Modern Engineering Thermodynamics

754 CHAPTER 18: Introduction to Statistical Thermodynamics
Table 18.9 Rotational Symmetry Number for Some Simple Materials
Material
σ
Any diatomic molecule with two different atoms (e.g., HCl, HI, or NO) 1
Any diatomic molecule with two identical atoms (e.g., H 2 ,O 2 ,orN 2 ) 2
Any triatomic molecule with two different atoms forming an isosceles triangle (such as H 2 O) or any linear triatomic
2
molecule (e.g., CO 2 or NO 2 ,)
Any quatratomic molecule with two different atoms forming an equilateral triangular pyramid (e.g., NH 3 ) 3
Any molecule forming a plane rectangle (e.g., C 2 H 4 ) 4
Any pentatomic molecule with two different atoms forming a regular tetrahedron with the carbon atom at the center of 12
mass (e.g., CCl 4 or CH 4 )
Thus,
u = u trans + u rot + u vib = ðu o Þ vib
+ 5 2 RT + RΘ v
½expðΘ v /TÞ− 1Š
(18.49d)
where
is the vibrational energy at absolute zero temperature. Similarly,
ðu o Þ vib
= RΘ v /2 (18.50)
h trans = 5 2 RT
(18.51a)
h rot = u rot = RT
(18.51b)
Thus,
RΘ
h vib = u vib = ðu o Þ vib +
v
½expðΘ v /TÞ− 1Š
h = h trans + h rot + h vib = ðu o Þ vib
+ 7 2 RT + RΘ v
½expðΘ v /TÞ− 1Š
(18.51c)
(18.51d)
Then, we can find
ðc v Þ trans
= 3 2 R
(18.52a)
ðc v Þ rot = R
(18.52b)
ðc v Þ vib
= R ð Θ v/TÞ 2 ½expðΘ v /TÞŠ
½expðΘ v /TÞ− 1Š 2 (18.52c)
Thus,
c v = ðc v Þ trans + ðc v Þ rot + ðc v Þ vib = 5 2 R + R ð Θ v/TÞ 2 ½expðΘ v /TÞŠ
½expðΘ v /TÞ− 1Š 2 (18.52d)
WHAT IS ABSOLUTE ZERO TEMPERATURE?
In the diatomic Maxwell-Boltzmann model, the concept of absolute zero temperature corresponds to the cessation of all
translational and rotational molecular motion, but vibrational motion is still allowed to occur. Thus, the internal energy
does not vanish at absolute zero temperature in this model.