Physics for Scientists and Engineers with Modern Physics

Paul Tipler, Gene Mosca

Chapter 22

Heat Engines, Entropy, and the Second Law of Thermodynamics - all with Video Answers

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Chapter Questions

Problem 1

A heat engine takes in 360 J of energy from a hot reservoir and performs 25.0 J of work in each cycle. Find (a) the efficiency of the engine and (b) the energy expelled to the cold reservoir in each cycle.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 2

A heat engine performs 200 J of work in each cycle and has an efficiency of 30.0%. For each cycle, how much energy is (a) taken in and (b) expelled by heat?

Sheh Lit Chang

University of Washington

Problem 3

A particular heat engine has a useful power output of 5.00 kW and an efficiency of 25.0%. The engine expels 8 000 J of exhaust energy in each cycle. Find (a) the energy taken in during each cycle and (b) the time interval for each cycle.

Sheh Lit Chang

University of Washington

Problem 4

Heat engine X takes in four times more energy by heat from the hot reservoir than heat engine Y. Engine X delivers two times more work, and it rejects seven times more energy by heat to the cold reservoir than heat engine Y. Find the efficiency of (a) heat engine X and (b) heat engine Y.

Sheh Lit Chang

University of Washington

Problem 5

A multicylinder gasoline engine in an airplane, operating at 2500 rev/min, takes in energy $7.89 \times 10^{3} \mathrm{J}$ and exhausts $4.58 \times 10^{3} \mathrm{J}$ for each revolution of the crankshaft. (a) How many liters of fuel does it consume in 1.00 $\mathrm{h}$ of operation if the heat of combustion is $4.03 \times 10^{7} \mathrm{J} / \mathrm{L}$ ? (b) What is the mechanical power output of the engine? Ignore friction and express the answer in horsepower. (c) What is the torque exerted by the crankshaft on the load? (d) What power must the exhaust and cooling system transfer out of the engine?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 6

Suppose a heat engine is connected to two energy reservoirs, one a pool of molten aluminum $\left(660^{\circ} \mathrm{C}\right)$ and the other a block of solid mercury $\left(-38.9^{\circ} \mathrm{C}\right)$ . The engine runs by freezing 1.00 $\mathrm{g}$ of aluminum and melting 15.0 $\mathrm{g}$ of mercury during each cycle. The heat of fusion of aluminum is $3.97 \times 10^{5} \mathrm{J} / \mathrm{kg} ;$ the heat of fusion of mercury is $1.18 \times 10^{4} \mathrm{J} / \mathrm{kg}$ . What is the efficiency of this engine?

Henrique Saito

Numerade Educator

Problem 7

A refrigerator has a coefficient of performance equal to 5.00. The refrigerator takes in 120 J of energy from a cold reservoir in each cycle.
Find (a) the work required in each cycle and (b) the energy expelled to the hot reservoir.

Sheh Lit Chang

University of Washington

Problem 8

A refrigerator has a coefficient of performance of $3.00 .$ The ice tray compartment is at $-20.0^{\circ} \mathrm{C},$ and the room temperature is $22.0^{\circ} \mathrm{C}$ . The refrigerator can convert 30.0 $\mathrm{g}$ of water at $22.0^{\circ} \mathrm{C}$ to 30.0 $\mathrm{g}$ of ice at $-20.0^{\circ} \mathrm{C}$ each minute. What input power is required? Give your answer in watts.

Sanjeev Kumar

Numerade Educator

Problem 9

In 1993 the federal government instituted a requirement that all room air conditioners sold in the United States must have an energy efficiency ratio (EER) of 10 or higher. The EER is defined as the ratio of the cooling capacity of the air conditioner, measured in Btu/h, to its electrical power requirement in watts. (a) Convert the EER of 10.0 to dimensionless form, using the conversion 1 Btu ! 1055 J. (b) What is the appropriate name for this dimensionless quantity? (c) In the 1970s it was common to find room air conditioners with EERs of 5 or lower. Compare the operating costs for 10 000-Btu/h air conditioners with EERs of 5.00 and 10.0. Assume that each air conditioner operates for 1 500 h during the summer in a city where electricity costs 10.0$€$ per $\mathrm{kWh}$.

Sheh Lit Chang

University of Washington

Problem 10

A Carnot engine has a power output of 150 $\mathrm{kW}$ . The engine operates between two reservoirs at $20.0^{\circ} \mathrm{C}$ and $500^{\circ} \mathrm{C}$. (a) How much energy does it take in per hour? (b) How much energy is lost per hour in its exhaust?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 11

One of the most efficient heat engines ever built is a steam turbine in the Ohio valley, operating between $430^{\circ} \mathrm{C}$ and $1870^{\circ} \mathrm{C}$ on energy from West Virginia coal to produce electricity for the Midwest. (a) What is its maximum theoretical efficiency? (b) The actual efficiency of the engine is 42.0$\%$ . How much useful power does the engine deliver if it takes in $1.40 \times 10^{5} \mathrm{J}$ of energy each second from its hot reservoir?

Sheh Lit Chang

University of Washington

Problem 12

A heat engine operating between $200^{\circ} \mathrm{C}$ and $80.0^{\circ} \mathrm{C}$ achieves
20.0$\%$ of the maximum possible efficiency. What energy
input will enable the engine to perform 10.0 $\mathrm{kJ}$ of work?

Sheh Lit Chang

University of Washington

Problem 13

An ideal gas is taken through a Carnot cycle. The isothermal expansion occurs at $250^{\circ} \mathrm{C}$ , and the isothermal compression takes place at $50.0^{\circ} \mathrm{C}$ . The gas takes in 1200 $\mathrm{J}$ of energy from the hot reservoir during the isothermal ex- pansion. Find (a) the energy expelled to the cold reservoir in each cycle and (b) the net work done by the gas in each cycle.

Sheh Lit Chang

University of Washington

Problem 14

The exhaust temperature of a Carnot heat engine is $300^{\circ} \mathrm{C} .$ What is the intake temperature if the efficiency of the engine is 30.0$\%$ ?

Sheh Lit Chang

University of Washington

Problem 15

A Carnot heat engine uses a steam boiler at $100^{\circ} \mathrm{C}$ as the high-temperature reservoir. The low-temperature reservoir is the outside environment at $20.0^{\circ} \mathrm{C}$ . Energy is exhausted to the low-temperature reservoir at the rate of 15.4 $\mathrm{W}$ .
(a) Determine the useful power output of the heat engine.
(b) How much steam will it cause to condense in the high-temperature reservoir in 1.00 $\mathrm{h}$ ?

Sheh Lit Chang

University of Washington

Problem 16

A power plant operates at a 32.0$\%$ efficiency during the summer when the sea water used for cooling is at $20.0^{\circ} \mathrm{C}$. The plant uses $350^{\circ} \mathrm{C}$ steam to drive turbines. If the plant's efficiency changes in the same proportion as the ideal effciency, what would be the plant's efficiency in the winter, when the sea water is $10.0^{\circ} \mathrm{C}$ ?

Sheh Lit Chang

University of Washington

Problem 17

Argon enters a turbine at a rate of 80.0 $\mathrm{kg} / \mathrm{min}$ , a temperature of $800^{\circ} \mathrm{C}$ and a pressure of 1.50 $\mathrm{MPa}$ . It expands adiabatically as it pushes on the turbine blades and exits at pressure 300 $\mathrm{kPa}$ (a) Calculate its temperature at exit. (b) Calculate the (maximum) power output of the turning turbine. (c) The turbine is one component of a model closed-cycle gas turbine engine. Calculate the maximum efficiency of the engine.

Henrique Saito

Numerade Educator

Problem 18

An electric power plant that would make use of the temperature gradient in the ocean has been proposed. The system is to operate between $20.0^{\circ} \mathrm{C}$ (surface water temperature) and $5.00^{\circ} \mathrm{C}$ (water temperature at a depth of about 1 $\mathrm{km}$ ). (a) What is the maximum efficiency of such a system? (b) If the useful power output of the plant is 75.0 MW, how much energy is taken in from the warm reservoir per hour? (c) In view of your answer to part (a), do you think such a system is worthwhile? Note that the “fuel” is free.

Sheh Lit Chang

University of Washington

Problem 19

Here is a clever idea. Suppose you build a two-engine device such that the exhaust energy output from one heat engine is the input energy for a second heat engine. We say that the two engines are running in series. Let $e_{1}$ and $e_{2}$ represent the efficiencies of the two engines. (a) The overall efficiency of the two-engine device is defined as the total work output divided by the energy put into the first engine by heat. Show that the overall efficiency is given by
$$e=e_{1}+e_{2}-e_{1} e_{2}$$
(b) What If? Assume the two engines are Carnot engines. Engine 1 operates between temperatures $T_{h}$ and $T_{i} .$ The gas in engine 2 varies in temperature between $T_{i}$ and $T_{c} .$ In terms of the temperatures, what is the efficiency of the combination engine? (c) What value of the intermedone by temperature $T_{i}$ will result in equal work being done by each of the two engines in series? (d) What value of $T_{i}$ will result in each of the two engines in series having the same efficiency?

Sheh Lit Chang

University of Washington

Problem 20

A 20.0%-efficient real engine is used to speed up a train from rest to 5.00 m/s. It is known that an ideal (Carnot) engine using the same cold and hot reservoirs would accelerate the same train from rest to a speed of 6.50 m/s using the same amount of fuel. The engines use air at 300 K as a cold reservoir. Find the temperature of the steam serving as the hot reservoir.

Sheh Lit Chang

University of Washington

Problem 21

A firebox is at 750 K, and the ambient temperature is 300 K. The efficiency of a Carnot engine doing 150 J of work as it transports energy between these constant-temperature baths is 60.0%. The Carnot engine must take in energy 150 J/0.600 ! 250 J from the hot reservoir and must put out 100 J of energy by heat into the environment. To follow Carnot’s reasoning, suppose that some other heat engine S could have efficiency 70.0%. (a) Find the energy input and wasted energy output of engine S as it does 150 J of work. (b) Let engine S operate as in part (a) and run the Carnot engine in reverse. Find the total energy the firebox puts out as both engines operate together, and the total energy transferred to the environment. Show that the Clausius statement of the second law of thermodynamics is violated. (c) Find the energy input and work output of engine S as it puts out exhaust energy of 100 J. (d) Let engine S operate as in (c) and contribute 150 J of its work output to running the Carnot engine in reverse. Find the total energy the firebox puts out as both engines operate together, the total work output, and the total energy transferred to the environment. Show that the Kelvin–Planck statement of the second law is violated. Thus our assumption about the efficiency of engine S must be false. (e) Let the engines operate together through one cycle as in part (d). Find the change in entropy of the Universe. Show that the entropy statement of the second law is violated.

Sheh Lit Chang

University of Washington

Problem 22

At point A in a Carnot cycle, 2.34 mol of a monatomic ideal gas has a pressure of 1 400 kPa, a volume of 10.0 L, and a temperature of 720 K. It expands isothermally to point B, and then expands adiabatically to point C where its volume is 24.0 L. An isothermal compression brings it to point D, where its volume is 15.0 L. An adiabatic process returns the gas to point A. (a) Determine all the unknown pressures, volumes and temperatures as you fill in the following table:
$$\begin{array}{|c|c|c|c|}\hline & {P} & {V} & {T} \\ \hline A & {1400 \mathrm{kPa}} & {10.0 \mathrm{L}} & {720 \mathrm{K}} \\ \hline B & {} & {} \\ \hline C & {} & {24.0 \mathrm{L}} \\ \hline D & {} & {15.0 \mathrm{L}} & {} \\ \hline\end{array}$$
(b) Find the energy added by heat, the work done by the
engine, and the change in internal energy for each of the steps $A \rightarrow B, B \rightarrow C, C \rightarrow D,$ and $D \rightarrow A .$ (c) Calculate the efficiency $W_{\text { net }} / Q_{h}$ . Show that it is equal to $1-T_{C} / T_{A}$ the Carnot efficiency.

Sheh Lit Chang

University of Washington

Problem 23

What is the coefficient of performance of a refrigerator that operates with Carnot efficiency between temperatures $-3.00^{\circ} \mathrm{C}$ and $+27.0^{\circ} \mathrm{C} ?$

Henrique Saito

Numerade Educator

Problem 24

What is the maximum possible coefficient of performance of a heat pump that brings energy from outdoors at $-3.00^{\circ} \mathrm{C}$ into a $22.0^{\circ} \mathrm{C}$ house? Note that the work done to run the heat pump is also available to warm up the house.

Henrique Saito

Numerade Educator

Problem 25

An ideal refrigerator or ideal heat pump is equivalent to a Carnot engine running in reverse. That is, energy $Q_{c}$ is taken in from a cold reservoir and energy $Q_{h}$ is rejected to a hot reservoir. (a) Show that the work that must be supplied to run the refrigerator or heat pump is
$$W=\frac{T_{h}-T_{c}}{T_{c}} Q_{c}$$
(b) Show that the coefficient of performance of the ideal refrigerator is
$$\operatorname{COP}=\frac{T_{c}}{T_{h}-T_{c}}$$

Sanjeev Kumar

Numerade Educator

Problem 26

A heat pump, shown in Figure $\mathrm{P} 22.26$ , is essentially an air conditioner installed backward. It extracts energy from colder air outside and deposits it in a warmer room. Sup- pose that the ratio of the actual energy entering the room to the work done by the device's motor is 10.0$\%$ of the theoretical maximum ratio. Determine the energy entering the room per joule of work done by the motor, given that the inside temperature is $20.0^{\circ} \mathrm{C}$ and the outside temperature is $-5.00^{\circ} \mathrm{C} .$

Sheh Lit Chang

University of Washington

Problem 27

How much work does an ideal Carnot refrigerator require to remove 1.00 J of energy from helium at 4.00 K and reject this energy to a room-temperature (293-K) environment?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 28

A refrigerator maintains a temperature of $0^{\circ} \mathrm{C}$ in the cold compartment with a room temperature of $25.0^{\circ} \mathrm{C} .$ It removes energy from the cold compartment at the rate of 8000 $\mathrm{kJ} / \mathrm{h}$ (a) What minimum power is required to operate the refrigerator? (b) The refrigerator exhausts energy into the room at what rate?

Sheh Lit Chang

University of Washington

Problem 29

If a 35.0$\%$ -efficient Carnot heat engine (Fig. 22.2$)$ is run in reverse so as to form a refrigerator (Fig. $22.5 ),$ what would be this refrigerator's coefficient of performance?

Sheh Lit Chang

University of Washington

Problem 30

Two Carnot engines have the same efficiency. One engine runs in reverse as a heat pump, and the other runs in reverse as a refrigerator. The coefficient of performance of the heat pump is 1.50 times the coefficient of performance of the refrigerator. Find (a) the coefficient of performance of the refrigerator, (b) the coefficient of performance of the heat pump, and (c) the efficiency of each heat engine.

Sheh Lit Chang

University of Washington

Problem 31

In a cylinder of an automobile engine, just after combustion, the gas is confined to a volume of 50.0 $\mathrm{cm}^{3}$ and has an initial pressure of $3.00 \times 10^{6} \mathrm{Pa}$ . The piston moves outward to a final volume of $300 \mathrm{cm}^{3},$ and the gas expands without energy loss by heat. (a) If $\gamma=1.40$ for the gas, what is the final pressure? (b) How much work is done by the gas in expanding?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 32

A gasoline engine has a compression ratio of 6.00 and uses a gas for which $\gamma=1.40 .$ (a) What is the efficiency of the engine if it operates in an idealized Otto cycle? (b) What If ? If the actual efficiency is 15.0%, what fraction of the fuel is wasted as a result of friction and energy losses by heat that could by avoided in a reversible engine? (Assume complete combustion of the air–fuel mixture.)

Sheh Lit Chang

University of Washington

Problem 33

A $1.60-\mathrm{L}$ gasoline engine with a compression ratio of 6.20 has a useful power output of 102 $\mathrm{hp}$ . Assuming the engine operates in an idealized Otto cycle, find the energy taken in and the energy exhausted each second. Assume the fuel-air mixture behaves like an ideal gas with $\gamma=1.40 .$

Sheh Lit Chang

University of Washington

Problem 34

The compression ratio of an Otto cycle, as shown in Figure $22.13,$ is $V_{A} / V_{B}=8.00 .$ At the beginning $A$ of the compression process, 500 $\mathrm{cm}^{3}$ of gas is at 100 $\mathrm{kPa}$ and $20.0^{\circ} \mathrm{C}$ . At the beginning of the adiabatic expansion the temperature is $T_{C}=750^{\circ} \mathrm{C}$ . Model the working fluid as an ideal gas with $E_{\text { int }}=n C_{V} T=2.50 n R T$ and $\gamma=1.40 .$ (a) Fill in the table below to follow the states of the gas:

Sheh Lit Chang

University of Washington

Problem 35

An ice tray contains 500 g of liquid water at $0^{\circ} \mathrm{C}$ . Calculate the change in entropy of the water as it freezes slowly and completely at $0^{\circ} \mathrm{C} .$

Sanjeev Kumar

Numerade Educator

Problem 36

At a pressure of 1 atm, liquid helium boils at 4.20 K. The latent heat of vaporization is 20.5 kJ/kg. Determine the entropy change (per kilogram) of the helium resulting from vaporization.

Sheh Lit Chang

University of Washington

Problem 37

Calculate the change in entropy of 250 $\mathrm{g}$ of water heated slowly from $20.0^{\circ} \mathrm{C}$ to $80.0^{\circ} \mathrm{C} .$ (Suggestion: Note that $d Q=m c d T . )$

Sheh Lit Chang

University of Washington

Problem 38

In making raspberry jelly, 900 g of raspberry juice is combined with 930 g of sugar. The mixture starts at room temperature, $23.0^{\circ} \mathrm{C},$ and is slowly heated on a stove until it reaches $220^{\circ} \mathrm{F}$ . It is then poured into heated jars and allowed to cool. Assume that the juice has the same specific heat as water. The specific heat of sucrose is 0.299 $\mathrm{cal} / \mathrm{g} \cdot^{\circ} \mathrm{C}$. Consider the heating process. (a) Which of the following terms describe(s) this process: adiabatic, isobaric, isothermal, isovolumetric, cyclic, reversible, isentropic? (b) How much energy does the mixture absorb? (c) What is the minimum change in entropy of the jelly while it is heated?

Rashmi Sinha

Numerade Educator

Problem 39

What change in entropy occurs when a $27.9-\mathrm{g}$ ice cube at $-12^{\circ} \mathrm{C}$ is transformed into steam at $115^{\circ} \mathrm{C}$ ?

Ashly Sunny

Numerade Educator

Problem 40

The temperature at the surface of the Sun is approximately 5 700 K, and the temperature at the surface of the Earth is approximately 290 K. What entropy change occurs when 1 000 J of energy is transferred by radiation from the Sun to the Earth?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 41

A 1 500-kg car is moving at 20.0 m/s. The driver brakes to a stop. The brakes cool off to the temperature of the surrounding air, which is nearly constant at $20.0^{\circ} \mathrm{C}$ . What is the total entropy change?

Sheh Lit Chang

University of Washington

Problem 42

A $1.00-\mathrm{kg}$ iron horseshoe is taken from a forge at $900^{\circ} \mathrm{C}$ and dropped into 4.00 $\mathrm{kg}$ of water at $10.0^{\circ} \mathrm{C}$ . Assuming that no energy is lost by heat to the surroundings, determine the total entropy change of the horseshoe-plus-water system.

Sanjeev Kumar

Numerade Educator

Problem 43

How fast are you personally making the entropy of the Universe increase right now? Compute an order-of-magnitude estimate, stating what quantities you take as data and the values you measure or estimate for them.

Sanjeev Kumar

Numerade Educator

Problem 44

A rigid tank of small mass contains 40.0 $\mathrm{g}$ of argon, initially at $200^{\circ} \mathrm{C}$ and 100 $\mathrm{kPa}$ . The tank is placed into a reservoir at $0^{\circ} \mathrm{C}$ and allowed to cool to thermal equilibrium. (a) Calculate the volume of the tank. (b) Calculate the change in internal energy of the argon. (c) Calculate the energy transferred by heat. (d) Calculate the change in entropy of the argon. (e) Calculate the change in entropy of the constant-temperature bath.

Sheh Lit Chang

University of Washington

Problem 45

A 1.00 -mol sample of $\mathrm{H}_{2}$ gas is contained in the left-hand side of the container shown in Figure $\mathrm{P} 22.45,$ which has equal volumes left and right. The right-hand side is evacuated. When the valve is opened, the gas streams into the right-hand side. What is the final entropy change of the gas? Does the temperature of the gas change?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 46

A 2.00 -L container has a center partition that divides it into two equal parts, as shown in Figure $\mathrm{P} 22.46$ . The left side contains $\mathrm{H}_{2}$ gas, and the right side contains $\mathrm{O}_{2}$ gas. Both gases are at room temperature and at at- mospheric pressure. The partition is removed, and the gases are allowed to mix. What is the entropy increase of the system?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 47

A 1.00 -mol sample of an ideal monatomic gas, initially at a pressure of 1.00 $\mathrm{atm}$ and a volume of $0.0250 \mathrm{m}^{3},$ is heated to a final state with a pressure of 2.00 $\mathrm{atm}$ and a volume of $0.0400 \mathrm{m}^{3} .$ Determine the change in entropy of the gas in this process.

Sheh Lit Chang

University of Washington

Problem 48

A 1.00-mol sample of a diatomic ideal gas, initially having pressure P and volume V, expands so as to have pressure 2P and volume 2V. Determine the entropy change of the gas in the process.

Sheh Lit Chang

University of Washington

Problem 49

If you toss two dice, what is the total number of ways in which you can obtain (a) a 12 and (b) a 7?

Sheh Lit Chang

University of Washington

Problem 50

Prepare a table like Table 22.1 for the following occurrence. You toss four coins into the air simultaneously and then record the results of your tosses in terms of the numbers of heads and tails that result. For example, HHTH and HTHH are two possible ways in which three heads and one tail can be achieved. (a) On the basis of your table, what is the most probable result of a toss? In terms of entropy, (b) what is the most ordered state and (c) what is the most disordered state?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 51

Repeat the procedure used to construct Table 22.1 (a) for the case in which you draw three marbles from your bag rather than four and (b) for the case in which you draw five rather than four.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 52

Every second at Niagara Falls (Fig. P22.52), some 5000 $\mathrm{m}^{3}$ of water falls a distance of $50.0 \mathrm{m} .$ What is the increase in entropy per second due to the falling water? Assume that the mass of the surroundings is so great that its temperature and that of the water stay nearly constant at $20.0^{\circ} \mathrm{C}$ . Suppose that a negligible amount of water evaporates.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 53

A house loses energy through the exterior walls and roof at a rate of 5 000 J/s ! 5.00 kW when the interior temperature is $22.0^{\circ} \mathrm{C}$ and the outside temperature is $-5.00^{\circ} \mathrm{C} .$ Calculate the electric power required to maintain the interior temperature at $22.0^{\circ} \mathrm{C}$ for the following two cases. (a) The electric power is used in electric resistance heaters (which convert all of the energy transferred in by electrical transmission into internal energy). (b) What If ? The electric power is used to drive an electric motor that operates the compressor of a heat pump, which has a coefficient of performance equal to 60.0% of the Carnot-cycle value.

Yaqub Khan

Numerade Educator

Problem 54

How much work is required, using an ideal Carnot refrigerator, to change 0.500 $\mathrm{kg}$ of tap water at $10.0^{\circ} \mathrm{C}$ into ice at $-20.0^{\circ} \mathrm{C}$ ? Assume the temperature of the freezer compartment is held at $-20.0^{\circ} \mathrm{C}$ and the refrigerator exhausts energy into a room at $20.0^{\circ} \mathrm{C} .$

Narayan Hari

Numerade Educator

Problem 55

A heat engine operates between two reservoirs at $T_{2}=600 \mathrm{K}$ and $T_{1}=350 \mathrm{K}$ . It takes in 1000 $\mathrm{J}$ of energy from the higher-temperature reservoir and performs 250 $\mathrm{J}$ of work. Find (a) the entropy change of the Universe $\Delta S_{U}$ for this process and (b) the work $W$ that could have been done by an ideal Carnot engine operating between these two reservoirs. (c) Show that the difference between the amounts of work done in parts (a) and (b) is $T_{1} \Delta S_{U} .$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 56

Two identically constructed objects, surrounded by thermal insulation, are used as energy reservoirs for a Carnot engine. The finite reservoirs both have mass $m$ and specific heat $c$ . They start out at temperatures $T_{h}$ and $T_{c}$ , where $T_{h}>T_{c} .$ (a) Show that the engine will stop working when the final temperature of each object is $\left(T_{h} T_{c}\right)^{1 / 2}$ (b) Show that the total work done by the Carnot engine is
$$W_{\text { eng }}=m c\left(T_{h}^{1 / 2}-T_{c}^{1 / 2}\right)^{2}$$

Sheh Lit Chang

University of Washington

Problem 57

In 1816 Robert Stirling, a Scottish clergyman, patented the Stirling engine, which has found a wide variety of applications ever since. Fuel is burned externally to warm one of the engine’s two cylinders. A fixed quantity of inert gas moves cyclically between the cylinders, expanding in the hot one and contracting in the cold one. Figure P22.57 represents a model for its thermodynamic cycle. Consider $n$ mol of an ideal monatomic gas being taken once through the cycle, consisting of two isothermal processes at temperatures 3$T_{i}$ and $T_{i}$ and two constant-volume processes. Determine, in terms of $n, R,$ and $T_{i}$ , (a) the net energy transferred by heat to the gas and (b) the efficiency of the engine. A Stirling engine is easier to manufacture than an internal combustion engine or a turbine. It can run on burning garbage. It can run on the energy of sunlight and produce no material exhaust.

Keshav Singh

Numerade Educator

Problem 58

An electric power plant has an overall efficiency of 15.0%. The plant is to deliver 150 MW of power to a city, and its turbines use coal as the fuel. The burning coal produces steam that drives the turbines. This steam is then condensed to water at $25.0^{\circ} \mathrm{C}$ by passing it through cooling coils in contact with river water. (a) How many metric tons of coal does the plant consume each day (1 metric ton $=10^{3} \mathrm{kg} )$ ? (b) What is the total cost of the fuel per year if the delivered price is $\$ 8.00 /$ metric ton? (c) If the river water is delivered at $20.0^{\circ} \mathrm{C}$ , at what minimum rate must it flow over the cooling coils in order that its temperature not exceed $25.0^{\circ} \mathrm{C}$ ? (Note: The heat of combustion of coal is $33.0 \mathrm{kJ} / \mathrm{g} . )$

Sheh Lit Chang

University of Washington

Problem 59

A power plant, having a Carnot efficiency, produces 1 000 MW of electrical power from turbines that take in steam at 500 K and reject water at 300 K into a flowing river. The water downstream is 6.00 K warmer due to the output of the power plant. Determine the flow rate of the river.

Pronoy Sinha

Numerade Educator

Problem 60

A power plant, having a Carnot efficiency, produces electric power $\mathscr{P}$ from turbines that take in energy from steam at temperature $T_{h}$ and discharge energy at temperature $T_{c}$ , through a heat exchanger into a flowing river. The water downstream is warmer by $\Delta T$ due to the output of the power plant. Determine the flow rate of the river.

Sanjeev Kumar

Numerade Educator

Problem 61

An athlete whose mass is 70.0 $\mathrm{kg}$ drinks 16 $\mathrm{oz}(453.6 \mathrm{g})$ of refrigerated water. The water is at a temperature of $35.0^{\circ} \mathrm{F}$ . (a) Ignoring the temperature change of the body that results from the water intake (so that the body is regarded as a reservoir always at $98.6^{\circ} \mathrm{F} ),$ find the entropy increase of the entire system. (b) What If? Assume that the entire body is cooled by the drink and that the average specific heat of a person is equal to the specific heat of liquid water. Ignoring any other energy transfers by heat and any metabolic energy release, find the athlete's temperature after she drinks the cold water, given an initial body temperature of $98.6^{\circ} \mathrm{F}$ . Under these assumptions, what is the entropy increase of the entire system? Compare this result with the one you obtained in part (a).

Sheh Lit Chang

University of Washington

Problem 62

A 1.00 -mol sample of an ideal monatomic gas is taken through the cycle shown in Figure $\mathrm{P} 22.62$ . The process $A \rightarrow B$ is a reversible isothermal expansion. Calculate (a) the net work done by the gas, (b) the energy added to the gas by heat, (c) the energy exhausted from the gas by heat, and (d) the efficiency of the cycle.

Penny Riley

Numerade Educator

Problem 63

A biology laboratory is maintained at a constant temperature of $7.00^{\circ} \mathrm{C}$ by an air conditioner, which is vented to the air outside. On a typical hot summer day the outside temperature is $27.0^{\circ} \mathrm{C}$ and the air conditioning unit emits energy to the outside at a rate of 10.0 $\mathrm{kW}$ . Model the unit as having a coefficient of performance equal to 40.0$\%$ of the coefficient of performance of an ideal Carnot device. (a) At what rate does the air conditioner remove energy from the laboratory? (b) Calculate the power required for the work input. (c) Find the change in entropy produced by the air conditioner in 1.00 h. (d) What If? The outside temperature increases to $32.0^{\circ} \mathrm{C} .$ Find the fractional change in the coefficient of performance of the air conditioner.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 64

A 1.00-mol sample of an ideal gas expands isothermally, doubling in volume. (a) Show that the work it does in expanding is $W=R T \ln 2 .$ (b) Because the internal energy $E_{\text { int }}$ of an ideal gas depends solely on its temperature, the change in internal energy is zero during the expansion. It follows from the first law that the energy input to the gas by heat during the expansion is equal to the energy output by work. Why does this conversion not violate the second law?

Sheh Lit Chang

University of Washington

Problem 65

A 1.00 -mol sample of a monatomic ideal gas is taken through the cycle shown in Figure $\mathrm{P} 2.65 .$ At point $A,$ the pressure, volume, and temperature are $P_{i}, V_{i},$ and $T_{i}$ , respectively. In terms of $R$ and $T_{i},$ find (a) the total energy entering the system by heat per cycle, (b) the total energy leaving the system by heat per cycle, (c) the efficiency of an engine operating in this cycle, and (d) the efficiency of an engine operating in a Carnot cycle between the same temperature extremes.

Ajay Singhal

Numerade Educator

Problem 66

A sample consisting of $n$ mol of an ideal gas undergoes a reversible isobaric expansion from volume $V_{i}$ to volume $3 V_{i} .$ Find the change in entropy of the gas by calculating $\int_{i}^{f} d Q / T$ where $d Q=n C_{P} d T .$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 67

A system consisting of $n$ mol of an ideal gas undergoes two reversible processes. It starts with pressure $P_{i}$ and volume $V_{i},$ expands isothermally, and then contracts adiabatically to reach a final state with pressure $P_{i}$ and volume 3$V_{i}$. (a) Find its change in entropy in the isothermal process. The entropy does not change in the adiabatic process. (b) What If ? Explain why the answer to part (a) must be the same as the answer to Problem 66.

Sheh Lit Chang

University of Washington

Problem 68

Suppose you are working in a patent office, and an inventor comes to you with the claim that her heat engine, which employs water as a working substance, has a thermodynamic efficiency of 0.61. She explains that it operates between energy reservoirs at $4^{\circ} \mathrm{C}$ and $0^{\circ} \mathrm{C} .$ It is a very complicated device, with many pistons, gears, and pulleys, and the cycle involves freezing and melting. Does her claim that $e=0.61$ warrant serious consideration? Explain.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 69

An idealized diesel engine operates in a cycle known as the air-standard diesel cycle, shown in Figure 22.14. Fuel is sprayed into the cylinder at the point of maximum compression, B. Combustion occurs during the expansion $B \rightarrow C,$ which is modeled as an isobaric process. Show that the efficiency of an engine operating in this idealized diesel cycle is
$$e=1-\frac{1}{\gamma}\left(\frac{T_{D}-T_{A}}{T_{C}-T_{B}}\right)$$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 70

A 1.00 -mol sample of an ideal gas $(\gamma=1.40)$ is carried through the Carnot cycle described in Figure 22.11. At point A, the pressure is 25.0 atm and the temperature is 600 K. At point C, the pressure is 1.00 atm and the temperature is 400 K. (a) Determine the pressures and volumes at points A, B, C, and D. (b) Calculate the net work done per cycle. (c) Determine the efficiency of an engine operating in this cycle.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 71

Suppose 1.00 $\mathrm{kg}$ of water at $10.0^{\circ} \mathrm{C}$ is mixed with 1.00 $\mathrm{kg}$ of water at $30.0^{\circ} \mathrm{C}$ at constant pressure. When the mixture has reached equilibrium, (a) what is the final temperature? (b) Take $c_{P}=4.19 \mathrm{kJ} / \mathrm{kg} \cdot \mathrm{K}$ for water and show that the entropy of the system increases by
$$\Delta S=4.19 \ln \left[\left(\frac{293}{283}\right)\left(\frac{293}{303}\right)\right] \mathrm{kJ} / \mathrm{K}$$
(c) Verify numerically that $\Delta S>0 .$ (d) Is the mixing an irreversible process?

Sheh Lit Chang

University of Washington