Modern Engineering Thermodynamics

18.11 Monatomic Maxwell-Boltzmann Gases 751
Then the number of microstates per macrostate for the three statistical models shown in Table 18.7 are
approximately the same; that is,
W BE ≈ W FD ≈ W MB = ∏
i
Also, under this condition, the most probable particle distribution of these three models are approximately the
same
ðN i Þ BE
mp
≈ ðN i Þ FD
mp
≈ ðN i Þ MB
mp
g Ni
i
N i !
= N
g i expð−ε i /kTÞ (18.43)
Z
These two results can be inserted into Eq. (18.4) to produce an equation for entropy as follows. First, we calculate
ln W mp from Table 18.7 as
We then use Stirling’s approximation:
for the factorial term to obtain
ln W mp = ∑
i
ln W mp ≈∑
i
Then, we use Eq. (18.43) to evaluate the term
ðg i /N i
ðN i ln g i − ln N i !
ln N! ≈ N ln N − N
Þ mp
h
i
ðN i Þ mp ln ðg i /N i Þ mp + 1
Þ mp = Z N exp ð ε i/kTÞ
which produces the result
which simplifies to
ln W mp ≈∑
i
ðN i Þ mp
½ln ðZ/NÞ+ ε i /kT + 1Š
The total entropy is now given by Eq. (18.40) as
ln W mp ≈ N½ln ðZ/NÞ+ 1Š+ U/kT
S = Nk½ln ðZ/NÞ+ 1Š+ U/T
and since Nk = NR/N o = nR = ðm/MÞR = mR, the specific entropy can be written as
s = R½ln ðZ/NÞ+ 1Š+ u/T (18.44)
The molecular model we consider here is that of a relatively simple molecule, in which the total molecular
energy can be separated into only three modes: translational, rotational, and vibrational. Then, the partition
function Z is made up of translational, rotational, and vibrational molecular energy storage mechanisms and
can be written as
Z = ðZ trans ÞðZ rot ÞðZ vib Þ
Consequently, we can determine the molecular translational, rotational, and vibrational contribution to each of
the properties, u, h, and s. Because these partition functions depend on the geometry of the molecule, we begin
their study with the simplest possible structure, a monatomic gas.
18.11 MONATOMIC MAXWELL-BOLTZMANN GASES
For a Maxwell-Boltzmann monatomic gas, it can be shown that
Z trans = Vð2πmkT/ħ 2 Þ 3/2 (18.45)