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2.1 Gas Kinetics 39

The number of collisions this molecule experiences during the time interval of
Dt is determined by the number molecules colliding with the molecule in the swept
volume, DV, which is

2
DVC N ¼ pd c A=B DtC N ð2:47Þ
3
where C N is the number of molecules per unit volume of the gas (#=m ). The
relative velocity is used because other molecules are also traveling within the space;
it is described in Eq. (2.13):
p ffiffiffi
c A=B ¼ 2 c

The distance it travels in Dt is cDt, then the mean free path, which is the distance
traveled divided by the number of collisions, can then be calculated as

cDt 1
k ¼ ¼ p ð2:48Þ
2 ffiffiffi 2
pd c A=B DtC N 2pd C N
There is no need to correct the average speed in the numerator for the calculation
of distance traveled, which is supposed to be calculated using the average speed of
the molecules.
Assuming ideal gas, the number concentration of the gas molecules can be
calculated as

N nN a P
C N ¼ ¼ ¼ N a ð2:49Þ
V V RT

where N is the total number of molecules in the container with a volume V. There is
a simple relationship between the gas density and molecule number concentration
as

q ¼ C N m ð2:50Þ

With Eq. (2.49) and Eq. (2.50), Eq. (2.48) becomes

RT M
k ¼ p ffiffiffi ¼ p ffiffiffi ð2:51Þ
2
2
2pN a d P 2pN a d q
Molar weight (M) and gas molecule diameter (d) are fixed for a gas at stable
state, therefore, the mean free path depends only on the density of the gas (q),
which depends on the pressure and temperature of the gas. For ambient conditions,
the mean free path increases with increasing temperature or decreasing pressure.