Physical Chemistry Third Edition

26.4 The Activated Complex Theory of Bimolecular Chemical Reaction Rates in Dilute Gases 1111

and the rate constant is given by

k

kBT

h

e−∆ε

‡

0 /kBT

z

‡′′

CDF

z′′CDz′′F

NAv

RT

h

e−∆ε

‡

0 /kBT

z

‡′′

CDF

z′′CDz′′F

(26.4-15)

where the concentrations [CD] and [F] are expressed in mol m−^3. Since thez‡′′factors

have units of m−^3 , the rate constant has the units m^3 mol−^1 s−^1. An additional factor

of 1000 L m−^3 would give the rate constant the units L mol−^1 s−^1.

Sometimes the expression shown in Eq. (26.4-15) is multiplied by atransmis-

sion coefficientκ, which represents the fraction of activated complexes that react.

The transmission coefficient is commonly used as a correction factor to make the

theory agree with experiment. It is probably better to omit this factor and to admit

that the theory is approximate and cannot be expected to give exact agreement with

experiment.

EXAMPLE26.12

Calculate the rate constant at 500.0 K for the gas-phase reaction

H+HBr→H 2 +Br

Assume a linear activated complex H···H···Br with the internuclear distances given by^9

rH−H 1. 50 × 10 −^10 m

rH−Br 1. 42 × 10 −^10 m

and the vibrational frequencies given by

ν ̃ 1  ̃ν(symmetric stretch)2340 cm−^1

ν ̃ 2  ̃ν(bend)460 cm−^1

The value of∆ε‡ 0 is 8.3× 10 −^21 J.

Solution

The rate constant is given by

k

kBT

h

e−∆ε

‡

0 /kBT

z‡HHBr′′

z′′Hz′′HBr

NAv

We assume that the electronic factors in the partition functions of HHBr‡and HBr can be

approximated by 1.000. The ground state of the hydrogen atom has a degeneracy of 2, so

we assume that its electronic factor can be approximated by 2.000. We also assume that the

vibrational factor in the HBr partition function can be approximated by 1.000. The partition

function of the H atom is

z′′Hz′′tr,Hz′′el,H

(

2 πmHkBT

h^2

) 3 / 2

(2)

(^9) K.J. Laidler,op. cit., p. 109 (note 8).