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

Appendix E 286
Appendix E: Laplace transforms
The concept of Laplace and inverse Laplace transforms is extremely useful in chemical
kinetics for two reasons:
(1) They connect microscopic molecular properties and statistically averaged quantities:
a) () ↔ ( ): Collision cross section vs. thermal rate constant for bimolecular
reactions (section ??),
b) () ↔ ( ): Specific rateconstantvs. thermal rate constant for unimolecular
reactions (section 8),
c) () ↔ ( ): Density of states vs. partition function in statistical rate
theories 8).
(2) They provide a convenient method for solving of differential equations (section
3.4.2).
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Definition E.1: The Laplace transform L [ ()] of a function () is defined as the
integral
Z ∞
() =L [ ()] = () −
(E.1)
where
0
• is a real variable,
• () is a real function of the variable with the property () =0for 0,
• is a complex variable,
• () =L [ ()] is a function of the variable .
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Definition E.2: The inverse Laplace transform L −1 [ ()] of the function () is
defined as the integral
() =L −1 [ ()] = 1 Z
2
+∞
−∞
()
(E.2)
where
• is an arbitrary real constant.