Modern Engineering Thermodynamics

750 CHAPTER 18: Introduction to Statistical Thermodynamics
Table 18.7 Formula for Computing the Number of Microstates in the ith Macrostate
for Various Statistical Models
Model
Maxwell-Boltzmann
Fermi-Dirac
Bose-Einstein
Note: Here, Z = ∑g i expð−ε i /kT
Number of Microstates per
Macrostate, W
N!
g
∏ i
N
i i !
∏
i
g i !
N i !ðg i − N i Þ!
ðg i + N i − 1Þ!
∏
N
i i !ðg i − 1Þ!
Most Probable Distribution (N i ) mp
N
Z
gi exp − ε
i
kT
h
g i B exp ε i −1
i
kT
+ 1
h
g i B exp ε
i
kT
− 1
Þ = partition function, and B = expð− μ/kTÞ, where μ is the molar chemical potential.
i −1
The problem we now face is that the total entropy S of a system is an additive property whereas W mp is not. In
the probability mathematics presented earlier, we found that the probability that independent events A and B
simultaneously occur is P (A and B) = P AB = P A P B , and since W mp is related to mathematical probability through
Eq. (18.38), we can write
but
Therefore, we must find a function f such that
WAand ð BÞ = W A W B
SAand ð BÞ = S A + S B
SAand ð BÞ = fðW A Þ+ fðW B Þ = fWAand ½ ð BÞŠ = fðW A W B Þ
The only general function that satisfies this relation is the logarithm, since
ln W A + ln W B = ln W A W B
Therefore, we choose to set S proportional to ln W mp . It can be shown that the constant of proportionality in
this relation is just Boltzmann’s constant k, so we end up with the following entropy-probability relation:
Thus, we see that entropy is a measure of the molecular order within a system.
S = k ln W mp (18.40)
18.10 MAXWELL-BOLTZMANN GASES
To limit the algebraic complexity of the resulting property formula, we restrict our attention to the Maxwell-
Boltzmann model. It can be shown that, for Maxwell-Boltzmann gases with N ≫ 1,
u = RT 2 ∂ ln Z
(18.41)
∂T
and
where
h = u + RT (18.42)
Z = ∑g i expð−ε i /kTÞ
Z is called the partition function of the system and g i is the degeneracy of the ith energy level ε i . At high temperatures,
the number of quantum states (or degeneracy levels g i ) available at any energy level is much larger than
the number of particles N i in that energy level, or
g i
N i
≫ 1