Physical Chemistry Third Edition

536 12 Chemical Reaction Mechanisms I: Rate Laws and Mechanisms

speed, the minimum relative speed that can lead to reaction:

v>vc (12.3-8)

This integration is carried out in Appendix D. The result is

Z 12 (reactive)πd 122 N 1 N 2 (8kBT /πμ 12 )^1 /^2

(

1 +

μv^2 c

2 kBT

)

e−μ^12 v

(^2) c/ 2 kBT
〈v 12 〉πd^212 N 1 N 2

(

1 +

μ 12 v^2 c

2 kBT

)

e−μ^12 v

(^2) c/ 2 kBT
(12.3-9)
where〈v 12 〉is the mean relative speed. Comparison of this equation with Eq. (12.1-4)
shows that we have an expression for the fraction of collisions that leads to reaction in
a bimolecular elementary process:
f

(

1 +

μ 12 v^2 c

2 kBT

)

e−μ^12 v

(^2) c/ 2 kBT
(12.3-10)
Thecritical energyis given by
εc

1

2

μ 12 v^2 c (12.3-11)

so that Eq. (12.3-9) can be written

Z 12 (reactive)〈v 12 〉πd 122 N 1 N 2

(

1 +

εc

kBT

)

e−εc/kBT (12.3-12)

which corresponds to

rate−

d[F 1 ]

dt

−

dN 1

dt



Z 12 (reactive)

NAv



1

NAv

〈v 12 〉πd 122

(

1 +

εc

kBT

)

e−εc/kBTN 1 N 2

NAv〈v 12 〉πd 122

(

1 +

εc

kBT

)

e−εc/kBT[F 1 ][F 2 ] (12.3-13)

where we have used the fact that

N 1 NAv[F 1 ] and N 2 NAv[F 2 ]

The reaction is second order with a rate constant given by

kNAvπd 122

(

8 kBT

πμ 12

) 1 / 2 (

1 +

Ec

RT

)

e−Ec/RT (12.3-14)

where we define amolar critical energy

EcNAvεc (12.3-15)