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2.1 Gas Kinetics 41

Assume one molecule collides with the wall only once during Dt; then the
number of collisions is the same as the number of molecules within the volume
formed by Dx and A. Then, the number of collisions is expressed as the corre-
sponding number of molecules that collide with the wall during Dt

n c ¼ C N c x ADt ð2:53Þ

where C N is the number of molecules per unit volume of gas, which is defined in
Eq. (2.49). From this equation, we can get the number of molecules colliding with
2
the wall per unit area per unit time (1=sm )
n c
j x ¼ ¼ C N c x ð2:54Þ
ADt
Then, the total number of collisions considering the molecule speed distribution
is determined by integration, assuming constant molecule number concentration C N
at steady state,
Z 1 Z 1
J x ¼ j x fc x c x fc x ð2:55Þ
ðÞdc x
ðÞdc x ¼ C N
0 0
Using the Maxwell-Boltzmann distribution described in Eq. (2.2), the integration
part in Eq. (2.55) can be manipulated following
1 1
1 1
Z Z
ðÞdc x ¼
c x ðÞdc x ¼ c x
c x fc x jjfc x ð2:56Þ
2 2
0 1
where c x is the average of the absolute value of c x . Then Eq. (2.55) becomes
jj
1
J x ¼ C N c x ð2:57Þ
2
A step further from the Boltzmann distribution, we can relate c x with average
jj
molecular speed that can be calculated from Eq. (2.2)
Z 1 Z 1
m mc x 1
3=2 2
ðÞdc x ¼
c x ¼ c x fc x exp c x dc x ¼ c ð2:58Þ
2pkT 2kT 2
0 0
Thus, the collision per unit time per unit area along x direction in terms of
average molecular speed is

1
J x ¼ C N c ð2:59Þ
4