Physical Chemistry Third Edition

1110 26 Equilibrium Statistical Mechanics. II. Statistical Thermodynamics

motion along the reaction coordinate and leads to dissociation to form either products

or reactants, since the potential energy is at a maximum instead of a minimum along

the reaction coordinate.

We write the partition function of the activated complex in the form

z′′CDFz‡CDF′′ zrc (26.4-9)

wherezrcis the factor in the partition function corresponding to motion along the

reaction coordinate and wherez

‡′′

CDFrepresents the translational factor, the rotational

factor, the electronic factor, and the three vibrational factors.

We adopt a nonrigorous approach to obtain a formula forzrc. We pretend that we

can change the potential energy surface so that the asymmetric stretch is like that of a

triatomic molecule in Figure 26.2. The partition function for the reaction coordinate is

now that for an asymmetric stretch:

zrc

1

1 −e−hνas/kBT

(26.4-10)

whereνasis the frequency of oscillation of the asymmetric stretch. We now let the

potential energy surface become flatter in theabcdirection. As its curvature in the

direction of the reaction coordinate becomes small,νasbecomes small, and we can use

the approximatione−x≈ 1 −x, which gives

zrc

1

1 −(1−hνas/kBT)



kBT

hνas

(26.4-11)

The frequency of the asymmetric stretch,νas, gives the number of oscillations per

second in the direction of the asymmetric stretch, which becomes the direction of the

reaction coordinate in this limit. We assume that we can identifyνaswithνpassage, and

write

νpassage

kBT

hzrc

(26.4-12)

This process of mentally distorting the potential energy surface is not very satisfactory

because the process must be continued until the curvature becomes negative in order

to recover the actual surface. There are better ways to derive Eq. (26.4-12) that lead to

the same result.^8

The concentration of the activated complex is given by Eqs. (26.4-8), (26.4-9), and

(26.4-12):

[CDF‡]e−∆ε

‡

0 /kBT

z

‡′′

CDFzrc

z′′CDz′′F

NAv[CD][F] (26.4-13)

The rate of the reaction is given by

Rateνpassage[CDF‡]

kBT

hzrc

e−∆ε

‡

0 /kBT

z

‡′′

CDFzrc

z′′CDzF

NAv[CD][F]



kBT

h

e−∆ε

‡

0 /kBT

z‡CDF′′

z′′CDz′′F

NAv[CD][F] (26.4-14)

(^8) K. J. Laidler,Chemical Kinetics, 3rd ed., HarperCollins, New York, 1987, p. 97ff.