Modern Engineering Thermodynamics

760 CHAPTER 18: Introduction to Statistical Thermodynamics
11. The statistical thermodynamic equations for a monatomic Maxwell-Boltzmann gas are
u = 3 9
2 RT
h = u + RT = 5 2 RT
c v = 3 2 R
>=
For monatomic gases only
c p = 5 2 R
n h
i
s = R ln ð2πm/ħ 2 Þ 3/2 ðkTÞ 5/2 /p + 5 o
>;
2
12. The statistical thermodynamic equations for a diatomic Maxwell-Boltzmann gas are
u = RΘ v
2 + 5 2 RT + RΘ v
½expðΘ v /TÞ−1Š
h = RΘ v
2 + 7 2 RT + RΘ v
½expðΘ v /TÞ−1Š
n h
i
s = R ln ð2πm/ħ 2 Þ 3/2 ðkTÞ 5/2 /p + 5 o
+ Rfln ½T/σ Θ r ÞŠ+ 1g
2
+ R ln ½1 − expð−Θ v /TÞŠ −1 + ðΘ v /TÞ/ ½expðΘ v /TÞ−1Š
c v = 5 2 R + R ð Θ v/TÞ 2 ½expðΘ v /TÞŠ
½expðΘ v /TÞ−1Š 2
c p = 7 2 R + R ð Θ v/TÞ 2 ½expðΘ v /TÞŠ
½expðΘ v /TÞ−1Š 2
13. The statistical thermodynamic equations for a linear polyatomic Maxwell-Boltzmann gas molecule are
u = 5 RΘ
RT + vi
∑3b−5
2
i=1
2 + Θ vi
R∑3b−5
i=1
½expðΘ vi /TÞ−1Š
RΘ vi
h = 7 RT + ∑3b−5
2
i=1
2 + R∑3b−5
i=1
½expðΘ vi /TÞ−1Š
n
s = R ln ð2πm/ħ 2 Þ 3/2 ðkTÞ 5/2 /p + 5 o
+ Rfln ½T/ðσ Θ r ÞŠ+ 1g
2
+ R∑ 3b−5
s vib = R∑ 3b−5
i=1
i=1
Θ vi
ln½1 − expð−Θ vi /TÞŠ −1 + ðΘ vi /TÞ/ ½expðΘ vi /TÞ−1Š
ln½1 − expð−Θ vi /TÞŠ −1 + ðΘ vi /TÞ/ ½expðΘ vi /TÞ−1Š
14. The statistical thermodynamic equations for a nonlinear polyatomic Maxwell-Boltzmann gas molecule are
the same as those for a linear molecule except the upper limit of 3b−5 in the summations is replaced by
3b−6.
Problems (* indicates problems in SI units)
1. Determine the number of diatomic nitrogen (N 2 ) molecules in
1in 3 at 70.0°F and a pressure of 1.00 × 10 −10 mm of mercury
absolute (a very high vacuum).
2.* Determine the mean free path and the collision frequency of
diatomic nitrogen at 1000. K and 10.0 MPa. The effective radius
of the nitrogen molecule is 1.10 × 10 −10 m.
3.* For 10 10 bromine (Br 2 ) molecules confined in a volume of
1.00 m 3 at a pressure of 1.00 Pascal, assume ideal gas kinetic
theory behavior and determine
a. The temperature in the container.
b. The mean free path between collisions.
c. The collision frequency F: