University Physics with Modern Physics

Hugh D. Young, Roger A. Freeman

Chapter 17

Temperature and Heat - all with Video Answers

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Chapter Questions

Problem 1

Convert the following Celsius temperatures to Fahrenheit: (a) $-62.8^{\circ} \mathrm{C},$ the lowest temperature ever recorded in North America (February $3,1947,$ Snag, Yukon); (b) $56.7^{\circ} \mathrm{C}$ , the highest temperature ever recorded in the United States (July $10,1913$ , Death Valley, California); (c) $31.1^{\circ} \mathrm{C}$ , the world's highest average annual temperature (Lugh Ferrandi, Somalia).

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Theodore Stenmark

Numerade Educator

Problem 2

Find the Celsius temperatures corresponding to (a) a winter night im Seattle $\left(41.0^{\circ} \mathrm{F}\right) ;(\mathrm{b})$ a hot summer day in Palm Springs $\left(107.0^{\circ} \mathrm{F}\right) ;(\mathrm{c})$ a cold winter day in northern Manitoba $\left(-18.0^{\circ} \mathrm{F}\right)$ .

Keshav Singh

Numerade Educator

Problem 3

While vacationing in Italy, you see on local TV one summer morning that temperature will rise from the current $18^{\circ} \mathrm{C}$ to a high of $39^{\circ} \mathrm{C}$ . What is the corresponding increase in the Fahrenheit temperature?

Keshav Singh

Numerade Educator

Problem 4

Two beakers of water, $A$ and $B$ , initially are at the same temperature. The temperature of the water in beaker $A$ is increased $10 F^{\circ},$ and the temperature of the water in beaker $B$ is increased
10 $\mathrm{K}$ . After these temperature changes, which beaker of water has the higher temperature? Explain.

Keshav Singh

Numerade Educator

Problem 5

You put a bottle of soft drink in a refrigerator and leave it until its temperature has dropped 10.0 $\mathrm{K}$ . What is its temperature change in (a) $\mathrm{F}^{\circ}$ and $(\mathrm{b}) \mathrm{C}^{\circ}$ ?

Keshav Singh

Numerade Educator

Problem 6

(a) On January $22,1943,$ the temperature in Spearfish, South Dakota, rose from $-4.0^{\circ} \mathrm{F}$ to $45.0^{\circ} \mathrm{F}$ in just 2 minutes. What was the temperature change in Celsius degrees? (b) The temperature in Browning, Montana, was $44.0^{\circ} \mathrm{F}$ on January $23,1916 .$ The next day the temperature plummeted to $-56^{\circ} \mathrm{C}$ . What was the temperature change in Celsius degrees?

Keshav Singh

Numerade Educator

Problem 7

(a) You feel sick and are told that you have a temperature of $40.2^{\circ} \mathrm{C}$ . What is your temperature in "F? Should you be concerned? (b) The morning weather report in Sydney gives a current
temperature of $12^{\circ} \mathrm{C}$ . What is this temperature in $^{\circ} \mathrm{F} ?$

Keshav Singh

Numerade Educator

Problem 8

(a) Calculate the one temperature at which Fahrenheit and Celsius thermometers agree with each other. (b) Calculate the one temperature at which Fahrenheit and Kelvin thermometers agree with each other.

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Tuan Pham

University of Wisconsin - Madison

Problem 9

Convert the following record-setting temperatures to the Kelvin scale: (a) the lowest temperature recorded in the 48 contiguous states $\left(-70.0^{\circ} \mathrm{F} \text { at Rogers Pass, Montana, on January } 20,\right.$ $1954 ) ;(6)$ Australia's highest temperature $\left(127.0^{\circ} \mathrm{F} \text { at Cloncurry, }\right.$ Queensland, on January $16,1889 ;$ (c) the lowest temperature recorded in the northern hemisphere $\left(-90.0^{\circ} \mathrm{F} \text { at Verkhoyansk, }\right.$ Siberia, in 1892 ).

Keshav Singh

Numerade Educator

Problem 10

Convert the following Kelvin temperatures to the Celsius and Fahrenheit scales: (a) the midday temperature at the surface of the moon $(400 \mathrm{K}) ;(\mathrm{b})$ temperature at the tops of the clouds in the atmosphere of Saturn $(95 \mathrm{K}) ;(\mathrm{c})$ the temperature at the center of the $\operatorname{sun}\left(1.55 \times 10^{7} \mathrm{K}\right) .$

Rachel Wellington

Rachel Wellington

University of Georgia

Problem 11

Why. Liquid nitrogen is a relatively inexpensive material that is often used to perform entertaining low-temperature physics demonstrations. Nitrogen gas liquefies at a temperature of $-346^{\circ} \mathrm{F}$ . Convert this temperature to (a) $^{\circ} \mathrm{C}$ and $(\mathrm{b}) \mathrm{K}$ .

Keshav Singh

Numerade Educator

Problem 12

A gas thermometer registers an absolute pressure corresponding to 325 $\mathrm{mm}$ of mercury when in contact with water at the triple point. What pressure does it read when in contact with water at the normal boiling point?

Keshav Singh

Numerade Educator

Problem 13

The pressure of a gas at the triple point of water is 1.35 atm. If its volume remains unchanged, what will its pressure be at the temperature at which $\mathrm{CO}_{2}$ solidifies?

Averell Hause

Carnegie Mellon University

Problem 14

Like the Kelvin scale, the Rankine scale is an absolute temperature scale: Absolute zero is zero degrees Rankine $\left(0^{\circ} \mathrm{R}\right)$ . However, the units of this scale are the same size as those of the Fahrenheit scale rather than the Celsius scale. What is the numerical value of the triple-point temperature of water on the Rankine scale?

TP

Tuan Pham

University of Wisconsin - Madison

Problem 15

An experimenter using a gas thermometer found the pressure at the triple point of water $\left(0.01^{\circ} \mathrm{C}\right)$ to be $4.80 \times 10^{4} \mathrm{Pa}$ and the pressure at the normal boiling point $\left(100^{\circ} \mathrm{C}\right)$ to be $6.50 \times 10^{4} \mathrm{Pa}$ , (a) Assuming that the pressure varies linearly with temperature, use these two data points to find the Celsius temperature at which the gas pressure would be zero (that is, find the Celsius temperature of absolute zero). (b) Does the gas in this thermometer obey Eq. $(17.4)$ precisely? If that equation were precisely obeyed and the pressure at $100^{\circ} \mathrm{C}$ were $6.50 \times 10^{4} \mathrm{Pa}$ , what pressure would the experimenter have measured at $0.01^{\circ} \mathrm{C} ?$ (As we will learn in Section $18.1, \mathrm{Eq} .(17.4)$ is accurate only for gases at very low density.)

Averell Hause

Carnegie Mellon University

Problem 16

The tallest building in the world, according to some architectural standards, is the Taipei 101 in Taiwan, at a height of 1671 feet. Assume that this height was measured on a cool spring day when the temperature was $15.5^{\circ} \mathrm{C}$ . You could use the building as a sort of giant thermometer on a hot summer day by carefully measuring its height. Suppose you do this and discover that the Taipei
101 is 0.471 foot taller than its official height. What is the temperature, assuming that the building is in thermal equilibrium with the air and that its entire frame is made of steel?

Keshav Singh

Numerade Educator

Problem 17

The Humber Bridge in England has the world's longest sin- gle span, 1410 $\mathrm{m}$ . Calculate the change in length of the steel deck of the span when the temperature increases from $-5.0^{\circ} \mathrm{C}$ to $18.0^{\circ} \mathrm{C}$ .

Keshav Singh

Numerade Educator

Problem 18

Aluminum rivets used in airplane construction are made slightly larger than the rivet holes and cooled
by "dry ice" (solid $\mathrm{CO}_{2} )$ before being driven. If the diameter of a hole is 4.500 $\mathrm{mm}$ , what should be the diameter of a rivet at $23.0^{\circ} \mathrm{C}$ , if its diameter is to equal that of the hole when the rivet is cooled to $-78.0^{\circ} \mathrm{C}$ , the temperature of dry ice? Assume that the expansion coefficient remains constant at the value given in Table $17.1 .$

Villiam Zelinsky

Villiam Zelinsky

Numerade Educator

Problem 19

A U.S. penny has a diameter of 1.9000 $\mathrm{cm}$ at $20.0^{\circ} \mathrm{C} .$ The coin is made of a metal alloy (mostly zinc) for which the coefficient of linear expansion is $2.6 \times 10^{-5} \mathrm{K}^{-1} .$ What would its diameter be on a hot day in Death Valley $\left(48.0^{\circ} \mathrm{C}\right) ?$ On a cold night in the mountains of Grecnland $\left(-53^{\circ} \mathrm{C}\right) ?$

Keshav Singh

Numerade Educator

Problem 20

A geodesic dome constructed with an aluminum framework is a nearly perfect hemisphere; its diameter measures 55.0 $\mathrm{m}$ on a winter day at a temperature of $-15^{\circ} \mathrm{C}$ . How much more interior space does the dome have in the summer, when the temperature is $35^{\circ} \mathrm{C} ?$

TP

Tuan Pham

University of Wisconsin - Madison

Problem 21

A metal rod is 40.125 $\mathrm{cm}$ long at $20.0^{\circ} \mathrm{C}$ and 40.148 $\mathrm{cm}$ long at $45.0^{\circ} \mathrm{C}$ . Calculate the average coefficient of linear expansion of the rod for this temperature range.

Keshav Singh

Numerade Educator

Problem 22

D. A copper cylinder is initially at $20.0^{\circ} \mathrm{C}$ . At what temperature will its volume be 0.150$\%$ larger than it is at $20.0^{\circ} \mathrm{C}$ ?

Keshav Singh

Numerade Educator

Problem 23

The density of water is 999.73 $\mathrm{kg} / \mathrm{m}^{3}$ at a temperature of $10^{\circ} \mathrm{C}$ and 958.38 $\mathrm{kg} / \mathrm{m}^{3}$ at a temperature of $100^{\circ} \mathrm{C} .$ Calculate the average coefficient of volume expansion for water in that range of temperature.

Keshav Singh

Numerade Educator

Problem 24

A steel tank is completely filled with 2.80 $\mathrm{m}^{3}$ of ethanol when both the tank and the ethanol are at a temperature of $32.0^{\circ} \mathrm{C} .$ When the tank and its contents have cooled to $18.0^{\circ} \mathrm{C},$ what additional volume of ethanol can be put into the tank?

Keshav Singh

Numerade Educator

Problem 25

A glass flask whose volume is 1000.00 $\mathrm{cm}^{3}$ at $0.0^{\circ} \mathrm{C}$ is completely filled with mercury at this temperature. When flask and mercury are warmed to $55.0^{\circ} \mathrm{C}, 8.95 \mathrm{cm}^{3}$ of mercury overflow. If the coefficient of volume expansion of mercury is $18.0 \times 10^{-5} \mathrm{K}^{-1}$ . compute the coefficient of volume expansion of the glass.

Averell Hause

Carnegie Mellon University

Problem 26

(a) If an area measured on the surface of a solid body is $A_{0}$ at some initial temperature and then changes by $\Delta A$ when the temperature changes by $\Delta T,$ show that
$$\Delta A=(2 \alpha) A_{0} \Delta T$$
where $\alpha$ is the coefficient of linear expansion. (b) A circular sheet of aluminum is 55.0 $\mathrm{cm}$ in diameter at $15.0^{\circ} \mathrm{C} .$ By how much does the area of one side of the sheet change when the temperature increases to $27.5^{\circ} \mathrm{C} ?$

Keshav Singh

Numerade Educator

Problem 27

A machinist bores a hole of diameter 1.35 $\mathrm{cm}$ in a steel plate at a temperature of $25.0^{\circ} \mathrm{C}$ . What is the cross-sectional area of the bole $(\mathrm{a})$ at $25.0^{\circ} \mathrm{C}$ and $(\mathrm{b})$ when the temperature of the plate is increased to $175^{\circ} \mathrm{C}$ ? Assume that the coefficient of linear expansion remains constant over this temperature range. (Hint: See Exercise 17.26 .

Keshav Singh

Numerade Educator

Problem 28

As a new mechanical engineer for Engines Inc., you have been assigned to design brass pistons to slide inside steel cylinders. The engines in which these pistons will be used will operate between $20.0^{\circ} \mathrm{C}$ and $150.0^{\circ} \mathrm{C}$ . Assume that the coefficients of expansion are constant over this temperature range. (a) If the piston just fits inside the chamber at $20.0^{\circ} \mathrm{C}$ , will the engines be able to run at higher temperatures? Explain. (b) If the cylindrical pistons are 25.000 $\mathrm{cm}$ in diameter at $20.0^{\circ} \mathrm{C}$ , what should be the minimum diameter of the cylinders at that temperature so the pistons will operate at $150.0^{\circ} \mathrm{C} ?$

Brandy Heflin

Numerade Educator

Problem 29

The outer diameter of a glass jar and the inner diameter of its iron lid are both 725 $\mathrm{mm}$ at room temperature $\left(20.0^{\circ} \mathrm{C}\right) .$ What will be the size of the mismatch between the lid and the jar if the lid is briefly held under hot water until its temperature rises to $50.0^{\circ} \mathrm{C},$ without changing the temperature of the glass?

Rachel Wellington

Rachel Wellington

University of Georgia

Problem 30

A brass rod is 185 $\mathrm{cm}$ long and 1.60 $\mathrm{cm}$ in diameter. What force must be applied to each end of the rod to prevent it from contracting when it is cooled from $120.0^{\circ} \mathrm{C}$ to $10.0^{\circ} \mathrm{C} ?$

OC

Olivia Cypull

Numerade Educator

Problem 31

(a) A wire that is 1.50 $\mathrm{m}$ long at $20.0^{\circ} \mathrm{C}$ is found to increase in length by 1.90 $\mathrm{cm}$ when warmed to $420.0^{\circ} \mathrm{C}$ . Compute its average coefficient of linear expansion for this temperature range. (b) The wire is stretched just (zero tension) at $420.0^{\circ} \mathrm{C}$ . Find the stress in the wire if it is cooled to $20.0^{\circ} \mathrm{C}$ without being allowed to contract. Young's modulus for the wire is $20 \times 10^{11} \mathrm{Pa}$ .

TS

Theodore Stenmark

Numerade Educator

Problem 32

Steel train rails are laid in $12.0-\mathrm{m}$ -long segments placed end to end. The rails are laid on a winter day when their temperature is $-2.0^{\circ} \mathrm{C} .$ (a) How much space must be left between adjacent rails if they are just to touch on a summer day when their temperature is $33.0^{\circ} \mathrm{C} ?$ (b) If the rails are originally laid in contact, what is the stress in them on a summer day when their temperature is $33.0^{\circ} \mathrm{C} ?$

Rachel Wellington

Rachel Wellington

University of Georgia

Problem 33

An aluminum tea kettle with mass 1.50 $\mathrm{kg}$ and containing 1.80 $\mathrm{kg}$ of water is placed on a stove. If no heat is lost to the surroundings, how much heat must be added to raise the temperature from $20.0^{\circ} \mathrm{C}$ to $85.0^{\circ} \mathrm{C} ?$

Mark Scythian

Numerade Educator

Problem 34

In an effort to stay awake for an all-night study session, a student makes a cup of coffee by first placing a $200-\mathrm{W}$ electric immersion heater in 0.320 $\mathrm{kg}$ of water. (a) How much heat must be added to the water to raise its temperature from $20.0^{\circ} \mathrm{C}$ to $80.0^{\circ} \mathrm{C}$ ? (b) How much time is required? Assume that all of the heater's power goes into heating the water.

Ajay Singhal

Numerade Educator

Problem 35

You are given a sample of metal and asked to determine its specific heat. You weigh the sample and find that its weight is 28.4 $\mathrm{N}$ . You carefully add $1.25 \times 10^{4} \mathrm{J}$ of heat energy to the sample and find that its temperature rises $18.0 \mathrm{C}^{\circ} .$ What is the sample's
specific heat?

Dading Chen

Numerade Educator

Problem 36

Heat Loss During Breathing. In very cold weather a significant mechanism for heat loss by the human body is energy expended in warming the air taken into the lungs with each breath. (a) On a cold winter day when the temperature is $-20^{\circ} \mathrm{C}$ , what amount of heat is needed to warm to body temperature $\left(37^{\circ} \mathrm{C}\right)$ the 0.50 $\mathrm{L}$ of air exchanged with each breath? Assume that the specific heat of air is 1020 $\mathrm{J} / \mathrm{kg} \cdot \mathrm{K}$ and that 1.0 $\mathrm{L}$ of air has mass $1.3 \times 10^{-3} \mathrm{kg}$ . (b) How much heat is lost per hour if the respiration rate is 20 breaths per minute?

James Nartey

Numerade Educator

Problem 37

While running, a $70-\mathrm{kg}$ student generates thermal energy at a rate of 1200 $\mathrm{W}$ . To maintain a constant body temperature of $37^{\circ} \mathrm{C},$ this energy must be removed by perspiration or other mechanisms. If these mechanisms failed and the heat could not flow out of the student's body, for what amount of time could a student run before irreversible body damage occurred? (Note: Protein structures in the body are irreversibly damaged if body temperature rises to $44^{\circ} \mathrm{C}$ or higher. The specific heat of a typical human body is 3480 $\mathrm{J} / \mathrm{kg} \cdot \mathrm{K}$ , shightly less than that of water. The difference is due to the presence of protein, fat, and minerals, which have lower specific heats.)

Rachel Wellington

Rachel Wellington

University of Georgia

Problem 38

While painting the top of an antenna 225 $\mathrm{m}$ in height, a worker accidentally lets a $1.00-\mathrm{L}$ water bottle fall from his lunchbox. The bottle lands in some bushes at ground level and does not break. If a quantity of heat equal to the magnitude of the change in mechanical energy of the water goes into the water, what is its increase in temperature?

Keshav Singh

Numerade Educator

Problem 39

A crate of fruit with mass 35.0 $\mathrm{kg}$ and specific heat 3650 $\mathrm{J} / \mathrm{kg} \cdot \mathrm{K}$ slides down a ramp inclined at $36.9^{\circ} \mathrm{C}$ below the horizontal. The ramp is 8.00 $\mathrm{m}$ long. (a) If the crate was at rest at the top of the incline and has a speed of 2.50 $\mathrm{m} / \mathrm{s}$ at the bottom, how much work was done on the crate by friction? (b) If an
amount of heat equal to the magnitude of the work done by friction goes into the crate of fruit and the fruit reaches a uniform final temperature, what is its temperature change?

Meghan Miholics

Meghan Miholics

Numerade Educator

Problem 40

A $25,000-\mathrm{kg}$ subway train initially traveling at 15.5 $\mathrm{m} / \mathrm{s}$ slows to a stop in a station and then stays there long enough for its brakes to cool. The station's dimensions are 65.0 $\mathrm{m}$ long by 20.0 $\mathrm{m}$ wide by 12.0 $\mathrm{m}$ high. Assuming all the work done by the brakes in stopping the train is transferred as heat uniformly to all the air in the station, by how much does the air temperature in the station rise? Take the density of the air to be 1.20 $\mathrm{kg} / \mathrm{m}^{3}$ and its specific heat to be 1020 $\mathrm{J} / \mathrm{kg} \cdot \mathrm{K}$ .

Rachel Wellington

Rachel Wellington

University of Georgia

Problem 41

A nail driven into a board increases in temperature. If we assume that 60$\%$ of the kinetic energy delivered by a $1.80-\mathrm{kg}$ hammer with a speed of 7.80 $\mathrm{m} / \mathrm{s}$ is transformed into heat that flows into the nail and does not flow out, what is the temperature increase of an 8.00 - g aluminum nail after it is struck ten times?

Keshav Singh

Numerade Educator

Problem 42

A technician measures the specific heat of an unidentified liquid by immersing an electrical resistor in it. Electrical energy is converted to heat transferred to the liquid for 120 $\mathrm{s}$ at a constant rate of 65.0 $\mathrm{W}$ . The mass of the liquid is 0.780 $\mathrm{kg}$ , and its temperature increases from $18.55^{\circ} \mathrm{C}$ to $22.54^{\circ} \mathrm{C}$ (a) Find the average specific heat of the liquid in this temperature range. Assume that negligible heat is transferred to the container that holds the liquid and that no heat is lost to the surroundings. (b) Suppose that in this experiment heat transfer from the liquid to the container or surroundings cannot be ignored. Is the result calculated in part (a) an overestimate or an underestimate of the average specific heat? Explain.

TP

Tuan Pham

University of Wisconsin - Madison

Problem 43

You add 8950 J of heat to 3.00 mol of iron. (a) What is the temperature increase of the iron? (b) If this same amount of heat is added to 3.00 $\mathrm{kg}$ of iron, what is the iron's temperature increase? (c) Explain the difference in your results for parts (a) and (b).

Dading Chen

Numerade Educator

Problem 44

As a physicist, you put heat into a 500.0 -g solid sample at the rate of 10.0 $\mathrm{kJ} / \mathrm{mim}$ , while recording its temperature as a function of time. You plot your data and obtain the graph shown in Fig. 17.30 . (a) What is the latent heat of fusion for this solid? (b) What are the specific heats of the liquid and solid states of the material?

TS

Theodore Stenmark

Numerade Educator

Problem 45

A 500.0 -g chunk of an unknown metal, which has been in boiling water for several minutes, is quickly dropped into an insulating Styrofoam beaker containing 1.00 $\mathrm{kg}$ of water at room temperature $\left(20.0^{\circ} \mathrm{C}\right)$ . After waiting and gently stirring for 5.00 minutes, you observe that the water's temperature has reached a constant value of $22.0^{\circ} \mathrm{C}$ (a) Assuming that the Styrofoam absorbs a negligibly small amount of heat and that no heat was lost to the surroundings, what is the specific heat of the metal? (b) Which is more useful for storing thermal energy: this metal or an equal weight of water? Explain. (c) What if the heat absorbed by the Styrofoam actually is not negligible. How would the specific heat you calculated in part (a) be in error? Would it be too large, too small, or still correct? Explain.

TS

Theodore Stenmark

Numerade Educator

Problem 46

Before going in for his annual physical, a $70.0-\mathrm{kg}$ man whose body temperature is $37.0^{\circ} \mathrm{C}$ consumes an entire $0.355-\mathrm{L}$ can of a soft drink (mostly water) at $12.0^{\circ} \mathrm{C}$ . (a) What will his body temperature be after equilibrium is attained? Ignore any heating bythe man's metabolism. The specific heat of the man's body is 3480 $\mathrm{J} / \mathrm{kg} \cdot \mathrm{K}$ . (b) Is the change in his body temperature great enough to be measured by a medical themometer?

James Nartey

Numerade Educator

Problem 47

In the situation described in Exercise $17.46,$ the man's metabolism will eventually return the temperature of his body (and of the soft drink that he consumed) to $37.0^{\circ} \mathrm{C}$ . If his body releases energy at a rate of $7.00 \times 10^{3} \mathrm{kJ} / \mathrm{day}$ (the basal metabolic rate, or BMR), how long does this take? Assume that all of the released energy goes into raising the temperature.

Averell Hause

Carnegie Mellon University

Problem 48

An ice-cube tray of negligible mass contains 0.350 $\mathrm{kg}$ of water at $18.0^{\circ} \mathrm{C}$ . How much heat must be removed to cool the water to $0.00^{\circ} \mathrm{C}$ and freeze it? Express your answer in joules, calories, and Btu.

Keshav Singh

Numerade Educator

Problem 49

How much heat is required to convert 12.0 $\mathrm{g}$ of ice at $-10.0^{\circ} \mathrm{C}$ to steam at $100.0^{\circ} \mathrm{C} 2$ Express your answer in joules, calories, and Btu.

James Nartey

Numerade Educator

Problem 50

An open container holds 0.550 $\mathrm{kg}$ of ice at $-15.0^{\circ} \mathrm{C} .$ The mass of the container can be ignored. Heat is supplied to the container at the constant rate of 800.0 $\mathrm{J} / \mathrm{min}$ for 500.0 $\mathrm{min}$ (a) After how many minutes does the ice start to melt? ( 6 ) After how many minutes, from the time when the heating is first started, does the temperature begin to rise above $0.0^{\circ} \mathrm{C} 2$ (c) Plot a curve showing the temperature as a function of the elapsed time.

Rachel Wellington

Rachel Wellington

University of Georgia

Problem 51

The capacity of commercial air conditioners is sometimes expressed in "tons," the number of tons of ice $(1 \text { ton }=2000 \mathrm{lb})$ that can be frozen from water at $0^{\circ} \mathrm{C}$ in 24 $\mathrm{h}$ by the unit. Express the capacity of a 2 -ton air conditioner in Btu/h and in watts.

Keshav Singh

Numerade Educator

Problem 52

Steam Burns Versus Water Rurns. What is the amount of heat input to your skin when it receives the hear released (a) by 25.0 $\mathrm{g}$ of steam initially at $100.0^{\circ} \mathrm{C}$ , when it is cooled to skin temperature $\left(34.0^{\circ} \mathrm{C}\right) ?\left(\text { b) By } 25.0 \text { g of water initially at } 100.0^{\circ} \mathrm{C}\right.$ when it is cooled to $34.0^{\circ} \mathrm{C} ?$ (c) What docs this tell you about the relative severity of steam and hot water burns?

Keshav Singh

Numerade Educator

Problem 53

What must the initial speed of a lead bullet be at a temperature of $25.0^{\circ} \mathrm{C}$ so that the heat developed when it is brought to rest will be just sufficient to melt it? Assume that all the initial mechanical energy of the bullet is converted to heat and that no heat flows from the bullet to its surroundings. (Typical rifles have muzzle speeds that exceed the speed of sound in air, which is 347 $\mathrm{m} / \mathrm{s}$ at $25.0^{\circ} \mathrm{C} . )$

James Nartey

Numerade Educator

Problem 54

Evaporation of sweat is an important mechanism for temperature control in some warm-blooded animals. (a) What mass of water must evaporate from the skin of a $70.0-\mathrm{kg}$ man to cool his body 1.00 $\mathrm{C}^{\circ}$ ? The heat of vaporization of water at body temperature $\left(37^{\circ} \mathrm{C}\right)$ is $2.42 \times 10^{6} \mathrm{J} / \mathrm{kg} \cdot \mathrm{K}$ . The specific heat of a typical human body is 3480 $\mathrm{J} / \mathrm{kg} \cdot \mathrm{K}$ (see Exercise $17.37 ) .$ (b) What volume of water must the man drink to replenish the evaporated water? Compare to the volume of a soft-drink can $\left(355 \mathrm{cm}^{3}\right) .$

James Nartey

Numerade Educator

Problem 55

"The Ship of the Desert" Camels require very little water because they are able to tolerate relatively large changes in their body temperature. While humans keep their body temperatures constant to within one or two Celsius degrees, a dehydrated camel permits its body temperature to drop to $34.0^{\circ} \mathrm{C}$ overnight and rise to $40.0^{\circ} \mathrm{C}$ during the day. To see how effective this mechanism is for saving water, calculate how many liters of water a 400 $\mathrm{kg}$ camel would have to drink if it attempted to keep its body temperature at a constant $34.0^{\circ} \mathrm{C}$ by evaporation of sweat during the day $\left(12 \text { hours) instead of letting it rise to } 40.0^{\circ} \mathrm{C} \text { . (Note: The }\right.$ specific heat of a camel or other mammal is about the same as that of a typical human, 3480 $\mathrm{J} / \mathrm{kg} \cdot \mathrm{K}$ . The heat of vaporization of water at $34^{\circ} \mathrm{C}$ is $2.42 \times 10^{6} \mathrm{J} / \mathrm{kg} .$ )

Dading Chen

Numerade Educator

Problem 56

An asteroid with a diameter of 10 $\mathrm{km}$ and a mass of $2.60 \times 10^{15} \mathrm{kg}$ impacts the earth at a speed of 32.0 $\mathrm{km} / \mathrm{s}$ , landing in the Pacific Ocean. If 1.00$\%$ of the asteroid's kinetic energy goes to boiling the ocean water (assume an initial water temperature of $10.0^{\circ} \mathrm{C} )$ , what mass of water will be boiled away by the collision?
(For comparison, the mass of water contained in Lake Superior is about $2 \times 10^{15} \mathrm{kg}$ .)

Dading Chen

Numerade Educator

Problem 57

A refrigerator door is opened and room-temperature air $\left(20.0^{\circ} \mathrm{C}\right)$ fills the $1.50-\mathrm{m}^{3}$ compartment. A $10.0-\mathrm{kg}$ turkey, also at room temperature, is placed in the refrigerator and the door is closed. The density of air is 1.20 $\mathrm{kg} / \mathrm{m}^{3}$ and its specific heat is 1020 $\mathrm{J} / \mathrm{kg} \cdot \mathrm{K}$ . Assume the specific heat of a turkey, like that of a human, is 3480 $\mathrm{J} / \mathrm{kg} \cdot \mathrm{K}$ . How much heat must the refrigerator remove from its compartment to bring the air and the turkey to thermal equilibrium at a temperature of $5.00^{\circ} \mathrm{C}$ ? Assume no heat exchange with the surrounding environment.

Keshav Singh

Numerade Educator

Problem 58

A laboratory technician drops a $0.0850-\mathrm{kg}$ sample of unknown material, at a temperature of $100.0^{\circ} \mathrm{C}$ , into a calorimeter. The calorimeter can, initially at $19.0^{\circ} \mathrm{C}$ , is made of 0.150 $\mathrm{kg}$ of copper and contains 0.200 $\mathrm{kg}$ of water. The final temperature of the calorimeter can and contents is $26.1^{\circ} \mathrm{C}$ . Compute the specific heat capacity of the sample.

Keshav Singh

Numerade Educator

Problem 59

An insulated beaker with negligible mass contains 0.250 $\mathrm{kg}$ of water at a temperature of $75.0^{\circ} \mathrm{C}$ . How many kilograms of ice at a temperature of $-20.0^{\circ} \mathrm{C}$ must be dropped into the water to make the final temperature of the system $30.0^{\circ} \mathrm{C} ?$

Keshav Singh

Numerade Educator

Problem 60

A glass vial containing a 16.0 -g sample of an enzyme is cooled in an ice bath. The bath contains water and 0.120 $\mathrm{kg}$ of ice. The sample has specific heat 2250 $\mathrm{J} / \mathrm{kg} \cdot \mathrm{K}$ ; the glass vial has mass 6.00 $\mathrm{g}$ and specific heat 2800 $\mathrm{J} / \mathrm{kg} \cdot \mathrm{K}$ . How much ice melts in cooling the enzyme sample from room temperature $\left(19.5^{\circ} \mathrm{C}\right)$ to the temperature of the ice bath?

Keshav Singh

Numerade Educator

Problem 61

A 4.00 tag silver ingot is taken from a furnace, where its temperature is $750.0^{\circ} \mathrm{C},$ and placed on a large block of ice at $0.0^{\circ} \mathrm{C} .$ Assuming that all the heat given up by the silver is used to melt the ice, how much ice is melted?

Banhishikha Sinha

Banhishikha Sinha

Numerade Educator

Problem 62

A copper calorimeter can with mass 0.100 kg contains 0.160 $\mathrm{kg}$ of water and 0.0180 $\mathrm{kg}$ of ice in thermal equilibrium at atmospheric pressure. If 0.750 $\mathrm{kg}$ of lead at a temperature of $255^{\circ} \mathrm{C}$ is dropped into the calorimeter can, what is the final temperature? Assume that no heat is lost to the surroundings.

Rashmi Sinha

Numerade Educator

Problem 63

A vessel whose walls are thermally insulated contains 2.40 $\mathrm{kg}$ of water and 0.450 $\mathrm{kg}$ of ice, all at a temperature of $0.0^{\circ} \mathrm{C}$ The outlet of a tube leading from a boiler in which water is boiling at atmospheric pressure is inserted into the water. How many grams of steam must condense inside the vessel (also at atmospheric pressure) to raise the temperature of the system to $28.0^{\circ} \mathrm{C}$ ? You can ignore the heat transferred to the container.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 64

Use Eq. $(17.21)$ to show that the SI units of thermal conductivity are $\mathrm{W} / \mathrm{m} \cdot \mathrm{K}$

Keshav Singh

Numerade Educator

Problem 65

Suppose that the rod in Fig. 17.23 $\mathrm{a}$ is made of copper, is 45.0 $\mathrm{cm}$ long, and has a cross-sectional area of 1.25 $\mathrm{cm}^{2}$ . Let $T_{\mathrm{H}}=100.0^{\circ} \mathrm{C}$ and $T_{\mathrm{C}}=0.0^{\circ} \mathrm{C} .$ (a) What is the final steady-state temperature gradient along the rod? (b) What is the heat current in the rod in the final steady state? (c) What is the final steady-state temperature at a point in the rod 12.0 $\mathrm{cm}$ from its left end?

James Nartey

Numerade Educator

Problem 66

One end of an insulated metal rod is maintained at $100.0^{\circ} \mathrm{C}$ and the other end is maintained at $0.00^{\circ} \mathrm{C}$ by an ice-water mixture. The rod is 60.0 $\mathrm{cm}$ long and has a cross-sectional area of 1.25 $\mathrm{cm}^{2}$ . The heat conducted by the rod melts 8.50 $\mathrm{g}$ of ic. 0 min. Find the thermal conductivity $k$ of the metal.

Zulfiqar Ali

Numerade Educator

Problem 67

A carpenter builds an exterior house wall with a layer of wood 3.0 $\mathrm{cm}$ thick on the outside and a layer of Styrofoam insulation 2.2 $\mathrm{cm}$ thick on the inside wall surface. The wood has
$k=0.080 \mathrm{W} / \mathrm{m} \cdot \mathrm{K}$ , and the Styrofoam has $k=0.010 \mathrm{W} / \mathrm{m} \cdot \mathrm{K}$ . The interior surface temperature is $19.0^{\circ} \mathrm{C}$ , and the exterior surface temperature is $-10.0^{\circ} \mathrm{C}$ (a) What is the temperature at the plane where the wood meets the Styrofoam? (b) What is the rate of heat flow per square meter through this wall?

James Nartey

Numerade Educator

Problem 68

An electric kitchen range has a total wall area of 1.40 $\mathrm{m}^{2}$ and is insulated with a layer of fiberglass 4.00 $\mathrm{cm}$ thick. The inside surface of the fiberglass has a temperature of $175^{\circ} \mathrm{C}$ , and its outside surface is at $35.0^{\circ} \mathrm{C}$ . The fiberglass has a thermal conductivity of 0.040 $\mathrm{W} / \mathrm{m} \cdot \mathrm{K}$ (a) What is the heat current through the insulation, assuming it may be treated as a flat slab with an area of 1.40 $\mathrm{m}^{2} ?(\mathrm{b})$ What electric-power input to the heating element is required to maintain this temperature?

James Nartey

Numerade Educator

Problem 69

The celling of a room has an area of 125 $\mathrm{ft}^{2}$ . The ceiling is insulated to an $R$ value of 30 (in units of $\mathrm{ft}^{2} \cdot \mathrm{F}^{\circ} \cdot \mathrm{h} / \mathrm{Btu} )$ . The surface in the room is maintained at $69^{\circ} \mathrm{F}$ , and the surface in the attic has a temperature of $35^{\circ} \mathrm{F}$ . What is the heat flow through the ceiling into the attic in 5.0 $\mathrm{h} ?$ Express your answer in Btu and in joules.

Keshav Singh

Numerade Educator

Problem 70

A long rod, insulated to prevent heat loss along its sides, is in perfect thermal contact with boiling water (at atmospheric pressure) at one end and with an ice-water mixture at the other (Fig. 17.31$)$ . The rod consists of a $1.00-\mathrm{m}$ section of copper (one end in boiling water) joined end to end to a length $L_{2}$ of steel (one end in the ice-water mixture). Both sections of the rod have crosssectional areas of 4.00 $\mathrm{cm}^{2}$ . The temperature of the copper-steel junction is 65.0" Cafter a steady state has been set up. (a) How much heat per second flows from the boiling water to the ice-water mixture? (b) What is the length $L_{2}$ of the steel section?

Keshav Singh

Numerade Educator

Problem 71

A pot with a steel bottom 8.50 $\mathrm{mm}$ thick rests on a hot stove.The area of the bottom of the pot is 0.150 $\mathrm{m}^{2}$ . The water inside the pot is at $100.0^{\circ} \mathrm{C},$ and 0.390 $\mathrm{kg}$ are evaporated every 3.00 $\mathrm{min}$ . Find the temperature of the lower surface of the pot, which is in contact with the stove.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 72

You are asked to design a cylindrical steel rod 50.0 $\mathrm{cm}$ long, with a circular cross section, that will conduct 150.0 $\mathrm{J} / \mathrm{s}$ from a furnace at $400.0^{\circ} \mathrm{C}$ to a container of boiling water under 1 atmosphere. What must the rod's diameter be?

Dading Chen

Numerade Educator

Problem 73

A picture window has dimensions of $1.40 \mathrm{m} \times 2.50 \mathrm{m}$ and is made of glass 5.20 $\mathrm{mm}$ thick. On a winter day, the outside temperature is $-20.0^{\circ} \mathrm{C}$ , while the inside temperature is a comfortable $19.5^{\circ} \mathrm{C}$ (a) At what rate is heat being lost through the window by conduction? (b) At what rate would heat be lost through the window if you covered it with a 0.750 -mm-thick layer of paper (thermal conductivity 0.0500)?

Keshav Singh

Numerade Educator

Problem 74

What is the rate of energy radiation per unit area of a black-body at a temperature of (a) 273 $\mathrm{K}$ and $(\mathrm{b}) 2730 \mathrm{K} ?$

Keshav Singh

Numerade Educator

Problem 75

What is the net rate of heat loss by radiation in Example 17.16 (Section 17.7$)$ if the temperature of the surroundings is $5.0^{\circ} \mathrm{C} ?$

Keshav Singh

Numerade Educator

Problem 76

The emissivity of tungsten is $0.350 .$ A tungsten sphere with radius 1.50 $\mathrm{cm}$ is suspended within a large evacuated enclosure whose walls are at 290.0 $\mathrm{K}$ . What power input is required to maintain the sphere at a temperature of 3000.0 $\mathrm{K}$ if heat conduction along the supports is neglected?

Keshav Singh

Numerade Educator

Problem 77

Size of a Light-Bulb Filament. The operating temperature of a tungsten filament in an incandescent light bulb is 2450 $\mathrm{K}$ , and its emissivity is 0.350 . Find the surface area of the filament of a $150-\mathrm{W}$ bulb if all the electrical energy consumed by the bulb is radiated by the filament as electromagnetic waves. (Only a fraction of the radiation appears as visible light)

AL

Alan Larson

Numerade Educator

Problem 78

The Sizes of Stars. The hot glowing surfaces of stars emit energy in the form of electromagnetic radiation. It is a good approximation to assume $e=1$ for these surfaces. Find the radii of the following stars (assumed to be spherical): (a) Rigel, the bright blue star in the constellation Orion, which radiates energy at a rate of $2.7 \times 10^{32} \mathrm{W}$ and has surface temperature $11,000 \mathrm{K} ;$ (b) Procyon $\mathrm{B}$ (visible only using a telescope), which radiates energy at a rate of $2.1 \times 10^{23} \mathrm{W}$ and has surface temperature $10,000 \mathrm{K}$ . (c) Compare your answers to the radius of the earth, the radius of the sun, and the distance between the earth and the sun. (Rigel is
an example of a supergiant star, and Procyon $\mathrm{B}$ is an example of a white dwarf star.)

Keshav Singh

Numerade Educator

Problem 79

You propose a new temperature scale with temperatures given in $^{\circ} \mathrm{M}$ . You define $0.0^{\circ} \mathrm{M}$ to be the normal melting point of mercury and $100.0^{\circ}$ to be the normal boiling point of mercury. (a) What is the normal boiling point of water in "M? (b) A temperature change of 10.0 $\mathrm{M}^{\circ}$ corresponds to how many $\mathrm{c}^{\circ} ?$

Averell Hause

Carnegie Mellon University

Problem 80

Suppose that a steel hoop could be constructed to fit just around the earth's equator at a temperature of $20.0^{\circ} \mathrm{C}$ . What would be the thickness of space between the hoop and the earth if the temperature of the hoop were increased by 0.500 $\mathrm{C}^{\circ} ?$

Zulfiqar Ali

Numerade Educator

Problem 81

At an absolute temperature $T_{0},$ a cube has sides of length $L_{0}$ and has density $\rho_{0}$ . The cube is made of a material with coefficient of volume expansion $\beta$ (a) Show that if the temperature increases to $T_{0}+\Delta T$ , the density of the cube becomes approximately
$$\rho \approx \rho_{0}(1-\beta \Delta T)$$
(Hint I Lise the expression $(1+x)^{n} \approx 1+n x$ , valid for $|x| \ll 1 . )$ Explain why this approximate result is valid only if $\Delta T$ is much less than $1 / \beta,$ and explain why you would expect this to be the case in most situations. (b) A copper cube has sides of length 1.25 $\mathrm{cm}$ at $20.0^{\circ} \mathrm{C}$ . Find the change in its volume and density when its temperature is increased to $70.0^{\circ} \mathrm{C}$ .

Keshav Singh

Numerade Educator

Problem 82

A $250-\mathrm{kg}$ weight is hanging from the ceiling by a thin copper wire. In its fundamental mode, this wire vibrates at the frequency of concert $\mathrm{A}(440 \mathrm{Hz}) .$ You then increase the temperature of the wire by $40 \mathrm{C}^{\circ} .$ (a) By how much will the fundamental frequency change? Will it increase or decrease? (b) By what percentage will the speed of a wave on the wire change? (c) By what percentage will the wavelength of the fundamental standing wave change? Will it increase or decrease?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 83

You are making pesto for your pasta and have a cylindrical measuring cup 10.0 cm high made of ordinary glass $\left[\beta=2.7 \times 10^{-5}\left(\mathrm{C}^{\circ}\right)^{-1}\right]$ that is filled with olive oil
$\left[\beta=6.8 \times 10^{-4}\left(\mathrm{C}^{\circ}\right)^{-1}\right]$ to a height of 1.00 $\mathrm{mm}$ below the top of the cup. Initially, the cup and oil are at room temperature
$\left(22.0^{\circ} \mathrm{C}\right) .$ You get a phone call and forget about the olive oil, which you inadvertently leave on the hot stove. The cup and oil heat up slowly and have a common temperature. At what temperature will the olive oil start to spill out of the cup?

James Nartey

Numerade Educator

Problem 84

Use Fig. 17.12 to find the approximate coefficient of volume expansion of water at $2.0^{\circ} \mathrm{C}$ and at $8.0^{\circ} \mathrm{C}$.

Keshav Singh

Numerade Educator

Problem 85

A Foucault pendulum consists of a brass sphere with a diameter of 35.0 $\mathrm{cm}$ suspended from a steel cable 10.5 $\mathrm{m}$ long (both measurements made at $20.0^{\circ} \mathrm{C} )$ . Due to a design oversight, the swinging sphere clears the floor by a distance of only 2.00 $\mathrm{mm}$ when the temperature is $20.0^{\circ} \mathrm{C}$ . At what temperature will the sphere begin to brush the floor?

Rachel Wellington

Rachel Wellington

University of Georgia

Problem 86

You pour 108 $\mathrm{cm}^{3}$ of ethanol, at a temperature of $-10.0^{\circ} \mathrm{C}$ into a graduated cylinder initially at $20.0^{\circ} \mathrm{C}$ , filling it to the very top. The cylinder is made of glass with a specific heat of 840 $\mathrm{J} / \mathrm{kg} \cdot \mathrm{K}$ and a coefficient of volume expansion of $1.2 \times 10^{-5} \mathrm{K}^{-1}$ ; its mass is 0.110 $\mathrm{kg}$ . The mass of the ethanol is 0.0873 $\mathrm{kg}$ . (a) What will be the final temperature of the cthanol, once thermal equilibrium is reached? (b) How much ethanol will overflow the cylinder before thermal equilibrium is reached?

Prashant Bana

Numerade Educator

Problem 87

A metal rod that is 30.0 $\mathrm{cm}$ long expands by 0.0650 $\mathrm{cm}$ when its temperature is raised from $0.0^{\circ} \mathrm{C}$ to $100.0^{\circ} \mathrm{C}$ . A rod of a different metal and of the same length expands by 0.0350 $\mathrm{cm}$ for the same rise in temperature. A third rod, also 30.0 $\mathrm{cm}$ long, is made up of pieces of each of the above metals placed end to end and expands 0.0580 $\mathrm{cm}$ between $0.0^{\circ} \mathrm{C}$ and $100.0^{\circ} \mathrm{C}$ . Find the
length of each portion of the composite rod.

Averell Hause

Carnegie Mellon University

Problem 88

On a cool $\left(4,0^{\circ} \mathrm{C}\right)$ Saturfay moming, a pilot fills the fuel tanks of her Pitts $S-2 C$ (a two-seat aerobatic airplane) to their full capacity of 106.0 L. Before flying on Sunday morning, when the temperature is again $4.0^{\circ} \mathrm{C}$ , she checks the fuel level and finds only 103.4 $\mathrm{L}$ of gasoline in the tanks. She realizes that it was hot on Saturday afternoon, and that thermal expansion of the gasoline caused the missing fuel to empty out of the tank's vent. (a) What was the maximum temperature (in "C) reached by the fuel and the tank on Saturday aftemoon? The coefficient of volume expansion of gasoline is $9.5 \times 10^{-4} \mathrm{K}^{-1}$ , and the tank is made of aluminum. (b) In order to have the maximum amount of fuel available for flight, when should the pilot have filled the fuel tanks?

Keshav Singh

Numerade Educator

Problem 89

(a) Equation $(17.12)$ gives the stress required to keep the length of a rod constant as its temperature changes. Show that if the length is permitted to change by an amount $\Delta L$ when its temperature changes by $\Delta T,$ the stress is equal to
$$\frac{F}{A}=Y\left(\frac{\Delta L}{L_{0}}-\alpha \Delta T\right)$$
where $F$ is the tension on the rod, $L_{0}$ is the original length of the rod, $A$ its cross-sectional area, $\alpha$ its coefficient of linear expansion, and $Y$ its Young's modulus. (b) A heavy brass bar has projections at its ends, as in Fig. 17.32 . Figure 17.32 Problem 17.89 . Two fine steel wires, fastened between the projections, are just taut (zero tension) when the whole system is at $20^{\circ} \mathrm{C}$ . What is the tensile stress in the steel wires when the temperature of the system is raised to $140^{\circ} \mathrm{C}$ ? Make any simplifying assumptions you think are justified, but state what they are.

Narayan Hari

Numerade Educator

Problem 90

A steel rod 0.350 $\mathrm{m}$ long and an aluminum rod 0.250 $\mathrm{m}$ long, both with the same diameter, are placed end to end between rigid supports with no initial stress in the rods. The temperature of $f$ the rods is now raised by 60.0 $\mathrm{C}^{\circ}$ . What is the stress in each rod? (Hint: The length of the combined rod remains the same, but the lengths of the individual rods do not. See Problem $17.89 .$ )

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 91

A steel ring with a 2.5000 -in. inside diameter at $20.0^{\circ} \mathrm{C}$ is to be warmed and slipped over a brass shaft with a 2.5020 -in. outside diameter at $20.0^{\circ} \mathrm{C}$ (a) To what temperature should the ring be warmed? (b) If the ring and the shart together are cooled by some means such as liquid air, at what temperature will the ring just slip off the shaft?

Carlos Henrique De Lima

Carlos Henrique De Lima

Numerade Educator

Problem 92

Bulk Stress Due to a Temperature Increase. (a) Prove that, if an object under pressure has its temperature raised but is not allowed to expand, the increase in pressure is
$$\Delta p=B \beta \Delta T$$
where the bulk modulus $B$ and the average coefficient of volume expansion $\beta$ are both assumed positive and constant. (b) What pressure is necessary to prevent a steel block from expanding when its temperature is increased from $20.0^{\circ} \mathrm{C}$ to $35.0^{\circ} \mathrm{C} ?$

Keshav Singh

Numerade Educator

Problem 93

A liquid is enclosed in a metal cylinder that is provided with a piston of the same metal. The system is originally at a pressure of 1.00 atm $\left(1.013 \times 10^{5} \mathrm{Pa}\right)$ and at a temperature of $30.0^{\circ} \mathrm{C} .$ The piston is forced down until the pressure on the liquid is increased by 50.0 atm, and then clamped in this position. Find the new temperature at which the pressure of the liquid is again 1.00 atm. Assume that the cylinder is sufficiently strong so that its volume is not altered by changes in pressure, but only by changes in temperature. Use the result derived in Problem 17.92 . (Hints See Section 11.4.)
Compressibility of liquid: $k=8.50 \times 10^{-10} \mathrm{Pa}^{-1}$
Coefficient of volume expansion of liquid: $\beta=4.80 \times 10^{-4} \mathrm{K}^{-1}$
Coefficient of volume expansion of metal $\beta=3.90 \times 10^{-5} \mathrm{K}^{-1}$

Lainey Roebuck

Numerade Educator

Problem 94

You cool a 100.0 - $\mathrm{g}$ shg of red-hot iron (temperature $745^{\circ} \mathrm{C} )$ by dropping it into an insulated cup of negligible mass containing 75.0 $\mathrm{g}$ of water at $20.0^{\circ} \mathrm{C}$ . Assuming no heat exchange with the surroundings, a) what is the final temperature of the water and b) what is the final mass of the iron and the remaining water?

Supratim Pal

Numerade Educator

Problem 95

Spacecraft Reentry. A spacecraft made of aluminum circles the earth at a speed of 7700 $\mathrm{m} / \mathrm{s}$ . (a) Find the ratio of its kinetic energy to the energy required to raise its temperature from
$0^{\circ} \mathrm{C}$ to $600^{\circ} \mathrm{C}$ . (The melting point of aluminum is $660^{\circ} \mathrm{C}$ . Assume a constant specific heat of $910 \mathrm{J} / \mathrm{kg} \cdot \mathrm{K} .$ (b) Discuss the bearing of your answer on the problem of the reentry of a manned space vehicle into the earth's atmosphere.

Keshav Singh

Numerade Educator

Problem 96

A capstan is a rotating drum or cylinder over which a rope or cond slides in order to provide a great amplification of the rope's tension while keeping both ends free (Fig. 17.33). Since the added tension in the rope is due to friction, the capstan generates thermal energy. (a) If the difference in tension between the two ends of the rope is 520.0 $\mathrm{N}$ and the capstan has a diameter of 10.0 $\mathrm{cm}$ and turns once in 0.900 $\mathrm{s}$ , find the rate at which thermal energy is generated. Why does the number of turns not matter? (b) If the capstan is made of iron and has mass 6.00 $\mathrm{kg}$ , at what rate does its temperature rise? Assume that the temperature in the capstan is uniform and that all the thermal energy generated flows into it.

Keshav Singh

Numerade Educator

Problem 97

Debye's $T^{3}$ Law. At very low temperatures the molar heat capacity of rock salt varies with temperature according to Debye's $T^{3}$ law:
$$C=k \frac{T^{3}}{\Theta^{3}}$$
where $k=1940 \mathrm{J} / \mathrm{mol} \cdot \mathrm{K}$ and $\theta=281 \mathrm{K}$ (a) How much heat is required to raise the temperature of 1.50 $\mathrm{mol}$ of rock salt from 10.0 $\mathrm{K}$ to 40.0 $\mathrm{K} ?$ (Hint: Use Eg. $(17.18)$ in the form $d Q=n C d T$ and integrate.) (b) What is the average molar heat capacity in this range? (c) What is the true molar heat capacity at 40.0 $\mathrm{K} ?$

Jackson Henningfield

Jackson Henningfield

Numerade Educator

Problem 98

A person of mass 70.0 $\mathrm{kg}$ is siting in the bathtub. The bathtub is 190.0 $\mathrm{cm}$ by 80.0 $\mathrm{cm}$ ; before the person got in, the water was 10.0 $\mathrm{cm}$ deep. The water is at a temperature of $37.0^{\circ} \mathrm{C}$ . Suppose that the water were to cool down spontaneously to form ice at $0.0^{\circ} \mathrm{C},$ and that all the energy released was used to launch the hapless bather vertically into the air. How high would the bather go? (As you will see in Chapter 20 , this event is allowed by energy conservation but is prohibited by the second law of thermodynamics.)

Keshav Singh

Numerade Educator

Problem 99

Hot Air in a Physics Lecture. (a) Atypical student listening attentively to a physics lecture has a heat output of 100 $\mathrm{W}$ . How much heat energy does a class of 90 physics students release into a lecture hall over the course of a 50 -min lecture? (b) Assume that all the heat energy in part (a) is transferred to the 3200 $\mathrm{m}^{3}$ of air in the room. The air has specific heat capacity 1020 $\mathrm{J} / \mathrm{kg} \cdot \mathrm{K}$ and density 1.20 $\mathrm{kg} / \mathrm{m}^{3}$ . If none of the heat escapes and the air conditioning system is off, how much will the temperature of the air in the room rise during the 50 -min lecture? (c) If the class is taking an exam, the heat output per student rises to 280 $\mathrm{W}$ . What is the temperature rise during 50 $\mathrm{min}$ in this case?

Rachel Wellington

Rachel Wellington

University of Georgia

Problem 100

The molar heat capacity of a certain substance varies with temperature according to the empirical equation
$$C=29.5 \mathrm{J} / \mathrm{mol} \cdot \mathrm{K}+\left(8.20 \times 10^{-3} \mathrm{J} / \mathrm{mol} \cdot \mathrm{K}^{2}\right) \mathrm{T}$$
How much heat is necessary to change the temperature of 3.00 mol of this substance from $27^{\circ} \mathrm{C}$ to $227^{\circ} \mathrm{C}$ ? (Hint: Use Eq. (17.18) in the form $d Q=n C d T$ and integrate. $)$

Keshav Singh

Numerade Educator

Problem 101

For your cabin in the wilderness, you decide to build a primitive refrigerator out of Styrofoam, planning to keep the interior cool with a block of ice that has an initial mass of 24.0 $\mathrm{kg}$ . The box has dimensions of $0.500 \mathrm{m} \times 0.800 \mathrm{m} \times 0.500 \mathrm{m}$ . Water from melting ice collects in the bottom of the box. Suppose the ice block is at $0.00^{\circ} \mathrm{C}$ and the outside temperature is $21.0^{\circ} \mathrm{C}$ . If the top of the empty box is never opened and you want the interior of the box to remain at $5.00^{\circ} \mathrm{C}$ for exactly one week, until all the ice
melts, what must be the thickness of the Styrofoam?

Keshav Singh

Numerade Educator

Problem 102

Hot Water Versus Steam Heating, In a houschold hotwater heating system, water is delivered to the radiators at $70.0^{\circ} \mathrm{C}$ $\left(158.0^{\circ} \mathrm{F}\right)$ and leaves at $28.0^{\circ} \mathrm{C}\left(82.4^{\circ} \mathrm{F}\right) .$ The system is to be replaced by a steam system in which steam at atmospheric pressure condenses in the radiators and the condensed steam leaves the
radiators at $35.0^{\circ} \mathrm{C}\left(95.0^{\circ} \mathrm{F}\right) .$ How many kilograms of steam will supply the same heat as was supplied by 1.00 $\mathrm{kg}$ of hot water in the first system?

Keshav Singh

Numerade Educator

Problem 103

A copper calorimeter can with mass 0.446 kg contains 0.0950 $\mathrm{kg}$ of ice. The system is initially at $0.0^{\circ} \mathrm{C}$ (a) If 0.0350 $\mathrm{kg}$ of steam at $100.0^{\circ} \mathrm{C}$ and 1.00 atm pressure is added to the can, what is the final temperature of the calorimeter can and its contents? (b) At the final temperature, how many kilograms are there of ice, how many of liquid water, and how many of steam?

Averell Hause

Carnegie Mellon University

Problem 104

A Styrofoam bucket of negligible mass contains 1.75 $\mathrm{kg}$ of water and 0.450 $\mathrm{kg}$ of ice. More ice, from a refrigerator at $-15.0^{\circ} \mathrm{C},$ is added to the mixture in the bucket, and when thermal equilibrium has been reached, the total mass of ice in the bucket is 0.778 $\mathrm{kg}$ . Assuming no hear exchange with the surroundings, what mass of ice was added?

Keshav Singh

Numerade Educator

Problem 105

In a container of negligible mass, 0.0400 $\mathrm{kg}$ of steam at $100^{\circ} \mathrm{C}$ and atmospheric pressure is added to 0.200 $\mathrm{kg}$ of water at $50.0^{\circ} \mathrm{C} .$ (a) If no heat is lost to the surroundings, what is the final temperature of the system?(b) At the final temperature, how many kilograms are there of steam and how many of liquid water?

Susan Hallstrom

Susan Hallstrom

Numerade Educator

Problem 106

A tube leads from a $0.150-\mathrm{kg}$ calorimeter to a flask in which water is boiling under atmospheric pressure. The calorimeter has specific heat capacity 420 $\mathrm{J} / \mathrm{kg} \cdot \mathrm{K}$ , and it originally contains 0.340 $\mathrm{kg}$ of water at $15.0^{\circ} \mathrm{C}$ . Steam is allowed to condense in the calorimeter at atmospheric pressure until the temperature of the calorimeter and contents reaches $71.0^{\circ} \mathrm{C}$ , at which point the total mass of the calorimeter and its contents is found to be 0.525 $\mathrm{kg}$ . Compute the heat of vaporization of water from these data.

Keshav Singh

Numerade Educator

Problem 107

A worker pours 1.250 kg of molten lead at a temperature of $365.0^{\circ} \mathrm{C}$ into 0.5000 $\mathrm{kg}$ of water at a temperature of $75.00^{\circ} \mathrm{C}$ in an insulated bucket of negligible mass. Assuming no heat loss to the surroundings, calculate the mass of lead and water remaining in the bucket when the materials have reached thermal equilibrium.

Laszlo Zalavari

Laszlo Zalavari

Numerade Educator

Problem 108

One experimental method of measuring an insulating material's thermal conductivity is to construct a box of the material and measure the power input to an electric heater inside the box that maintains the interior at a measured temperature above the outside surface. Suppose that in such an apparatus a power input of 180 $\mathrm{W}$ is required to keep the interior surface of the box 65.0 $\mathrm{C}^{\circ}$ (about 120 $\mathrm{F}^{\circ}$ ) above the temperature of the outer surface. The total area of the box is 2.18 $\mathrm{m}^{2}$ , and the wall thickness is 3.90 $\mathrm{cm}$ . Find the thermal conductivity of the material in SI units.

Dading Chen

Numerade Educator

Problem 109

Effect of a Window in a Door. A carpenter builds a solid wood door with dimensions $2.00 \mathrm{m} \times 0.95 \mathrm{m} \times 5.0 \mathrm{cm} .$ Its thermal conductivity is $k=0.120 \mathrm{W} / \mathrm{m} \cdot \mathrm{K}$ . The air films on the inner and outer surfaces of the door have the same combined thermal resistance as an additional $1.8-\mathrm{cm}$ thickness of solid wood. The inside air temperature is $20.0^{\circ} \mathrm{C}$ , and the outside air temperature is $-8.0^{\circ} \mathrm{C} .$ (a) What is the rate of heat flow through the door? (b) By what factor is the heat flow increased if a window 0.500 $\mathrm{m}$ on a side is inserted in the door? The glass is 0.450 $\mathrm{cm}$ thick, and the glass has a thermal conductivity of 0.80 $\mathrm{W} / \mathrm{m} \cdot \mathrm{K}$ . The air films on the two sides of the glass have a total thermal resistance that is the same as an additional 12.0 $\mathrm{cm}$ of glass.

Jackson Henningfield

Jackson Henningfield

Numerade Educator

Problem 110

A wood ceiling with thermal resistance $R_{1}$ is covered with a layer of insulation with thermal resistance $R_{2} .$ Prove that the effective thermal resistance of the combination is $R=R_{1}+R_{2}$ .

Keshav Singh

Numerade Educator

Problem 111

Compute the ratio of the rate of heat loss through a singlepane window with area 0.15 $\mathrm{m}^{2}$ to that for a double-pane window with the same area. The glass of a single pane is 4.2 $\mathrm{mm}$ thick, and the air space between the two panes of the double-pane window is 7.0 $\mathrm{mm}$ thick. The glass has thermal conductivity 0.80 $\mathrm{W} / \mathrm{m} \cdot \mathrm{K}$ . The air films on the room and outdoor surfaces of either window have a combined thermal resistance of 0.15 $\mathrm{m}^{2} \cdot \mathrm{K} / \mathrm{W}$ .

Narayan Hari

Numerade Educator

Problem 112

Rods of copper, brass, and steel are welded together to form a Y-shaped figure. The cross-sectional area of each rod is $2.00 \mathrm{cm}^{2} .$ The free end of the copper rod is maintained at $100.0^{\circ} \mathrm{C}$ , and the free ends of the brass and steel rods at $0.0^{\circ} \mathrm{C}$ . Assume there is no heat loss from the surfaces of the rods. The lengths of the rods are: copper, $13.0 \mathrm{cm} ;$ brass, $18.0 \mathrm{cm} ;$ steel, $24.0 \mathrm{cm} .$ (a) What is the temperature of the junction point? (b) What is the heat current in each of the three rods?

Keshav Singh

Numerade Educator

Problem 113

Time Needed for a Lake to Freeze Over. (a) When the air temperature is below $0^{\circ} \mathrm{C}$ , the water at the surface of a lake freezes to form an ice sheet. Why doesn't freezing occur throughout the entire volume of the lake? (b) Show that the thickness of the ice sheet formed on the surface of a lake is proportional to the square root of the time if the heat of fusion of the water freezing on the underside of the ice sheet is conducted through the sheet. (c) Assuming that the upper surface of the ice sheet is at $-10^{\circ} \mathrm{C}$ and the bottom surface is at $0^{\circ} \mathrm{C}$ , calculate the time it will take to form an ice sheet 25 $\mathrm{cm}$ thick. (d) If the lake in part ( $\mathrm{c} )$ is uniformly 40 $\mathrm{m}$ deep, how long would it take to freeze all the water in the lake? Is this likely to occur?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 114

Arod is initially at a uniform temperature of $0^{\circ} \mathrm{C}$ throughout. One end is kept at $0^{\circ} \mathrm{C}$ , and the other is brought into contact with a steam bath at $100^{\circ} \mathrm{C}$ . The surface of the rod is insulated so that heat can flow only lengthwise along the rod. The cross-sectional area of the rod is $2.50 \mathrm{cm}^{2},$ its length is 120 $\mathrm{cm}$ , its thermal conductivity is $380 \mathrm{W} / \mathrm{m} \cdot \mathrm{K},$ its density is $1.00 \times 10^{4} \mathrm{kg} / \mathrm{m}^{3},$ and its specific heat is 520 $\mathrm{J} / \mathrm{kg} \cdot \mathrm{K}$ . Consider a short cylindrical element of the rod 1.00 $\mathrm{cm}$ in length. (a) If the temperature gradient at the cooler end of this element is 140 $\mathrm{C}^{\circ} / \mathrm{m}$ , how many joules of heat energy flow across this end per second? (b) If the average temperature of the element is increasing at the rate of 0.250 $\mathrm{C} \%$ /s, what is the temperature gradient at the other end of the element?

Narayan Hari

Numerade Educator

Problem 115

A rustic cabin has a floor area of $3.50 \mathrm{m} \times 3.00 \mathrm{m} .$ Its walls, which are 2.50 $\mathrm{m}$ tall, are made of wood (thermal conductivity 0.0600 $\mathrm{W} / \mathrm{m} \cdot \mathrm{K} ) 1.80 \mathrm{cm}$ thick and are further insulated with 1.50 $\mathrm{cm}$ of a synthetic material. When the outside temperature is $2.00^{\circ} \mathrm{C},$ it is found necessary to heat the room at a rate of 1.25 $\mathrm{kW}$ to maintain its temperature at $19.0^{\circ} \mathrm{C}$ . Calculate the thermal conductivity of the insulating material. Neglect the heat lost through the ceiling and floor. Assume the inner and outer surfaces of the wall have the same temperature as the air inside and outside the cabin.

Vishal Gupta

Numerade Educator

Problem 116

The rate at which radiant energy from the sun reaches the earth's upper atmosphere is about 1.50 $\mathrm{kW} / \mathrm{m}^{2}$ . The distance from the earth to the sun is $1.50 \times 10^{11} \mathrm{m},$ and the radius of the sun is $6.96 \times 10^{8} \mathrm{m}$ (a) What is the rate of radiation of energy per unit area from the sun's surface? (b) If the sun radiates as an ideal black-body, what is the temperature of its surface?

Keshav Singh

Numerade Educator

Problem 117

A Thermos for Liquid Helium. A physicist uses a cylindrical metal can 0.250 $\mathrm{m}$ high and 0.090 $\mathrm{m}$ in diameter to store liquid helium at 4.22 $\mathrm{K}$ ; at that temperature the heat of vaporization of helium is $2.09 \times 10^{4} \mathrm{J} / \mathrm{kg}$ . Completely surrounding the metal can are walls maintained at the temperature of liquid nitrogen, 77.3 $\mathrm{K}$ , with vacuum between the can and the surrounding walls. How much helium is lost per hour? The emissivity of the metal can is 0.200 . The only heat transfer between the metal can and the surrounding walls is by radiation.

Dading Chen

Numerade Educator

Problem 118

Thermal Expansion of an Ideal Gas. (a) The pressure $p$ , volume $V,$ number of moles $n,$ and Kelvin temperature $T$ of an ideal gas are related by the equation $p V=n R T$ , where $R$ is a constant. Prove that the coefficient of volume expansion for an ideal gas is equal to the reciprocal of the Kelvin temperature if the expansion occurs at constant pressure. (b) Compare the coeffcients of volume expansion of copper and air at temperature of $20^{\circ} \mathrm{C}$ . Assume that air may be treated as an ideal gas and that the pressure remains constant.

Keshav Singh

Numerade Educator

Problem 119

An engineer is developing an electric water heater to provide a continuous supply of hot water. One trial design is shown in Fig. 17.34 . Water is flowing at the rate of 0.500 $\mathrm{kg} / \mathrm{min}$ , the inlet thermometer registers $18.0^{\circ} \mathrm{C}$ , the voltmeter reads 120 $\mathrm{V}$ , and the ammeter reads 15.0 $\mathrm{A}$ [corresponding to a power input of $(120 \mathrm{V}) \times(15.0 \mathrm{A})=1800 \mathrm{W} \mathrm{J}$ (a) When a steady state is finally reached, what is the reading of the outlet thermometer? (b) Why is it unnecessary to take into account the heat capacity $m c$ of the apparatus itself?

Keshav Singh

Numerade Educator

Problem 120

Food Intake of a Hamster. The energy output of an animal engaged in an activity is called the basal metabolic rate (BMR) and is a measure of the conversion of food energy into other forms of energy. A simple calorimeter to measure the BMR consists of an insulated box with a thermometer to measure the temperature of the air. The air has density 1.20 $\mathrm{kg} / \mathrm{m}^{3}$ and specific heat 1020 $\mathrm{J} / \mathrm{kg} \cdot \mathrm{K}$ . A 50.0 -g hamster is placed in a calorimeter that contains 0.0500 $\mathrm{m}^{3}$ of air at room temperature. (a) When the hamster is running in a wheel, the temperature of the air in the calorimeter rises 1.60 $\mathrm{C}^{\circ}$ per hour. How much heat does the running hamster generate in an hour? Assume that all this heat goes into the air in the calorimeter. You can ignore the heat that goes into the walls of the box and into the thermometer, and assume that no heat is lost to the surroundings. (b) Assuming that the hamster converts seed into heat with an efficiency of 10$\%$ and that hamster seed has a food energy value of 24 $\mathrm{J} / \mathrm{g}$ , how many grams of seed must the hamster eat per hour to supply this energy?

Keshav Singh

Numerade Educator

Problem 121

The icecaps of Greenland and Antarctica contain about 1.75$\%$ of the total water (by mass) on the earth's surface; the occens contain about $97.5 \%,$ and the other 0.75$\%$ is mainly groundwater. Suppose the icecaps, currently at an average temperature of about $-30^{\circ} \mathrm{C},$ somehow slid into the ocean and melted. What would be the resulting temperature decrease of the ocean? Assume
that the average temperature of ocean water is currently $5.00^{\circ} \mathrm{C}$ .

Rachel Wellington

Rachel Wellington

University of Georgia

Problem 122

(a) A spherical shell has inner and outer radii $a$ and $b$ , respectively, and the temperatures at the inner and outer surfaces are $T_{2}$ and $T_{1}$ . The thermal conductivity of the material of which the shell is made is $k$ . Derive an equation for the total heat current through the shell. (b) Derive an equation for the temperature variation within the shell in part (a); that is, calculate $T$ as a function of $r$ , the distance from the center of the shell. (c) A hollow cylinder has length $L,$ inner radius $a$ , and outer radius $b$ , and the temperatures at the inner and outer surfaces are $T_{2}$ and $T_{1}$ . (The cylinder could represent an insulated hot-water pipe, for example.) The thermal conductivity of the material of which the cylinder is made is $k$ . Derive an equation for the total heat current through the walls of the cylinder. (d) For the cylinder of part (c), derive an equation for the temperature variation inside the cylinder walls. (e) For the spherical shell of part (a) and the hollow cylinder of part (c), show that the equation for the total heat current in each case reduces to Eq. $(17.21)$ for linear heat flow when the shell or cylinder is very thin.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 123

A steam pipe with a radius of $2.00 \mathrm{cm},$ carrying steam at $140^{\circ} \mathrm{C} .$ is surrounded by a cylindrical jacket with inner and outer radii 2.00 $\mathrm{cm}$ and 4.00 $\mathrm{cm}$ and made of a type of cork with thermal conductivity $4.00 \times 10^{-2} \mathrm{W} / \mathrm{m} \cdot \mathrm{K}$ . This in turn is surrounded by a cylindrical jacket made of a brand of Styrofoam with thermal conductivity 1.00 $\times 10^{-2} \mathrm{W} / \mathrm{m} \cdot \mathrm{K}$ and having inner and outer radii 4.00 $\mathrm{cm}$ and 6.00 $\mathrm{cm}$ (Fig. 17.35). The outer surface of the Styrofoam is in contact with air at $15^{\circ} \mathrm{C}$ . Assume that this outer surface has a temperature of $15^{\circ} \mathrm{C}$ . (a) What is the temperature at a radius of $4.00 \mathrm{cm},$ where the two insulating layers meet? (b) What is the total rate of transfer of heat out of a 2.00 -m length of pipe? (Hint: Use the expression derived in part (c) of Challenge Problem 17.122 .

Lainey Roebuck

Numerade Educator

Problem 124

Suppose that both ends of the rod in Fig. 17.23 are kept at a temperature of $0^{\circ} \mathrm{C},$ and that the initial temperature distribution along the rod is given by $T=\left(100^{\circ} \mathrm{C}\right) \sin \pi x / L,$ where $x$ is measured from the left end of the rod. Let the rod be copper, with length
$L=0.100 \mathrm{m}$ and cross-sectional area $1.00 \mathrm{cm}^{2} .$ (a) Show the initial temperature distribution in a diagram. (b) What is the final temperature distribution after a very long time has elapsed? (c) Sketch curves that you think would represent the temperature distribution at intermediate times. (d) What is the initial temperature gradient at the ends of the rod?(e) What is the initial heat current from the ends of the rod into the bodies making contact with its ends? ( $\mathrm{f}$ ) What is the initial hear current at the center of the rod? Explain. What is the heat current at this point at any later time? (g) What is the value of the thermal diffusivity $k / \rho c$ for copper, and in what unit is it expressed? (Here $k$ is the thermal conductivity, $\rho=8.9 \times 10^{3} \mathrm{kg} / \mathrm{m}^{3}$ is the density, and $c$ is the specific heat, (h) What is the initial time rate of change of temperature at the center of the rod? (i) How much time would be required for the center of the rod to reach its final temperature if the temperature continued to decrease at this rate? (This time is called the relaxation time of the rod.) () From the graphs in part (c), would you expect the magnitude of the rate of temperature change at the midpoint to remain constant, increase, or decrease as a function of time? ( $\mathrm{k} )$ What is the initial rate of change of temperature at a point in the rod 2.5 $\mathrm{cm}$ from its left end?

Narayan Hari

Numerade Educator

Problem 125

Temperature Change in a Clock. A pendulum clock is designed to tick off one second on each side-to-side swing of the pendulum (two ticks per complete period). (a) Will a pendulum clock gain time in hot weather and lose it in cold, or the reverse? Explain your reasoning. (b) A particular pendulum clock keeps correct time at $20.0^{\circ} \mathrm{C}$ . The pendulum shaft is steel, and its mass can be ignored compared with that of the bob. What is the fractional change in the length of the shaft when it is cooled to $10.0^{\circ} \mathrm{C} ?$ (c) How many seconds per day will the clock gain or lose at $10.0^{\circ} \mathrm{C} ?$ (d) How closely must the temperature be controlled if the clock is not to gain or lose more than 1.00 $\mathrm{s}$ a day? Does the answer depend on the period of the pendulum?

Keshav Singh

Numerade Educator

Problem 126

One end of a solid cylindrical copper rod 0.200 $\mathrm{m}$ long is maintained at a temperature of 20.00 $\mathrm{K}$ . The other end is blackencl and exposed to thermal radiation from surrounding walls at 500.0 $\mathrm{K}$ . The sides of the rod are insulated, so no energy is lost or gained except at the ends of the rod. When equilibrium is reached, what is the temperature of the blackened end? (Hint: Since copper is a very good conductor of heat at low temperature, with $k=1670 \mathrm{W} / \mathrm{m} \cdot \mathrm{K}$ at 20 $\mathrm{K}$ , the temperature of the blackened end is only slightly higher than $20.00 \mathrm{K} .$ )

Keshav Singh

Numerade Educator

Problem 127

A Walk in the Sun. Consider a poor lost soul waking at 5 $\mathrm{km} / \mathrm{h}$ on a hot day in the desert, wearing only a bathing suit. This person's skin temperature tends to rise due to four mechanisms: (i) energy is generated by metabolic reactions in the body at a rate of 280 $\mathrm{W}$ , and almost all of this energy is converted to heat that flows to the skin; (ii) heat is delivered to the skin by convection from the outside air at a rate equal to $k^{\prime} A_{\mathrm{skin}}$ $\left(T_{\text { air }}-T_{\text { skin }}\right),$ where $k^{\prime}$ is 54 $\mathrm{J} / \mathrm{h} \cdot \mathrm{C}^{\circ} \cdot \mathrm{m}^{2}$ the exposed skin area $A_{\mathrm{skin}}$ is $1.5 \mathrm{m}^{2},$ the air temperature $T_{\text { air }}$ is $47^{\circ} \mathrm{C}$ the exposed skin area $A_{\mathrm{skin}}$ is $36^{\circ} \mathrm{C}$ (iii) the skin absorbs radiant energy from the sun at a rate of 1400 $\mathrm{W} / \mathrm{m}^{2}$ , (iv) the skin absorbs radiant energy from the environment, which has temperature $47^{\circ} \mathrm{C} .$ (a) Calculate the net rate (in wats) at which the person's skin is heated by all four of these mechanisms. Assume that the emissivity of the skin is $e=1$ and that the skin temperature is initially $36^{\circ} \mathrm{C} .$ Which mechanism is the most important? (b) At what rate (in $L / h )$ must perspiration evaporate from this person's skin to maintain a constant skin temperature? (The heat of vaporization of water at $36^{\circ} \mathrm{C}$ is $2.42 \times 10^{6} \mathrm{J} / \mathrm{kg} . )(\mathrm{c})$ Suppose instead the person is protected by light-colored clothing $(e \approx 0)$ so that the exposed skin area is only $0.45 \mathrm{m}^{2} .$ What rate of perspiration is required now? Discuss the usefulness of the traditional clothing worn by desert peoples.

Matthew Miranda

Matthew Miranda

Numerade Educator