|| Prove that the work done in an adiabatic process $\text{i}\to \text{f}$ is $W_{\text{s}}=(p_{\text{f}}{V}_{\text{f}}-p_{\text{i}}{V}_{\text{i}})\text{/}(1-\gamma )$.

|| Prove that the coefficient of performance of a Carnot refrigerator is $K_{\text{Carnot}}=T_{\text{C}}\text{/}(T_{\text{H}}-T_{\text{C}})$.

|| FIGURE P21.37 shows a heat engine going through one cycle. The gas is diatomic. The masses are such that when the pin is removed, in steps 3 and 6, the piston does not move.

1. Draw the pV diagram for this heat engine.

2. How much work is done per cycle?

3. What is this engine’s thermal efficiency?

FIGURE P21.37

|| INT The engine that powers a crane burns fuel at a flame temperature of $2000\text{°C}$. It is cooled by $20\text{°C}$ air. The crane lifts a 2000 kg steel girder 30 m upward. How much heat energy is transferred to the engine by burning fuel if the engine is 40% as efficient as a Carnot engine?

|| INT A Carnot refrigerator operates between energy reservoirs at $0\text{°C}$ and $250\text{°C}$. A 2.4-cm-diameter, 50-cm-long copper bar connects the two energy reservoirs. At what rate, in W, must work be done on the refrigerator to remove heat from the cold reservoir at the same rate that it arrives through the copper bar?

|| INT At fairly slow speeds, a car’s resistance to motion is primarily due to rolling friction. (Air drag becomes significant only at higher speeds.) Suppose a 2200 kg car with rubber tires is driving at $11\text{ mph}=5.0\text{ m/s}$. The coefficient of rolling friction is 0.020.

1. What propulsion power is needed to keep the car moving at a steady speed?

2. The car’s drivetrain efficiency (from the motor to the wheels) is 85% and the thermal efficiency of its engine is 20%. The engine burns gasoline, which has an energy density of $130\text{MJ}\text{/}\text{gal}$. What is the car’s mpg (miles per gallon) at this speed?

|| INT An ideal refrigerator utilizes a Carnot cycle operating between $0\text{°}\text{C}$ and $25\text{°}\text{C}$. To turn 10 kg of liquid water at $0\text{°}\text{C}$ into 10 kg of ice at $0\text{°}\text{C}$, (a) how much heat is exhausted into the room and (b) how much energy must be supplied to the refrigerator?

|| INT A freezer with a coefficient of performance 30% that of a Carnot refrigerator keeps the inside temperature at $-22\text{°}\text{C}$ in a $25\text{°}\text{C}$ room. 3.0 L of water at $20\text{°}\text{C}$ are placed in the freezer. How long does it take for the water to freeze if the freezer’s compressor does work at the rate of 200 W while the water is freezing?

|| There has long been an interest in using the vast quantities of thermal energy in the oceans to run heat engines. A heat engine needs a temperature difference, a hot side and a cold side. Conveniently, the ocean surface waters are warmer than the deep ocean waters. Suppose you build a floating power plant in the tropics where the surface water temperature is  $\approx 30\text{°}\text{C}$. This would be the hot reservoir of the engine. For the cold reservoir, water would be pumped up from the ocean bottom where it is always $\approx 5\text{°}\text{C}$. What is the maximum possible efficiency of such a power plant?

|| INT A Carnot heat engine operates between reservoirs at $182\text{°}\text{C}$ and $0\text{°}\text{C}$. If the engine extracts 25 J of energy from the hot reservoir per cycle, how many cycles will it take to lift a 10 kg mass a height of 10 m?

||| A Carnot engine operates between temperatures of $5\text{°}\text{C}$ and $500\text{°}\text{C}$. The output is used to run a Carnot refrigerator operating between $-5\text{°}\text{C}$ and $25\text{°}\text{C}$. How many joules of heat energy does the refrigerator exhaust into the room for each joule of heat energy used by the heat engine?

| FIGURE P21.46 shows a Carnot heat engine driving a Carnot refrigerator.

1. Determine $Q_2\text{,}{Q}_3$, and $Q_4$.

2. Is $Q_3$ greater than, less than, or equal to $Q_1\text{?}$

3. Do these two devices, when operated together in this way, violate the second law?

FIGURE P21.46

|| A heat engine running backward is called a refrigerator if its purpose is to extract heat from a cold reservoir. The same engine running backward is called a heat pump if its purpose is to exhaust warm air into the hot reservoir. Heat pumps are widely used for home heating. You can think of a heat pump as a refrigerator that is cooling the already cold outdoors and, with its exhaust heat $Q_{\text{H}}$, warming the indoors. Perhaps this seems a little silly, but consider the following. Electricity can be directly used to heat a home by passing an electric current through a heating coil. This is a direct, 100% conversion of work to heat. That is, 15 kW of electric power (generated by doing work at the rate of $15\text{kJ}\text{/}\text{s}$ at the power plant) produces heat energy inside the home at a rate of $15\text{kJ}\text{/}\text{s}$. Suppose that the neighbor’s home has a heat pump with a coefficient of performance of 5.0, a realistic value. Note that “what you get” with a heat pump is heat delivered, $Q_{\text{H}}$, so a heat pump’s coefficient of performance is defined as $K=Q_{\text{H}}/W_{\text{in}}$.

1. How much electric power (in kW) does the heat pump use to deliver $15\text{kJ}\text{/}\text{s}$ of heat energy to the house?

2. An average price for electricity is about 40 MJ per dollar. A furnace or heat pump will run typically 250 hours per month during the winter. What does one month’s heating cost in the home with a 15 kW electric heater and in the home of the neighbor who uses a heat pump?

|| Home air conditioners in the United States have their power specified in the truly obscure units of tons, where 1 ton is the power needed to melt 1 ton (2000 lb or 910 kg) of ice in 24 hours. A modest-size house typically has a 4.0 ton air conditioner. If a 4.0 ton air conditioner has a coefficient of performance of 2.5, a typical value, at what rate in kW is heat energy removed from the house?

|| A car’s internal combustion engine can be modeled as a heat engine operating between a combustion temperature of $1500\text{°}\text{C}$ and an air temperature of $20\text{°}\text{C}$ with 30% of the Carnot efficiency. The heat of combustion of gasoline is 47 kJ/g. What mass of gasoline is burned to accelerate a 1500 kg car from rest to a speed of 30 m/s?

||| Consider a 1.0 MW power plant (this is the useful output in the form of electric energy) that operates between $30\text{°}\text{C}$ and $450\text{°}\text{C}$ at 65% of the Carnot efficiency. This is enough electric energy for about 750 homes. One way to use energy more efficiently would be to use the $30\text{°}\text{C}$ “waste” energy to heat the homes rather than releasing that heat energy into the environment. This is called cogeneration, and it is used in some parts of Europe but rarely in the United States. The average home uses 70 GJ of energy per year for heating. For estimating purposes, assume that all the power plant’s exhaust energy can be transported to homes without loss and that home heating takes place at a steady rate for half a year each year. How many homes could be heated by the power plant?

|| A typical coal-fired power plant burns 300 metric tons of coal every hour to generate 750 MW of electricity. $1\text{ metric ton}=1000\text{kg}$. The density of coal is $1500\text{kg}\text{/}{\text{m}}^3$ and its heat of combustion is $28\text{MJ/kg}$. Assume that all heat is transferred from the fuel to the boiler and that all the work done in spinning the turbine is transformed into electric energy.

1. Suppose the coal is piled up in a $10\text{m}\times 10\text{m}$ room. How tall must the pile be to operate the plant for one day?

2. What is the power plant’s thermal efficiency?

|| INT A nuclear power plant generates 3000 MW of heat energy from nuclear reactions in the reactor’s core. This energy is used to boil water and produce high-pressure steam at $300\text{°C}$. The steam spins a turbine, which produces 1000 MW of electric power, then the steam is condensed and the water is cooled to $25\text{°C}$ before starting the cycle again.

1. What is the maximum possible thermal efficiency of the power plant?

2. What is the plant’s actual efficiency?

3. Cooling water from a river flows through the condenser (the low-temperature heat exchanger) at the rate of $1.2\times {10}^8\text{L}\text{/}\text{h}(\approx 30\text{\hspace{0.17em}million\hspace{0.17em}gallons\hspace{0.17em}per\hspace{0.17em}hour})$. If the river water enters the condenser at $18\text{°C}$, what is its exit temperature?

|| INT The electric output of a power plant is 750 MW. Cooling water flows through the power plant at the rate $1.0\times {10}^8\text{L}\text{/}\text{h}$. The cooling water enters the plant at $16\text{°C}$ and exits at $27\text{°C}$. What is the power plant’s thermal efficiency?

|| CALC Suppose the hot side of a Carnot engine is attached not to a large energy reservoir but to an object with mass M, specific heat c, and initial temperature $T_{\text{H}}$. Each small amount of heat energy drawn from the object to run the engine slightly decreases the object’s temperature. The engine will continue running, with lower and lower efficiency, until the object’s temperature is reduced to $T_{\text{C}}$, the temperature of the cold reservoir, which we assume is a true energy reservoir. Find an expression for the total work done by the engine.

|| A heat engine using a diatomic gas follows the cycle shown in FIGURE P21.55. Its temperature at point 1 is $20\text{°C}$.

1. Determine $W_{\text{s}}$, Q, and $\text{Δ}{E}_{\text{th}}$ for each of the three processes in this cycle. Display your results in a table.

2. What is the thermal efficiency of this heat engine?

3. What is the power output of the engine if it runs at 500 rpm?

FIGURE P21.55

|| A heat engine using 1.0 mol of a monatomic gas follows the cycle shown in FIGURE P21.56. 3750 J of heat energy is transferred to the gas during process $1\to 2$.

1. Determine $W_{\text{s}}$, Q, and $\text{Δ}{E}_{\text{th}}$ for each of the four processes in this cycle. Display your results in a table.

2. What is the thermal efficiency of this heat engine?

FIGURE P21.56

|| FIGURE P21.57 shows the cycle for a heat engine that uses a gas having $\gamma =1.25$. The initial temperature is $T_1=300\text{K}$, and this engine operates at 20 cycles per second.

1. What is the power output of the engine?

2. What is the engine’s thermal efficiency?

FIGURE P21.57

|| A heat engine using a monatomic gas follows the cycle shown in FIGURE P21.58.

1. Find $W_{\text{s}}$, Q, and $\text{Δ}{E}_{\text{th}}$ for each process in the cycle. Display your results in a table.

2. What is the thermal efficiency of this heat engine?

FIGURE P21.58

|| A heat engine uses a diatomic gas that follows the pV cycle in FIGURE P21.59.

1. Determine the pressure, volume, and temperature at point 2.

2. Determine $\text{Δ}{E}_{\text{th}}$, $W_{\text{s}}$, and Q for each of the three processes. Put your results in a table for easy reading.

3. How much work does this engine do per cycle and what is its thermal efficiency?

FIGURE P21.59

|| A heat engine using 120 mg of helium as the working substance follows the cycle shown in FIGURE P21.60.

1. Determine the pressure, temperature, and volume of the gas at points 1, 2, and 3.

2. What is the engine’s thermal efficiency?

3. What is the maximum possible efficiency of a heat engine that operates between $T_{\text{max}}$ and $T_{\text{min}}\text{?}$

FIGURE P21.60

|| FIGURE P21.61 is the pV diagram of Example 21.2, but now the device is operated in reverse.

1. During which processes is heat transferred into the gas?

2. Is this $Q_{\text{H}}$, heat extracted from a hot reservoir, or $Q_{\text{C}}$, heat extracted from a cold reservoir? Explain.

3. Determine the values of $Q_{\text{H}}$ and $Q_{\text{C}}$.

   Hint: The calculations have been done in Example 21.2 and do not need to be repeated. Instead, you need to determine which processes now contribute to $Q_{\text{H}}$ and which to $Q_{\text{C}}$.

4. Is the area inside the curve $W_{\text{in}}$ or $W_{\text{out}}\text{?}$ What is its value?

5. The device is now being operated in a ccw cycle. Is it a refrigerator? Explain.

FIGURE P21.61

|| The heat engine shown in FIGURE P21.62 uses 2.0 mol of a monatomic gas as the working substance.

1. Determine $T_1\text{,}{T}_2$, and $T_3$.

2. Make a table that shows $\text{Δ}{E}_{\text{th}}\text{,}{W}_{\text{s}}$, and Q for each of the three processes.

3. What is the engine’s thermal efficiency?

FIGURE P21.62

|| The heat engine shown in FIGURE P21.63 uses 0.020 mol of a diatomic gas as the working substance.

1. Determine $T_1\text{,}{T}_2$, and $T_3$.

2. Make a table that shows $\text{Δ}{E}_{\text{th}}\text{,}{W}_{\text{s}}$, and Q for each of the three processes.

3. What is the engine’s thermal efficiency?

FIGURE P21.63

||| A heat engine with 0.20 mol of a monatomic ideal gas initially fills a $2000{\text{cm}}^3$ cylinder at 600 K. The gas goes through the following closed cycle:

• Isothermal expansion to $4000{\text{cm}}^3$.

• Isochoric cooling to 300 K.

• Isothermal compression to $2000{\text{cm}}^3$.

• Isochoric heating to 600 K.

How much work does this engine do per cycle and what is its thermal efficiency?

||| 100 mL of water at $15\text{°C}$ is placed in the freezer compartment of a refrigerator with a coefficient of performance of 4.0. How much heat energy is exhausted into the room as the water is changed to ice at $-15\text{°}\text{C}\text{?}$

||| FIGURE CP21.70 shows two insulated compartments separated by a thin wall. The left side contains 0.060 mol of helium at an initial temperature of 600 K and the right side contains 0.030 mol of helium at an initial temperature of 300 K. The compartment on the right is attached to a vertical cylinder, above which the air pressure is 1.0 atm. A 10-cm-diameter, 2.0 kg piston can slide without friction up and down the cylinder. Neither the cylinder diameter nor the volumes of the compartments are known.

1. What is the final temperature?

2. How much heat is transferred from the left side to the right side?

3. How high is the piston lifted due to this heat transfer?

4. What fraction of the heat is converted into work?

FIGURE CP21.70

||| A refrigerator using helium gas operates on the reversed cycle shown in FIGURE CP21.71. What are the refrigerator’s (a) coefficient of performance and (b) power input if it operates at 60 cycles per second?

FIGURE CP21.71

||| A heat engine using a diatomic ideal gas goes through the following closed cycle:

• Isothermal compression until the volume is halved.

• Isobaric expansion until the volume is restored to its initial value.

• Isochoric cooling until the pressure is restored to its initial value.

What are the thermal efficiencies of (a) this heat engine and (b) a Carnot engine operating between the highest and lowest temperatures reached by this engine?

||| The gasoline engine in your car can be modeled as the Otto cycle shown in FIGURE CP21.73. A fuel-air mixture is sprayed into the cylinder at point 1, where the piston is at its farthest distance from the spark plug. This mixture is compressed as the piston moves toward the spark plug during the adiabatic compression stroke. The spark plug fires at point 2, releasing heat energy that had been stored in the gasoline. The fuel burns so quickly that the piston doesn’t have time to move, so the heating is an isochoric process. The hot, high-pressure gas then pushes the piston outward during the power stroke. Finally, an exhaust value opens to allow the gas temperature and pressure to drop back to their initial values before starting the cycle over again.

1. Analyze the Otto cycle and show that the work done per cycle is

   $$W_{\text{out}}={\displaystyle \frac{nR}{1-\gamma }}(T_2-T_1+T_4-T_3)$$

2. Use the adiabatic connection between $T_1$ and $T_2$ and also between $T_3$ and $T_4$ to show that the thermal efficiency of the Otto cycle is

   $$\eta =1-{\displaystyle \frac{1}{r^{(\gamma -1)}}}$$

   where $r=V_{\text{max}}\text{/}{V}_{\text{min}}$ is the engine’s compression ratio.

3. Graph $\eta$ versus r out to $r=30$ for a diatomic gas.

FIGURE CP21.73

||| FIGURE CP21.74 shows the Diesel cycle. It is similar to the Otto cycle (see Problem 21.73), but there are two important differences. First, the fuel is not admitted until the air is fully compressed at point 2. Because of the high temperature at the end of an adiabatic compression, the fuel begins to burn spontaneously. (There are no spark plugs in a diesel engine!) Second, combustion takes place more slowly, with fuel continuing to be injected. This makes the ignition stage a constant-pressure process. The cycle shown, for one cylinder of a diesel engine, has a displacement $V_{\text{max}}-V_{\text{min}}$ of $1000{\text{cm}}^3$ and a compression ratio $r=V_{\text{max}}\text{/}{V}_{\text{min}}=21$. These are typical values for a diesel truck. The engine operates with intake air $(\gamma =1.40)$ at $25\text{°C}$ and 1.0 atm pressure. The quantity of fuel injected into the cylinder has a heat of combustion of 1000 J.

1. Find p, V, and T at each of the four corners of the cycle. Display your results in a table.

2. What is the net work done by the cylinder during one full cycle?

3. What is the thermal efficiency of this engine?

4. What is the power output in kW and horsepower $(1\text{ hp }=746\text{ W})$ of an eight-cylinder diesel engine running at 2400 rpm?

FIGURE CP21.74