solucionario capitulo 19 - paul e. tippens

Chapter 19. Thermal Properties of Matter Physics, 6th Edition

Chapter 19. Thermal Properties of Matter

General Gas Laws: Initial and Final States

19-1. An ideal gas occupies a volume of 4.00 m3 at 200 kPa absolute pressure. What will be the

new pressure if the gas is slowly compressed to 2.00 m3 at constant temperature?

31 1

1 1 2 2 2 32

(200 kPa)(4.00 m ); (2.00 m )

PVPV PV PV

P2 = 400 kPa

19-2. The absolute pressure of a sample of ideal gas is 300 kPa at a volume of 2.6 m3. If the

pressure decreases to 101 kPa at constant temperature, what is the new volume?

31 1

1 1 2 2 22

(300 kPa)(2.6 m ); V(101 kPa)

PVPV PVP

V2 = 7.72 m3

19-3. Two hundred cubic centimeters of an ideal gas at 200C expands to a volume of 212 cm3 at

constant pressure. What is the final temperature. [ T1 = 200 + 2730 = 293 K ]

31 2 1 2

2 31 2 1

(293 K)(212 cm ); 200 cm

V V TVTT T V

; T2 = 310.6 K

tC = 310.60 – 2730 = 37.60C; tC = 37.60C

19-4. The temperature of a gas sample decreases from 550C to 250C at constant pressure. If the

initial volume was 400 mL, what is the final volume?

258


T1 = 550 + 2730 = 328 K; T2 = 250 + 2730 = 298 K; V1 = 400 mL

1 2 2 12

1 2 1

(298 K)(400 mL); V328 K

V V T VT T T

V2 = 363 mL

19-5. A steel cylinder contains an ideal gas at 270C. The gauge pressure is 140 kPa. If the

temperature of the container increases to 790C, what is the new gauge pressure?

T1 = 270 + 2730 = 300 K; T2 = 790 + 2730 = 352 K;

P1 = 140 kPa + 101.3 kPa = 241.3 kPa;

1 2 1 22

1 2 1

(241.3 kPa)(352 K); 283.1 kPa300 K

P P PTPT T T

Gauge Pressure = 283.1 kPa – 101.3 kPa; P2 = 182 kPa

19-6. The absolute pressure of a sample of gas initially at 300 K doubles as the volume remains

constant. What is the new temperature? [ P2 = 2P1 ]

1 2 2 1 12

1 2 1 1

(2 )(300 K); P P PT PTT T P P

; T2 = 600 K

19-7. A steel cylinder contains 2.00 kg of an ideal gas. Overnight the temperature and volume

remain constant, but the absolute pressure decreases from 500 kPa down to 450 kPa. How

many grams of the gas leaked out over night?

259


1 1 2 2 1 22

1 1 2 2 1

(2.00 kg)(450 kPa); 500 kPa

PV PV m Pmm T m T P

; m2 = 1.80 kg

Lost = 2.00 kg – 1.80 kg = 0.200 kg; Amt. Leaked = 200 g

19-8. Five liters of a gas at 250C has an absolute pressure of 200 kPa. If the absolute pressure

reduces to 120 kPa and the temperature increases to 600C, what is the final volume?

T1 = 250 + 2730 = 298 K; T2 = 600 + 2730 = 333 K

1 1 2 2 1 1 22

1 2 1 2

(200 kPa)(5 L)(333 K); (298 K)(120 kPa)

PV PV PV TVT T T P

V2 = 9.31 L

19-9. An air compressor takes in 2 m3 of air at 200C and one atmosphere (101.3 kPa) pressure.

If the compressor discharges into a 0.3-m3 tank at an absolute pressure of 1500 kPA, what

is the temperature of the discharged air? [ T1 = 200 + 2730 = 293 K ]

31 1 2 2 2 2 1

2 31 2 1 1

(1500 kPa)(0.3 m )(293 K); T(101.3 kPa)(2.0 m )

PV PV PV TT T PV

; T2 = 651 K

19-10. A 6-L tank holds a sample of gas under an absolute pressure of 600 kPa and a temperature

of 570C. What will be the new pressure if the same sample of gas is placed into a 3-L

container at 70C. [ T1 = 570 + 2730 = 330 K; T2 = 70 + 2730 = 280 K ]

1 1 2 2 1 1 22

1 2 1 2

(600 kPa)(6 L)(280 K); (330 K)(3 L)

PV PV PV TPT T TV

; P2 = 1020 kPa

260


19-11. If 0.8 –L of a gas at 100C is heated to 900C at constant pressure. What is the new volume?

T1 = 100 + 2730 = 283 K; T2 = 900 + 2730 = 363 K

1 2 2 12

1 2 1

(363 K)(0.8 L); V283 K

V V T VT T T

; V2 = 1.03 L

19-12. The inside of an automobile tire is under a gauge pressure of 30 lb/in.2 at 40C. After

several hours, the inside air temperature rises to 500C. Assuming constant volume, what

is the new gauge pressure?

P1 = 30 lb/in.2 + 14.7 lb/in.2 = 44.7 lb/in.2

T1 = 2730 + 40 = 277 K; T2 = 2730 + 500 = 323 K

221 2 1 2

21 2 1

(44.7 lb/in. )(323 K); 52.1 lb/in.(277 K)

P P PTPT T T

P2 = 52.1 lb/in.2 – 14.7 lb/in.2 = 37.4 lb/in.2; P2 = 37.4 lb/in.2

19-13. A 2-L sample of gas has an absolute pressure of 300 kPa at 300 K. If both pressure and

volume experience a two-fold increase, what is the final temperature?

1 1 2 2 2 2 12

1 2 1 1

(600 kPa)(4 L)(300 K); T(300 kPa)(2 L)

PV PV PV TT T PV

T2 = 1200 K

Molecular Mass and the Mole

19-14. How many moles are contained in 600 g of air (M = 29 g/mole)?

261


600 g29 g/mol

mnM

; n = 20.7 mol

19-15. How many moles of gas are there in 400 g of nitrogen gas (M = 28 g/mole)? How many

molecules are in this sample?

400 g28 g/mol

mnM

; n = 14.3 mol A

NnN

23 (14.3 mol)(6.023 x 10 molecules/mol)AN nN ;

N = 8.60 x 1024 molecules

19-16. What is the mass of a 4-mol sample of air (M = 29 g/mol)?

; (4 mol)(29 g/mol);mn m nMM

m = 116 g

19-17. How many grams of hydrogen gas (M = 2 g/mol) are there in 3.0 moles of hydrogen?

How many grams of air (M = 29 g/mol) are there in 3.0 moles of air?


m = 6.00 g


m = 87.0 g

262


*19-18. How many molecules of hydrogen gas (M = 2 g/mol) are needed to have the same mass

as 4 g of oxygen (M = 32 g/mol)? How many moles are in each sample?

23(4 g)(6.023 x 10 molecules/mol); 2 g/mol

H AH

A H

m N m Nn NM N M

NH = 1.20 x 1024 molecules of H2

4 g ;2 g/mol

HH

H

mnM

nH = 2 mol

4 g ;32 g/mol

OO

O

mnM

nH = 0.125 mol

*19-19. What is the mass of one molecule of oxygen (M = 32 g/mol)?

23

(1 molecule)(32 g/mol); 6.023 x 10 molecules/molA A

m N NMn mM N N

; m = 5.31 x 10-23 g

-23 1 kg5.31 x 10 g1000 g

m

; m = 5.31 x 10-26 kg

*19-20. The molecular mass of CO2 is 44 g/mol. What is the mass of a single molecule of CO2?

23

(1 molecule)(44 g/mol); 6.023 x 10 molecules/molA A

m N NMn mM N N

; m = 7.31 x 10-23 g

263


-23 1 kg7.31 x 10 g1000 g

m

; m = 7.31 x 10-26 kg

The Ideal Gas Law

19-21. Three moles of an ideal gas have a volume of 0.026 m3 and a pressure of 300 kPa. What is

the temperature of the gas in degrees Celsius?

3(300,000 Pa)(0.026 m ); (3 mol)(8.314 J/mol K)

PVPV nRT TnR

T = 313 K; tC = 3130 – 2730; tC = 39.70C

19-22. A 16-L tank contains 200 g of air (M = 29 g/mol) at 270C. What is the absolute pressure

of this sample? [ T = 270 + 2730 = 300 K; V = 16 L = 16 x 10-3 m3 ]

-3 3

(200 g)(8.314 J/mol K)(300 K); (29 g/mol)(16 x 10 m )

m mRTPV RT PM MV

; P = 1.08 x 106 Pa

19-23. How many kilograms of nitrogen gas (M = 28 g/mol) will occupy a volume of 2000 L at

an absolute pressure of 202 kPa and a temperature of 800C? [ T = (80 + 273) = 353 K ]

3(28 g/mol)(202,000 Pa)(2 m ); (353 K)(8.314 J/mol K)

m MPVPV RT mM RT

;

m = 3854 g; m = 3.85 kg

264


19-24. What volume is occupied by 8 g of nitrogen gas (M = 28 g/mol) at standard temperature

and pressure (STP)? [ T = 273 K, P = 101.3 kPa ]

(8 g)(8.314 J/mol K)(273 K); (28 g/mol)(101,300 Pa)

m mRTPV RT VM MP

; V = 6.40 x 10-3 m3

V = 6.40 x 10-3 m3; V = 6.40 L

19-25. A 2-L flask contains 2 x 1023 molecules of air (M = 29 g/mol) at 300 K. What is the

absolute gas pressure?

23

23 -3 3

(2 x 10 molecules)(8.314 J/mol K)(300 K); (6.023 x 10 molecules/mol)(2 x 10 m )A A

N NRTPV RT PN N V

P = 414 kPa

19-26. A 2 m3 tank holds nitrogen gas (M = 28 g/mole) under a gauge pressure of 500 kPa. If

the temperature is 270C, what is the mass of gas in the tank? [ T = 270 + 2730 = 300 K ]

3(28 g/mol)(500,000 Pa)(2 m ); (300 K)(8.314 J/mol K)

m MPVPV RT mM RT

; m = 11.2 kg

19-27. How many moles of gas are contained in a volume of 2000 cm3 at conditions of standard

temperature and pressure (STP)?

-3 3(101,300 Pa)(2 x 10 m ); n(273 K)(8.314 J/mol K)

PVPV nRTRT

; n = 0.0893 mol

265


19-28. A 0.30 cm3 cylinder contains 0.27 g of water vapor (M = 18 g/mol) at 3400C. What is its

absolute pressure assuming the water vapor is an ideal gas? [ T = 3400 + 2730 = 613 K ]

-6 3

(0.27 g)(8.314 J/mol K)(613 K); (18 g/mol)(0.3 x 10 m )

m mRTPV RT PM MV

P = 2.55 x 108 Pa

Humidity

19-29. If the air temperature is 200C and the dew point is 120C, what is the relative humidity?

Remember that the actual vapor pressure at a given temperature is the same as the

saturated vapor pressure for the dew-point temperature. The table values for 200C and

120C are used here, and the relative humidity is found as follows:

Relative Humidity =

10.64 mmHg 0.608;17.4 mmHg

60.8%

19-30. The dew point is 200C. What is the relative humidity when the air temperature is 240C?

17.5 mmHgRe Humidity = 22.4 mmHg

lative; Rel. hum. = 78.1%

19-31. The relative humidity is 77 percent when the air temperature is 280C. What is the dew

point? From the table, at 280C, saturation vapor pressure is 28.3 mmHg.

vapor pressure = 0.77(28.3 mmHg) = 21.8 mmHg

Dew point = 23.50C From Table

266


19-32. What is the pressure of water vapor in the air on a day when the temperature is 860F and

the relative humidity is 80 percent? [ For 860F the sat. vapor pressure is 31.8 mm Hg ]

0.8031.8 mmHg

x

; x = 25.4 mm Hg

19-33. The air temperature in a room during the winter is 280C. What is the relative humidity if

moisture first starts forming on a window when the temperature of its surface is at 200C?

17.5 mmHg 0.61828.3 mmHg

Rel Hum. = 61.8%

Challenge Problems

19-34. A sample of gas occupies 12 L at 70C and at an absolute pressure of 102 kPa. Find its

temperature when the volume reduces to 10 L and the pressure increases to 230 kPa.

V1 = 12 L = 12 x 10-3 m3; V2 = 10 L = 10 x 10-3 m3; T1 = 70 + 2730 = 280 K

1 1 2 2 2 2 12 3

1 2 1 1

(230 kPa)(10 L)(280 K); T(102 kPa)(12 L m )

PV PV PV TT T PV

; T2 = 526 K

19-35. A tractor tire contains 2.8 ft3 of air at a gauge pressure of 70 lb/in.2. What volume of air

at one atmosphere of pressure is required to fill this tire if there is no change in

temperature or volume? [ P2 = 70 lb/in.2 + 14.7 lb/in.2 = 84.7 lb/in.2 ]

2 32 2

1 1 2 2 2 21

(84.7 lb/in. )(2.8 ft ); 14.7 lb/in.

PVPV PV VP

; V2 = 16.1 ft3.

267


19-36. A 3-L container is filled with 0.230 mol of an ideal gas at 300 K. What is the pressure of

the gas? How many molecules are in this sample of gas?

-3 3

(0.230 mol)(8.314 J/mol K)(300 K)3 x 10 m

nRTPV

; P = 191 kPa A

NnN

N = nNA = (0.230 mol)(6.023 x 1023 molecules/mol) = 1.39 x 1023 molecules

19-37. How many moles of helium gas (M = 4 g/mol) are there in a 6-L tank when the pressure

is 2 x 105 Pa and the temperature is 270C? What is the mass of the helium?

6 -3 3(2 x 10 Pa)(6 x 10 m ) ;(8.314 J/mol K)(300 K)

PVnRT

n = 0.481 mol

mnM

; m = nM = (0.481 mol)(4 g/mol) ; m = 1.92 g

19-38. How many grams of air (M = 29 g/mol) must be pumped into an automobile tire if it is to

have a gauge pressure of 31 lb/in.2 Assume the volume of the tire is 5000 cm3 and its

temperature is 270C?

P = 31 lb/in.2 + 14.7 lb/in.2 = 45.7 lb/in.2; V = 5000 cm3 = 5 L

2 3 3(45.7 lb/in. )(5 x 10 m ); (8.314 J/mol K)(300 K)

m PVMPV RT mM RT

;

m = 9.16 x 10-5 g or m = 9.16 x 10-8 kg

268


19-39. The air temperature inside a car is 260C. The dew point is 240C. What is the relative

humidity inside the car?

22.4 mmHg 0.88925.2 mmHg

x Rel. hum. = 88.9%

19-40. The lens in a sensitive camera is clear when the room temperature is 71.60F and the

relative humidity is 88 percent. What is the lowest temperature of the lens if it is not to

become foggy from moisture?

At 71.60C, saturated pressure is 19.8 mmHg

Actual pressure = 0.88(19.8 mmHg) = 17.4 mmHg

Thus, from tables, the dewpoint is: 67.80F

*19-41. What is the density of oxygen gas (M = 32 g/mol) at a temperature of 230C and

atmospheric pressure?

3(101,300 Pa)(32 g/mol); 1320 g/m(8.314 J/mol K)(296 K)

m m PMPV RTM V RT

= 1.32 kg/m3

*19-42. A 5000-cm3 tank is filled with carbon dioxide (M = 44 g/mol) at 300 K and 1 atm of

pressure. How many grams of CO2 can be added to the tank if the maximum absolute

pressure is 60 atm and there is no change in temperature? [ 5000 cm3 = 5 L ]

At 1 atm:

(1 atm)(5 L)(44 g/mol)(0.0821 L atm/mol K)(300 K)

PVMmRT

8.932 g

269


At 60 atm:

(60 atm)(5 L)(44 g/mol)(0.0821 L atm/mol K)(300 K)

PVMmRT

535.9 g

Mass added = 535.9 g – 8.932 g; Added = 527 g

*19-43. The density of an unknown gas at STP is 1.25 kg/m3. What is the density of this gas at

18 atm and 600C?

P1 = 1 atm; T1 = 273 K; P2 = 18 atm; T2 = 600 + 2730 = 333 K

1 1 1 1 2

2 2 2 2 1

/; /

m m PM MP RT PTPV RTM V RT MP RT PT

1 1 2 1 2 12

2 2 1 1 2

; PT PTPT PT

31 2 1

21 2

(1.25 kg/m )(18 atm)(273 K)(1 atm)(333 K)

PTPT

; = 18.4 kg/m3

Critical Thinking Questions

*19-44. A tank with a capacity of 14 L contains helium gas at 240C under a gauge pressure of

2700 kPa. (A) What will be the volume of a balloon filled with this gas if the helium

expands to an internal absolute pressure of 1 atm and the temperature drops to -350C?

270


(b) Now suppose the system returns to its original temperature (240C). What is the final

volume of the balloon?

(a) P1 = 2700 kPa + 101.3 kPa = 2801.3 kPa; P2 = 101.3 kPa; V1 = 14 L

T1 = 240 + 2730 = 297 K; T2 = -350 + 2730 = 238 K

1 1 2 2 1 1 22

1 2 2 1

(2801 kPa)(14 L)(238 K); (101.3 kPa)(297 K)

PV PV PV TVT T PT

; V2 = 310 L

(b) P2 = P3 = 1 atm; T2 = 238 K, V2 = 310 L, T3 = 297 K

3 3 2 32 23

2 3 2

(310 L)(297 K); 238 K

PV V TPV VT T T

; V3 = 387 L

*19-45 A steel tank is filled with oxygen. One evening when the temperature is 270C, the gauge

at the top of the tank indicates a pressure of 400 kPa. During the night a leak develops in

the tank. The next morning it is noticed that the gauge pressure is only 300 kPa and that

the temperature is 150C. What percentage of the original gas remains in the tank?

P1 = 400 kPa + 101.3 kPa = 501.3 kPa; P2 = 300 kPa + 101.3 kPa = 401.3 kPa

T1 = 2730 + 270 = 300 K; T2 = 2730 + 150 = 288 K; V1 = V2

1 1 2 2 2 2 1

1 1 2 2 1 1 2

(401.3 kPa)(300 K); ;(501.3 kPa)(288 K)

PV PV m PTm T m T m PT

2

1

0.834mm

; Mass remaining = 83.4%

271


*19-46. A 2-L flask is filled with nitrogen (M = 28 g/mol) at 270C and 1 atm of absolute

pressure. A stopcock at the top of the flask is opened to the air and the system is heated to

a temperature of 1270C. The stopcock is then closed and the system is allowed to return

to 270C. What mass of nitrogen is in the flask? What is the final pressure?

(a) The mass remaining in the flask after heating to 1270C is found as follows:

T = 1270 + 2730 = 400 K; R = 0.0821 atm L/mol K

(1 atm)(2 L)(28 g/mol); (0.0821 L atm/mol K)(400 K)

m PVMPV RT mM RT

; m = 1.705 g

(b) This same mass remains when it returns to T = 270+ 2730 = 300 K, therefore,

(1.705 g)(0.0821 L atm/mol K)(300 K)(28 g/mol)(2 L)

mRTPMV

; P = 0.750 atm

*19-47. What is the volume of 8 g of sulfur dioxide (M = 64 g/mol) if it has an absolute pressure

of 10 atm and a temperature of 300 K? If 1020 molecules leak from this volume every

second, how long will it take to reduce the pressure by one-half?

(8 g )(0.0821 L atm/mol K)(300 K); V(64 g/mol)(10 atm)

m mRTPV RTM MP

; V = 0.308 L

231

1(6.023 x 10 molecules/mol)(8 g);

64 g/molA

A

m N N mn NM N M

N1 = 7.529 x 1022 molecules in original sample

272


Now, the pressure P is proportional to the number of molecules so that:

22121 1 1 1 1

22 2 1

( ) 7.529 x 10 molecules and 2 2

P N N P NNP N P

N2 = 3.765 x 1022 molecules; Molecules lost = N1 – N2 = 3.765 x 1022 molecules

22

22

3.764 x 10 molecules ;10 molecules/s

time time = 371 s

*19-48. A flask contains 2 g of helium (M = 4 g/mol) at 570C and 12 atm absolute pressure. The

temperature then decreases to 170C and the pressure falls to 7 atm. How many grams of

helium have leaked out of the container? [ T1 = 570 + 2730 = 330 K; T2 = 290 K ]

11

1

(2 g)(0.0821 L atm/mol K)(330 K) ;(4 g/mol)(12 atm)

mRTVMP

V1 = 1.13 L

2 22

2

(4 g/mol)(7 atm)(1.129 L)(0.0821 L atm/mol K)(290 K)

MPVmRT

; m2 = 1.328 g

m1 = 2.00 g; mass lost = 2.00 – 1.328 = 0.672 g

*19-49. What must be the temperature of the air in a hot-air balloon in order that the mass of the

air be 0.97 times that of an equal volume of air at a temperature of 270C?

m2 = 0.97m1; T2 = 270 + 2730 = 300 K; Assume V1 = V2 T1 = ?

Note: The pressure inside a balloon will always be equal to the pressure of the

atmosphere, since that is the pressure applied to the surface of the balloon, so P1 = P2.

273


1 1 1 1 1 21 2 1 2

1 1 2

(0.97 ); ; 0.97m RT m RT m RTV V T TMP MP MP

12

300 K0.97 0.97TT

; T2 = 309 K

274

solucionario capitulo 19 - paul e. tippens

Documents

water vapor

constant pressure

relative humidity

absolute pressure

ideal gas

v1 v2

thermal properties

steel cylinder