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Physics

James S. Walker

Chapter 16

Temperature and Heat

Educators


Problem 1

$\cdot$ CE Predict/Explain The temperature inside a freezer is $22^{\circ} \mathrm{F}$
and the temperature outside is $42^{\circ} \mathrm{F}$ . The temperature difference
is 20 $\mathrm{F}^{\circ} .$ (a) Is the temperature difference $\Delta T$ in degrees Celsius
greater than, less than, or equal to 20 $\mathrm{C}^{\circ}$ (b) Choose the best
explanation from among the following:
I The temperature difference is equal to 20 $\mathrm{C}^{\circ}$ because temperature differences are the same in all temperature scales.
II. The temperature difference is less than 20 $\mathrm{C}^{\circ}$ because
$\Delta T_{\mathrm{c}}=\frac{5}{9}\left(20^{\circ} \mathrm{F}\right)=11^{\circ} \mathrm{C} .$
III. The temperature difference is greater than 20 $\mathrm{C}^{\circ}$ because
$\Delta T_{\mathrm{C}}=\frac{9}{5}\left(20^{\circ} \mathrm{F}\right)+32=68^{\circ} \mathrm{C}$

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Problem 2

$\cdot$ Lowest Temperature on Earth The official record for the lowest temperature ever recorded on Earth was set at Vostok, Antarctica, on
July $21,1983 .$ The temperature on that day fell to $-89.2^{\circ} \mathrm{C}$ , well
below the temperature of dry ice. What is this temperature in
degrees Fahrenheit?

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Problem 3

$\cdot$ Incandescent lightbulbs heat a tungsten filament to a temperature of about $4580^{\circ} \mathrm{F}$ , which is almost half as hot as the surface of
the Sun. What is this temperature in degrees Celsius?

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Problem 4

$\cdot$ Normal body temperature for humans is $98.6^{\circ} \mathrm{F} .$ What is the corresponding temperature in (a) degrees Celsius and (b) kelvins?

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Problem 5

$\cdot$ The temperature at the surface of the Sun is about 6000 $\mathrm{K}$ . Convert this temperature to the (a) Celsius and (b) Fahrenheit scales.

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Problem 6

$\cdot$ One day you notice that the outside temperature increased by
17 $\mathrm{F}^{\circ}$ between your early morning jog and your lunch at noon.
What is the corresponding change in temperature in the (a) Celsius
and (b) Kelvin scales?

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Problem 7

$\cdot$ The gas in a constant-volume gas thermometer has a pressure of
93.5 $\mathrm{kPa}$ at $105^{\circ} \mathrm{C} .$ (a) What is the pressure of the gas at $47.5^{\circ} \mathrm{C}$ ?
(b) At what temperature does the gas have a pressure of 105 $\mathrm{kPa}$ ?

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Problem 8

$\because$ Predict/Calculate $A$ constant-volume gas thermometer has
a pressure of 80.3 $\mathrm{kPa}$ at $-10.0^{\circ} \mathrm{C}$ and a pressure of 86.4 $\mathrm{kPa}$ at $10.0^{\circ} \mathrm{C} .$ (a) At what temperature does the pressure of this system
extrapolate to zero? (b) What are the pressures of the gas at the
freezing and boiling points of water? (c) In general terms, how
would your answers to parts (a) and (b) change if a different constant-volume gas thermometer is used? Explain.

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Problem 9

$\bullet$ Greatest Change in Temperature A world record for the greatest
change in temperature was set in Spearfish, SD, on January $22,$
1943. At 730 A.M. the temperature was $-4.0^{\circ} \mathrm{F}$ two minutes later
the temperature was $45^{\circ} \mathrm{F}$ . Find the average rate of temperature
change during those two minutes in kelvins per second.

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Problem 10

$\bullet$ We know that $-40^{\circ} \mathrm{C}$ corresponds to $-40^{\circ} \mathrm{F} .$ What temperature
has the same value in both the Fahrenheit and Kelvin scales?

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Problem 11

$\because$ At what temperature is the reading on a Fahrenheit scale twice
the reading on a Celsius scale?

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Problem 12

$\cdots$ When the bulb of a constant-volume gas thermometer is placed
in a beaker of boiling water at $100^{\circ} \mathrm{C},$ the pressure of the gas is
227 $\mathrm{mm} \mathrm{Hg} .$ When the bulb is moved to an ice-salt mixture, the pressure of the gas drops to 162 $\mathrm{mm} \mathrm{g}$ . Assuming ideal behavior, as in
Figure $16-3,$ what is the Celsius temperature of the ice-salt mixture?

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Problem 13

$\cdot$ CE Bimetallic strip A is made of copper and steel; bimetallic strip
Bis made of aluminum and steel. (a) Referring to Table $16-1,$ which
strip bends more for a given change in temperature? (b) Which of
the metals listed in Table $16-1$ would give the greatest amount of
bend when combined with steel in a bimetallic strip?

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Problem 14

$\cdot$ CE Referring to Table $16-1,$ which would be more accurate for all-season outdoor use: a tape measure made of steel or one made of
aluminum?

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Problem 15

$\cdot$ CE Predict/Explain A brass plate has a circular hole whose diameter is slightly smaller than the diameter of an aluminum ball. If
the ball and the plate are always kept at the same temperature,
(a) should the temperature of the system be increased or decreased
in order for the ball to fit through the hole? (b) Choose the best
explanation from among the following:
I. The aluminum ball changes its diameter more with temperature than the brass plate, and therefore the temperature
should be decreased.
II. Changing the temperature won't change the fact that the ball
is larger than the hole.
III. Heating the brass plate makes its hole larger, and that will
allow the ball to pass through.

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Problem 16

$\cdot$ CE FIGURE $16-25$ shows five metal plates, all at the same temperature and all made from the same material. They are all placed in
an oven and heated by the same amount. Rank the plates in order
of increasing expansion in (a) the vertical and (b) the horizontal
direction. Indicate ties where appropriate.

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Problem 17

$\cdot$ Longest Suspension Bridge The world's longest suspension bridge is
the Akashi Kaikyo Bridge in Japan. The bridge is 3910 $\mathrm{m}$ long and
is constructed of steel. How much longer is the bridge on a warm
summer day $\left(30.0^{\circ} \mathrm{C}\right)$ than on a cold winter day $\left(-5.00^{\circ} \mathrm{C}\right) ?$

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Problem 18

$\because$ A vinyl siding panel for a house is installed on a day when the
temperature is $15.6^{\circ} \mathrm{C} .$ If the coefficient of thermal expansion for
vinyl siding is $55.8 \times 10^{-6} \mathrm{K}^{-1}$ , how much room (in mm) should
the installer leave for expansion of a $3.66-\mathrm{m}$ length if the sunlit
temperature of the siding could reach $48.9^{\circ} \mathrm{C} ?$

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Problem 19

$\because$ A cylinder bore in an aluminum engine block has a diameter of
96.00 $\mathrm{mm}$ at $20.00^{\circ} \mathrm{C}$ (a) What is the diameter of the bore when
the engine operates at $119.0^{\circ} \mathrm{C} ?$ (b) At what temperature is the
diameter of the hole equal to 95.85 $\mathrm{mm}$ ?

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Problem 20

$\bullet$ Predict/Calculate It is desired to slip an aluminum ring over a
steel bar (FIGURE $16-26 ) .$ At $10.00^{\circ} \mathrm{C}$ the inside diameter of the ring
is 4.000 $\mathrm{cm}$ and the diameter of the rod is 4.040 $\mathrm{cm} .$ (a) In order for the ring to slip over the bar, should the ring be heated or cooled?
Explain. (b) Find the temperature of the ring at which it fits over
the bar. The bar remains at $10.0^{\circ} \mathrm{C}$ .

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Problem 21

$\because$ At $18.75^{\circ} \mathrm{C}$ a brass sleeve has an inside diameter of 2.21988 $\mathrm{cm}$
and a steel shaft has a diameter of 2.22258 $\mathrm{cm} .$ It is desired to
shrink-fit the sleeve over the steel shaft. (a) To what temperature
must the sleeve be heated in order for it to slip over the shaft?
(b) Alternatively, to what temperature must the shaft be cooled
before it is able to slip through the sleeve?

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Problem 22

$\bullet$ Early in the morning, when the temperature is $5.5^{\circ} \mathrm{C},$ gasoline
is pumped into a car's 53 -L steel gas tank until it is filled to the top.
Later in the day the temperature rises to $27^{\circ} \mathrm{C}$ . Since the volume of
gasoline increases more for a given temperature increase than the
volume of the steel tank, gasoline will spill out of the tank. How
much gasoline spills out in this case?

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Problem 23

.. Some cookware has a stainless steel interior $\left(\alpha=17.3 \times 10^{-6} \mathrm{K}^{-1}\right)$
and a copper bottom $\left(\alpha=17.0 \times 10^{-6} \mathrm{K}^{-1}\right)$ for better heat distribution. Suppose an 8.0 -in. pot of this construction is heated to $610^{\circ} \mathrm{C}$ on the stove. If the initial temperature of the pot is $22^{\circ} \mathrm{C},$ what is
the difference in diameter change for the copper and the steel?

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Problem 24

$\because$ Predict/Calculate You construct two wire-frame cubes, one
using copper wire, the other using aluminum wire. At $23^{\circ} \mathrm{C}$ the
cubes enclose equal volumes of 0.016 $\mathrm{m}^{3} .$ (a) If the temperature of the cubes is increased, which cube encloses the greater volume?
(b) Find the difference in volume between the cubes when their
temperature is $97^{\circ} \mathrm{C}$ .

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Problem 25

$\because$ A metal ball that is 1.2 $\mathrm{m}$ in diameter expands by 2.2 $\mathrm{mm}$ when
the temperature is increased by $95^{\circ} \mathrm{C} .$ (a) What is the coefficient of thermal expansion for the ball? (b) What substance is this ball
likely made from? (Refer to Table 16-1.)

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Problem 26

$\cdots$ A copper ball with a radius of 1.7 $\mathrm{cm}$ is heated until its diameter has increased by 0.16 mm. Assuming an initial temperature of
$21^{\circ} \mathrm{C},$ find the final temperature of the ball.

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Problem 27

$\cdots$ Predict/Calculate An aluminum saucepan with a diameter
of 23 $\mathrm{cm}$ and a height of 6.0 $\mathrm{cm}$ is filled to the brim with water.
The initial temperature of the pan and water is $19^{\circ} \mathrm{C} .$ The pan is
now placed on a stove burner and heated to $88^{\circ} \mathrm{C}$ (a) Will water
overflow from the pan, or will the water level in the pan decrease?
Explain. (b) Calculate the volume of water that overflows or the
drop in water level in the pan, whichever is appropriate.

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Problem 28

$\cdot$ BIO Sleeping Metabolic Rate When people sleep, their metabolic
rate is about $2.6 \times 10^{-4} \mathrm{C} /(\mathrm{s} \cdot \mathrm{kg}) .$ How many Calories does an $81-\mathrm{kg}$
person metabolize while getting a good night's sleep of 8.5 $\mathrm{h}$ ?

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Problem 29

$\because$ BIO An exercise machine indicates that you have worked off
2.1 Calories in a minute-and-a-half of running in place. What was
your power output during this time? Give your answer in both
watts and horsepower.

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Problem 30

$\because$ BIO A certain sandwich cookie contains 53 $\mathrm{C}$ of nutritional
energy. (a) For what amount of time must you swim in order to
work off three cookies if swimming consumes 540 $\mathrm{C} / \mathrm{h} ?$ (b) Instead
of swimming you decide to run for 45 min at a pace that works off
850 $\mathrm{C} / \mathrm{h} .$ How many cookies have you worked off?

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Problem 31

$\cdots$ BIO During a workout, a person repeatedly lifts a $16-1 \mathrm{b}$ barbell through a distance of 1.1 $\mathrm{ft}$ . How many "reps" of this lift are
required to work off 150 $\mathrm{C}$ ?

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Problem 32

$\bullet$ Predict/Calculate Consider the apparatus that Joule used in
his experiments on the mechanical equivalent of heat, shown in
Figure $16-12$ . Suppose both blocks have a mass of 0.95 $\mathrm{kg}$ and that
they fall through a distance of 0.48 $\mathrm{m} .$ (a) Find the expected rise
in temperature of the water, given that 6200 $\mathrm{J}$ are needed for every
1.0 $\mathrm{C}^{\circ}$ increase. Give your answer in Celsius degrees. (b) Would the
temperature rise in Fahrenheit degrees be greater than or less than
the result in part (a)? Explain. (c) Find the rise in temperature in
Fahrenheit degrees.

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Problem 33

$\because$ BIO It was shown in Example $16-18$ that a typical person radiates about 62 $\mathrm{W}$ of power at room temperature. Given this result,
how much time does it take for a person to radiate away the energy
acquired by consuming a $230-\mathrm{C}$ doughnut?

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Problem 34

CE Predict//Explain Two objects are made of the same material
but have different temperatures. Object 1 has a mass $m,$ and object
2 has a mass 2$m .$ If the objects are brought into thermal contact,
(a) is the temperature change of object 1 greater than, less than, or
equal to the temperature change of object 2$?$ (b) Choose the best
explanation from among the following:
I. The larger object gives up more heat, and therefore its temperature change is greatest.
II. The heat given up by one object is taken up by the other object. since the objects have the same heat capacity, the temperature changes are the same.
III. One object loses heat of magnitude $Q$ , the other gains heat of
magnitude $Q .$ With the same magnitude of heat involved, the
smaller object has the greater temperature change.

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Problem 35

$\cdot$ CE Predict/Explain A certain amount of heat is transferred to
2 kg of aluminum, and the same amount of heat is transferred to
1 $\mathrm{kg}$ of ice. Referring to Table $16-2,(\mathrm{a})$ is the increase in temperature of the aluminum greater than, less than, or equal to the
increase in temperature of the ice? (b) Choose the best explanation
from among the following:
I. Twice the specific heat of aluminum is less than the specific
heat of ice, and hence the aluminum has the greater temperature change.
II. The aluminum has the smaller temperature change since its
mass is less than that of the ice.
III. The same heat will cause the same change in temperature.

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Problem 36

$\cdot$ Suppose 72.3 $\mathrm{J}$ of heat are added to a $101-\mathrm{g}$ piece of aluminum at
$20.5^{\circ} \mathrm{C} .$ What is the final temperature of the aluminum?

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Problem 37

$\cdot$ Estimate the heat required to heat a 0.15 -kg apple from $12^{\circ} \mathrm{C}$ to
$36^{\circ} \mathrm{C}$ . (Assume the apple is mostly water.)

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Problem 38

$\cdot$ A 9.7 -g lead bullet is fired into a fence post. The initial speed of
the bullet is $720 \mathrm{m} / \mathrm{s},$ and when it comes to rest, half its kinetic
energy goes into heating the bullet. How much does the bullet's
temperature increase?

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Problem 39

$\cdots$ Thermal energy is added to 150 $\mathrm{g}$ of water at the rate of 55 $\mathrm{J} / \mathrm{s}$ for
2.5 $\mathrm{min} .$ How much does the temperature of the water increase?

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Problem 40

$\bullet$ Predict/Calculate Silver pellets with a mass of 1.0 $\mathrm{g}$ and a temperature of $85^{\circ} \mathrm{C}$ are added to 220 $\mathrm{g}$ of water at $14^{\circ} \mathrm{C}$ (a) How
many pellets must be added to increase the equilibrium temperature of the system to $25^{\circ} \mathrm{C} ?$ Assume no heat is exchanged with the surroundings. (b) If copper pellets are used instead, does the
required number of pellets increase, decrease, or stay the same?
Explain. (c) Find the number of copper pellets that are required.

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Problem 41

$\because \mathrm{A} 225$ -g lead ball at a temperature of $81.2^{\circ} \mathrm{C}$ is placed in a light
calorimeter containing 155 $\mathrm{g}$ of water at $20.3^{\circ} \mathrm{C} .$ Find the equilibrium temperature of the system.

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Problem 42

$\cdots$ If 2200 $\mathrm{J}$ of heat are added to a $190-\mathrm{g}$ object, its temperature
increases by 12 $\mathrm{C}^{\circ}$ (a) What is the heat capacity of this object?
(b) What is the object's specific heat?

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Problem 43

$\bullet$ Chips by the Ton Tortilla chips are manufactured by submerging
baked and partially cooled masa at $41^{\circ} \mathrm{C}$ into corn oil at $182^{\circ} \mathrm{C}$ .
(a) If the specific heat of masa is 1850 $\mathrm{J} /(\mathrm{kg} \cdot \mathrm{K})$ , how much thermal
energy is required to warm 0.454 $\mathrm{kg}$ of masa to the temperature of
the oil? (b) If the production line makes 227 $\mathrm{kg}$ of chips every hour,
at what rate (in $\mathrm{kW}$ ) must thermal energy be supplied to the corn
oil, assuming that all of the energy goes into warming the masa?

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Problem 44

$\because \mathrm{A} 117$ -g lead ball is dropped from rest from a height of 7.62 $\mathrm{m}$ .
The collision between the ball and the ground is totally inelastic.
Assuming all the ball's kinetic energy goes into heating the ball,
find its change in temperature.

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Problem 45

$\cdots$ To determine the specific heat of an object, a student heats it
to $100^{\circ} \mathrm{C}$ in boiling water. She then places the 34.5 object in a
151 -g aluminum calorimeter containing 114 $\mathrm{g}$ of water. The aluminum and water are initially at a temperature of $20.0^{\circ} \mathrm{C},$ and are thermally insulated from their surroundings. If the final temperature is $23.6^{\circ} \mathrm{C},$ what is the specific heat of the object? Referring to
Table $16-2,$ identify the material in the object.

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Problem 46

$\cdots$ Predict/Calculate A student drops a $0.33-\mathrm{kg}$ piece of steel at
$42^{\circ} \mathrm{C}$ into a container of water at $22^{\circ} \mathrm{C}$ The student also drops a
$0.51-\mathrm{kg}$ chunk of lead into the same container at the same time.
The temperature of the water remains the same. (a) Was the temperature of the lead greater than, less than, or equal to $22^{\circ} \mathrm{C} ?$
Explain. (b) What was the temperature of the lead?

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Problem 47

$\cdots$ A ceramic coffee cup, with $m=116 \mathrm{g}$ and $c=1090 \mathrm{J} /(\mathrm{kg} \cdot \mathrm{K})$
is initially at room temperature $\left(24.0^{\circ} \mathrm{C}\right) .$ If 225 $\mathrm{g}$ of $80.3^{\circ} \mathrm{C}$ coffee and 12.2 g of $5.00^{\circ} \mathrm{C}$ cream are added to the cup, what is
the equilibrium temperature of the system? Assume that no heat
is exchanged with the surroundings, and that the specific heat of
coffee and cream are the same as the specific heat of water.

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Problem 48

$\cdot$ cE Predict/Explain In a popular lecture demonstration, a sheet
of paper is wrapped around a rod that is made from wood on one
half and metal on the other half. If held over a flame, the paper on
one half of the rod is burned while the paper on the other half is
unaffected. (a) Is the burned paper on the wooden half of the rod,
or on the metal half of the rod? (b) Choose the best explanation
from among the following:
I. The metal will be hotter to the touch than the wood; therefore the metal side will be burnt.
II. The metal conducts heat better than the wood, and hence the
paper on the metal half is unaffected.
III. The metal has the smaller specific heat; hence it heats up
more and burns the paper on its half of the rod.

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Problem 49

CE FIGURE $16-27$ shows a composite slab of three different materials with equal thickness but different thermal conductivities. The
opposite sides of the composite slab are held at the fixed temperatures $T_{1}$ and $T_{2} .$ Given that $k_{\mathrm{B}}>k_{\mathrm{A}}>k_{\mathrm{C}},$ rank the materials in
order of the temperature difference across them, starting with the
smallest. Indicate ties where appropriate.

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Problem 50

$\cdot$ CE Heat is transferred from an area where the temperature is $20^{\circ} \mathrm{C}$
to an area where the temperature is $0^{\circ} \mathrm{C}$ through a composite slab
consisting of four different materials, each with the same thickness.
The temperatures at the interface between each of the materials are
given in FlGURE $16-28 .$ Rank the four materials in order of increasing
thermal conductivity. Indicate ties where appropriate.

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Problem 51

CE Predict/Explain Two identical bowls of casserole need to be
kept warm. The cook covers bowl 1 with transparent plastic wrap
and bowl 2 with shiny aluminum foil, in each case trapping a layer
of air between the covering and the casserole. (a) After a few minutes, will the temperature of casserole 1 be greater than, less than,
or equal to the temperature of casserole 2$?$ (b) Choose the best
explanation from among the following:
I. The lower thermal conductivity of the plastic wrap over casserole 1 will suppress heat loss by conduction better than aluminum foil.
II. In each case heat loss by convection is suppressed, so the final
temperatures will be the same.
III. The high reflectivity of the aluminum foil over casserole 2
will suppress heat loss by radiation, while the heat losses
by conduction and convection for casseroles 1 and 2 will be
similar.

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Problem 52

$\cdot$ CE Two bowls of soup with identical temperatures are placed on
a table. Bowl 1 has a metal spoon in it; bowl 2 does not. After a few
minutes, is the temperature of the soup in bowl 1 greater than, less
than, or equal to the temperature of the soup in bowl 2$?$

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Problem 53

$\cdot$ A glass window 0.33 $\mathrm{cm}$ thick measures 81 $\mathrm{cm}$ by 39 $\mathrm{cm} .$ How
much heat flows through this window per minute if the inside and
outside temperatures differ by 13 $\mathrm{C}^{\circ} ?$

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Problem 54

$\cdot$ BIO Assuming your skin temperature is $37.2^{\circ} \mathrm{C}$ and the tem-
perature of your surroundings is $20.0^{\circ} \mathrm{C}$ , determine the length of
time required for you to radiate away the energy gained by eating a
278 -Cice cream cone. Let the emissivity of your skin be 0.915 and
its area be 1.78 $\mathrm{m}^{2} .$

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Problem 55

$\cdot$ Find the heat that flows in 1.0 s through a lead brick 25 $\mathrm{cm}$ long if
the temperature difference between the ends of the brick is 8.5 $\mathrm{C}^{\circ} .$
The cross-sectional area of the brick is 14 $\mathrm{cm}^{2} .$

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Problem 56

$\bullet$ Consider a double-paned window consisting of two panes of
glass, each with a thickness of 0.500 $\mathrm{cm}$ and an area of 0.725 $\mathrm{m}^{2}$ ,
separated by a layer of air with a thickness of 1.75 $\mathrm{cm} .$ The temperature on one side of the window is $0.00^{\circ} \mathrm{C} ;$ the temperature on the
other side is $20.0^{\circ} \mathrm{C}$ In addition, note that the thermal conductivity of glass is roughly 36 times greater than that of air. (a) Approximate the heat transfer through this window by ignoring the glass. That is, calculate the heat flow per second through 1.75 $\mathrm{cm}$ of air
with a temperature difference of 20.0 $\mathrm{C}^{\circ} .$ The exact result for the
complete window is 19.1 $\mathrm{J} / \mathrm{s} .$ (b) Use the approximate heat flow
found in part (a) to find an approximate temperature difference
across each pane of glass. (The exact result is 0.157 $\mathrm{C}^{\circ} . )$

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Problem 57

$\bullet$ Predict/Calculate Two metal rods of equal length-one aluminum, the other stainless steel-are connected in parallel with
a temperature of $20.0^{\circ} \mathrm{C}$ at one end and and $118^{\circ} \mathrm{C}$ at the other end.
Both rods have a circular cross section with a diameter of 3.50 $\mathrm{cm} .$
(a) Determine the length the rods must have if the combined rate
of heat flow through them is to be 27.5 $\mathrm{J}$ per second. (b) If the
length of the rods is doubled, by what factor does the rate of heat
flow change?

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Problem 58

$\bullet$ Two cylindrical metal rods - one copper, the other lead-are
connected in parallel with a temperature of $21.0^{\circ} \mathrm{C}$ at one end and
$112^{\circ} \mathrm{Cat}$ the otherend. Both rods are 0.650 $\mathrm{min}$ length, and the lead
rod is 2.76 $\mathrm{cm}$ in diameter. If the combined rate of heat flow through
the two rods is $33.2 \mathrm{J} / \mathrm{s},$ what is the diameter of the copper rod?

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Problem 59

$\bullet$ Predict/Calculate Two metal rods-one lead, the other copper-are connected in series, as shown in FIGURE $16-29$ . These are
the same two rods that were connected in parallel in Example
$16-16 .$ Note that each rod is 0.525 $\mathrm{m}$ in length and has a square
cross section 1.50 $\mathrm{cm}$ on a side. The temperature at the lead end
of the rods is $2.00^{\circ} \mathrm{C}$ ; the temperature at the copper end is $106^{\circ} \mathrm{C}$ . (a) The average temperature of the two ends is $54.0^{\circ} \mathrm{C}$ . Is the temperature in the middle, at the lead-copper interface, greater than, less than, or equal to $54.0^{\circ} \mathrm{C} ?$ Explain. (b) Given that the heat flow
through each of these rods in 1.00 s is $1.41 \mathrm{J},$ find the temperature
at the lead-copper interface.

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Problem 60

$\bullet$ Predict/Calculate Consider two cylindrical metal rods with
equal cross section-one lead, the other aluminum-connected in
series. The temperature at the lead end of the rods is $22.5^{\circ} \mathrm{C}$ ; the
temperature at the aluminum end is $85.0^{\circ} \mathrm{C}$ (a)Given that the temperature at the lead-aluminum interface is $52.5^{\circ} \mathrm{C}$ and that the lead
rod is 14.5 $\mathrm{cm}$ long, what condition can you use to find the length of
the aluminum rod? (b) Find the length of the aluminum rod.

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Problem 61

$\because$ A copper rod 85 $\mathrm{cm}$ long is used to poke a fire. The hot end of
the rod is maintained at $115^{\circ} \mathrm{C}$ and the cool end has a constant
temperature of $22^{\circ} \mathrm{C} .$ What is the temperature of the rod 25 $\mathrm{cm}$
from the cool end?

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Problem 62

$\bullet$ Two identical objects are placed in a room at $24^{\circ} \mathrm{C} .$ Object 1 has
a temperature of $99^{\circ} \mathrm{C},$ and object 2 has a temperature of $27^{\circ} \mathrm{C}$ .
What is the ratio of the net power emitted by object 1 to that radiated by object 2 ?

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Problem 63

$\cdots$ A block has the dimensions $L, 2 L,$ and 3$L .$ When one of the
$L \times 2 L$ faces is maintained at the temperature $T_{1}$ and the other
$L \times 2 L$ face is held at the temperature $T_{2}$ , the rate of heat conduction through the block is $P$ . Answer the following questions in terms of $P$ . (a) What is the rate of heat conduction in this block if one of the
$L \times 3 L$ faces is held at the temperature $T_{1}$ and the other $L \times 3 L$ face
is held at the temperature $T_{2} ?$ (b) What is the rate of heat conduction
in this block if one of the 2$L \times 3 L$ faces is held at the temperature $T_{1}$
and the other 2$L \times 3 L$ face is held at the temperature $T_{2} ?$

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Problem 64

CE Predict/Explain A pendulum is made from an aluminum
rod with a mass attached to its free end. If the pendulum is cooled,
(a) does the pendulum's period increase, decrease, or stay the
same? (b) Choose the best explanation from among the following:
I. The period of a pendulum depends only on its length and the
acceleration of gravity. It is independent of mass and temperature.
II. Cooling makes everything move more slowly, and hence the
period of the pendulum increases.
III. Cooling shortens the aluminum rod, which decreases the period of the pendulum.

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Problem 65

$\cdot$ CE A copper ring stands on
edge with a metal rod placed
inside it, as shown in FlGURE
$16-30$ . As this system is heated,
will the rod ever touch the
top of the ring? Answer yes or
no for the case of a rod that is
made of (a) copper, (b) aluminum, and (c) steel.

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Problem 66

$\cdot$ CE Referring to the copper ring in the previous problem, imagine
that initially the ring is hotter than room temperature, and that
an aluminum rod that is colder than room temperature fits snugly
inside the ring. When this system reaches thermal equilibrium at
room temperature, is the rod firmly wedged in the ring (A) or can
it be removed easily ( $\mathrm{B}$ )?

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Problem 67

CE Predict/Explain The specific heat of alcohol is about half
that of water. Suppose you have 0.5 kg of alcohol at the temperature $20^{\circ} \mathrm{C}$ in one container, and 0.5 $\mathrm{kg}$ of water at the temperature
$30^{\circ} \mathrm{C}$ in a second container. When these fluids are poured into the
same container and allowed to come to thermal equilibrium, (a) is
the final temperature greater than, less than, or equal to $25^{\circ} \mathrm{C} ?$
Choose the best explanation from among the following:
I. The low specific heat of alcohol pulls in more heat, giving a
$\quad$ final temperature that is less than $25^{\circ} \mathrm{C} .$
II. More heat is required to change the temperature of water than
to change the temperature of alcohol. Therefore, the final
temperature will be greater than $25^{\circ} \mathrm{C}$ .
III. Equal masses are mixed together; therefore, the final temperature will be $25^{\circ} \mathrm{C},$ the average of the two initial temperatures.

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Problem 68

$\cdot$ Making Steel Sheets In the continuous-caster process, steel sheets
25.4 $\mathrm{cm}$ thick, 2.03 $\mathrm{m}$ wide, and 10.0 $\mathrm{m}$ long are produced at a
temperature of $872^{\circ} \mathrm{C}$ . What are the dimensions of a steel sheet
once it has cooled to $20.0^{\circ} \mathrm{C} ?$

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Problem 69

$\cdot$ The Coldest Place in the Universe The Boomerang nebula holds the
distinction of having the lowest recorded temperature in the uni-
verse, a frigid $-272^{\circ} \mathrm{C} .$ What is this temperature in kelvins?

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Problem 70

$\cdot$ BIO The Hottest Living Things From the surreal realm of deep-sea
hydrothermal vents 200 miles offshore from Puget Sound, comes
a newly discovered hyperthermophilic-or extreme heat-loving-microbe that holds the record for the hottest existence known to science. This microbe is tentatively known as Strain 121 for the
temperature at which it thrives: $121^{\circ} \mathrm{C}$ (At sea level, water at this
temperature would boil vigorously, but the extreme pressures at
the ocean floor prevent boiling from occurring. What is this temperature in degrees Fahrenheit?

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Problem 71

$\bullet$ Thermal energy is added to 180 $\mathrm{g}$ of water at a constant rate for
$3.5 \mathrm{min},$ resulting in an increase in temperature of 12 $\mathrm{C}^{\circ} .$ What is
the heating rate, in joules per second?

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Problem 72

$\bullet$ You wish to chill 12 cans of soda at $25.0^{\circ} \mathrm{C}$ down to $5.0^{\circ} \mathrm{C}$ before
serving them to guests. Each can has a mass of 0.354 $\mathrm{kg}$ , and the
specific heat of soda is the same as that of water. (a) How much
thermal energy must be removed in order to chill all 12 cans?
(b) If the chilling process requires $4.0 \mathrm{h},$ what is the average rate
(in $\mathrm{W} )$ at which thermal energy is removed from the soda?

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Problem 73

$\cdot \bullet$ BIO Brain Power As you read this problem, your brain is consuming about 22 $\mathrm{W}$ of power. (a) If your mass is $65 \mathrm{kg},$ how
many steps with a height of 21 $\mathrm{cm}$ must you climb to expend a
mechanical energy equivalent to one hour of brain operation?
(b) A typical human brain, which is 77$\%$ water, has a mass of
1.4 kg. Assuming that the 22 W of brain power is converted to
heat, what temperature rise would you estimate for the brain in
one hour of operation? Ignore the significant heat transfer that
occurs between a human head and its surroundings, as well as
the 23$\%$ of the brain that is not water.

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Problem 74

$\because$ BIO Brain Food Your brain consumes about 22 $\mathrm{W}$ of power,
and avocados have been shown to promote brain health. If each
avocado contains $310 \mathrm{C},$ and your brain were powered entirely
by the energy from the avocados, how many must you eat each
day?

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Problem 75

$\because$ BIO The Cricket Thermometer The rate of chirping of the snowy
tree cricket (Oecanthus fultoni Walker) varies with temperature in
a predictable way. A linear relationship provides a good match to
the chirp rate, but an even more accurate relationship is the following:
$$N=\left(5.63 \times 10^{10}\right) e^{-(6290 \mathrm{K}) / T}$$
In this expression, $N$ is the number of chirps in 13.0 s and $T$ is the
temperature in kelvins. If a cricket is observed to chirp 185 times in
60.0 s, what is the temperature in degrees Fahrenheit?

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Problem 76

$\cdots$ Predict/Calculate A pendulum consists of a large weight suspended by a steel wire that is 0.9500 m long.(a) If the temperature
increases, does the period of the pendulum increase, decrease, or
stay the same? Explain. (b) Calculate the change in length of the
pendulum if the temperature increase is 150.0 $\mathrm{C}^{\circ} .$ (c) Calculate
the period of the pendulum before and after the temperature
increase. (Assume that the coefficient of linear expansion for the
wire is $12.00 \times 10^{-6} \mathrm{K}^{-1}$ , and that $g=9.810 \mathrm{m} / \mathrm{s}^{2}$ at the location
of the pendulum.)

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Problem 77

$\bullet$ Predict/Calculate Once the aluminum ring in Problem 20 is
slipped over the bar, the ring and bar are allowed to equilibrate at
a temperature of $22^{\circ} \mathrm{C}$ . The ring is now stuck on the bar. (a) If the
temperatures of both the ring and the bar are changed together,
should the system be heated or cooled to remove the ring? (b) Find
the temperature at which the ring can be removed.

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Problem 78

$\because \mathrm{A} 256$ -kg rock sits in full sunlight on the edge of a cliff 5.75 $\mathrm{m}$
high. The temperature of the rock is $33.2^{\circ} \mathrm{C}$ . If the rock falls from
the cliff into a pool containing 6.00 $\mathrm{m}^{3}$ of water at $15.5^{\circ} \mathrm{C},$ what is
the final temperature of the rock-water system? Assume that the
specific heat of the rock is 1010 $\mathrm{J} /(\mathrm{kg} \cdot \mathrm{K})$ .

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Problem 79

$\bullet$ Water going over Iguacu Falls on the border of Argentina and
Brazil drops through a height of about 72 $\mathrm{m}$ . Suppose that all the
gravitational potential energy of the water goes into raising its
temperature. Find the increase in water temperature at the bottom
of the falls as compared with the top.

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Problem 80

$\bullet$ Thermal Storage Solar heating of a house is much more efficient
if there is a way to store the thermal energy collected during the
day to warm the house at night. Suppose one solar-heated home
utilizes a concrete slab of area 12 $\mathrm{m}^{2}$ and 25 $\mathrm{cm}$ thick. (a) If the
density of concrete is $2400 \mathrm{kg} / \mathrm{m}^{3},$ what is the mass of the slab?
(b) The slab is exposed to sunlight and absorbs energy at a rate of
$1.4 \times 10^{7} \mathrm{J} / \mathrm{h}$ for 10 $\mathrm{h}$ . If it begins the day at $22^{\circ} \mathrm{C}$ and has a specific
heat of $750 \mathrm{J} /(\mathrm{kg} \cdot \mathrm{K}),$ what is its temperature at sunset? (c) Model the concrete slab as being surrounded on both sides (contact area
24 $\mathrm{m}^{2}$ ) with a 2.0 -thick layer of air in contact with a surface that
is $5.0^{\circ} \mathrm{C}$ cooler than the concrete. At sunset, what is the rate at
which the concrete loses thermal energy by conduction through
the air layer? (d) Model the concrete slab as having a surface area
of 24 $\mathrm{m}^{2}$ and surrounded by an environment $5.0^{\circ} \mathrm{C}$ cooler than
the concrete. If its emissivity is $0.94,$ what is the rate at which the
concrete loses thermal energy by radiation at sunset?

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Problem 81

$\cdots$ Pave lt over Suppose city 1 leaves an entire block $(100 \mathrm{m} \times 100 \mathrm{m})$
as a park with trees and grass (emissivity 0.96) while city 2 paves
the same area over with asphalt (emissivity 1.0$) .$ Sunlight heats
each surface to $40.0^{\circ} \mathrm{C}$ by sunset, and then the surface radiates
its heat into a cube of air 100 $\mathrm{m}$ on a side and at $30.0^{\circ} \mathrm{C}$ (a) At
what rate does the park in city 1 deliver energy to the air at sunset?
(b) At what rate does the asphalt in city 2 deliver energy to the air
at sunset? (c) If each city block maintains the same radiated power
for 2.0 h and there are no other energy losses, what are the final
temperatures of the cubes of air above each city block? The density
of air at $30.0^{\circ} \mathrm{C}$ is 1.16 $\mathrm{kg} / \mathrm{m}^{3} .$ Although this example is oversimplified, a more sophisticated analysis recently showed that a city
park can cool the air that passes through it by more than $4^{\circ} \mathrm{C}$

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Problem 82

$\because$ BIO Suppose you could convert the 495 $\mathrm{C}$ in the cheeseburger
you ate for lunch into mechanical energy with 100$\%$ efficiency.
(a) How high could you throw a $141-\mathrm{kg}$ baseball with the energy
contained in the cheeseburger? (b) How fast would the ball be
moving at the moment of release?

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Problem 83

$\because$ You turn a crank on a device similar to that shown in Figure $16-12$ and produce a power of 0.16 hp. If the paddles are
immersed in 0.75 $\mathrm{kg}$ of water, for what length of time must
you turn the crank to increase the temperature of the water by
5.5 $\mathrm{C}^{\circ} ?$

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Problem 84

$\because$ Predict/Calculate B1O Heat Transport in the Human Body The core
temperature of the human body is $37.0^{\circ} \mathrm{C},$ and the skin, with a
surface area of 1.40 $\mathrm{m}^{2}$ , has a temperature of $34.0^{\circ} \mathrm{C}$ . (a) Find the
rate of heat transfer out of the body under the following assumptions: (i) The average thickness of tissue between the core and the
skin is $1.20 \mathrm{cm} ;$ (ii) the thermal conductivity of the tissue is that of
water. (b) Without repeating the calculation of part (a), what rate
of heat transfer would you expect if the skin temperature were to
fall to $31.0^{\circ} \mathrm{C}$ ? Explain.

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Problem 85

$\bullet$ The Solar Constant The surface of the Sun has a temperature of
$5500^{\circ} \mathrm{C} .$ (a) Treating the Sun as a perfect blackbody, with an emissivity of $1.0,$ find the power that it radiates into space. The radius of
the Sun is $7.0 \times 10^{8} \mathrm{m},$ and the temperature of space can be taken
to be 3.0 $\mathrm{K}$ . (b) The solar constant is the number of watts of sunlight
power falling on a square meter of the Earth's upper atmosphere.
Use your result from part (a) to calculate the solar constant, given
that the distance from the Sun to the Earth is $1.5 \times 10^{11} \mathrm{m}$ .

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Problem 86

$\cdots \bullet$ Bars of two different metals are bolted together, as shown in
FlGuRE $16-31 .$ Show that the distance $D$ does not change with temperature if the lengths of the two bars have the following ratio:
$L_{\mathrm{A}} / L_{\mathrm{B}}=\alpha_{\mathrm{B}} / \alpha_{\mathrm{A}}$

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Problem 87

$\cdots$ A grandfather clock has a simple brass pendulum of length $L$
One night, the temperature in the house is $25.0^{\circ} \mathrm{C}$ and the period
of the pendulum is 1.00 $\mathrm{s}$ . The clock keeps correct time at this temperature. If the temperature in the house quickly drops to $17.1^{\circ} \mathrm{C}$
just after 10 $\mathrm{P.M.}$ , and stays at that value, what is the actual time
when the clock indicates that it is 10 $\mathrm{A}$ . . the next morning?

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Problem 88

$\cdots$ Predict/Calculate A sheet of aluminum has a circular hole
with a diameter of 10.0 $\mathrm{cm} . \mathrm{A} 9.99$ -cm-long steel rod is placed
inside the hole, along a diameter of the circle, as shown in FIGURE
$16-32 .$ It is desired to change the temperature of this system until
the steel rod just touches both sides of the circle. (a) Should the
temperature of the system be increased or decreased? Explain. (b)
By how much should the temperature be changed?

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Problem 89

$\cdots$ A layer of ice has formed on a small pond. The air just above
the ice is at $-5.4^{\circ} \mathrm{C}$ , the water-ice interface is at $0^{\circ} \mathrm{C},$ and the water
at the bottom of the pond is at $4.0^{\circ} \mathrm{C}$ . If the total depth from the
top of the ice to the bottom of the pond is 1.4 $\mathrm{m},$ how thick is the
layer of ice? Note: The thermal conductivity of ice is 1.6 $\mathrm{W} /\left(\mathrm{m} \cdot \mathrm{C}^{\circ}\right)$
and that of water is 0.60 $\mathrm{W} /\left(\mathrm{m} \cdot \mathrm{C}^{\circ}\right) .$

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Problem 90

$\cdots$ A Double-Paned Window An energy-efficient double-paned window consists of two panes of glass, each with thickness $L_{1}$ and
thermal conductivity $k_{1},$ separated by a layer of air of thickness
$L_{2}$ and thermal conductivity $k_{2} .$ Show that the equilibrium rate of
heat flow through this window per unit area, $A,$ is
$$\frac{Q}{A t}=\frac{\left(T_{2}-T_{1}\right)}{2 L_{1} / k_{1}+L_{2} / k_{2}}$$
In this expression, $T_{1}$ and $T_{2}$ are the temperatures on either side of
the window.

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Problem 91

Cool Medicine In situations in which the brain is deprived of oxygen, such as in a heart attack or traumatic brain injury, irreversible
damage to the brain can happen very quickly. It's been proposed
that cooling the body of a patient after cardiac arrest or traumatic
brain injury may help protect the brain against long-term damage, perhaps by reducing the metabolic rate and thus the oxygen demand of the brain. In order to achieve some protection for the brain without causing other undesired physiological effects such as disruptions to heart rhythms, the goal is to reduce the core body temperature
from its normal value of $37^{\circ} \mathrm{C}$ to between $30^{\circ} \mathrm{C}$ and $35^{\circ} \mathrm{C} .$ Animal experiments have shown some promising results, and human trials are being done. In one trial, 95$\%$ of patients had their core
temperature reduced to $34^{\circ} \mathrm{C}$ in 2.0 hours. To cool the body this
quickly, each patient was placed between two cooled blankets, one above the patient and one below. Water and alcohol were sprayed
on the patient, and any skin that wasn't covered by the top blanket
was exposed to the environment.

$\cdot$ Which method of cooling the patient in this study relies primarily on conduction?
A. Exposing skin to the environment
B. Spraying water and alcohol on the patient
C. Using cooled blankets above and below
D. None of these

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Problem 92

Cool Medicine In situations in which the brain is deprived of oxygen, such as in a heart attack or traumatic brain injury, irreversible
damage to the brain can happen very quickly. It's been proposed
that cooling the body of a patient after cardiac arrest or traumatic
brain injury may help protect the brain against long-term damage, perhaps by reducing the metabolic rate and thus the oxygen demand of the brain. In order to achieve some protection for the brain without causing other undesired physiological effects such as disruptions to heart rhythms, the goal is to reduce the core body temperature
from its normal value of $37^{\circ} \mathrm{C}$ to between $30^{\circ} \mathrm{C}$ and $35^{\circ} \mathrm{C} .$ Animal experiments have shown some promising results, and human trials are being done. In one trial, 95$\%$ of patients had their core
temperature reduced to $34^{\circ} \mathrm{C}$ in 2.0 hours. To cool the body this
quickly, each patient was placed between two cooled blankets, one above the patient and one below. Water and alcohol were sprayed
on the patient, and any skin that wasn't covered by the top blanket
was exposed to the environment.

$. .$ What total amount of heat must be removed to drop the whole
body temperature of a typical 65 -kg patient in this study by $1.0^{\circ} \mathrm{C}$ ?
Because the human body is mostly water, the average specific heat
of the human body is relatively high, 3.5 $\times 10^{3} \mathrm{J} /(\mathrm{kg} \cdot \mathrm{K})$ .
$$\begin{array}{ll}{\text { A. } 3.5 \times 10^{3} \mathrm{J}} & {\text { B. } 2.3 \times 10^{5} \mathrm{J}} \\ {\text { C. } 5.4 \times 10^{6} \mathrm{J}} & {\text { D. } 6.2 \times 10^{7} \mathrm{J}}\end{array}$$

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Problem 93

Cool Medicine In situations in which the brain is deprived of oxygen, such as in a heart attack or traumatic brain injury, irreversible
damage to the brain can happen very quickly. It's been proposed
that cooling the body of a patient after cardiac arrest or traumatic
brain injury may help protect the brain against long-term damage, perhaps by reducing the metabolic rate and thus the oxygen demand of the brain. In order to achieve some protection for the brain without causing other undesired physiological effects such as disruptions to heart rhythms, the goal is to reduce the core body temperature
from its normal value of $37^{\circ} \mathrm{C}$ to between $30^{\circ} \mathrm{C}$ and $35^{\circ} \mathrm{C} .$ Animal experiments have shown some promising results, and human trials are being done. In one trial, 95$\%$ of patients had their core
temperature reduced to $34^{\circ} \mathrm{C}$ in 2.0 hours. To cool the body this
quickly, each patient was placed between two cooled blankets, one above the patient and one below. Water and alcohol were sprayed
on the patient, and any skin that wasn't covered by the top blanket
was exposed to the environment.

$\bullet$ During the cooling of a typical 65 -kg patient in this study, what
is the average rate of heat loss, assuming the whole body's temperature changes by the same amount as the core temperature does?
$$\begin{array}{ll}{\text { A. } 95 \mathrm{W}} & {\text { B. } 200 \mathrm{W}} \\ {\text { C. } 100 \mathrm{W}} & {\text { D. } 8700 \mathrm{W}}\end{array}$$

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Problem 94

Cool Medicine In situations in which the brain is deprived of oxygen, such as in a heart attack or traumatic brain injury, irreversible
damage to the brain can happen very quickly. It's been proposed
that cooling the body of a patient after cardiac arrest or traumatic
brain injury may help protect the brain against long-term damage, perhaps by reducing the metabolic rate and thus the oxygen demand of the brain. In order to achieve some protection for the brain without causing other undesired physiological effects such as disruptions to heart rhythms, the goal is to reduce the core body temperature
from its normal value of $37^{\circ} \mathrm{C}$ to between $30^{\circ} \mathrm{C}$ and $35^{\circ} \mathrm{C} .$ Animal experiments have shown some promising results, and human trials are being done. In one trial, 95$\%$ of patients had their core
temperature reduced to $34^{\circ} \mathrm{C}$ in 2.0 hours. To cool the body this
quickly, each patient was placed between two cooled blankets, one above the patient and one below. Water and alcohol were sprayed
on the patient, and any skin that wasn't covered by the top blanket
was exposed to the environment.

$\because$ For a patient in a typical hospital setting, the most important
mechanism of heat loss is radiation. How does the power radiated
by a patient change when the skin temperature drops from $34^{\circ} \mathrm{C}$
to $33^{\circ} \mathrm{C},$ assuming no other changes?
A. It's now 99$\%$ of what it originally was.
B. It's now 97$\%$ of what it originally was.
C. It's now 89$\%$ of what it originally was.
D. It doesn't change.

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Problem 95

$\because$ REFERRING TO EXAMPLE $16-12$ Suppose the mass of the block is to
be increased enough to make the final temperature of the system
equal to $22.5^{\circ} \mathrm{C}$ . What is the required mass? Everything else in
Example $16-12$ remains the same.

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Problem 96

$. .$ REFERRING TO EXAMPLE $16-12$ Suppose the initial temperature of the
block is to be increased enough to make the final temperature of
the system equal to $22.5^{\circ} \mathrm{C} .$ What is the required initial temperature? Everything else remains the same as in Example $16-12$ .

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Problem 97

$\cdots$ Predict/Calculate REFERRING TO EXAMPLE $16-16$ Suppose the lead
rod is replaced with a second copper rod. (a) Will the heat that
flows in 1.00 s increase, decrease, or stay the same? Explain.
(b) Find the heat that flows in 1.00 s with two copper rods. Everything else remains the same as in Example $16-16 .$

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Problem 98

$\bullet$ Predict/Calculate REFERRING TO EXAMPLE $16-16$ Suppose the temperature of the hot plate is to be changed to give a total heat flow of 25.2 $\mathrm{Jin} 1.00 \mathrm{s}$ . (a) Should the new temperature of the hot plate
be greater than or less than $106^{\circ} \mathrm{C} ?$ Explain. (b) Find the required
temperature of the hot plate. Everything else is the same as in
Example $16-16 .$

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