Modern Engineering Thermodynamics

734 CHAPTER 18: Introduction to Statistical Thermodynamics
EXAMPLE 18.2
Determine the collision frequency and mean free path for neon at 273 K and 0.113 MPa. The molecular mass of neon is
20.183 kg/kgmole.
Solution
For neon,
m = M N o
=
20:183 kg/kgmole
6:022 × 10 26 molecules/kgmole = 3:35 × 10−26 kg/molecule
Then, the root mean square velocity of the neon molecules is given by Eq. (18.13) as
V rms =
3kT 1/2
31:38 × 10 −23 1/2
½
J/ðmolecule⋅KÞŠð273 KÞ
=
m
3:35 × 10 − 26 = 581 m/s
kg/molecule
From Table 18.1, we find that the radius of the neon molecule is 1.30 × 10 −10 m,
so the collision cross-section is
and the collision frequency is
σ = 4πr 2 = 4πð1:30 × 10 −10 mÞ 2 = 2:12 × 10 −19 m 2
where
F = σV
N 8 1/2
rms
V 3π
N
V = p
kT = 0:113 × 10 6 N/m 2
½1:380 × 10 −23 N⋅m/ðmolecule⋅KÞŠð273 KÞ = 3:00 × 1025 molecules/m 3
then,
F = σV avg
N
V
8 1/2
= ð3:00 × 10 25 molecules/m 3 Þ
3π
= 3:40 × 10 9 collisions/s
8
3π
1/2ð2:12
× 10 −19 m 2 Þð581 m/sÞ
so that the molecular mean free path is given by Eq. (18.15) as
λ =
1
ðN/VÞσ = 1
ð3:00 × 10 25 molecules/m 3 Þð2:12 × 10 −19 m 2 Þ = 1:57 × 10−7 m
Exercises
4. If the temperature of the neon in Example 18.2 is increased from 273 K to 1000. K, determine the new root mean
square velocity and collision frequency of the neon molecules. Answer: V rms = 112 m/s and F = 6:5 × 10 9 collisions/s:
5. Determine the collision cross-sectional area of methane molecules. Answer: σ = 5.38 × 10 −19 m 2 .
6. Compute the molecular mean free path for carbon dioxide molecules at 300. K and 1.50 kPa. Answer: λ = 4.15 × 10 −6 m.
18.5 MOLECULAR VELOCITY DISTRIBUTIONS
Theories attempting to explain population behavior in living systems often begin with the following simple differential
equation for the time rate of change of the population N:
dN
=±αN (18.16)
dt
which says that the rate of change of the population N depends directly on the instantaneous value of the population.
If α is a constant, Eq. (18.16) can be integrated to give
N = N 0 e ±αt (18.17)
where N 0 is the initial population at time zero. This equation predicts an exponential growth or decay in population
depending on the sign of α (see Figure 18.4). In biological systems, Eq. (18.17) is usually inaccurate over
long time intervals, because α is not constant but instead depends on a number of variables and is often dependent
upon N itself.