College Physics 2013

Eugenia Etkina, Michael Gentle, Alan Van Heuvelen

Chapter 13

Second Law of Thermodynamics

Educators


Problem 1

Types of energy and reversibility of a process Describe the types of energy that change, the work done on the system, and the energy transferred through heating during the following processes. Indicate whether a reverse process can occur. (a) Water at the top of Niagara Falls cascades onto the blades of an electric generator near the bottom of the falls, rotating the blades and generating an electric current that causes a light- bulb to glow. The water, generator, lightbulb, and Earth are the
system. (b) Each second, your body converts 100 J of metabolic energy (converting complex molecules from food) to thermal energy transferred to the air surrounding your body.
The system is your body and the surrounding air. (c) The hot gas in a cylinder pushes a piston, which causes the blades of an electric generator to turn, which in turn causes a light bulb to glow briefly. The system is the original hot gas (which cools while pushing the piston), the generator, and the light bulb (which first glows and then stops glowing and cools down).

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Problem 2

For the following processes, choose the initial and final states and describe the process using the physical quantities internal energy, work, and heating. Explain why the process is irreversible. (a) A large foam ball is moving vertically up at speed $v$ and reaches a maximum height $h^{\prime}$ somewhat less than $\sqrt{v^{2} / 2 g .}$ The ball, Earth, and air are the system. (b) Two cups of water, one cold and the other hot, are mixed in an insulated bowl. The mixture reaches an intermediate temperature. The water in the cups is the system.

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Problem 3

Hourglass An hourglass starts with all of the sand in the top bulb. During the next hour, the sand slowly leaks into the bottom bulb. Describe the energy changes in a system that includes the glass, sand, and Earth. Is this a reversible or irreversible process? Explain.

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Problem 4

Car hits tree Your car slides on ice and runs into a tree, causing the front of the car to become slightly hotter and crumpled. The car and ice are the system. Indicate what object does the work on the system. Indicate whether heating occurs. Identify the types of energy that change. Are these quantities
positive or negative? Explain.

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Problem 5

Human metabolism A 60 -kg person consumes about 2000 kcal of food in one day. If 10$\%$ of this food energy is converted to thermal energy and cannot leave the body, estimate the temperature change of the person.[Note: 1 kcal $=4180 \mathrm{J}$ .] Is this a reversible or irreversible process?

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Problem 6

(a) Identify all of the macrostate distributions for five atoms located in a box with two halves. (b) Determine the number of microstates for each macrostate. (c) Determine the entropy of each state.

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Problem 7

'Repeat the previous problem for a system with six atoms.

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Problem 8

Determine the ratio of the number of microstates (count) of a system of cight atoms when in a macrostate with four atoms on the left side of a container and four on the right side and
when in a macrostate with seven atoms on the left and one on the right. Which state has the greatest entropy? Explain.

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Problem 9

Person lost on island The probability that a lost person wandering about an island will be on the north part is one-half and on the south part is also one-half. (a) Determine the probability that three lost people wandering about independently will all be on the south half. (b) Repeat part (a) for the probability of six lost people all being on the south half of the island.

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Problem 10

Parachutists landing on island Parachutists have an equal chance of landing on the south half of a small island or the north half. If eight parachutists jump at one time, what is the ratio of the probability that six land on the north half and two on the south half to the probability that four land on each half?

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Problem 11

Determine the ratio of the counts of a system of 20 atoms when in a macrostate with 10 atoms on the left half of a box and 10 on the right half and when in a macrostate with 18 at- oms on the left and 2 on the right. (b) Do the same for a system with 10 coins for the states with 5 coins on the left and 5 on
the right compared to 9 coins on the left and 1 on the right. (c) When you compare your answers to parts (a) and (b), what do you infer about a similar ratio for a system with $10,000$ atoms?

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Problem 12

Nine numbered balls are dropped randomly into three boxes. The numbers of balls falling into each box are labeled $n_{1}, n_{2},$ and $n_{3} .$ (a) Identify five of the many possible arrangements or macrostates of the balls. (b) Determine the ratio of the count for the equal distribution $\left(n_{1}=3, n_{2}=3, \text { and }\right.$ $n_{3}=3 )$ and for the $0,0,9$ distribution. (c) Determine the ratio of the count for the equal distribution and for the $2,3,$ 4 distribution. INote: The count is given by
$$
W=\frac{n !}{n_{1} ! n_{2} ! n_{3} !}
$$
where $n$ is the total number of balls.]

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Problem 13

Rolling dice Two dice are rolled. Macrostates of these dice are distinguished by the total number for each roll (that is, $2,$ $3,4, \ldots, 12 ) .$ (a) Determine the number of microstates for each macrostate. For example, there are three microstates for macrostate $4 :(2,2),(3,1),$ and $(1,3)$ . (b) What is the macro-
state with greatest entropy? (c) What is the macrostate with least entropy? Explain.

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Problem 14

(a) Apply your knowledge of probability to explain why a drop of food coloring in a glass of clear water spreads out so that all of the water has an even color after some time. (b) Dis- cuss whether after the food coloring spreads evenly in a glass of clear water it could condense back to an original droplet.

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Problem 15

Explain using your knowledge of probability why a gas always occupies the entire volume of its container.

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Problem 16

Estimate the total change in entropy of two containers of water. One container holds 0.1 $\mathrm{kg}$ of water at $70^{\circ} \mathrm{C}$ and is warmed to $90^{\circ} \mathrm{C}$ by heating from contact with the other container. The other container, also holding 0.1 $\mathrm{kg}$ of water, cools from $30^{\circ} \mathrm{C}$ to $10^{\circ} \mathrm{C}$ . Is this energy transfer process allowed by the first law of thermodynamics? By the second?

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Problem 17

(a) You add 0.1 $\mathrm{kg}$ of water at $0^{\circ} \mathrm{C}$ to 0.3 $\mathrm{kg}$ of iced tea at $70^{\circ} \mathrm{C}$ . Determine the final temperature of the mixture after it reaches equilibrium. The specific heat of iced tea is the same as water. (b) Estimate the entropy change of this system during this prcess. Is it allowed by the second law of thermodynamics? Explain.

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Problem 18

Entropy change of a house A house at $20^{\circ} \mathrm{C}$ transfers $1.0 \times 10^{5}$ J of thermal energy to the outside air, which has a temperature of $-15^{\circ} \mathrm{C}$ . Determine the entropy change of thehouse-outside air system. Is this process allowed by the second law of thermodynamics?

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Problem 19

Barrel of water in cellar in winter A barrel containing 200 $\mathrm{kg}$ of water sits in a cellar during the winter. On a cold day, the water freezes, releasing thermal energy to the room. This energy passes from the cellar to the outside air, which has a temperature of $-20^{\circ} \mathrm{C}$ . Determine the entropy change for this process if the cellar remains at $0^{\circ} \mathrm{C}$ .

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Problem 20

(a) Determine the final temperature when 0.1 $\mathrm{kg}$ of wa-ter at $10^{\circ} \mathrm{C}$ is added to 0.3 $\mathrm{kg}$ of soup at $50^{\circ} \mathrm{C} .$ What assumptions did you make? (b) Estimate the entropy change of this water-soup system during the process. Does the second law of
thermodynamics allow this process?

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Problem 21

A 5.0 -kg block slides on a level surface and stops because of friction. Its initial speed is 10 $\mathrm{m} / \mathrm{s}$ and the temperature of the surface is $20^{\circ} \mathrm{C}$ . Determine the entropy change of the block, which is the system in this process.

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Problem 22

A 5.0 -kg block slides from an initial speed of 8.0 $\mathrm{m} / \mathrm{s}$ to a final speed of zero. It travels 12 $\mathrm{m}$ down a plane inclined at $15^{\circ}$ with the horizontal. Determine the entropy change of the block-inclined plane-Earth system for this process if originally the block and the inclined plane were at $27^{\circ} \mathrm{C}$ . Why did the block stop?

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Problem 23

Maximum efficiencies Determine the maximum efficiencies of the thermodynamic engines described below. (a) Burning coal heats the gas in the turbine of an electric power plant to 700 $\mathrm{K}$ . After turning the blades of the generator, the gas is cooled in cooling towers to 350 $\mathrm{K}$ . (b) An inventor claims to have a thermodynamic engine that attaches to a car's exhaust system. The temperature of the exhaust gas is $90^{\circ} \mathrm{C}$ and the temperature of the output of this proposed heat engine is $20^{\circ} \mathrm{C} .(\mathrm{c})$ Near Bermuda, ocean water is about $24^{\circ} \mathrm{C}$ at the surface and about $10^{\circ} \mathrm{C}$ at a depth of 800 $\mathrm{m} .$

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Problem 24

Efficiency of woman walking $\mathrm{A}$ 60-kg woman walking on level ground at 1 $\mathrm{m} / \mathrm{s}$ metabolizes energy at a rate of 230 $\mathrm{W}$ . When she walks up a $5^{\circ}$ incline at the same speed, her metabolic rate increases to 370 $\mathrm{W}$ . Determine her efficiency at converting chemical energy into gravitational potential energy.

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Problem 25

Nuclear power plant A nuclear power plant operates between a high-temperature heat reservoir at $560^{\circ} \mathrm{C}$ and a lowtemperature stream at $20^{\circ} \mathrm{C}$ . (a) Determine the maximum possible efficiency of this thermodynamic engine. (b) Determine the heating rate $(\mathrm{J} / \mathrm{s})$ from the high-temperature heat reservoir to the power plant so that
it produces 1000 $\mathrm{MW}$ of power (work/time).

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Problem 26

A cyclic process in volving 1 mole of ideal gas is shown in Figure P13. 26 . (a) Determine the work done on the gas by the environment during cach step $(A, B, C, \text { and } D)$ of the cycle. (b) Determine the net work done on the gas $W_{\text { Fnvon Gas }}$ (c) Find the work that the gas does on the environment; it is cqual to $W_{\text { Gas on Env }}=-W_{\text { Envon Gas. }}(d)$ Use the ideal gas law to determine the temperature of the gas at each corner of the process $(1,2,3, \text { and } 4) .(\mathrm{e})$ Use the temperatures found in $(\mathrm{d})$ to determine the thermal energy of the gas at each corner of the process and the change in internal thermal energy during each step of the process.(f) Use the results recorded in the previous parts and the first law of thermodynamics to determine
the heating during each step of the process. (g) Determine the efficiency of the process.

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Problem 27

A cyclic process in - $F$ volving 1 mole of ideal gas is shown in Figure P13.27 (a) Determine the
work done on the gas by the environment during each step $(A, B, \text { and } C)$ of
the cycle. (b) Determine the net work done on the gas $W_{\text { Env on Gas. }}(c)$ Find the work that the gas does on the environment equal to $W_{\text { Gas on Einv }}=-W_{\text { Env on Gas. }}$ (d) Use the ideal gas law to determine the temperature of the gas at each corner of the process $(1,2, \text { and } 3) .$ (e) Use the temperatures found in (d) to determine the thermal energy of the gas
at cach corner of the process and the change in internal thermal energy during each step of the process. (f) Use the results re- corded in the previous parts and the first law of thermodynamics to determine the heating during each step of the process. (g) Determine the efficiency of the process.

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Problem 28

Home thermodynamic pump A heat pump collects thermal energy from outside air at $5^{\circ} \mathrm{C}$ and delivers it into a house at $40^{\circ} \mathrm{C}$ . (a) Determine the maximum coefficient of performance (the maximum coefficient is determined in the same way as for a thermodynamic engine). (b) If the motor of the heating pump uses 1000 $\mathrm{J}$ of electrical energy to do work during a certain time interval, how much thermal energy is delivered into
the house through heating, assuming the heating pump works at the maximum coefficient of performance? (c) Repeat (b) for a coefficient of performance of $2.0 .$

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Problem 29

Ice-making machine An ice-making machine needs to convert 0.20 $\mathrm{kg}$ of water at $0^{\circ} \mathrm{C}$ to 0.20 $\mathrm{kg}$ of ice at $0^{\circ} \mathrm{C}$ . The room temperature surrounding the ice machine is $20^{\circ} \mathrm{C} .(\mathrm{a})$ How much thermal energy must be removed from the water? (b) Determine the minimum work needed to extract this energy by the ice-making machine. (c) How much energy is deposited in the room?

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Problem 30

"Automobile engine An automobile engine has a power out- put for doing work of 150 $\mathrm{kW}$ (about 200 $\mathrm{hp} )$ . The efficiency of the engine is 0.32 . Determine the heating input per second to the engine by burning gasoline and the heating rate of the engine to the environment.

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Problem 31

Diesel car engine A diesel engine in a car does 1000 $\mathrm{J}$ of work due to 2800 $\mathrm{J}$ of heating caused by the combustion of diesel fuel in its cylinders. Determine the efficiency of the engine and the thermal energy emitted by the engine to the environment.

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Problem 32

Gas used in car's engine During each cycle of an automobile's gasoline engine operation, the gasoline burned provides $12,000 \mathrm{J}$ of energy through heating to the engine and the engine does 3600 $\mathrm{J}$ of work. Gasoline provides $4.4 \times 10^{7} \mathrm{J}$ of energy for each kilogram of gasoline burned. (a) Determine the efficiency of the engine. (b) Determine the thermal energy exhausted from the engine during each cycle. ( ) Determine the mass of the gasoline burned during each cycle. (d) If the engine has 80 cycles/s, how much gasoline does the engine use in 1 $\mathrm{h}$ in kilograms and in gallons? The density of gasoline is 737 $\mathrm{kg} / \mathrm{m}^{3}$ .

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Problem 33

Nuclear power plant A nuclear power plant does useful work generating electric energy at a rate of 500 $\mathrm{MW}$ . The energy transfer rate to the electric generator from the high-temperature nuclear fuel (the hot reservoir) is 1200 $\mathrm{MW}$ . Determine the efficiency of the power plant and the rate at which the working substance in the plant transfers energy through heating to the cold water (the cool reservoir).

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Problem 34

Nuclear power plant A nuclear power plant warms water to $500^{\circ} \mathrm{C}$ and emits it at $100^{\circ} \mathrm{C}$ . You want to get useful work done by the plant at a rate of $1.0 \times 10^{9} \mathrm{J} / \mathrm{s}$ . Determine the rate at which the nuclear fuel must provide energy to the
working substance and the rate of thermal energy exhausted from the plant to the environment.

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Problem 35

Body efficiency-experiment design Describe an experiment that can be performed to estimate the average efficiency of a human body.

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Problem 36

The following equations represent the four parts $(A, B,$ C, and D) of a cyclic process with a gas. In this case we consider the work done by the system on the environment and consequently $Q-W=\Delta U_{\text { int }} .$
$$
\begin{aligned} W=&\left(3.0 \times 10^{5} \mathrm{N} / \mathrm{m}^{2}\right)\left(0.020 \mathrm{m}^{3}-0.010 \mathrm{m}^{3}\right)+0 \\ &+\left(1.0 \times 10^{5} \mathrm{N} / \mathrm{m}^{2}\right)\left(0.010 \mathrm{m}^{3}-0.020 \mathrm{m}^{3}\right)+0 \end{aligned}
$$
$$
\begin{aligned} \Delta U_{\mathrm{int}}=&(3 / 2)(1.0 \mathrm{mole})(8.3 \mathrm{J} / \mathrm{mole} \cdot \mathrm{K})[(700 \mathrm{K}-360 \mathrm{K})\\ &+(480 \mathrm{K}-700 \mathrm{K})+(240 \mathrm{K}-480 \mathrm{K}) \\ &+(360 \mathrm{K}-240 \mathrm{K}) | \\ Q=& Q_{\mathrm{A}}+Q_{\mathrm{B}}+Q_{\mathrm{C}}+Q_{\mathrm{C}} \end{aligned}
$$
(a) Draw a P-versus-V graph for the process with labeled axes (including a scale).
(b) Determine the net change in the internal energy during the entire cycle.
(c) Determine the heating of the system for each of the four parts of the process.

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Problem 37

Assuming a $200-\mathrm{N}$ drag force when traveling at 22 $\mathrm{m} / \mathrm{s}$ through air of density $1.3 \mathrm{kg} / \mathrm{m}^{3},$ what is the closest value to the product $C A$ in the drag force equation for the vehicle?
$$
0.50 \mathrm{m}^{2} \quad \text { (b) } 0.62 \mathrm{m}^{2} \quad \text { (c) } 0.86 \mathrm{m}^{2}
$$
$$
1.1 \mathrm{m}^{2} \quad \text { (e) } 1.5 \mathrm{m}^{2}
$$

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Problem 38

At 22 $\mathrm{m} / \mathrm{s}$ the magnitude of the resistive force that air exerts on the car is about 200 $\mathrm{N}$ . Which answer below is closest to the magnitude of the drag force when the car is traveling at 31 $\mathrm{m} / \mathrm{s} ?$
$$
\begin{array}{llll}{\text { (a) } 100 \mathrm{N}} & {\text { (b) } 140 \mathrm{N}} & {\text { (c) } 280 \mathrm{N}}\end{array}
$$
$$
400 \mathrm{N} \text { (c) } 520 \mathrm{N}
$$

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Problem 39

The amount of fucl used to counter air resistance should do what?
(a) Increase in proportion to the speed squared
(b) Increase in proportion to the speed
(c) Be the same independent of the speed
(d) Decrease in proportion to the inverse of the speed
(e) Decrease in proportion to the inverse of the speed

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Problem 40

The $200-\mathrm{N}$ resistive force of the air in this problem is closest to which answer below?
$$
\begin{array}{lll}{\text { (a) } 800 \mathrm{lb}} & {\text { (b) } 90 \mathrm{lb}} & {\text { (c) } 60 \mathrm{lb}}\end{array}
$$
$$
45 \mathrm{lb} \quad \text { (e) } 30 \mathrm{lb}
$$

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Problem 41

Why is the resistive force of the air on a Hummer $\mathrm{H} 3$ traveling at 22 $\mathrm{m} / \mathrm{s}$ about 600 $\mathrm{N}$ instead of 200 $\mathrm{N} ?$
(a) The Hummer has a bulky, less streamlined shape.
(b) The Hummer has a greater cross-sectional area along the line of motion.
(c) The Hummer has greater mass.
(d) a and b
(e) a, b, and c

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Problem 42

The value of CA for a Ford Escape Hybrid is 1.08 $\mathrm{m}^{2}$ . Which answer below is closest to the drag force on this car when raveling at 22 $\mathrm{m} / \mathrm{s} ?$
$$
\begin{array}{ll}{\text { (a) } 130 \mathrm{N}} & {\text { (b) } 180 \mathrm{N}} \\ {\text { (d) } 350 \mathrm{N}} & {\text { (e) } 440 \mathrm{N}}\end{array}
$$

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Numerade Educator