the formation of products. The
variable , called the transmission coefficient,is defined as the proportionality
constant, yielding
k * (20.73)

as our expression for the unimolecular rate constant for step b. Transmission
coefficients are usually assumed to be 1, and k is simply equal to .
We turn our attention to Kc by applying some results from Chapter 18: we
can use statistical thermodynamics to determine a numerical value for the
equilibrium constant Kc. For the reaction

k

A B →C*

the equilibrium constant expression in terms of the partition functions of A,
B, and C* is

Kc* 

(qA/

q

V

C

)

*

(

/

q

V

B/V)

 (20.74)

(see equation 18.62). Each individual partition function can be separated into
five parts: translational, electronic, vibrational, rotational, and nuclear. Let us
consider two of these parts. First, using equation 18.8, the electronic parts of
the partition function can be written as



qel,

q

A

el



,C

q

*

el,B



e(D

(e

e,

D

A

e/k

D

T

e

)

,B

C

)/

*

kT (20.75)

where the De’s in the exponentials refer to the dissociation energies of the re-
spective species. The difference between Deof C and (De,ADe,B) is simply
the difference in the electronic energies of C and species (A and B), collec-
tively. We will use  to represent the difference in the electronic energies;
a negative sign is used by convention and means that it refers in part to the
transition state. Equation 20.75 becomes



qel,

q

A

el



,C

q

*

el,B

e*/kT (20.76)

If we use q to represent the remaining partition functions within each q,equa-
tion 20.74 becomes

Kc* 

(q (^) A/
q
V
(^) C
)

• (

/

q

V

(^) B/V)
e/kT
(eDe/kT)C
(eDe/kT)A(eDe/kT)B
20.10 Transition-State Theory 721