Modern Engineering Thermodynamics

Problems 761
4. Find the temperature T at which V mp = c (the velocity of light)
for the neon atoms discussed in Examples 18.2 and 18.3.
5.* Hydrogen (H 2 ) at 3.00 MPa and 1000. K is confined in a
volume of 1.00 m 3 . Determine V avg , V rms , V mp , and the collision
frequency F:
6.* Oxygen (O 2 ) at 300. K and 1.00 atm pressure is confined in a
volume of 1.00 × 10 −4 m 3 . Determine
a. The number of oxygen molecules present.
b. The average molecular velocity.
c. The rms molecular velocity.
d. The most probable molecular velocity.
e. The number of molecules with a velocity in the range of 0 to
10 − 4 m/s:
f. The number of molecules with velocities greater than
10 6 m/s:
7. Using the following infinite series expansion
expð−x 2 Þ = 1 − x 2 /1! + x 4 /2! − x 6 /3! + x 8 /4! − …
show that the equivalent expansion for the error function is
erfðÞ=
x
2
pffiffiffi
x − x3
π 3ð1!Þ + x5
5ð2!Þ − x7
7ð3!Þ + …
8. Show that Eq. (18.26) can be written as
NðV ! ∞Þ
= 2 Z ∞
pffiffiffi
e −x2 dx + xe −x2
N π
where x = V/V mp :
9. Show that if a ≫ 1, then
Z ∞
a
e −x2 dx ≈ 1 2a e−a2
(Hint: Set x = a + y, then dx = dy and the integral over dy have
limits from 0 to ∞.)
10. Using the results from Problems 8 and 9, show that for x ≫ 1,
NðV ! ∞Þ
N
x
= 2
pffiffiffi
x + 1 π 2x
11. Using the equations of kinetic theory, it can be shown that the
number of molecules per unit time that leak out of an
isothermal pressurized container of volume V through a small
hole of area A is
_N leak = ðN/VÞðA/4
ÞV avg
e −x2
Show that the pressure in the container then decays according to
p = p 0 exp −AV avg t/4V
where V avg = ð8kT/πmÞ 1/2 , and p 0 is the pressure in the container
at time t = 0.
12.* Using the information given in Problem 11 and the equations
of kinetic theory, calculate the mass rate of separation of the
isotope U 235 from a gaseous mixture of U 238 and U 235 by
a molecular sieve. The sieve is a porous pipe 0.0300 m in
diameter and 2000. m long. The area of each pore is
2.60 × 10 −19 m 2 , and there are 10 9 pores per meter of pipe
length. The internal sieve temperature is 1000. K, and the partial
pressure difference of the U 235 across the sieve is 10.0 Pa.
13. The probability of failure of a space shuttle primary computer
system is 1.50 × 10 −3 . This computer system has a secondary
backup computer system with the same failure probability. What
is the probability of simultaneous failure of both the primary
and secondary computer systems?
14. An eight-cylinder engine has one bad spark plug. If the
mechanic removes two spark plugs at random, what is the
probability that the defective spark plug is found on the
first try?
15. A single die is tossed. What is the probability that it will come
up with a value greater than 4?
16. Two dice are tossed. Determine the probability that their sum
will be greater than (a) 2, (b) 4, (c) 6, (d) 8, and (e) 10.
17. Two cards are to be drawn from a standard deck of 52 cards.
Calculate the probability that these two cards are an ace and a
10, drawn in any order.
18. A coin is flipped twice. Determine the probability that only one
head results.
19. Two coins are flipped simultaneously. Determine the probability
that at least one is a head.
20. If three coins are simultaneously flipped, what is the probability
of getting (a) at least two heads and (b) exactly two heads.
21. Show that C N R ≡ CN N−R for any N and R.
22. An electronic component is available from five suppliers. How
many different ways can two suppliers be chosen from the five
available?
23. How many different ten-digit phone numbers can be made from
the digits 0 through 9 if the first three digits must be 414?
24. How many different six-digit automobile license plates can be
made using only the digits 0 through 9 if the digits may be
repeated?
25. How many different nine-digit social security numbers can be
made if
a. No digit is allowed to be repeated.
b. The digits can be repeated.
26. How many different three-letter “words” can be made from the
26 letters of the English alphabet without regard to vowels if the
letters can be used more than once per word?
27. A component subassembly consists of five pieces which can be
assembled in any order. A production test is to be designed to
determine the minimum time required to assemble the
subassembly. If each sequence of assembly is to be tested once,
how many tests need to be conducted?
28. A person is shopping for a new car. One dealer offers a choice
of five body styles, four engine types, ten color combinations,
three transmissions, and three accessory packages. How many
different cars are there to choose from?
29. Determine the collision probability of the bromine molecules in
Problem 4.
30.* A typical galaxy occupies a volume of about 1.00 × 10 61 m 3
and contains 1.00 × 10 11 stars, each with an effective radius of
1.00 × 10 9 m. Determine
a. The collision probability of two stars within the galaxy.
b. The number of stars that will have experienced a collision by
the time they have traveled a distance of 0.0100 mean free
paths.
31. Show that, for a Maxwell-Boltzmann gas, the total enthalpy H is
given by
H = −N∂ð ln ZÞ/∂β + NkTV½∂ð ln ZÞ/∂VŠ
where β = 1/kT and the pressure is p = NkTð∂ ln Z/∂VÞ: