396 C HAPTER 12 Th e rma l Prope rt ies of Matte r

SUMMARY The goal of Chapter 12 has been to use the atomic model of matter to explain and explore many macroscopic phenomena associated with heat. temperature. and the properties of matter.

GENERAL PRINCIPLES

Atomic Model Atomic Model of a Gas Ideal-Gas Law We model matter as being made of simple basic parti cles. The relationship of these particles (0 each other defines the phase.

Gas Sol id

o • Macroscopic properties of gases can be explained in tenns of the atomic model of the gas. The speed of the particles is related to the temperature:

_ )3kBT "mlS -

III

The atomic model explains thermal expan­sion, specific heat, and heat transfer.

IMPORTANT CONCEPTS

Effects of heat transfer

The colli sions of particles with each other and wi th the walls of the container determine the pressure.

A system that is heated can either change temperature or change phase.

The specific heat c of a materi al is the heat required to rai se I kg by I K.

The heat of transformation is the energy necessary to change the phase of 1 kg of a substance. Heat is added (0 change a solid to a liquid or a liquid to a gas; heat is removed to reverse these changes.

Q = Mc lH

{ ± MLr (mel tlfreeze)

Q= ± M Lv (boi IIcondense)

The molar specific heat of a gas depends on (he process. {

For a constant-volume process: Q = /lev tl T For a constant-pressure process: Q = nCp dT

m The ideal gas law relates the pressure. volume, and temperatu re in a sample of gas. We can express the law in terms of the number of atoms or the number of moles in the sample:

I

p V = nRT [[J For a gas process in a sealed container,

Pi Vi pfVr T; Tr

Mechanisms of heat transfer

An object can transfer heat to other objects or to its environment:

Conduction is the transfer of heat by direct physical contact.

Convection is the transfer of heat by the motion of a fluid .

Radiation is the transfer of heat by electromagnetic ..rIJ'­waves.

..

Q - = cCTAT4 /;.1

A pV (pressure-volume) diagram is a useful means of looking at a process involving a gas.

A constant-volume process has no change in vo lume.

An isobaric process happens at a constant pressure.

p'LLP

tf ~P ; f P I

1', i

V V V v; VI

APPLICATIONS

Thermal expansion Objects experi ence an increase in volume and an increase in length when their temperature changes:

An isothermal process happens at a constant temperature .

p

p,

Isothenn

/ p,

An adiabatic process involves no transfer of heat; the temperature changes.

P

~. P, L_~ ____ ~-v

V, v,

The work done by a gas is the area under the graph.

p

W = area '----1'-----'1-- v

Calorimetry When two or more systems interact thermally, they come to a common final temperature determined by

The number of moles is

M (in grams) " =

Q~, ~ Q, + Q, + Q, + ... ~ 0 M mol

r,:;;t For homewo~k assjg~ed on MasteringPhysics. go to ~ www.mastertngphys lcs.com

Problem difficulty is labeled as I (straightforward) to nm (challenging),

QUESTIONS

Conceptual Questions

I. Which has morc mass, a mole of Ne gas or a mole of N2 gas? 2. If YO LI launch a projectile upward with a high enough speed. its

kinetic energy is sufficient to allow it to escape the earth 's gravity-i t will go up and nO( come back down. Given enough time, hydrogen and helium gas aloms in the earth 's atmosphere wi ll escape, so these e lements are nOI present in our almos· phere. Explain why hydrogen and helium atoms have the neces­

sary speed to escape bUl why other e lements. such as oxygen and nitrogen. do not.

3. You may have not iced thaI latex hel ium balloons tend to shrink rather quick ly; a ba lloon filled with air lasts a lot longer. Ba l­loons shrink because gas diffuses out of them. The rate of diffu­sion is faster for smaller particles and for panictes of higher speed. Diffusion is also faster when there is a large difference in concentration between two sides of a membrane. Given these facts, ex plain why an air-filled balloon lasts longer than a helium balloon.

4. If you buy a scaled bag of potato chips in Mi ami and dri ve with it to Denver, where the atmospheric pressure is lower, you will find that the bag gets very ··puffy. " Explain why.

5. If yo u doub le the typical speed of the molecules in a gas, by what factor does the pressure change? Give a simple explana­ti on why the pressure changes by thi s factor.

6. Two gases have the same number of molecules pe r cubic meter (NIV) and the same rms speed. The molecules of gas 2 are more mass ive than the molecules of gas I. a. Do the two gases have the same pressure? If not , which is

larger? b. Do the two gases have the same temperature? If not, which

is larger? 7. a. Which contains more particles, a mole of he li um gas or a

mole of oxygen gas? Explain. b. Which contains more particles, a gram of he lium gas or a

gram of oxygen gas? Explain. 8. You have 100 g of aluminum and 100 g of lead.

a. Which contains a larger num ber of moles? Explain . b. Which contains more atoms? Explain .

9. Suppose you could suddenly increase the speed of every atom in a gas by a factoro f 2. u. Docs the thermal energy of the gas change? If so, by what

factor? I f not, why not? b. Does the molar speci fi c heat change? If so, by what factor?

If not, why not? 10. A gas cylinder con tains 1.0 mol of he lium at a temperature of

20°C. A second identical cyli nder conta ins 1.0 mol of neon at 20°C. The he li um atoms arc moving wi th a higher average speed, but the gas pressure in the two conta iners is the same. Eltp lain how this is poss ible.

II . A gas is in a sealed container. By what factor does the gas pres­sure change if

Questions 397

Problems labeled !NT integrate significant material from ear lier

chapters; BK) are of biological or medical interest.

a. The volume is doubled and the temperature is tripled? b. The volume is halved and the temperature is tripled?

12. A gas is in a sea led container. By what factor does the gas tem­perature change if a. The volume is doubled and the pressure is tripled? b. The volume is halved and the pressure is tripled?

13. What is the max imum amount of work that a gas can do during a constant-volume process?

14. You need to precise ly measure the dimensions of a large wood panel fo r a construction projec t. Your metal tape measure was left ou tside for hours in the sun on a hot summer day, and now the tape is so hot it's painfu l to pick up. How wi ll your measure­ments differ from those taken by your coworker, whose tape stayed in the shade? Explain.

15. Your car's radiator is made of sleel and is filled with water. You fill the rad iator to the very top with cold water, then drive off withou t remembe ri ng to replace the cap. As the water and the stee l rad iator heat up, wi ll the leve l of water drop or wi ll it ri se and overflow? Expla in.

16. Materials A and B have equal densities, but A has a larger spec ific heat than B. You have 100 g cubes of each material. Cube A, ini­tiallyat O°C, is placed in good thermal contact with cube B, in i­tiall y at 200°C. The cubes are inside a well-insulated conta iner where they don ' t interact with their surroundings. Is their fin al temperature greater than. less than , or equal 10 100°C? Ex plain.

17. Two contai ners hold equal masses of nitrogen gas at equal tem­peratures. YOLI supply 10 J of heat to container A wh ile not allow­ing its volume to change , and you supply 10 J of heat to con tainer B while not a llowing its pressure to change. Afterward, is tem­perature TA greater than, less than, or equal to To? Explain.

18. YOLI need to raise the temperature of a gas by 10°C. To use the smallest amount of heat ene rgy, should you heal the gas at con­stant pressure or at constant vo lume? Eltp lain .

19. A sample of ideal gas is in a cylinder with a movable piston. 600 J of heat is added to the gas in an isothermal process. As the gas expands, pushing against the piston, how much work does it do?

20. A student is healing chocolate in a pan on the stove. He uses a cooking thermometer to measure the temperature of the choco­late and sees it varies as shown in Figure QI2.20. Describe what is happening to the chocolate in each of the three portions of the graph.

The temp- then ~ t aylo . and then CrJ.tu rc fi ni l cOlllotant fur increases increil!tCs ... a whi le .

. ,~( ,

FIGURE Q12.20

398 CHAPTER 12 Thermal Properties of Matter

21. If you bake a cake at high elevation, where atmospheric pres­sure is lower than at sea level , you will need to adjust the recipe. You will need to cook the cake for a longer time, and you will need to add less baking powder. (Baking powder is a leavening age nt. As it heats, it releases gas bubbles that cause the cake to rise.) Explain why those adjustments are necessary.

22. The spec ific heat of aluminum is higher than that of iron. I kg blocks of iron and aluminum are heated to 10OOC, and each is then dropped into its own 1 L beaker of 20°C water. Which beaker will end up with the warmer water? Explain.

23. A student is asked to ske tch a p

pV diagram for a gas that goes through a cycle consist ing of (a) an isobaric expansion. (b) a constant- volume reduction in temperature, and (c) an isother­mal process that returns the gas to its initial state. The student draws the diagram shown in Figure Q 12.23. What, if any-

L---------_______ v

thing, is wrong with the stu- FIGURE 012.23

dent's diagram? 24. If you have two spoons of the same size, one silver and one

stainless stee l, there is a quick test to tell which is which. Hold the end of a spoon in each hand , then lower them both into a cup of very hot water. One spoon will feel hot first. Is that the s ilver spoon or the stainless steel spoon? Explain.

25. If you li ve somewhere with cold , clear nights, you may have noticed some mornings when there was frost on open patches of ground but not under trees. This is because the ground under trees does not get as cold as open ground. Explain how tree cover keeps the ground under trees wanner.

Multiple-Choice Questions

26. I A tin is inflated to a gauge pressure of 35 psi. The abso lute pressure in the tire is A. Less than 35 psi. B. Equal to 35 psi. e. Greater than 35 psi.

27. I The number of atoms in a con tainer is increased by a factor of2 while the temperature is held constant. The pressure A. Decreases by a factor of 4. B. Decreases by a factor of 2. e. Stays the same. D. Increases by a factorof2. E. Increases by a factor of 4.

28. II A gas is compressed by an isothermal process that decreases its volume by a factor of2.ln thi s process , the pressure A. Does not change. B. Increases by a factor of less than 2. e. Increases by a factorof2. D. Increases by a factor of more than 2.

29. III A gas is compressed by an adiabatic process that decreases its vo lume by a factor of 2. In this process, the pressure A. Does not change. B. Increases by a factor of less than 2. e. Increases by a factor of 2. D. Increases by a factor of more than 2.

30. I Suppose you do a calorimetry experiment to measure the specific heat of a penny. You take a number of pennies, measure the ir mass, heat them to a known temperature, and then drop them into a container of water at a known temperature. You then deduce the spec ific heat of a penny by measuring the tempera­ture change of the water. Unfortunately. yo u didn ' t realize that you dropped one penny on the floor while transferring them to the water. This will A. Cause you to underestimate the spec ific heat. B. Cause you to overestimate the spec ific heat. C. Not affect your calculation of spec ific heat.

3 1. I Acup of water is heated with a heating co il that delivers looW of heal. In one minute, the temperature of the water ri ses by 20"e. What is the mass of the water? A.72g B.140g C. 720 g D. 1.4 kg

32. I Three identical beakers each hold 1000 g of water at 20"C. 100 g of liquid water at O"C is added to the first beaker, 100 g of ice at OOC is added to the second beaker, and the third beaker gets 100 g of aluminum at O"e. The conten ts of wh ich container end up at the lowest final temperature? A. The first beaker. B. The second beaker. C. The third beaker. D. All end up at the same temperature.

33. 1111 100 g of ice at O°C and 100 g of steam at 100°C interact ther­mally in a well-insulated con tainer. The final state of the system is A. An ice-water mixture at O°e. B. Water at a temperature between O"C and 50"C. e. Water at 50°C. D. Water at a temperature between 50"C and I OO"C. E. A water-steam mixture at loo°e.

34. I Suppose the 600 W of radiation emitted in a microwave oven is absorbed by 250 g of water in a very lightweight cup. Approx­imate ly how long will it take to heat the water from 200C to 80"C? A. 50s B. 100 s C. 150 s D. 200 s

35. II 40,000 J of heat is added to 1.0 kg of ice at - IOOe. How much ice melts?

36. BID

A. 0.012 kg C. 0.12 kg II Steam at 100°C causes 100°C. This is because

B. 0.057 kg D. 1.0 kg

worse burns than liquid water at

A. The steam is hOlier than the water. B. Heat is transferred to the skin as steam condenses. C. Steam has a higher specific heat than water. D. Evaporation of liquid water on the skin causes cool ing.

PROBLEMS

Section 12.1 The Atomic Model of Matter

I. 1 Whic h contains the most moles: 10 g of hydrogen gas, 100 g of carbon, or 50 g of lead?

2. 1111 How many grams of water (H20 ) have the same num ber of oxygen atoms as 1.0 mol of oxygen gas?

3. 1111 How many ato ms of hydrogen are in 100 g of hydrogen per­ox ide (H 20 z)?

4. II How Jllany cubic millimeters (mm3) are in I L? 5. 11 A box is 200 em wide, 40 em deep, and 3.0 em high. What is

its volume in m3?

Section 12.2 The Atomic Model of an Ideal Gas

6. III Dry ice is frozen carbon diox ide. If you have 1.0 kg of dry ice, what vo lume will it occupy i f you heal it e nough to turn it in to a gas at a temperature of 200C?

7. J What is the absol ute pressure of the air in your car 's tires, in psi, whe n your pressure gauge ind icates they are inflated to 35 .0 psi? Assume you are at sea leve l.

8. 1111 Total lung capac ity of a typical adu lt is approx imately 5.0 L. BIO Approx imately 20% of the a ir is oxygen . At sea level and at an

average body temperatu re of 37°C, how many moles of oxygen do the lungs con ta in at the end of an inhalati on?

9. 1111 Many cultu res around the wor ld st ill use a s imple weapon BIO called a blowgun , a tube with a dart that fits ti ghtl y inside. A

sharp breath into the end of the tube launches the dart. When exhaling forcefull y, a healthy person can supp ly a ir at a gauge pressure o f 6.0 kPa. What force does thi s pressure exert on a dart in a 1.5-cm-diameter tube?

10. III When you st ifle a sneeze, you can damage delicate ti ssues BIO because the pressure of the air that is not allowed to escape may

ri se by up to 45 kPa. If thi s extra pressure ac ts on the ins ide of your 8A-mm-di ameter eardrum, what is the outward force?

II . 1111 7.5 mol of hel ium are in a 15 L cylinder. The pressure gauge on the cylinder reads 65 psi. What are (a) the temperature of the gas in °C and (b) the average ki neti c energy of a helium atom?

12. II Mars has an atmosphere composed almost en ti rely of carbon diox ide, with an average temperature of -63°e. What is the fin s speed of a molecule in Mars 's atmosphere?

13. II 3 .0 mol of gas at a tempe rature o f - 1200C fill s a 2.0 L con­taine r. What is the gas pressure?

14. I 265 mls is a typica l crui s ing speed for ajet airl iner. At what temperature (in °C) do the molecules of nitrogen gas have an fins speed of 265 m/s?

15. 1111 10 g o f liquid wate r is placed in a fl ex ible bag, the air is excluded, and the bag is sealed. It is then placed in a microwave oven where the water is boil ed to make steam at !OO°e. What is the volume of the bag afte r all the water has bo iled? Assume that the pressure inside the bag is equal to atmospheri c pressure.

Section 12.3 Ideal-Gas Processes

16. II A cylinder contains 3.0 L of oxygen at 300 K and 2.4 atm. The gas is heated, causing a piston in the cylinder to move outward. The healing causes the temperature to rise to 600 K and the vol ume of the cylinder to increase to 9.0 L. What is the fi nal gas pressure?

17. III A gas with ini tial cond itions Pi, Vi, and Ti expands isothe r­mally until Vr = 2Vi • What are (a) l j- and (b) Pr?

Problems 399

18. III 0. 10 mol of argon gas is admitted to an evacuated 50 cm3

conta ine r at 20°e. The gas the n undergoes heat ing at constant volume to a te mperature of 300°e. a. What is the fin al pressure of the gas? b. Show the process on a p V d iagram . Include a proper scale on

both axes. 19. 0. 10 mol of argon gas is ad mitted to an evacuated 50 cm3

conta ine r at 20°e. The gas the n undergoes a n isobaric heating to a temperature o f 300°e. a. What is the fin al vol ume of the gas? b. Show the process on a pV diagram. Include a proper scale on

both axes. 20. III 0. 10 mol of argon gas is admitted to an evacuated 50 cm3

con tainer at 20°e. The gas then undergoes an isothermal expan­sion to a vol ume of 200 em3•

a. W hat is the final pressure o f the gas? b. Show the process on a P V d iagram. Inc lude a proper scale on

both axes. 2 1. II 0.0040 mol o f gas undergoes the process shown in

Figure P 12.2 1.

22.

23.

a. What type of process is thi s? b. What are the ini tial and final te mperatures?

p (aIm) p(allll)

3 3

2 2

100 200 300

FIGURE P12.21 FIGURE P12.22

II I 0.0040 mol of gas fo 1.1 ows the hyperboli c trajectory shown in Figure P 12.22. a. What lype of process is thi s? b. What are the initial and final te mperatures? c. What is the fin al volume \'r?

A gas with an initi al te mpe rature o f 900°C undergoes the process shown in Figure P 12.23. a. What lype of process is thi s? b. What is the fi nal temperature? c . How many moles o f gas are there?

p(atm)

p (kPa)

::1 f , ;

O~V(''''' ) o 100 200 300

FIGURE P12.23 FIGURE P12.24

24. III How much work is done on the gas in the process shown in Figure P1 2.24?

400 CHAPTER 12 Thermal Properties of Matter

25. It is poss ible to make a thermometer by sealing gas in a rigid conta iner and measuring the absolute pressure. Such a constant­volume gas thermometer is placed in an ice-water bath at 0.00°e. After reaching thermal equilibrium, the gas pressure is recorded as 55.9 kPa. The thermometer is then placed in contact with a sample of unknown temperature. After the thermometer reaches a new equilibrium, the gas pressure is 65.1 kPa. What is the temperature of thi s sample?

26. II A 1.0 cm3 air bubble is re leased from the sandy bottom of a warm, shallow sea, where the gauge pressure is 1.5 atm. The bubble ri ses slowly enough that the air inside remains at the same constant temperature as the water. a. What is the volume of the bubble as it reaches the surface? b. As the bubble rises, is heat energy transferred from the

water to the bubble or from the bubble to the water? Explain.

27. A weather balloon rises through the atmosphere, its volume expanding from 4.0 m3 tol2 m3 as the temperature drops from 200C to - lODe. If the initial gas pressure inside the balloon is 1.0 atm, what is the final pressure?

Section 12.4 Thermal Expansion

28. III A straight rod consists of a 1.2-cm-long piece of aluminum attached to a 2.0-cm-long piece of steel. By how much will the length of this rod change if its temperalUre is increased from 20°C to 40°C?

29. II The length of a steel beam increases by 0.73 mm when it s temperature is raised from 22°C to 35°e. What is the length of the beam at 22°C?

30. II Older railroad tracks in the U.S. are made of 12-m-Iong pieces of stee l. When the tracks are laid, gaps are left between the sec­tions to prevent buckling when the steel thermally expands. If a track is laid at 16°C, how large should the gaps be if the track is not to buckle when the temperature is as high as 50°C?

31. 11111 The temperature of an aluminum disk is increased by 120°e. By what percentage does its volume increase?

Section 12.5 Specific He~lt and He~lt of Transformation

32. III How much energy must be removed from a 200 g block of ice to cool it from ooe to - 30°C?

33. 1111 How much heat is needed to change 20 g of mercury at 20°C into mercury vapor at the boil ing point?

34. II a. 100 J of heat energy are transferred to 20 g of mercury. By how much does the temperature increase?

b. How much heat is needed to raise the temperature of 20 g of water by the same amount?

35. II The maximum amount of water an adult in temperate BID cl imates can perspire in one hour is typically 1.8 L. However,

after several weeks in a tropical climate the body can adapt, increasi ng the maximum perspiration rate to 3.5 Uh. At what rate, in watts, is ene rgy being removed when perspiring that rapidly? Assume all of the perspired water evaporates. At body temperature, the heat of vaporization of water is Lv = 24 X I CT' J/kg.

36. II Alligators and other reptil es don·t use enough metabolic BID energy to keep their body temperatures constant. They cool off

at night and must warm lip in the sun in the morning. Suppose a

300 kg alligator with an early ­morning body temperature of 25°C is absorbing radiation from the sun at a rate of 1200 W. How long will the alligator need to warm up to a more favorable 30°C? (Assume that the specific heat of the reptilian body is the same as that of the mammalian body.)

37. II When air is inhaled, it quickly becomes saturated with water BID vapor as it passes through the moist upper airways. When

breathing dry air, about 25 mg of water are exhaled with each breath. At 12 breaths/min , what is the rate of energy loss due to evaporation? Express your answer in both watts and Calories per day. At body temperature, the heat of vaporization of water is L , = 24 X 10' J/kg.

38. III It is important for the body to have mechani sms to e ffect ively BID cool itself; if not, moderate exerc ise could easi ly increase body

temperatures to dangerous levels. Suppose a 70 kg man runs on a u·eadmill for 30 min , using a metabolic power of 1000 W. Assume that all of thi s power goes to thermal energy in the body. If he couldn 't perspire or otherwise cool hi s body, by how much wou ld hi s body temperature rise during this exerc ise?

39. III What minimum heat is needed 10 bring 100 g of water at 200e to the boiling point and completely boi l it away?

Section 12.6 Calorimetry

40. III 30 g of copper pellets are removed from a 300°C oven and immediately dropped into 100 IT\L of water at 20°C in an insu­lated cup. What wi111he new water temperature be?

41 . III A copper block is removed from a 300°C oven and dropped into 1.00 kg of water at 20.0°e. The water quickly reaches 25.5 °e and then remains at that temperature. What is the mass of the copper block?

42. 1111 A 750 g aluminum pan is removed from the stove and plunged into a sink filled with 10.0 kg of water at 20.COe. The water temperature quickly ri ses to 24.0°e. What was the initial temperature of the pan?

43. II A 500 g metal sphere is heated to 300°C, then dropped into a beaker containing 4.08 kg of mercury at 20.0oe. A short time later the mercury temperature stabili zes at 99.0°e. Identify the metal.

44. II Brewed coffee is often too hot to drink right away. You can cool it with an ice cube, but thi s dilutes it. Or you can buy a device that will cool your coffee without dilution-a 200 g alu­minum cylinder that yo u take from your freezer and place in a mug of hot coffee. If the cy linder is cooled to - 20°C, a typical freezer temperature. and then dropped into a large cup of coffee (essen ti all y water, with a mass of 500 g) at 85°C, what is the final temperat ure of the coffee?

45 . 11111 Marianne really likes coffee, but on summer days she doesn' t want to drink a hot beverage. If she is served 200 mL of coffee at 800C in a well-insulated container, how much ice at O°C should she add to obtain a final temperature of 300e?

46. II IJ a person has a dangerously high fever, submerging her in BID ice water is a bad idea, but an ice pack can help to quickly bring

her body temperature down. How many grams of ice at DoC will be melted in bri nging down a 60 kg patient's fever from 40°C to

39°C?

Section 12.7 Thermal Properties of Gases

47. A container holds 1.0 g of argon at a pressure ofS.O atm. a. How much heat is required to increase the temperature by

100°C at constant vo lume? b. How much will the temperature increase if thi s amount of

heat energy is transferred to the gas at constant pressure? 4S. II A conta iner holds 1.0 g of oxygen at a pressure of 8.0 atm.

a. How much heat is required to increase the temperature by 100°C at constant pressure?

b. How much will the temperature increase if thi s amount of heat energy is transferred to the gas at constant vo lume?

49. I What is the temperature change of 1.0 mol of a monatomic gas if its thermal energy is increased by 1.0 J?

50. II The temperature of 2.0 g of helium is increased at constan t volume by tlT. What mass of oxygen can have its temperature inc reased by the same amount at constant volume using the same amount of heat?

51. III How much work is done per cycle by a gas following the p V

trajectory of Figure P12.51 ?

,, (aim)

p (kPa) 3

[7 2

tOO

o V (cmJ) 200 400 600 o 200 400 600 800

FIGURE P12.51 FIGURE P12.52

52. III A gas following the pV trajectory of Figure P12.52 does 60 J of work per cycle. What is Pmax ?

Section 12.8 Heat Transfer

53. III A I .S-cm-thick wood floor covers a 4.0 m X 5.5 m room. The subfloor on which the flooring sits is at a temperature of 16.2°C, while the air in the room is at 19.6°C. What is the rate of heat conduction through the floor?

54. 11111 A copper-bottomed kettle, its boltom 24 em in diameter and 3.0 mm thick , sits on a burner. The kettle holds boiling water, and energy flows into the water from the kettle bottom at 800 W. What is the temperature of the bottom surface of the kettle?

55. 11111 What is the greatest possible rate of energy transfer by radia­tion from a meta l cube 2.0 cm on a side that is at 700°C? Its emissivity is 0.20.

56. III What is the greatest poss ible rate of energy transfer by radia­tion for a 5.0-em-diameter sphere that is at 100°C?

57. III Seals may cool themselves by BID using tltermal windolVs, patches on

their bodies with much hi gher than average surface temperature. Sup­pose a seal has a 0.030 m2 thermal window at a temperature of 30°C. If the seal's surroundings are a frosty - 10°C, what is the net rate of energy loss by radiation? Assume an emissivity equaJ to that of a human.

5S. II Electronics and inhabitants of the Lnternational Space Station generate a significant amount of thermal energy that the station must get rid of. The only way that the station can exhaust thelmal energy is by radiation , which it does using thin , 1. 8-m-by-3.6-m

Problems 401

panels that have a working temperature of about 6°C. How much power is radiated from each panel? Assume that the panels are in the shade so that the absorbed radiation will be negligible. Assume that the emissivity of the panels is 1.0. Hint: Don ' t forget that the panels have two sides!

59. II The glowing filament in a lamp is radiating energy at a rate of 60 W. At the filament 's temperature of I 500°C, the emiss iv­ity is 0.23. What is the surface area of the filament?

60. III If you li e on the ground at night with no cover, you get cold BID rather quickly. Much of this is due to energy loss by radiation.

At night in a dry climate, the temperature of the s J...'Y can drop to -40°C. If you are lying on the ground with thin clothing that provides little insulat ion, the surface temperature of your skin and clothes wiLl be about 300C. Estimate the net rate at which your body loses energy by radiation to the night sky under these conditions. Hint: What area should you use?

General Problems

61. II A rigid container holds 2.0 mol of gas at a pressure of 1.0 atm and a temperature of 30°C. a. What is the container's volume? b. What is the pressure if the temperature is rai sed to 130°C?

62. II A J 5-cm-diameter compressed-a ir tank is 50 cm tall. The pressure at 20°C is 150 atm. a. How many moles of air are in the tank? b. What volume would this air occupy at STP?

63. 1111 A 10-cm-diameter cylinder of helium gas is 30 cm long and at 20°C. The pressure gauge reads 120 psi. a. How many helium atoms are in the cylinder? b. What is the mass of the he lium?

64. II Party stores sell smalJ tanks containing 30 g ofheliulll gas. If you use sllch a tank to fill 0.010 m3 foil balloons (which don't stretch, and so have an internal ~ressure that is very close to atmospheric pressure), how many balloons can you expect to fill ?

65. 11111 Suppose you take and hold a deep breath on a chilly day, BID inhaling 3.0 L of air at O°C. Assume that air pressure is 1.0 atm.

a. How much heat must your body supply to warm the air to your internal body temperature of 37°C?

b. I-I ow much does the air 's volume increase as it is warmed? 66. II On average, each person in the industriali zed world is respon­

sible for the emission of 10,000 kg of carbon dioxide (C0 2) every year. This includes CO2 that you generate directly, by burning foss il fue ls to operate your car or your furnace, as well as CO2

generated on your behalf by electric generating stati ons and man­ufacturing plants. CO2 is a greenhouse gas that contributes to global warming. If you were to store your yearly CO2 emjssions in a cube at STP. how long would each edge of the cube be?

67. ili On a cool morning, when the temperature is 15°C, you mea­sure the pressure in your car tires to be 30 psi. After driving 20 l11i on the freeway, the temperature of your tires is 45°C. What pressure will your tire gauge now show?

68. III Suppose you inflate your car tires to 35 psi on a 20°C day. Later, the temperature drops to O°C. What is the pressure in your tires now?

69. I The volume in a constant-pres.wre gas thermometer is directly proportional to the absolute temperature. A constant-pressure thermometer is calibrated by adj usting its volume to 1000 mL while it is in contact with a reference cell at 0.01 0C. The vol­ume increases to 1638 mL when the thermometer is placed in contact with a sample. What is the sample's temperature in °C?

402 CHAPTER 12 Therm a l Properties of Matte r

70. A compressed-air cyl inde r is known to fail if the pressure exceeds 11 0 atm. A cylinder that was fiUed to 25 atm at 20°C is stored in a warehouse. Unfortunately, the warehouse catches fi re and the temperature reac hes 950°C. Does the cy li nder explode?

7 1. II A rigid sphere has a valve that can be opened or closed. The sphere with the valve open is placed in boiling water in a room where the a ir pressure is 1.0 atm. After a long period of time has elapsed, the valve is closed. What will be the pressure inside the sphere if it is then placed in (a) a mixtu re of ice and water and (b) an insulated box fi lled wi th dry ice, which is at - 78 .5°C?

72. III 80 J of work" are done on the gas in the process shown in Figure PI 2.72. What is Vf in cm3?

p (kPa) p (kPa)

,ao! ~-o~v o vf 2Vf 3Vf

:1 /"c o V (cmJ)

o 100 200 300

FIGURE P12.12 FIGURE P12.73

73. III How much work is do ne by the gas in the process shown in Figure P I2.73?

74. I 0. 10 mol of gas undergoes the process 1- 2 shown in Figure P12 .74. a. What are temperatures Tl and T2 ? b. What lype of process is this? c. The gas undergoes constant-vo lume heat ing from point 2

unti l the press ure is restored to the value it had at point I. What is the fi nal temperature of the gas?

p (alm)

2 I~ 2

o V (cm' ) FIGURE P12 .74 0 1000 2000 3000

75. II 10 g of dry ice (solid CO2) is placed in a 10,000 cm3 con­tai ner, then all lhe air is quickly pum ped out and the container sealed. The container is warmed to DoC, a temperature at which CO2 is a gas. a. What is the gas pressure? Give your answer in atm. The gas then undergoes an isothermal compress ion unt il the pressure is 3.0 atm, immediate ly followed by an isobar ic com­press ion until the volume is 1000 cmJ

.

b. What is the final temperature of the gas? c. Show the process on a pV diagram.

76. III! A large freshwater fish has a swim bladder with a volume of BIO 5.0 X 1O- 4 m3. The fish descends from a dep th where the

absolute pressure is 3.0 atm to deeper water where the swi m bladder is compressed to 60% of its ini tial vo lume. As the fi sh descends, the gas pressure in the swim bladder is always equal to the water pressure , and the temperature of the gas remains at the internal temperature of the fi sh's body. To adapt to its new locat ion, the fish must add gas to re inflate its swim bladder to the origina l vo lume. Thi s takes energy to accomplish. What's the mi nimum amount of work requi red to expand the swi m bladder back to its origina l vo lume?

77. III A 5.0-m-diameter garden pond holds 5.9 X 103 kg of water. Solar energy is incident on the pond at an average rate of 400 W 1m2• I f the water absorbs all the solar energy and does not exchange energy with its surround ings, how many hours wi ll it take to warm rrom 15°C to 25°C?

78. 11 111 0.030 mo l or an ideal monatomic gas undergoes an ad iabatic compress ion that ra ises its temperature from lOoC to 50°C. How much work is done on the gas to compress it?

79. 11111 0. 15 mol of an ideal monatom ic gas undergoes an ad iabatic expans ion, cooling from 20°C to _ 10°C. How much work is done by the gas during the expans ion?

80. 1111 Susan, whose mass is 68 kg, cl imbs 59 m to the to p of the BlO Cape Hatteras lighthouse.

INT a. Duri ng (he cl imb, by how much does her potential energy increase?

b. For a typical efficiency of25%, what metabol ic energy does she require to complete the cl imb?

c . When exerc ising , the body must persp ire and use other mechanisms to cool itse lf to avo id potenti a ll y da ngerous increases on body temperature. If we assume that Susan doesn' t perspire or otherwise cool herself and that all of the " lost" energy goes into increasing her body temperature, by how much would her body temperature increase during thi s climb?

8 1. III A typ ical nuclear reactor generates 1000 MW of e lec tr ic energy. In doing so, it produces "waste heat" at a rate or 2000 MW, and thi s heat mustbe removed rrom the reac tor. Many reactors are sited nex t to large bodies of water so that they can use the water fo r cool ing. Consider a reactor where the intake water is at 18°C, State regulat ions limi t' the temperature of the outpu t water to 30°C so as not to harm aquat ic-organisms. How many kilograms of cooling wate r have to be pumped through the reac tor each minu te?

82. 1111 A 68 kg woman cycles at a constant 15 kmlh. All of the meta­BIO bolic energy that does not go to fo rward propuls ion is converted INT to thermal energy in her body. If the only way her body has to

keep cool is by evaporat ion, how many kilograms of water must she lose to perspi rat ion each hour to keep her body temperature constant?

83. III A 1200 kg car trave ling at 60 mph quickly brakes to a halt. INT The kinet ic energy of the car is converted to thermal energy of

the di sk brakes. T he brake disks (one per wheel) are iro n disks with a mass of 4.0 kg. Est imate the temperature ri se in each disk as the car SLOpS.

84. III A 5000 kg African elephant BlO has a resting metabolic rate of INT 2500 W. On a hOI day, the e le-

phant's environment is likely to be nearl y the same tempera­ture as the ani mal itsel f, so cooling by radiation is not effecti ve. The on ly plausible way to keep cool is by evaporation, and e lephants spray water on the ir body to accomplish thi s. If thi s is the only possible means of cool ing, how many kilograms of water per hour must be evaporated from an elephant 's skin to keep it at a constant temperatu re?

85. III Suppose you drop a water ba ll oon from a he ight o r 10 m. l r INT the balloon doesn ' t break on impact, its kineti c energy will be

converted to thermal energy. Est imate the temperat ure rise of the water. Is thi s likely to be not iceable?

86. III What is the maximum mass of lead you could melt with 1000 J of heat, starting from 200 e?

87. II An experiment measures the temperature of a 200 g sub­stance while steadily supplying heat to it. Figure P12.87 shows the results of the exper iment. What are (a) the specific heat of the liquid phase and (b) the heat of vapori zat ion?

T(QC)

1800

1500 1200 900

600 300

0 80 Q (kJ) FIGURE P12.87 0 20 40 60

88. 11111 10 g of aluminum at 200°C and 20 g of copper are dropped into 50 cm) of ethyl alcohol at 15°e. The temperature quickly comes to 25°e. What was the initial temperature of the copper?

89. III A 100 g ice cube at - 10°C is placed in an aluminum cup whose initial temperature is 70°C. The system comes to an equil ibrium temperature of 20°C. What is the mass o(the cup?

90. III A 50.0 g thermometer is used to measure the temperature of 200 g of water. The specific heat of the thermometer, which is mostly glass. is 750 J/kg . K. and it reads 20.0°C while lying on the tab le. After being complete ly immersed in the water, the thermometer's reading stabilizes at 71.2°C. What was the actual water temperature before it was measured?

91. 1111 Your 300 mL cup of coffee is too hot to drink when se rved at 90°e. What is the mass of an ice cube, taken from a - 200C freezer, that will cool your coffee to a pleasant 60°C?

92. II A gas is compressed from 600 cm) to 200 cm] at a constant pressure of 400 kPa. At the same time, 100 J of heat energy is transferred oul of the gas. What is the change in thermal energy of the gas during this process?

93. 1111 An expandable cube, initially 20 cm on each side. contains 3.0 g of helium at 20°e. 1000 J of heat energy are transferred to this gas. What are (a) the final pressure if the process is at con­stant volume and (b) the final volume if the process is at con­stant pressure?

94. III 0.10 mol of a monatomic gas follows the process shown in Figure PI2.94.

95.

a. How much heat energy is transferred to or from the gas dur­ing process I ~ 2?

b. How much heat energy is transferred to or from the gas dur­ing process 2 -) 3?

c. What is the total change in thermal energy of the gas?

p (atm)

1 2 4 t 2

o +0---8~OO---I6~()()-V (emJ)

FIGURE P12.94

"tm

) 'SJ' I , , , 2 "

/ ' lOOQC Isotherm ... -_

3

o l--~-_--~V (emJ) o 100 200 300

FIGURE P12.95

II A monatomic gas follows the process I -) 2 -) 3 shown in Figure P 12.95. How much heat is needed for (a) process I -) 2 and (b) process 2 -) 3?

Problems 403

96. III What are (a) the heat QH extracted from the hot reservoir and INT (b) the efficiency for a heat engine described by the pV diagram

of Figure PI2.96?

p (kPa)

300

200

100

180 J

100J 4r"

o l--~-_'--~V (em) FIGURE P12.96 o 200 400 600

97. III The top layer of you r goose down sleeping bag has a thick­ness of 5.0 cm and a surface area of 1.0 m2

. When the ou tside temperature is - 200C, you lose 25 Callh by heat conduction through the bag (which remains at a cozy 35°C inside). Assume that you ' re sleepi ng on an insulated pad that eliminates heat conduction to the ground beneath you. What is the thennal con­ductivity of the goose down?

98. 1111 Suppose you go outside in your fiber-filled jacket on a windless but very cold day. The thickness of the jacket is 2.5 cm, and it covers 1.1 m2 of your body. The purpose of fiber- or down-filled jackets is to trap a layer of air, and it 's really the air layer that provides the insulation. If your sk in temperature is 34°C while the air temperature is -20°C, at what rate is heat being conducted through the jacket and away from your body?

99. 11111 Two thin, square copper plates are radiating energy at the same rate. The edge length of plate 2 is four times that of plate I. What is the ratio of absolute temperatures T] IT2 of the plates?

100. 11 The surface area of an adult human is about L.8 m2. Suppose

BID a person with a ski n temperature of 34°C is standing with bare skin in a room where the air is 25°C but the walls are 17°e. a. There is a "dead-air" layer next to your skin that acts as

insulation. If the dead-air layer is 5.0 mm thick, what is the person 's rate of heat loss by conduction?

b. What is the person' s net radiation loss to the walls? The emissivity of skin is 0.97.

c. Does conduction or radiation contribute more to the LOtal rate of energy loss?

d. If the person is metaboli zi ng food at a rate of 155 W, does the person feel comfortable, chilly, or too warm?

Passage Problems

Thermal Properties of the Oceans

Seasonal temperature changes in the ocean only affect the top layer of water, to a depth of 500 m or so. This "m ixed" layer is thermally isolated from the cold, deep water below. The ave rage temperature of this top layer of the world 's oceans, which has area 3.6 X 108 km2

,

is approximately 17°C. In addition to seasonaJ temperature changes, the oceans have

experienced an overall warming trend over the last century that is expected to continue as the earth 's climate changes. A warmer ocean means a larger vo lume of wa ter; the oceans will rise. Suppose the average temperature of the top layer of the world's oceans were to increase from a temperature T-, to a temperature 1j- . The area of the oceans will not change, as this is fixed by the size of the ocean basin,

404 CHAPTER 12 Thermal Properties of Matter

so any thermal expansion of the water will cause the water level to rise, as shown in Figure P12.101. The original vo lume is the product of the original depth and the surface area, Vi = Ad;. The change in volume is given by tiV ~ A tid.

Change in depth

D'P'h,~d~ "'r --~- -- fL2V \ /

Arc<lA

FIGURE P12 .101

101. 1 Lfthe (OP 500 m of ocean water increased in temperature from 17°C to 18°C, what would be the resulting rise in ocean height? A. 0.11 m B. 0.22 m C. 0.44 m D. 0.88 m

102. I Approximately how much energy would be required to rai se the temperature of the top layer of tbe oceans by 1°C? ( I m3 of water has a mass of 1000 kg.) A. 1 X 10" J B. 1 X 10" J C. 1 X 10" J D. 1 X IO " J

7

Stop to Think 12.1: A. Both helium and neon are monatomic gases, where the basic particles are atoms. 5 mol of hel ium contain 5 times as many atoms as I mol of neon, though both samples have the same mass.

Stop to Think 12.2: A, B. An increase in temperature means that the atoms have a larger average kinetic energy and will thus have a larger rms speed. Because the thermal energy of the gas is simply the total kinetic energy of the atoms, thi s must increase as well. The pressure couid change, but it' s not requ ired; T and V could increase by the same factor, which would keep p constanl. The container is sealed, so the number of molecules does not change.

Stop to Think 12.3: B. The product pVIT is constant. During the process, p V decreases by a factor of 2, so T must decrease by a factor of2 as well.

Stop to Think 12.4: A. The thermal expansion coefficients of alu­minum are greater than those of iron. Heating the rod and the ring will expand the outer diameter of the rod and the inner diameter of the ring, but the ring's expansion will be greater.

I 03. I Water 's coefficient of expansion varies with temperature. For water at 2°C, an increase in temperature of I °C would cause the volume to A. Increase. B. Stay the same. C. Decrease.

104. 1 The ocean is mostly heated from the top, by light from the sun. The warmer surface wate r doesn ' t mix much with the colder deep ocean water. This lack of mixing can be ascribed to a lack of A. Conduction. B. Convection. C. Radiation. D. Evaporation.

Stop to Think 12.5: B. To solidify the lead, heat must be removed; thi s heat boils the water. The heat of vaporization of water is 10 limes the heat of fusion of lead, so much less than 1 kg of water vapori zes as I kg of lead sol idifies.

Stop to Think 12.6: A.The lead cools and the water warms as heat is transferred from the lead to the water. The speci fi c heat of water is much larger than that of lead, so the temperature change of the water is much less than that of the lead.

Stop to Think 12.7: C. With a sealed suit and no matter around you, there is no way to transfe r heat to the environment except by radiation.