Modern Engineering Thermodynamics

762 CHAPTER 18: Introduction to Statistical Thermodynamics
32. Compute the percent error in Stirling’s approximation,
ln N! ≈ N ln N − N, for the following values of N: (a) N = 5,
(b) N = 10, and (c) N = 50.
33.* Use the Maxwell-Boltzmann formulae for diatomic gases and
calculate the value of c p for molecular iodine I 2 at 0.00°C.
34. Use the Maxwell-Boltzmann formulae for diatomic gases and
calculate the value of c p /R at 2000.°C for (a) hydrogen,
(b) carbon monoxide, and (c) oxygen. Use the characteristic
vibrational temperatures found in Table 18.8. Compare your
results with the values given in Table 18.3.
35. An experimentally determined relation for the temperature
dependence of the constant pressure specific heat of molecular
oxygen O 2 over a wide temperature range is
pffiffiffi
c p = 0:3598 + 47:81/T − 5:406/ T
where c p is in Btu/ ðlbm.RÞ and T is in R. Use the Maxwell-
Boltzmann equations for a diatomic molecule and attempt to
predict the coefficients of the first two terms (i.e., 0.3598 and
47.81) of the equation. (Hint: At very high temperatures,
expðΘ v /TÞ≈ 1 + Θ v /TÞ: Explain why your results may not be very
accurate.
36. Use the Maxwell-Boltzmann formulae for diatomic gases and
calculate the value of c p for water vapor at 400.°F and 1.00 psia.
The characteristic temperatures of H 2 O are Θ r = 0.337 R,
Θ v1 = 4131 R, Θ v2 = 9459 R, and Θ v3 = 9720 R:
37.* The experimentally measured value for c v /R for ammonia, NH 3 ,
at 15.0°C is 3.42. Use the Maxwell-Boltzmann formulae for
nonlinear polyatomic gases to calculate the value of c v /R for
ammonia, then compute the percent error in your result. The
characteristic vibrational temperatures of ammonia are
Θ v1 = 1367 K, Θ v2 = Θ v3 = 2341 K, Θ v4 = 4801 K, and
Θ v5 = Θ v6 = 4955 K:
38.* The experimentally measured value for c v /R for methane, CH 4 ,
at 300. K is 3.2553. Use the Maxwell-Boltzmann formulae for
nonlinear polyatomic gases to calculate the value of c v /R for
methane, and then compute the percent error in your result. The
characteristic vibrational temperatures of methane are
Θ v1 = Θ v2 = Θ v3 = 1879 K,Θ v4 = Θ v5 = 2207 K,Θ v6 = 4197 K, and
Θ v7 = Θ v8 = Θ v9 = 4344 K:
39.* Calculate the specific entropy of HF at 0.00°C and atmospheric
pressure.
40.* Calculate the change in specific entropy as HCl gas is heated at
a constant pressure of 2.00 atm from 300. to 3000. K.
41. Determine the heat transfer required to heat 11.3 lbm of HBr
gas from 100. to 1000.°F in a closed, rigid, 1.50 ft 3 container.
42.* Determine the power produced as 0.300 kg/s of HI passes
through a steady state, adiabatic turbine from 2000. K, at
50.0 atm, to 1000. K at 1.00 atm pressure.
43.* Determine the entropy production rate for the turbine in
Problem 42.
Design Problems
The following are open-ended design problems. The objective is to
carryoutapreliminarydesignasindicated. A detailed design with
working drawings is not required unless otherwise specified by the
instructor. These problems have no specific answers, so each student’s
design is unique.
44. Design a heater that raises the temperature of diatomic chlorine
gas from 70°F to 2500°F without changing the pressure
significantly. Do not assume ideal gas behavior. Use Eqs.
(18.49a) through (18.54d) to calculate the necessary
thermodynamic properties of chlorine. Provide an assembly
drawing of your design along with all the relevant
thermodynamic and design calculations.
45. Design a flow meter that uses a measurement of one or more of
the thermodynamic u, h, ors to calculate the mass flow rate _m :
For example, we can construct an open system such that _m can
be calculated from _Q , h 1 , and h 2 as _m = _Q /ðh 2 − h 1 Þ: Provide an
assembly drawing of your design along with all the relevant
thermodynamic and design calculations.
46. Using the equation given in Problem 11, design an
instrument that determines the porosity of a small, square,
flat test sample of material. Be sure to explain how to
calculate the porosity from the measurements taken. If
possible, set up an experiment and test your technique with
samples of known porosity. Provide an assembly drawing of
your design along with all the relevant thermodynamic and
design calculations.
47. Design a spring-loaded throttling valve that isothermally
throttles 8:30 lbm/s of molecular bromine gas from 1000. psia,
70.0°F, to atmospheric pressure using choked flow conditions
(see Chapter 16). Provide an assembly drawing of your design
along with all the relevant thermodynamic and design
calculations.
48.* Design a system that increases the temperature of 0.350 kg of
diatomic sodium gas Na 2 from 1500. K at 20.0 kPa to 3000. K
simply by compressing it by some mechanism in a closed
system. No auxiliary heaters or coolers may be used. Provide an
assembly drawing of your design along with all the relevant
thermodynamic and design calculations.
Computer Problems
The following open-ended computer problems are designed to be
done on a personal computer using a spreadsheet, equation solver,
or programming language.
49. Using the equations from the kinetic theory of gases, write an
interactive computer program that returns values for u, h, and s
when p, T,M, and b are input by the user from the keyboard.
Allow the user to choose either the English or the SI units
system for the input values, and output values in the same units
system.
50.* Using the data on NO given in Example 18.9, plot curves of c p
and c v for NO vs. temperature for 0 ≤ T ≤ 5000 K: Compute at
least 100 points for each curve.
51. Using the equations for the thermodynamic properties of
diatomic gases given in the text, write an interactive computer
program that returns values for u, h, and s when p and T are
input by the user from the keyboard for one or more
(instructor’s choice) of the gases listed in Table 18.8. Assume
u 0 = 0, and allow the user to choose to work in either the
English or the SI units system. Output values in the same units
system that was chosen for input values.
52.* Using the appropriate equations from the text, plot k = c p /c v vs.
T for molecular oxygen over the range 0 ≤ T ≤ 5000 K: