Modern Engineering Thermodynamics

18.12 Diatomic Maxwell-Boltzmann Gases 753
23. Alternatively, Dorkmann in Example 18.8 might choose to lower the final compression pressure to 1.00 MPa. Would
the process work under this condition, if all the other variables remain unchanged? Answer: No, because now
1(S P ) 2 = −0.358 kJ/kg·K, which, being negative, violates the second law of thermodynamics.
24. However, instead of choosing either of the alternatives proposed in Exercises 22 and 23, Dorkmann in Example 18.8, for
some unknown reason, chooses to change the gas from krypton to helium. Will his original process work with this new
gas? Answer: No, since now 1 (S P ) 2 = −32.8 kJ/k·K, which, being negative, violates the second law of thermodynamics.
18.12 DIATOMIC MAXWELL-BOLTZMANN GASES
For a diatomic Maxwell-Boltzmann gas, it can be shown that the translational partition function is the same as
that for a monatomic gas, Eq. (18.45). However, now, 2 rotational and 1 vibrational degrees of freedom are present.
In this case, the rotational and vibrational partition functions are
and
Z rot =
T
σ Θ r
(18.47)
Z vib = ½1 − expð−Θ v /TÞŠ −1 (18.48)
where σ is the rotational symmetry number (the number of axes about which the molecule can be rotated 180°
and be indistinguishable from the original configuration), Θ r is called the characteristic rotational temperature, and
Θ v is called the characteristic vibrational temperature. Tables 18.8 and 18.9 give values for Θ r , Θ v , and σ for various
substances. The various components to the resulting specific property equations then become
u trans = 3 2 RT
(18.49a)
u rot = RT
(18.49b)
RΘ
u vib = ðu o Þ vib +
v
½expðΘ v /TÞ− 1Š
(18.49c)
Table 18.8 Characteristic Vibrational and Rotational Temperatures
of Some Common Diatomic Materials
Material Θ v ðKÞ Θ r ðKÞ
H 2 6140 85.5
HF 5954 30.3
OH 5360 27.5
HCl 4300 15.3
CH 4100 20.7
N 2 3340 2.86
HBr 3700 12.1
HI 3200 9.0
Co 3120 2.77
NO 2740 2.47
O 2 2260 2.09
Cl 2 810 0.35
Br 2 470 0.12
I 2 309 0.05
Na 2 230 0.22
K 2 140 0.08
Source: From Lee, John F., Sears, Francis W., Turcotte, Donald L. Statistical Thermodynamics, © 1963. Addison-
Wesley Publishing Co., Inc, Reading, MA. Adapted from Table 10-1 on page 204. Reprinted with permission.