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Fundamentals of Physics

David Halliday, Robert Resnick, Jearl Walker

Chapter 20

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

Educators


Chapter Questions

02:38

Problem 1

Suppose $4.00 \mathrm{~mol}$ of an ideal gas undergoes a reversible isothermal expansion from volume $V_{1}$ to volume $V_{2}=2.00 V_{1}$ at temperature $T=400 \mathrm{~K}$. Find (a) the work done by the gas and (b) the entropy change of the gas. (c) If the expansion is reversible and adiabatic instead of isothermal, what is the entropy change of the gas?

Sri Datta Vikas Buchemmavari
Sri Datta Vikas Buchemmavari
Numerade Educator
01:17

Problem 2

An ideal gas undergoes a reversible isothermal expansion at $77.0^{\circ} \mathrm{C}$, increasing its volume from $1.30 \mathrm{~L}$ to $3.40 \mathrm{~L}$. The entropy change of the gas is $22.0 \mathrm{~J} / \mathrm{K}$. How many moles of gas are present?

Averell Hause
Averell Hause
Carnegie Mellon University
01:08

Problem 3

A $2.50 \mathrm{~mol}$ sample of an ideal gas expands reversibly and isothermally at $360 \mathrm{~K}$ until its volume is doubled. What is the increase in entropy of the gas?

Manish Kumar ( Iit K )
Manish Kumar ( Iit K )
Numerade Educator
00:53

Problem 4

How much energy must be transferred as heat for a reversible isothermal expansion of an ideal gas at $132^{\circ} \mathrm{C}$ if the entropy of the gas increases by $46.0 \mathrm{~J} / \mathrm{K} ?$

Averell Hause
Averell Hause
Carnegie Mellon University
02:45

Problem 5

Find (a) the energy absorbed as heat and (b) the change in entropy of a $2.00 \mathrm{~kg}$ block of copper whose temperature is increased reversibly from $25.0^{\circ} \mathrm{C}$ to $100^{\circ} \mathrm{C}$. The specific heat of copper is $386 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}$

Sri Datta Vikas Buchemmavari
Sri Datta Vikas Buchemmavari
Numerade Educator
02:23

Problem 6

(a) What is the entropy change of a $12.0 \mathrm{~g}$ ice cube that melts completely in a bucket of water whose temperature is just above the freezing point of water? (b) What is the entropy change of a $5.00 \mathrm{~g}$ spoonful of water that evaporates completely on a hot plate whose temperature is slightly above the boiling point of water?

Averell Hause
Averell Hause
Carnegie Mellon University
06:19

Problem 7

A $50.0 \mathrm{~g}$ block of copper whose temperature is $400 \mathrm{~K}$ is placed in an insulating box with a $100 \mathrm{~g}$ block of lead whose temperature is $200 \mathrm{~K}$. (a) What is the equilibrium temperature of the twoblock system? (b) What is the change in the internal energy of the system between the initial state and the equilibrium state? (c) What is the change in the entropy of the system? (See Table $18-3 .$ )

Vipender Yadav
Vipender Yadav
Numerade Educator
01:10

Problem 8

At very low temperatures, the molar specific heat $C_{V}$ of many solids is approximately $C_{V}=A T^{3}$, where $A$ depends on the particular substance. For aluminum, $A=3.15 \times 10^{-5} \mathrm{~J} / \mathrm{mol} \cdot \mathrm{K}^{4} .$ Find the entropy change for $4.00 \mathrm{~mol}$ of aluminum when its temperature is raised from $5.00 \mathrm{~K}$ to $10.0 \mathrm{~K}$.

Averell Hause
Averell Hause
Carnegie Mellon University
07:06

Problem 9

A $10 \mathrm{~g}$ ice cube at $-10^{\circ} \mathrm{C}$ is placed in a lake whose temperature is $15^{\circ} \mathrm{C}$. Calculate the change in entropy of the cube-lake system as the ice cube comes to thermal equilibrium with the lake. The specific heat of ice is $2220 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}$. (Hint: Will the ice cube affect the lake temperature?)

Sri Datta Vikas Buchemmavari
Sri Datta Vikas Buchemmavari
Numerade Educator
02:15

Problem 10

A $364 \mathrm{~g}$ block is put in contact with a thermal reservoir. The block is initially at a lower temperature than the reservoir. Assume that the consequent transfer of energy as heat from the reservoir to the block is reversible. Figure $20-22$ gives the change in entropy $\Delta S$ of the block until thermal equilibrium is reached. The scale of the horizontal axis is set by $T_{a}=280 \mathrm{~K}$ and $T_{b}=380 \mathrm{~K}$. What is the specific heat of the block?

Averell Hause
Averell Hause
Carnegie Mellon University
03:54

Problem 11

- In an experiment, $200 \mathrm{~g}$ of aluminum (with a specific heat of $900 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}$ ) at $100^{\circ} \mathrm{C}$ is mixed with $50.0 \mathrm{~g}$ of water at $20.0^{\circ} \mathrm{C}$, with the mixture thermally isolated. (a) What is the equilibrium temperature? What are the entropy changes of (b) the aluminum, (c) the water, and (d) the aluminum-water system?

Sri Datta Vikas Buchemmavari
Sri Datta Vikas Buchemmavari
Numerade Educator
02:19

Problem 12

A gas sample undergoes a reversible isothermal expansion. Figure $20-23$ gives the change $\Delta S$ in entropy of the gas versus the final volume $V_{f}$ of the gas. The scale of the vertical axis is set by $\Delta S_{s}=64 \mathrm{~J} / \mathrm{K}$. How many moles are in the sample?

Vipender Yadav
Vipender Yadav
Numerade Educator
09:00

Problem 13

In the irreversible process of Fig. $20-5$, let the initial temperatures of the identical blocks $L$ and $R$ be $305.5$ and $294.5 \mathrm{~K}$, respectively, and let $215 \mathrm{~J}$ be the energy that must be transferred between the blocks in order to reach equilibrium. For the reversible processes of Fig. $20-6$, what is $\Delta S$ for (a) block $L,(\mathrm{~b})$ its reservoir, (c) block $R,(\mathrm{~d})$ its reservoir, $(\mathrm{e})$ the two-block system, and (f) the system of the two blocks and the two reservoirs?

Vipender Yadav
Vipender Yadav
Numerade Educator
10:08

Problem 14

(a) For $1.0 \mathrm{~mol}$ of $\mathrm{a}$ monatomic ideal gas taken through the cycle in Fig. $20-24$, where $V_{1}=$ $4.00 V_{0}$, what is $W / p_{0} V_{0}$ as the gas goes from state $a$ to state $c$ along path $a b c ?$ What is $\Delta E_{\text {int }} / p_{0} V_{0}$ in going (b) from $b$ to $c$ and (c) through one full cycle? What is $\Delta S$ in going (d) from $b$ to $c$ and (e) through one full cycle?

Vipender Yadav
Vipender Yadav
Numerade Educator
03:35

Problem 15

A mixture of $1773 \mathrm{~g}$ of water and $227 \mathrm{~g}$ of ice is in an initial equilibrium state at $0.000^{\circ} \mathrm{C}$. The mixture is then, in a reversible process, brought to a second equilibrium state where the water-ice ratio, by mass, is $1.00: 1.00$ at $0.000^{\circ} \mathrm{C}$. (a) Calculate the entropy change of the system during this process. (The heat of fusion for water is $333 \mathrm{~kJ} / \mathrm{kg} .$ (b) The system is then returned to the initial equilibrium state in an irreversible process (say, by using a Bunsen burner). Calculate the entropy change of the system during this process. (c) Are your answers consistent with the second law of thermodynamics?

Sri Datta Vikas Buchemmavari
Sri Datta Vikas Buchemmavari
Numerade Educator
05:56

Problem 16

An $8.0 \mathrm{~g}$ ice cube at $-10^{\circ} \mathrm{C}$ is put into a Thermos flask containing $100 \mathrm{~cm}^{3}$ of water at $20^{\circ} \mathrm{C} .$ By how much has the entropy of the cube-water system changed when equilibrium is reached? The specific heat of ice is $2220 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}$.

Averell Hause
Averell Hause
Carnegie Mellon University
10:52

Problem 17

In Fig. 20-25, where $V_{23}=$ $3.00 V_{1}, n$ moles of a diatomic ideal gas are taken through the cycle with the molecules rotating but not oscillating. What are (a) $p_{2} / p_{1}$, (b) $p_{3} / p_{1}$, and (c) $T_{3} / T_{1}$ ? For path $1 \rightarrow 2$, what
are
(d) $W / n R T_{1}$,
(e) $Q \ln R T_{1}$,
(f) $\Delta E_{\text {int }} / n R T_{1}$, and
(g) $\Delta \operatorname{Sln} R ?$ For
path $2 \rightarrow 3$, what are (h) $W / n R T_{1}$,
(i) $Q / n R T_{1}$,
(j) $\Delta E_{\text {int }} / n R T_{1}, \quad(\mathrm{k})$
$\Delta S / n R ?$ For path $3 \rightarrow 1$, what are
(l) $W / n R T_{1}$,
$(\mathrm{m}) \quad Q / n R T_{1}$,
(n) $\Delta E_{\mathrm{int}} / n R T_{1}$, and (o) $\Delta S / n R ?$

Keshav Singh
Keshav Singh
Numerade Educator
04:19

Problem 18

A $2.0 \mathrm{~mol}$ sample of an ideal monatomic gas undergoes the reversible process shown in Fig. $20-26 .$ The scale of the vertical axis is set by $T_{s}=400.0 \mathrm{~K}$ and the scale of the horizontal axis is set by $S_{s}=20.0 \mathrm{~J} / \mathrm{K}$. (a) How much energy is absorbed as heat by the gas?
(b) What is the change in the internal energy of the gas? (c) How much work is done by the gas?

Vipender Yadav
Vipender Yadav
Numerade Educator
12:11

Problem 19

Suppose $1.00$ mol of a monatomic ideal gas is taken from initial pressure $p_{1}$ and volume $V_{1}$ through two steps: $(1)$ an isothermal expansion to volume $2.00 V_{1}$ and $(2)$ a pressure increase to $2.00 p_{1}$ at constant volume. What is $Q / p_{1} V_{1}$ for (a) step 1 and (b) step 2 ? What is $W / p_{1} V_{1}$ for (c) step 1 and (d) step 2? For the full process, what are
(e) $\Delta E_{\text {int }} / p_{1} V_{1}$ and (f) $\Delta S ?$ The gas is returned to its initial state and again taken to the same final state but now through these two steps:
(1) an isothermal compression to pressure $2.00 p_{1}$ and $(2)$ a volume increase to $2.00 V_{1}$ at constant pressure. What is $Q / p_{1} V_{1}$ for $(\mathrm{g})$ step 1 and (h) step 2 ? What is $W / p_{1} V_{1}$ for (i) step 1 and (j) step 2? For the full process, what are (k) $\Delta E_{\text {int }} / p_{1} V_{1}$ and (1) $\Delta S$ ?

Keshav Singh
Keshav Singh
Numerade Educator
08:14

Problem 20

Expand $1.00$ mol of an monatomic gas initially at $5.00 \mathrm{kPa}$ and $600 \mathrm{~K}$ from initial volume $V_{i}=1.00 \mathrm{~m}^{3}$ to final volume $V_{f}=$ $2.00 \mathrm{~m}^{3}$. At any instant during the expansion, the pressure $p$ and volume $V$ of the gas are related by $p=5.00 \exp \left[\left(V_{i}-V\right) / a\right]$, with $p$ in kilopascals, $V_{i}$ and $V$ in cubic meters, and $a=1.00 \mathrm{~m}^{3}$. What are the final (a) pressure and (b) temperature of the gas? (c) How much work is done by the gas during the expansion? (d) What is $\Delta S$ for the expansion? (Hint: Use two simple reversible processes to find $\Delta S .$ )

Vipender Yadav
Vipender Yadav
Numerade Educator
07:44

Problem 21

Energy can be removed from water as heat at and even below the normal freezing point $\left(0.0^{\circ} \mathrm{C}\right.$ at atmospheric pressure) without causing the water to freeze; the water is then said to be supercooled. Suppose a $1.00 \mathrm{~g}$ water drop is supercooled until its temperature is that of the surrounding air, which is at $-5.00^{\circ} \mathrm{C}$. The drop then suddenly and irreversibly freezes, transferring energy to the air as heat. What is the entropy change for the drop? (Hint: Use a three-step reversible process as if the water were taken through the normal freezing point.) The specific heat of ice is $2220 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}$.

Eduard Sanchez
Eduard Sanchez
Numerade Educator
11:13

Problem 22

An insulated Thermos contains $130 \mathrm{~g}$ of water at $80.0^{\circ} \mathrm{C}$. You put in a $12.0 \mathrm{~g}$ ice cube at $0^{\circ} \mathrm{C}$ to form a system of ice $+$ original water. (a) What is the equilibrium temperature of the system? What are the entropy changes of the water that was originally the ice cube (b) as it melts and (c) as it warms to the equilibrium temperature? (d) What is the entropy change of the original water as it cools to the equilibrium temperature? (e) What is the net entropy change of the ice $+$ original water system as it reaches the equilibrium temperature?

Vipender Yadav
Vipender Yadav
Numerade Educator
02:30

Problem 23

A Carnot engine whose low-temperature reservoir is at $17^{\circ} \mathrm{C}$ has an efficiency of $40 \%$. By how much should the temperature of the high-temperature reservoir be increased to increase the efficiency to $50 \%$ ?

Vipender Yadav
Vipender Yadav
Numerade Educator
01:38

Problem 24

A Carnot engine absorbs $52 \mathrm{~kJ}$ as heat and exhausts $36 \mathrm{~kJ}$ as heat in each cycle. Calculate (a) the engine's efficiency and (b) the work done per cycle in kilojoules.

Vipender Yadav
Vipender Yadav
Numerade Educator
02:27

Problem 25

A Carnot engine has an efficiency of $22.0 \%$. It operates between constant-temperature reservoirs differing in temperature by $75.0 \mathrm{C}^{\circ} .$ What is the temperature of the (a) lower-temperature and
(b) higher-temperature reservoir?

Vipender Yadav
Vipender Yadav
Numerade Educator
01:27

Problem 26

In a hypothetical nuclear fusion reactor, the fuel is deuterium gas at a temperature of $7 \times 10^{8} \mathrm{~K}$. If this gas could be used to operate a Carnot engine with $T_{\mathrm{L}}=100^{\circ} \mathrm{C}$, what would be the engine's efficiency? Take both temperatures to be exact and report your answer to seven significant figures.

Vipender Yadav
Vipender Yadav
Numerade Educator
02:34

Problem 27

A Carnot engine operates between $235^{\circ} \mathrm{C}$ and $115^{\circ} \mathrm{C}$, absorbing $6.30 \times 10^{4} \mathrm{~J}$ per cycle at the higher temperature. (a) What is the efficiency of the engine? (b) How much work per cycle is this engine capable of performing?

Vipender Yadav
Vipender Yadav
Numerade Educator
03:52

Problem 28

In the first stage of a two-stage Carnot engine, energy is absorbed as heat $Q_{1}$ at temperature $T_{1}$, work $W_{1}$ is done, and energy is expelled as heat $Q_{2}$ at a lower temperature $T_{2}$. The second stage absorbs that energy as heat $Q_{2}$, does work $W_{2}$, and expels energy as heat $Q_{3}$ at a still lower temperature $T_{3}$. Prove that the efficiency of the engine is $\left(T_{1}-T_{3}\right) / T_{1}$.

Vipender Yadav
Vipender Yadav
Numerade Educator
11:20

Problem 29

Figure $20-27$ shows a reversible cycle through which $1.00 \mathrm{~mol}$ of a monatomic ideal gas is taken. Assume that $p=2 p_{0}, V=2 V_{0}, p_{0}=$ $1.01 \times 10^{5} \mathrm{~Pa}$, and $V_{0}=0.0225 \mathrm{~m}^{3} .$ Calculate (a) the work done during the cycle, (b) the energy added as heat during stroke $a b c$, and $(\mathrm{c})$ the efficiency of the cycle. (d) What is the efficiency of a Carnot engine operating between the highest and lowest temperatures that occur in the cycle? (e) Is this greater than or less than the efficiency calculated in (c)?

Vipender Yadav
Vipender Yadav
Numerade Educator
03:44

Problem 30

A $500 \mathrm{~W}$ Carnot engine operates between constanttemperature reservoirs at $100^{\circ} \mathrm{C}$ and $60.0^{\circ} \mathrm{C}$. What is the rate at which energy is (a) taken in by the engine as heat and (b) exhausted by the engine as heat?

Vipender Yadav
Vipender Yadav
Numerade Educator
03:30

Problem 31

The efficiency of a particular car engine is $25 \%$ when the engine does $8.2 \mathrm{~kJ}$ of work per cycle. Assume the process is reversible. What are (a) the energy the engine gains per cycle as heat $Q_{\text {gain }}$ from the fuel combustion and (b) the energy the engine loses per cycle as heat $Q_{\text {lost }} ?$ If a tune-up increases the efficiency to $31 \%$, what are (c) $Q_{\text {gain }}$ and (d) $Q_{\text {lost }}$ at the same work value?

Vipender Yadav
Vipender Yadav
Numerade Educator
03:39

Problem 32

A Carnot engine is set up to produce a certain work $W$ per cycle. In each cycle, energy in the form of heat $Q_{\mathrm{H}}$ is transferred to the working substance of the engine from the higher-temperature thermal reservoir, which is at an adjustable temperature $T_{\mathrm{H}}$. The lower-temperature thermal reservoir is maintained at temperature $T_{\mathrm{L}}=250 \mathrm{~K}$. Figure $20-28$ gives $Q_{\mathrm{H}}$ for a range of $T_{\mathrm{H}}$. The scale of the vertical axis is set by $Q_{\mathrm{Hs}}=6.0 \mathrm{~kJ}$. If $T_{\mathrm{H}}$ is set at $550 \mathrm{~K}$, what is $Q_{\mathrm{H}}$ ?

Vipender Yadav
Vipender Yadav
Numerade Educator
12:08

Problem 33

Figure $20-29$ shows a reversible cycle through which $1.00$ mol of a monatomic ideal gas is taken. Volume $V_{c}=8.00 V_{b} .$ Process $b c$ is an adiabatic expansion, with $p_{b}$ $=10.0 \mathrm{~atm}$ and $V_{b}=1.00 \times 10^{-3} \mathrm{~m}^{3} .$ For the cycle, find (a) the energy added to the gas as heat, (b) the energy leaving the gas as heat, (c) the net work done by the gas, and (d) the efficiency of the cycle.

Vipender Yadav
Vipender Yadav
Numerade Educator
10:08

Problem 34

An ideal gas $(1.0 \mathrm{~mol})$ is the working substance in an engine that operates on the cycle shown in Fig. 20-30. Processes $B C$ and $D A$ are reversible and adiabatic. (a) Is the gas monatomic, diatomic, or polyatomic? (b) What is the engine efficiency?

Vipender Yadav
Vipender Yadav
Numerade Educator
11:15

Problem 35

The cycle in Fig. 20-31 represents the operation of a gasoline internal combustion engine. Volume $V_{3}=4.00 V_{1} . \quad$ Assume the gasoline-air intake mixture is an ideal gas with $\gamma=1.30$. What are the ratios (a) $T_{2} / T_{1}$, (b) $T_{3} / T_{1}$, (c) $T_{4} / T_{1}$, (d) $p_{3} / p_{1}$, and (e) $p_{4} / p_{1}$ ? (f) What is the engine efficiency?

Keshav Singh
Keshav Singh
Numerade Educator
04:10

Problem 36

How much work must be done by a Carnot refrigerator to transfer $1.0$ J as heat (a) from a reservoir at $7.0^{\circ} \mathrm{C}$ to one at $27^{\circ} \mathrm{C},(\mathrm{b})$ from a reservoir at $-73^{\circ} \mathrm{C}$ to one at $27^{\circ} \mathrm{C},(\mathrm{c})$ from a reservoir at $-173^{\circ} \mathrm{C}$ to one at $27^{\circ} \mathrm{C}$, and $(\mathrm{d})$ from a reservoir at $-223^{\circ} \mathrm{C}$ to one at $27^{\circ} \mathrm{C}$ ?

Vipender Yadav
Vipender Yadav
Numerade Educator
01:59

Problem 37

A heat pump is used to heat a building. The external temperature is less than the internal temperature. The pump's coefficient of performance is $3.8$, and the heat pump delivers $7.54$ MJ as heat to the building each hour. If the heat pump is a Carnot engine working in reverse, at what rate must work be done to run it?

Vipender Yadav
Vipender Yadav
Numerade Educator
02:38

Problem 38

The electric motor of a heat pump transfers energy as heat from the outdoors, which is at $-5.0^{\circ} \mathrm{C}$, to a room that is at $17^{\circ} \mathrm{C}$. If the heat pump were a Carnot heat pump (a Carnot engine working in reverse), how much energy would be transferred as heat to the room for each joule of electric energy consumed?

Vipender Yadav
Vipender Yadav
Numerade Educator
02:18

Problem 39

A Carnot air conditioner takes energy from the thermal energy of a room at $70^{\circ} \mathrm{F}$ and transfers it as heat to the outdoors, which is at $96^{\circ} \mathrm{F}$. For each joule of electric energy required to operate the air conditioner, how many joules are removed from the room?

Vipender Yadav
Vipender Yadav
Numerade Educator
02:24

Problem 40

To make ice, a freezer that is a reverse Carnot engine extracts $42 \mathrm{~kJ}$ as heat at $-15^{\circ} \mathrm{C}$ during each cycle, with coefficient of performance $5.7 .$ The room temperature is $30.3^{\circ} \mathrm{C}$. How much (a) energy per cycle is delivered as heat to the room and (b) work per cycle is required to run the freezer?

Vipender Yadav
Vipender Yadav
Numerade Educator
02:59

Problem 41

An air conditioner operating between $93^{\circ} \mathrm{F}$ and $70^{\circ} \mathrm{F}$ is rated at 4000 Btu/h cooling capacity. Its coefficient of performance is $27 \%$ of that of a Carnot refrigerator operating between the same two temperatures. What horsepower is required of the air conditioner motor?

Vipender Yadav
Vipender Yadav
Numerade Educator
02:44

Problem 42

The motor in a refrigerator has a power of $200 \mathrm{~W}$. If the freezing compartment is at $270 \mathrm{~K}$ and the outside air is at $300 \mathrm{~K}$, and assuming the efficiency of a Carnot refrigerator, what is the maximum amount of energy that can be extracted as heat from the freezing compartment in $10.0 \mathrm{~min} ?$

Vipender Yadav
Vipender Yadav
Numerade Educator
04:43

Problem 43

Figure $20-32$ represents a Carnot engine that works between temperatures $T_{1}=400 \mathrm{~K}$ and $T_{2}=150 \mathrm{~K}$ and drives a Carnot refrigerator that works between temperatures $T_{3}=325 \mathrm{~K}$ and $T_{4}=$ $225 \mathrm{~K}$. What is the ratio $Q_{3} / Q_{1} ?$

Vipender Yadav
Vipender Yadav
Numerade Educator
03:57

Problem 44

(a) During each cycle, a Carnot engine absorbs $750 \mathrm{~J}$ as heat from a high-temperature reservoir at $360 \mathrm{~K}$, with the low-temperature reservoir at $280 \mathrm{~K}$. How much work is done per cycle? (b) The engine is then made to work in reverse to function as a Carnot refrigerator between those same two reservoirs. During each cycle, how much work is required to remove $1200 \mathrm{~J}$ as heat from the low-temperature reservoir?

Vipender Yadav
Vipender Yadav
Numerade Educator
07:08

Problem 45

Construct a table like Table $20-1$ for eight molecules.

Vipender Yadav
Vipender Yadav
Numerade Educator
07:59

Problem 46

A box contains $N$ identical gas molecules equally divided between its two halves. For $N=50$, what are (a) the multiplicity $W$ of the central configuration, (b) the total number of microstates, and (c) the percentage of the time the system spends in the central configuration? For $N=100$, what are (d) $W$ of the central configuration, (e) the total number of microstates, and (f) the percentage of the time the system spends in the central configuration? For $N=200$, what are (g) $W$ of the central configuration, (h) the total number of microstates, and (i) the percentage of the time the system spends in the central configuration? (j) Does the time spent in the central configuration increase or decrease with an increase in $N$ ?

Averell Hause
Averell Hause
Carnegie Mellon University
04:12

Problem 47

A box contains $N$ gas molecules. Consider the box to be divided into three equal parts. (a) By extension of Eq. $20-20$, write a formula for the multiplicity of any given configuration. (b) Consider two configurations: configuration $A$ with equal numbers of molecules in all three thirds of the box, and configuration $B$ with equal numbers of molecules in each half of the box divided into two equal parts rather than three. What is the ratio $W_{A} / W_{B}$ of the multiplicity of configuration $A$ to that of configuration $B ?(\mathrm{c})$ Evaluate $W_{A} / W_{B}$ for $N=100$. (Because 100 is not evenly divisible by 3, put 34 molecules into one of the three box parts of configuration $A$ and 33 in each of the other two parts.)

Keshav Singh
Keshav Singh
Numerade Educator
03:15

Problem 48

Four particles are in the insulated box of Fig. 20-17. What are (a) the least multiplicity, (b) the greatest multiplicity,
(c) the least entropy, and
(d) the greatest entropy of the four-particle system?

Eduard Sanchez
Eduard Sanchez
Numerade Educator
06:20

Problem 49

A cylindrical copper rod of length $1.50 \mathrm{~m}$ and radius $2.00 \mathrm{~cm}$ is insulated to prevent heat loss through its curved surface. One end is attached to a thermal reservoir fixed at $300^{\circ} \mathrm{C} ;$ the other is attached to a thermal reservoir fixed at $30.0^{\circ} \mathrm{C}$. What is the rate at which entropy increases for the rod-reservoirs system?

Eduard Sanchez
Eduard Sanchez
Numerade Educator
04:43

Problem 50

Suppose $0.550 \mathrm{~mol}$ of an ideal gas is isothermally and reversibly expanded in the four situations given below. What is the change in the entropy of the gas for each situation?
$$
\begin{array}{lcccl}
\hline \text { Situation } & \text { (a) } & \text { (b) } & \text { (c) } & \text { (d) } \\
\hline \text { Temperature (K) } & 250 & 350 & 400 & 450 \\
\text { Initial volume }\left(\mathrm{cm}^{3}\right) & 0.200 & 0.200 & 0.300 & 0.300 \\
\text { Final volume }\left(\mathrm{cm}^{3}\right) & 0.800 & 0.800 & 1.20 & 1.20 \\
\hline
\end{array}
$$

Eduard Sanchez
Eduard Sanchez
Numerade Educator
05:23

Problem 51

As a sample of nitrogen gas $\left(\mathrm{N}_{2}\right)$ undergoes a temperature increase at constant volume, the distribution of molecular speeds increases. That is, the probability distribution function $P(v)$ for the molecules spreads to higher speed values, as suggested in Fig. $19-8 b$. One way to report the spread in $P(v)$ is to measure the difference $\Delta v$ between the most probable speed $v_{P}$ and the rms speed $v_{\mathrm{rms}} .$ When $P(v)$ spreads to higher speeds, $\Delta v$ increases. Assume that the gas is ideal and the $\mathrm{N}_{2}$ molecules rotate but do not oscillate. For $1.5 \mathrm{~mol}$, an initial temperature of $250 \mathrm{~K}$, and a final temperature of $500 \mathrm{~K}$, what are (a) the initial difference $\Delta v_{i},(\mathrm{~b})$ the final difference $\Delta v_{f}$, and $(\mathrm{c})$ the entropy change $\Delta S$ for the gas?

Keshav Singh
Keshav Singh
Numerade Educator
05:47

Problem 52

Suppose $1.0 \mathrm{~mol}$ of a monatomic ideal gas initially at $10 \mathrm{~L}$ and $300 \mathrm{~K}$ is heated at constant volume to $600 \mathrm{~K}$, allowed to expand isothermally to its initial pressure, and finally compressed at constant pressure to its original volume, pressure, and temperature. During the cycle, what are (a) the net energy entering the system (the gas) as heat and (b) the net work done by the gas? (c) What is the efficiency of the cycle?

Keshav Singh
Keshav Singh
Numerade Educator
06:35

Problem 53

Suppose that a deep shaft were drilled in Earth's crust near one of the poles, where the surface temperature is $-40^{\circ} \mathrm{C}$, to a depth where the temperature is $800^{\circ} \mathrm{C}$. (a) What is the theoretical limit to the efficiency of an engine operating between these temperatures? (b) If all the energy released as heat into the lowtemperature reservoir were used to melt ice that was initially at $-40^{\circ} \mathrm{C}$, at what rate could liquid water at $0^{\circ} \mathrm{C}$ be produced by a 100 MW power plant (treat it as an engine)? The specific heat of ice is $2220 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K} ;$ water's heat of fusion is $333 \mathrm{~kJ} / \mathrm{kg}$. (Note that the engine can operate only between $0^{\circ} \mathrm{C}$ and $800^{\circ} \mathrm{C}$ in this case. Energy exhausted at $-40^{\circ} \mathrm{C}$ cannot warm anything above $-40^{\circ} \mathrm{C}$.)

Keshav Singh
Keshav Singh
Numerade Educator
03:27

Problem 54

What is the entropy change for $3.20 \mathrm{~mol}$ of an ideal monatomic gas undergoing a reversible increase in temperature from $380 \mathrm{~K}$ to $425 \mathrm{~K}$ at constant volume?

Eduard Sanchez
Eduard Sanchez
Numerade Educator
03:34

Problem 55

A $600 \mathrm{~g}$ lump of copper at $80.0^{\circ} \mathrm{C}$ is placed in $70.0 \mathrm{~g}$ of water at $10.0^{\circ} \mathrm{C}$ in an insulated container. (See Table $18-3$ for specific heats.)
(a) What is the equilibrium temperature of the copperwater system? What entropy changes do (b) the copper, (c) the water, and (d) the copper-water system undergo in reaching the equilibrium temperature?

Keshav Singh
Keshav Singh
Numerade Educator
04:10

Problem 56

Figure $20-33$ gives the force magnitude $F$ versus stretch distance $x$ for a rubber band, with the scale of the $F$ axis set by $F_{s}=1.50 \mathrm{~N}$ and the scale of the $x$ axis set by $x_{s}=3.50 \mathrm{~cm}$. The temperature is $2.00^{\circ} \mathrm{C}$. When the rubber band is stretched by $x=1.70 \mathrm{~cm}$, at what rate does the entropy of the rubber band change during a small additional stretch?

Eduard Sanchez
Eduard Sanchez
Numerade Educator
03:43

Problem 57

The temperature of $1.00 \mathrm{~mol}$ of a monatomic ideal gas is raised reversibly from $300 \mathrm{~K}$ to $400 \mathrm{~K}$, with its volume kept constant. What is the entropy change of the gas?

Eduard Sanchez
Eduard Sanchez
Numerade Educator
03:53

Problem 58

Repeat Problem 57 , with the pressure now kept constant.

Eduard Sanchez
Eduard Sanchez
Numerade Educator
03:43

Problem 59

A $0.600 \mathrm{~kg}$ sample of water is initially ice at temperature $-20^{\circ} \mathrm{C}$. What is the sample's entropy change if its temperature is increased to $40^{\circ} \mathrm{C}$ ?

Keshav Singh
Keshav Singh
Numerade Educator
02:48

Problem 60

A three-step cycle is undergone by $3.4$ mol of an ideal diatomic gas: (1) the temperature of the gas is increased from $200 \mathrm{~K}$ to $500 \mathrm{~K}$ at constant volume; (2) the gas is then isothermally expanded to its original pressure; (3) the gas is then contracted at constant pressure back to its original volume. Throughout the cycle, the molecules rotate but do not oscillate. What is the efficiency of the cycle?

Keshav Singh
Keshav Singh
Numerade Educator
01:14

Problem 61

An inventor has built an engine $X$ and claims that its efficiency $\varepsilon_{\mathrm{X}}$ is greater than the efficiency $\varepsilon$ of an ideal engine operating between the same two temperatures. Suppose you couple engine $\mathrm{X}$ to an ideal refrigerator (Fig. $20-34 a$ ) and adjust the cycle of engine $\mathrm{X}$ so that the work per cycle it provides equals the work per cycle required by the ideal refrigerator. Treat this combination as a single unit and show that if the inventor's claim were true (if $\left.\varepsilon_{\mathrm{X}}>\varepsilon\right)$, the combined unit would act as a perfect refrigerator (Fig. 20-34b), transferring energy as heat from the low-temperature reservoir to the high-temperature reservoir without the need for work.

Keshav Singh
Keshav Singh
Numerade Educator
05:47

Problem 62

Suppose $2.00 \mathrm{~mol}$ of a diatomic gas is taken reversibly around the cycle shown in the $T$ $S$ diagram of Fig. $20-35$, where $S_{1}=6.00 \mathrm{~J} / \mathrm{K}$ and $S_{2}=8.00 \mathrm{~J} / \mathrm{K}$ The molecules do not rotate or oscillate. What is the energy transferred as heat $Q$ for (a) path $1 \rightarrow 2,(\mathrm{~b})$ path $2 \rightarrow 3$, and $(\mathrm{c})$ the full cycle? (d) What is the work $W$ for the isothermal process? The volume $V_{1}$ in state 1 is $0.200 \mathrm{~m}^{3}$. What is the volume in
(e) state 2 and $(\mathrm{f})$ state $3 ?$What is the change $\Delta E_{\text {int }}$ for $(\mathrm{g})$ path $1 \rightarrow 2,(\mathrm{~h})$ path $2 \rightarrow 3$, and (i) the full cycle? (Hint: (h) can be done with one or two lines of calculation using Module $19-7$ or with a page of calculation using Module 19-9.) (j) What is the work $W$ for the adiabatic process?

Keshav Singh
Keshav Singh
Numerade Educator
03:09

Problem 63

A three-step cycle is undergone reversibly by $4.00$ mol of an ideal gas: (1) an adiabatic expansion that gives the gas $2.00$ times its initial volume, (2) a constant-volume process, (3) an isothermal compression back to the initial state of the gas. We do not know whether the gas is monatomic or diatomic; if it is diatomic, we do not know whether the molecules are rotating or oscillating. What are the entropy changes for
(a) the cycle, (b) process $1,(\mathrm{c})$ process 3, and $(\mathrm{d})$ process $2 ?$

Keshav Singh
Keshav Singh
Numerade Educator
03:32

Problem 64

(a) A Carnot engine operates between a hot reservoir at $320 \mathrm{~K}$ and a cold one at $260 \mathrm{~K}$. If the engine absorbs $500 \mathrm{~J}$ as heat per cycle at the hot reservoir, how much work per cycle does it deliver? (b) If the engine working in reverse functions as a refrigerator between the same two reservoirs, how much work per cycle must be supplied to remove $1000 \mathrm{~J}$ as heat from the cold reservoir?

Eduard Sanchez
Eduard Sanchez
Numerade Educator
03:21

Problem 65

A $2.00 \mathrm{~mol}$ diatomic gas initially at $300 \mathrm{~K}$ undergoes this cycle: It is (1) heated at constant volume to $800 \mathrm{~K},(2)$ then allowed to expand isothermally to its initial pressure, $(3)$ then compressed at constant pressure to its initial state. Assuming the gas molecules neither rotate nor oscillate, find (a) the net energy transferred as heat to the gas, (b) the net work done by the gas, and (c) the efficiency of the cycle.

Keshav Singh
Keshav Singh
Numerade Educator
02:07

Problem 66

An ideal refrigerator does $150 \mathrm{~J}$ of work to remove $560 \mathrm{~J}$ as heat from its cold compartment. (a) What is the refrigerator's coefficient of performance? (b) How much heat per cycle is exhausted to the kitchen?

Eduard Sanchez
Eduard Sanchez
Numerade Educator
05:14

Problem 67

Suppose that $260 \mathrm{~J}$ is conducted from a constant-temperature reservoir at $400 \mathrm{~K}$ to one at (a) $100 \mathrm{~K}$, (b) $200 \mathrm{~K}$, (c) $300 \mathrm{~K}$, and (d) $360 \mathrm{~K}$. What is the net change in entropy $\Delta S_{\text {net }}$ of the reservoirs in each case? (e) As the temperature difference of the two reservoirs decreases, does $\Delta S_{\text {net }}$ increase, decrease, or remain the same?

Eduard Sanchez
Eduard Sanchez
Numerade Educator
01:27

Problem 68

An apparatus that liquefies helium is in a room maintained a $300 \mathrm{~K}$. If the helium in the apparatus is at $4.0 \mathrm{~K}$, what is the minimum ratio $Q_{\mathrm{to}} / Q_{\text {from }}$, where $Q_{\mathrm{to}}$ is the energy delivered as heat to the room and $Q_{\text {from }}$ is the energy removed as heat from the helium?

Vipender Yadav
Vipender Yadav
Numerade Educator
03:35

Problem 69

A brass rod is in thermal contact with a constant-temperature reservoir at $130^{\circ} \mathrm{C}$ at one end and a constant-temperature reservoir at $24.0^{\circ} \mathrm{C}$ at the other end. (a) Compute the total change in entropy of the rod-reservoirs system when $5030 \mathrm{~J}$ of energy is conducted through the rod, from one reservoir to the other. (b) Does the entropy of the rod change?

Eduard Sanchez
Eduard Sanchez
Numerade Educator
03:41

Problem 70

A $45.0 \mathrm{~g}$ block of tungsten at $30.0^{\circ} \mathrm{C}$ and a $25.0 \mathrm{~g}$ block of silver at $-120^{\circ} \mathrm{C}$ are placed together in an insulated container. (See Table 18-3 for specific heats.) (a) What is the equilibrium temperature? What entropy changes do (b) the tungsten, (c) the silver, and (d) the tungsten-silver system undergo in reaching the equilibrium temperature?

Keshav Singh
Keshav Singh
Numerade Educator
05:47

Problem 71

A $45.0 \mathrm{~g}$ block of tungsten at $30.0^{\circ} \mathrm{C}$ and a $25.0 \mathrm{~g}$ block of silver at $-120^{\circ} \mathrm{C}$ are placed together in an insulated container. (See Table 18-3 for specific heats.) (a) What is the equilibrium temperature? What entropy changes do (b) the tungsten, (c) the silver, and (d) the tungsten-silver system undergo in reaching the equilibrium temperature?

Keshav Singh
Keshav Singh
Numerade Educator
04:02

Problem 72

Calculate the efficiency of a fossil-fuel power plant that consumes 380 metric tons of coal each hour to produce useful work at the rate of $750 \mathrm{MW}$. The heat of combustion of coal (the heat due to burning it) is $28 \mathrm{MJ} / \mathrm{kg}$.

Eduard Sanchez
Eduard Sanchez
Numerade Educator
03:20

Problem 73

A Carnot refrigerator extracts $35.0 \mathrm{~kJ}$ as heat during each cycle, operating with a coefficient of performance of $4.60$. What are (a) the energy per cycle transferred as heat to the room and (b) the work done per cycle?

Keshav Singh
Keshav Singh
Numerade Educator
03:06

Problem 74

A Carnot engine whose high-temperature reservoir is at 400 $\mathrm{K}$ has an efficiency of $30.0 \%$. By how much should the temperature of the low-temperature reservoir be changed to increase the efficiency to $40.0 \%$ ?

Eduard Sanchez
Eduard Sanchez
Numerade Educator
03:26

Problem 75

System $A$ of three particles and system $B$ of five particles are in insulated boxes like that in Fig. $20-17$. What is the least multiplicity $W$ of (a) system $A$ and (b) system $B ?$ What is the greatest multiplicity $W$ of (c) $A$ and (d) $B$ ? What is the greatest entropy of (e) $A$ and $(\mathrm{f}) B ?$

Keshav Singh
Keshav Singh
Numerade Educator
04:54

Problem 76

Figure $20-36$ shows a Carnot cycle on a $T-S$ diagram, with a scale set by $S_{s}=0.60 \mathrm{~J} / \mathrm{K} .$ For a full cycle, find (a) the net heat transfer and (b) the net work done by the system.

Eduard Sanchez
Eduard Sanchez
Numerade Educator
02:26

Problem 77

Find the relation between the efficiency of a reversible ideal heat engine and the coefficient of performance of the reversible refrigerator obtained by running the engine backwards.

Eduard Sanchez
Eduard Sanchez
Numerade Educator
02:47

Problem 78

A Carnot engine has a power of $500 \mathrm{~W}$. It operates between heat reservoirs at $100^{\circ} \mathrm{C}$ and $60.0^{\circ} \mathrm{C}$. Calculate (a) the rate of heat input and (b) the rate of exhaust heat output.

Keshav Singh
Keshav Singh
Numerade Educator
02:10

Problem 79

In a real refrigerator, the low-temperature coils are at $-13^{\circ} \mathrm{C}$, and the compressed gas in the condenser is at $26^{\circ} \mathrm{C}$. What is the theoretical coefficient of performance?

Eduard Sanchez
Eduard Sanchez
Numerade Educator