Heat Engines and the Second Law of Thermodynamics chapter 22.pdf

Chapter 22

Heat Engines, Entropy, and the

Second Law of Thermodynamics

CHAPTER OUTLINE

22.1 Heat Engines and the Second

Law of Thermodynamics

22.2 Heat Pumps and Refrigerators

22.3 Reversible and Irreversible

Processes

22.4 The Carnot Engine

22.5 Gasoline and Diesel Engines

22.6 Entropy

22.7 Entropy Changes in

Irreversible Processes

22.8 Entropy on a Microscopic

Scale

! This cutaway image of an automobile engine shows two pistons that have work done on

them by an explosive mixture of air and fuel, ultimately leading to the motion of the

automobile. This apparatus can be modeled as a heat engine, which we study in this chapter.

(Courtesy of Ford Motor Company)

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T he ﬁrst law of thermodynamics, which we studied in Chapter 20, is a statement of

conservation of energy. This law states that a change in internal energy in a system can

occur as a result of energy transfer by heat or by work, or by both. As was stated in

Chapter 20, the law makes no distinction between the results of heat and the results of

work—either heat or work can cause a change in internal energy. However, there is an

important distinction between heat and work that is not evident from the ﬁrst law. One

manifestation of this distinction is that it is impossible to design a device that, operat-

ing in a cyclic fashion, takes in energy by heat and expels an equal amount of energy by

work. A cyclic device that takes in energy by heat and expels a fraction of this energy by

work is possible and is called a heat engine .

Although the first law of thermodynamics is very important, it makes no distinc-

tion between processes that occur spontaneously and those that do not. However,

only certain types of energy-conversion and energy-transfer processes actually take

place in nature. The second law of thermodynamics , the major topic in this chapter,

establishes which processes do and which do not occur. The following are examples

of processes that do not violate the principle of conservation of energy if they pro-

ceed in either direction, but are observed to proceed in only one direction, governed

by the second law:

Lord Kelvin

British physicist and

mathematician (1824–1907)

• When two objects at different temperatures are placed in thermal contact with

each other, the net transfer of energy by heat is always from the warmer object to

the cooler object, never from the cooler to the warmer.

• A rubber ball dropped to the ground bounces several times and eventually comes

to rest, but a ball lying on the ground never gathers internal energy from the

ground and begins bouncing on its own.

• An oscillating pendulum eventually comes to rest because of collisions with air mol-

ecules and friction at the point of suspension. The mechanical energy of the system

is converted to internal energy in the air, the pendulum, and the suspension; the

reverse conversion of energy never occurs.

Born William Thomson in Belfast,

Kelvin was the ﬁrst to propose

the use of an absolute scale of

temperature. The Kelvin

temperature scale is named in

his honor. Kelvin’s work in

thermodynamics led to the idea

that energy cannot pass

spontaneously from a colder

object to a hotter object.

(J. L. Charmet/SPL/Photo

Researchers, Inc.)

All these processes are irreversible —that is, they are processes that occur naturally in

one direction only. No irreversible process has ever been observed to run backward—if

it were to do so, it would violate the second law of thermodynamics. 1

From an engineering standpoint, perhaps the most important implication of the

second law is the limited efﬁciency of heat engines. The second law states that a ma-

chine that operates in a cycle, taking in energy by heat and expelling an equal amount

of energy by work, cannot be constructed.

1 Although we have never observed a process occurring in the time-reversed sense, it is possible for it to

occur. As we shall see later in the chapter, however, the probability of such a process occurring is

inﬁnitesimally small. From this viewpoint, we say that processes occur with a vastly greater probability in

one direction than in the opposite direction.

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SECTION 22.1 • Heat Engines and the Second Law of Thermodynamics

669

22.1 Heat Engines and the Second Law

of Thermodynamics

A heat engine is a device that takes in energy by heat 2 and, operating in a cyclic

process, expels a fraction of that energy by means of work. For instance, in a typical

process by which a power plant produces electricity, coal or some other fuel is burned,

and the high-temperature gases produced are used to convert liquid water to steam.

This steam is directed at the blades of a turbine, setting it into rotation. The mechani-

cal energy associated with this rotation is used to drive an electric generator. Another

device that can be modeled as a heat engine—the internal combustion engine in an

automobile—uses energy from a burning fuel to perform work on pistons that results

in the motion of the automobile.

A heat engine carries some working substance through a cyclic process during

which (1) the working substance absorbs energy by heat from a high-temperature en-

ergy reservoir, (2) work is done by the engine, and (3) energy is expelled by heat to a

lower-temperature reservoir. As an example, consider the operation of a steam engine

(Fig. 22.1), which uses water as the working substance. The water in a boiler absorbs

energy from burning fuel and evaporates to steam, which then does work by expand-

ing against a piston. After the steam cools and condenses, the liquid water produced

returns to the boiler and the cycle repeats.

It is useful to represent a heat engine schematically as in Figure 22.2. The engine

absorbs a quantity of energy ! Q h ! from the hot reservoir. For this discussion of heat en-

gines, we will use absolute values to make all energy transfers positive and will indicate

the direction of transfer with an explicit positive or negative sign. The engine does

work W eng (so that negative work W !" W eng is done on the engine), and then gives up

a quantity of energy ! Q c ! to the cold reservoir. Because the working substance goes

Hot reservoir at T h

Q h

Figure 22.1 This steam-driven

locomotive runs from Durango to

Silverton, Colorado. It obtains its

energy by burning wood or coal.

The generated energy vaporizes

water into steam, which powers the

locomotive. (This locomotive must

take on water from tanks located

along the route to replace steam

lost through the funnel.) Modern

locomotives use diesel fuel instead

of wood or coal. Whether old-

fashioned or modern, such

locomotives can be modeled as

heat engines, which extract energy

from a burning fuel and convert a

fraction of it to mechanical energy.

W eng

Engine

Q c

Cold reservoir at T c

Active Figure 22.2 Schematic

representation of a heat engine.

The engine does work W eng . The

arrow at the top represents energy

Q h # 0 entering the engine. At the

bottom, Q c $ 0 represents energy

leaving the engine.

2 We will use heat as our model for energy transfer into a heat engine. Other methods of energy

transfer are also possible in the model of a heat engine, however. For example, the Earth’s atmosphere

can be modeled as a heat engine, in which the input energy transfer is by means of electromagnetic

radiation from the Sun. The output of the atmospheric heat engine causes the wind structure in the

atmosphere.

At the Active Figures link

at http://www.pse6.com, you

can select the efﬁciency of the

engine and observe the

transfer of energy.

670

CHAPTER 22 • Heat Engines, Entropy, and the Second Law of Thermodynamics

through a cycle, its initial and ﬁnal internal energies are equal, and so % E int ! 0.

Hence, from the ﬁrst law of thermodynamics, % E int ! Q & W ! Q " W eng , and with

no change in internal energy, the net work W eng done by a heat engine is equal

to the net energy Q net transferred to it. As we can see from Figure 22.2,

Q net ! | Q h | " | Q c |; therefore,

Area = W eng

W eng ! ! Q h ! " ! Q c !

(22.1)

If the working substance is a gas, the net work done in a cyclic process is the

area enclosed by the curve representing the process on a PV diagram. This is

shown for an arbitrary cyclic process in Figure 22.3.

The thermal efﬁciency e of a heat engine is deﬁned as the ratio of the net work

done by the engine during one cycle to the energy input at the higher temperature

during the cycle:

Figure 22.3 PV diagram for an

arbitrary cyclic process taking place

in an engine. The value of the net

work done by the engine in one

cycle equals the area enclosed by

the curve.

e ! W eng

! Q h ! " ! Q c !

! Q h !

! Q c !

! Q h !

! Q h ! !

! 1 "

(22.2)

Thermal efﬁciency of a heat

engine

We can think of the efﬁciency as the ratio of what you gain (work) to what you give

(energy transfer at the higher temperature). In practice, all heat engines expel only a

fraction of the input energy Q h by mechanical work and consequently their efﬁciency

is always less than 100%. For example, a good automobile engine has an efﬁciency of

about 20%, and diesel engines have efﬁciencies ranging from 35% to 40%.

Equation 22.2 shows that a heat engine has 100% efﬁciency ( e ! 1) only if

! Q c ! ! 0—that is, if no energy is expelled to the cold reservoir. In other words, a heat

engine with perfect efﬁciency would have to expel all of the input energy by work. On

the basis of the fact that efﬁciencies of real engines are well below 100%, the

Kelvin–Planck form of the second law of thermodynamics states the following:

Hot reservoir at T h

Q h

It is impossible to construct a heat engine that, operating in a cycle, produces no

effect other than the input of energy by heat from a reservoir and the performance

of an equal amount of work.

W eng

Engine

This statement of the second law means that, during the operation of a heat engine,

W eng can never be equal to ! Q h !, or, alternatively, that some energy ! Q c ! must be

rejected to the environment. Figure 22.4 is a schematic diagram of the impossible

“perfect” heat engine.

Cold reservoir at T c

Quick Quiz 22.1 The energy input to an engine is 3.00 times greater than

the work it performs. What is its thermal efficiency? (a) 3.00 (b) 1.00 (c) 0.333

(d) impossible to determine

The impossible engine

Figure 22.4 Schematic diagram of

a heat engine that takes in energy

from a hot reservoir and does an

equivalent amount of work. It is

impossible to construct such a

perfect engine.

Quick Quiz 22.2 For the engine of Quick Quiz 22.1, what fraction of the en-

ergy input is expelled to the cold reservoir? (a) 0.333 (b) 0.667 (c) 1.00 (d) impossible

to determine

Example 22.1 The Efﬁciency of an Engine

An engine transfers 2.00 ' 10 3 J of energy from a hot reser-

voir during a cycle and transfers 1.50 ' 10 3 J as exhaust to a

cold reservoir.

Solution The efﬁciency of the engine is given by Equation

22.2 as

e ! 1 " ! Q c !

! Q h ! ! 1 " 1.50 ' 10 3 J

(A) Find the efﬁciency of the engine.

2.00 ' 10 3 J !

0.250, or 25.0%

SECTION 22.2 • Heat Pumps and Refrigerators

671

(B) How much work does this engine do in one cycle?

Answer No, you do not have enough information. The

power of an engine is the rate at which work is done by the

engine. You know how much work is done per cycle but you

have no information about the time interval associated with

one cycle. However, if you were told that the engine oper-

ates at 2 000 rpm (revolutions per minute), you could relate

this rate to the period of rotation T of the mechanism of the

engine. If we assume that there is one thermodynamic cycle

per revolution, then the power is

Solution The work done is the difference between the

input and output energies:

W eng ! ! Q h ! " ! Q c ! ! 2.00 ' 10 3 J " 1.50 ' 10 3 J

5.0 ' 10 2 J

What If? Suppose you were asked for the power output of

this engine? Do you have sufﬁcient information to answer

this question?

! ! W eng

! 5.0 ' 10 2 J

" 1 min

# ! 1.7 ' 10 4 W

22.2 Heat Pumps and Refrigerators

! PITFALL PREVENTION

22.1 The First and Second

Laws

Notice the distinction between

the ﬁrst and second laws of

thermodynamics. If a gas under-

goes a one-time isothermal process

% E int ! Q & W ! 0. Therefore,

the ﬁrst law allows all energy in-

put by heat to be expelled by

work. In a heat engine, however,

in which a substance undergoes a

cyclic process, only a portion of

the energy input by heat can be

expelled by work according to

the second law.

In a heat engine, the direction of energy transfer is from the hot reservoir to the cold

reservoir, which is the natural direction. The role of the heat engine is to process the

energy from the hot reservoir so as to do useful work. What if we wanted to transfer en-

ergy from the cold reservoir to the hot reservoir? Because this is not the natural direc-

tion of energy transfer, we must put some energy into a device in order to accomplish

this. Devices that perform this task are called heat pumps or refrigerators. For exam-

ple, we cool homes in summer using heat pumps called air conditioners. The air condi-

tioner transfers energy from the cool room in the home to the warm air outside.

In a refrigerator or heat pump, the engine takes in energy ! Q c ! from a cold reser-

voir and expels energy ! Q h ! to a hot reservoir (Fig. 22.5). This can be accomplished

only if work is done on the engine. From the ﬁrst law, we know that the energy given up

to the hot reservoir must equal the sum of the work done and the energy taken in from

the cold reservoir. Therefore, the refrigerator or heat pump transfers energy from a

colder body (for example, the contents of a kitchen refrigerator or the winter air out-

side a building) to a hotter body (the air in the kitchen or a room in the building). In

practice, it is desirable to carry out this process with a minimum of work. If it could be

accomplished without doing any work, then the refrigerator or heat pump would be

“perfect” (Fig. 22.6). Again, the existence of such a device would be in violation of the

second law of thermodynamics, which in the form of the Clausius statement 3

states:

Hot reservoir at T h

Q h

Heat pump

Q c

Cold reservoir at T c

Active Figure 22.5 Schematic diagram of a heat pump,

which takes in energy Q c # 0 from a cold reservoir and

expels energy Q h $ 0 to a hot reservoir. Work W is done

on the heat pump. A refrigerator works the same way.

At the Active Figures link

at http://www.pse6.com, you

can select the COP of the heat

pump and observe the transfer

of energy.

First expressed by Rudolf Clausius (1822–1888).

2 000 min #

60 s

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