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

5.2 Kinetic gas theory 103
I Velocity distribution in 3 dimensions: Since the velocities in direction are
independent, the probability that a molecule has a velocity between and + ,
and + ,and and + is given be the product of ( ) , ( ) ,
and ( ) :
( )
= ( ) ( ) ( )
µ Ã
32
=
× exp − ¡ ¢ !
2 + 2 + 2
(5.62)
2
2
We are, however, not interested in the distribution of velocities pointing into a specific
direction ( ), but in the probability that a molecule has a certain speed || in
any direction, i.e., that the velocity vector ends somewhere on a sphere with radius
q
|| = 2 + 2 + 2 (5.63)
To obtain this distribution, we must integrate over all polar angles and . This
corresponds to replacing the volume element with the volume 4 2 of
a spherical shell with radius and thickness y
I The 3D Maxwell-Boltzmann speed distribution (Fig. 5.4):
() =
µ 32
× 4 2 × exp
µ− 2 (5.64)
2
2
Note that this distribution is also normalized for
Z ∞
() =1 We check this by
substituting
y
Z ∞
0
r
r
µ 12
=
2 y = 2 y = 2
(5.65)
= 2
2 (5.66)
2
() =
µ 32
× 4
2
0
Z ∞
0
0
µ
2 2
2 −2
12
(5.67)
µ 32 Z∞
µ 32 1 1
=4 × 2 −2 =4 × × 1 √ =1 (5.68)
4