• Home
  • Textbooks
  • Principles of Physics
  • Entropy and the Second Law of Thermodynamics

Principles of Physics

David Halliday , Robert Resnick , Jearl Walker

Chapter 20

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

Educators

Chapter Questions

02:27

Problem 1

A Carnot engine has an efficiency of $15.0 \%$. It operates between constant-temperature reservoirs differing in temperature by $55.0 \mathrm{C}^{\circ}$ What is the temperature of the lower-temperature reservoir?

Vipender Yadav
Vipender Yadav
Numerade Educator
08:14

Problem 2

Expand $1.00 \mathrm{~mol}$ of an monatomic gas initially at $8.00 \mathrm{kPa}$ 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
05:35

Problem 3

Figure 20-19 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 $V_0=0.0335 \mathrm{~m}^3$. Calculate
(a) the work done during the cycle,
(b) the energy added as heat during stroke $a b c$, and (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)?

Keshav Singh
Keshav Singh
Numerade Educator
02:44

Problem 4

The motor in a refrigerator has a power of $200 \mathrm{~W}$. If the freezing compartment is at $260 \mathrm{~K}$ 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 $15.0 \mathrm{~min}$ ?

Vipender Yadav
Vipender Yadav
Numerade Educator
02:30

Problem 5

A Carnot engine whose high-temperature reservoir is at $483 \mathrm{~K}$ has an efficiency of $40 \%$. By how much should the temperature of the low-temperature reservoir be decreased to increase the efficiency to $50 \%$ ?

Vipender Yadav
Vipender Yadav
Numerade Educator
06:38

Problem 6

The electric motor of a heat pump transfers energy as heat from the outdoors, which is at $-10^{\circ} \mathrm{C}$, 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?
a configuration of the system. The number of microstates in a configuration is the multiplicity $W$ of the configuration.

For a system of $N$ molecules that may be distributed between the two halves of a box, the multiplicity is given by
$$
W=\frac{N !}{n_{1} ! n_{2} !},
$$
in which $n_1$ is the number of molecules in one half of the box and $n_2$ is the number in the other half. A basic assumption of statistical mechanics is that all the microstates are equally probable. Thus, configurations with a large multiplicity occur most often.

The multiplicity $W$ of a configuration of a system and the entropy $S$ of the system in that configuration are related by Boltzmann's entropy equation:
$$
S=k \ln W,
$$
where $k=1.38 \times 10^{-23} \mathrm{~J} / \mathrm{K}$ is the Boltzmann constant.

Ravindra Yadav
Ravindra Yadav
Numerade Educator
03:30

Problem 7

The efficiency of a particular car engine is $20 \%$ 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
11:13

Problem 8

An insulated Thermos contains $150 \mathrm{~g}$ 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
02:18

Problem 9

A Carnot air conditioner is keeping a room cool at a temperature of $65^{\circ} \mathrm{F}$ by transferring energy as heat to the outdoors, which is at a temperature of $101^{\circ} \mathrm{F}$. For each watt of electric energy powering the air conditioner, what is the rate at which energy is removed from the room?

Vipender Yadav
Vipender Yadav
Numerade Educator
01:27

Problem 10

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 $T_{\mathrm{L}}=50^{\circ} \mathrm{C}$, what would be the engine's efficiency? Take both temperatures to be exact and report your answer to eight significant figures.

Vipender Yadav
Vipender Yadav
Numerade Educator
04:43

Problem 11

Figure 20-20 represents a Carnot engine that works between temperatures $T_1=500 \mathrm{~K}$ and $T_2=130 \mathrm{~K}$ and drives a Carnot refrigerator that works between temperatures $T_3=360 \mathrm{~K}$ and $T_4=200 \mathrm{~K}$. What is the ratio $Q_3 / Q_1$ ?

Vipender Yadav
Vipender Yadav
Numerade Educator
04:19

Problem 12

A $2.5 \mathrm{~mol}$ sample of an ideal monatomic gas undergoes the reversible process shown in Fig. 20-21. 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
04:18

Problem 13

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 $B$ ? (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
02:24

Problem 14

An ice-cream maker is kept cold by a reverse Carnot engine that removes $28.0 \mathrm{~kJ}$ as heat per cycle, with coefficient of performance 6.90. Per cycle, what are (a) the energy delivered as heat to the room and (b) the work done?

Vipender Yadav
Vipender Yadav
Numerade Educator
02:57

Problem 15

Construct a table like Table 20-1 for four molecules and give the entropy for (a) the first configuration, (b) the second configuration, and (c) the third configuration.

Sonia Tripathy
Sonia Tripathy
Numerade Educator
07:59

Problem 16

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
06:19

Problem 17

A $50.0 \mathrm{~g}$ block of copper whose temperature is $200 \mathrm{~K}$ is placed in an insulating box with a $100 \mathrm{~g}$ block of lead whose temperature is $400 \mathrm{~K}$. (a) What is the equilibrium temperature of the two-block 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
10:08

Problem 18

An ideal gas $(3.0 \mathrm{~mol})$ is the working substance in an engine that operates on the cycle shown in Fig. 20-22. 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
03:57

Problem 19

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 $1.60 \mathrm{~g}$ water drop is supercooled until its temperature is that of the surrounding air, which is at $-10.0^{\circ} \mathrm{C}$. 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 $2220 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}$.

Keshav Singh
Keshav Singh
Numerade Educator
03:57

Problem 20

(a) During each cycle, a Carnot engine absorbs $730 \mathrm{~J}$ as heat from a high-temperature reservoir at $360 \mathrm{~K}$, with the lowtemperature 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
10:52

Problem 21

In Fig. 20-23, where $V_{23}=4.00 V_1, n$ 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) $Q / n R T_1$, (f) $\Delta E_{\text {int }} / n R T_1$, and (g) $\Delta S / n R$ ? For path $2 \rightarrow 3$, what are (h) $W / n R T_1$, (i) $Q / n R T_1$, (j) $\Delta E_{\text {int }} / n R T_1$, (k) $\Delta S / n R$ ? For path $3 \rightarrow 1$, what are (l) $W / n R T_1$, (m) $Q / n R T_1$, (n) $\Delta E_{\text {int }} / n R T_1$, and (o) $\Delta S / n R ?$

Keshav Singh
Keshav Singh
Numerade Educator
04:10

Problem 22

How much work must be done by a Carnot refrigerator to transfer $3.5 \mathrm{~J}$ as heat (a) from a reservoir at $7.0^{\circ} \mathrm{C}$ to one at $35^{\circ} \mathrm{C}$, (b) from a reservoir at $-73^{\circ} \mathrm{C}$ to one at $35^{\circ} \mathrm{C}$, (c) from a reservoir at $-173^{\circ} \mathrm{C}$ to one at $35^{\circ} \mathrm{C}$, and (d) from a reservoir at $-223^{\circ} \mathrm{C}$ to one at $35^{\circ} \mathrm{C}$ ?

Vipender Yadav
Vipender Yadav
Numerade Educator
01:08

Problem 23

A $3.90 \mathrm{~mol}$ sample of an ideal gas expands reversibly and isothermally at $360 \mathrm{~K}$ until its volume is $3.40$ times larger. What is the increase in entropy of the gas?

Manish Kumar ( Iit K )
Manish Kumar ( Iit K )
Numerade Educator
10:08

Problem 24

(a) For $2.5 \mathrm{~mol}$ of 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
12:08

Problem 25

Figure 20-25 shows a reversible cycle through which $1.00 \mathrm{~mol}$ of a monatomic ideal gas is taken. Volume $V_c=8.00 V_b$. Process $b c$ is an adiabatic expansion, with $p_b=5.00 \mathrm{~atm}$ 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
02:07

Problem 26

A 600 W Carnot engine operates between constant-temperature 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?

Keshav Singh
Keshav Singh
Numerade Educator
02:59

Problem 27

An air conditioner operating between $93^{\circ} \mathrm{F}$ and $70^{\circ} \mathrm{F}$ is rated at $5200 \mathrm{Btu} / \mathrm{h}$ cooling capacity. Its coefficient of performance is $15 \%$ 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
01:10

Problem 28

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 decreased from $8.00 \mathrm{~K}$ to $5.00 \mathrm{~K}$.

Averell Hause
Averell Hause
Carnegie Mellon University
01:59

Problem 29

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 $3.30$, and the heat pump delivers $7.54 \mathrm{MJ}$ 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
03:39

Problem 30

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 $20-26$ gives $Q_{\mathrm{H}}$ for a range of $T_{\mathrm{H}}$. The scale of the vertical axis is set by $Q_{\mathrm{Hs}}=12.0 \mathrm{~kJ}$. If $T_{\mathrm{H}}$ is set at $550 \mathrm{~K}$, what is $Q_{\mathrm{H}}$ ?

Vipender Yadav
Vipender Yadav
Numerade Educator
02:46

Problem 31

Find (a) the energy absorbed as heat and (b) the change in entropy of a $3.30 \mathrm{~kg}$ block of copper whose temperature is increased reversibly from $25.0^{\circ} \mathrm{C}$ to $170^{\circ} \mathrm{C}$. The specific heat of copper is $386 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}$

Manish Kumar ( Iit K )
Manish Kumar ( Iit K )
Numerade Educator
02:19

Problem 32

A gas sample undergoes a reversible isothermal expansion. Figure 20-27 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 $\Delta S_s=128 \mathrm{~J} / \mathrm{K}$. How many moles are in the sample?

Vipender Yadav
Vipender Yadav
Numerade Educator
03:18

Problem 33

A mixture of $1773 \mathrm{~g}$ of water and $513 \mathrm{~g}$ 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?

Ma Ednelyn Lim
Ma Ednelyn Lim
Numerade Educator
01:17

Problem 34

An ideal gas undergoes a reversible isothermal expansion at $77.0^{\circ} \mathrm{C}$, increasing its volume from $1.30 \mathrm{~L}$ to $3.90 \mathrm{~L}$. 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
00:55

Problem 36

How much energy must be transferred as heat for a reversible isothermal expansion of an ideal gas at $180^{\circ} \mathrm{C}$ if the entropy of the gas increases by $46.0 \mathrm{~J} / \mathrm{K}$ ?

Manish Kumar ( Iit K )
Manish Kumar ( Iit K )
Numerade Educator
11:15

Problem 37

The cycle in Fig. 20-28 represents the operation of a gasoline internal combustion engine. Volume $V_3=4.00 V_1$. 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
01:38

Problem 38

A Carnot engine absorbs $52 \mathrm{~kJ}$ as heat and exhausts $30 \mathrm{~kJ}$ 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
03:54

Problem 39

In an experiment, $400 \mathrm{~g}$ 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
01:03

Problem 40

A two-stage Carnot engine consists of two Carnot cycles: In stage 1 , a Carnot cycle absorbs heat at temperature $T_1=500 \mathrm{~K}$ and discharges heat at temperature $T_2=400 \mathrm{~K}$. In stage 2, a Carnot cycle absorbs that discharged heat at $T_2$ and then discharges heat at temperature $T_3=300 \mathrm{~K}$. What is the efficiency of the engine?

Penny Riley
Penny Riley
Numerade Educator
09:00

Problem 41

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 $350 \mathrm{~J}$ 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 $L$, (b) its reservoir, (c) block $R$, (d) its reservoir, (e) the two-block system, and (f) the system of the two blocks and the two reservoirs?

Vipender Yadav
Vipender Yadav
Numerade Educator
02:15

Problem 42

A $270 \mathrm{~g}$ 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 20-29 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
02:34

Problem 43

A Carnot engine operates between $235^{\circ} \mathrm{C}$ and $115^{\circ} \mathrm{C}$, absorbing $3.00 \times 10^4 \mathrm{~J}$ per cycle at the higher temperature. (a) How much energy per cycle is discharged at the lower temperature? (b) How much work per cycle is this engine capable of performing?

Vipender Yadav
Vipender Yadav
Numerade Educator
05:56

Problem 44

An $6.0 \mathrm{~g}$ 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
02:38

Problem 45

Suppose $2.00 \mathrm{~mol}$ of an ideal gas undergoes a reversible isothermal compression from volume $V_1$ to volume $V_2=0.500 V_1$ 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 compression 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
02:23

Problem 46

(a) What is the entropy change of a $15.0 \mathrm{~g}$ 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
06:51

Problem 47

A $20 \mathrm{~g}$ ice cube at $-15^{\circ} \mathrm{C}$ 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?)

Keshav Singh
Keshav Singh
Numerade Educator