Physics

Robert Resnick, David Halliday, Kenneth S. Krane

Chapter 24

Entropy And The Second Law Of Thermodynamics - all with Video Answers

Educators

Chapter Questions

Problem 1

An ideal gas undergoes a reversible isothermal expansion at $132^{\circ} \mathrm{C}$. The entropy of the gas increases by $46.2 \mathrm{~J} / \mathrm{K}$. How much heat is absorbed?

Eduard Sanchez

Eduard Sanchez

Numerade Educator

Problem 2

. In Fig. $24-18$, suppose that the change in entropy of the system in passing from state $a$ to state $b$ along path 1 is $+0.60$ $\mathrm{J} / \mathrm{K}$. What is the entropy change in passing $(a)$ from state $a$ to $b$ along path 2 and $(b)$ from state $b$ to $a$ along path 2 ?

Eduard Sanchez

Numerade Educator

Problem 3

For the Carnot cycle shown in Fig. $24-19$, calculate $(a)$ the heat that enters and $(b)$ the work done on the system.

Eduard Sanchez

Numerade Educator

Problem 4

Four moles of an ideal gas are caused to expand from a volume $V_{1}$ to a volume $V_{2}=3.45 V_{1} .(a)$ If the expansion is isothermal at the temperature $T=410 \mathrm{~K}$, find the work done on the expanding gas. ( $b$ ) Find the change in entropy, if any. ( $c$ ) If the expansion is reversibly adiabatic instead of isothermal, what is the entropy change?

Eduard Sanchez

Numerade Educator

Problem 5

Find $(a)$ the heat absorbed and $(b)$ the change in entropy of a $1.22-\mathrm{kg}$ block of copper whose temperature is increased reversibly from $25.0$ to $105^{\circ} \mathrm{C}$

Eduard Sanchez

Numerade Educator

Problem 6

Heat can be transferred from water at $0^{\circ} \mathrm{C}$ and atmospheric pressure without causing the water to freeze, if done with little disturbance of the water. Suppose the water is cooled to $-5.0^{\circ} \mathrm{C}$ before ice begins to form. Find the change in entropy occurring during the sudden freezing of $1.0 \mathrm{~g}$ of water that then takes place.

Eduard Sanchez

Numerade Educator

Problem 7

An ideal gas undergoes an isothermal expansion at $77^{\circ} \mathrm{C}$ increasing its volume from $1.3$ to $3.4 \mathrm{~L}$. The entropy change of the gas is $24 \mathrm{~J} / \mathrm{K}$. How many moles of gas are present?

Eduard Sanchez

Numerade Educator

Problem 8

Suppose that the same amount of heat energy-say, $260 \mathrm{~J}-$ is transferred by conduction from a heat reservoir at a temperature of $400 \mathrm{~K}$ to another reservoir, the temperature of which is $(a) 100 \mathrm{~K}$, (b) $200 \mathrm{~K},(c) 300 \mathrm{~K}$, and $(d) 360 \mathrm{~K}$. Calculate the changes in entropy and discuss the trend.

Eduard Sanchez

Numerade Educator

Problem 9

A brass rod is in thermal contact with a heat reservoir at $130^{\circ} \mathrm{C}$ at one end and a heat reservoir at $24.0^{\circ} \mathrm{C}$ at the other end. ( $a$ ) Compute the total change in the entropy arising from the process of conduction of $1200 \mathrm{~J}$ of heat through the rod.
(b) Does the entropy of the rod change in the process?

Eduard Sanchez

Numerade Educator

Problem 10

A $50.0-\mathrm{g}$ block of copper having a temperature of $400 \mathrm{~K}$ is placed in an insulating box with a $100-\mathrm{g}$ block of lead having a temperature of $200 \mathrm{~K}$. ( $a$ ) What is the equilibrium temperature of this two-block system? ( $b$ ) What is the change in the internal energy of the two-block system as it changes from the initial condition to the equilibrium condition? ( $c$ ) What is the change in the entropy of the two-block system? (See Table $23-2 .)$

Eduard Sanchez

Numerade Educator

Problem 11

A mixture of $1.78 \mathrm{~kg}$ of water and $262 \mathrm{~g}$ of ice at $0{ }^{\circ} \mathrm{C}$ is, in a reversible process, brought to a final equilibrium state where the water/ice ratio, by mass, is $1: 1$ at $0^{\circ} \mathrm{C} .(a)$ Calculate the entropy change of the system during this process. (b) The system is then returned to the first equilibrium state, but in an irreversible way (by using a Bunsen burner, for instance). Calculate the entropy change of the system during this process.
(c) Show that your answer is consistent with the second law of thermodynamics.

Eduard Sanchez

Numerade Educator

Problem 12

In a specific heat experiment, $196 \mathrm{~g}$ of aluminum at $107^{\circ} \mathrm{C}$ is mixed with $52.3 \mathrm{~g}$ of water at $18.6^{\circ} \mathrm{C}$. ( $a$ ) Calculate the equilibrium temperature. Find the entropy change of $(b)$ the aluminum and $(c)$ the water. $(d)$ Calculate the entropy change of the system. (Hint: See Sample Problem 24-3.)

Eduard Sanchez

Numerade Educator

Problem 13

A heat engine absorbs $52.4 \mathrm{~kJ}$ of heat and exhausts $36.2 \mathrm{~kJ}$ of heat each cycle. Calculate $(a)$ the efficiency and $(b)$ the work done by the engine per cycle.

Eduard Sanchez

Numerade Educator

Problem 14

A car engine delivers $8.18 \mathrm{~kJ}$ of work per cycle. (a) Before a tune-up, the efficiency is $25.0 \% .$ Calculate, per cycle, the heat absorbed from the combustion of fuel and the heat exhausted to the atmosphere. (b) After a tune-up, the efficiency is $31.0 \%$. What are the new values of the quantities calculated in $(a) ?$

Eduard Sanchez

Numerade Educator

Problem 15

Calculate the efficiency of a fossil-fuel power plant that consumes 382 metric tons of coal each hour to produce useful work at the rate of $755 \mathrm{MW}$. The heat of combustion of coal is $28.0 \mathrm{MJ} / \mathrm{kg}$.

Eduard Sanchez

Numerade Educator

Problem 16

Engine $\mathrm{A}$, compared to engine $\mathrm{B}$, produces, per cycle, five times the work but receives three times the heat input and exhausts out twice the heat. Determine the efficiency of each engine.

Eduard Sanchez

Numerade Educator

Problem 17

In a Carnot cycle, the isothermal expansion of an ideal gas takes place at $412 \mathrm{~K}$ and the isothermal compression at $297 \mathrm{~K}$. During the expansion, $2090 \mathrm{~J}$ of heat energy are transferred to the gas. Determine $(a)$ the work performed by the gas during the isothermal expansion, ( $b$ ) the heat rejected from the gas during the isothermal compression, and $(c)$ the work done on the gas during the isothermal compression.

Eduard Sanchez

Numerade Educator

Problem 18

A Carnot engine has an efficiency of $22 \%$. It operates between heat reservoirs differing in temperature by $75 \mathrm{C}^{\circ} .$ Find the temperatures of the reservoirs.

Eduard Sanchez

Numerade Educator

Problem 19

For the Carnot cycle illustrated in Fig. $24-8$, show that the work done by the gas during process $B C$ has the same absolute value as the work done on the gas during process $D A$.

Eduard Sanchez

Numerade Educator

Problem 20

(a) In a two-stage Carnot heat engine, a quantity of heat $\left|Q_{1}\right|$ is absorbed at a temperature $T_{1}$, work $\left|W_{1}\right|$ is done, and a quantity of heat $\left|Q_{2}\right|$ is expelled at a lower temperature $T_{2}$, by the first stage. The second stage absorbs the heat expelled by the first, does work $\left|W_{2}\right|$, and expels a quantity of heat $\left|Q_{3}\right|$ at a lower temperature $T_{3}$. Prove that the efficiency of the combination is $\left(T_{1}-T_{3}\right) / T_{1} .(b)$ A combination mercury-steam turbine takes saturated mercury vapor from a boiler at $469^{\circ} \mathrm{C}$ and exhausts it to heat a steam boiler at $238^{\circ} \mathrm{C}$. The steam turbine receives steam at this temperature and exhausts it to a condenser at $37.8^{\circ} \mathrm{C}$. Calculate the maximum efficiency of the combination.

Eduard Sanchez

Numerade Educator

Problem 21

In a steam locomotive, steam at a boiler pressure of $16.0$ atm enters the cylinders, is expanded adiabatically to $5.60$ times its original volume, and then is exhausted to the atmosphere. Calculate (a) the steam pressure after expansion and $(b)$ the greatest possible efficiency of the engine.

Eduard Sanchez

Numerade Educator

Problem 22

One mole of an ideal monatomic gas is used as the working substance of an engine that operates on the cycle shown in Fig. 24-20. Calculate ( $a$ ) the work done by the engine per cycle, $(b)$ the heat added per cycle during the expansion stroke $a b c$, and $(c)$ the engine efficiency. $(d)$ What is the Carnot efficiency of an engine operating between the highest and lowest temperatures present in the cycle? How does this compare to the efficiency calculated in $(c) ?$ Assume that $p_{1}=2 p_{0}$ $V_{1}=2 V_{0}, p_{0}=1.01 \times 10^{5} \mathrm{~Pa}$, and $V_{0}=0.0225 \mathrm{~m}^{3}$

Susan Hallstrom

Susan Hallstrom

Numerade Educator

Problem 23

To make some ice, a freezer extracts $185 \mathrm{~kJ}$ of heat at $-12.0^{\circ} \mathrm{C} .$ The freezer has a coefficient of performance of 5.70. The room temperature is $26.0^{\circ} \mathrm{C}$. (a) How much heat is delivered to the room? ( $b$ ) How much work is required to run the freezer?

Eduard Sanchez

Numerade Educator

Problem 24

A refrigerator does $153 \mathrm{~J}$ of work to transfer $568 \mathrm{~J}$ of heat from its cold compartment. (a) Calculate the refrigerator's coefficient of performance. ( $b$ ) How much heat is exhausted to the kitchen?

Eduard Sanchez

Numerade Educator

Problem 25

How much work must be done to extract $10.0 \mathrm{~J}$ of heat $(a)$ from a reservoir at $7^{\circ} \mathrm{C}$ and transfer it to one at $27^{\circ} \mathrm{C}$ by means of a refrigerator using a Carnot cycle; $(b)$ from one at $-73^{\circ} \mathrm{C}$ to one at $27^{\circ} \mathrm{C} ;(c)$ from one at $-173^{\circ} \mathrm{C}$ to one at $27^{\circ} \mathrm{C} ;$ and $(d)$ from one at $-223^{\circ} \mathrm{C}$ to one at $27^{\circ} \mathrm{C} ?$

Eduard Sanchez

Numerade Educator

Problem 26

Apparatus that liquefies helium is in a laboratory at $296 \mathrm{~K}$. The helium in the apparatus is at $4.0 \mathrm{~K}$. If $150 \mathrm{~mJ}$ of heat is transferred from the helium, find the minimum amount of heat delivered to the laboratory.

Eduard Sanchez

Numerade Educator

Problem 27

An air conditioner takes air from a room at $70^{\circ} \mathrm{F}$ and transfers it to the outdoors, which is at $95^{\circ} \mathrm{F}$. For each joule of electrical energy required to run the refrigerator, how many joules of heat are transferred from the room?

Eduard Sanchez

Numerade Educator

Problem 28

An inventor claims to have created a heat pump that draws heat from a lake at $3.0^{\circ} \mathrm{C}$ and delivers heat at a rate of $20 \mathrm{~kW}$ to a building at $35^{\circ} \mathrm{C}$, while using only $1.9 \mathrm{~kW}$ of electrical power. How would you judge the claim?

Eduard Sanchez

Numerade Educator

Problem 29

(a) A Carnot engine operates between a hot reservoir at $322 \mathrm{~K}$ and a cold reservoir at $258 \mathrm{~K}$. If it absorbs $568 \mathrm{~J}$ of heat per cycle at the hot reservoir, how much work per cycle does it deliver? (b) If the same engine, working in reverse, functions as a refrigerator between the same two reservoirs, how much work per cycle must be supplied to transfer $1230 \mathrm{~J}$ of heat from the cold reservoir?

Eduard Sanchez

Numerade Educator

Problem 30

A heat pump is used to heat a building. The outside temperature is $-5.0^{\circ} \mathrm{C}$ and the temperature inside the building is to be maintained at $22^{\circ} \mathrm{C}$. The coefficient of performance is $3.8$, and the pump delivers $7.6 \mathrm{MJ}$ of heat to the building each hour. At what rate must work be done to run the pump?

Eduard Sanchez

Numerade Educator

Problem 31

In a refrigerator the low-temperature coils are at a temperature of $-13^{\circ} \mathrm{C}$ and the compressed gas in the condenser has a temperature of $25^{\circ} \mathrm{C}$. Find the coefficient of performance of a Carnot refrigerator operating between these temperatures.

Eduard Sanchez

Numerade Educator

Problem 32

The motor in a refrigerator has a power output of $210 \mathrm{~W}$. The freezing compartment is at $-3.0^{\circ} \mathrm{C}$ and the outside air is at $26^{\circ} \mathrm{C}$. Assuming that the efficiency is $85 \%$ of the ideal, calculate the amount of heat that can be extracted from the freezing compartment in $15 \mathrm{~min}$.

Eduard Sanchez

Numerade Educator

Problem 33

A Carnot engine works between temperatures $T_{1}$ and $T_{2}$. It drives a Carnot refrigerator that works between two different temperatures $T_{3}$ and $T_{4}$ (see Fig. $24-21$ ). Find the ratio $\left|Q_{3}\right||| Q_{1} \mid$ in terms of the four temperatures.

Ma Ednelyn Lim

Numerade Educator

Problem 34

a) Derive Stirling's approximation (Eq. $24-21$ ) by substituting an integral for the sum in the expression
$$\ln N !=\sum_{x=1}^{N} \ln x \approx \int_{1}^{N} \ln x d x$$
(b) For what values of $N$ is the error in Stirling's approximation less than $1 \%, 0.1 \%$, and $1 \times 10^{-6}$ ?

Tyler Moulton

Numerade Educator

Problem 35

Consider a container that is divided into two sections. ( $a$ ) Initially $N$ molecules of a gas are in one section, and the other side is empty. Compute the multiplicity of this initial state. ( $b$ ) After a hole is punched in the partition, the gas fills the entire container uniformly, with $N / 2$ molecules on each side of the partition. Find the multiplicity of the final state. (c) Show that the change in entropy is $\Delta S=k N \ln 2 .(d)$ Compare this result with the result of Sample Problem $24-4$ for the entropy change in a free expansion, and explain the similarities of the two results.

Ajay Singhal

Numerade Educator