Page 55 - Air pollution and greenhouse gases from basic concepts to engineering applications for air emission control

2.1 Gas Kinetics 29

In order to complete the integration, we need to know that

Z 1
1
3 2
x exp ax dx ¼ ð2:5Þ
2a 2
0
For this specific problem, a ¼ m=2kT, and the integration term can be deter-
mined as

Z 1 2 2
mc 1 kT
3
c exp dc ¼ 2 ¼ 2 ð2:6Þ
2kT 2 m=2kTÞ m
ð
0
Substituting Eq. (2.6) into Eq. (2.4) leads to
1=2
8kT
c ¼ ð2:7Þ
pm
By similar approaches, we can get the root-mean-square speed (v rms ), which is
the square root of the average squared speed:

s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
r ffiffiffiffiffiffiffiffi
Z 1 3kT
2
c rms ¼ c fcðÞdc ¼ ð2:8Þ
m
0
Comparison between Eqs. (2.7) and (2.8) shows that c rms [ c because c rms
contains a factor of 3 and c contains a factor of 8=p 2:55. This is resulted from
2
the fact that greater speeds are weighted more heavily in the integration based on c .
Average relative velocity is another molecular speed needed in our analysis that
follows. From engineering dynamics, we have learned that the relative velocity of
any two molecules A and B which behave like particles is

~ c A=B ¼~ c A ~ c B ð2:9Þ
where ~ c A=B is the velocity of molecule A relative to molecule B (m/s), and the
magnitude of the relative velocity is the square root of the scale product of itself:

c 2 ð ð
A=B ¼~ c A=B ~ c A=B ¼ ~ c A ~ c B Þ ~ c A ~ c B Þ ¼~ c A ~ c A 2~ c A ~ c B þ~ c B ~ c B
ð2:10Þ

Replacing the speeds in the above equation with the average speeds gives

c 2 ¼ ~ c A ~ c A Þ 2 ~ c A ~ c B Þ þ ~ c B ~ c B Þ ð2:11Þ
ð
ð
ð
A=B ave ave ave