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

5.1 Hardspherecollisiontheory 99
Rule-of-thumb:
Time between collisions:
=03mbary =03mm (5.43)
=1000mbar (5.44)
1 1
=
2 ×
(5.45)
=15 × 10 −10 s (5.46)
≈ 01ns (5.47)
¤
5.1.2 Allowance for a threshold energy for reaction
I The threshold energy E 0 : The majority of reactions exhibit a threshold energy, below
which the reaction does not occur (quantum mechanical tunneling is excluded here).
The molecules must have a minimum energy min to overcome this reaction threshold energy
0 . For our structureless spherical molecules, we consider only translational energy
: 36 0 : no reaction (5.48)
0 : reaction (5.49)
I Boltzmann distribution: Fraction of molecules with translational energy 0 :
µ
( 0 )
=exp − µ
0
=exp −
0
(5.50)
I
Resulting rate constant:
or
µ 12
max = 2 8
×
× − 0
(5.51)
µ 12
max = 2 8
×
× − 0
(5.52)
I Arrhenius activation energy: Applying the definition of the apparent (Arrhenius)
activation energy ,wefind that
= 2 ln
= 0 + 1 (5.53)
2
As we see, is not constant but (slightly) -dependent: The slope of a plot of ln
max
vs. −1 increases as →∞(or −1 → 0). The Arrhenius preexponential factor
would not be constant as well, it is also slightly -dependent.
36 We use and 0 to indicate the energies in molecular units, and and 0 for the respective
values in molar units.