Solved: (II) It is observed that 55.50 mL of water at 20°C

1. (I) How many atoms are there in a 3.4-gram copper penny?

2. (I) How does the number of atoms in a 26.5-gram gold ring compare to the number in a silver ring of the same mass?

3. (I) (a) Room temperature" is often taken to be 68F. What is this on the Celsius scale? (b) 1'he temperature of the filament in a lightbulb is about 1800C. What is this on the Fahrenheit scale?

4. (I) Among the highest and lowest temperatures recorded are 136F in the Libyan desert and -129F in Antarctica. What are these temperatures on the Celsius scale?

5. (I) {a) 15 below zero on the Celsius scale is what Fahrenheit temperature? (/>) 15 below zero on the Fahrenheit scale is what Celsius temperature?

6. (II) In an alcohol-in-glass thermometer, the alcohol column has length 11.82 cm at 0.0C and length 22.85 cm at 100.0G What is the temperature if the column has length (a) 16.70 cm, and (b) 20.50cm?

7. (I) A concrete highway is built of slabs 12 m long (20nC). How wide should the expansion cracks between the slabs be (at 20C) to prevent buckling if the range of temperature is 30C to +50C?

8. (I) Super Invar, an alloy of iron and nickel, is a strong material with a very low coefficient of linear expansion |0.2 X 10"6 (C)-1]. A 2.0-m-long tabletop made of this alloy is used for sensitive laser measurements w'here extremely high tolerances arc required. How much will this table expand along its length if the temperature increases 5.0 Cc? Compare to tabletops made of steel.

9. (1) The Riffel Tower (Fig. 13-29) is built of wrought iron approximately 300 m tall. Estimate how much its height changes between July (average temperature of 25C) and January1 (average temperature of 2C). Ignore the angles of the iron beams, and treat the tower as a vertical beam.

10. (II) To make a secure fit, rivets that are larger than the rivet hole are often used and the rivet is cooled (usually in dry ice) before it is placed in the hole. A steel rivet 1.871 cm in diameter is to be placed in a hole 1.869 cm in diameter at 20C. To what temperature must the rivet be cooled if it is to fit in the hole?

11. (II) The density of water at 4C is 1.00 x 103kg/m\ What is water's density at 94C?

12. (II) A quartz sphere is 8.75 cm in diameter. What will be its change in volume if it is heated from 30nC to 200C?

13. (II) An ordinary glass is filled to the brim with 350.0mL of water at 100.0X. If the temperature decreased to 20.0C, how much water could be added to the glass?

14. (II) It is observed that 55.50 mL of water at 20C completely fills a container to the brim. When the container and the wrater are heated to 60C. 0.35 g of wrater is lost, (a) What is the coefficient of volume expansion of the container? (b) What is the most likely material of the container? The density of w ater at 60C is 0.98324 g/mL.

15. (II) (a) A brass plug is to be placed in a ring made of iron. At 20WC, the diameter of the plug is 8.753 cm and that of the inside of the ring is 8.743 cm. They must both be brought to what common temperature in order to fit? (b) What if the plug wrere iron and the ring brass?

16. (II) If a fluid is contained in a long, narrow vessel so it can expand in essentially one direction only, show that the effective coefficient of linear expansion a is approximately equal to the coefficient of volume expansion /3.

17. (II) (a) Show- that the change in the density p of a substance, when the temperature changes by A T. is given by Ap = -pp A T. (b) What is the fractional change in density of a lead sphere w'hose temperature decreases from 25C to -40C?

18. (II) A uniform rectangular plate of length / and width w has coefficient of linear expansion a. Showr that, if we neglect very small quantities, the change in area of the plate due to a temperature change A T is

19. (Ill) Showr that for an isotropic solid, p = 3a. if the amount of expansion is small, p and a arc the coefficients of volume and linear expansion, respectively. [Hint. Consider a cubical solid, and neglect very' small quantities. See also Problem 18 and Fig. 1330.]

20. (Ill) The pendulum in a grandfather clock is made of brass and keeps perfect time at 17C. Howr much time is gained or lost in a year if the clock is kept at 25C? (Assume the frequency dependence on length for a simple pendulum applies.)

21. (Ill) (a) Hie lube of a mercury thermometer has an inside diameter of 0.140 mm. The bulb has a volume of 0.255 cm3. Howr far will the thread of mercury move when the temperature changes from 11.5C to 33.0C? lake into account expansion of the Pyrex glass. (/>) Determine a formula for the change in length of the mercury column in terms of relevant variables. Ignore tube volume compared to bulb volume.

22. (Ill) A 23.4-kg solid aluminum cylindrical wheel of radius 0.41 m is rotating about its axle on frictionless bearings with angular velocity w = 32.8rad/s. If its temperature is now raised from 20.0C to 75.0CC. wrhat is the fractional change in </?

23. (II) An aluminum bar has the desired length when at 15C. How much stress is required to keep it at this length if the temperature increases to 35C?

24. (II) (a) A horizontal steel I-beam of cross-sectional area 0.041 m2 is rigidly connected to two vertical steel girders. If the beam was installed when the temperature was 30C, what stress is developed in the beam when the temperature drops to -30"C? (b) Is the ultimate strength of the steel exceeded? (c) What stress is developed if the beam is concrete and has a cross-sectional area of 0.13 m2? Will it fracture?

25. (Ill) A barrel of diameter 134.122cm at 20UC is to be enclosed by an iron band. The circular band has an inside diameter of 134.110 cm at 20C It is 7.4 cm wide and 0.65 cm thick, (a) To what temperature must the band be heated so that it will fit over the barrel? (/?) What will be the tension in the band when it cools to 20UC?

26. (I) What arc the following temperatures on the Kelvin scale: () 86C. (b) 78F. (c) -100UC. (d) 5500C (e) -459F?

27. (I) Absolute zero is what temperature on the Fahrenheit scale?

28. (II) Typical temperatures in the interior of the Earth and Sun are about 4000C and 15 X 106 C, respectively. (a) What are these temperatures in kelvins? (6) What percent error is made in each case if a person forgets to change C to K ?

29. (I) If 3.00 m3 of a gas initially at STP is placed under a pressure of 3.20 atm. the temperature of the gas rises to 38.0C. What is the volume?

30. (I) In an internal combustion engine, air at atmospheric pressure and a temperature of about 20C is compressed in the cylinder by a piston to | of its original volume (compression ratio = 9.0). Estimate the temperature of the compressed air. assuming the pressure reaches 40 atm.

31. (II) Calculate the density of oxygen at STP using the ideal gas law.

32. (II) A storage tank contains 21.6 kg of nitrogen (N2) at an absolute pressure of 3.65 atm. What will the pressure be if the nitrogen is replaced by an equal mass of C02?

33. (II) A storage tank at STP contains 18.5 kg of nitrogen (N2). (a) What is the volume of the tank? (b) What is the pressure if an additional 15.0 kg of nitrogen is added without changing the temperature?

34. (II) If 18.75 mol of helium gas is at 10.0C and a gauge pressure of 0.350 atm. (a) calculate the volume of the helium gas under these conditions. (/?) Calculate the temperature if the gas is compressed to precisely half the volume at a gauge pressure of 1.00 atm.

35. (II) What is the pressure inside a 35.0-L container holding 105.0 kg of argon gas at 385 K?

36. (II) A tank contains 26.0 kg of O? gas at a gauge pressure of 8.70 atm. If the oxygen is replaced by helium, how many kilograms of the latter will be needed to produce a gauge pressure of 7.00atm?

37. (II) A hot-air balloon achieves its buoyant lift by heating the air inside the balloon, which makes it less dense than the air outside. Suppose the volume of a balloon is 1800 m3 and the required lift is 2700 N (rough estimate of the weight of the equipment and passenger). Calculate the temperature of the air inside the balloon which will produce the required lift. Assume that the outside air temperature is 0!>C and that air is an ideal gas under these conditions. What factors limit the maximum altitude attainable by this method for a given load? (Neglect variables like wind.)

38. (II) A tire is filled with air at 15C to a gauge pressure of 220 kPa. If the tire reaches a temperature of 38cC. what fraction of the original air must be removed if the original pressure of 220 kPa is to be maintained?

39. (II) If 61.5 L of oxygen at 18.0C and an absolute pressure of 2.45 atm are compressed to 48.8 L and at the same time the temperature is raised to 50.0C. what will the new pressure be?

40. (Ill) A helium-filled balloon escapes a childs hand at sea level and 20.0C. When it reaches an altitude of 3000 m, where the temperature is 5.0' C and the pressure is only 0.70 atm, how will its volume compare to that at sea level?

41. (I) Calculate the number of molecules/m3 in an ideal gas at STP.

42. (I) How many moles of water arc there in 1.000 L? How many molecules?

43. (II) Estimate the number of (a) moles, and (b) molecules of water in all the Earth's oceans. Assume water covers 75% of the Earth to an average depth of 3 km.

44. (II) A cubic box of volume 5.1 X 10 2 m3 is filled with air at atmospheric pressure at 20C. 'ITie box is closed and heated to 180C What is the net force on each side of the box?

45. (Ill) Estimate how many molecules of air are in each 2.0-L breath you inhale that were also in the last breath Galileo took. [Hint: Assume the atmosphere is about 10 km high and of constant density.]

46. (I) (a) What is the average translational kinetic energy of an oxygen molecule at STP? (b) What is the total translational kinetic energy of 2.0 mol of 02 molecules at 20C?

47. (I) Calculate the rms speed of helium atoms near the surface of the Sun at a temperature of about 6000 K.

48. (I) By what factor will the rms speed of gas molecules increase if the temperature is increased from 0C to 100C?

49. (I) A gas is at 20C. To what temperature must it be raised to double the rms speed of its molecules?

50. (I) Twelve molecules have the following speeds, given in units of km/s: 6, 2. 4, 6, 0, 4. 1. 8. 5. 3. 7. and 8. Calculate the rms speed.

51. (II) The rms speed of molecules in a gas at 20.0C is to be increased by 1.0%. To what temperature must it be raised ?

52. (II) If the pressure of a gas is doubled while its volume is held constant, by what factor does urms change?

53. (II) Show that the rms speed of molecules in a gas is given by t?rim = \3P/p. where P is the pressure in the gas. and p is the gas density.

54. (II) Show that for a mixture of two gases at the same temperature, the ratio of their rms speeds is equal to the inverse ratio of the square roots of their molecular masses.

55. (II) What is the rms speed of nitrogen molecules contained in an 8.5-m3 volume at 2.1 atm if the total amount of nitrogen is 1300 mol?

56. (II) Calculate (tf) the rms speed of an oxygen molecule at 0C and (b) determine how many times per second it would move back and forth across a 7.0-m-long room on the average, assuming it made very few collisions with other molecules.

57. (II) What is the average distance between nitrogen molecules at STP?

58. (II) (a) Estimate the rms speed of an amino acid whose molecular mass is 89 u in a living cell at 37cC. (b) What would be the rms speed of a protein of molecular mass 50.000 u at 37C?

59. (II) Show that the pressure P of a gas can be written P = \pv2, where p is the density of the gas and v is the rms speed of the molecules.

60. (Ill) 'Ihe two isotopes of uranium. and 238U (the superscripts refer to their atomic mass), can be separated by a gas-diffusion process by combining them with fluorine to make the gaseous compound UF6. Calculate the ratio of the rms speeds of these molecules for the two isotopes, at constant T.

61. (I) (a) At atmospheric pressure, in what phases can CO2 exist? (b) For what range of pressures and temperatures can CO2 be a liquid? Refer to Fig. 13-21.

62. (I) Water is in which phase when the pressure is 0.01 atm and the temperature is (a) 90C. (b) -20"C?

63. (I) What is the dew point (approximately) if the humidity is 50% on a day when the temperature is 25C?

64. (I) What is the air pressure at a place where water boils at 90C?

65. (I) If the air pressure at a particular place in the mountains is 0.72 atm. estimate the temperature at which water boils.

66. (I) What is the temperature on a day when the partial pres sure of water is 530 Pa and the relative humidity is 40% ?

67. (I) What is the partial pressure of water on a day when the temperature is 25C and the relative humidity is 35%?

68. (I) What is the approximate pressure inside a pressure cooker if the water is boiling at a temperature of 120"C? Assume no air escaped during the heating process, which started at 20C.

69. (II) If the humidity in a room of volume 680 m3 at 25C is 80%. what mass of water can still evaporate from an open pan ?

70. (Ill) Air that is at its dew point of 5C is drawn into a building where it is heated to 25C. What will be the relative humidity at this temperature? Assume constant pressure of l.O atm.Take into account the expansion of the air.

71. (II) Estimate the time needed for a glycine molecule (see Table 13-4) to diffuse a distance of 15 pm in water at 20CC if its concentration varies over that distance from 1.00 mol/m3 to 0.40mol/mJ. Compare this speed to its rms (thermal) speed. The molecular mass of glycine is alxiut 75 u.

72. (II) Oxygen diffuses from the surface of insects to the interior through tiny tubes called tracheae. An average trachea is about 2 mm long and has cross-sectional area of 2 X 10-I>m2. Assuming the concentration of oxygen inside is half what it is outside in the atmosphere. (a) show that the concentration of oxygen in the air (assume 21%. is oxygen) at 20C is about 8.7 mol/m3. then (b) calculate the diffusion rate ./. and (c) estimate the average time for a molecule to diffuse in. Assume the diffusion constant is 1 x 10~5m2/s.

73. A precise steel tape measure has been calibrated at 20C. At 34C. (a) will it read high or low. and (b) what will be the percentage error?

74. A Pyrcx measuring cup was calibrated at normal room temperature. How much error will be made in a recipe calling for 300 mL of cool water, if the water and the cup are hot, at 80<!C, instead of at 20"C? Neglect the glass expansion.

75. The gauge pressure in a helium gas cylinder is initially 28 atm. After many balloons have been blown up. the gauge pressure has decreased to 5 atm. What fraction of the original gas remains in the cylinder?

76. Estimate the number of air molecules in a room of length 6.5 m. width 3.1 m, and height 2.5 m. Assume the temperature is 22C. How many moles does that correspond to?

77. In outer space the density of matter is about one atom per cm3, mainly hydrogen atoms, and the temperature is about 2.7 K. Calculate the rms speed of these hydrogen atoms, and the pressure (in atmospheres).

78. The lowest pressure attainable using the best available vacuum techniques is about 10_,2N/m2. At such a pressure. how many molecules are there per cm3 at 0C?

79. If a scuba diver fills his lungs to full capacity of 5.5 L when 10 m below the surface, to what volume would his lungs expand if he quickly rose to the surface? Is this advisable?

80. A space vehicle returning from the Moon enters Earth's atmosphere at a speed of about 40.000 km/h. Molecules (assume nitrogen) striking the nose of the vehicle with this speed correspond to what temperature? (Because of this high temperature, the nose of a space vehicle must be made of special materials; indeed, part of it docs vaporize, and this is seen as a bright blaze upon reentry.)

81. The temperature of an ideal gas is increased from 110C to 360"C while the volume and the number of moles stay constant. By what factor does the pressure change? By what factor does i>rim change?

82. A house has a volume of 770 m \ (a) What is the total mass of air inside the house at 20C? (b) If the temperature drops to 10C\ what mass of air enters or leaves the house?

83. From the known value of atmospheric pressure at the surface of the Earth, estimate the total number of air molecules in the Earth's atmosphere.

84. What is the rms speed of nitrogen molecules contained in a 7.6-m3 volume at 4.2 atm if the total amount of nitrogen is 1800 mol?

85. A standard cylinder of oxygen used in a hospital has gauge pressure = 2000 psi (13,800 kPa) and volume = 16 L (0.016 m3) at T = 295 K. How long will the cylinder last if the flow rate, measured at atmospheric pressure, is constant at 2.4 L/min?

86. An iron cube floats in a bowl of liquid mercury at 0C. (a) If the temperature is raised to 25C, will the cube float higher or lower in the mercury? (b) By what percent will the fraction of volume submerged change?

87. The density of gasoline at 0C is 0.68 X 103 kg/m3. What is the density on a hot day. when the temperature is 38C? What is the percentage change?

88. If a steel band were to fit snugly around the Earth's equator at 25C but then wras heated to 45C how high above the Earth would the band be (assume equal everywhere)?

89. A brass lid screw-s tightly onto a glass jar at 20"C. To help open the jar. it can be placed into a bath of hot water. After this treatment, the temperatures of the lid and the jar are both 60"C. The inside diameter of the lid is 8.0 cm at 20C. Find the size of the gap (difference in radius) that develops by this procedure.

90. The first length standard, adopted in the 18,h century. wras a platinum bar with two very fine marks separated by what was defined to be exactly 1 m. If this standard bar was to be accurate to within 1.0/xm, how carefully wrould the trustees have needed to control the temperature? 'ITie coefficient of linear expansion for platinum is 9 X 10"6Co_I.

91. A scuba tank, when fully charged, has a pressure of 195 atm at 20C. The volume of the tank is 11.3 L. (a) What wrould the volume of the air be at 1.00 atm and at the same temperature? (6) Before entering the water, a person consumes 2.0 L of air in each breath, and breathes 12 times a minute. At this rate, how long wrould the tank last? (c) At a depth of 20.0 m of sea water and temperature of 10WC. howr long would the same tank last assuming the breathing rate does not change?

92. The escape speed from the Earth is 1.12 x 104 m/s, so a gas molecule travelling away from Earth near the outer boundary of the Earths atmosphere would, at this speed, be able to escape from the Earth's gravitational field. At wfhat temperature is the average speed of () oxygen molecules, and (b) helium atoms equal to 1.12 X 104m/s? (c) Can you see wrhy our atmosphere contains oxygen but not helium?

93. A 1.0-kg trash-can lid is suspended against gravity by tennis balls thrown vertically upward at it. Howr many tennis balls per second must rebound from the lid elastically. assuming they have a mass of 0.060 kg and are thrown at 12 m/s?

94. A scuba diver releases a 3.00-cm-diameter (spherical) bubble of air from a depth of 14.0 m in a lake. Assume the temperature is constant at 298 K. and the air behaves as a perfect gas. How large is the bubble wrhcn it reaches the surface?

95. Calculate the total w ater vapor pressure in the air on the following two days: (a) a hot summer day. with the temperature 30C and the relative humidity at 40%; () a cold winter day. with the temperature 5CC and the relative humidity at 80%.

96. A sauna has 7.0 m3 of air volume, and the temperature is 90C. The air is perfectly dry. How much water (in kg) should be evaporated if we want to increase the relative humidity from 0% to 10%? (See Table 13-3.)

97. Estimate the percent difference in the density of iron at STP. and wrhen it is a solid deep in the Earth where the temperature is 2000C and under 5000 atm of pressure. Assume the bulk modulus (90 x 109N/m2) and the coefficient of volume expansion do not vary with temperature and arc the same as at STP.

98. (a) Use the ideal gas lawr to show that, for an ideal gas at constant pressure, the coefficient of volume expansion is equal to /3 = l/7\ where T is the temperature in kelvins. Compare to Table 13-1 for gases at / = 293 K. (6) Show that the bulk modulus (Section 9-5) for an ideal gas held at constant temperature is B = P, where P is the pressure.

99. In humid climates, people constantly dehumidify their cellars to prevent rot and mildew. If the cellar in a house (kept at 20CC) has 95 m2 of floor space and a ceiling height of 2.8 m, wrhat is the mass of water that must be removed from it to drop the humidity from 95%: to a more reasonable 30%?

Problem 15P (a) A brass plug is to be placed in a ring made of iron. At 20°C, the diameter of the plug is 8.753 cm and that of the inside of the ring is 8.743 cm. They must both be brought to what common temperature in order to fit? (b) What if the plug were iron and the ring brass?

Problem 15Q Freezing a can of soda will cause its bottom and top to bulge so badly the can will not stand up. What has happened?

Problem 16P If a fluid is contained in a long, narrow vessel so it can expand in essentially one direction only, show that the effective coefficient of linear expansion ? is approximately equal to the coefficient of volume expansion ?.

Problem 16Q When a gas is rapidly compressed (say, by pushing down a piston), its temperature increases. When a gas expands against a piston, it cools. Explain these changes in temperature using the kinetic theory, in particular noting what happens to the momentum of molecules when they strike the moving piston.

Problem 17P (a) Show that the change in the density ? of a substance, when the temperature changes by ?T, is given by ?? = –?? ?T. (b) What is the fractional change in density of a lead sphere whose temperature decreases from 25°C to –40°C?

Problem 55P What is the rms speed of nitrogen molecules contained in an 8.5-m3 volume at 2.1 atm if the total amount of nitrogen is 1300 mol?

Problem 56P Calculate (a) the rms speed of an oxygen molecule at 0°C and (b) determine how many times per second it would move back and forth across a 7.0-m-long room on the average, assuming it made very few collisions with other molecules.

Problem 57P What is the average distance between nitrogen molecules at STP?

Problem 59P Show that the pressure P of a gas can be written where ? is the density of the gas and v is the rms speed of the molecules.

Problem 58P (a) Estimate the rms speed of an amino acid whose molecular mass is 89 u in a living cell at 37°C. (b) What would be the rms speed of a protein of molecular mass 50,000 u at 37°C?

Problem 17Q Will the buoyant force on an aluminum sphere submerged in water increase, decrease, or remain the same, if the temperature is increased from 20°C to 40°C? Explain.

(II) A uniform rectangular plate of length l and width w has coefficient of linear expansion ?. Show that, if we neglect very small quantities, the change in area of the plate due to a temperature change ?T is ?A = 2?lw ?T. See Fig. 13–30.

Problem 60P (III) The Two isotopes of uranium, 235U and 238U (the superscripts refer to their atomic mass), can be separated by a gas diffusion process by combining them with fluorine to make the gaseous compound UF6. Calculate the ratio of the rms speeds of these molecules for the two isotopes, at constant T.

(I) (a) At atmospheric pressure, in what phases can CO2 exist? (b) For what range of pressures and temperatures can CO2 be a liquid? Refer to Fig. 13–21.

Problem 12P A quartz sphere is 8.75 cm in diameter. What will be its change in volume if it is heated from 30°C to 200°C?

Problem 12Q A glass container may break if one part of it is heated or cooled more rapidly than adjacent parts. Explain.

Problem 13Q The principal virtue of Pyrex glass is that its coefficient of linear expansion is much smaller than that for ordinary glass (Table 13–1). Explain why this gives rise to the higher heat resistance of Pyrex.

Problem 14Q Will a grandfather clock, accurate at 20°C, run fast or slow on a hot day (30°C)? Explain. The clock uses a pendulum supported on a long, thin brass rod.

Problem 50P Twelve molecules have the following speeds, given in units of km/s: 6, 2, 4, 6, 0, 4, 1, 8, 5, 3, 7, and 8. Calculate the rms speed.

Problem 49P (I) A gas is at 20°C. To what temperature must it be raised to triple the rms speed of its molecules?

Problem 51P The rms speed of molecules in a gas at 20.0°C is to be increased by 1.0%. To what temperature must it be raised?

Problem 52P If the pressure of a gas is doubled while its volume is held constant, by what factor does vrmschange?

(II) Show that the rms speed of molecules in a gas is given by rms = where P is the pressure in the gas and is the gas density.

Problem 54P Show that for a mixture of two gases at the same temperature, the ratio of their rms speeds is equal to the inverse ratio of the square roots of their molecular masses.

Problem 13P An ordinary glass is filled to the brim with 350.0 mL of water at 100.0°C. If the temperature decreased to 20.0°C, how much water could be added to the glass?

Problem 14P (II) It is observed that 55.50 mL of water at 20°C completely fills a container to the brim. When the container and the water are heated to 60°C, 0.35 g of water is lost. (a) What is the coefficient of volume expansion of the container? (b) What is the most likely material of the container? Density of water at 60°C is 0.98324 g/mL.

Problem 18Q Explain in words how Charles’s law follows from kinetic theory and the relation between average kinetic energy and the absolute temperature.

(III) Show that for an isotropic solid, , if the amount of expansion is small. and are the coefficients of volume and linear expansion, respectively. [Hint: Consider a cubical solid, and neglect very small quantities. See also Problem 18 and Fig. 13-30.]

Problem 20P The pendulum in a grandfather clock is made of brass and keeps perfect time at 17°C. How much time is gained or lost in a year if the clock is kept at 25°C? (Assume the frequency dependence on length for a simple pendulum applies.)

Problem 20Q As you go higher in the Earth’s atmosphere, the ratio of N2 molecules to O2 molecules increases. Why?

Problem 62P (I) Water is in which phase when the pressure is 0.01 atm and the temperature is (a) 90°C, (b) -20°C?

Problem 63P What is the dew point (approximately) if the humidity is 50% on a day when the temperature is 25°C?

Problem 64P What is the air pressure at a place where water boils at 90°C?

Problem 65P If the air pressure at a particular place in the mountains is 0.72 atm, estimate the temperature at which water boils.

Problem 66P What is the temperature on a day when the partial pressure of water is 530 Pa and the relative humidity is 40%?

Problem 19Q Explain in words how Gay-Lussac’s law follows from kinetic theory.

Problem 21P (a) The tube of a mercury thermometer has an inside diameter of 0.140 mm. The bulb has a volume of 0.255 cm3. How far will the thread of mercury move when the temperature changes from 11.5°C to 33.0°C? Take into account expansion of the Pyrex glass. (b) Determine a formula for the change in length of the mercury column in terms of relevant variables. Ignore tube volume compared to bulb volume.

Problem 21Q Escape velocity for the Earth refers to the minimum speed an object must have to leave the Earth and never return. The escape velocity for the Moon is about one-fifth what it is for the Earth due to the Moon’s smaller mass. Explain why the Moon has practically no atmosphere.

Problem 22P A 23.4-kg solid aluminum cylindrical wheel of radius 0.41 m is rotating about its axle on frictionless bearings with angular velocity ? = 32.8rad/s. If its temperature is now raised from 20.0°C to 75.0°C, what is the fractional change in ??

Problem 22Q Alcohol evaporates more quickly than water at room temperature. What can you infer about the molecular properties of one relative to the other?

Problem 23P An aluminum bar has the desired length when at 15°C. How much stress is required to keep it at this length if the temperature increases to 35°C?

Problem 67P What is the partial pressure of water on a day when the temperature is 25°C and the relative humidity is 35%?

Problem 68P What is the approximate pressure inside a pressure cooker if the water is boiling at a temperature of 120°C? Assume no air escaped during the heating process, which started at 20°C.

Problem 69P If the humidity in a room of volume 680 m3 at 25°C is 80%, what mass of water can still evaporate from an open pan?

Problem 70P Air that is at its dew point of 5°C is drawn into a building where it is heated to 25°C. What will be the relative humidity at this temperature? Assume constant pressure of 1.0 atm. Take into account the expansion of the air.

Problem 71P Estimate the time needed for a glycine molecule (see Table 13–4) to diffuse a distance of 15 ?m in water at 20°C if its concentration varies over that distance from 1.00 mol/m3 to 0.40 mol/m3. Compare this “speed” to its rms (thermal) speed. The molecular mass of glycine is about 75 u. Table 13–4 Diffusion Constants, D (20ºC, 1 atm) Diffusing Molecules Medium D(m2/s) H2 Air 6.3 × 10–5 O2 Air 1.8 × 10–5 O2 Water 100 × 10–11 Blood hemoglobin Water 100 × 10–11 Glycine (an amino acid) Water 95 × 10–11 DNA (mass 6 × 106 u) Water 0.13 × 10–11

Problem 23Q Explain why a hot humid day is far more uncomfortable than a hot dry day at the same temperature.

Problem 24P (a) A horizontal steel I-beam of cross-sectional area 0.041 m2 is rigidly connected to two vertical steel girders. If the beam was installed when the temperature was 30°C, what stress is developed in the beam when the temperature drops to –30°C? (b) Is the ultimate strength of the steel exceeded? (c) What stress is developed if the beam is concrete and has a cross-sectional area of 0.13 m2? Will it fracture?

Problem 24Q Is it possible to boil water at room temperature (20°C) without heating it? Explain.

Problem 72P (II) Oxygen diffuses from the surface of insects to the interior through tiny tubes called tracheae. An average trachea is about 2 mm long and has cross-sectional area of 2X 10-9 m2 Assuming the concentration of oxygen inside is half what it is outside in the atmosphere, (a) show that the concentration of oxygen in the air (assume 21% is oxygen) at 20°C is about 8.7 mol/m3 then (b) calculate the diffusion rate J, and (c) estimate the average time for a molecule to diffuse in. Assume the diffusion constant is 1X 10-5 m2/s.

Problem 25P A barrel of diameter 134.122 cm at 20°C is to be enclosed by an iron band. The circular band has an inside diameter of 134.110 cm at 20°C. It is 7.4 cm wide and 0.65 cm thick. (a) To what temperature must the band be heated so that it will fit over the barrel? (b) What will be the tension in the band when it cools to 20°C?

Problem 25Q Consider two days when the air temperature is the same but the humidity is different. Which is more dense, the dry air or the humid air at the same T? Explain.

Problem 73GP A precise steel tape measure has been calibrated at 20°C. At 34°C, (a) will it read high or low, and (b) what will be the percentage error?

Problem 74GP A Pyrex measuring cup was calibrated at normal room temperature. How much error will be made in a recipe calling for 300 mL of cool water, if the water and the cup are hot, at 80°C, instead of at 20°C? Neglect the glass expansion.

Problem 76GP Estimate the number of air molecules in a room of length 6.5 m, width 3.1 m, and height 2.5 m. Assume the temperature is 22°C. How many moles does that correspond to?

Problem 75GP The gauge pressure in a helium gas cylinder is initially 28 atm. After many balloons have been blown up, the gauge pressure has decreased to 5 atm. What fraction of the original gas remains in the cylinder?

Problem 1P How many atoms are there in a 3.4-gram copper penny?

Problem 1Q Which has more atoms: 1 kg of iron or 1 kg of aluminum? See the Periodic Table or Appendix B.

Problem 26P Gas Laws; Absolute Temperature What are the following temperatures on the Kelvin scale; (a) 86°C, (b) 78°F, (c) –100°C, (d) 5500°C, (e) –459°F?

Problem 26Q Explain why it is dangerous to open the radiator cap of an overheated automobile engine.

Problem 27P (I) Absolute zero is what temperature on the Fahrenheit scale?

Problem 27Q Why does exhaled air appear as a little white cloud in the winter (Fig. 13–30)?

Problem 28P (II) Typical temperatures in the interior of the Earth and Sun are about 4000°C and 15 X 106 °C respectively. (a) What are these temperatures in kelvins? (b) What percent error is made in each case if a person forgets to change °C to K?

Problem 77GP In outer space the density of matter is about one atom per cm3 mainly hydrogen atoms, and the temperature is about 2.7 K. Calculate the rms speed of these hydrogen atoms, and the pressure (in atmospheres).

Problem 78GP (II) The lowest pressure attainable using the best available vacuum techniques is about 10-12N/m2, At such a pressure, how many molecules are there per cm3 at 0°C?

Problem 79GP If a scuba diver fills his lungs to full capacity of 5.5 L when 10 m below the surface, to what volume would his lungs expand if he quickly rose to the surface? Is this advisable?

Problem 80GP A space vehicle returning from the Moon enters Earth’s atmosphere at a speed of about 40,000 km/h. Molecules (assume nitrogen) striking the nose of the vehicle with this speed correspond to what temperature? (Because of this high temperature, the nose of a space vehicle must be made of special materials; indeed, part of it does vaporize, and this is seen as a bright blaze upon reentry.)

Problem 81GP The temperature of an ideal gas is increased from 110°C to 360°C while the volume and the number of moles stay constant. By what factor does the pressure change? By what factor does vrms change?

Problem 2P How does the number of atoms in a 26.5-gram gold ring compare to the number in a silver ring of the same mass?

Problem 2Q Name several properties of materials that could be used to make a thermometer.

Problem 3P (a) “Room temperature” is often taken to be 68°F. What is this on the Celsius scale? (b) The temperature of the filament in a lightbulb is about 1800°C. What is this on the Fahrenheit scale?

Problem 3Q Which is larger, 1 C° or 1 F°? Explain why.

Problem 4P (I) Among the highest and lowest natural air temperatures claimed are 136°F in the Libyan desert and-129°F in Antarctica. What are these temperatures on the Celsius scale?

Problem 29P (I) If 3.50 m3 of a gas initially at STP is placed under a pressure of 3.20 atm, the temperature of the gas rises to 38.0°C. What is the volume?

(I) In an internal combustion engine, air at atmospheric pressure and a temperature of about 20°C is compressed in the cylinder by a piston to of its original volume (compression ratio = 9.0). Estimate the temperature of the compressed air, assuming the pressure reaches 40 atm.

Problem 31P Calculate the density of oxygen at STP using the ideal gas law.

Problem 32P A storage tank contains 21.6 kg of nitrogen (N2) at an absolute pressure of 3.65 atm. What will the pressure be if the nitrogen is replaced by an equal mass of CO2 ?

Problem 33P A storage tank at STP contains 18.5 kg of nitrogen (N2). (a) What is the volume of the tank? (b) What is the pressure if an additional 15.0 kg of nitrogen is added without changing the temperature?

Problem 82GP A house has a volume of 770 m3. (a) What is the total mass of air inside the house at 20°C? (b) If the temperature drops to –10°C, what mass of air enters or leaves the house?

Problem 83GP From the known value of atmospheric pressure at the surface of the Earth, estimate the total number of air molecules in the Earth’s atmosphere.

Problem 84GP What is the rms speed of nitrogen molecules contained in a 7.6-m3 volume at 4.2 atm if the total amount of nitrogen is 1800 mol?

Problem 85GP A standard cylinder of oxygen used in a hospital has gauge pressure = 2000 psi (13,800 kPa) and volume = 16 L (0.016 m3) at T – 295 K. How long will the cylinder last if the flow rate, measured at atmospheric pressure, is constant at 2.4 L/min?

Problem 86GP An iron cube floats in a bowl of liquid mercury at 0°C. (a) If the temperature is raised to 25°C, will the cube float higher or lower in the mercury? (b) By what percent will the fraction of volume submerged change?

Problem 37P A hot-air balloon achieves its buoyant lift by heating the air inside the balloon, which makes it less dense than the air outside. Suppose the volume of a balloon is 1800 m3 and the required lift is 2700 N (rough estimate of the weight of the equipment and passenger). Calculate the temperature of the air inside the balloon which will produce the required lift. Assume that the outside air temperature is 0°C and that air is an ideal gas under these conditions. What factors limit the maximum altitude attainable by this method for a given load? (Neglect variables like wind.)

Problem 38P A tire is filled with air at 15°C to a gauge pressure of 220 kPa. If the tire reaches a temperature of 38°C, what fraction of the original air must be removed if the original pressure of 220 kPa is to be maintained?

Problem 87GP The density of gasoline at 0°C is 0.68 Times 103 kg/m3. What is the density on a hot day, when the temperature is 38°C? What is the percentage change?

Problem 88GP If a steel band were to fit snugly around the Earth’s equator at 25°C, but then was heated to 45°C, how high above the Earth would the band be (assume equal everywhere)?

Problem 89GP A brass lid screws tightly onto a glass jar at 20°C. To help open the jar, it can be placed into a bath of hot water. After this treatment, the temperatures of the lid and the jar are both 60°C. The inside diameter of the lid is 8.0 cm at 20°C. Find the size of the gap (difference in radius) that develops by this procedure.

Problem 90GP The first length standard, adopted in the 18th century, was a platinum bar with two very fine marks separated by what was defined to be exactly 1m. If this standard bar was to be accurate to within ± 1.0 ?m, how carefully would the trustees have needed to control the temperature? The coefficient of linear expansion for platinum is 9 × 10–6 C°–1.

Problem 91GP A scuba tank, when fully charged, has a pressure of 195 atm at 20°C. The volume of the tank is 11.3 L. (a) What would the volume of the air be at 1.00 atm and at the same temperature? (b) Before entering the water, a person consumes 2.0 L of air in each breath, and breathes 12 times a minute. At this rate, how long would the tank last? (c) At a depth of 20.0 m of sea water and temperature of 10°C, how long would the same tank last assuming the breathing rate does not change?

Problem 4Q If system A is in thermal equilibrium with system B, but B is not in thermal equilibrium with system C, what can you say about the temperatures of A, B, and C?

Problem 5P (I) (a) 18° below zero on the Celsius scale is what Fahrenheit temperature? (b) 18° below zero on the Fahrenheit scale is what Celsius temperature?

A flat bimetallic strip consists of aluminum riveted to a strip of iron. When heated, the strip will bend. Which metal will be on the outside of the curve? [Hint: See Table 13–1.] Why?

Problem 6P In an alcohol-in-glass thermometer, the alcohol column has length 11.82 cm at 0.0°C and length 22.85 cm at 100.0°C. What is the temperature if the column has length (a) 16.70 cm, and (b) 20.50 cm?

Problem 6Q In the relation ?L = ?L0 ?T, should L0 be the initial length, the final length, or does it matter? Explain.

Problem 34P If 18.75 mol of helium gas is at 10.0°C and a gauge pressure of 0.350 atm, (a) calculate the volume of the helium gas under these conditions. (b) Calculate the temperature if the gas is compressed to precisely half the volume at a gauge pressure of 1.00 atm.

Problem 35P What is the pressure inside a 35.0-L container holding 105.0 kg of argon gas at 385 K?

Problem 36P A tank contains 26.0 kg of O2 gas at a gauge pressure of 8.70 atm. If the oxygen is replaced by helium, how many kilograms of the latter will be needed to produce a gauge pressure of 7.00 atm?

Problem 7P A concrete highway is built of slabs 12 m long (20°C). How wide should the expansion cracks between the slabs be (at 20°C) to prevent buckling if the range of temperature is –30°C to +50°C?

Problem 7Q The units for the coefficient of linear expansion ? are (Co)-1 and there is no mention of a length unit such as meters. Would the expansion coefficient change if we used feet or millimeters instead of meters? Explain.

Figure 13–27 shows a diagram of a simple bimetallic thermostat used to control a furnace (or other heating or cooling system). The electric switch (attached to the bimetallic strip) is a glass vessel containing liquid mercury that conducts electricity when it touches both contact wires. Explain how this device controls the furnace and how it can be set at different temperatures.

(I) The Eiffel Tower (Fig. 13–29) is built of wrought iron approximately 300 m tall. Estimate how much its height changes between January (average temperature of 2°C) and July (average temperature of 25°C). Ignore the angles of the iron beams and treat the tower as a vertical beam. FIGURE 13-29 Problem 9. The Eiffel Tower in Paris

Problem 39P If 61.5 L of oxygen at 18.0°C and an absolute pressure of 2.45 atm are compressed to 48.8 L and at the same time the temperature is raised to 50.0°C, what will the new pressure be?

Problem 40P A helium-filled balloon escapes a child’s hand at sea level and 20.0°C. When it reaches an altitude of 3000 m, where the temperature is 5.0°C and the pressure is only 0.70 atm, how will its volume compare to that at sea level?

Problem 41P (I) Calculate the number of molecules/m3 in an ideal gas at STP.

Problem 43P (II) Estimate the number of (a) moles and (b) molecules of water in all the Earth’s oceans. Assume water covers 75% of the Earth to an average depth of 3 km.

Problem 42P How many moles of water are there in 1.000 L? How many molecules?

Problem 92GP The escape speed from the Earth is 1.12*104 m/s, so that a gas molecule traveling away from Earth near the outer boundary of the Earth’s atmosphere would, at this speed, be able to escape from the Earth’s gravitational field and be lost to the atmosphere. At what temperature is the rms speed of (a) oxygen molecules, and (b) helium atoms equal to1.12*104 m/s? (c) Can you explain why our atmosphere contains oxygen but not helium?

Problem 93GP A 1.0-kg trash-can lid is suspended against gravity by tennis balls thrown vertically upward at it. How many tennis balls per second must rebound from the lid elastically, assuming they have a mass of 0.060 kg and are thrown at 12 m/s?

Problem 94GP A scuba diver releases a 3.00-cm-diameter (spherical) bubble of air from a depth of 14.0 m in a lake. Assume the temperature is constant at 298 K, and the air behaves as a perfect gas. How large is the bubble when it reaches the surface?

Problem 96GP A sauna has 7.0 m3 of air volume, and the temperature is 90°C. The air is perfectly dry. How much water (in kg) should be evaporated if we want to increase the relative humidity from 0% to 10%? (See Table 13–3) Table 13–3 Saturated Vapor Pressure of Water Temperature (ºC) Saturated Vapor Pressure torr (= mm-Hg) Pa (= N/m2) –50 0.030 4.0 –10 1.95 2.60 × 102 0 4.58 6.11 × 102 5 6.54 8.72 × 102 10 9.21 1.23 × 103 15 12.8 1.71 × 103 20 17.5 2.33 × 103 25 23.8 3.17 × 103 30 31.8 4.24 × 103 40 55.3 7.37 × 103 50 92.5 1.23 × 104 60 149 1.99 × 104 70† 234 3.12 × 104 80 355 4.73 × 104 90 526 7.01 × 104 100‡ 760 1.01 × 104 120 1489 1.99 × 104 150 3570 4.76 × 104 †Boiling point on summit of Mt. Everest. ‡Boiling point at sea level.

Problem 95GP Calculate the total water vapor pressure in the air on the following two days: (a) a hot summer day, with the temperature 30°C and the relative humidity at 40%; (b) a cold winter day, with the temperature 5°C and the relative humidity at 80%.

Problem 8P (I) Super Invar™, an alloy of iron and nickel, is a strong material with a very low coefficient of thermal expansion (0.20 X 10-6/C°). A 1.8-m-long tabletop made of this alloy is used for sensitive laser measurements where extremely high tolerances are required. How much will this alloy table expand along its length if the temperature increases 6.0 C°? Compare to tabletops made of steel.

Problem 9Q Long steam pipes that are fixed at the ends often have a section in the shape of a U. Why?

Problem 10P To make a secure fit, rivets that are larger than the rivet hole are often used and the rivet is cooled (usually in dry ice) before it is placed in the hole. A steel rivet 1.871 cm in diameter is to be placed in a hole 1.869 cm in diameter at 20°C. To what temperature must the rivet be cooled if it is to fit in the hole?

Problem 10Q A flat, uniform cylinder of lead floats in mercury at 0°C. Will the lead float higher or lower when the temperature is raised? Explain.

Problem 11P The density of water at 4°C is 1.00 × 103 kg/m3. What is water’s density at 94°C?

Problem 11Q When a cold mercury-in-glass thermometer is first placed in a hot tub of water, the mercury initially descends a bit and then rises. Explain.

Problem 45P Estimate how many molecules of air are in each 2.0-L breath you inhale that were also in the last breath Galileo took. [Hint: Assume the atmosphere is about 10 km high and of constant density.]

Problem 46P (a) What is the average translational kinetic energy of an oxygen molecule at STP? (b) What is the total translational kinetic energy of 2.0 mol of O2 molecules at 20°C?

Problem 47P (I) Calculate the rms speed of helium atoms near the surface of the Sun at a temperature of about 6000 K.

Problem 48P By what factor will the rms speed of gas molecules increase if the temperature is increased from 0°C to 100°C?

Problem 97GP Estimate the percent difference in the density of iron at STP, and when it is a solid deep in the Earth where the temperature is 2000°C and under 5000 atm of pressure. Assume the bulk modulus (90 × 109 N/m2) and the coefficient of volume expansion do not vary with temperature and are the same as at STP.

Problem 98GP (a)Use the ideal gas law to show that, for an ideal gas at constant pressure, the coefficient of volume expansion is equal to ? = 1/T, where T is the temperature in kelvins. Compare to Table 13–1 for gases at T = 293 K. (b) Show that the bulk modulus (Section 9–5) for an ideal gas held at constant temperature is B = P, where P is the pressure. Table 13–1 Material Coefficient of Linear Expansion, ? (C°)-1 Coefficient of Volume Expansion, ? (C°)-1 Solids Aluminum 25 × 10–6 75 × 10–6 Brass 19 × 10–6 56 × 10–6 Copper 17 × 10–6 50 × 10–6 Gold 14 × 10–6 42 × 10–6 Iron or steel 12 × 10–6 35 × 10–6 Lead 29 × 10–6 87 × 10–6 Glass (Pyrex®) 3 × 10–6 9 × 10–6 Glass (ordinary) 25 × 10–6 27 × 10–6 Quartz 0.4 × 10–6 1 × 10–6 Concrete and brick ?12 × 10–6 ?36 × 10–6 Marble 1.4 – 3.5 × 10–6 4–10 × 10–6 Liquids Gasoline 950 × 10–6 Mercury 180 × 10–6 Ethyl alcohol 1100 × 10–6 Glycerin 500 × 10–6 Water 210 × 10–6 Gases Air (and most other gases at atmospheric pressure) 3400 × 10–6

Problem 99GP In humid climates, people constantly dehumidify their cellars to prevent rot and mildew. If the cellar in a house (kept at 20°C) has 95 m2 of floor space and a ceiling height of 2.8 m, what is the mass of water that must be removed from it to drop the humidity from 95% to a more reasonable 30%?

Problem 44P A cubic box of volume 5.1 × 10–2 m3 is filled with air at atmospheric pressure at 20°C. The box is closed and heated to 180°C. What is the net force on each side of the box?