Physical Chemistry 3: — Chemical Kinetics — - Christian-Albrechts ...

5.4 Advanced collision theory 123
5.4.5 Bimolecular rate constants from collision theory
I Transformation from velocity to energy space: Taking our results from kinetic
gas theory, we are now in the position to apply the fundamental equation of collision
theory (Eq. 5.140)
( )=
Z ∞
0
() ( ) (5.156)
using various functional forms for . Towards these ends, we first transform from to
by inserting () from Eq. 5.64 (with the reduced mass instead of ). We thus
obtain
µ
( )=4
2
32
∞
As done above, we substitute for according to
and obtain
( )=
Z
0
3 ()exp
µ− 2 (5.157)
2
= 1 2 2 (5.158)
µ 12 2
y =
(5.159)
y
µ 12 2 1
=
2
µ 1
2
y =
µ 12 µ 1 2
µ 12 µ 12 µ 12 1 1 1
=
(5.160)
2
12 µ 1
12
(5.161)
32
∞
Z
0
( )exp
µ
−
(5.162)
The integration from 0 to ∞ takes into account that a reaction can occur only if
0 .
This expression is interpreted as follows:
• The function − describes the fraction of molecules with energy .
• The function ( ) in the integrand is called the excitation function 38 . This
can, in general, be a rather complicated (and not necessarily smooth) function
(see Fig. 5.17).
• The complete integrand ( ) − is called the reaction function 39 ,
because it describes the reactive fraction of the molecules.
• Therateconstant ( ) corresponds to the area under the reaction function.
38 “Anregungsfunktion”
39 “Reaktionsfunktion”