Physical Chemistry Third Edition

D Some Derivations of Formulas and Methods 1265

D.5 An Integration for the Collision Theory of

Bimolecular Reactions

In Section 12.3 we stated the result of an integration over all relative velocities satisfying

the constant thatv<vc. In order to carry out this integration, we change variables,

expressing the kinetic energy of the pair of particles in terms of the velocityvcof the

center of mass and the relative velocityvof the two molecules. The kinetic energy is

expressed in terms of the kinetic energy of the center of mass and the kinetic energy of

relative motion, as shown in Appendix E:

K 

1

2

Mv^2 c+

1

2

μv^2 (D-27)

whereMm 1 +m 2 , whereμm 1 m 2 /Mis the reduced mass of the two particles,

wherevcis the speed of the center of mass of the two particles, and wherevis the

relative speed of one particle relative to the other. Using Eq. (9.3-40) for the probability

distributions, we write

dZ 12 πd 122 N 1 N 2

(

m 1

2 πkBT

) 3 / 2 (

m 2

2 πkBT

) 3 / 2

ve−MV

(^2) / 2 kBT
e−μv
(^2) / 2 kBT
d^3 vcxd^3 v
(D-28)
We integrate Eq. (D-28) over all values ofvcx,vcy, andvcz. Integration overvcxis
just like the integration in Eq. (9.3-20), and gives a factor of (2πkBT/M)^3 /^2 :
Z 12 (reactive)πd 122 N 1 N 2

(

2 πkBT

M

) 3 / 2 (

m 1

2 πkBT

) 3 / 2 (

m 2

2 πkBT

) 3 / 2 ∫

ve−μv

(^2) / 2 kBT
d^3 v
(D-29)
We now integrate over the values ofvthat satisfy the condition that the relative speed
exceeds the critical valuevc. The integration in this equation is carried out in spherical
polar coordinates in the relative velocity space ofv,θ, andφ. Integration over the angles
θandφgives a factor of 4π. The integration must include only values ofvsatisfying
the condition thatv<vc. We use a tabulated indefinite integral to obtain
∫∞
vc
e−μv
(^2) / 2 kBT
v^3 dv

1

2

e−μv

(^2) c/ 2 kBT

(

2 kBT

μ

)(

v^2 c+

2 kBT

μ

)

(D-30)

The final result is

Z 12 (reactive)πd 122 N 1 N 2 (8kBT/πμ)^1 /^2 (1+μv^2 c/ 2 kBT)e−μv

(^2) c/ 2 kBT
(D-31)