Physics For Scientists And Engineers 6E - part 175

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 " 20.0°C each minute.
What input power is required? Give your answer in watts.

9. In 1993 the federal government instituted a requirement

that  all  room  air  conditioners  sold  in  the  United  States
must have an energy efficiency ratio (EER) of 10 or higher.
The EER is defined as the ratio of the cooling capacity of
the  air  conditioner,  measured  in  Btu/h,  to  its  electrical
power requirement in watts. (a) Convert the EER of 10.0 to
dimensionless  form,  using  the  conversion  1 Btu ! 1 055 J.
(b)  What  is  the  appropriate  name  for  this  dimensionless
quantity? (c) In the 1970s it was common to find room air
conditioners with EERs of 5 or lower. Compare the operat-
ing  costs  for  10 000-Btu/h  air  conditioners  with  EERs  of
5.00  and  10.0.  Assume  that  each  air  conditioner  operates
for  1 500 h  during  the  summer  in  a  city  where  electricity
costs 10.0¢ per kWh.

Section 22.3 Reversible and Irreversible Processes
Section 22.4 The Carnot Engine

10. A Carnot engine has a power output of 150 kW. The en-

gine operates between two reservoirs at 20.0°C and 500°C.
(a) How much energy does it take in per hour? (b) How
much energy is lost per hour in its exhaust?

One of the most efficient heat engines ever built is a steam
turbine in the Ohio valley, operating between 430°C and
1 870°C on energy from West Virginia coal to produce
electricity for the Midwest. (a) What is its maximum theo-
retical efficiency? (b) The actual efficiency of the engine is
42.0%. How much useful power does the engine deliver if
it takes in 1.40 ' 10

J of energy each second from its hot

reservoir?

12.

A heat engine operating between 200°C and 80.0°C achieves
20.0% of the maximum possible efficiency. What energy
input will enable the engine to perform 10.0 kJ of work?

An ideal gas is taken through a Carnot cycle. The

isothermal expansion occurs at 250°C, and the isothermal
compression takes place at 50.0°C. The gas takes in 1 200 J
of energy from the hot reservoir during the isothermal ex-
pansion. Find (a) the energy expelled to the cold reservoir
in each cycle and (b) the net work done by the gas in each
cycle.

14. The exhaust temperature of a Carnot heat engine is

300°C. What is the intake temperature if the efficiency of
the engine is 30.0%?

15.

A Carnot heat engine uses a steam boiler at 100°C as the
high-temperature reservoir. The low-temperature reservoir
is the outside environment at 20.0°C. Energy is exhausted
to the low-temperature reservoir at the rate of 15.4 W.
(a) Determine the useful power output of the heat engine.
(b) How much steam will it cause to condense in the high-
temperature reservoir in 1.00 h?

16. A power plant operates at a 32.0% efficiency during the

summer when the sea water 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 proportion as the ideal effi-

13.

11.

ciency, what would be the plant’s efficiency in the winter,
when the sea water is 10.0°C?

17.

Argon enters a turbine at a rate of 80.0 kg/min, a temper-
ature of 800°C and a pressure of 1.50 MPa. It expands adi-
abatically  as  it  pushes  on  the  turbine  blades  and  exits  at
pressure  300 kPa.  (a)  Calculate  its  temperature  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.

18.

An  electric  power  plant  that  would  make  use  of  the  tem-
perature  gradient  in  the  ocean  has  been  proposed.  The
system is to operate between 20.0°C (surface water temper-
ature) 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  taken  in  from  the  warm
reservoir  per  hour?  (c)  In  view  of  your  answer  to  part
(a), do  you  think  such  a  system  is  worthwhile?  Note  that
the “fuel” is free.

19.

Here is a clever idea. Suppose you build a two-engine
device such that the exhaust energy output from one heat
engine is the input energy for a second heat engine. We
say that the two engines are running in series. Let e

and e

represent  the  efficiencies  of  the  two  engines.  (a)  The
overall  efficiency  of  the  two-engine  device  is  defined  as
the  total  work  output  divided  by  the  energy  put  into  the
first  engine  by  heat.  Show  that  the  overall  efficiency  is
given by

e ! e

(b) What If? Assume the two engines are Carnot engines.
Engine 1 operates between temperatures T

and T

. The

gas in engine 2 varies in temperature between T

and T

In terms of the temperatures, what is the efficiency of the
combination engine? (c) What value of the intermediate
temperature T

will result in equal work being done by

each of the two engines in series? (d) What value of T

will

result in each of the two engines in series having the same
efficiency?

20.

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 accel-
erate the same train from rest to a speed of 6.50 m/s using
the same amount of fuel. The engines use air at 300 K as a
cold reservoir. Find the temperature of the steam serving
as the hot reservoir.

21.

A firebox is at 750 K, and the ambient temperature is 300 K.
The  efficiency  of  a  Carnot  engine  doing  150 J  of  work  as
it transports  energy  between  these  constant-temperature
baths  is  60.0%.  The  Carnot  engine  must  take  in  energy
150 J/0.600 ! 250 J  from  the  hot  reservoir  and  must  put
out 100 J of energy by heat into the environment. To follow
Carnot’s reasoning, suppose that some other heat engine S
could have efficiency 70.0%. (a) Find the energy input and
wasted energy output of engine S as it does 150 J of work.
(b) Let engine S operate as in part (a) and run the Carnot
engine in reverse. Find the total energy the firebox puts out
as both engines operate together, and the total energy trans-

Problems

697

ferred  to  the  environment.  Show  that  the  Clausius  state-
ment  of  the  second  law  of  thermodynamics  is  violated.
(c) Find the energy input and work output of engine S as it
puts out exhaust energy of 100 J. (d) Let engine S operate
as in (c) and contribute 150 J of its work output to running
the Carnot engine in reverse. Find the total energy the fire-
box  puts  out  as  both  engines  operate  together,  the  total
work  output,  and  the  total  energy  transferred  to  the  envi-
ronment. Show that the Kelvin–Planck statement of the sec-
ond  law  is  violated.  Thus  our  assumption  about  the  effi-
ciency  of  engine  S  must  be  false.  (e)  Let  the  engines
operate together through one cycle as in part (d). Find the
change  in  entropy  of  the  Universe.  Show  that  the  entropy
statement of the second law is violated.

22.

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.  It  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:

26.

A heat pump, shown in Figure P22.26, is essentially an air
conditioner installed backward. It extracts energy from
colder air outside and deposits it in a warmer room. Sup-
pose that the ratio of the actual energy entering the room
to the work done by the device’s motor is 10.0% of the the-
oretical maximum ratio. Determine the energy entering
the room per joule of work done by the motor, given that
the inside temperature is 20.0°C and the outside tempera-
ture is " 5.00°C.

How much work does an ideal Carnot refrigerator

require to remove 1.00 J of energy from helium at 4.00 K
and reject this energy to a room-temperature (293-K)
environment?

28. A refrigerator maintains a temperature of 0°C in the cold

compartment  with  a  room  temperature  of  25.0°C.  It
removes energy from the cold compartment at the rate of
8 000 kJ/h.  (a)  What  minimum  power  is  required  to
operate  the  refrigerator?  (b)  The  refrigerator  exhausts
energy into the room at what rate?

29. If a 35.0%-efficient Carnot heat engine (Fig. 22.2) is run in

reverse so as to form a refrigerator (Fig. 22.5), what would
be this refrigerator’s coefficient of performance?

30.

Two Carnot engines have the same efficiency. One engine
runs in reverse as a heat pump, and the other runs in reverse
as a refrigerator. The coefficient of performance of the heat
pump is 1.50 times the coefficient of performance of the
refrigerator. Find (a) the coefficient of performance of the
refrigerator, (b) the coefficient of performance of the heat
pump, and (c) the efficiency of each heat engine.

Section 22.5 Gasoline and Diesel Engines

In a cylinder of an automobile engine, just after combus-
tion, the gas is confined to a volume of 50.0 cm

and has

an initial pressure of 3.00 ' 10

Pa. The piston moves out-

ward to a final volume of 300 cm

, and the gas expands

without energy loss by heat. (a) If * ! 1.40 for the gas,
what is the final pressure? (b) How much work is done by
the gas in expanding?

32. A gasoline engine has a compression ratio of 6.00 and

uses a gas for which * ! 1.40. (a) What is the efficiency

31.

27.

698

C H A P T E R 2 2 • Heat Engines, Entropy, and the Second Law of Thermodynamics

Inside

Outside

Heat

pump

Figure P22.26

1 400 kPa

10.0 L

720 K

24.0 L

15.0 L

(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 : B, B : C, C : D, and D : A. (c) Calculate the
efficiency W

net

. Show that it is equal to 1 " T

the Carnot efficiency.

23. What is the coefficient of performance of a refrigerator

that operates with Carnot efficiency between temperatures
"

3.00°C and & 27.0°C?

24. What is the maximum possible coefficient of performance

of a heat pump that brings energy from outdoors at
"

3.00°C into a 22.0°C house? Note that the work done to

run the heat pump is also available to warm up the house.
An ideal refrigerator or ideal heat pump is equivalent to a
Carnot engine running in reverse. That is, energy Q

taken in from a cold reservoir and energy Q

is rejected to

a hot reservoir. (a) Show that the work that must be sup-
plied to run the refrigerator or heat pump is

(b) Show that the coefficient of performance of the ideal
refrigerator is

COP !

W !

25.

Q (input)

W (output)

int

ABCDA

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  losses  by  heat  that  could  by  avoided  in  a  re-
versible  engine?  (Assume  complete  combustion  of  the
air–fuel mixture.)

33. A 1.60-L gasoline engine with a compression ratio of 6.20

has  a  useful  power  output  of  102 hp.  Assuming  the  en-
gine  operates  in  an  idealized  Otto  cycle,  find  the  energy
taken  in  and  the  energy  exhausted  each  second.  Assume
the  fuel–air  mixture  behaves  like  an  ideal  gas  with
* !

1.40.

34.

The compression ratio of an Otto cycle, as shown in Figure
22.13, is V

8.00. At the beginning A of the compres-

sion process, 500 cm

of gas is at 100 kPa and 20.0°C. At

the beginning of the adiabatic expansion the temperature
is T

750°C. Model the working fluid as an ideal gas

with E

int

T ! 2.50nRT and * ! 1.40. (a) Fill in the

table below to follow the states of the gas:

38. In making raspberry jelly, 900 g of raspberry juice is com-

bined with 930 g of sugar. The mixture starts at room tem-
perature,  23.0°C,  and  is  slowly  heated  on  a  stove  until  it
reaches  220°F.  It  is  then  poured  into  heated  jars  and  al-
lowed to cool. Assume that the juice has the same specific
heat as water. The specific heat of sucrose is 0.299 cal/g ( °C.
Consider  the  heating  process.  (a)  Which  of  the  following
terms  describe(s)  this  process:  adiabatic,  isobaric,  isother-
mal,  isovolumetric,  cyclic,  reversible,  isentropic?  (b)  How
much energy does the mixture absorb? (c) What is the mini-
mum change in entropy of the jelly while it is heated?

39.

What change in entropy occurs when a 27.9-g ice cube at
"

12°C is transformed into steam at 115°C?

Section 22.7 Entropy Changes in Irreversible

Processes

40.

The temperature at the surface of the Sun is approxi-
mately 5 700 K , and the temperature at the surface of the
Earth is approximately 290 K. What entropy change occurs
when 1 000 J of energy is transferred by radiation from the
Sun to the Earth?

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?

42. 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. Assuming
that no energy is lost by heat to the surroundings, deter-
mine the total entropy change of the horseshoe-plus-water
system.

43.

How fast are you personally making the entropy of the
Universe increase right now? Compute an order-of-magni-
tude estimate, stating what quantities you take as data and
the values you measure or estimate for them.

44. A rigid tank of small mass contains 40.0 g of argon, initially

at 200°C and 100 kPa. The tank is placed into a reservoir
at 0°C and allowed to cool to thermal equilibrium. (a) Cal-
culate the volume of the tank. (b) Calculate the change in
internal energy of the argon. (c) Calculate the energy
transferred by heat. (d) Calculate the change in entropy of
the argon. (e) Calculate the change in entropy of the con-
stant-temperature bath.
A 1.00-mol sample of H

gas is contained in the left-hand

side  of  the  container  shown  in  Figure  P22.45,  which  has
equal volumes left and right. The right-hand side is evacu-
ated.  When  the  valve  is  opened,  the  gas  streams  into  the
right-hand  side.  What  is  the  final  entropy  change  of  the
gas? Does the temperature of the gas change?

45.

41.

Problems

699

Valve

Vacuum

Figure P22.45

T (K)

P (kPa)

V (cm

)

int

293

100

500

B
C

1 023

D
A

, the energy exhaust Q

and the net output work W

eng

. (d) Calculate the thermal

efficiency.  (e)  Find  the  number  of  crankshaft  revolutions
per  minute  required  for  a  one-cylinder  engine  to  have
an output  power  of  1.00 kW ! 1.34 hp.  Note  that  the
thermodynamic cycle involves four piston strokes.

Section 22.6 Entropy

35. An ice tray contains 500 g of liquid water at 0°C. Calculate

the change in entropy of the water as it freezes slowly and
completely at 0°C.

36. At a pressure of 1 atm, liquid helium boils at 4.20 K . The

latent  heat  of  vaporization  is  20.5 kJ/kg.  Determine  the
entropy  change  (per  kilogram)  of  the  helium  resulting
from vaporization.
Calculate the change in entropy of 250 g of water heated
slowly  from  20.0°C  to  80.0°C.  (Suggestion: Note  that
dQ ! mc dT.)

37.

(b) Fill in the table below to follow the processes:

700

C H A P T E R 2 2 • Heat Engines, Entropy, and the Second Law of Thermodynamics

46. A 2.00-L container has a center partition that divides

it into two equal parts, as shown in Figure P22.46. The
left side contains H

gas, and the right side contains

gas. Both gases are at room temperature and at at-

mospheric pressure. The partition is removed, and the
gases are allowed to mix. What is the entropy increase of
the system?

47. A 1.00-mol sample of an ideal monatomic gas, initially at a

pressure of 1.00 atm and a volume of 0.025 0 m

, is heated

to a final state with a pressure of 2.00 atm and a volume of
0.040 0 m

. Determine the change in entropy of the gas in

this process.

48.

A 1.00-mol sample of a diatomic ideal gas, initially having
pressure P and volume V, expands so as to have pressure
2P and volume 2V. Determine the entropy change of the
gas in the process.

Section 22.8 Entropy on a Microscopic Scale

49. If you toss two dice, what is the total number of ways in

which you can obtain (a) a 12 and (b) a 7?

50. Prepare a table like Table 22.1 for the following occur-

rence. You toss four coins into the air simultaneously and
then record the results of your tosses in terms of the num-
bers  of  heads  and  tails  that  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 en-
tropy,  (b)  what  is  the  most  ordered  state  and  (c)  what  is
the most disordered state?
Repeat the procedure used to construct Table 22.1 (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 rather than four.

Additional Problems

52. Every second at Niagara Falls (Fig. P22.52), some

5 000 m

of water falls a distance of 50.0 m. What is the

increase in entropy per second due to the falling water?
Assume that the mass of the surroundings is so great that
its temperature and that of the water stay nearly constant
at 20.0°C. Suppose that a negligible amount of water
evaporates.

51.

A house loses energy through the exterior walls and

roof at a rate of 5 000 J/s ! 5.00 kW when the interior
temperature is 22.0°C and the outside temperature is
"

5.00°C. Calculate the electric power required to main-

tain  the  interior  temperature  at  22.0°C  for  the  following
two  cases.  (a)  The  electric  power  is  used  in  electric  resis-
tance heaters (which convert all of the energy transferred
in  by  electrical  transmission  into  internal  energy).
(b) What If ? The electric power is used to drive an electric
motor that operates the compressor of a heat pump, which
has  a  coefficient  of  performance  equal  to  60.0%  of  the
Carnot-cycle value.

54. How much work is required, using an ideal Carnot refrig-

erator, to change 0.500 kg of tap water at 10.0°C into ice at
"

20.0°C? Assume the temperature of the freezer compart-

ment is held at " 20.0°C and the refrigerator exhausts en-
ergy into a room at 20.0°C.

55.

A heat engine operates between two reservoirs at
T

600 K and T

350 K . It takes in 1 000 J of energy

from the higher-temperature reservoir and performs
250 J of work. Find (a) the entropy change of the Uni-
verse %S

for this process and (b) the work W that could

have been done by an ideal Carnot engine operating be-
tween these two reservoirs. (c) Show that the difference
between the amounts of work done in parts (a) and (b) is
T

56.

Two  identically  constructed  objects,  surrounded  by  ther-
mal insulation, are used as energy reservoirs for a Carnot
engine.  The  finite  reservoirs  both  have  mass  m and  spe-
cific  heat  c.  They  start  out  at  temperatures  T

and T

where T

. (a) Show that the engine will stop work-

ing when the final temperature of each object is
(T

)

1/2

. (b) Show that the total work done by the

53.

Figure P22.52

Niagara Falls, a popular tourist attraction.

CORBIS/Stock Market

0.044 mol

Figure P22.46

Content .. 173 174 175 176 ..