Problems

Note to Instructors: Most of the homework problems in this chapter are available for assignment via an online homework management program such as WileyPLUS or WebAssign, and those marked with the icon are presented in a guided tutorial format that provides enhanced interactivity. See Preface for additional details.
ssm Solution is in the Student Solutions Manual.
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Section 15.3The First Law of Thermodynamics

| 1 | ssm A system does 164 J of work on its environment and gains 77 J of heat in the process. Find the change in the internal energy of (a) the system and (b) the environment.

| 2 | A jogger’s internal energy changes because he performs of work and gives off of heat. However, to cause the same change in his internal energy while walking, he must do of work. Determine the magnitude of the heat given off while walking.

| 3 | In moving out of a dormitory at the end of the semester, a student does of work. In the process, his internal energy decreases by . Determine each of the following quantities (including the algebraic sign): (a) W, (b) and (c) Q.

| 4 | Three moles of an ideal monatomic gas are at a temperature of 345 K. Then, 2438 J of heat is added to the gas, and 962 J of work is done on it. What is the final temperature of the gas?

| 5 | ssm When one gallon of gasoline is burned in a car engine, of internal energy is released. Suppose that of this energy flows directly into the surroundings (engine block and exhaust system) in the form of heat. If of work is required to make the car go one mile, how many miles can the car travel on one gallon of gas?

| 6 | In exercising, a weight lifter loses 0.150 kg of water through evaporation, the heat required to evaporate the water coming from the weight lifter’s body. The work done in lifting weights is . (a) Assuming that the latent heat of vaporization of perspiration is , find the change in the internal energy of the weight lifter. (b) Determine the minimum number of nutritional Calories of food () that must be consumed to replace the loss of internal energy.
Section 15.4 Thermal Processes

| 7 | A system undergoes a two-step process. In the first step, the internal energy of the system increases by 228 J when 166 J of work is done on the system. In the second step, the internal energy of the system increases by 115 J when 177 J of work is done on the system. For the overall process, find the heat. What type of process is the overall process? Explain.

| 8 | A gas, while expanding under isobaric conditions, does 480 J of work. The pressure of the gas is , and its initial volume is . What is the final volume of the gas?

| 9 | ssm When a .22-caliber rifle is fired, the expanding gas from the burning gunpowder creates a pressure behind the bullet. This pressure causes the force that pushes the bullet through the barrel. The barrel has a length of 0.61 m and an opening whose radius is . A bullet () has a speed of 370 m/s after passing through this barrel. Ignore friction and determine the average pressure of the expanding gas.

| 10 | The pressure and volume of a gas are changed along the path ABCA. Using the data shown in the graph, determine the work done (including the algebraic sign) in each segment of the path: (a) A to B, (b) B to C, and (c) C to A.

| 11 | Using the data presented in the accompanying pressure-versus-volume graph, estimate the magnitude of the work done when the system changes from A to B to C along the path shown. (b) Determine whether the work is done by the system or on the system and, hence, whether the work is positive or negative.

| 12 | Refer to Multiple-Concept Example 3 to see how the concepts pertinent to this problem are used. The pressure of a gas remains constant while the temperature, volume, and internal energy of the gas increase by 53.0 C°, , and 939 J, respectively. The mass of the gas is 24.0 g, and its specific heat capacity is 1080 J/(kg · C°). Determine the pressure.

| 13 | ssm A system gains 1500 J of heat, while the internal energy of the system increases by 4500 J and the volume decreases by 0.010 m3. Assume the pressure is constant and find its value.

* | 14 | Refer to the drawing that accompanies problem 11. When a system changes from A to B along the path shown on the pressure-versus-volume graph, it gains 2700 J of heat. What is the change in the internal energy of the system?

* | 15 | ssm www A monatomic ideal gas expands isobarically. Using the first law of thermodynamics, prove that the heat Q is positive, so that it is impossible for heat to flow out of the gas.

* | 16 | A piece of aluminum has a volume of . The coefficient of volume expansion for aluminum is . The temperature of this object is raised from 20 to 320 °C. How much work is done by the expanding aluminum if the air pressure is ?

** | 17 | Water is heated in an open pan where the air pressure is one atmosphere. The water remains a liquid, which expands by a small amount as it is heated. Determine the ratio of the work done by the water to the heat absorbed by the water. Section 15.5 Thermal Processes Using an Ideal Gas

| 18 | The temperature of a monatomic ideal gas remains constant during a process in which 4700 J of heat flows out of the gas. How much work (including the proper + or − sign) is done?

| 19 | ssm One-half mole of a monatomic ideal gas absorbs 1200 J of heat while 2500 J of work is done by the gas. (a) What is the temperature change of the gas? (b) Is the change an increase or a decrease?

| 20 | Five moles of a monatomic ideal gas expand adiabatically, and its temperature decreases from 370 to 290 K. Determine (a) the work done (including the algebraic sign) by the gas, and (b) the change in its internal energy.

| 21 | The pressure of a monatomic ideal gas () doubles during an adiabatic compression. What is the ratio of the final volume to the initial volume?

| 22 | Heat is added isothermally to 2.5 mol of a monatomic ideal gas. The temperature of the gas is 430 K. How much heat must be added to make the volume of the gas double?

| 23 | ssm A monatomic ideal gas has an initial temperature of 405 K. This gas expands and does the same amount of work whether the expansion is adiabatic or isothermal. When the expansion is adiabatic, the final temperature of the gas is 245 K. What is the ratio of the final to the initial volume when the expansion is isothermal?

* | 24 | A monatomic ideal gas expands from point A to point B along the path shown in the drawing. (a) Determine the work done by the gas. (b) The temperature of the gas at point A is 185 K. What is its temperature at point B? (c) How much heat has been added to or removed from the gas during the process?

* | 25 | The pressure and volume of an ideal monatomic gas change from A to B to C, as the drawing shows. The curved line between A and C is an isotherm. (a) Determine the total heat for the process and (b) state whether the flow of heat is into or out of the gas.

* | 26 | The drawing refers to one mole of a monatomic ideal gas and shows a process that has four steps, two isobaric (A to B, C to D) and two isochoric (B to C, D to A). Complete the following table by calculating , W, and Q (including the algebraic signs) for each of the four steps.

|  | W | Q
A to B |  |  |
B to C |  |  |
C to D |  |  |
D to A |  |  |

* | 27 | ssm A monatomic ideal gas () is contained within a perfectly insulated cylinder that is fitted with a movable piston. The initial pressure of the gas is . The piston is pushed so as to compress the gas, with the result that the Kelvin temperature doubles. What is the final pressure of the gas?

** | 28 | One mole of a monatomic ideal gas has an initial pressure, volume, and temperature of P0, V0, and 438 K, respectively. It undergoes an isothermal expansion that triples the volume of the gas. Then, the gas undergoes an isobaric compression back to its original volume. Finally, the gas undergoes an isochoric increase in pressure, so that the final pressure, volume, and temperature are P0, V0, and 438 K, respectively. Find the total heat for this three-step process, and state whether it is absorbed by or given off by the gas.

** | 29 | ssm The work done by one mole of a monatomic ideal gas () in expanding adiabatically is 825 J. The initial temperature and volume of the gas are 393 K and 0.100 m3. Obtain (a) the final temperature and (b) the final volume of the gas. Section 15.6 Specific Heat Capacities

| 30 | Heat is added to two identical samples of a monatomic ideal gas. In the first sample the heat is added while the volume of the gas is kept constant, and the heat causes the temperature to rise by 75 K. In the second sample, an identical amount of heat is added while the pressure (but not the volume) of the gas is kept constant. By how much does the temperature of this sample increase?

| 31 | ssm The temperature of 2.5 mol of a monatomic ideal gas is 350 K. The internal energy of this gas is doubled by the addition of heat. How much heat is needed when it is added at (a) constant volume and (b) constant pressure?

| 32 | Three moles of a monatomic ideal gas are heated at a constant volume of 1.50 m3. The amount of heat added is . (a) What is the change in the temperature of the gas? (b) Find the change in its internal energy. (c) Determine the change in pressure.

| 33 | Heat Q is added to a monatomic ideal gas at constant pressure. As a result, the gas does work W. Find the ratio Q/W.

| 34 | Under constant-volume conditions, 3500 J of heat is added to 1.6 moles of an ideal gas. As a result, the temperature of the gas increases by 75 K. How much heat would be required to cause the same temperature change under constant-pressure conditions? Do not assume anything about whether the gas is monatomic, diatomic, etc.

| 35 | ssm How much heat is required to change the temperature of 1.5 mol of a monatomic ideal gas by 77 K if the pressure is held constant?

* | 36 | Even at rest, the human body generates heat. The heat arises because of the body’s metabolism—that is, the chemical reactions that are always occurring in the body to generate energy. In rooms designed for use by large groups, adequate ventilation or air conditioning must be provided to remove this heat. Consider a classroom containing 200 students. Assume that the metabolic rate of generating heat is 130 W for each student and that the heat accumulates during a fifty-minute lecture. In addition, assume that the air has a molar specific heat of and that the room (, , and ) is sealed shut. If all the heat generated by the students were absorbed by the air, by how much would the air temperature rise during a lecture?

* | 37 | Interactive Solution 15.37 at offers one approach to this problem. A fifteen-watt heater is used to heat a monatomic ideal gas at a constant pressure of . During the process, the volume of the gas increases by 25.0%. How long was the heater on?

* | 38 | Suppose a monatomic ideal gas is contained within a vertical cylinder that is fitted with a movable piston. The piston is frictionless and has a negligible mass. The area of the piston is , and the pressure outside the cylinder is . Heat (2093 J) is removed from the gas. Through what distance does the piston drop?

** | 39 | One mole of neon, a monatomic gas, starts out at conditions of standard temperature and pressure. The gas is heated at constant volume until its pressure is tripled, then further heated at constant pressure until its volume is doubled. Assume that neon behaves as an ideal gas. For the entire process, find the heat added to the gas.
Section 15.8 Heat Engines

| 40 | Due to a tune-up, the efficiency of an automobile engine increases by 5.0%. For an input heat of 1300 J, how much more work does the engine produce after the tune-up than before?

| 41 | ssm Multiple-Concept Example 6 deals with the concepts that are important in this problem. In doing 16 600 J of work, an engine rejects 9700 J of heat. What is the efficiency of the engine?

| 42 | Engine 1 has an efficiency of 0.18 and requires 5500 J of input heat to perform a certain amount of work. Engine 2 has an efficiency of 0.26 and performs the same amount of work. How much input heat does the second engine require?

| 43 | Multiple-Concept Example 6 provides a review of the concepts that play roles here. An engine has an efficiency of 64% and produces 5500 J of work. Determine (a) the input heat and (b) the rejected heat.

* | 44 | Due to design changes, the efficiency of an engine increases from 0.23 to 0.42. For the same input heat |QH|, these changes increase the work done by the more efficient engine and reduce the amount of heat rejected to the cold reservoir. Find the ratio of the heat rejected to the cold reservoir for the improved engine to that for the original engine.

** | 45 | ssm www Engine A receives three times more input heat, produces five times more work, and rejects two times more heat than engine B. Find the efficiency of (a) engine A and (b) engine B. Section 15.9 Carnot’s Principle and the Carnot Engine

| 46 | Five thousand joules of heat is put into a Carnot engine whose hot and cold reservoirs have temperatures of 500 and 200 K, respectively. How much heat is converted into work?

| 47 | A Carnot engine operates with an efficiency of 27.0% when the temperature of its cold reservoir is 275 K. Assuming that the temperature of the hot reservoir remains the same, what must be the temperature of the cold reservoir in order to increase the efficiency to 32.0%?

| 48 | A Carnot engine has an efficiency of 0.40. The Kelvin temperature of its hot reservoir is quadrupled, and the Kelvin temperature of its cold reservoir is doubled. What is the efficiency that results from these changes?

| 49 | ssm A Carnot engine has an efficiency of 0.700, and the temperature of its cold reservoir is 378 K. (a) Determine the temperature of its hot reservoir. (b) If 5230 J of heat is rejected to the cold reservoir, what amount of heat is put into the engine?

| 50 | A Carnot 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. When 6800 J of heat is put into the engine and the engine produces work, how many kilograms of ice in the tub are melted due to the heat delivered to the cold reservoir?

| 51 | Concept Simulation 15.1 at illustrates the concepts pertinent to this problem. A Carnot engine operates between temperatures of 650 and 350 K. To improve the efficiency of the engine, it is decided either to raise the temperature of the hot reservoir by 40 K or to lower the temperature of the cold reservoir by 40 K. Which change gives the greatest improvement? Justify your answer by calculating the efficiency in each case.

* | 52 | The hot reservoir for a Carnot engine has a temperature of 890 K, while the cold reservoir has a temperature of 670 K. The heat input for this engine is 4800 J. The 670-K reservoir also serves as the hot reservoir for a second Carnot engine. This second engine uses the rejected heat of the first engine as input and extracts additional work from it. The rejected heat from the second engine goes into a reservoir that has a temperature of 420 K. Find the total work delivered by the two engines.

* | 53 | ssm A power plant taps steam superheated by geothermal energy to 505 K (the temperature of the hot reservoir) and uses the steam to do work in turning the turbine of an electric generator. The steam is then converted back into water in a condenser at 323 K (the temperature of the cold reservoir), after which the water is pumped back down into the earth where it is heated again. The output power (work per unit time) of the plant is 84 000 kilowatts. Determine (a) the maximum efficiency at which this plant can operate and (b) the minimum amount of rejected heat that must be removed from the condenser every twenty-four hours.

* | 54 | Suppose the gasoline in a car engine burns at 631 °C, while the exhaust temperature (the temperature of the cold reservoir) is 139 °C and the outdoor temperature is 27 °C. Assume that the engine can be treated as a Carnot engine (a gross oversimplification). In an attempt to increase mileage performance, an inventor builds a second engine that functions between the exhaust and outdoor temperatures and uses the exhaust heat to produce additional work. Assume that the inventor’s engine can also be treated as a Carnot engine. Determine the ratio of the total work produced by both engines to that produced by the first engine alone.

** | 55 | ssm A nuclear-fueled electric power plant utilizes a so-called “boiling water reactor.” In this type of reactor, nuclear energy causes water under pressure to boil at 285 °C (the temperature of the hot reservoir). After the steam does the work of turning the turbine of an electric generator, the steam is converted back into water in a condenser at 40 °C (the temperature of the cold reservoir). To keep the condenser at 40 °C, the rejected heat must be carried away by some means-for example, by water from a river. The plant operates at three-fourths of its Carnot efficiency, and the electrical output power of the plant is . A river with a water flow rate of is available to remove the rejected heat from the plant. Find the number of Celsius degrees by which the temperature of the river rises.

** | 56 | The drawing (not to scale) shows the way in which the pressure and volume change for an ideal gas that is used as the working substance in a Carnot engine. The gas begins at point a () and expands isothermally at temperature TH until point b () is reached. During this expansion, the input heat of magnitude |QH| enters the gas from the hot reservoir of the engine. Then, from point b to point c (), the gas expands adiabatically. Next, the gas is compressed isothermally at temperature TC from point c to point d (). During this compression, heat of magnitude |QC| is rejected to the cold reservoir of the engine. Finally, the gas is compressed adiabatically from point d to point a, where the gas is back in its initial state. The overall process a to b to c to d is called a Carnot cycle. Prove for this cycle that .

Section 15.10 Refrigerators, Air Conditioners, and Heat Pumps

| 57 | A refrigerator operates between temperatures of 296 and 275 K. What would be its maximum coefficient of performance?

| 58 | See Multiple-Concept Example 10 to review the concepts that are important in this problem. The water in a deep underground well is used as the cold reservoir of a Carnot heat pump that maintains the temperature of a house at 301 K. To deposit 14 200 J of heat in the house, the heat pump requires 800 J of work. Determine the temperature of the well water.

| 59 | ssm www The temperatures indoors and outdoors are 299 and 312 K, respectively. A Carnot air conditioner deposits of heat outdoors. How much heat is removed from the house?

| 60 | A Carnot refrigerator is used in a kitchen in which the temperature is kept at 301 K. This refrigerator uses 241 J of work to remove 2561 J of heat from the food inside. What is the temperature inside the refrigerator?

| 61 | A heat pump removes 2090 J of heat from the outdoors and delivers 3140 J of heat to the inside of a house. (a) How much work does the heat pump need? (b) What is the coefficient of performance of the heat pump?

| 62 | A Carnot air conditioner maintains the temperature in a house at 297 K on a day when the temperature outside is 311 K. What is the coefficient of performance of the air conditioner?

* | 63 | Review Conceptual Conceptual Example 9 before attempting this problem. A window air conditioner has an average coefficient of performance of 2.0. This unit has been placed on the floor by the bed, in a futile attempt to cool the bedroom. During this attempt of heat is pulled in the front of the unit. The room is sealed and contains 3800 mol of air. Assuming that the molar specific heat capacity of the air is , determine the rise in temperature caused by operating the air conditioner in this manner.

* | 64 | How long would a 3.00-kW space heater have to run to put into a kitchen the same amount of heat as a refrigerator () does when it freezes 1.50 kg of water at 20.0 °C into ice at 0.0 °C?

* | 65 | ssm www A Carnot refrigerator transfers heat from its inside (6.0 °C) to the room air outside (20.0 °C). (a) Find the coefficient of performance of the refrigerator. (b) Determine the magnitude of the minimum work needed to cool 5.00 kg of water from 20.0 to 6.0 °C when it is placed in the refrigerator.

* | 66 | An air conditioner keeps the inside of a house at a temperature of 19.0 °C when the outdoor temperature is 33.0 °C. Heat, leaking into the house at the rate of 10 500 joules per second, is removed by the air conditioner. Assuming that the air conditioner is a Carnot air conditioner, what is the work per second that must be done by the electrical energy in order to keep the inside temperature constant?

** | 67 | ssm A Carnot engine uses hot and cold reservoirs that have temperatures of 1684 and 842 K, respectively. The input heat for this engine is |QH|. The work delivered by the engine is used to operate a Carnot heat pump. The pump removes heat from the 842-K reservoir and puts it into a hot reservoir at a temperature T. The amount of heat removed from the 842-K reservoir is also |QH|. Find the temperature T.
Section 15.11 Entropy

| 68 | On a cold day, 24 500 J of heat leaks out of a house. The inside temperature is 21 °C, and the outside temperature is −15 °C. What is the increase in the entropy of the universe that this heat loss produces?

| 69 | Heat Q flows spontaneously from a reservoir at 394 K into a reservoir that has a lower temperature T. Because of the spontaneous flow, thirty percent of Q is rendered unavailable for work when a Carnot engine operates between the reservoir at temperature T and a reservoir at 248 K. Find the temperature T.

| 70 | Consider three engines that each use 1650 J of heat from a hot reservoir (). These three engines reject heat to a cold reservoir (). Engine I rejects 1120 J of heat. Engine II rejects 990 J of heat. Engine III rejects 660 J of heat. One of the engines operates reversibly, and two operate irreversibly. However, of the two irreversible engines, one violates the second law of thermodynamics and could not exist. For each of the engines determine the total entropy change of the universe, which is the sum of the entropy changes of the hot and cold reservoirs. On the basis of your calculations, identify which engine operates reversibly, which operates irreversibly and could exist, and which operates irreversibly and could not exist.

| 71 | ssm A process occurs in which the entropy of a system increases by 125 J/K. During the process, the energy that becomes unavailable for doing work is zero. (a) Is this process reversible or irreversible? Give your reasoning. (b) Determine the change in the entropy of the surroundings.

* | 72 | (a) Find the equilibrium temperature that results when one kilogram of liquid water at 373 K is added to two kilograms of liquid water at 283 K in a perfectly insulated container. (b) When heat is added to or removed from a solid or liquid of mass m and specific heat capacity c, the change in entropy can be shown to be , where Ti and Tf are the initial and final Kelvin temperatures. Use this equation to calculate the entropy change for each amount of water. Then combine the two entropy changes algebraically to obtain the total entropy change of the universe. Note that the process is irreversible, so the total entropy change of the universe is greater than zero. (c) Assuming that the coldest reservoir at hand has a temperature of 273 K, determine the amount of energy that becomes unavailable for doing work because of the irreversible process.

* | 73 | Refer to Interactive Solution 15.73 at to review a method by which this problem can be solved, (a) After 6.00 kg of water at 85.0 °C is mixed in a perfect thermos with 3.00 kg of ice at 0.0 °C, the mixture is allowed to reach equilibrium. Using the expression [see problem 46] and the change in entropy for melting, find the change in entropy that occurs, (b) Should the entropy of the universe increase or decrease as a result of the mixing process? Give your reasoning and state whether your answer in part (a) is consistent with your answer here.

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