Download CP7e: Ch. 12 Problems

Chapter 12 Problems
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challenging
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Section 12.1 Work in Thermodynamic
Processes
1.
The only form of
energy possessed by molecules of a
monatomic ideal gas is translational kinetic
energy. Using the results from the
discussion of kinetic theory in Section 10.5,
show that the internal energy of a
monatomic ideal gas at pressure P and
occupying volume V may be written as U =
3/2 PV.
2.
Sketch a PV diagram and find the
work done by the gas during the following
stages: (a) A gas is expanded from a volume
of 1.0 L to 3.0 L at a constant pressure of 3.0
atm. (b) The gas is then cooled at constant
volume until the pressure falls to 2.0 atm.
(c) The gas is then compressed at a constant
pressure of 2.0 atm from a volume of 3.0 L
to 1.0 L. (Note: Be careful of signs.) (d) The
gas is heated until its pressure increases
from 2.0 atm to 3.0 atm at a constant
volume. (e) Find the net work done during
the complete cycle.
3.
A container of volume 0.40 m3
contains 3.0 mol of argon gas at 30°C.
Assuming argon behaves as an ideal gas,
find the total internal energy of the gas.
(Hint: See Problem 1.)
4.
A 40.0-g projectile is launched by the
expansion of hot gas in an arrangement
shown in Figure P12.4a. The cross-sectional
area of the launch tube is 1.0 cm2, and the
length that the projectile travels down the
tube after starting from rest is 32 cm. As the
gas expands, the pressure varies as shown
in Figure P12.4b. The values for the initial
pressure and volume are Pi = 11 × 105 Pa
and Vi = 8.0 cm3 while the final values are Pf
= 1.0 × 105 Pa and Vf = 40.0 cm3. Friction
between the projectile and the launch tube
is negligible. (a) If the projectile is launched
into a vacuum, what is the speed of the
projectile as it leaves the launch tube? (b) If
instead the projectile is launched into air at
a pressure of 1.0 × 105 Pa, what fraction of
the work done by the expanding gas in the
tube is spent by the projectile pushing air
out of the way as it proceeds down the
tube?
expanded at constant pressure P2 to a final
volume V2. (c) In which of the processes is
more work done by the gas? Why?
7.
Gas in a container is at a pressure of
1.5 atm and a volume of 4.0 m3. What is the
work done on the gas (a) if it expands at
constant pressure to twice its initial
volume? (b) if it is compressed at constant
pressure to one-quarter its initial volume?
Figure P12.4
5.
A gas expands from I to F along the
three paths indicated in Figure P12.5.
Calculate the work done on the gas along
paths (a) IAF, (b) IF, and (c) IBF.
8.
A movable piston having a mass of
8.00 kg and a cross-sectional area of 5.00
cm2 traps 0.200 moles of an ideal gas in a
vertical cylinder. If the piston slides
without friction in the cylinder, how much
work is done on the gas when its
temperature is increased from 20°C to
300°C?
9.
One mole of an ideal gas initially at a
temperature of Ti = 0°C undergoes an
expansion at a constant pressure of 1.00 atm
to four times its original volume. (a)
Calculate the new temperature Tf of the gas.
(b) Calculate the work done on the gas
during the expansion.
Figure P12.5 (Problems 5 and 15)
6.
Sketch a PV diagram of the following
processes: (a) A gas expands at constant
pressure P1 from volume V1 to volume V2. It
is then kept at constant volume while the
pressure is reduced to P2. (b) A gas is
reduced in pressure from P1 to P2 while its
volume is held constant at V1. It is then
10.
(a) Determine the work done on a
fluid that expands from i to f as indicated in
Figure P12.10. (b) How much work is done
on the fluid if it is compressed from f to i
along the same path?
Figure P12.10
Section 12.2 The First Law of
Thermodynamics
11.
A container is placed in a water bath
and held at constant volume as a mixture of
fuel and oxygen is burned inside it. The
temperature of the water is observed to rise
during the burning. (The water is also held
at constant volume. (a) Consider the
burning mixture to be the system. What are
the signs of Q, ΔU, and W? (b) What are the
signs of these quantities if the water bath is
considered to be the system?
12.
A quantity of a monatomic ideal gas
undergoes a process in which both its
pressure and volume are doubled as shown
in Figure P12.12. What is the energy
absorbed by heat into the gas during this
process? (Hint: See Problem 1.)
Figure P12.12
13.
A gas is compressed at a constant
pressure of 0.800 atm from 9.00 L to 2.00 L.
In the process, 400 J of energy leaves the gas
by heat. (a) What is the work done on the
gas? (b) What is the change in its internal
energy?
14.
A monatomic ideal gas undergoes
the thermodynamic process shown in the
PV diagram of Figure P12.14. Determine
whether each of the values ΔU, Q, and W
for the gas is positive, negative, or zero.
(Hint: See Problem 1.)
Figure P12.14
15.
A gas expands from
I to F in Figure P12.5. The energy added to
the gas by heat is 418 J when the gas goes
from I to F along the diagonal path. (a)
What is the change in internal energy of the
gas? (b) How much energy must be added
to the gas by heat for the indirect path IAF
to give the same change in internal energy?
16.
A gas is taken through the cyclic
process described by Figure P12.16. (a) Find
the net energy transferred to the system by
heat during one complete cycle. (b) If the
cycle is reversed—that is, the process
follows the path ACBA—what is the net
energy transferred by heat per cycle?
decreases by 8.00 J, find the amount of heat
removed from the system by heat during
the compression.
18.
Consider the cyclic process described
by Figure P12.16. If Q is negative for the
process BC and ΔU is negative for the
process CA, determine the signs of Q, W,
and ΔU associated with each process.
19.
One gram of water changes to ice at
a constant pressure of 1.00 atm and a
constant temperature of 0°C. In the process,
the volume changes from 1.00 cm3 to 1.09
cm3. (a) Find the work done on the water
and (b) the change in the internal energy of
the water.
20.
A thermodynamic system undergoes
a process in which its internal energy
decreases by 500 J. If at the same time 220 J
of work is done on the system, find the
energy transferred to or from it by heat.
21.
A 5.0-kg block of aluminum is
heated from 20°C to 90°C at atmospheric
pressure. Find (a) the work done by the
aluminum, (b) the amount of energy
transferred to it by heat, and (c) the increase
in its internal energy.
Figure P12.16 (Problems 16 and 18)
17.
A gas is enclosed in a container fitted
with a piston of cross-sectional area 0.150
m2. The pressure of the gas is maintained at
6 000 Pa as the piston moves inward 20.0
cm. (a) Calculate the work done by the gas.
(b) If the internal energy of the gas
22.
One mole of gas initially at a
pressure of 2.00 atm and a volume of 0.300
L has an internal energy equal to 91.0 J. In
its final state, the gas is at a pressure of 1.50
atm and a volume of 0.800 L, and its
internal energy equals 180 J. For the paths
IAF, IBF, and IF in Figure P12.22, calculate
(a) the work done on the gas and (b) the net
energy transferred to the gas by heat in the
process.
in each cycle, find (a) the energy absorbed
in each cycle and (b) the time required to
complete each cycle.
27.
One of the most efficient engines
ever built is a coal-fired steam turbine
engine in the Ohio River valley, driving an
electric generator as it operates between 1
870°C and 430°C. (a) What is its maximum
theoretical efficiency? (b) Its actual
efficiency is 42.0%. How much mechanical
power does the engine deliver if it absorbs
1.40 × 105 J of energy each second from the
hot reservoir?
Figure P12.22
Section 12.3 Heat Engines and the Second
Law of Thermodynamics
23.
A heat engine operates between two
reservoirs at temperatures of 20°C and
300°C. What is the maximum efficiency
possible for this engine?
24.
A steam engine has a boiler that
operates at 300°F, and the temperature of
the exhaust is 150°F. Find the maximum
efficiency of this engine.
25.
The energy absorbed by an engine is
three times greater than the work it
performs. (a) What is its thermal efficiency?
(b) What fraction of the energy absorbed is
expelled to the cold reservoir?
26.
A particular engine has a power
output of 5.00 kW and an efficiency of
25.0%. If the engine expels 8 000 J of energy
28.
A gun is a heat engine. In particular,
it is an internal combustion piston engine
that does not operate in a cycle, but comes
apart during its adiabatic expansion
process. A certain gun consists of 1.80 kg of
iron. It fires one 2.40-g bullet at 320 m/s
with an energy efficiency of 1.10%. Assume
that the body of the gun absorbs all of the
energy exhaust and increases uniformly in
temperature for a short time before it loses
any energy by heat into the environment.
Find its temperature increase.
29.
An engine absorbs 1 700 J from a hot
reservoir and expels 1 200 J to a cold
reservoir in each cycle. (a) What is the
engine’s efficiency? (b) How much work is
done in each cycle? (c) What is the power
output of the engine if each cycle lasts for
0.300 s?
30.
A power plant has been proposed
that would make use of the temperature
gradient in the ocean. The system is to
operate between 20.0°C (surface water
temperature) and 5.00°C (water
temperature at a depth of about 1 km). (a)
What is the maximum efficiency of such a
system? (b) If the useful power output of
the plant is 75.0 MW, how much energy is
absorbed per hour? (c) In view of your
answer to (a), do you think such a system is
worthwhile (considering that there is no
charge for fuel)?
31.
In one cycle, a heat engine absorbs
500 J from a high-temperature reservoir and
expels 300 J to a low-temperature reservoir.
If the efficiency of this engine is 60% of the
efficiency of a Carnot engine, what is the
ratio of the low temperature to the high
temperature in the Carnot engine?
32.
A heat engine operates in a Carnot
cycle between 80.0°C and 350°C. It absorbs
21 000 J of energy per cycle from the hot
reservoir. The duration of each cycle is 1.00
s. (a) What is the mechanical power output
of this engine? (b) How much energy does
it expel in each cycle by heat?
33.
A nuclear power plant has an
electrical power output of 1 000 MW and
operates with an efficiency of 33%. If excess
energy is carried away from the plant by a
river with a flow rate of 1.0 × 106 kg/s, what
is the rise in temperature of the flowing
water?
34.
A 20.0%-efficient real engine is used
to speed up a train from rest to 5.00 m/s. It
is known that an ideal (Carnot) engine
using the same cold and hot reservoirs
would accelerate the same train from rest to
a speed of 6.50 m/s using the same amount
of fuel. If the engines use air at 300 K as a
cold reservoir, find the temperature of the
steam serving as the hot reservoir.
Section 12.4 Entropy
35.
A freezer is used to
freeze 1.0 L of water completely into ice.
The water and the freezer remain at a
constant temperature of T = 0°C. Determine
(a) the change in the entropy of the water
and (b) the change in the entropy of the
freezer.
36.
What is the change in entropy of 1.00
kg of liquid water at 100°C as it changes to
steam at 100°C?
37.
A 70-kg log falls from a height of 25
m into a lake. If the log, the lake, and the air
are all at 300 K, find the change in entropy
of the Universe during this process.
38.
Two 2 000-kg cars, both traveling at
20 m/s, undergo a head-on collision and
stick together. Find the change in entropy
of the Universe resulting from the collision
if the temperature is 23°C.
39.
The surface of the Sun is
approximately at 5 700 K, and the
temperature of the Earth’s surface is
approximately 290 K. What entropy change
occurs when 1 000 J of energy is transferred
by heat from the Sun to the Earth?
40.
Repeat the procedure used to
construct Table 12.3 (a) for the case in
which you draw three marbles rather than
four from your bag and (b) for the case in
which you draw five rather than four.
41.
Prepare a table like Table 12.3 for the
following occurrence: You toss four coins
into the air simultaneously and record all
the possible results of the toss in terms of
the numbers of heads and tails that can
result. (For example, HHTH and HTHH are
two possible ways in which three heads
and one tail can be achieved.) (a) On the
basis of your table, what is the most
probable result of a toss? In terms of
entropy, (b) what is the most ordered state,
and (c) what is the most disordered?
45.
A Carnot heat engine extracts energy
Qh from a hot reservoir at constant
temperature Th and rejects energy Qc to a
cold reservoir at constant temperature Tc.
Find the entropy changes of (a) the hot
reservoir, (b) the cold reservoir, (c) the
engine, and (d) the complete system.
46.
One end of a copper rod is in
thermal contact with a hot reservoir at T =
500 K, and the other end is in thermal
contact with a cooler reservoir at T = 300 K.
If 8 000 J of energy is transferred from one
end of the rod to the other, with no change
in the temperature distribution, find the
entropy change of each reservoir and the
total entropy change of the Universe.
42.
Consider a standard deck of 52
playing cards that has been thoroughly
shuffled. (a) What is the probability of
drawing the ace of spades in one draw? (b)
What is the probability of drawing any ace?
(c) What is the probability of drawing any
spade?
47.
Find the change in temperature of a
river due to the exhausted energy from a
nuclear power plant. Assume that the input
power to the boiler in the plant is 25 × 108
W, the efficiency of use of this power is
30%, and the river flow rate is 9.0 × 106
kg/min.
Additional Problems
48.
A Carnot engine operates between
100°C and 20°C. How much ice can the
engine melt from its exhaust after it has
done 5.0 × 104 J of work?
43.
A student claims that she has
constructed a heat engine that operates
between the temperatures of 200 K and 100
K with 60% efficiency. The professor does
not give her credit for the project. Why not?
44.
A Carnot engine operates between
the temperatures Th = 100°C and Tc = 20°C.
By what factor does the theoretical
efficiency increase if the temperature of the
hot reservoir is increased to 550°C?
49.
A 1500-kW heat engine operates at
25% efficiency. The heat energy expelled at
the low temperature is absorbed by a
stream of water that enters the cooling coils
at 20°C. If 60 L flows across the coils per
second, determine the increase in
temperature of the water.
50.
When a gas follows path 123 on the
PV diagram in Figure P12.50, 418 J of
energy flows into the system by heat and
–167 J of work is done on the gas. (a) What is
the change in the internal energy of the
system? (b) How much energy Q flows into
the system if the gas follows path 143? The
work done on the gas along this path is
–63.0 J. What net work would be done on or
by the system if the system followed (c)
path 12341? (d) path 14321? (e) What is the
change in internal energy of the system in
the processes described in parts (c) and (d)?
during process BC? (c) What is the net
energy input Q during this cycle?
Figure P12.51
52.
A power plant having a Carnot
efficiency produces 1 000 MW of electrical
power from turbines that take in steam at
500 K and eject water at 300 K into a
flowing river. The water downstream is
6.00 K warmer due to the output of the
plant. Determine the flow rate of the river.
Figure P12.50
51.
A substance undergoes the cyclic
process shown in Figure P12.51. Work
output occurs along path AB while work
input is required along path BC, and no
work is involved in the constant volume
process CA. Energy transfers by heat occur
during each process involved in the cycle.
(a) What is the work output during process
AB? (b) How much work input is required
53.
A 100-kg steel support rod in a
building has a length of 2.0 m at a
temperature of 20°C. The rod supports a
hanging load of 6 000 kg. Find (a) the work
done on the rod as the temperature
increases to 40°C, (b) the energy Q added to
the rod (assume the specific heat of steel is
the same as that for iron), and (c) the
change in internal energy of the rod.
54.
An ideal gas initially at pressure P0,
volume V0, and temperature T0 is taken
through the cycle described in Figure
P12.54. (a) Find the net work done by the
gas per cycle in terms of P0 and V0. (b) What
is the net energy Q added to the system per
cycle? (c) Obtain a numerical value for the
net work done per cycle for 1.00 mol of gas
initially at 0°C. (Hint: Recall that the work
done by the system equals the area under a
PV curve.)
Figure P12.54
55.
One mole of neon gas is heated from
300 K to 420 K at constant pressure.
Calculate (a) the energy Q transferred to the
gas, (b) the change in the internal energy of
the gas, and (c) the work done on the gas.
Note that neon has a molar specific heat of c
= 20.79 J/mol ∙ K for a constant-pressure
process.
56.
A 1.0-kg block of aluminum is
heated at atmospheric pressure so that its
temperature increases from 22°C to 40°C.
Find (a) the work done on the aluminum,
(b) the energy Q added to the aluminum,
and (c) the change in internal energy of the
aluminum.
57.
Suppose a heat
engine is connected to two energy
reservoirs, one a pool of molten aluminum
at 660°C and the other a block of solid
mercury at –38.9°C. The engine runs by
freezing 1.00 g of aluminum and melting
15.0 g of mercury during each cycle. The
latent heat of fusion of aluminum is 3.97 ×
105 J/kg, and that of mercury is 1.18 × 104
J/kg. (a) What is the efficiency of this
engine? (b) How does the efficiency
compare with that of a Carnot engine?
58.
One mole of an ideal gas is taken
through the cycle shown in Figure P12.58.
At point A, the pressure, volume, and
temperature are P0, V0, and T0. In terms of R
and T0, find (a) the total energy entering the
system by heat per cycle, (b) the total
energy leaving the system by heat per cycle,
(c) the efficiency of an engine operating in
this cycle, and (d) the efficiency of an
engine operating in a Carnot cycle between
the temperature extremes for this process.
(Hint: Recall that work done on the gas is
the negative of the area under a PV curve.)
the boiling point. (b)When the water is
boiled, it becomes 1 671 cm3 of steam.
Calculate the change in internal energy for
this process. Assume the steam vapor
doesn’t mix with the surrounding air and
that it expands at atmospheric pressure.
Figure P12.58
59.
An electrical power plant has an
overall efficiency of 15%. The plant is to
deliver 150 MW of electrical power to a city,
and its turbines use coal as fuel. The
burning coal produces steam at 190°C,
which drives the turbines. The steam is
condensed into water at 25°C by passing
through coils that are in contact with river
water. (a) How many metric tons of coal
does the plant consume each day (1 metric
ton = 1 × 103 kg)? (b) What is the total cost of
the fuel per year if the delivery price is $8
per metric ton? (c) If the river water is
delivered at 20°C, at what minimum rate
must it flow over the cooling coils in order
that its temperature not exceed 25°C? (Note:
The heat of combustion of coal is 7.8 × 103
cal/g.)
60.
At atmospheric pressure (1.013 × 105
Pa) and 20.0°C, 1.00 g of water occupies a
volume of 1.00 cm3. (a) Find the change in
internal energy when the water is heated to
61.
A gas is enclosed in a container fitted
with a piston of cross-sectional area 0.10 m2.
The pressure of the gas is maintained at 8
000 Pa while energy is slowly added by
heat; as a result, the piston is pushed up a
distance of 4.0 cm. (Recall that any process
in which the pressure remains constant is
an isobaric process.) (a) If 42 J of energy is
added to the system by heat during the
expansion, what is the change in internal
energy of the system? (b) If 42 J of energy is
added by heat to the system with the piston
clamped in a fixed position, what is the
work done by the gas? What is the change
in its internal energy?
62.
Hydrothermal vents deep on the
ocean floor spout water at temperatures as
high as 570°C. This temperature is below
the boiling point of water because of the
immense pressure at that depth. Since the
surrounding ocean temperature is at 4.0°C,
an organism could use the temperature
gradient as a source of energy. (a)
Assuming the specific heat of water under
these conditions is 1.0 cal/g ∙ °C, how much
energy is released when 1.0 liter of water is
cooled from 570°C to 4.0°C? (b) What is the
maximum useable energy an organism can
extract from this energy source? (Assume
that the organism has some internal type of
heat engine acting between the two
temperature extremes.) (c) Water from
these vents contains hydrogen sulfide (H2S)
at a concentration of 0.90 mmole/liter.
Oxidation of one mole of H2S produces 310
kJ of energy. How much energy is available
through H2S oxidation of 1.0 L of water?
63.
Suppose you spend 30.0 minutes on
a stair-climbing machine, climbing at a rate
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of 90.0 steps per minute, with each step 8.00
inches high. If you weigh 150 lb and the
machine reports that 600 kcal have been
burned at the end of the workout, what
efficiency is the machine using in obtaining
this result? If your actual efficiency is 0.18,
how many kcal did you really burn?