Solved: A monatomic ideal gas is taken around the cycle

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

An aircraft engine takes in 9000 J of heat and discards 6400 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. A gasoline engine takes in of heat and delivers 3700 J of work per cycle. The heat is obtained by burning gasoline with a heat of combustion of (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 kW (about 241 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?

The pV-diagram in Fig. E20.5 shows a cycle of a heat engine that uses 0.250 mole of an ideal gas having g = 1.40. The curved part ab f the cycle is adiabatic. (a) Find the pressure of the gas at point a. (b) How much heat enters this gas per cycle, and where does it happen? (c) How much heat leaves this gas in a cycle, and where does it occur? (d) How much work does this engine do in a cycle? (e) What is the thermal efficiency of the engine?

(a) Calculate the theoretical efficiency for an Otto-cycle engine with and (b) If this engine takes in 10,000 J of heat from burning its fuel, how much heat does it discard to the outside air?

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 GT2 has a slightly higher compression ratio of 9.6. How much increase in the ideal efficiency results from this increase in the compression ratio?

The coefficient of performance is a dimensionless quantity. Its value is independent of the units used for H and P, as long as the same units, such as watts, are used for both quantities. However, it is common practice to express H in Btu h and P in watts. When these mixed units are used, the ratio is called the energy efficiency rating (EER). If a room air conditioner has a coefficient of performance , what is its EER?

A refrigerator has a coefficient of performance of 2.10. In each cycle it absorbs \(3.40 x 10^{4} J\) 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? Text Transcription: 3.40 x 10^4 J

A room air conditioner has a coefficient of performance of 2.9 on a hot day and uses 850 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.

A refrigerator has a coefficient of performance of 2.25, runs on an input of 95 W of electrical power, and keeps its inside compartment at 5C. If you put a dozen 1.0-L plastic bottles of water at 31C into this refrigerator, how long will it take for them to be cooled down to 5C? (Ignore any heat that leaves the plastic.)

A freezer has a coefficient of performance of 2.40. The freezer is to convert 1.80 kg of water at \(25.0^{\circ} \mathrm{C}\) to 1.80 kg of ice at \(-5.0^{\circ} \mathrm{C}\) in one 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?

A Carnot engine whose high-temperature reservoir is at 620 K takes in 550 J of heat at this temperature in each cycle and gives up 335 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?

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

A Carnot engine has an efficiency of 59% and performs 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 What is the temperature of its heat source?

An ice-making machine operates in a Carnot cycle. It takes heat from water at and rejects heat to a room at Suppose that 85.0 kg of water at are converted to ice at (a) How much heat is discharged into the room? (b) How much energy must be supplied to the device?

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

A certain brand of freezer is advertised to use of energy per year. (a) Assuming the freezer operates for 5 hours each day, how much power does it require while operating? (b) If the freezer keeps its interior at a temperature of in a room, what is its theoretical maximum performance coeffi- cient? (c) What is the theoretical maximum amount of ice this freezer could make in an hour, starting with water a

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

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 kg of ice. During this time, how much work W is performed by the engine?

You design an engine that takes in of heat at 650 K in each cycle and rejects heat at a temperature of 350 K. The engine completes 240 cycles in 1 minute. What is the theoretical maximum power output of your engine, in horsepower?

A 4.50-kg block of ice at 0.00C falls into the ocean and melts. The average temperature of the ocean is 3.50C, including all the deep water. By how much does the melting of this ice change the entropy of the world? Does it make it larger or smaller? (Hint: Do you think that the ocean will change temperature appreciably as the ice melts?)

A sophomore with nothing better to do adds heat to 0.350 kg of ice at 0.0°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°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?

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 kg of \(30.0^{\circ} \mathrm{C}\) water and attempt to warm it further by pouring in 5.00 kg of boiling water from the stove. (a) Is this a reversible or an irreversible 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.

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).

You make tea with 0.250 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 isothermal 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? Text Transcription: 85.0^circ C (20.0^circ C)

Three moles of an ideal gas undergo a reversible isothermal compression at \(20.0^{\circ} \mathrm{C}\). During this compression, 1850 J of work is done on the gas. What is the change of entropy of the gas?

What is the change in entropy of 0.130 kg of helium gas at the normal boiling point of helium when it all condenses isothermally to 1.00 L of liquid helium? (Hint: See Table 17.4 in Section 17.6.)

(a) Calculate the change in entropy when 1.00 kg of water at is vaporized and converted to steam at (see Table 17.4). (b) Compare your answer to the change in entropy when 1.00 kg of ice is melted at calculated in Example 20.5 (Section 20.7). Is the change in entropy greater for melting or for vaporization? Interpret your answer using the idea that entropy is a measure of the randomness of a system

(a) Calculate the change in entropy when 1.00 mol of water (molecular mass ) at evaporates to form water vapor at (b) Repeat the calculation of part (a) for 1.00 mol of liquid nitrogen, 1.00 mol of silver, and 1.00 mol of mercury when each is vaporized at its normal boiling point. (See Table 17.4 for the heats of vaporization, and Appendix D for the molar masses. Note that the nitrogen molecule is ) (c) Your results in parts (a) and (b) should be in relatively close agreement. (This is called the rule of Drepez and Trouton.) Explain why this should be so, using the idea that entropy is a measure of the randomness of a system.

A 10.0-L gas tank containing 3.20 moles of ideal He gas at 20.0C is placed inside a completely evacuated, insulated bell jar of volume 35.0 L. A small hole in the tank allows the He to leak out into the jar until the gas reaches a final equilibrium state with no more leakage. (a) What is the change in entropy of this system due to the leaking of the gas? (b) Is the process reversible or irreversible? How do you know?

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? (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?

CALC Two moles of an ideal gas occupy a volume V. The gas expands isothermally and reversibly to a volume 3V. (a) Is the velocity distribution changed by the isothermal expansion? Explain. (b) Use Eq. (20.23) to calculate the change in entropy of the gas. (c) Use Eq. (20.18) to calculate the change in entropy of the gas. Compare this result to that obtained in part (b).

CALC A lonely party balloon with a volume of 2.40 L and containing 0.100 mol of air is left behind to drift in the temporarily uninhabited and depressurized International 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 Calculate the entropy change of the air during the expansion.

CP An ideal Carnot engine operates between and with a heat input of 250 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-kg rock through a height of 100 m?

You are designing a Carnot engine that has 2 mol of as its working substance; the gas may be treated as ideal. The gas is to have a maximum temperature of and a maximum pressure of 5.00 atm. With a heat input of 400 J per cycle, you want 300 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 using only the heat rejected by the engine?

CP A certain heat engine operating on a Carnot cycle absorbs 150 J of heat per cycle at its hot reservoir at 135C and has a thermal efficiency of 22.0%. (a) How much work does this engine do per cycle? (b) How much heat does the engine waste each cycle? (c) What is the temperature of the cold reservoir? (d) By how much does the engine change the entropy of the world each cycle? (e) What mass of water could this engine pump per cycle from a well 35.0 m deep?

Entropy of Metabolism. An average sleeping person metabolizes at a rate of about 80 W by digesting food or burning fat. Typically, 20% of this energy goes into bodily functions, such as cell repair, pumping blood, and other uses of mechanical energy, while the rest goes to heat. Most people get rid of all this excess heat by transferring it (by conduction and the flow of blood) to the surface of the body, where it is radiated away. The normal internal temperature of the body (where the metabolism takes place) is 37C, and the skin is typically 7 C cooler. By how much does the persons entropy change per second due to this heat transfer?

Entropy Change from Digesting Fat. Digesting fat produces 9.3 food calories per gram of fat, and typically 80% of this energy goes to heat when metabolized. (One food calorie is 1000 calories and therefore equals 4186 J.) The body then moves all this heat to the surface by a combination of thermal conductivity and motion of the blood. The internal temperature of the body (where digestion occurs) is normally 37C, and the surface is usually about 30C. By how much do the digestion and metabolism of a 2.50-g pat of butter change your bodys entropy? Does it increase or decrease?

A heat engine takes 0.350 mol of a diatomic ideal gas around the cycle shown in the pV-diagram of Fig. P20.40. Process 1 S 2 is at constant volume, process is adiabatic, and process is at a constant pressure of 1.00 atm. The value of for this gas is 1.40. (a) Find the pressure and volume at points 1, 2, and 3. (b) Calculate Q, W, and for each of the three processes. (c) Find the net work done by the gas in the cycle. (d) Find the net heat flow into the engine in one cycle. (e) What is the thermal efficiency of the engine? How does this compare to the efficiency of a Carnot-cycle engine operating between the same minimum and maximum temperatures T1 T2?

CALC You build a heat engine that takes 1.00 mol of an ideal diatomic gas through the cycle shown in Fig. P20.41. (a) Show that segment ab is an isothermal compression. (b) During which segment(s) of the cycle is heat absorbed by the gas? During which segment(s) is heat rejected? How do you know? (c) Calculate the temperature at points a, b, and c. (d) Calculate the net heat exchanged with the surroundings and the net work done by the engine in one cycle. (e) Calculate the thermal efficiency of the engine

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 and the inside temperature is 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 Show that the heat 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.

A heat engine operates using the cycle shown in Fig. P20.43. The working substance is 2.00 mol of helium gas, which reaches a maximum temperature of \(327^{\circ} \mathrm{C}\). Assume the helium can be treated as an ideal gas. Process bc is isothermal. The pressure in states a and c is \(1.00 \times 10^{5} \mathrm{~Pa}\), and the pressure in state b is \(3.00 \times 10^{5} \mathrm{~Pa}\). (a) How much heat enters the gas and how much leaves the gas each cycle? (b) How much work does the engine do each cycle, and what is its efficiency? (c) Compare this engine’s efficiency with the maximum possible efficiency attainable with the hot and cold reservoirs used by this cycle.

CP As a budding mechanical engineer, you are called upon to design a Carnot engine that has 2.00 mol of a monatomic ideal gas as its working substance and operates from a hightemperature reservoir at The engine is to lift a 15.0-kg weight 2.00 m per cycle, using 500 J of heat input. The gas in the engine chamber can have a minimum volume of 5.00 L during the cycle. (a) Draw a pV-diagram for this cycle. Show in your diagram where heat enters and leaves the gas. (b) What must be the temperature of the cold reservoir? (c) What is the thermal efficiency of the engine? (d) How much heat energy does this engine waste per cycle? (e) What is the maximum pressure that the gas chamber will have to withstand?

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 and respectively. (a) What is the maximum theoretical efficiency of this power plant? (b) If the power plant is to produce 210 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 What must be the flow rate of cold water through the system? Give your answer in and in

What is the thermal efficiency of an engine that operates by taking n moles of diatomic ideal gas through the cycle shown in Fig. P20.46?

A cylinder contains oxygen at a pressure of 2.00 atm. The volume is 4.00 L, and the temperature is 300 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 K. (ii) Cooled at constant volume to 250 K (state 3). (iii) Compressed at constant temperature to a volume of 4.00 L (state 4). (iv) Heated at constant volume to 300 K, which takes the system back to state 1. (a) Show these four processes in a pV-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 in the complete cycle. (d) What is the efficiency of this device as a heat engine? How does this compare to the efficiency of a Carnot-cycle engine operating between the same minimum and maximum temperatures of 250 K and 450 K?

CP BIO Human Entropy. A person who has skin of surface area and temperature 30.0C is resting in an insulated room where the ambient air temperature is 20.0C. In this state, a person gets rid of excess heat by radiation. By how much does the person change the entropy of the air in this room each second? (Recall that the room radiates back into the person and that the emissivity of the skin is 1.00.)

P BIO A Human Engine. You decide to use your body as a Carnot heat engine. The operating gas is in a tube with one end in your mouth (where the temperature is 37.0C) and the other end at the surface of your skin, at 30.0C. (a) What is the maximum efficiency of such a heat engine? Would it be a very useful engine? (b) Suppose you want to use this human engine to lift a 2.50-kg box from the floor to a tabletop 1.20 m above the floor. How much must you increase the gravitational potential energy, and how much heat input is needed to accomplish this? (c) If your favorite candy bar has 350 food calories (1 food calorie 4186 J) and 80% of the food energy goes into heat, how many of these candy bars must you eat to lift the box in this way?

Entropy Change Due to the Sun. Our sun radiates from a surface at 5800 K (with an emissivity of 1.0) into the near-vacuum of space, which is at a temperature of 3 K. (a) By how much does our sun change the entropy of the universe every second? (Consult Appendix F.) (b) Is the process reversible or irreversible? Is your answer to part (a) consistent with this conclusion? Explain

A monatomic ideal gas is taken around the cycle shown in Fig. P20.51 in the direction shown in the figure. The path for process \(c \rightarrow a\) is a straight line in the pV-diagram. (a) Calculate Q, W, and \(\Delta U\) for each process \(a \rightarrow b\), \(b \rightarrow c\), and \(c \rightarrow a\). (b) What are Q, W, and \(\Delta U\) for one complete cycle? (c) What is the efficiency of the cycle?

A Stirling-Cycle Engine. The Stirling cycle is similar to 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 dosolar, 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, Stirling 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. P20.52): (i) Compressed isothermally at temperature from the initial state a to state b, with a compression ratio r. (ii) Heated at constant volume to state c at temperature (iii) Expanded isothermally at 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 is independent of temperature). (a) Calculate Q, W, and for each of the processes and (b) In the Stirling cycle, the heat transfers in the processes and 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 also absorbs heat back from the gas in the process Hence the heat transfers and do not play a role in determining the efficiency of the engine. Explain this last statement by comparing the expressions for and calculated in part (a). (c) Calculate the efficiency of a Stirlingcycle engine in terms of the temperatures and 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.

A Carnot engine operates between two heat reservoirs at temperatures \(T_{H}\) and \(T_{C}\). An inventor proposes to increase the efficiency by running one engine between \(T_{H}\) and an intermediate temperature \(T^{\prime}\) and a second engine between \(T^{\prime}\) and \(T_{C}\) using as input the heat expelled by the first engine. Compute the efficiency of this composite system, and compare it to that of the original engine.

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 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 rivers temperature is before it reaches the power plant and after it has received the plants waste heat. Calculate the rivers flow rate, in cubic meters per second. (e) By how much does the rivers entropy increase each second?

Automotive Thermodynamics. A Volkswagen Passat has a six-cylinder Otto-cycle engine with compression ratio The diameter of each cylinder, called the bore of the engine, is 82.5 mm. The distance that the piston moves during the compression in Fig. 20.5, called the stroke of the engine, is 86.4 mm. The initial pressure of the airfuel mixture (at point a in Fig. 20.6) is and the initial temperature is 300 K (the same as the outside air). Assume that 200 J of heat is added to each cylinder in each cycle by the burning gasoline, and that the gas has and (a) Calculate the total work done in one cycle in each cylinder of the engine, and the heat released when the gas is cooled to the temperature of the outside air. (b) Calculate the volume of the airfuel mixture at point a in the cycle. (c) Calculate the pressure, volume, and temperature of the gas at points b, c, and d in the cycle. In a pV-diagram, show the numerical values of p, V, and T for each of the four states. (d) Compare the efficiency of this engine with the efficiency of a Carnot-cycle engine operating between the same maximum and minimum temperatures.

An air conditioner operates on 800 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)?

Unavailable Energy. The discussion of entropy and the second law that follows Example 20.10 (Section 20.7) says that the increase in entropy in an irreversible process is associated with energy becoming less available. Consider a Carnot cycle that uses a low-temperature reservoir with Kelvin temperature This is a true reservoirthat is, large enough not to change temperature when it accepts heat from the engine. Let the engine accept heat from an object of temperature where The object is of finite size, so it cools as heat is extracted from it. The engine continues to operate until (a) Show that the total magnitude of heat rejected to the low-temperature reservoir is where is the change in entropy of the high-temperature reservoir. (b) Apply the result of part (a) to 1.00 kg of water initially at a temperature of 373 K as the heat source for the engine and How much total mechanical work can be performed by the engine until it stops? (c) Repeat part (b) for 2.00 kg of water at 323 K. (d) Compare the amount of work that can be obtained from the energy in the water of Example 20.10 before and after it is mixed. Discuss whether your result shows that energy has become less available.

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 (Fig. P20.58). This cylinder has diameter d, length L = vt and density \(\rho\). (b) The Mod-5B wind turbine at Kahaku on the Hawaiian island of Oahu has a blade diameter of 97 m (slightly longer than a football field) and sits atop a 58-m tower. It can produce 3.2 MW of electric power. Assuming 25% efficiency, what wind speed is required to produce this amount of power? Give your answer in m/s and in km/h. (c) Commercial wind turbines are commonly located in or downwind of mountain passes. Why?

CALC (a) For the Otto cycle shown in Fig. 20.6, calculate the changes in entropy of the gas in each of the constant-volume processes and in terms of the temperatures and and the number of moles Td n and the heat capacity of the gas. (b) What is the total entropy change in the engine during one cycle? (Hint: Use the relationships between and and between and ) (c) The processes and occur irreversibly in a real Otto engine. Explain how can this be reconciled with your result in part (b).

CALC A TS-Diagram. (a) Graph a Carnot cycle, plotting Kelvin temperature vertically and entropy horizontally. This is called a temperatureentropy diagram, or TS-diagram. (b) Show that the area under any curve representing a reversible path in a temperatureentropy diagram represents the heat absorbed by the system. (c) Derive from your diagram the expression for the thermal efficiency of a Carnot cycle. (d) Draw a temperatureentropy diagram for the Stirling cycle described in Problem 20.52. Use this diagram to relate the efficiencies of the Carnot and Stirling cycles.

A physics student immerses one end of a copper rod in boiling water at and the other end in an icewater mixture at The sides of the rod are insulated. After steady-state conditions have been achieved in the rod, 0.120 kg of ice melts in a certain time interval. For this time interval, find (a) the entropy change of the boiling water; (b) the entropy change of the icewater mixture; (c) the entropy change of the copper rod; (d) the total entropy change of the entire system.

To heat 1 cup of water \(\left(250\mathrm{\ cm}^3\right)\) to make coffee, you place an electric heating element in the cup. As the water temperature increases from \(20^{\circ} \mathrm{C}\) to \(78^{\circ} \mathrm{C}\), the temperature of the heating element remains at a constant \(120^{\circ} \mathrm{C}\). Calculate the change in entropy of (a) the water; (b) the heating element; (c) the system of water and heating element. (Make the same assumption about the specific heat of water as in Example 20.10 in Section 20.7, and ignore the heat that flows into the ceramic coffee cup itself.) (d) Is this process reversible or irreversible? Explain.

An object of mass \(m_{1}\), specific heat \(c_{1}\), and temperature \(T_{1}\) is placed in contact with a second object of mass \(m_{2}\), specific heat \(c_{2}\) 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’. (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. Text Transcription: m_1 c_1 T_1 m_2 c_2 T_2 > T_1 Delta S = m_1c_1 ln T/T_1 + m_2c_2 ln T’/T_2 m_1c_1(T - T_1) = m_2c_2(T_2 - T’) Delta S

Consider a Diesel cycle that starts (at point a in Fig. 20.7) with air at temperature The air may be treated as an ideal gas. (a) If the temperature at point c is derive an expression for the efficiency of the cycle in terms of the compression ratio r. (b) What is the efficiency if and r = 21.0?

Problem 1DQ A pot is half-filled with water, and a lid is placed on it, forming a tight seal so that no water vapor can escape. The pot is heated on a stove, forming water vapor inside the pot. The heat is then turned off and the water vapor condenses back to liquid. Is this cycle reversible or irreversible? Why?

Problem 1E A diesel engine performs 2200 J of mechanical work and discards 4300 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?

Problem 2E An aircraft engine takes in 9000 J of heat and discards 6400 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?

What irreversible processes occur in a gasoline engine? Why are they irreversible?

Problem 3E A Gasoline Engine. A gasoline engine takes in 1.61 X 104 J of heat and delivers 3700 J of work per cycle. The heat is obtained by burning gasoline with a heat of combustion of 4.60 X 104 J/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?

Problem 4DQ Suppose you try to cool the kitchen of your house by leaving the refrigerator door open. What happens? Why? Would the result be the same if you left open a picnic cooler full of ice? Explain the reason for any differences.

Problem 4E A gasoline engine has a power output of 180 kW (about 241 hp). Its thermal efficiency is 28.0%. (a) How much heat must be supplied to the engine per second? (b) How much heat is dis-carded by the engine per second?

A member of the U.S. Congress proposed a scheme to produce energy as follows. Water molecules (H\(^{2}\)O) are to be broken apart to produce hydrogen and oxygen. The hydrogen is then burned (that is, combined with oxygen), releasing energy in the process. The only product of this combustion is water, so there is no pollution. In light of the second law of thermodynamics, what do you think of this energy-producing scheme?

The pV-diagram in Fig. E20.5 shows a cycle of a heat engine that uses 0.250 mole of an ideal gas having \(\gamma=1.40\). The curved part \(a b\) of the cycle is adiabatic. (a) Find the pressure of the gas at point. (b) How much heat enters this gas per cycle, and where does it happen? (c) How much heat leaves this gas in a cycle, and where does it occur? (d) How much work does this engine do in a cycle? (e) What is the thermal efficiency of the engine? Equation Transcription: Text Transcription: gamma=1.40 ab p(atm) V(m^3)

Is it a violation of the second law of thermodynamics to convert mechanical energy completely into heat? To convert heat completely into work? Explain your answers.

(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 discard to the outside air?

Problem 7DQ Imagine a special air filter placed in a window of a house. The tiny holes in the filter allow only air molecules moving faster than a certain speed to exit the house, and allow only air molecules moving slower than that speed to enter the house from outside. Explain why such an air filter would cool the house, and why the second law of thermodynamics makes building such a filter an impossible task.

Problem 7E 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 ? = 1.40. (b) The engine in a Dodge Viper GT2 has a slightly higher compression ratio of 9.6. How much increase in the ideal efficiency results from this increase in the compression ratio?

An electric motor has its shaft coupled to that of an electric generator. The motor drives the generator, and some current from the generator is used to run the motor. The excess current is used to light a home. What is wrong with this scheme?

Problem 8E The coefficient of performance K = H/P is a dimensionless quantity. Its value is independent of the units used for H and P, as long as the same units, such as watts, are used for both quantities. However, it is common practice to express H in Btu/h and P in watts. When these mixed units are used, the ratio H/P is called the energy efficiency ratio (EER). If a room air conditioner has K = 3.0, what is its EER?

Problem 9DQ When a wet cloth is hung up in a hot wind in the desert, it is cooled by evaporation to a temperature that may be 20 Co or so below that of the air. Discuss this process in light of the second law of thermodynamics.

A refrigerator has a coefficient of performance of 2.10. In each cycle it absorbs \(3.40 \times 10^{4} J\) 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 heal is discarded to the high-temperature reservoir?

Compare the \(p V \text {-diagram }\) for the Otto cycle in Fig. 20.6 with the diagram for the Carnot heat engine in Fig. 20.13. Explain some of the important differences between the two cycles. Equation Transcription: Text Transcription: pV-diagram

Problem 10E A room air conditioner has a coefficient of performance of 2.9 on a hot day and uses 850 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.

If no real engine can be as efficient as a Carnot engine operating between the same two temperatures, what is the point of developing and using Eq. (20.14)?

A refrigerator has a coefficient of performance of 2.25, runs on an input of 95 W of electrical power, and keeps its inside compartment at 5\(^\circ\)C. If you put a dozen 1.0-L, plastic bottles of water at 31\(^\circ\)C into this refrigerator, how long will it lake for them to be cooled down to 5\(^\circ\)C? (ignore any heat that leaves the plastic.)

The efficiency of heat engines is high when the temperature difference between the hot and cold reservoirs is large. Refrigerators, on the other hand, work better when the temperature difference is small. Thinking of the mechanical refrigeration cycle shown in Fig. 20.9, explain in physical terms why it takes less work to remove heat from the working substance if the two reservoirs (the inside of the refrigerator and the outside air) are at nearly the same temperature, than if the outside air is much warmer than the interior of the refrigerator.

A freezer has a coefficient of performance of 2.40. The freezer is to convert 1.80 kg of water at 25.0\(^\circ\)C to 1.80 kg of ice at - 5.0\(^\circ\)C in one hour. (a) What amount of heat must be removed from the water at 25.0\(^\circ\)C to convert it to ice at - 5.0\(^\circ\)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?

Problem 13E A Carnot engine whose high-temperature reservoir is at 620 K takes in 550 J of heat at this temperature in each cycle and gives up 335 J to the low-temperature reservoir. (a) How much mechanical work does the engine perform during each cycle? What is (b) the temperature of the low-temperature reservoir; (c) the thermal efficiency of the cycle?

Real heat engines, like the gasoline engine in a car, always have some friction between their moving parts, although lubricants keep the friction to a minimum. Would a heat engine with completely frictionless parts be 100% efficient? Why or why not? Does the answer depend on whether or not the engine runs on the Carnot cycle? Again, why or why not?

Problem 14E A Carnot engine is operated between two heat reservoirs at temperatures of 520 K and 300 K. (a) If the engine receives 6.45 kJ of heat energy from the reservoir at 520 K in each cycle, how many joules per cycle does it discard to the reservoir at 300 K? (b) How much mechanical work is performed by the engine during each cycle? (c) What is the thermal efficiency of the engine?

Problem 15DQ Does a refrigerator full of food consume more power if the room temperature is 20o C than if it is 15o C? Or is the power consumption the same? Explain your reasoning.

Problem 15E A Carnot engine has an efficiency of 59% and performs 2.5 × 104 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°C). What is the temperature of its heat source?

In Example 20.4, a Carnot refrigerator requires a work input of only 230 J to extract 346 J of heat from the cold reservoir. Doesn’t this discrepancy imply a violation of the law of conservation of energy? Explain why or why not.

An ice-making machine operates in a Carnot 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 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? Text Transcription: 0.0^circ C 24.0^circ C

Explain why each of the following processes is an example of increasing disorder or randomness: mixing hot and cold water; free expansion of a gas; irreversible heat flow; developing heat by mechanical friction. Are entropy increases involved in all of these? Why or why not?

Problem 17E A Carnot refrigerator is operated between two heat reservoirs at temperatures of 320 K and 270 K. (a) If in each cycle the refrigerator receives 415 J of heat energy from the reservoir at 270 K, how many joules of heat energy does it deliver to the reservoir at 320 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?

Problem 18DQ The free expansion of a gas is an adiabatic process and so no heat is transferred. No work is done, so the internal energy does not change. Thus, \(Q / T=0\), yet the disorder of the system and thus its entropy have increased after the expansion. Why does Eq. (20.19) not apply to this situation? Equation Transcription: Text Transcription: Q/T=0

Problem 18E A certain brand of freezer is advertised to use 730 kW ? h of energy per year. (a) Assuming the freezer operates for 5 hours each day, how much power does it require while operating? (b) If the freezer keeps its interior at - 5.0o C in a 20.0o C room, what is its theoretical maximum performance coefficient? (c) What is the theoretical maximum amount of ice this freezer could make in an hour, starting with water at 20.0o C?

Problem 19DQ Are the earth and sun in thermal equilibrium? Are there entropy changes associated with the transmission of energy from the sun to the earth? Does radiation differ from other modes of heat transfer with respect to entropy changes? Explain your reasoning.

Problem 19E A Carnot heat engine has a thermal efficiency of 0.600, and the temperature of its hot reservoir is 800 K. If 3000 J of heat is rejected to the cold reservoir in one cycle, what is the work output of the engine during one cycle?

Problem 20DQ Discuss the entropy changes involved in the preparation and consumption of a hot fudge sundae.

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 kg of ice. During this time, how much work W is performed by the engine?

Problem 21E You design an engine that takes in 1.50 × 104 J of heat at 650 K in each cycle and rejects heat at a temperature of 350 K. The engine completes 240 cycles in 1 minute. What is the theoretical maximum power output of your engine, in horsepower Section 20.7 Entropy

Problem 22E A 4.50-kg block of ice al 0.00°C falls into the ocean and melts. The average temperature of the ocean is 3.50°C, including all the deep water. By how much does the melting of this ice change the entropy of the world? Does it make it larger or smaller? (Hint: Do you think that the ocean will change temperature appreciably as the ice melts?)

Problem 23E A sophomore with nothing better to do adds heat to 0.350 kg of ice at 0.0o 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 25.0o 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?

Problem 24E 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 kg of 30.0°C water and attempt to warm it further by pouring in 5.00 kg of boiling water from the stove. (a) Is this a reversible or an irreversible 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.

Problem 25E A 15.0-kg block of ice at 0.0o C melts to liquid water at 0.0o C inside a large room at 20.0o 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).

Problem 26E CALC You make tea with 0.250 kg of 85.0o C water and let it cool to room temperature (20.0o C). (a) Calculate the entropy change of the water while it cools. (b) The cooling process is essentially isothermal for the air in your kitchen. Calculate the change in entropy of the air while the tea cools, assuming that all of the heat lost by the water goes into the air. What is the total entropy change of the system tea + air?

Problem 27E Three moles of an ideal gas undergo a reversible isothermal compression at 20.0o C. During this compression, 1850 J of work is done on the gas. What is the change of entropy of the gas?

Problem 28E What is the change in entropy of 0.130 kg of helium gas at the normal boiling point of helium when it all condenses isothermally to 1.00 L of liquid helium? (Hint: See Table 17.4 in Section 17.6.)

Problem 29E (a) Calculate the change in entropy when 1.00 kg of water at \(100^{\circ} \mathrm{C}\) is vaporized and converted to steam at \(100^{\circ} \mathrm{C}\) (see Table 17.4). (b) Compare your answer to the change in entropy when 1.00 kg of ice is melted at \(0^{\circ} \mathrm{C}\), calculated in Example 20.5 (Section 20.7). Is the change in entropy greater for melting or for vaporization? Interpret your answer using the idea that entropy is a measure of the randomness of a system. Equation Transcription: Text Transcription: 100degC 100degC 0degC

(a) Calculate the change in entropy when 1.00 mol of water (molecular mass 18.0g/mol ) at \(100^\circ C\) evaporates to form water vapor at \(100^\circ C\). (b) Repeat the calculation of part (a) for 1.00 mol of liquid nitrogen, 1.00 mol of silver, and 1.00 mol of mercury when each is vaporized at its normal boiling point. (See Table 17.4 for the heats of vaporization, and Appendix D for the molar masses. Note that the nitrogen molecule is \(N_{2}\).) (c) Your results in parts (a) and (b) should be in relatively close agreement. (This is called the rule of Drepez and Trouton.) Explain why this should be so, using the idea that entropy is a measure of the randomness of a system.

Problem 31E A 10.0-L gas tank containing 3.20 moles of ideal He gas at 20.0°C is placed inside, completely evacuated, insulated bell jar of volume 35.0 L. A small hole in the tank allows the He to leak out into the jar until the gas reaches a final equilibrium state with no more leakage. (a) What is the change in entropy of this system due to the leaking of the gas? (b) Is the process reversible or irreversible How do you know?

Problem 32E 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? (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?

Problem 34E CALC A lonely party balloon with a volume of 2.40 L and containing 0.100 mol of air is left behind to drift in the temporarily uninhabited and depressurized International 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 m3. Calculate the entropy change of the air during the expansion.

Problem 35P CP An ideal Carnot engine operates between 500o C and 100o C with a heat input of 250 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-kg rock through a height of 100 m?

Problem 36P You are designing a Carnot engine that has 2 mol of CO2 as its working substance; the gas may be treated as ideal. The gas is to have a maximum temperature of 527o C and a maximum pressure of 5.00 atm. With a heat input of 400 J per cycle, you want 300 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 0.0o C, using only the heat rejected by the engine?

Problem 37P A certain heat engine operating on a Carnot cycle absorbs 150 J of heat per cycle at its hot reservoir al 135°C and has a thermal efficiency or 22.0%. (a) How much work does this engine do per cycle? (b) How much heat does the engine waste each cycle? (c) What is the temperature of the cold reservoir? (d) By how much does the engine change the entropy of the world each cycle? (e) What mass of water could this engine pump per cycle from a well 35.0 m deep?

Problem 38P BIO Entropy of Metabolism. An average sleeping person metabolizes at a rate of about 80 W by digesting food or burning fat. Typically, 20% of this energy goes into bodily functions, such as cell repair, pumping blood, and other uses of mechanical energy, while the rest goes to heat. Most people get rid of all this excess heat by transferring it (by conduction and the flow of blood) to the surface of the body, where it is radiated away. The normal internal temperature of the body (where the metabolism takes place) is 37°C, and the skin is typically 7 C° cooler. By how much does the person’s entropy change per second due to this heat transfer?

Problem 39P BIO Entropy Change from Digesting Fat. Digesting fat produces 9.3 food calories per gram of fat, and typically 80% of this energy goes to heat when metabolized. (One food calorie is 1000 calories and therefore equals 4186 J.) The body then moves all this heat to the surface by a combination of thermal conductivity and motion of the blood. The internal temperature of the body (where digestion occurs) is normally 37°C, and the surface is usually about 30°C. By how much do the digestion and metabolism of a 2.50-g pat of butter change your body’s entropy? Does it increase or decrease?

Problem 40P A heat engine takes 0.350 mol of a diatomic ideal gas around the cycle shown in the pV-diagram of Fig. P20.40. Process \(1 \rightarrow 2\) is at constant volume, process \(2 \rightarrow 3\) is adiabatic, and \(3 \rightarrow 1\) is at a constant pressure of 1.00 atm. The value of ? for this gas is 1.40. (a) Find the pressure and volume at points 1, 2, and 3. (b) Calculate Q, W, and \(\Delta U\) for each of the three processes. (c) Find the net work done by the gas in the cycle. (d) Find the net heat flow into the engine in one cycle. (e) What is the thermal efficiency of the engine? How does this compare to the efficiency of a Carnot-cycle engine operating between the same minimum and maximum temperatures \(\mathrm{T}_{1}) and \(\mathrm{T}_{2}\)? Equation Transcription: Text Transcription: 1-2 2-3 3-1 deltaU T_1 T_2 T_2 T_1 T_3

Problem 41P CALC You build a heat engine that takes 1.00 mol of an ideal diatomic gas through the cycle shown in Fig. P20.41. (a) Show that segment ab is an isothermal compression. (b) During which segment(s) of the cycle is heat absorbed by the gas? During which segment(s) is heat rejected? How do you know? (c) Calculate the temperature at points a, b, and c. (d) Calculate the net heat exchanged with the surroundings and the net work done by the engine in one cycle. (e) Calculate the thermal efficiency of the engine. Equation Transcription: Text Transcription: p(Pa) V(m^3)

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 sutuner 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\(^\circ\)C and the inside temperature is 17.0\(^\circ\)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\(^\circ\)F. Show that the heat 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.

Problem 43P CALC A heat engine operates using the cycle shown in Fig. P20.43. The working substance is 2.00 mol of helium gas, which reaches a maximum temperature of \(327^{\circ} \mathrm{C}\). Assume the helium can be treated as an ideal gas. Process bc is isothermal. The pressure in states a and c is \(1.00 \times 10^{5}\) Pa, and the pressure in state b is \(3.00 \times 10^{5}\) Pa. (a) How much heat enters the gas and how much leaves the gas each cycle? (b) How much work does the engine do each cycle, and what is its efficiency? (c) Compare this engine’s efficiency with the maximum possible efficiency attainable with the hot and cold reservoirs used by this cycle. Equation Transcription: Text Transcription: 327degC 1.00x10^5 3.00x10^5

Problem 44P CP As a budding mechanical engineer, you are called upon to design a Carnot engine that has 2.00 mol of a monatomic ideal gas as its working substance and operates from a high- temperature reservoir at 500o C. The engine is to lift a 15.0-kg weight 2.00 m per cycle, using 500 J of heat input. The gas in the engine chamber can have a minimum volume of 5.00 L during the cycle. (a) Draw a pV-diagram for this cycle. Show in your diagram where heat enters and leaves the gas. (b) What must be the temperature of the cold reservoir? (c) What is the thermal efficiency of the engine? (d) How much heat energy does this engine waste per cycle? (e) What is the maximum pressure that the gas chamber will have to withstand?

Problem 45P 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 270c and 60c respectively. (a) what is the maximum theoretical efficiency of this power plant? (b) If the powerplant is to produce 210 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 100c. What must be the flow rate of cold water through the system? Give your answer in kg/h and in L/h.

Problem 46P What is the thermal efficiency of an engine that operates by taking n moles of diatomic ideal gas through the cycle \(1 \rightarrow 2 \rightarrow 3 \rightarrow 4 \rightarrow 1\) shown in Fig. P20.46? Equation Transcription: Text Transcription: 1-2-3-4-1 2p_0 p_0 v_0 2V_0

CALC A cylinder contains oxygen at a pressure of 2.00 atm. The volume is 4.00 L, and the temperature is 300 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 K (ii) Cooled at constant volume to 250 K (state 3) (iii) Compressed at constant temperature to a volume of 4.00 L (state 4).(iv) Heated at constant volume to 300 K, which takes the system back to state 1. (a) Show these four processes in a pV-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 in the complete cycle. (d) What is the efficiency of this device as a heat engine? How does this compare to the efficiency of a Carnot-cycle engine operating between the same minimum and maximum temperatures of 250 K and 450 K?

Problem 48P BIO Human Entropy. A person with skin of surface area 1.85 m2 and temperature 30.0°C is resting in an insulated room where the ambient air temperature is 20.0°C. In this state, a person gets rid of excess heat by radiation. By how much does the person change the entropy of the air in this room each second? (Recall that the room radiates back into the person and that the emissivity of the skin is 1.00.)

Problem 49P CP BIO A Human Engine. You decide to use your body as a Carnot heat engine. The operating gas is in a tube with one end in your mouth (where the temperature is 37.0°C) and the other end at the surface of your skin, at 30.0°C. (a) What is the maximum efficiency of such a heat engine? Would it be a very useful engine? (b) Suppose you want to use this human engine to lift a 2.50-kg box from the floor to a tabletop 1.20 m above the floor. How much must you increase the gravitational potential energy, and how much heat input is needed to accomplish this? (c) If your favorite candy bar has 350 food calories (1 food calorie = 4186 J) and 80% of the food energy goes into heat, how many of these candy bars must you eat to lift the box in this way?

Entropy Change Due to the Sun. Our sun radiates from a surface at 5800 K (with an emissivity of 1) into the near-vacuum of space, which is at a temperature of 3 K. (a) By how much does our sun change the entropy of the universe every second? (Consult Appendix E) (b) Is the process reversible or irreversible? Is your answer to part (a) consistent with this conclusion? Explain.

Problem 51P A monatomic ideal gas is taken around the cycle shown in Fig. P20.51 in the direction shown in the figure. The path for process \(\mathrm{c} \rightarrow \mathrm{a}\) is a straight line in the \(p V \text {-diagram }\). (a) Calculate Q, W, and \(\Delta U\) for each process \(a \rightarrow b, b \rightarrow c\), and \(c \rightarrow a\). (b) What are Q, W, and \(\Delta U\) for one complete cycle? (c) What is the efficiency of the cycle? Equation Transcription: Text Transcription: c-a pV-diagram deltaU a-b,b-c, c-a deltaU 3.00x10^5 Pa 1.00x10^5 Pa 0.500 m^3 0.800 m^3

Problem 52P A Stirling-Cycle Engine. The Stirling cycle is similar to 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, Stirling 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. P20.52): (i) Compressed isothermally at temperature \(\mathrm{T}_{1}\) from the initial state a to state b, with a compression ratio r. (ii) Heated at constant volume to state c at temperature \(\mathrm{T}_{2}\). (iii) Expanded isothermally at \(\mathrm{T}_{2}\) 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 \(a \rightarrow b, b \rightarrow c, c \rightarrow d, d \rightarrow a\). (b) In the Stirling cycle, the heat transfers in the processes\(\mathrm{b} \rightarrow \mathrm{c}\) and \(\mathrm{d} \rightarrow \mathrm{a}\) 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 \(\mathrm{d} \rightarrow \mathrm{a}\). 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 \(\mathrm{T}_{1}\) and \(\mathrm{T}_{2}\). 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. Equation Transcription: Text Transcription: T_1 T_2 T_2 C_V deltaU a-b,b-c,c-d,d-a b-c d-a b-c d-a Q_b-c Q_d-a Q_b-c Q_d-a T_1 T_2 T_2 T_1 V_b=V_a/r V_a

Problem 53P A Carnot engine operates between two heat reservoirs at temperatures TH and TC. An inventor proposes to increase the efficiency by running one engine between TH and an intermediate temperature T and a second engine between T and TC, using as input the heat expelled by the first engine. Compute the efficiency of this composite system, and compare it to that of the original engine.

Problem 54P 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 2.65 X 107 J/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 is at 18.0o C before it reaches the power plant and 18.5o 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?

Problem 55P Automotive Thermodynamics. A Volkswagen Passat has a six-cylinder Otto-cycle engine with compression ratio \(r=10.6\). The diameter of each cylinder, called the bore of the engine, is 82.5 mm. The distance that the piston moves during the compression in Fig. 20.5, called the stroke of the engine, is 86.4 mm. The initial pressure of the air–fuel mixture (at point a in Fig. 20.6) is \(8.50 \times 10^{4}\) Pa, and the initial temperature is 300 K (the same as the outside air). Assume that 200 J of heat is added to each cylinder in each cycle by the burning gasoline, and that the gas has \(C_{v}=20.5 \mathrm{~J} / \mathrm{mol} \cdot \mathrm{K}\) and \(\gamma=1.40\). (a) Calculate the total work done in one cycle in each cylinder of the engine, and the heat released when the gas is cooled to the temperature of the outside air. (b) Calculate the volume of the air–fuel mixture at point a in the cycle. (c) Calculate the pressure, volume, and temperature of the gas at points b, c, and d in the cycle. In a \(p V \text {-diagram }\), show the numerical values of p, V, and T for each of the four states. (d) Compare the efficiency of this engine with the efficiency of a Carnot-cycle engine operating between the same maximum and minimum temperatures. Equation Transcription: Text Transcription: r=10.6 8.50x10^4 C_v=20.5 J/mol.K gamma=1.40 pV-diagram

Problem 57P Unavailable Energy. The discussion of entropy and the second law that follows Example 20.10 (Section 20.7) says that the increase in entropy in an irreversible process is associated with energy becoming less available. Consider a Carnot cycle that uses a low-temperature reservoir with Kelvin temperature \(T_{c}\). This is a true reservoir—that is, large enough not to change temperature when it accepts heat from the engine. Let the engine accept heat from an object of temperature \(\mathrm{T}^{\prime}\), where \(\mathrm{T}^{\prime}>\mathrm{T}_{\mathrm{c}}\). The object is of finite size, so it cools as heat is extracted from it. The engine continues to operate until \(T^{\prime}=T_{c}\). (a) Show that the total magnitude of heat rejected to the low-temperature reservoir is \(T_{\mathrm{c}}\left|\Delta S_{\mathrm{h}}\right|\), where \(\Delta S_{h}\) is the change in entropy of the high-temperature reservoir. (b) Apply the result of part (a) to 1.00 kg of water initially at a temperature of 373 K as the heat source for the engine and \(T_{C}=273\) K. How much total mechanical work can be performed by the engine until it stops? (c) Repeat part (b) for 2.00 kg of water at 323 K. (d) Compare the amount of work that can be obtained from the energy in the water of Example 20.10 before and after it is mixed. Discuss whether your result shows that energy has become less available. Equation Transcription: Text Transcription: T_C T’ T’>T_C T’=T_C T_C|deltaS_h| deltaS_h T_C=273

Problem 59P (a) For the Otto cycle shown in Fig. 20.6, calculate the changes in entropy of the gas in each of the constant-volume processes \(b \rightarrow c\) and \(d \rightarrow a\) in terms of the temperatures T_{a}, T_{b}, T_{c}, and \(T_{d}\) and the number of moles n and the heat capacity \(C_{V}\) of the gas. (b) What is the total entropy change in the engine during one cycle? (Hint: Use the relationships between \(T_{a}\) and \(T_{b}\) and between \(T_{d}\) and \(T_{c}\).) (c) The processes \(b \rightarrow c\) and \(d \rightarrow a\) occur irreversibly in a real Otto engine. Explain how can this be reconciled with your result in part (b). Equation Transcription: Text Transcription: b-c d-a T_a,T_b,T_c T-d C_V T_a T_b T_d T_c b-c d-a

Problem 60P A TS-Diagram. (a) Graph a Carnot cycle, plotting Kelvin temperature vertically and entropy horizontally. This is called a temperature-entropy diagram, or T.S-diagram. (b) Show that the area under any curve representing a reversible path in a temperature-entropy diagram represents the heal absorbed by the system. (c) Derive from your diagram the expression for the thermal efficiency of a Carnot cycle. (d) Draw a temperature-entropy diagram for the Stirling cycle described. Use this diagram to relate the efficiencies of the Carnot and Stirling cycles.

Problem 61P A physics student immerses one end of a copper rod in boiling water at 100°C and the other end in an ice-water mixture at 0°C. The sides of the rod are insulated. After steady-state conditions have been achieved in the rod, 0.120 kg of ice melts in a certain time interval. For this time interval, find (a) the entropy change of the boiling water; (b) the entropy change of the ice-water mixture; (c) the entropy change of the copper rod; (d) the total entropy change of the entire system.

To heat 1 cup of water \(\left(250 \mathrm{~cm}^3\right)\) to make coffee, you place an electric heating element in the cup. As the water temperature increases from \(20^{\circ} \mathrm{C}\) to \(78^{\circ} \mathrm{C}\), the temperature of the heating element remains at a constant \(120^{\circ} \mathrm{C}\). Calculate the change in entropy of (a) the water; (b) the heating element; (c) the system of water and heating element. (Make the same assumption about the specific heat of water as in Example 20.10 in Section 20.7, and ignore the heat that flows into the ceramic coffee cup itself.) (d) Is this process reversible or irreversible? Explain.

Consider a Diesel cycle that starts (at point a in Fig. 20.7) with air at temperature \(T_{a}\). The air may be treated as an ideal gas. (a) If the temperature at point c is \(T_{c}\), derive an expression for the efficiency of the cycle in terms of the compression ratio r. (b) What is the efficiency if \(T_a=300\mathrm{\ K},\ T_c=950\mathrm{\ K},\ \gamma=1.40, \text { and } r=21.0\)? Text Transcription: T_a T_c T_a = 300 K, T_c = 950 K, gamma = 1.40, and r = 21.0

Problem 2DQ Give two examples of reversible processes and two examples of irreversible processes in purely mechanical systems, such as blocks sliding on planes, springs, pulleys, and strings. Explain what makes each process reversible or irreversible.

Problem 21DQ If you run a movie film backward, it is as if the direction of time were reversed. In the time-reversed movie, would you see processes that violate conservation of energy? Conservation of linear momentum? Would you see processes that violate the second law of thermodynamics? In each case, if law-breaking processes could occur, give some examples.

Some critics of biological evolution claim that it violates the second law of thermodynamics, since evolution involves simple life forms developing into more complex and more highly ordered organisms. Explain why this is not a valid argument against evolution.

Problem 23DQ BIO A growing plant creates a highly complex and organized structure out of simple materials such as air, water, and trace minerals. Does this violate the second law of thermodynamics? Why or why not? What is the plant’s ultimate source of energy? Explain.

CALC Two moles of an ideal gas occupy a volume V. The gas expands isothermally and reversibly to a volume 3V. (a) Is the velocity distribution changed by the isothermal expansion? Explain. (b) Use Eq. (20.23) to calculate the change in entropy of the gas. (c) Use Eq. (20.18) to calculate the change in entropy of the gas. Compare this result to that obtained in part (b).

Problem 56P An air conditioner operates on 800 W of power and has a performance coefficient of 2.80 with a room temperature of 21.0o C and an outside temperature of 35.0o 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)?

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 (Fig. P20.58). This cylinder has diameter d, length L = vt, and density \(\rho\). (b) The Mod-5B wind turbine at Kahaku on the Hawaiian island of Oahu has a blade diameter of 97 m (slightly longer than a football field) and sits atop a 58-m tower. It can produce 3.2 MW of electric power. Assuming 25% efficiency, what wind speed is required to produce this amount of power? Give your answer in m/s and in km/h. (c) Commercial wind turbines are commonly located in or downwind of mountain passes. Why? Text Transcription: P = kd^2v^3 k = 0.5 W cdot s^3/m^5 rho

An object of mass \(m_1\), specific heat \(c_1\), and temperature \(T_1\) is placed in contact with a second object of mass \(m_2\), specific heat \(c_2\), 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’. (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.