Page 54 - Air pollution and greenhouse gases from basic concepts to engineering applications for air emission control

28 2 Basic Properties of Gases

2.1.1 Speeds of Gas Molecules

An understanding of the gas properties requires a good understanding of the
molecular velocities. In engineering dynamics analysis, we describe a particle
velocity with its magnitude and its direction. Similarly, a gas molecule velocity
vector ~ c is described using its three directional components in a rectangular x-y-z
coordinate system as

^
^
~ c ¼ c x i þ c y j þ c z k ^ ð2:1Þ
Maxwell-Boltzmann distribution is the most commonly used for molecular
speed distribution. The distribution of one-dimensional velocity 1 \ c i \ 1 is

m mc i
3=2 2
fc i ¼ exp i ¼ x; y; z ð2:2Þ
ðÞ
2pkT 2kT
This is a normal distribution with a mean of 0 and a variance of kT=m: It also
applies to the other two velocity components.
In engineering applications, total speeds of molecules are of more interest than
their components. The Maxwell-Boltzmann distribution describes the probability of
molecular speed [16].

m mc
3=2 2
2
f ðcÞ¼ 4pc exp ð2:3Þ
2pkT 2kT
where the Boltzmann constant k ¼ 1:3807 10 23 ð J/KÞ; c is the molecular
speed of a molecule, m is the mass of the molecule, and T is the temperature of the
gas.
In air pollution, we are interested in the distributions of the molecular
mean speed, root-mean-square speed and mean relative speed. As to be seen
shortly, they are useful parameters in molecular kinetics that lead us to microscopic
properties like pressure, viscosity, diffusivity, and so on. These speeds can be
computed from the Maxwell–Boltzmann distribution of molecular speed described
in Eq. (2.3).
The mean molecular speed ( c) is the mathematical average of the speed distri-
bution and it can be calculated by integration
Z 1 Z 1
m 3=2 3 mc 2
c ¼ cf cðÞdc ¼ 4p c exp dc ð2:4Þ
2pkT 2kT
0 0