Modern Engineering Thermodynamics

Summary 759
In this chapter, we summarize the essential concepts of statistical thermodynamics and present them in a simple
enough manner to make the results useful. This subject is very effective when experimental data are not available
and values of various thermodynamic properties are needed to carry out an engineering analysis. The interested
reader is encouraged to fill in the various theoretical gaps in the material presented in this chapter by additional
reading in this area.
Some of the more important equations introduced in this chapter follow. Do not attempt to use them blindly
without understanding their limitations. Please refer to the text material where they were introduced to gain an
understanding of their use and limitations.
1. The collision frequency F and mean free path λ of a moving gas molecule in a volume V containing N
molecules are given by
F = σ V rms
N
V
8 1/2
λ =
3π
1
ðN/VÞσ
where σ = 4πrmolecule 2 is the collision cross-section.
2. The average, root mean square and most probable molecular velocities are given by
rffiffiffiffiffiffiffiffiffiffi
rffiffiffiffiffiffiffiffi
rffiffiffiffiffiffiffiffi
V
8kT
avg =
πm , V
3kT
rms = , V
2kT
mp =
m
m
3. The fraction of molecules with velocities between V 1 and V 2 is given by
NðV 1 ! V 2 Þ
N
= erfðx 2 Þ − erfðx 1 Þ − p 2 ffiffiffi ðx 2 e −x2 2 − x1 e −x2 1 Þ
π
x 1 = V 1 /V mp , x 2 = V 2 /V mp
4. The compound probability that both independent events A and B will occur is given by
PAand ð BÞ = P AB = P A P B
5. The probability that either event A or B will occur is given by
PAor ð BÞ = P A + P B − P AB
6. The number of different ordered arrangements (permutations) of N things taken R at a time without
repetition is given by
P N R = N!
ðN − RÞ!
7. The number of different ordered arrangements (permutations) of N things taken R at a time allowing
repetition is given by
P N N = NR
8. The number of different unordered arrangements (combinations) of N things taken R at a time without
repetition is given by
C N R = PN R
R!
=
N!
ðN − RÞ!R!
9. The number of unordered arrangements (combinations) of N things taken R at a time allowing repetition is
given by
C N R = PN R
R!
=
ðN + R − 1Þ!
ðN − 1Þ!R!
10. The number of permutations or combinations of N things, R 1 of one kind, R 2 of a second kind, …, and
R k of a kth kind is given by
P N R 1,R 2 ,…,R k
= C N R 1,R 2,…,R k
=
N!
R 1 !R 2 !R 3 !⋯, ...,R k !