Modern Engineering Thermodynamics

18.3 Kinetic Theory of Gases 729
3. All the molecules behave like rigid elastic spheres.
4. The molecules exert no forces on each other except when they collide (i.e., there are no long-range forces).
5. All molecular collisions are perfectly elastic.
6. The molecules are always distributed uniformly in their container.
7. The molecular velocities range continuously between zero and infinity. 1
8. The laws of classical mechanics govern the behavior of all molecules in the system.
Each of the N molecules has its own unique velocity V i . Using this velocity, we can define the following
concepts for the system of molecules.
The average molecular velocity : V avg = ∑ N
i=1
V i
N
(18.2)
The root mean square molecular velocity : V rms =
∑ N
V 2 !1/2
i
i=1
N
(18.3)
and in the limit as N → ∞, we can extend the summations in Eqs. (18.2) and (18.3) into integrals as follows
Z ∞
V
V avg = dN V (18.4)
N
0
Z ∞
V rms =
V 2 1/2
dN V
(18.5)
N
0
where dN V is the number of molecules with velocities between V and V + dV. Wealsodefinethetranslational total
internal energy U trans as the sum of the kinetic energies of all the molecules in the system, or 2
U trans = ∑ N
i=1
m i V 2 i /2 (18.6)
Since assumption 1 requires that all the molecules have identical mass, we can set m i = m, and Eq. (18.6) becomes
!
U trans = m 2
∑N Vi
2
i=1
= 1 2 Nm
Consider now a spherical shell of radius R containing N ≫ 1 molecules. The radial force F r on the shell due to a
single molecular collision is (see Figure 18.1)
ðF r Þ per
= ma r = m dV r
≈ m ΔV r
dt Δt
molecule
where a r and V r are the radial components of the acceleration and velocity. Using the geometry shown in
Figure 18.1, this equation becomes
ðF r Þ per
≈ m V i cos θ − ð−V i cos θÞ
= 2mV i cos θ
Δt
Δt
molecule
and the total radial force on the shell due to collisions by all N molecules is
ðF r Þ total = ∑ N
i=1
ð
F r
Þ per
= ∑ N
2mV i cos θ
molecule i=1
Δt
The internal pressure inside the shell can now be computed from
p =
ð F rÞ total
= 1 2mV i cos θ
Area 4πR 2 ∑N
i=1
Δt
(18.7)
1 Clearly, no molecule can have a velocity greater than the speed of light, but allowing the velocities to range to infinity is of
tremendous mathematical value in the development of this theory. Though fundamentally wrong, we find that this assumption adds
little error to the results.
2 Since the formulae presented in this chapter were developed by physicists using the SI units system wherein g c = 1, I elected to set
g c = 1 in all the relevant equations in this chapter to simplify them somewhat. Thus, we write mV 2 /2 instead of mV 2 /2g ð c Þ for kinetic
energy and so forth.