Chapter 20

A diesel engine performs 2200 \(\mathrm{J}\) of mechanical work and discards 4300 \(\mathrm{J}\) of heat each cycle. (a) How much heat must be supplied to the engine in each cycle? (b) What is the thermal efficiency of the engine?

An aircraft engine takes in 9000 \(\mathrm{J}\) of heat and discards 6400 \(\mathrm{J}\) each cycle. (a) What is the mechanical work output of the engine during one cycle? (b) What is the thermal efficiency of the engine?

A Gasoline Engine. Agasoline engine takes in \(1.61 \times 10^{4} \mathrm{J}\) of heat and delivers 3700 \(\mathrm{J}\) of work per cycle. The heat is obtained by burning gasoline with a heat of combustion of \(4.60 \times 10^{4} \mathrm{J} / \mathrm{g}\) . (a) What is the thermal efficiency? (b) How much heat is discarded in each cycle? (c) What mass of fuel is burned in each cycle? (d) If the engine goes through 60.0 cycles per second, what is its power output in kilowatts? In horsepower?

A gasoline engine has a power output of 180 \(\mathrm{kW}\) (about 241 \(\mathrm{hp}\) ). Its thermal efficiency is 28.0\(\%\) . (a) How much heat must be supplied to the engine per second? (b) How much heat is discarded by the engine per second?

20.5. A certain nuclear-power plant has a mechanical-power output (used to drive an electric generator) of 330 \(\mathrm{MW}\) . Its rate of heat input from the nuclear reactor is 1300 \(\mathrm{MW}\) . (a) What is the thermal efficiency of the system? (b) At what rate is heat discarded by the system?

20.6. (a) Calculate the theoretical efficiency for an Otto cycle engine with \(\gamma=1.40\) and \(r=9.50 .\) (b) If this engine takes in \(10,000\) J of heat from burning its fuel, how much heat does it dis- card to the outside air?

20\. 07. What compression ratio \(r\) must an Otto cycle have to achieve an ideal efficiency of 65.0\(\%\) if \(\gamma=1.40 ?\)

20.8. The Otto-cycle engine in a Mercedes-Benz SLK230 has a compression ratio of 8.8 . (a) What is the ideal efficiency of the engine? Use \(\gamma=1.40 .\) (b) The engine in a Dodge Viper G.2 has a slightly higher compression ratio of 9.6 . How much increase in the ideal efficiency results from this increase in the compression ratio?

20.9. A refrigerator has a coefficient of performance of \(2.10 .\) In cach cycle it absorbs \(3.40 \times 10^{4} \mathrm{J}\) of heat from the cold reservoir. (a) How much mechanical energy is required each cycle to operate the refrigerator? (b) During each cycle, how much heat is discarded to the high-temperature reservoir?

20.10. A room air conditioner has a coefficient of performance of 29 on a hot day, and uses 850 \(\mathrm{W}\) of electrical power. (a) How many joules of heat does the air conditioner remove from the room in one minute? (b) How many joules of heat does the air conditioner deliver to the hot outside air in one minute? (c) Explain why your answers to parts (a) and (b) are not the same.

20.11. A window air-conditioner unit absorbs \(9.80 \times 10^{4} \mathrm{J}\) of heat per minute from the room being cooled and in the same time period deposits \(1.44 \times 10^{5} \mathrm{J}\) of heat into the outside air (a) What is the power consumption of the unit in watts? (b) What is the energy efficiency rating of the unit?

20.12. A freezer has a coefficient of performance of \(2.40 .\) The freezer is to convert 1.80 \(\mathrm{kg}\) of water at \(25.0^{\circ} \mathrm{C}\) to 1.80 \(\mathrm{kg}\) of ice at \(-5.0^{\circ} \mathrm{C}\) in hour. (a) What amount of heat must be removed from the water at \(25.0^{\circ} \mathrm{C}\) to convert it to ice at \(-5.0^{\circ} \mathrm{C} ?\) (b) How much electrical energy is consumed by the freezer during this hour? (c) How much wasted heat is delivered to the room in which the freezer sits?

20.13. A Camot engine whose high-temperature reservoir is at 620 \(\mathrm{K}\) takes in 550 \(\mathrm{J}\) of heat at this temperature in each cycle and gives up 335 \(\mathrm{J}\) to the low-temperature reservoir. (a) How much mechanical work does the engine perform during each cycle? (b) What is the temperature of the low-temperature reservoir? (c) What is the thermal efficiency of the cycle?

20.14. A Carmot engine is operated between two heat reservoirs at temperatures of 520 \(\mathrm{K}\) and 300 \(\mathrm{K}\) (a) If the engine receives 6.45 \(\mathrm{kJ}\) of heat energy from the reservoir at 520 \(\mathrm{K}\) in each cycle, how many joules per cycle does it discard to the reservoir at 300 \(\mathrm{K}\) ? (b) How much mechanical work is performed by the engine during each cycle? (c) What is the thermal efficiency of the engine?

20.15. A Carnot engine has an efficiency of 59\% and performs \(2.5 \times 10^{4} \mathrm{J}\) of work in each cycle. (a) How much heat does the engine extract from its heat source in each cycle? (b) Suppose the engine exhausts heat at room temperature ( \(20.0^{\circ} \mathrm{C} )\) . What is the temperature of its heat source?

20.16. An ice-making machine operates in a Camot cycle. It takes heat from water at \(0.0^{\circ} \mathrm{C}\) and rejects heat to a room at \(24.0^{\circ} \mathrm{C}\) . Suppose that 85.0 \(\mathrm{kg}\) of water at \(0.0^{\circ} \mathrm{C}\) are converted to ice at \(0.0^{\circ} \mathrm{C}\) . (a) How much heat is discharged into the room? (b) How much energy must be supplied to the device?

20.17. A Carnot refrigerator is operated between two heat reservoirs at temperatures of 320 \(\mathrm{K}\) and 270 \(\mathrm{K}\) (a) If in each cycle the refrigerator receives 415 \(\mathrm{J}\) of heat energy from the reservoir at 270 \(\mathrm{K}\) , how many joules of heat energy does it deliver to the reservoir at 320 \(\mathrm{K} ?\) (b) If the refrigerator completes 165 cycles each minute, what power input is required to operate it? (c) What is the coefficient of performance of the refrigerator?

20.18. A Carnot device extracts 5.00 \(\mathrm{kJ}\) of heat from a body at \(-10.0^{\circ} \mathrm{C} .\) How much work is done if the device exhausts heat into the environment at \((a) 25.0^{\circ} \mathrm{C} ;(\mathrm{b}) 0.0^{\circ} \mathrm{C} ;(\mathrm{c})-25.0^{\circ} \mathrm{C} ;\) In each case, is the device acting as an engine or as a refrigerator?

20.20. An ideal Carnot engine operates between \(500^{\circ} \mathrm{C}\) and \(100^{\circ} \mathrm{C}\) with a heat input of 250 \(\mathrm{J}\) per cycle. (a) How much heat is delivered to the cold reservoir in each cycle? (b) What minimum number of cycles is necessary for the engine to lift a \(500-\mathrm{kg}\) rock through a height of 100 \(\mathrm{m} ?\)

20.21. A Carnot heat engine has a thermal efficiency of 0.600 , and the temperature of its hot reservoir is 800 \(\mathrm{K}\) . If 3000 \(\mathrm{J}\) of heat is rejected to the cold reservoir in one cycle, what is the work output of the engine during one cycle?

20.22. A Carnot heat engine uses a hot reservoir consisting of a large amount of boiling water and a cold reservoir consisting of a large tub of ice and water. In 5 minutes of operation, the heat rejected by the engine melts 0.0400 \(\mathrm{kg}\) of ice. During this time, how much work \(W\) is performed by the engine?

20.24. (a) Show that the efficiency e of a Carnot engine and the coefficient of performance \(K\) of a Carnot refrigerator are related by \(K=(1-e) / e\) . The engine and refrigerator operate between the same hot and cold reservoirs. (b) What is \(K\) for the limiting values \(e \rightarrow 1\) and \(e \rightarrow 0 ?\) Explain.

20.25. A sophomore with nothing better to do adds heat to 0.350 \(\mathrm{kg}\) of ice at \(0.0^{\circ} \mathrm{C}\) until it is all melted. (a) What is the change in entropy of the water? (b) The source of heat is a very massive body at a temperature of \(25.0^{\circ} \mathrm{C}\) . What is the change in entropy of this body? (c) What is the total change in entropy of the water and the heat source?

20.26. You decide to take a nice hot bath but discover that your thoughtless roommate has used up most of the hot water. You fill the tub with 270 \(\mathrm{kg}\) of \(30.0^{\circ} \mathrm{C}\) water and attempt to warm it further by pouring in 5.00 \(\mathrm{kg}\) of boiling water from the stove. (a) Is this a reversible or an imeversible process? Use physical reasoning to explain. (b) Calculate the final temperature of the bath water. (c) Calculate the net change in entropy of the system (bath water + boiling water), assuming no heat exchange with the air or the tub itself.

20.27 A 15.0-kg block of ice at \(0.0^{\circ} \mathrm{C}\) melts to liquid water at \(0.0^{\circ} \mathrm{C}\) inside a large room that has a temperature of \(20.0^{\circ} \mathrm{C}\) . Treat the ice and the room as an isolated system, and assume that the room is large enough for its temperature change to be ignored. (a) Is the melting of the ice reversible or irreversible? Explain, using simple physical reasoning without resorting to any equations. (b) Calculate the net entropy change of the system during this process. Explain whether or not this result is consistent with your answer to part (a).

20.28. You make tea with 0.250 \(\mathrm{kg}\) of \(85.0^{\circ} \mathrm{C}\) water and let it cool to room temperature \(\left(20.0^{\circ} \mathrm{C}\right)\) before drinking it. (a) Calculate the entropy change of the water while it cools. (b) The cooling process is essentially isothernal for the air in your kitchen. Calculate the change in entropy of the air while the tea cools, assuming that all the heat lost by the water goes into the air. What is the total entropy change of the system tea \(+\) air?

20.29. You make tea with 0.250 \(\mathrm{kg}\) of \(85.0^{\circ} \mathrm{C}\) water and let it cool to room temperature \(\left(20.0^{\circ} \mathrm{C}\right)\) before drinking it. (a) Calculate the entropy change of the water while it cools. (b) The cooling process is essentially isothernal for the air in your kitchen. Calculate the change in entropy of the air while the tea cools, assuming that all the heat lost by the water goes into the air. What is the total entropy change of the system tea \(+\) air?

20.34 . A box is separated by a partition into two parts of equal volume. The left side of the box contains 500 molecules of nitrogen gas; the right side contains 100 molecules of oxygen gas. The two gases are at the same temperature. The partition is punctured, and equilibrium is eventually attained. Assume that the volume of the box is large enough for each gas to undergo a free expansion and not change temperature, (a) On average, how many molecules of each type will there be in either half of the box? (b) What is the change in entropy of the system when the partition is punctured? the (c) What is the probability that the molecules will be found in the same distribution as they were before the partition was punctured - that is, 500 nitrogen molecules in the left half and 100 oxygen molecules in the right half?

20.36 . A lonely party balloon with a volume of 2.40 \(\mathrm{L}\) and containing 0.100 \(\mathrm{mol}\) of air is left behind to drift in the temporarily uninhabited and depressurized Intermational Space Station. Sunlight coming through a porthole heats and explodes the balloon, causing the air in it to undergo a free expansion into the empty station, whose total volume is 425 \(\mathrm{m}^{3}\) . Calculate the entropy change of the air during the expansion.

20.37. You design a Carnot engine that operates between temperatures of 500 \(\mathrm{K}\) and 400 \(\mathrm{K}\) and produces 2000 \(\mathrm{J}\) of work in each cycle. (a) Calculate your engine's efficiency. (b) Calculate the amount of heat discarded during the isothermal compression at 400 \(\mathrm{K}\) . (c) Sketch the 500 \(\mathrm{K}\) and 400 \(\mathrm{K}\) isotherms on apV-diagram (no calculations); then sketch the Carnot cycle followed by your engine. (d) On the same diagram, sketch the 300 \(\mathrm{K}\) isotherm; then sketch, in a different color if possible, the Carnot cycle starting at the same point on the 500 \(\mathrm{K}\) isotherm but operating in a cycle between the 500 \(\mathrm{K}\) and 300 \(\mathrm{K}\) isotherms. (e) Compare the areas inside the loops (the net work done) for the two cycles. Notice that the same amount of heat is extracted from the hot reservoir in both cases. Can you explain why less heat is "wasted" during the 300 \(\mathrm{K}\) isothermal compression than during the 400 \(\mathrm{K}\) compression?

20.38. You are designing a Carnot engine that has 2 \(\mathrm{mol}\) of \(\mathrm{CO}_{2}\) as its working substance; the gas may be treated as ideal. The gas is to have a maximum temperature of \(527^{\circ} \mathrm{C}\) and a maximum pressure of 5.00 \(\mathrm{atm}\) . With a heat input of 400 \(\mathrm{J}\) per cycle, you want 300 \(\mathrm{J}\) of useful work (a) Find the temperature of the cold reservoir. (b) For how many cycles must this engine run to melt completely a 10.0 kg block of ice originally at \(0.0^{\circ} \mathrm{C}\) , using only the beat rejected by the engine?

20.39. A Carnot engine whose low-temperature reservoir is at \(-90.0^{\circ} \mathrm{C}\) has an efficiency of 40.0\(\%\) . An engineer is assigned the problem of increasing this to 45.0\(\%\) . (a) By how many Celsius degrees must the temperature of the high-temperature reservoir be increased if the temperature of the low-temperature reservoir remains constant? (b) By how many Celsius degrees must the temperature of the low-temperature reservoir be decreased if the temperature of the high-temperature reservoir remains constant?

20.42. Heat Pump. A heat pump is a heat engine run in reverse. In winter it pumps heat from the cold air outside into the warmer air inside the building, maintaining the building at a comfortable temperature. In summer it pumps heat from the cooler air inside the building to the warmer air outside, acting as an air conditioner. (a) If the outside temperature in winter is \(-5.0^{\circ} \mathrm{C}\) and the inside temperature is \(17.0^{\circ} \mathrm{C}\) , how many joules of heat will the heat pump deliver to the inside for each joule of electrical energy used to run the unit, assuming an ideal Carnot cycle? ( b) Suppose you have the option of using electrical resistance heating rather than a heat pump. How much electrical energy would you need in order to deliver the same amount of heat to the inside of the house as in part (a)? Consider a Carnot heat pump delivering heat to the inside of a house to maintain it at \(68^{\circ} \mathrm{F}\) . Show that the beat pump delivers less heat for each joule of electrical energy used to operate the unit as the outside temperature decreases. Notice that this behavior is opposite to the dependence of the efficiency of a Carnot heat engine on the difference in the reservoir temperatures. Explain why this is so.

20.45. An experimental power plant at the Natural Energy Laboratory of Hawaii generates electricity from the temperature gradient of the ocean. The surface and deep-water temperatures are \(27^{\circ} \mathrm{C}\) and \(6^{\circ} \mathrm{C}\) , respectively. (a) What is the maximum theoretical effciency of this power plant? (b) If the power plant is to produce 210 \(\mathrm{kW}\) of power, at what rate must heat be extracted from the warm water? At what rate must heat be absorbed by the cold water? Assume the maximum theoretical efficiency. (c) The cold water that enters the plant leaves it at a temperature of \(10^{\circ} \mathrm{C}\) . What must be the flow rate of cold water through the system? Give your answer in \(\mathrm{kg} / \mathrm{h}\) and \(\mathrm{L} / \mathrm{h}\) .

20.42 A cylinder contains oxygen at a pressure of 2.00 atm. The volume is 4.00 \(\mathrm{L}\) , and the temperature is 300 \(\mathrm{K}\) . Assume that the oxygen may be treated as an ideal gas. The oxygen is carried through the following processes: (i) Heated at constant pressure from the initial state (state 1) to state \(2,\) which has \(T=450 \mathrm{K}\) . (ii) Cooled at constant volume to 250 \(\mathrm{K}\) (state 3). (iii) Compressed at constant temperature to a volume of 4.00 \(\mathrm{L}\) (state 4\()\) . (iv) Heated at constant volume to 300 \(\mathrm{K}\) , which takes the system back to state 1. (a) Show these four processes in a \(p V\) -diagram, giving the numerical values of \(p\) and \(V\) in each of the four states. (b) Calculate \(Q\) and \(W\) for each of the four processes. (c) Calculate the net work done by the oxygen. (d) What is the efficiency of this device as a heal engine? How does this compare to the efficiency of a Carnot-cycle engine operating between the same minimum and maximum tem- peratures of 250 \(\mathrm{K}\) and 450 \(\mathrm{K} ?\)

20.50. A stirling-cycle Engine. the Otto cycle, except that the compression and expansion of the gas are done at constant temperature, not adiabatically as in the Otto cycle. The Stirling cycle is used in external combustion engines (in fact, burning fuel is not necessary; any way of producing a temperature difference will do -solar, geothermal, ocean temperature gradient, etc. \(.\) which means that the gas inside the cylinder is not used in the combustion process. Heat is supplied by burning fuel steadily outside the cylinder, instead of explosively inside the cylinder as in the Otto cycle. For this reason Stirling-cycle engines are quieter than Otto-cycle engines, since there are no intake and exhaust valves (a major source of engine noise). While small Stirling engines are used for a variety of purposes, Stiring engines for automobiles have not been successful because they are larger, heavier, and more expensive than conventional automobile engines. In the cycle, the working fluid goes through the following sequence of steps (Fig. 20.30\()\) : (i) Compressed isothermally at temperature \(T_{1}\) from the initial state \(a\) to state \(b\) , with a compression ratio \(r .\) (ii) Heated at constant volume to state \(c\) at temperature \(T_{2}\) . (iii) Expanded isothermally at \(T_{2}\) to state \(d\) . (iv) Cooled at constant volume back to the initial state \(a\) . Assume that the working fluid is \(n\) moles of an ideal gas (for which \(C_{V}\) is independent of temperature). (a) Calculate \(Q, W,\) and \(\Delta U\) for each of the processes \(a \rightarrow b, b \rightarrow c, c \rightarrow d,\) and \(d \rightarrow a\) . (b) In the Stirling cycle, the heat transfers in the processes \(b \rightarrow c\) and \(d \rightarrow a\) do not involve external heat sources but rather use regeneration: The same substance that transfers heat to the gas inside the cylinder in the process \(b \rightarrow c\) also absorbs heat back from the gas in the process \(d \rightarrow a\) . Hence the heat transfers \(Q_{b \rightarrow c}\) and \(Q_{d \rightarrow a}\) do not play a role in determining the efficiency of the engine. Explain this last statement by comparing the expressions for \(Q_{b \rightarrow c}\) and \(Q_{d \rightarrow a}\) calculated in part (a). (c) Calculate the efficiency of a Stirling-cycle engine in terms of the temperatures \(T_{1}\) and \(T_{2}\) . How does this compare to the efficiency of a Carnot-cycle engine operating between these same two temperatures? (Historically, the Stirling cycle was devised before the Carnot cycle.) Does this result violate the second law of thermodynamics? Explain. Unfortunately, actual Stirling-cycle engines cannot achieve this efficiency due to problems with the heat-transfer processes and pressure losses in the engine.

20.52. A typical coal-fired power plant generates 1000 MW of usable power at an overall thermal efficiency of 40\(\%\) (a) What is the rate of heat input to the plant? (b) The plant burns anthracite coal, which has a heat of combustion of \(265 \times 10^{7} \mathrm{J} / \mathrm{kg}\) . How much coal does the plant use per day, if it operates continuously? (c) At what rate is heat ejected into the cool reservoir, which is the nearby river?(d) The river's temperature is \(18.0^{\circ} \mathrm{C}\) before it reaches the power plant and \(18.5^{\circ} \mathrm{C}\) after it has received the plant's waste heat. Calculate the river's flow rate, in cubic meters per second. (e) By how much does the river's entropy increase each second?

20.54. An air conditioner operates on 800 \(\mathrm{W}\) of power and has a performance coefficient of 2.80 with a room temperature of \(21.0^{\circ} \mathrm{C}\) and an outside temperature of \(35.0^{\circ} \mathrm{C}\) (a) Calculate the rate of heat removal for this unit. (b) Calculate the rate at which heat is discharged to the outside air. (c) Calculate the total entropy change in the room if the air conditioner runs for 1 hour. Calculate the total entropy change in the outside air for the same time period. (d) What is the net change in entropy for the system (room + outside air)?

20.56. The maximum power that can be extracted by a wind turbine from an air stream is approximately $$ P=k d^{2} v^{3} $$ where \(d\) is the blade diameter \(v\) is the wind speed, and the constant \(k=0.5 \mathrm{W} \cdot \mathrm{s}^{3} / \mathrm{m}^{5} .\) (a) Explain the dependence of \(P\) on \(d\) and on \(v\) by considering a cylinder of air that passes over the turbine blades in time \(t(\text { Fig. } 20.31)\) . This cylinder has diameter \(d .\) length \(L=v t\) and density \(\rho .\) (b) The Mod-SB wind turbine at Kahaku on the Hawaiian island of Oahu has a blade diameter of 97 \(\mathrm{m}\) (slightly longer than a football field sits atop a \(58-\mathrm{m}\) tower. It can produce 3.2 \(\mathrm{MW}\) of electric power. Assuming 25\(\%\) efficiency, what wind speed is required to produce this amount of power? Give your answer in \(\mathrm{m} / \mathrm{s}\) and in \(\mathrm{km} / \mathrm{h}\) . (c) Commercial wind turbines are commonly located in or downwind of mountain passes. Why?

20.57. (a) How much work must a Carnot refrigerator do on a hot day to transfer 1000 \(\mathrm{J}\) of heat from its interior at \(10^{\circ} \mathrm{C}\) to the out- side air at \(35.0^{\circ} \mathrm{C}\) ? (b) How much work must the same refrigerator do to transfer the same amount of heat if the interior temperature is the same, but the outside air is at only \(15.0^{\circ} \mathrm{C} ?\) (c) Sketch \(p V_{-}\) diagrams for these two situations. Can you explain in physical terms why more work must be done when the temperature difference between the two isothermal stages is greater?

20.58. A \(0.0500-\mathrm{kg}\) cube of ice at an initial temperature of \(-15.0^{\circ} \mathrm{C}\) is placed in 0.600 \(\mathrm{kg}\) of water at \(T=45.0^{\circ} \mathrm{C}\) in an insulated container of negligible mass. (a) Calculate the final temperature of the water once the ice has melted. (b) Calculate the change in entropy of the system.

20.63. An object of mass \(m_{1}\) , specific heat capacity \(c_{1}\) , and temperature \(T_{1}\) is placed in contact with a second object of mass \(m_{2},\) specific heat capacity \(c_{2},\) and temperature \(T_{2}>T_{1} .\) As a result, the temperature of the first object increases to \(T\) and the temperature of the second object decreases to \(T^{\prime} .\) (a) Show that the entropy increase of the system is $$ \Delta S=m_{1} c_{1} \ln \frac{T}{T_{1}}+m_{2} c_{2} \ln \frac{T^{\prime}}{T_{2}} $$ and show that energy conservation requires that $$ m_{1} c_{1}\left(T-T_{1}\right)=m_{2} c_{2}\left(T_{2}-T^{\prime}\right) $$ (b) Show that the entropy change \(\Delta S\) , considered as a function of \(T,\) is a maximum if \(T=T,\) which is just the condition of thermodynamic equilibrium. (c) Discuss the result of part (b) in terms of the idea of entropy as a measure of disorder.