58. (d) What is the coefficient of performance of an ideal

1. (I) An ideal gas expands isothermally, performing 3.40 x 101 2 J of work in the process. Calculate (a) the change in internal energy of the gas. and (6) the heat absorbed during this expansion.

2. Use the conservation of energy to explain why the temperature of a gas increases when it is quickly compressed, whereas the temperature decreases when the gas expands.

3. In an isothermal process. 3700 J of work is done by an ideal gas. Is this enough information to tell how much heat has been added to the system? If so. how much?

4. (I) Sketch a PV diagram of the following process: 2.0 L of ideal gas at atmospheric pressure are cooled at constant pressure to a volume of 1.0 L. and then expanded isothermally back to 2.0 L, whereupon the pressure is increased at constant volume until the original pressure is reached.

5. (II) A 1.0-L volume of air initially at 4.5 atm of (absolute) pressure is allowed to expand isothermally until the pressure is 1.0 atm. It is then compressed at constant pressure to its initial volume, and lastly is brought back to its original pressure by heating at constant volume. Draw the process on a PV diagram, including numbers and labels for the axes.

6. (II) The pressure in an ideal gas is cut in half slowly, while being kept in a container with rigid walls. In the process, 265 kJ of heat left the gas. (a) How much work was done during this process? (/>) What was the change in internal energy of the gas during this process?

7. (II) In an engine, an almost ideal gas is compressed adia-batically to half its volume. In doing so, 1850 J of work is done on the gas. (a) How much heat flows into or out of the gas? (b) What is the change in internal energy of the gas? (c) Does its temperature rise or fall?

8. (II) An ideal gas expands at a constant total pressure of 3.0 atm from 400 mL to 660 mL. Heat then flows out of the gas at constant volume, and the pressure and temperature are allowed to drop until the temperature reaches its original value. Calculate (r/) the total work done by the gas in the process, and (b) the total heat flow into the gas.

9. (II) One and one-half moles of an ideal monatomic gas expand adiabatically. performing 7500 J of work in the process. What is the change in temperature of the gas during this expansion?

10. (II) Consider the following two-step process. Heat is allowed to flow out of an ideal gas at constant volume so that its pressure drops from 2.2 atm to 1.4 atm. Then the gas expands at constant pressure, from a volume of 6.8 L to 9.3 L. where the temperature reaches its original value. See Fig. 15-22. Calculate (a) the total work done by the gas in the process. (b) the change in internal energy of the gas in the process, and (c) the total heat flow into or out of the gas.

11. (II) The PV diagram in Fig. 15-23 shows two possible states of a system containing 1.35 moles of a monatomic-ideal gas. (P\ Pi 455 N/m2. Vx = 2.00 m3, V2 = 8.00 m3.) (a) Draw the process which depicts an isobaric expansion from state 1 to state 2, and label this process A. (/>) Find the work done by the gas and the change in internal energy of the gas in process A. (c) Draw the two-step process which depicts an isothermal expansion from state 1 to the volume V2, followed by an isovolumetric increase in temperature to state 2. and label this process B. (d) Find the change in internal energy of the gas for the two-step process B.

12. (Ill) When a gas is taken from a to c along the curved path in Fig. 15-24. the work done by the gas is W = 35 J and the heat added to the gas is Q = 63 J. Along path abc, the work done is W = -48 J. (a) What is Q for path abc? (b) If Pc = \ P^. what is W for path eda? (c) What is Q for path eda? (d) What is Ua - f/c ? (e) If /d - Uc = 5 J, what is Q for path da?

13. (Ill) In the process of taking a gas from state a to state c along the curved path shown in Fig. 15- 24. 80 J of heat leaves the system and 55 J of work is done on the system. (a) Determine the change in internal energy. UA - Uc. (b) When the gas is taken along the path eda. the work done by the gas is IV' = 38 J. How much heat Q is added to the gas in the process eda? (c) If PA = 2.5P^. how much work is done by the gas in the process abc? (d) What is Q for path abc? (e) If UH - Ut> = 10 J. what is Q for the process be? Here is a summary of what is given: <2a-c = -80 J = -55 J Wcda = 38 J ua - Ub = 10 J Pa = 2.5Pd.

14. (I) How much energy would the person of Example 15-8 transform if instead of working 11.0 h she took a noontime break and ran for 1.0 h?

15. (I) Calculate the average metabolic rate of a person who sleeps 8.0 h. sits at a desk 8.0 h. engages in light activity 4.0 h, watches television 2.0 h, plays tennis 1.5 h. and runs 0.5 h daily.

16. (II) A person decides to lose weight by sleeping one hour less per day. using the time for light activity. How much weight (or mass) can this person expect to lose in 1 year, assuming no change in food intake? Assume that 1 kg of fat stores about 40,000 kJ of energy.

17. (I) A heat engine exhausts 8200J of heat while performing 3200 J of useful work. What is the efficiency of this engine?

18. (I) A heat engine does 9200 J of work per cycle while absorbing 22.0 kcal of heal from a high- temperature reservoir. What is the efficiency of this engine?

19. (I) What is the maximum efficiency of a heat engine whose operating temperatures are 580C and 380C?

20. (I) The exhaust temperature of a heat engine is 230G What must be the high temperature if the Carnot efficiency is to be 28% ?

21. (II) A nuclear power plant operates at 75% of its maximum theoretical (Carnot) efficiency between temperatures of 625C and 350C. If the plant produces electric energy at the rate of 1.3 GW. how much exhaust heat is discharged per hour?

22. (II) It is not necessary that a heat engine's hot environment be hotter than ambient temperature. Liquid nitrogen (77 K) is about as cheap as bottled water. What would be the efficiency of an engine that made use of heat transferred from air at room temperature (293 K) to the liquid nitrogen fuel (Fig. 15-25)?

23. (II) A Carnot engine performs work at the rate of 440 kW while using 680kcal of heat per second. If the temperature of the heat source is 570G at what temperature is the waste heat exhausted?

24. (II) A Carnot engines operating temperatures are 2l0C and 45CC. The engines power output is 950 W. Calculate the rate of heat output.

25. (II) A certain power plant puls out 550 MW of electric power. Estimate the heat discharged per second, assuming that the plant has an efficiency of 38%.

26. (II) A heat engine utilizes a heat source at 550nC and has an ideal (Carnot) efficiency of 28%. To increase the ideal efficiency to 35%, what must be the temperature of the heat source?

27. (II) A heat engine exhausts its heat at 350C and has a Carnot efficiency of 39%. What exhaust temperature would enable it to achieve a Carnot efficiency of 49%?

28. (Ill) At a steam power plant, steam engines work in pairs, the output of heat from one being the approximate heat input of the second. The operating temperatures of the first are 670C and 440C, and of the second 430C and 290C. If the heat of combustion of coal is 2.8 X 107 J/kg, at what rate must coal be burned if the plant is to pul out 1100 MW of power? Assume the efficiency of the engines is 60% of the ideal (Carnot) efficiency.

29. (I) The low temperature of a freezer cooling coil is -15C, and the discharge temperature is 30C. What is the maximum theoretical coefficient of performance?

30. (II) An ideal refrigerator-freezer operates with a COP = 7.0 in a 24C room. What is the temperature inside the freezer?

31. (II) A restaurant refrigerator has a coefficient of performance of 5.0. If the temperature in the kitchen outside the refrigerator is 29"C. what is the lowest temperature that could be obtained inside the refrigerator if it were ideal?

32. (II) A heat pump is used to keep a house warm at 22CC. How much work is required of the pump to deliver 2800 J of heat into the house if the outdoor temperature is (a) 0C, (b) 15C? Assume ideal (Carnot) behavior.

33. (II) What volume of water at 0C can a freezer make into ice cubes in 1.0 hour, if the coefficient of performance of the cooling unit is 7.0 and the power input is 1.0 kilowatt?

34. (II) An ideal (Carnot) engine has an efficiency of 35%. If it were possible to run it backward as a heat pump, what would be its coefficient of performance?

35. (I) What is the change in entropy of 250 g of steam at 100C when it is condensed to water at 100C?

36. (I) One kilogram of water is heated from 0C to 100C. Estimate the change in entropy of the water.

37. (I) What is the change in entropy of 1.00 m3 of water at 0C when it is frozen to ice at 0C?

38. (II) If 1.00 m3 of water at 0C is frozen and cooled to 10C by being in contact with a great deal of ice at 10C, what would be the total change in entropy of the process?

39. (II) A 10.0-kg box having an initial speed of 3.0 m/s slides along a rough table and comes to rest. Estimate the total change in entropy of the universe. Assume all objects are at room temperature (293 K).

40. (II) A falling rock has kinetic energy ke just before striking the ground and coming to rest. What is the total change in entropy of the rock plus environment as a result of this collision?

41. (II) An aluminum rod conducts 7.50 cal/s from a heat source maintained at 240C to a large body of water at 27C. Calculate the rate entropy increases per unit lime in this process.

42. (II) 1.0 kg of water at 30CC is mixed with 1.0 kg of water at 60C in a well-insulated container. Estimate the net change in entropy of the system.

43. (II) A 3.8-kg piece of aluminum at 30"C is placed in 1.0 kg of water in a Styrofoam container at room temperature (20C). Calculate the approximate net change in entropy of the system.

44. (Ill) A real heat engine working between heat reservoirs at 970 K and 650 K produces 550.1 of work per cycle for a heat input of 2200 J. (r/) Compare the efficiency of this real engine to that of an ideal (Carnot) engine. (b) Calculate the total entropy change of the universe per cycle of the real engine, (c) Calculate the total entropy-change of the universe per cycle of a Carnot engine operating between the same two temperatures.

45. (II) Calculate the probabilities, when you throw two dice, of obtaining (a) a 5, and (/>) an 11.

46. (II) Rank the following five-card hands in order of increasing probability: (a) four aces and a king: (b) six of hearts, eight of diamonds, queen of clubs, three of hearts, jack of spades: (c) two jacks, two queens, and an ace: and (d) any hand having no two equal-value cards. Discuss your ranking in terms of microstates and macrostates.

47. (II) Suppose that you repeatedly shake six coins in your hand and drop them on the floor. Construct a table showing the number of microstates that correspond to each macrostate. What is the probability of obtaining (<?) three heads and three tails, and (b) six heads?

48. (I) Solar cells (Fig. 15-26) can produce about 40 W of electricity per square meter of surface area if directly facing the Sun. Howr large an area is required to supply the needs of a house that requires 22kWh/day? Would this fit on the roof of an average house? (Assume the Sun shines about 9 h/day.)

49. (II) Energy may be stored for use during peak demand by pumping water to a high reservoir when demand is low and then releasing it to drive turbines when needed. Suppose water is pumped to a lake 135 m above the turbines at a rate of 1.00 X 105kg/s for 10.0 h at night, (r/) How' much energy (kWh) is needed to do this each night? (b) If all this energy is released during a 14-h day. at 75% efficiency. wrhat is the average power output?

50. (II) Water is stored in an artificial lake created by a dam (Fig. 15-27). The water depth is 45 m at the dam. and a steady flow rate of 35 m3/s is maintained through hydroelectric turbines installed near the base of the dam. How-much electrical power can be produced?

51. An inventor claims to have designed and built an engine that produces 1.50 MW of usable wrork while taking in 3.00 MW of thermal energy at 425 K. and rejecting 1.50 MW of thermal energy at 215 K. Is there anything fishy about his claim? F.xplain.

52. When 5.30 x 105 J of heat is added to a gas enclosed in a cylinder fitted with a light frictionless piston maintained at atmospheric pressure, the volume is observed to increase from 1.9 m3 to 4.1 ml Calculate (a) the work done by the gas, and (b) the change in internal energy of the gas. (c) Graph this process on a PV diagram.

53. A 4-cylinder gasoline engine has an efficiency of 0.25 and delivers 220 J of work per cycle per cylinder. When the engine fires at 45 cycles per second, (a) wrhat is the work done per second? (b) What is the total heat input per second from the fuel? (c) If the energy content of gasoline is 35 MJ per liter, howr long does one liter last?

54. A Carnot* refrigerator (the reverse of a Carnot engine) absorbs heat from the freezer compartment at a temperature of -17C and exhausts it into the room at 25C. (a) How much work must be done by the refrigerator to change 0.50 kg of water at 25C into ice at -17C? (b) If the compressor output is 210 W, what minimum time is needed to accomplish this?

55. It has been suggested that a heat engine could be developed that made use of the temperature difference between water at the surface of the ocean and that several hundred meters deep. In the tropics, the temperatures may be 27C and 4C. respectively, (a) What is the maximum efficiency such an engine could have? (/>) Why-might such an engine be feasible in spite of the lowr efficiency? (c) Can you imagine any adverse environmental effects that might occur?

56. Two 1100-kg cars are traveling 95 km/h in opposite directions when they collide and are brought to rest. Estimate the change in entropy of the universe as a result of this collision. Assume T = 20C.

57. A 120-g insulated aluminum cup at 15C is filled with 140 g of water at 50C. After a few minutes, equilibrium is reached, (a) Determine the final temperature, and (b) estimate the total change in entropy.

58. (d) What is the coefficient of performance of an ideal heat pump that extracts heat from 6C air outside and deposits heat inside your house at 24C? (b) If this heat pump operates on 1200W of electrical power, what is the maximum heat it can deliver into your house each hour?

59. The burning of gasoline in a car releases about 3.0 X 104 kcal/gal. If a car averages 41 km/gal when driving 90 km/h. which requires 25 hp. what is the efficiency of the engine under those conditions?

60. A Carnot engine has a lower operating temperature Tl = 20C and an efficiency of 30%. By how many kelvins should the high operating temperature Tu be increased to achieve an efficiency of 40%?

61. Calculate the work done by an ideal gas in going from state A to state C in Fig. 15-28 for each of the following processes: (a) ADC. (b) ABC. and (c) AC directly.

62. A 33%; efficient power plant puts out 850 MW of electrical power. Cooling towers are used to take away the exhaust heat, (a) If the air temperature is allowed to rise 7.0 C. estimate what volume of air (km3) is heated per day. Will the local climate be heated significantly? (b) If the heated air were to form a layer 200 m thick, estimate how large an area it would cover for 24 h of operation. Assume the air has density 1.2kg/m3 and that its specific heat is about 1.0kJ/kg*C at constant pressure.

63. Suppose a power plant delivers energy at 980 MW using steam turbines. The steam goes into the turbines superheated at 625 K and deposits its unused heat in river water at 285 K. Assume that the turbine operates as an ideal Carnot engine, (a) If the river flow rate is 37 m3/s, estimate the average temperature increase of the river water immediately downstream from the power plant. (/>) What is the entropy increase per kilogram of the downstream river water in J/kg-K?

64. A 100-hp car engine operates at about 15% efficiency. Assume the engines water temperature of 85C is its cold-temperature (exhaust) reservoir and 495C is its thermal intake" temperature (the temperature of the exploding gas-air mixture), (tf) What is the ratio of its efficiency relative to its maximum possible (Carnot) efficiency? (b) Estimate how much power (in watts) goes into moving the car. and how much heat, in joules and in keal. is exhausted to the air in 1.0 h.

65. An ideal gas is placed in a tall cylindrical jar of cross-sectional area 0.080 m2. A frictionless 0.10- kg movable piston is placed vertically into the jar such that the pistons weight is supported by the gas pressure in the jar. When the gas is heated (at constant pressure) from 25 X to 55C, the piston rises 1.0 cm. How much heat was required for this process? Assume atmospheric pressure outside.

66. Metabolizing 1.0 kg of fat results in about 3.7 X 107J of internal energy in the body, (a) In one day. how much fat docs the body burn to maintain the body temperature of a person staying in bed and metabolizing at an average rate of 95 W? (/>) How long would it take to burn 1.0-kg of fat this way assuming there is no food intake?

67. An ideal air conditioner keeps the temperature inside a room at 21C when the outside temperature is 32C. If 5.3 kW of power enters a room through the windows in the form of direct radiation from the Sun. how much electrical power would be saved if the windows were shaded so that the amount of radiation were reduced to 500 W?

68. A dehumidifier is essentially a refrigerator with an open door. The humid air is pulled in by a fan and guided to a cold coil, where the temperature is less than the dew point, and some of the airs water condenses. After this water is extracted, the air is warmed back to its original temperature and sent into the room. In a well-designed dehumidifier, the heat is exchanged between the incoming and outgoing air. This way the heat that is removed by the refrigerator coil mostly comes from the condensation of water vapor to liquid. Estimate how much water is removed in 1.0 h by an ideal dehumidifier, if the temperature of the room is 25C. the water condenses at 8C. and the dehumidifier does work at the rate of 600 W of electrical power.

Problem 14P How much energy would the person of Example 15–8 transform if instead of working 11.0 h she took a noontime break and ran for 1.0 h? Example 15–8 Energy transformation in the body. How much energy is transformed in 24 h by a 65-kg person who spends 8.0 h sleeping, 1.0 h at moderate physical labor, 4.0 h in light activity, and 11.0 h working at a desk or relaxing? Approach The energy transformed during each activity equals the metabolic rate (Table 15–2) multiplied by the time. SOLUTION Table 15–2 gives the metabolic rate in watts (J/s). Since there are 3600 s in an hour, the total energy transformed is [(8.0h) (70J/s) + (1.0h) (460J/s) + (4.0 h) (230 J/s) + (11.0 h) (115 J/s)] (3600 s/h) = 1.15 × 107 J. NOTE Since 4.186 × 103 J = 1 kcal, this is equivalent to 2800 kcal; a food intake of 2800 Cal would compensate for this energy output. A 65-kg person who wanted to lose weight would have to eat less than 2800 Cal a day, or increase his or her level of activity. Table 15–2 Metabolic Rates (65-kg human) Metabolic Rate (approximate) Activity kcal/h watts Sleeping 60 70 Sitting upright 100 115 Light activity (eating, dressing, household chores) 200 230 Moderate work (tennis, walking) 400 460 Running (15 km/h) 1000 1150 Bicycling (race) 1100 1270

Problem 14Q Give three examples, other than those mentioned in this Chapter, of naturally occurring processes in which order goes to disorder. Discuss the observability of the reverse process.

Problem 15P Calculate the average metabolic rate of a person who sleeps 8.0 h, sits at a desk 8.0 h, engages in light activity 4.0 h, watches television 2.0 h, plays tennis 1.5 h, and runs 0.5 h daily.

Problem 35P What is the change in entropy of 250 g of steam at 100°C when it is condensed to water at 100°C?

Problem 36P (I) 1.0 kg of water is heated from 0°C to 100°C. Estimate the change in entropy of the water.

Problem 37P (I) What is the change in entropy of of water at 1.00 m3 0°C when it is frozen to ice at 0°C?

Problem 65GP An ideal gas is placed in a tall cylindrical jar of cross-sectional area 0.080 m2. A frictionless 0.10-kg movable piston is placed vertically into the jar such that the piston’s weight is supported by the gas pressure in the jar. When the gas is heated (at constant pressure) from 25°C to 55°C, the piston rises 1.0 cm. How much heat was required for this process? Assume atmospheric pressure outside.

Problem 66GP Metabolizing 1.0 kg of fat results in about 3.7*107 J of internal energy in the body. (a) In one day, how much fat does the body burn to maintain the body temperature of a person staying in bed and metabolizing at an average rate of 95 W? (b) How long would it take to burn 1.0 kg of fat this way assuming there is no food intake?

Problem 67GP An ideal air conditioner keeps the temperature inside a room at 21°C when the outside temperature is 32°C. If 5.3 kW of power enters a room through the windows in the form of direct radiation from the Sun, how much electrical power would be saved if the windows were shaded so that the amount of radiation were reduced to 500 W?

Problem 1P An ideal gas expands isothermally, performing 3.40 × 103J of work in the process. Calculate (a) the change in internal energy of the gas, and (b) the heat absorbed during this expansion.

Problem 15Q Which do you think has the greater entropy, 1 kg of solid iron or 1 kg of liquid iron? Why?

Problem 16P A Person decides to lose weight by sleeping one hour less per day, using the time for light activity. How much weight (or mass) can this person expect to lose in 1 year, assuming no change in food intake? Assume that 1 kg of fat stores about 40,000 kJ of energy.

Problem 16Q (a) What happens if you remove the lid of a bottle containing chlorine gas? (b) Does the reverse process ever happen? Why or why not? (c) Can you think of two other examples of irreversibility?

Problem 39P A 10.0-kg box having an initial speed of 3.0 m/s slides along a rough table and comes to rest. Estimate the total change in entropy of the universe. Assume all objects are at room temperature (293 K).

Problem 40P (II) A falling rock has kinetic energy KE just before striking the ground and coming to rest. What is the total change in entropy of rock plus environment as a result of this collision?

Problem 38P If 1.00 m3 of water at 0°C is frozen and cooled to –10°C by being in contact with a great deal of ice at –10°C, what would be the total change in entropy of the process?

Problem 68GP A dehumidifier removes water vapor from air and has been referred to as a “refrigerator with an open door.” The humid air is pulled in by a fan and passes over a cold coil, whose temperature is less than the dew point, and some of the air’s water condenses. After this water is extracted, the air is warmed back to its original temperature and sent into the room. In a well-designed dehumidifier, the heat that is removed by the cooling coil mostly comes from the condensation of water vapor to liquid, and this heat is used to re-warm the air. Estimate how much water is removed in 1.0 h by an ideal dehumidifier, if the temperature of the room is 25°C, the water condenses at 8°C, and the dehumidifier does work at the rate of 600 W of electrical power. (See Sections 15–6, 13–12, and 14–5.)

Problem 1Q What happens to the internal energy of water vapor in the air that condenses on the outside of a cold glass of water? Is work done or heat exchanged? Explain.

Problem 2P A gas is enclosed in a cylinder fitted with a light frictionless piston and maintained at atmospheric pressure. When 1400 kcal of heat is added to the gas, the volume is observed to increase slowly from 12.0 m3 to 18.2 m3. Calculate (a) the work done by the gas and (b) the change in internal energy of the gas.

Problem 2Q Use the conservation of energy to explain why the temperature of a gas increases when it is quickly compressed, whereas the temperature decreases when the gas expands.

A heat engine exhausts 8200 J of heat while performing 3200 J of useful work. What is the efficiency of this engine?

Problem 17Q You are asked to test a machine that the inventor calls an “in-room air conditioner”: a big box, standing in the middle of the room, with a cable that plugs into a power outlet. When the machine is switched on, you feel a stream of cold air coming out of it. How do you know that this machine cannot cool the room?

Problem 18P A heat engine does 9200 J of work per cycle while absorbing 22.0 kcal of heat from a high-temperature reservoir. What is the efficiency of this engine?

Problem 41P An aluminum rod conducts 7.50 cal/s from a heat source maintained at 240°C to a large body of water at 27°C. Calculate the rate entropy increases per unit time in this process.

Problem 42P 1.0 kg of water at 30°C is mixed with 1.0 kg of water at 60°C in a well-insulated container. Estimate the net change in entropy of the system.

Problem 43P A 3.8-kg piece of aluminum at 30°C is placed in 1.0 kg of water in a Styrofoam container at room temperature (20°C). Calculate the approximate net change in entropy of the system.

Problem 3P (I) One liter of air is cooled at constant pressure until its volume is halved, and then it is allowed to expand isothermally back to its original volume. Draw the process on a PV diagram.

Problem 3Q In an isothermal process, 3700 J of work is done by an ideal gas. Is this enough information to tell how much heat has been added to the system? If so, how much? If not, why not?

Problem 4P Sketch a PV diagram of the following process: 2.0 L of ideal gas at atmospheric pressure are cooled at constant pressure to a volume of 1.0 L, and then expanded isothermally back to 2.0 L, whereupon the pressure is increased at constant volume until the original pressure is reached.

Problem 18Q Think up several processes (other than those already mentioned) that would obey the first law of thermodynamics, but, if they actually occurred, would violate the second law.

Problem 19P What is the maximum efficiency of a heat engine whose operating temperatures are 580°C and 380°C?

Problem 19Q Suppose a lot of papers are strewn all over the floor; then you stack them neatly. Does this violate the second law of thermodynamics? Explain.

Problem 44P A real heat engine working between heat reservoirs at 970 K and 650 K produces 550 J of work per cycle for a heat input of 2200 J. (a) Compare the efficiency of this real engine to that of an ideal (Carnot) engine. (b) Calculate the total entropy change of the universe per cycle of the real engine. (c) Calculate the total entropy change of the universe per cycle of a Carnot engine operating between the same two temperatures.

Problem 45P Calculate the probabilities, when you throw two dice, of obtaining (a) a 5, and (b) an 11.

Problem 46P (III) Rank the following five-card hands in order of increasing probability: (a) four aces and a king; (b) six of hearts, eight of diamonds, queen of clubs, three of hearts, jack of spades; (c) two jacks, two queens, and an ace; and (d) any hand having no two equal-value cards (no pairs, etc.). Discuss your ranking in terms of microstates and macrostates.

Problem 5P A 1.0-L volume of air initially at 4.5 atm of (absolute) pressure is allowed to expand isothermally until the pressure is 1.0 atm. It is then compressed at constant pressure to its initial volume, and lastly is brought back to its original pressure by heating at constant volume. Draw the process on a PV diagram, including numbers and labels for the axes.

Problem 4Q Is it possible for the temperature of a system to remain constant even though heat flows into or out of it? If so, give one or two examples.

Problem 5Q Explain why the temperature of a gas increases when it is compressed adiabatically.

Problem 20P The exhaust temperature of a heat engine is 230°C. What must be the high temperature if the Carnot efficiency is to be 28%?

Problem 20Q The first law of thermodynamics is sometimes whimsically stated as, “You can’t get something for nothing,” and the second law as, “You can’t even break even.” Explain how these statements could be equivalent to the formal statements.

Problem 21P A nuclear power plant operates at 75% of its maximum theoretical (Carnot) efficiency between temperatures of 625°C and 350°C. If the plant produces electric energy at the rate of 1.3 GW, how much exhaust heat is discharged per hour?

Problem 47P (II) Suppose that you repeatedly shake six coins in your hand and drop them on the floor. Construct a table showing the number of microstates that correspond to each macrostate. What is the probability of obtaining (a) three heads and three tails, and (b) six heads?

(I) Solar cells (Fig. 15-26) can produce about of electricity per square meter of surface area if directly facing the Sun. How large an area is required to supply the needs of a house that requires day? Would this fit on the roof of an average house? (Assume the Sun shines about day.

Problem 49P Energy may be stored for use during peak demand by pumping water to a high reservoir when demand is low and then releasing it to drive turbines when needed Suppose water is pumped to a lake 135 m above the turbines at a rate of 1.00 × 105 kg/s for 10.0 h at night. (a) How much energy (kWh) is needed to do this each night? (b) If all this energy is released during a 14-h day, at 75% efficiency, what is the average power output?

Problem 6P The pressure in an ideal gas is cut in half slowly, while being kept in a container with rigid walls. In the process, 265 kJ of heat left the gas. (a) How much work was done during this process? (b) What was the change in internal energy of the gas during this process?

Problem 6Q Can mechanical energy ever be transformed completely into heat or internal energy? Can the reverse happen? In each case, if your answer is no, explain why not; if yes, give one or two examples.

Problem 7P In an engine, an almost ideal gas is compressed adiabatically to half its volume. In doing so, 1850 J of work is done on the gas. (a) How much heat flows into or out of the gas? (b) What is the change in internal energy of the gas? (c) Does its temperature rise or fall?

Problem 21Q Entropy is often called “time’s arrow” because it tells us in which direction natural processes occur. If a movie were run backward, name some processes that you might see that would tell you that time was “running backward.”

Problem 22P It is not necessary that a heat engine’s hot environment be hotter than ambient temperature. Liquid nitrogen (77 K) is about as cheap as bottled water. What would be the efficiency of an engine that made use of heat transferred from air at room temperature (293 K) to the liquid nitrogen “fuel” (Fig. 15–25)? Figure 15–25

Water is stored in an artificial lake created by a dam (Fig. 15–27). The water depth is 45 m at the dam, and a steady flow rate of 35 m3/s is maintained through hydroelectric turbines installed near the base of the dam. How much electrical power can be produced?

Problem 22Q Living organisms, as they grow, convert relatively simple food molecules into a complex structure. Is this a violation of the second law of thermodynamics? Explain your answer.

Problem 51GP An inventor claims to have designed and built an engine that produces 1.50 MW of usable work while taking in 3.00 MW of thermal energy at 425 K, and rejecting 1.50 MW of thermal energy at 215 K. Is there anything fishy about his claim? Explain.

Problem 52GP When 5.30 × 105 J of heat is added to a gas enclosed in a cylinder fitted with a light frictionless piston maintained at atmospheric pressure, the volume is observed to increase from 1.9 m3 to 4.1m3. Calculate (a) the work done by the gas, and (b) the change in internal energy of the gas. (c) Graph this process on a PV diagram.

Problem 7Q Can you warm a kitchen in winter by leaving the oven door open? Can you cool the kitchen on a hot summer day by leaving the refrigerator door open? Explain.

Problem 8P An ideal gas expands at a constant total pressure of 3.0 atm from 400 mL to 660 mL. Heat then flows out of the gas at constant volume, and the pressure and temperature are allowed to drop until the temperature reaches its original value. Calculate (a) the total work done by the gas m the process, and (b) the total heat flow into the gas.

Problem 8Q Would a definition of heat engine efficiency as e = W/QL be useful? Explain.

Problem 23P A Carnot engine performs work at the rate of 440 kW while using 680 kcal of heat per second. If the temperature of the heat source is 570°C, at what temperature is the waste heat exhausted?

Problem 24P A Carnot engine’s operating temperatures are 210°C and 45°C. The engine’s power output is 950 W. Calculate the rate of heat output.

Problem 25P A certain power plant puts out 550 MW of electric power. Estimate the heat discharged per second, assuming that the plant has an efficiency of 38%.

Problem 53GP A 4-cylinder gasoline engine has an efficiency of 0.25 and delivers 220 J of work per cycle per cylinder. When the engine fires at 45 cycles per second, (a) what is the work done per second? (b) What is the total heat input per second from the fuel? (c) If the energy content of gasoline is 35 MJ per liter, how long does one liter last?

Problem 54GP A “Carnot” refrigerator (the reverse of a Carnot engine) absorbs heat from the freezer compartment at a temperature of –17°C and exhausts it into the room at 25°C. (a) How much work must be done by the refrigerator to change 0.50 kg of water at 25°C into ice at – 17°C? (b) If the compressor output is 210 W, what minimum time is needed to accomplish this?

Problem 55GP It has been suggested that a heat engine could be developed that made use of the temperature difference between water at the surface of the ocean and water several hundred meters deep. In the tropics, the temperatures may be 27°C and 4°C, respectively. (a) What is the maximum efficiency such an engine could have? (b) Why might such an engine be feasible in spite of the low efficiency? (c) Can you imagine any adverse environmental effects that might occur?

Problem 9P One and one-half moles of an ideal monatomic gas expand adiabatically, performing 7500 J of work in the process. What is the change in temperature of the gas during this expansion?

Problem 9Q What plays the role of high-temperature and low-temperature areas in (a) an internal combustion engine, and (b) a steam engine?

(II) Consider the following two-step process. Heat is allowed to flow out of an ideal gas at constant volume so that its pressure drops from 2.2 atm to 1.4 atm. Then the gas expands at constant pressure, from a volume of 6.8 L to 9.3 L, where the temperature reaches its original value. See Fig. 15–22. Calculate (a) the total work done by the gas in the process, (b) the change in internal energy of the gas in the process, and (c) the total heat flow into or out of the gas.

Problem 26P A heat engine utilizes a heat source at 550°C and has an ideal (Carnot) efficiency of 28%. To increase the ideal efficiency to 35%, what must be the temperature of the heat source?

Problem 27P A heat engine exhausts its heat at 350°C and has a Carnot efficiency of 39%, What exhaust temperature would enable it to achieve a Carnot efficiency of 49%?

Problem 28P At a steam power plant, steam engines work in pairs, the output of heat from one being the approximate heat input of the second. The operating temperatures of the first are 670°C and 440°C, and of the second 430°C and 290°C. If the heat of combustion of coal is 2.8 × 107 J/kg, at what rate must coal be burned if the plant is to put out 1100 MW of power? Assume the efficiency of the engines is 60% of the ideal (Carnot) efficiency.

Problem 56GP Two 1100-kg cars are traveling 95 km/h in opposite directions when they collide and are brought to rest. Estimate the change in entropy of the universe as a result of this collision. Assume T = 20°C.

Problem 57GP A 120-g insulated aluminum cup at 15°C is filled with 140 g of water at 50°C. After a few minutes, equilibrium is reached. (a) Determine the final temperature, and (b) estimate the total change in entropy.

(II) (a) What is the coefficient of performance of an ideal heat pump that extracts heat from 6°C air outside and deposits heat inside a house at 24°C? (b) If this heat pump operates on 1200 W of electrical power, what is the maximum heat it can deliver into the house each hour?

The PV diagram in Fig. 15–23 shows two possible states of a system containing 1.35 moles of a monatomic ideal gas, (P? = P? = 455 N/m³, V? = 2.00 m3, V? = 8.00m³.) (a) Draw the process which depicts an isobaric expansion from state 1 to state 2, and label this process A. (b) Find the work done by the gas and the change in internal energy of the gas in process A. (c) Draw the two-step process which depicts an isothermal expansion from state 1 to the volume V?, followed by an isovolumetric increase in temperature to state 2, and label this process B. (d) Find the change in internal energy of the gas for the two-step process B. ________________

Problem 11Q The oceans contain a tremendous amount of thermal (internal) energy. Why, in general, is it not possible to put this energy to useful work?

(III) When a gas is taken from a to along the curved path in Fig. , the work done by the gas is and the heat added to the gas is . Along path abc, the work done is What is for path abc? If , what is for path cda? (c) What is for path cda? What is (e) If , what is for path da?

Problem 29P The low temperature of a freezer cooling coil is –15°C, and the discharge temperature is 30°C. What is the maximum theoretical coefficient of performance?

Problem 30P An ideal refrigerator-freezer operates with a COP = 7.0 in a 24°C room. What is the temperature inside the freezer?

Problem 31P A restaurant refrigerator has a coefficient of performance of 5.0. If the temperature in the kitchen outside the refrigerator is 29°C, what is the lowest temperature that could be obtained inside the refrigerator if it were ideal?

Problem 59GP The burning of gasoline in a car releases about 3.0 × 104kcal/gal. If a car averages 41 km/gal when driving 90 km/h, which requires 25 hp, what is the efficiency of the engine under those conditions?

Problem 60GP A Carnot engine has a lower operating temperature TL = 20°C and an efficiency of 30%. By how many kelvins should the high operating-temperature TH be increased to achieve an efficiency of 40%?

Calculate the work done by an ideal gas in going from state to state in Fig. for each of the following processes: (a) , and directly.

Problem 12Q A gas is allowed to expand (a) adiabatically and (b) isothermally. In each process, does the entropy increase, decrease, or stay the same? Explain.

(III) In the process of taking a gas from state a to state along the curved path shown in Fig. of heat leaves the system and of work is done on the system. (a) Determine the change in internal energy, . (b) When the gas is taken along the path cda, the work done by the gas is . How much heat is added to the gas in the process cda? (c) If , how much work is done by the gas in the process abc? ( ) What is for path abc? (e) If , what is for the process be? Here is a summary of what is given:

Problem 13Q A gas can expand to twice its original volume either adiabatically or isothermally. Which process would result in a greater change in entropy? Explain.

Problem 32P A heat pump is used to keep a house warm at 22°C. How much work is required of the pump to deliver 2800 J of heat into the house if the outdoor temperature is (a) 0°C, (b) –15°C? Assume ideal (Carnot) behavior.

Problem 33P What volume of water at 0°C can a freezer make into ice cubes in 1.0 hour, if the coefficient of performance of the cooling unit is 7.0 and the power input is 1.0 kilowatt?

Problem 34P An ideal (Carnot) engine has an efficiency of 35%. If it were possible to run it backward as a heat pump, what would be its coefficient of performance?

Problem 62GP A 33% efficient power plant puts out 850 MW of electrical power. Cooling towers are used to take away the exhaust heat. (a) If the air temperature is allowed to rise 7.0 C°, estimate what volume of air (km3) is heated per day. Will the local climate be heated significantly? (b) If the heated air were to form a layer 200 m thick, estimate how large an area it would cover for 24 h of operation. Assume the air has density 1.2kg/m3 and that its specific heat is about 1.0kJ/kg · C° at constant pressure.

Problem 63GP Suppose a power plant delivers energy at 980 MW using steam turbines. The steam goes into the turbines superheated at 625 K and deposits its unused heat in river water at 285 K. Assume that the turbine operates as an ideal Carnot engine. (a) If the river flow rate is 37m3/s, estimate the average temperature increase of the river water immediately downstream from the power plant. (b) What is the entropy increase per kilogram of the downstream river water in J/kg · K?

Problem 64GP A 100-hp car engine operates at about 15% efficiency. Assume the engine’s water temperature of 85°C is its cold-temperature (exhaust) reservoir and 495°C is its thermal “intake” temperature (the temperature of the exploding gas–air mixture). (a) What is the ratio of its efficiency relative to its maximum possible (Carnot) efficiency? (b) Estimate how much power (in watts) goes into moving the car, and how much heat, in joules and in kcal, is exhausted to the air in 1.0 h.