by using the thermal window instead of the single-pane
window? Include the contributions of inside and out-
side stagnant air layers.
54. At our distance from the Sun, the intensity of solar
radiation is 1 370 W/m2. The temperature of the
Earth is affected by the greenhouse effect of the atmo-
sphere. This phenomenon describes the effect of
absorption of infrared light emitted by the surface so
as to make the surface temperature of the Earth
higher than if it were airless. For comparison, consider
a spherical object of radius r with no atmosphere at
the same distance from the Sun as the Earth. Assume
its emissivity is the same for all kinds of electromag-
netic waves and its temperature is uniform over its sur-
face. (a) Explain why the projected area over which it
absorbs sunlight is pr2 and the surface area over
which it radiates is 4pr2. (b) Compute its steady-state
temperature. Is it chilly?
55. A bar of gold (Au) is in ther-
mal contact with a bar of silver
(Ag) of the same length and
area (Fig. P20.55). One end
of the compound bar is main-
tained at 80.0°C, and the oppo-
site end is at 30.0°C. When the
energy transfer reaches steady
state, what is the temperature
at the junction?
56. For bacteriological testing of
water supplies and in medical
clinics, samples must routinely be incubated for 24h at
37°C. Peace Corps volunteer and MIT engineer Amy
Smith invented a low-cost, low-maintenance incuba-
tor. The incubator consists of a foam-insulated box
containing a waxy material that melts at 37.0°C inter-
spersed among tubes, dishes, or bottles containing
the test samples and growth medium (bacteria food).
Outside the box, the waxy material is first melted by a
stove or solar energy collector. Then the waxy material
is put into the box to keep the test samples warm as
the material solidifies. The heat of fusion of the phase-
change material is 205 kJ/kg. Model the insulation as
a panel with surface area 0.490 m2, thickness 4.50 cm,
and conductivity 0.012 0 W/m ? °C. Assume the exte-
rior temperature is 23.0°C for 12.0 h and 16.0°C for
12.0 h. (a) What mass of the waxy material is required
to conduct the bacteriological test? (b) Explain why
your calculation can be done without knowing the
mass of the test samples or of the insulation.
57. A large, hot pizza floats in outer space after being jet-
tisoned as refuse from a spacecraft. What is the order
of magnitude (a) of its rate of energy loss and (b)of
its rate of temperature change? List the quantities you
estimate and the value you estimate for each.
58. A gas expands from I to F in Figure P20.58 (page 622).
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
47. The tungsten filament of a certain 100-W lightbulb
radiates 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 mm2 and an emissivity of 0.950.
Find the filament’s temperature. (The melting point of
tungsten is 3 683 K.)
48. At high noon, the Sun delivers 1 000 W to each square
meter of a blacktop road. If the hot asphalt trans-
fers energy only by radiation, what is its steady-state
49. Two lightbulbs have cylindrical filaments much greater
in length than in diameter. The evacuated bulbs are
identical except that one operates at a filament tem-
perature of 2 100°C and the other operates at 2 000°C.
(a) Find the ratio of the power emitted by the hot-
ter lightbulb to that emitted by the cooler lightbulb.
(b) With the bulbs operating at the same respective
temperatures, the cooler lightbulb is to be altered by
making its filament thicker so that it emits the same
power as the hotter one. By what factor should the
radius of this filament be increased?
50. The human body must maintain its core temperature
inside a rather narrow range around 37°C. Metabolic
processes, notably muscular exertion, convert chemical
energy into internal energy deep in the interior. From
the interior, energy must flow out to the skin or lungs
to be expelled to the environment. During moderate
exercise, an 80-kg man can metabolize food energy at
the rate 300 kcal/h, do 60 kcal/h of mechanical work,
and put out the remaining 240 kcal/h of energy by
heat. Most of the energy is carried from the body inte-
rior out to the skin by forced convection (as a plumber
would say), whereby blood is warmed in the interior
and then cooled at the skin, which is a few degrees
cooler than the body core. Without blood flow, living
tissue is a good thermal insulator, with thermal con-
ductivity about 0.210 W/m · °C. Show that blood flow
is essential to cool the man’s body by calculating the
rate of energy conduction in kcal/h through the tissue
layer under his skin. Assume that its area is 1.40 m2, its
thickness is 2.50 cm, and it is maintained at 37.0°C on
one side and at 34.0°C on the other side.
51. A copper rod and an aluminum rod of equal diameter
are joined end to end in good thermal contact. The
temperature of the free end of the copper rod is held
constant at 100°C and that of the far end of the alumi-
num rod is held at 0°C. If the copper rod is 0.150 m
long, what must be the length of the aluminum rod so
that the temperature at the junction is 50.0°C?
52. A box with a total surface area of 1.20 m2 and a wall
thickness of 4.00 cm is made of an insulating material.
A 10.0-W electric heater inside the box maintains the
inside temperature at 15.0°C above the outside temper-
ature. Find the thermal conductivity k of the insulating
53. (a) Calculate the R-value of a thermal window made of
two single panes of glass each 0.125 in. thick and sepa-
rated by a 0.250-in. air space. (b) By what factor is the
transfer of energy by heat through the window reduced
chapter 20 the First Law of thermodynamics
stream of the liquid while energy is added by heat
at a known rate. A liquid of density 900 kg/m3 flows
through the calorimeter with volume flow rate of
2.00 L/min. At steady state, a temperature difference
3.50°C is established between the input and output
points when energy is supplied at the rate of 200 W.
What is the specific heat of the liquid?
64. A flow calorimeter is an apparatus used to measure the
specific heat of a liquid. The technique of flow calo-
rimetry involves measuring the temperature differ-
ence between the input and output points of a flowing
stream of the liquid while energy is added by heat at
a known rate. A liquid of density r flows through the
calorimeter with volume flow rate R. At steady state, a
temperature difference DT is established between the
input and output points when energy is supplied at the
rate P. What is the specific heat of the liquid?
65. Review. Following a collision between a large space-
craft and an asteroid, a copper disk of radius 28.0 m
and thickness 1.20 m at a temperature of 850°C is
floating in space, rotating about its symmetry axis
with an angular speed of 25.0 rad/s. As the disk radi-
ates infrared light, its temperature falls to 20.0°C. No
external torque acts on the disk. (a)Find the change
in kinetic energy of the disk. (b) Find the change in
internal energy of the disk. (c) Find the amount of
energy it radiates.
66. An ice-cube tray is filled with 75.0 g of water. After
the filled tray reaches an equilibrium temperature of
20.0°C, it is placed in a freezer set at 28.00°C to make
ice cubes. (a) Describe the processes that occur as
energy is being removed from the water to make ice.
(b) Calculate the energy that must be removed from
the water to make ice cubes at 28.00°C.
67. 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 copper at –12.0°C. Your pocket already con-
tains 14.0 g of silver 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?
68. The rate at which a resting person converts food energy
is called one’s basal metabolic rate (BMR). Assume that
the resulting internal energy leaves a person’s body
by radiation and convection of dry air. When you jog,
most of the food energy you burn above your BMR
becomes internal energy that would raise your body
temperature if it were not eliminated. Assume that
evaporation of perspiration is the mechanism for
eliminating this energy. Suppose a person is jogging
for “maximum fat burning,” converting food energy at
the rate 400 kcal/h above his BMR, and putting out
energy by work at the rate 60.0 W. Assume that the heat
of evaporation of water at body temperature is equal
to its heat of vaporization at 100°C. (a) Determine the
hourly rate at which water must evaporate from his
skin. (b) When you metabolize fat, the hydrogen atoms
much energy must be added to the gas by heat along
the indirect path IAF?
59. Gas in a container is at a pressure of 1.50 atm and a
volume of 4.00 m3. What is the work done on the gas
(a) if it expands at constant pressure to twice its initial
volume, and (b) if it is compressed at constant pressure
to one-quarter its initial volume?
60. Liquid nitrogen has a boiling point of 77.3 K and a
latent heat of vaporization of 2.01 3 105 J/kg. A 25.0-W
electric heating element is immersed in an insulated
vessel containing 25.0 L of liquid nitrogen at its boil-
ing point. How many kilograms of nitrogen are boiled
away in a period of 4.00 h?
61. An aluminum rod 0.500 m in length and with a cross-
sectional area of 2.50 cm2 is inserted into a thermally
insulated vessel containing liquid helium at 4.20 K.
The rod is initially at 300 K. (a) If one-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 cir-
cular surface of the upper end of the rod is maintained
at 300 K, what is the approximate boil-off rate of liq-
uid helium in liters per second after the lower half has
reached 4.20 K? (Aluminum has thermal conductivity
of 3 100 W/m · K at 4.20 K; ignore its temperature vari-
ation. The density of liquid helium is 125 kg/m3.)
62. Review. Two speeding lead bullets, one of mass 12.0g
moving to the right at 300 m/s and one of mass 8.00 g
moving to the left at 400 m/s, collide head-on, and all
the material sticks together. Both bullets are originally
at temperature 30.0°C. Assume the change in kinetic
energy of the system appears entirely as increased
internal energy. We would like to determine the tem-
perature and phase of the bullets after the collision.
(a)What two analysis models are appropriate for the
system of two bullets for the time interval from before
to after the collision? (b)From one of these models,
what is the speed of the combined bullets after the
collision? (c)How much of the initial kinetic energy
has transformed to internal energy in the system after
the collision? (d)Does all the lead melt due to the col-
lision? (e) What is the temperature of the combined
bullets after the collision? (f) What is the phase of the
combined bullets after the collision?
63. A flow calorimeter is an apparatus used to measure the
specific heat of a liquid. The technique of flow calo-
rimetry involves measuring the temperature difference
between the input and output points of a flowing
rial of the meteoroid rises momentarily to the same
final temperature. Find this temperature. Assume the
specific heat of liquid and of gaseous aluminum is
1 170 J/kg ? °C.
74. Why is the following situation impossible? A group of camp-
ers arises at 8:30 a.m. and uses a solar cooker, which
consists of a curved, reflecting surface that concen-
trates sunlight onto the object to be warmed (Fig.
P20.74). During the day, the maximum solar intensity
reaching the Earth’s surface at the cooker’s location
is I 5 600 W/m2. The cooker faces the Sun and has a
face diameter of d 5 0.600 m. Assume a fraction f of
40.0% of the incident energy is transferred to 1.50L
of water in an open container, initially at 20.0°C. The
water comes to a boil, and the campers enjoy hot cof-
fee for breakfast before hiking ten miles and returning
by noon for lunch.
75. During periods of high activity, the Sun has more sun-
spots than usual. Sunspots are cooler than the rest of
the luminous layer of the Sun’s atmosphere (the pho-
tosphere). Paradoxically, the total power output of the
active Sun is not lower than average but is the same
or slightly higher than average. Work out the details
of the following crude model of this phenomenon.
Consider a patch of the photosphere with an area of
5.10 3 1014 m2. Its emissivity is 0.965. (a) Find the power
it radiates if its temperature is uniformly 5 800 K,
corresponding to the quiet Sun. (b) To represent a
sunspot, assume 10.0% of the patch area is at 4 800 K
and the other 90.0% is at 5 890 K. Find the power
output of the patch. (c) State how the answer to part
(b) compares with the answer to part (a). (d) Find
the average temperature of the patch. Note that this
cooler temperature results in a higher power output.
76. (a) In air at 0°C, a 1.60-kg copper block at 0°C is set
sliding at 2.50 m/s over a sheet of ice at 0°C. Friction
brings the block to rest. Find the mass of the ice that
melts. (b) As the block slows down, identify its energy
input Q, its change in internal energy DE
, and the
change in mechanical energy for the block–ice system.
(c) For the ice as a system, identify its energy input Q
and its change in internal energy DE
. (d) A 1.60-kg
block of ice at 0°C is set sliding at 2.50m/s over a sheet
of copper at 0°C. Friction brings the block to rest.
Find the mass of the ice that melts. (e)Evaluate Q and
for the block of ice as a system and DE
block–ice system. (f) Evaluate Q and DE
for the metal
in the fat molecule are transferred to oxygen to form
water. Assume that metabolism of 1.00 g of fat gener-
ates 9.00 kcal of energy and produces 1.00 g of water.
What fraction of the water the jogger needs is provided
by fat metabolism?
69. An iron plate is held against an iron wheel so that a
kinetic friction force of 50.0 N acts between the two
pieces of metal. The relative speed at which the two sur-
faces slide over each other is 40.0 m/s. (a) Calculate the
rate at which mechanical energy is converted to internal
energy. (b) The plate and the wheel each have a mass of
5.00 kg, and each receives 50.0% of the internal energy.
If the system is run as described for 10.0 s and each
object is then allowed to reach a uniform internal tem-
perature, what is the resultant temperature increase?
70. A resting adult of average size converts chemical energy
in food into internal energy at the rate 120 W, called
her basal metabolic rate. To stay at constant temperature,
the body must put out energy at the same rate. Several
processes exhaust energy from your body. Usually, the
most important is thermal conduction into the air in
contact with your exposed skin. If you are not wear-
ing a hat, a convection current of warm air rises verti-
cally from your head like a plume from a smokestack.
Your body also loses energy by electromagnetic radia-
tion, by your exhaling warm air, and by evaporation of
perspiration. In this problem, consider still another
pathway for energy loss: moisture in exhaled breath.
Suppose you breathe out 22.0 breaths per minute, each
with a volume of 0.600 L. Assume you inhale dry air
and exhale air at 37.0°C containing water vapor with a
vapor pressure of 3.20 kPa. The vapor came from evap-
oration of liquid water in your body. Model the water
vapor as an ideal gas. Assume its latent heat of evapora-
tion at 37.0°C is the same as its heat of vaporization at
100°C. Calculate the rate at which you lose energy by
exhaling humid air.
71. A 40.0-g ice cube floats in 200 g of water in a 100-g
copper cup; all are at a temperature of 0°C. A piece of
lead at 98.0°C is dropped into the cup, and the final
equilibrium temperature is 12.0°C. What is the mass of
72. One mole of an ideal gas is contained in a cylinder
with a movable piston. The initial pressure, volume,
and temperature are P
, and T
, respectively. Find
the work done on the gas in the following processes.
In operational terms, describe how to carry out each
process and show each process on a PV diagram.
(a) an isobaric compression in which the final volume
is one-half the initial volume (b) an isothermal com-
pression in which the final pressure is four times the
initial pressure (c) an isovolumetric process in which
the final pressure is three times the initial pressure
73. Review. A 670-kg meteoroid happens to be composed
of aluminum. When it is far from the Earth, its tem-
perature is 215.0°C and it moves at 14.0 km/s relative
to the planet. As it crashes into the Earth, assume the
internal energy transformed from the mechanical
energy of the meteoroid–Earth system is shared equally
between the meteoroid and the Earth and all the mate-
chapter 20 the First Law of thermodynamics
Mass of water:
Mass of calorimeter:
Specific heat of calorimeter:
0.63 kJ/kg ? °C
Initial temperature of aluminum:
Mass of aluminum:
Final temperature of mixture:
(a) Use these data to determine the specific heat of
aluminum. (b) Explain whether your result is within
15% of the value listed in Table 20.1.
81. Consider the piston–
cylinder apparatus shown
in Figure P20.81. The bot-
tom of the cylinder con-
tains 2.00kg of water at
just under 100.0°C. The
cylinder has a radius of
r 5 7.50cm. The piston of
mass m 5 3.00kg sits on
the surface of the water.
An electric heater in the
cylinder base transfers
energy into the water at a rate of 100W. Assume the
cylinder is much taller than shown in the figure, so
we don’t need to be concerned about the piston reach-
ing the top of the cylinder. (a) Once the water begins
boiling, how fast is the piston rising? Model the steam
as an ideal gas. (b) After the water has completely
turned to steam and the heater continues to transfer
energy to the steam at the same rate, how fast is the
82. A spherical shell has inner radius 3.00 cm and outer
radius 7.00 cm. It is made of material with thermal
conductivity k 5 0.800 W/m ? °C. The interior is main-
tained at temperature 5°C and the exterior at 40°C.
After an interval of time, the shell reaches a steady
state with the temperature at each point within it
remaining constant in time. (a)Explain why the rate
of energy transfer P must be the same through each
spherical surface, of radius r, within the shell and must
(b) Next, prove that
where T is in degrees Celsius and r is in meters.
(c) Find the rate of energy transfer through the shell.
(d) Prove that
where T is in degrees Celsius and r is in meters.
(e) Find the temperature within the shell as a func-
tion of radius. (f)Find the temperature at r 5 5.00 cm,
halfway through the shell.
sheet as a system. (g) A thin, 1.60-kg slab of copper at
20°C is set sliding at 2.50 m/s over an identical station-
ary slab at the same temperature. Friction quickly stops
the motion. Assuming no energy is transferred to the
environment by heat, find the change in temperature
of both objects. (h)Evaluate Q and DE
for the slid-
ing slab and DE
for the two-slab system. (i) Evalu-
ate Q and DE
for the stationary slab.
77. Water in an electric teakettle is boiling. The power
absorbed by the water is 1.00 kW. Assuming the pres-
sure of vapor in the kettle equals atmospheric pres-
sure, determine the speed of effusion of vapor from
the kettle’s spout if the spout has a cross-sectional area
of 2.00 cm2. Model the steam as an ideal gas.
78. The average thermal conductivity of the walls (includ-
ing the windows) and roof of the house depicted in
Figure P20.78 is 0.480 W/m ? °C, and their average
thickness is 21.0 cm. The house is kept warm with
natural gas having a heat of combustion (that is, the
energy provided per cubic meter of gas burned) of
9 300 kcal/m3. How many cubic meters of gas must be
burned each day to maintain an inside temperature of
25.0°C if the outside temperature is 0.0°C? Disregard
radiation and the energy transferred by heat through
79. A cooking vessel on a slow burner contains 10.0 kg of
water and an unknown mass of ice in equilibrium at
0°C at time t 5 0. The temperature of the mixture is
measured at various times, and the result is plotted in
Figure P20.79. During the first 50.0 min, the mixture
remains at 0°C. From 50.0min to 60.0 min, the tem-
perature increases to 2.00°C. Ignoring the heat capac-
ity of the vessel, determine the initial mass of the ice.
80. A student measures the following data in a calorimetry
experiment designed to determine the specific heat of
Initial temperature of water
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