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

5.1 Hardspherecollisiontheory 96
• mean speed of A and B:
=
=
r r
8 8
=
(5.4)
r r
8 8
=
(5.5)
• mean relative speed (2 different molecules, i.e., A 6= B): 35
=
r 8
(5.9)
with the reduced mass
=
+
(5.10)
• mean relative speed (single type of molecules, i.e., A = B):
y
=
+
= 1 2 (5.11)
= √ 2 ×
r
8
(5.12)
I
Hard sphere gas kinetic collision frequency:
• We want to determine the number of collisions of a single molecule A with
molecules B in time . This is given by the product of the (volume that A has
swept in ) × (number density of B):
— volume of cylinder that A has swept in time :
volume = cross section × (5.13)
y
2 × (5.14)
35 The mean relative speed of two molecules with velocities u and u is found by looking at the
square of the difference speed
2 = |u − u | 2 = | | 2 + | | 2 − 2 | || | cos (5.6)
The third term averaged over cos is 0, therefore
Since and
2 = | | 2 + | | 2 = 3
+ 3
= 3 ( + )
= 3
³
2´12 differ only by a constant factor, we also find that
=
s
8
(5.7)
(5.8)