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

5.2 Kinetic gas theory 109
I
Correct derivation of the gas pressure and the ideal gas equation:
• Likewise, we take the differential number of wall collisions for specific speeds(Eq.
5.83) to compute the momentum change ( ) in time
( )=2 | |×| | × ( ) ( ) ( )
(5.93)
• Inserting again the 1D-Maxwell-Boltzmann distributions and integrating over all
positive and all ,wefind the total momentum change in time :
µ 32
=
× 2×
2
Z ∞ Z ∞ Z ∞ µ
2 exp − 2
2
0 −∞ −∞
µ µ
exp − 2
exp − 2
2 2
• Pressure for = µ 12
again using the substitution =
2
y
y
=
=
=
µ 12
× 2
2
µ 1
12
2 2
Z ∞
0
Z ∞
0
(5.94)
µ
2 exp − 2
(5.95)
2
2 −2 =
(5.96)
= (5.97)
Thus, we have fixed our earlier simple derivations by appropriately integrating over the
Maxwell-Boltzmann velocity distribution of the molecules.