College Physics

Hugh D. Young Philip W. Adams

Chapter 16

The Second Law of Thermodynamics - all with Video Answers

Educators

Chapter Questions

Problem 1

A coal-fired power plant that operates at an efficiency of $38 \%$ generates $750 \mathrm{MW}$ of electric power. How much heat does the plant discharge to the environment in one day?

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 2

Each cycle, a certain heat engine expels $250 \mathrm{~J}$ of heat when you put in $325 \mathrm{~J}$ of heat. Find the efficiency of this engine and the amount of work you get out of the $325 \mathrm{~J}$ heat input.

Shahab Ullah

Numerade Educator

Problem 3

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

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 4

An aircraft engine has a heat efficiency of $e=0.3 .$ It discards 6400 J each cycle. (a) How much heat is supplied to the engine each cycle?
(b) How much work energy does the engine produce each cycle?

Shahab Ullah

Numerade Educator

Problem 5

A certain nuclear power plant has a thermal efficiency $e=0.25$. Its rate of heat input from the nuclear reactor is $1300 \mathrm{MW}$. What would be the reduction in the rate of discarded heat if the plant's efficiency were increased to $e=0.3 ?$

Shahab Ullah

Numerade Educator

Problem 6

Figure 16.14 shows a $p V$ diagram for a heat engine that uses 1.40 moles of an ideal diatomic gas. The internal energy of the gas changes by the following amounts: $\Delta U_{a \rightarrow b}=+5050 \mathrm{~J}$, $\Delta U_{b \rightarrow c}=-6060 \mathrm{~J}, \Delta U_{c \rightarrow d}=-1010 \mathrm{~J},$ and $\Delta U_{d \rightarrow a}=+2020 \mathrm{~J}$ (a) How much heat goes into this gas per cycle, and where in the cycle does it occur? (b) How much heat is ejected by the gas per cycle, and at what point in the cycle does this occur? (c) How much work does this engine do each cycle?
(d) What is the thermal efficiency of the engine?

Linda Winkler

Numerade Educator

Problem 7

The $p V$ diagram in Figure 16.15 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. The internal energy of the gas changes by the following amounts: $\Delta U_{a \rightarrow b}=-2800 \mathrm{~J}$, $\Delta U_{b \rightarrow c}=-2650 \mathrm{~J},$ and $\Delta U_{c \rightarrow a}=+5450 \mathrm{~J} .$ (a) Find the pressure of the gas at point $a$. (b) How much heat enters this gas per cycle, and at what point in the cycle does this occur? (c) How much heat leaves this gas in a cycle, and at what point in the cycle does this occur? (d) How much work does this engine do in a cycle? (e) What is the thermal efficiency of the engine?

David Morabito

Numerade Educator

Problem 8

A gasoline engine takes in $1.61 \times 10^{4} \mathrm{~J}$ of heat and delivers $3700 \mathrm{~J}$ of work per cycle. The heat is obtained by burning gasoline with a heat of combustion of $4.60 \times 10^{4} \mathrm{~J} / \mathrm{g}$. (a) What is the thermal efficiency of the engine? (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?

Ethan Ludecker

Numerade Educator

Problem 9

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

Ryan Hood

Numerade Educator

Problem 10

In one cycle, a freezer uses $785 \mathrm{~J}$ of electrical energy in order to remove $1750 \mathrm{~J}$ of heat from its freezer compartment at $10^{\circ} \mathrm{F}$. (a) What is the coefficient of performance of this freezer? (b) How much heat does it expel into the room during this cycle?

Ryan Hood

Numerade Educator

Problem 11

A refrigerator has a coefficient of performance of $K=2.0 .$ Each cycle, it absorbs $3.40 \times 10^{4} \mathrm{~J}$ of heat from the cold reservoir. The refrigerator is driven by a Carnot engine that has an efficiency of $e=0.5 .$ (a) How much mechanical energy is required each cycle to operate the refrigerator? (b) During each cycle, how much heat flows into the Carnot engine?

Shahab Ullah

Numerade Educator

Problem 12

A window air-conditioner unit absorbs $9.80 \times 10^{4} \mathrm{~J}$ of heat per minute from the room being cooled and in the same period deposits $1.44 \times 10^{5} \mathrm{~J}$ of heat into the outside air. What is the power consumption of the unit in watts?

Ryan Hood

Numerade Educator

Problem 13

A freezer has a coefficient of performance of $2.40 .$ The freezer is to convert $1.80 \mathrm{~kg}$ of water at $25.0^{\circ} \mathrm{C}$ to $1.80 \mathrm{~kg}$ of ice at $-5.0^{\circ} \mathrm{C}$ in 1 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 rejected to the room in which the freezer sits?

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 14

A cooling unit for chilling the water of an aquarium gives specifications of $1 / 10 \mathrm{hp}$ and $1270 \mathrm{Btu} / \mathrm{h}$. Assuming the unit produces its $1 / 10$ hp at $70.0 \%$ efficiency, calculate its performance coefficient.

Linda Winkler

Numerade Educator

Problem 15

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

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 16

A heat engine is to be built to extract energy from the temperature gradient in the ocean. If the surface and deepwater temperatures are $25^{\circ} \mathrm{C}$ and $8^{\circ} \mathrm{C},$ respectively, what is the maximum efficiency such an engine can have?

Shahab Ullah

Numerade Educator

Problem 17

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

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 18

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

Ryan Hood

Numerade Educator

Problem 19

An ice-making machine operates as a Carnot refrigerator. It takes heat from water at $0.0^{\circ} \mathrm{C}$ and exhausts the heat into a room at $24.0^{\circ} \mathrm{C}$. Suppose that it converts $85.0 \mathrm{~kg}$ of water at $0.0^{\circ} \mathrm{C}$ into ice at $0.0^{\circ} \mathrm{C}$. (a) How much heat must be removed from the water? (b) How much work energy must be supplied to the refrigerator?

Linda Winkler

Numerade Educator

Problem 20

A Carnot freezer that runs on electricity removes heat from the freezer compartment, which is at $-10^{\circ} \mathrm{C},$ and expels it into the room at $20^{\circ} \mathrm{C}$. You put an ice-cube tray containing $375 \mathrm{~g}$ of water at $18^{\circ} \mathrm{C}$ into the freezer. (a) What is the coefficient of performance of this freezer? (b) How much energy is needed to freeze this water? (c) How much electrical energy must be supplied to the freezer to freeze the water? (d) How much heat does the freezer expel into the room while freezing the ice?

Ryan Hood

Numerade Educator

Problem 21

The $p V$ diagram in Figure 16.16 shows a general Carnot cycle for an engine, with the hot and cold thermal reservoir segments identified. Segments $a b$ and $c d$ are isothermal, while the other two are adiabatic. (a) If this engine is used as a heat engine, what is the direction of the cycle, clockwise or counterclockwise? In which segments does heat enter the gas, and in which ones does it leave the gas? (b) If the engine is used as a refrigerator, what is the direction of the cycle? In which segments does heat enter the gas, and in which ones does it leave the gas? Also, which segments take place at the inside of the refrigerator, and which ones occur in the air of the room in which the refrigerator is operating? (c) If the engine is used as a heat pumpair conditioner, what is the direction of the cycle, and in which segments does heat enter the gas and in which ones does it leave the gas? Which segments take place inside of the house, and which ones occur outside? (d) If the engine is used as a heat pump-house heater, what is the direction of the cycle? In which segments does heat enter the gas, and in which ones does it leave the gas? Which segments take place inside of the house, and which ones occur outside?

David Morabito

Numerade Educator

Problem 22

A sophomore with nothing better to do adds heat to $0.350 \mathrm{~kg}$ of ice at $0.00^{\circ} \mathrm{C}$ until it is all melted. (a) What is the change in entropy of the water? (b) The source of the heat is a very massive body at a temperature of $25.0^{\circ} \mathrm{C}$. What is the change in entropy of this body? (c) What is the total change in entropy of the water and the heat source?

Ryan Hood

Numerade Educator

Problem 23

A $4.50 \mathrm{~kg}$ block of ice at $0.00^{\circ} \mathrm{C}$ falls into the ocean and melts. The average temperature of the ocean is $3.50^{\circ} \mathrm{C},$ 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?)

Linda Winkler

Numerade Educator

Problem 24

You decide to take a nice hot bath but discover that your thoughtless roommate has used up most of the hot water. You fill the tub with $270 \mathrm{~kg}$ of $30.0^{\circ} \mathrm{C}$ water and attempt to warm it further by pouring in $5.00 \mathrm{~kg}$ of boiling water from the stove. (a) Is this a reversible or an 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.

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 25

A crucible contains $0.1 \mathrm{~kg}$ of liquid lead that is at its melting point $(600 \mathrm{~K}) .$ What is the change in entropy of the lead if it freezes into a solid? Is the change positive or negative? (Use Table $14.4 .)$

Shahab Ullah

Numerade Educator

Problem 26

Three moles of an ideal gas undergo a reversible isothermal compression at $20.0^{\circ} \mathrm{C}$. During this compression, $1850 \mathrm{~J}$ of work is done

Linda Winkler

Numerade Educator

Problem 27

Premium gasoline produces $1.23 \times 10^{8} \mathrm{~J}$ of heat per gallon when it is burned at a temperature of approximately $400^{\circ} \mathrm{C}$ (although the amount can vary with the fuel mixture). If the car's engine is $25 \%$ efficient, three-fourths of that heat is expelled into the air, typically at $20^{\circ} \mathrm{C}$. If your car gets 35 miles per gallon of gas, by how much does the car's engine change the entropy of the world when you drive 1.0 mile? Does it decrease or increase it?

Linda Winkler

Numerade Educator

Problem 28

A typical doughnut contains approximately 200 food calories (kilocalories), of which about $80 \%$ is metabolized into heat by your body. Assume that the internal temperature of your body (where digestion occurs) is normally $37^{\circ} \mathrm{C}$ and that this does not significantly change after you digest the doughnut. By how much does your body's entropy change after you eat a doughnut?

Linda Winkler

Numerade Educator

Problem 29

A well-insulated house of moderate size in a temperate climate requires an average heat input rate of $20.0 \mathrm{~kW}$. If this heat is to be supplied by a solar collector with an average (night and day) energy input of $300 \mathrm{~W} / \mathrm{m}^{2}$ and a collection efficiency of $60.0 \%,$ what area of solar collector is required?

Linda Winkler

Numerade Educator

Problem 30

A solar power plant is to be built with an average power output capacity of $2500 \mathrm{MW}$ in a location where the average power from the sun's radiation is $200 \mathrm{~W} / \mathrm{m}^{2}$ at the earth's surface. What land area (in $\mathrm{km}^{2}$ and $\mathrm{mi}^{2}$ ) must the solar collectors occupy if they are (a) photocells with $42 \%$ efficiency, (b) mirrors that generate steam for a turbine generator unit with an overall efficiency of $21 \% ?$

Linda Winkler

Numerade Educator

Problem 31

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

Linda Winkler

Numerade Educator

Problem 32

A solar water heater for domestic hot-water supply uses solar collecting panels with a collection efficiency of $50 \%$ in a location where the average solar-energy input is $200 \mathrm{~W} / \mathrm{m}^{2}$ If the water comes into the house at $15.0^{\circ} \mathrm{C}$ and is to be heated to $60.0^{\circ} \mathrm{C},$ what volume of water can be heated per hour if the area of the collector is $30.0 \mathrm{~m}^{2} ?$

Ryan Hood

Numerade Educator

Problem 33

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

Katie Mcalpine

Numerade Educator

Problem 34

A heat engine takes 0.350 mol of an ideal diatomic gas around the cycle shown in the $\underline{p V}$ diagram of Figure 16.17 . Process $1 \rightarrow 2$ is at constant volume, process $2 \rightarrow 3$ is adiabatic, and process $3 \rightarrow 1$ is at a constant pressure of 1.00 atm. The value of $\gamma$ for this gas is $1.40 .$ The magnitude of the change in internal energy for each process is $\left|\Delta U_{1 \rightarrow 2}\right|=2180 \mathrm{~J},\left|\Delta U_{2 \rightarrow 3}\right|=785 \mathrm{~J},$ and $\left|\Delta U_{3 \rightarrow 1}\right|=1396 \mathrm{~J}$ (a) Find the pressure and volume at points $1,2,$ and $3 .$ (b) Calculate $Q$ and $W$ 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 efficiency compare with that of a Carnot-cycle engine operating between the same minimum and maximum temperatures $T_{1}$ and $T_{2} ?$

David Morabito

Numerade Educator

Problem 35

As a budding mechanical engineer, you are called upon to design a Carnot engine that has 2.00 moles of He gas (see Table 15.4 ) as its working substance and that operates from a high-temperature reservoir at $500^{\circ} \mathrm{C}$. The engine is to lift a $15.0 \mathrm{~kg}$ weight $2.00 \mathrm{~m}$ per cycle, using $500 \mathrm{~J}$ of heat input. The gas in the engine chamber can have a minimum volume of $5.00 \mathrm{~L}$ during the cycle. (a) Draw a $p V$ diagram for this cycle. In your diagram, show 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?

Linda Winkler

Numerade Educator

Problem 36

The Kwik-Freez Appliance Co. wants you to design a food freezer that will keep the freezing compartment at $-5.0^{\circ} \mathrm{C}$ and will operate in a room at $20.0^{\circ} \mathrm{C}$. The freezer is to make $5.00 \mathrm{~kg}$ of ice at $0.0^{\circ} \mathrm{C}$, starting with water at $20.0^{\circ} \mathrm{C}$. Find the least possible amount of electrical energy needed to make this ice and the smallest possible amount of heat expelled into the room.

Linda Winkler

Numerade Educator

Problem 37

A Carnot engine operates between two heat reservoirs at temperatures $T_{\mathrm{H}}$ and $T_{\mathrm{C}}$. An inventor proposes to increase the efficiency of the engine by increasing both $T_{\mathrm{H}}$ and $T_{\mathrm{C}}$ by a factor of $2 .$ Will this plan work? Why or why not?

Shahab Ullah

Numerade Educator

Problem 38

An engineer is working with a Carnot engine that has an unknown cold-reservoir temperature $\left(T_{\mathrm{C}}\right)$ but a known and controllable hot-reservoir temperature $\left(T_{\mathrm{H}}\right) .$ He measures the efficiency of the engine as a function of the hot-reservoir temperature and produces the following data set:
$$
\begin{array}{cc}
\hline T_{\mathrm{H}}(\mathrm{K}) & e \\
\hline 300 & 0.133 \\
370 & 0.298 \\
425 & 0.390 \\
483 & 0.461 \\
535 & 0.513 \\
\hline
\end{array}
$$
Produce a linearized plot of the engine efficiency as a function of the hot-reservoir temperature. Using a "best fit" to the data, determine the cold-reservoir temperature.

Linda Winkler

Numerade Educator

Problem 39

A person having skin of surface area $1.85 \mathrm{~m}^{2}$ and temperature $30.0^{\circ} \mathrm{C}$ is resting in an insulated room where the ambient air temperature is $20.0^{\circ} \mathrm{C}$. 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 $1.00 .)$

Salamat Ali

Numerade Educator

Problem 40

A typical coal-fired power plant generates $1000 \mathrm{MW}$ 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 $2.65 \times 10^{7} \mathrm{~J} / \mathrm{kg} .$ How much coal does the plant use per day, if it operates continuously? (c) At what rate is heat ejected into the cool reservoir, which is the nearby river? (d) The river's temperature is $18.0^{\circ} \mathrm{C}$ before it reaches the power plant and $18.5^{\circ} \mathrm{C}$ after it has received the plant's waste heat. Calculate the river's flow rate, in cubic meters per second. (e) By how much does the river's entropy increase each second?

Ravindra Yadav

Numerade Educator

Problem 41

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 $37.0^{\circ} \mathrm{C}$ ) and the other end at the surface of your skin, at $30.0^{\circ} \mathrm{C}$. (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 $2.50 \mathrm{~kg}$ box from the floor to a tabletop $1.20 \mathrm{~m}$ above the floor. How much must you increase the gravitational potential energy, and how much heat input is needed to accomplish this? (c) How many 350 calorie (those are food calories, remember) candy bars must you eat to lift the box in this way? Recall that $80 \%$ of the food energy goes into heat.

Linda Winkler

Numerade Educator

Problem 42

One end of a copper rod is immersed in boiling water at $100^{\circ} \mathrm{C}$ and the other end in an ice-water mixture at $0^{\circ} \mathrm{C}$. The sides of the rod are insulated. After steady-state conditions have been achieved in the rod, $0.160 \mathrm{~kg}$ 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.

David Morabito

Numerade Educator

Problem 43

The $p V$ diagram in Figure 16.18 shows a heat engine operating on $0.850 \mathrm{~mol}$ of $\mathrm{H}_{2}$. (See Table 15.4.) Segment $c a$ is isothermal, and the change in internal energy associated with the $a b$ segment is $\Delta U_{a \rightarrow b}=12,190 \mathrm{~J}$. (a) Find the temperature at $a, b,$ and $c .(\mathrm{b})$ Without doing any calculations, identify the segments during which heat enters the gas and those during which it leaves the gas. Explain your reasoning. (c) Calculate the thermal efficiency of the engine, assuming the gas is ideal.

Linda Winkler

Numerade Educator

Problem 44

If the power plant uses a Carnot cycle and the desired theoretical efficiency is $6.5 \%,$ from what depth must cold water be brought?
A. $100 \mathrm{~m}$
B. $400 \mathrm{~m}$
C. $800 \mathrm{~m}$
D. Deeper than $1000 \mathrm{~m}$

Linda Winkler

Numerade Educator

Problem 45

What is the change in entropy of the ammonia vaporized per second in the $10 \mathrm{MW}$ power plant, assuming an ideal Carnot efficiency of $6.5 \% ?$
A. $+6 \times 10^{6} \mathrm{~J} / \mathrm{K}$ per second
B. $+5 \times 10^{5} \mathrm{~J} / \mathrm{K}$ per second
C. $+1 \times 10^{5} \mathrm{~J} / \mathrm{K}$ per second
D. 0

Linda Winkler

Numerade Educator

Problem 46

Compare the entropy change of the warmer water to that of the colder water during one cycle of the heat engine, assuming an ideal Carnot cycle.
A. The entropy does not change during one cycle in either case.
B. The entropy of both increases, but the entropy of the colder water increases more because its initial temperature is lower.
C. The entropy of the warmer water decreases more than the entropy of the colder water increases because some of the heat removed from the warmer water goes to the work done by the engine.
D. The entropy of the warmer water decreases by the same amount that the entropy of the colder water increases.

Shital Rijal

Numerade Educator

Problem 47

If the proposed plant is built and produces $10 \mathrm{MW}$ but the rate at which waste heat is exhausted to the cold water is $165 \mathrm{MW}$, what is the plant's actual efficiency?
A. $5.7 \%$
B. $6.1 \%$
C. $6.5 \%$
D. $16.5 \%$

Shahab Ullah

Numerade Educator