Physics For Scientists And Engineers 6E - part 159

Problems

633

tank is filled with 200 g of water. What is the increase in
the temperature of the water after the blocks fall through
a distance of 3.00 m?

Section 20.2 Specific Heat and Calorimetry

The temperature of a silver bar rises by 10.0°C when it ab-
sorbs 1.23 kJ of energy by heat. The mass of the bar is
525 g. Determine the specific heat of silver.

4. A 50.0-g sample of copper is at 25.0°C. If 1 200 J of energy

is added to it by heat, what is the final temperature of the
copper?

5. Systematic use of solar energy can yield a large saving in

the cost of winter space heating for a typical house in the
north central United States. If the house has good insula-
tion, you may model it as losing energy by heat steadily at
the rate 6 000 W on a day in April when the average exte-
rior temperature is 4°C, and when the conventional heat-
ing system is not used at all. The passive solar energy col-
lector can consist simply of very large windows in a room
facing south. Sunlight shining in during the daytime is ab-
sorbed by the floor, interior walls, and objects in the room,
raising their temperature to 38°C. As the sun goes down,
insulating draperies or shutters are closed over the win-
dows. During the period between 5:00

. and 7:00

the temperature of the house will drop, and a sufficiently
large “thermal mass” is required to keep it from dropping
too far. The thermal mass can be a large quantity of stone
(with specific heat 850 J/kg $ °C) in the floor and the inte-
rior walls exposed to sunlight. What mass of stone is re-
quired if the temperature is not to drop below 18°C
overnight?

6. The Nova laser at Lawrence Livermore National Labora-

tory in California is used in studies of initiating controlled
nuclear fusion (Section 45.4). It can deliver a power of
1.60 % 10

W over a time interval of 2.50 ns. Compare its

energy output in one such time interval to the energy re-
quired to make a pot of tea by warming 0.800 kg of water
from 20.0°C to 100°C.

A 1.50-kg iron horseshoe initially at 600°C is dropped

into a bucket containing 20.0 kg of water at 25.0°C. What
is the final temperature? (Ignore the heat capacity of the
container, and assume that a negligible amount of water
boils away.)

An aluminum cup of mass 200 g contains 800 g of water
in  thermal  equilibrium  at  80.0°C.  The  combination  of
cup  and  water  is  cooled  uniformly  so  that  the  tempera-
ture decreases by 1.50°C per minute. At what rate is en-
ergy  being  removed  by  heat?  Express  your  answer  in
watts.

An  aluminum  calorimeter  with  a  mass  of  100 g  contains
250 g  of  water.  The  calorimeter  and  water  are  in  thermal
equilibrium at 10.0°C. Two metallic blocks are placed into
the water. One is a 50.0-g piece of copper at 80.0°C. The
other block has a mass of 70.0 g and is originally at a tem-
perature  of  100°C.  The  entire  system  stabilizes  at  a  final
temperature of 20.0°C. (a) Determine the specific heat of
the  unknown  sample.  (b)  Guess  the  material  of  the  un-
known, using the data in Table 20.1.

10.

A 3.00-g copper penny at 25.0°C drops 50.0 m to the
ground. (a) Assuming that 60.0% of the change in poten-
tial energy of the penny–Earth system goes into increasing
the internal energy of the penny, determine its final tem-
perature. (b) What If ? Does the result depend on the mass
of the penny? Explain.

11.

A combination of 0.250 kg of water at 20.0°C, 0.400 kg of
aluminum  at  26.0°C,  and  0.100 kg  of  copper  at  100°C  is
mixed  in  an  insulated  container  and  allowed  to  come  to
thermal  equilibrium.  Ignore  any  energy  transfer  to  or
from  the  container  and  determine  the  final  temperature
of the mixture.

12.

If water with a mass m

at temperature T

is poured into an

aluminum cup of mass m

containing mass m

of water at

, where T

(

, what is the equilibrium temperature of

the system?

13.

A water heater is operated by solar power. If the solar col-
lector has an area of 6.00 m

and the intensity delivered by

sunlight is 550 W/m

, how long does it take to increase

the temperature of 1.00 m

of water from 20.0°C to

60.0°C?

14.

Two thermally insulated vessels are connected by a narrow
tube fitted with a valve that is initially closed. One vessel, of
volume 16.8 L, contains oxygen at a temperature of 300 K
and a pressure of 1.75 atm. The other vessel, of volume
22.4 L, contains oxygen at a temperature of 450 K and a
pressure of 2.25 atm. When the valve is opened, the gases
in the two vessels mix, and the temperature and pressure
become uniform throughout. (a) What is the final temper-
ature? (b) What is the final pressure?

Section 20.3 Latent Heat

15. How much energy is required to change a 40.0-g ice cube

from ice at &10.0°C to steam at 110°C?

16. A 50.0-g copper calorimeter contains 250 g of water at

20.0°C. How much steam must be condensed into the wa-
ter if the final temperature of the system is to reach
50.0°C?
A 3.00-g lead bullet at 30.0°C is fired at a speed of 240 m/s
into a large block of ice at 0°C, in which it becomes em-
bedded. What quantity of ice melts?

18.

Steam at 100°C is added to ice at 0°C. (a) Find the amount
of ice melted and the final temperature when the mass of
steam is 10.0 g and the mass of ice is 50.0 g. (b) What If ?
Repeat when the mass of steam is 1.00 g and the mass of
ice is 50.0 g.

19.

A 1.00-kg block of copper at 20.0°C is dropped into a large
vessel of liquid nitrogen at 77.3 K. How many kilograms of
nitrogen boil away by the time the copper reaches 77.3 K?
(The specific heat of copper is 0.092 0 cal/g $ °C. The la-
tent heat of vaporization of nitrogen is 48.0 cal/g.)

20. Assume that a hailstone at 0°C falls through air at a uni-

form temperature of 0°C and lands on a sidewalk also at
this temperature. From what initial height must the hail-
stone fall in order to entirely melt on impact?

In an insulated vessel, 250 g of ice at 0°C is added to

600 g of water at 18.0°C. (a) What is the final temperature

21.

17.

of the system? (b) How much ice remains when the system
reaches equilibrium?

22.

Review problem. Two speeding lead bullets, each of mass
5.00 g, and at temperature 20.0°C, collide head-on at
speeds of 500 m/s each. Assuming a perfectly inelastic col-
lision and no loss of energy by heat to the atmosphere, de-
scribe the final state of the two-bullet system.

Section 20.4 Work and Heat in Thermodynamic

Processes

A sample of ideal gas is expanded to twice its original

volume of 1.00 m

in a quasi-static process for which

P ! ,V

, with , ! 5.00 atm/m

, as shown in Figure

P20.23. How much work is done on the expanding gas?

23.

24.

(a) Determine the work done on a fluid that expands from
i to f as indicated in Figure P20.24. (b) What If? How
much work is performed on the fluid if it is compressed
from f to i along the same path?

An ideal gas is enclosed in a cylinder with a movable

piston on top of it. The piston has a mass of 8 000 g and
an area of 5.00 cm

and is free to slide up and down, keep-

ing the pressure of the gas constant. How much work is
done on the gas as the temperature of 0.200 mol of the gas
is raised from 20.0°C to 300°C?

26. An ideal gas is enclosed in a cylinder that has a movable

piston on top. The piston has a mass m and an area A and
is free to slide up and down, keeping the pressure of the
gas constant. How much work is done on the gas as the
temperature of n mol of the gas is raised from T

to T

25.

27.

One mole of an ideal gas is heated slowly so that it goes
from the PV state (P

, V

) to (3P

, 3V

) in such a way that the

pressure is directly proportional to the volume. (a) How
much work is done on the gas in the process? (b) How is
the temperature of the gas related to its volume during this
process?

Section 20.5 The First Law of Thermodynamics

28. 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?
A thermodynamic system undergoes a process in which its
internal energy decreases by 500 J. At the same time, 220 J
of work is done on the system. Find the energy transferred
to or from it by heat.

30. A gas is taken through the cyclic process described in Fig-

ure P20.30. (a) Find the net energy transferred to the sys-
tem by heat during one complete cycle. (b) What If? If the
cycle is reversed—that is, the process follows the path
ACBA—what is the net energy input per cycle by heat?

29.

31. Consider the cyclic process depicted in Figure P20.30. If Q

is negative for the process BC and #E

int

is negative for the

process CA, what are the signs of Q , W, and #E

int

that are

associated with each process?

32.

A sample of an ideal gas goes through the process shown
in  Figure  P20.32.  From  A to  B,  the  process  is  adiabatic;
from  B to  C,  it  is  isobaric  with  100 kJ  of  energy  entering
the system by heat.  From C to D, the process is isothermal;
from D to A, it is isobaric with 150 kJ of energy leaving the
system  by  heat.  Determine  the  difference  in  internal  en-
ergy E

int, B

int, A

634

C H A P T E R 2 0 • Heat and the First Law of Thermodynamics

P =

2.00 m

1.00 m

Figure P20.23

× 10

P(Pa)

× 10

V(m

)

Figure P20.24

P(kPa)

V(m

)

Figure P20.30 Problems 30 and 31.

1.0

3.0

P(atm)

0.090 0.20

0.40

1.2

V(m

)

Figure P20.32

Problems

635

33.

A sample of an ideal gas is in a vertical cylinder fitted with
a  piston.  As  5.79 kJ  of  energy  is  transferred  to  the  gas  by
heat  to  raise  its  temperature,  the  weight  on  the  piston  is
adjusted so that the state of the gas changes from point A
to  point  B along  the  semicircle  shown  in  Figure  P20.33.
Find the change in internal energy of the gas.

Section 20.6 Some Applications of the First Law

of Thermodynamics

34.

One mole of an ideal gas does 3 000 J of work on its sur-
roundings as it expands isothermally to a final pressure of
1.00 atm and volume of 25.0 L. Determine (a) the initial
volume and (b) the temperature of the gas.
An ideal gas initially at 300 K undergoes an isobaric expan-
sion at 2.50 kPa. If the volume increases from 1.00 m

3.00 m

and 12.5 kJ is transferred to the gas by heat, what

are (a) the change in its internal energy and (b) its final
temperature?

36.

A 1.00-kg block of aluminum is heated at atmospheric pres-
sure so that its temperature increases from 22.0°C to 40.0°C.
Find  (a)  the  work  done  on  the  aluminum,  (b)  the  energy
added to it by heat, and (c) the change in its internal energy.
How  much  work  is  done  on  the  steam  when  1.00 mol  of
water  at  100°C  boils  and  becomes  1.00 mol  of  steam  at
100°C at 1.00 atm pressure? Assuming the steam to behave
as an ideal gas, determine the change in internal energy of
the material as it vaporizes.

38.

An ideal gas initially at P

, V

, and T

is taken through a cy-

cle as in Figure P20.38. (a) Find the net work done on the
gas per cycle. (b) What is the net energy added by heat 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.

37.

35.

A  2.00-mol  sample  of  helium  gas  initially  at  300 K  and
0.400 atm  is  compressed  isothermally  to  1.20 atm.  Noting
that the helium behaves as an ideal gas, find (a) the final
volume  of  the  gas,  (b)  the  work  done  on  the  gas,  and
(c) the energy transferred by heat.

40.

In Figure P20.40, the change in internal energy of a gas

that is taken from A to C is " 800 J. The work done on the
gas along path ABC is &500 J. (a) How much energy must
be added to the system by heat as it goes from A through B
to  C ?  (b)  If  the  pressure  at  point  A is  five  times  that  of
point  C, what  is  the  work  done  on  the  system  in  going
from  C to  D?  (c)  What  is  the  energy  exchanged  with  the
surroundings by heat as the cycle goes from C to A along
the green path? (d) If the change in internal energy in go-
ing from point D to point A is " 500 J, how much energy
must be added to the system by heat as it goes from point
C to point D ?

39.

Section 20.7 Energy-Transfer Mechanisms

41.

A box with a total surface area of 1.20 m

and a wall thick-

ness of 4.00 cm is made of an insulating material. A 10.0-W
electric heater inside the box maintains the inside temper-
ature at 15.0°C above the outside temperature. Find the
thermal conductivity k of the insulating material.

42. A glass window pane has an area of 3.00 m

and a thick-

ness of 0.600 cm. If the temperature difference between its
faces is 25.0°C, what is the rate of energy transfer by con-
duction through the window?
A bar of gold is in thermal contact with a bar of silver of
the same length and area (Fig. P20.43). One end of the
compound bar is maintained at 80.0°C while the opposite
end is at 30.0°C. When the energy transfer reaches steady
state, what is the temperature at the junction?

43.

44. A thermal window with an area of 6.00 m

is constructed

of two layers of glass, each 4.00 mm thick, and separated
from each other by an air space of 5.00 mm. If the inside

P(kPa)

500

300

1.2

3.6

6.0 V(L)

400

200

100

Figure P20.33

Figure P20.38

Figure P20.40

Insulation

30.0

°C

80.0

°C

Figure P20.43

636

C H A P T E R 2 0 • Heat and the First Law of Thermodynamics

surface is at 20.0°C and the outside is at &30.0°C, what
is the rate of energy transfer by conduction through the
window?

45. A power transistor is a solid-state electronic device. Assume

that  energy  entering  the  device  at  the  rate  of  1.50 W  by
electrical  transmission  causes  the  internal  energy  of  the
device to increase. The surface area of the transistor is so
small that it tends to overheat. To prevent overheating, the
transistor is attached to a larger metal heat sink with fins.
The  temperature  of  the  heat  sink  remains  constant  at
35.0°C under steady-state conditions. The transistor is elec-
trically insulated from the heat sink by a rectangular sheet
of mica measuring 8.25 mm by 6.25 mm, and 0.085 2 mm
thick.  The  thermal  conductivity  of  mica  is  equal  to
0.075 3 W/m $ °C.  What  is  the  operating  temperature  of
the transistor?

46.

Calculate the R value of (a) a window made of a single pane
of flat glass  in. thick, and (b) a thermal window made of
two single panes each  in. thick and separated by a  -in. air
space.  (c)  By  what  factor  is  the  transfer  of  energy  by  heat
through the window reduced by using the thermal window
instead of the single pane window?

47. The surface of the Sun has a temperature of about

5 800 K. The radius of the Sun is 6.96 % 10

m. Calculate

the total energy radiated by the Sun each second. Assume
that the emissivity of the Sun is 0.965.

48.

A large hot pizza floats in outer space. What is the order of
magnitude of (a) its rate of energy loss? (b) its rate of tem-
perature change? List the quantities you estimate and the
value you estimate for each.

49. The tungsten filament of a certain 100-W light bulb radi-

ates 2.00 W of light. (The other 98 W is carried away by
convection and conduction.) The filament has a surface
area of 0.250 mm

and an emissivity of 0.950. Find the fila-

ment’s temperature. (The melting point of tungsten is
3 683 K.)

50.

At high noon, the Sun delivers 1 000 W to each square me-
ter of a blacktop road. If the hot asphalt loses energy only
by radiation, what is its equilibrium temperature?

51.

The intensity of solar radiation reaching the top of the
Earth’s atmosphere is 1 340 W/m

. The temperature of

the Earth is affected by the so-called greenhouse effect of
the atmosphere. That effect makes our planet’s emissivity
for visible light higher than its emissivity for infrared light.
For comparison, consider a spherical object with no atmos-
phere,  at  the  same  distance  from  the  Sun  as  the  Earth.
Assume that its emissivity is the same for all kinds of elec-
tromagnetic  waves  and  that  its  temperature  is  uniform
over  its  surface.  Identify  the  projected  area  over  which  it
absorbs  sunlight  and  the  surface  area  over  which  it  radi-
ates. Compute its equilibrium temperature.  Chilly, isn’t it?
Your calculation applies to (a) the average temperature of
the  Moon,  (b)  astronauts  in  mortal  danger  aboard  the
crippled  Apollo  13 spacecraft,  and  (c)  global  catastrophe
on  the  Earth  if  widespread  fires  should  cause  a  layer  of
soot to accumulate throughout the upper atmosphere, so
that  most  of  the  radiation  from  the  Sun  were  absorbed
there rather than at the surface below the atmosphere.

Additional Problems

52. Liquid nitrogen with a mass of 100 g at 77.3 K is stirred

into a beaker containing 200 g of 5.00°C water. If the nitro-
gen leaves the solution as soon as it turns to gas, how much
water freezes? (The latent heat of vaporization of nitrogen
is 48.0 cal/g, and the latent heat of fusion of water is
79.6 cal/g.)

53.

A  75.0-kg  cross-country  skier  moves  across  the  snow  (Fig.
P20.53).  The  coefficient  of  friction  between  the  skis  and
the snow is 0.200. Assume that all the snow beneath his skis
is at 0°C and that all the internal energy generated by fric-
tion  is  added  to  the  snow,  which  sticks  to  his  skis  until  it
melts.  How  far  would  he  have  to  ski  to  melt  1.00 kg  of
snow?

54.

On  a  cold  winter  day  you  buy  roasted  chestnuts  from  a
street vendor. Into the pocket of your down parka you put
the change he gives you—coins constituting 9.00 g of cop-
per at &12.0°C. Your pocket already contains 14.0 g of sil-
ver coins at 30.0°C. A short time later the temperature of
the  copper  coins  is  4.00°C  and  is  increasing  at  a  rate  of
0.500°C/s. At this time, (a) what is the temperature of the
silver coins, and (b) at what rate is it changing?
An  aluminum  rod  0.500 m  in  length  and  with  a  cross-
sectional area of 2.50 cm

is inserted into a thermally insu-

lated vessel containing liquid helium at 4.20 K. The rod is
initially at 300 K. (a) If half of the rod is inserted into the
helium, how many liters of helium boil off by the time the
inserted half cools to 4.20 K ? (Assume the upper half does
not yet cool.) (b) If the upper end of the rod is maintained
at 300 K, what is the approximate boil-off rate of liquid he-
lium after the lower half has reached 4.20 K? (Aluminum
has thermal conductivity of 31.0 J/s $ cm $ K at 4.2 K; ignore
its temperature variation. Aluminum has a specific heat of
0.210 cal/g $ °C and density of 2.70 g/cm

. The density of

liquid helium is 0.125 g/cm

56.

A copper ring (with mass of 25.0 g, coefficient of linear
expansion of 1.70 % 10

(°C)

, and specific heat of

9.24 % 10

cal/g $ °C) has a diameter of 5.00 cm at its

temperature of 15.0°C. A spherical aluminum shell (with
mass 10.9 g, coefficient of linear expansion 2.40 % 10

(°C)

, and specific heat 0.215 cal/g$°C) has a diameter of

5.01 cm at a temperature higher than 15.0°C. The sphere
is placed on top of the horizontal ring, and the two are
allowed to come to thermal equilibrium without any

55.

Figure P20.53

Nathan Bilow/Leo de Wys, Inc.

Content .. 157 158 159 160 ..