Page 57 - Air pollution and greenhouse gases from basic concepts to engineering applications for air emission control

2.1 Gas Kinetics 31

Before
mc 1 Δ
impact F t
Wall
l
After mc 2 x
impact

l

Fig. 2.1 Principle of impulse and linear momentum analysis on a single molecule

~
m~ c 1 þ FDt ¼ m~ c 2 ð2:16Þ
~
where the term FDt is the impulse during the collision [9]. Both force and velocity
are vector quantities defined by their magnitudes and directions. Along one of the
rectangular coordinates, say x direction as depicted in Fig. 2.1, the above equation
can be written in terms of magnitudes,

mc 1x F x Dt ¼ mc 2x ð2:17Þ
Reorganize it and one gets,

F x Dt ¼ mc 1x þ c 2x Þ ð2:18Þ
ð
Since the impact between the surface and the gas molecules is elastic, we have
c 1x ¼ c 2x ¼ c x and the above equation becomes

F x Dt ¼ 2mc x ð2:19Þ
With a constant speed of c x , the molecule will impact on the same wall once
every 2l=c x time units (for a round trip), where l is the distance between two
opposite walls of the container. Then, the force on the wall produced by the same
molecule along x direction is:

mc 2
2mc x x
F x ¼ ¼ ð2:20Þ
2l=c x l