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

5.4 Advanced collision theory 121
I 5 → 6:Thermal averaging over all ² :
( )= X
× ( )
= X −
× ( )
Z
−
=
× ( )
I
3 → 5 and 4 → 6:Laplace Transformation:
Z ∞
( ) ∝ × ( ) × − (5.152)
0
This expression follows from the fundamental equation (Eq. 5.140) by substituting
for . The exact transformation (giving the correct factors before the integral) will be
carried out below (see Eq. 5.162).
5.4.4 Laplace transforms in chemistry
Formally, the transformation (Eq. 5.152) is a Laplace transformation (see Appendix
E). A Laplace transform generally described the transformation from a microscopic
(“microcanonical”) quantity such as ( ) to a macroscopic (“macrocanonical”, i.e.,
dependent) quantity such as ( ).
I Definition of the Laplace Transformation: The Laplace transform L [ ()] of a
function () is defined as the integral
() =L [ ()] =
Z ∞
0
() − (5.153)
I
Inverse Laplace Transformation:
with an arbitrary real number .
() =L −1 [ ()] = 1
2
Z
+∞
−∞
() (5.154)