chapter 22 heat engines, entropy, and the Second Law of thermodynamics
has a thermodynamic efficiency of 0.110. Although
this efficiency is low compared with typical automo-
bile engines, she explains that her engine operates
between an energy reservoir at room temperature and
a water–ice mixture at atmospheric pressure and there-
fore requires no fuel other than that to make the ice.
The patent is approved, and working prototypes of the
engine prove the inventor’s efficiency claim.
17. A Carnot engine has a power output of 150 kW. The
engine operates between two reservoirs at 20.0°C and
500°C. (a)How much energy enters the engine by heat
per hour? (b) How much energy is exhausted by heat
18. A Carnot engine has a power output P. The engine
operates between two reservoirs at temperature T
. (a) How much energy enters the engine by heat in a
time interval Dt? (b) How much energy is exhausted by
heat in the time interval Dt?
19. What is the coefficient of performance of a refrigera-
tor that operates with Carnot efficiency between tem-
peratures 23.00°C and 127.0°C?
20. An ideal refrigerator or ideal heat pump is equivalent
to a Carnot engine running in reverse. That is, energy
| is taken in from a cold reservoir and energy |Q
is rejected to a hot reservoir. (a) Show that the work
that must be supplied to run the refrigerator or heat
(b) Show that the coefficient of performance (COP) of
the ideal refrigerator is
21. What is the maximum possible coefficient of perfor-
mance of a heat pump that brings energy from outdoors
at 23.00°C into a 22.0°C house? Note: The work done to
run the heat pump is also available to warm the house.
22. How much work does an ideal Carnot refrigerator
require to remove 1.00 J of energy from liquid helium
at 4.00 K and expel this energy to a room-temperature
23. If a 35.0%-efficient Carnot heat engine (Fig. 22.2) is run
in reverse so as to form a refrigerator (Fig. 22.4), what
would be this refrigerator’s coefficient of performance?
24. A power plant operates at a 32.0% efficiency during
the summer when the seawater used for cooling is at
20.0°C. The plant uses 350°C steam to drive turbines.
If the plant’s efficiency changes in the same propor-
tion as the ideal efficiency, what would be the plant’s
efficiency in the winter, when the seawater is at 10.0°C?
25. A heat engine is being designed to have a Carnot effi-
ciency of 65.0% when operating between two energy
reservoirs. (a) If the temperature of the cold reservoir
is 20.0°C, what must be the temperature of the hot res-
fusion of aluminum is 3.97 3 105 J/kg; the heat of
fusion of mercury is 1.18 3 104J/kg. What is the effi-
ciency of this engine?
Section 22.2 heat Pumps and Refrigerators
8. A refrigerator has a coefficient of performance equal
to 5.00. The refrigerator takes in 120 J of energy from a
cold reservoir in each cycle. Find (a) the work required
in each cycle and (b) the energy expelled to the hot
9. During each cycle, a refrigerator ejects 625 kJ of energy
to a high-temperature reservoir and takes in 550 kJ of
energy from a low-temperature reservoir. Determine
(a) the work done on the refrigerant in each cycle and
(b) the coefficient of performance of the refrigerator.
10. A heat pump has a coefficient of performance of 3.80
and operates with a power consumption of 7.03 3 103 W.
(a)How much energy does it deliver into a home dur-
ing 8.00 h of continuous operation? (b) How much
energy does it extract from the outside air?
11. A refrigerator has a coefficient of performance of 3.00.
The ice tray compartment is at 220.0°C, and the room
temperature is 22.0°C. The refrigerator can convert
30.0 g of water at 22.0°C to 30.0 g of ice at 220.0°C
each minute. What input power is required? Give your
answer in watts.
12. A heat pump has a coefficient of performance equal to
4.20 and requires a power of 1.75 kW to operate. (a) How
much energy does the heat pump add to a home in one
hour? (b) If the heat pump is reversed so that it acts
as an air conditioner in the summer, what would be its
coefficient of performance?
13. A freezer has a coefficient of performance of 6.30. It is
advertised as using electricity at a rate of 457 kWh/yr.
(a)On average, how much energy does it use in a single
day? (b) On average, how much energy does it remove
from the refrigerator in a single day? (c) What maximum
mass of water at 20.0°C could the freezer freeze in a sin-
gle day? Note: One kilowatt-hour (kWh) is an amount of
energy equal to running a 1-kW appliance for one hour.
Section 22.3 Reversible and irreversible Processes
Section 22.4 The Carnot Engine
14. A heat engine operates between a reservoir at 25.0°C
and one at 375°C. What is the maximum efficiency
possible for this engine?
15. One of the most efficient heat engines ever built is a
coal-fired steam turbine in the Ohio River valley, oper-
ating between 1 870°C and 430°C. (a) What is its maxi-
mum theoretical efficiency? (b) The actual efficiency
of the engine is 42.0%. How much mechanical power
does the engine deliver if it absorbs 1.40 3 105 J of
energy each second from its hot reservoir?
16. Why is the following situation impossible? An inventor
comes to a patent office with the claim that her heat
engine, which employs water as a working substance,
perature at exit. (b) Calculate the (maximum) power
output of the turning turbine. (c) The turbine is one
component of a model closed-cycle gas turbine engine.
Calculate the maximum efficiency of the engine.
32. At point A in a Carnot cycle, 2.34 mol of a monatomic
ideal gas has a pressure of 1 400 kPa, a volume of
10.0 L, and a temperature of 720 K. The gas expands
isothermally to point B and then expands adiabatically
to point C, where its volume is 24.0 L. An isothermal
compression brings it to point D, where its volume is
15.0 L. An adiabatic process returns the gas to point A.
(a) Determine all the unknown pressures, volumes,
and temperatures as you fill in the following table:
1 400 kPa
(b) Find the energy added by heat, the work done by the
engine, and the change in internal energy for each of
the steps A S B, B S C, C S D, and D S A. (c) Cal-
culate the efficiency W
|. (d) Show that the effi-
ciency is equal to 1 2 T
, the Carnot efficiency.
33. An electric generating station is designed to have
an electric output power of 1.40 MW using a turbine
with two-thirds the efficiency of a Carnot engine. The
exhaust energy is transferred by heat into a cooling
tower at 110°C. (a) Find the rate at which the station
exhausts energy by heat as a function of the fuel com-
bustion temperature T
. (b) If the firebox is modified to
run hotter by using more advanced combustion technol-
ogy, how does the amount of energy exhaust change?
(c) Find the exhaust power for T
5 800°C. (d) Find the
value of T
for which the exhaust power would be only
half as large as in part (c). (e) Find the value of T
which the exhaust power would be one-fourth as large
as in part (c).
34. An ideal (Carnot) freezer in a kitchen has a constant
temperature of 260 K, whereas the air in the kitchen
has a constant temperature of 300 K. Suppose the insu-
lation for the freezer is not perfect but rather conducts
energy into the freezer at a rate of 0.150 W. Determine
the average power required for the freezer’s motor to
maintain the constant temperature in the freezer.
35. A heat pump used for heating shown in Figure P22.35
is essentially an air conditioner installed backward. It
ervoir? (b)Can the actual efficiency of the engine be
equal to 65.0%? Explain.
26. A Carnot heat engine operates between temperatures
. (a) If T
5 500 K and T
5 350 K, what is
the efficiency of the engine? (b) What is the change
in its efficiency for each degree of increase in T
500 K? (c) What is the change in its efficiency for each
degree of change in T
? (d) Does the answer to part
(c) depend on T
27. An ideal gas is taken through a Carnot cycle. The iso-
thermal expansion occurs at 250°C, and the isother-
mal compression takes place at 50.0°C. The gas takes
in 1.20 3 103J of energy from the hot reservoir during
the isothermal expansion. Find (a) the energy expelled
to the cold reservoir in each cycle and (b) the net work
done by the gas in each cycle.
28. An electric power plant that would make use of the
temperature gradient in the ocean has been proposed.
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 electric power
output of the plant is 75.0MW, how much energy is
taken in from the warm reservoir per hour? (c) In view
of your answer to part (a), explain whether you think
such a system is worthwhile. Note that the “fuel” is free.
29. 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 out-
put of this engine? (b) How much energy does it expel
in each cycle by heat?
30. Suppose you build a two-engine device with the exhaust
energy output from one heat engine supplying the input
energy for a second heat engine. We say that the two
engines are running in series. Let e
efficiencies of the two engines. (a) The overall efficiency
of the two-engine device is defined as the total work out-
put divided by the energy put into the first engine by
heat. Show that the overall efficiency e is given by
e 5 e
What If? For parts (b) through (e) that follow, assume
the two engines are Carnot engines. Engine 1 operates
between temperatures T
. The gas in engine
2 varies in temperature between T
. In terms
of the temperatures, (b) what is the efficiency of the
combination engine? (c) Does an improvement in net
efficiency result from the use of two engines instead of
one? (d) What value of the intermediate temperature
results in equal work being done by each of the two
engines in series? (e) What value of T
results in each of
the two engines in series having the same efficiency?
31. Argon enters a turbine at a rate of 80.0 kg/min, a
temperature of 800°C, and a pressure of 1.50 MPa. It
expands adiabatically as it pushes on the turbine blades
and exits at pressure 300 kPa. (a) Calculate its tem-
chapter 22 heat engines, entropy, and the Second Law of thermodynamics
neously and then record the results of your tosses in
terms of the numbers of heads (H) and tails (T) that
result. For example, HHTH and HTHH are two pos-
sible ways in which three heads and one tail can be
achieved. (b) On the basis of your table, what is the
most probable result recorded for a toss?
41. If you roll two dice, what is the total number of ways in
which you can obtain (a) a 12 and (b) a 7?
Section 22.7 Changes in Entropy for Thermodynamic Systems
Section 22.8 Entropy and the Second Law
42. An ice tray contains 500 g of liquid water at 0°C. Cal-
culate the change in entropy of the water as it freezes
slowly and completely at 0°C.
43. A Styrofoam cup holding 125 g of hot water at 100°C
cools to room temperature, 20.0°C. What is the change
in entropy of the room? Neglect the specific heat of the
cup and any change in temperature of the room.
44. A 1.00-kg iron horseshoe is taken from a forge at 900°C
and dropped into 4.00 kg of water at 10.0°C. Assum-
ing that no energy is lost by heat to the surroundings,
determine the total entropy change of the horseshoe-
plus-water system. (Suggestion: Note that dQ 5 mc dT.)
45. A 1 500-kg car is moving at 20.0 m/s. The driver brakes
to a stop. The brakes cool off to the temperature of
the surrounding air, which is nearly constant at 20.0°C.
What is the total entropy change?
46. Two 2.00 3 103-kg cars both traveling at 20.0 m/s
undergo a head-on collision and stick together. Find
the change in entropy of the surrounding air result-
ing from the collision if the air temperature is 23.0°C.
Ignore the energy carried away from the collision by
47. A 70.0-kg log falls from a height of 25.0 m into a lake. If
the log, the lake, and the air are all at 300 K, find the
change in entropy of the air during this process.
48. A 1.00-mol sample of H
gas is contained in the left
side of the container shown in Figure P22.48, which
has equal volumes on the left and right. The right side
is evacuated. When the valve is opened, the gas streams
into the right side. (a) What is the entropy change of
the gas? (b) Does the temperature of the gas change?
Assume the container is so large that the hydrogen
behaves as an ideal gas.
49. A 2.00-L container has a center partition that divides it
into two equal parts as shown in Figure P22.49. The
left side contains 0.044 0 mol of H
gas, and the right
side contains 0.044 0 mol of O
gas. Both gases are at
extracts energy from colder air outside and deposits it in
a warmer room. Suppose the ratio of the actual energy
entering the room to the work done by the device’s
motor is 10.0% of the theoretical maximum ratio.
Determine the energy entering the room per joule of
work done by the motor given that the inside tempera-
ture is 20.0°C and the outside temperature is 25.00°C.
Section 22.5 Gasoline and Diesel Engines
Note: For problems in this section, assume the gas in
the engine is diatomic with g 5 1.40.
36. A gasoline engine has a compression ratio of 6.00.
(a) What is the efficiency of the engine if it operates
in an idealized Otto cycle? (b) What If? If the actual
efficiency is 15.0%, what fraction of the fuel is wasted
as a result of friction and energy transfers by heat that
could be avoided in a reversible engine? Assume com-
plete combustion of the air–fuel mixture.
37. In a cylinder of an automobile engine, immediately
after combustion the gas is confined to a volume of
50.0cm3 and has an initial pressure of 3.00 3 106 Pa.
The piston moves outward to a final volume of 300 cm3,
and the gas expands without energy transfer by heat.
(a) What is the final pressure of the gas? (b) How much
work is done by the gas in expanding?
38. An idealized diesel engine operates in a cycle known as
the air-standard diesel cycle shown in Figure P22.38. Fuel
is sprayed into the cylinder at the point of maximum
compression, B. Combustion occurs during the expan-
sion BSC, which is modeled as an isobaric process.
Show that the efficiency of an engine operating in this
idealized diesel cycle is
Section 22.6 Entropy
39. Prepare a table like Table 22.1 by using the same proce-
dure (a) for the case in which you draw three marbles
from your bag rather than four and (b) for the case in
which you draw five marbles rather than four.
40. (a) Prepare a table like Table 22.1 for the following
occurrence. You toss four coins into the air simulta-
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