Modern Engineering Thermodynamics

732 CHAPTER 18: Introduction to Statistical Thermodynamics
EXAMPLE 18.1 (Continued )
Since the total mass of gas present is m total = m molecule N, where N is the number of molecules present, and m molecule =
molecular mass/Avogadro’s number = M/N o ; for nitrogen,
then,
m molecule = M N o
=
N =
28:0 kg/kgmole
6:022 × 10 26 molecules/kgmole = 4:65 × 10−26 kg/molecule
m
1:00 kg
=
m molecule 4:65 × 10 −26 kg/molecule = 2:15 × 1025 molecules
The total translational internal energy in 1.00 kg of nitrogen gas is
U trans = 3 2 NkT = 3 h
i
2 ð2:15 × 1025 moleculesÞ 1:380 × 10 −23 J
ð20:0 + 273:15 KÞ
molecule.K
= 131,000 J = 131 kJ
Note that the kinetic theory internal energy is independent of gas pressure and depends only on gas temperature, which
is a characteristic of ideal gas behavior.
Exercises
1. Determine the root mean square velocity of the nitrogen gas in Example 18.1 when the temperature is increased from
20.0°C to 200.°C. Answer: V rms = 648 m/s.
2. Suppose the nitrogen gas in Example 18.1 is replaced with water vapor. Determine the root mean square velocity and
mass of a water vapor molecule at 20.0°C. Answer: (m molecule ) water = 2.99 × 10 −26 kg and (V rms ) water = 637 m/s.
3. Determine the number of water molecules in 1.00 m 3 of saturated water vapor at 100.°C. Answer: N water = 2.00 × 10 25
water molecules.
18.4 INTERMOLECULAR COLLISIONS
To better understand molecule-molecule collisions, imagine that all the molecules except one are frozen in
space. Then, the moving molecule travels through this stationary forest of molecules colliding with them at
random. This model can be further simplified if the moving molecule is enlarged to twice its normal diameter
while all the stationary molecules are reduced to points of zero diameter (see Figure 18.3). The area swept out
by the motion of the enlarged molecule is called its molecular collision cross-section, given by
σ = πð2rÞ 2 = 4πr 2 :
Moving molecule
enlarged to twice
its normal diameter
Stationary molecules whose
diameters have been shrunk
to a point
The moving molecule’s
collision cross-section
FIGURE 18.3
A simplified model illustrating the molecular collision cross-section.