Page 58 - Air pollution and greenhouse gases from basic concepts to engineering applications for air emission control

32 2 Basic Properties of Gases

Now, consider all the N molecules in the container. The total force acting on the
wall due to N molecules with the same mass of m is
!
N
m X 2
F ¼ c ix ð2:21Þ
l
i¼1
Recall the assumptions of kinetic theory, there are a large number of molecules
moving randomly and constantly, therefore, Eq. (2.21) is applicable to any direc-
tions in the container. And the force on each wall can be considered same in
magnitude. Consider the force acting only on one wall and the magnitude of the
velocity can be calculated using
2
2
2
c ¼ c þ c þ c 2 iz ð2:22Þ
ix
iy
i
Since x, y, and z are randomly chosen and the molecular motion is random and
uniform along any direction, we get
1 2
2
2
2
c ¼ c ¼ c ¼ c ð2:23Þ
ix iy iz i
3
Now, the total force exerted by the molecules on one wall described in Eq. (2.21)
can be expressed in terms of the total speed instead of a single component of the
velocity,
!
N
m X 2
F ¼ c i ð2:24Þ
3l
i¼1
According to the definition of root-mean-square speed described in Eq. (2.8),
N
X
Nc 2 ¼ c 2 ð2:25Þ
rms i
i¼1
Then the total force on the wall of the container can be written as:
Nmc 2 rms
F ¼ ð2:26Þ
3l
The resultant pressure, which is force per unit area of the wall, of the gas can
then be written as

F Nmc 2 rms Nm 2
P ¼ ¼ ¼ c rms ð2:27Þ
A 3lA 3V