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

5.2 Kinetic gas theory 107
• Momentum change due to wall collision, force on wall, and pressure:
=2 × 1 (5.79)
6
• Gas pressure:
y
= = 1
= 1 × 2 × 1 (5.80)
6
= 1 3 2 (5.81)
• Result for 1mol ( =
with =6022 1 × 10 23 mol −1 and kin = 2
2 ):
= 1 3 2 = 2 3 kin = 2 3 × 3 2 = (5.82)
I
Correct derivation of the gas kinetic wall collision frequency:
• Number of molecules with velocities in a volume of with =
p
2 + 2 + 2 (cf. Fig. 5.7) that will hit the area in the time :
( )=| | ( )
( )
( )
(5.83)
• To find the number of all molecules hitting in , we have to integrate over all
positive and over all (positive and negative) and , using the 1D-Maxwell-
Boltzmann distributions ( ) = ( )
,etc.:
µ 32
=
×
2
Z ∞ Z ∞ Z ∞ µ
| | exp − 2
2
0 −∞ −∞
µ µ
exp − 2
exp − 2
2 2
(5.84)
We know that the distributions over and are normalized to 1. Hence, we
only have to solve the integral over , which we do by the substitution
µ 12 µ 12 µ 12
2
=
y =
y =
(5.85)
2
2
y
Z ∞
0
µ
| | exp − 2
=
2
Z ∞
0
= 2
µ 2
Z ∞
0
12
|| −2 µ 2
−2 = 2
1
2 =
12
(5.86)
(5.87)