Physics For Scientists And Engineers 6E - part 167

acting toward the center of the circular path on a given
particle is m+

r. (a) Discuss how a gas centrifuge can be

used to separate particles of different mass. (b) Show that
the density of the particles as a function of r is

65.

Verify Equations 21.27 and 21.28 for the rms and average
speed of the molecules of a gas at a temperature T. Note
that the average value of v

Use the table of definite integrals in Appendix B (Table B.6).

66.

On the PV diagram for an ideal gas, one isothermal curve
and one adiabatic curve pass through each point. Prove
that the slope of the adiabat is steeper than the slope of
the isotherm by the factor (.

67.

A sample of monatomic ideal gas occupies 5.00 L at atmos-
pheric pressure and 300 K (point A in Figure P21.67). It is
heated at constant volume to 3.00 atm (point B). Then it is
allowed to expand isothermally to 1.00 atm (point C) and
at last compressed isobarically to its original state. (a) Find
the number of moles in the sample. (b) Find the tempera-
ture at points B and C and the volume at point C. (c) As-
suming that the molar specific heat does not depend on
temperature, so that E

int

3nRT/2, find the internal en-

ergy at points A, B, and C. (d) Tabulate P, V, T, and E

int

for

the states at points A, B, and C. (e) Now consider the
processes A : B, B : C, and C : A. Describe just how to
carry out each process experimentally. (f ) Find Q , W, and
"

int

for each of the processes. (g) For the whole cycle

A : B : C : A find Q , W, and "E

int

68.

This problem can help you to think about the size of mole-
cules.  In  the  city  of  Beijing  a  restaurant  keeps  a  pot  of
chicken broth simmering continuously. Every morning it is
topped  up  to  contain  10.0 L  of  water,  along  with  a  fresh
chicken,  vegetables,  and  spices.  The  soup  is  thoroughly
stirred.  The  molar  mass  of  water  is  18.0 g/mol.  (a)  Find

n(r) # n

/2k

(a) Explain why the negative sign in this expression
ensures that / is always positive. (b) Show that if an ideal
gas is compressed isothermally, its compressibility is given
by /

1/P. (c) What If? Show that if an ideal gas is

compressed adiabatically, its compressibility is given by
/

1/(P. (d) Determine values for /

and /

for a

monatomic ideal gas at a pressure of 2.00 atm.

60.

Review problem. (a) Show that the speed of sound in an
ideal gas is

where M is the molar mass. Use the general expression for
the speed of sound in a fluid from Section 17.1, the defini-
tion of the bulk modulus from Section 12.4, and the result
of  Problem  59  in  this  chapter.  As  a  sound  wave  passes
through a gas, the compressions are either so rapid or so
far apart that thermal conduction is prevented by a negli-
gible  time  interval  or  by  effective  thickness  of  insulation.
The compressions and rarefactions are adiabatic. (b) Com-
pute  the  theoretical  speed  of  sound  in  air  at  20°C  and
compare  it  with  the  value  in  Table  17.1.  Take  M #
28.9 g/mol. (c) Show that the speed of sound in an ideal
gas is

where m is the mass of one molecule. Compare it with the
most probable, average, and rms molecular speeds.

61.

Model air as a diatomic ideal gas with M # 28.9 g/mol. A
cylinder with a piston contains 1.20 kg of air at 25.0°C and
200 kPa. Energy is transferred by heat into the system as it
is allowed to expand, with the pressure rising to 400 kPa.
Throughout the expansion, the relationship between pres-
sure and volume is given by

where C is a constant. (a) Find the initial volume. (b) Find
the final volume. (c) Find the final temperature. (d) Find the
work done on the air. (e) Find the energy transferred by heat.

62.

Smokin’! A  pitcher  throws  a  0.142-kg  baseball  at  47.2 m/s
(Fig. P21.62). As it travels 19.4 m, the ball slows to a speed
of  42.5 m/s  because  of  air  resistance.  Find  the  change  in
temperature of the air through which it passes. To find the
greatest  possible  temperature  change,  you  may  make  the
following  assumptions:  Air  has  a  molar  specific  heat  of
C

7R/2 and an equivalent molar mass of 28.9 g/mol.

The process is so rapid that the cover of the baseball acts
as thermal insulation, and the temperature of the ball it-
self does not change. A change in temperature happens
initially only for the air in a cylinder 19.4 m in length and
3.70 cm in radius. This air is initially at 20.0°C.

For a Maxwellian gas, use a computer or programma-

ble calculator to find the numerical value of the ratio
N

(v)/N

) for the following values of v: v # (v

/50),

/10), (v

/2), v

, 2v

, 10v

, and 50v

. Give your

results to three significant figures.

64.

Consider the particles in a gas centrifuge, a device used to
separate particles of different mass by whirling them in a
circular path of radius r at angular speed +. The force

63.

P # CV

1/2

v #

√

(

v #

√

(

Problems

665

Figure P21.62 John Lackey, the first rookie to win a World Series

game 7 in 93 years, pitches for the Anaheim Angels during the

final game of the 2002 World Series.

AP/W

orld Wide Photos

666

CHAPTE R 21 • The Kinetic Theory of Gases

the number of molecules of water in the pot. (b) During a
certain month, 90.0% of the broth was served each day to
people who then emigrated immediately. Of the water
molecules in the pot on the first day of the month, when
was the last one likely to have been ladled out of the pot?
(c) The broth has been simmering for centuries, through
wars, earthquakes, and stove repairs. Suppose the water
that was in the pot long ago has thoroughly mixed into the
Earth’s hydrosphere, of mass 1.32 % 10

kg. How many of

the water molecules originally in the pot are likely to be
present in it again today?

69.

Review problem. (a) If it has enough kinetic energy, a mol-
ecule at the surface of the Earth can “escape the Earth’s
gravitation,” in the sense that it can continue to move away
from the Earth forever, as discussed in Section 13.7. Using
the principle of conservation of energy, show that the mini-
mum kinetic energy needed for “escape” is mgR

, where m

is the mass of the molecule, g is the free-fall acceleration at
the surface, and R

is the radius of the Earth. (b) Calculate

the temperature for which the minimum escape kinetic en-
ergy is ten times the average kinetic energy of an oxygen
molecule.

70.

Using multiple laser beams, physicists have been able to cool
and trap sodium atoms in a small region. In one experiment
the  temperature  of  the  atoms  was  reduced  to  0.240 mK.
(a)  Determine  the  rms  speed  of  the  sodium  atoms  at  this
temperature.  The  atoms  can  be  trapped  for  about  1.00  s.
The  trap  has  a  linear  dimension  of  roughly  1.00 cm.
(b) Approximately how long would it take an atom to wan-
der out of the trap region if there were no trapping action?

Answers to Quick Quizzes

21.1 (b). The average translational kinetic energy per mole-

cule is a function only of temperature.

21.2 (a). Because there are twice as many molecules and the

temperature of both containers is the same, the total en-
ergy in B is twice that in A.

21.3 (b). Because both containers hold the same type of gas,

the rms speed is a function only of temperature.

21.4 (a). According to Equation 21.10, E

int

is a function of

temperature only. Because the temperature increases, the
internal energy increases.

21.5 (c). Along an isotherm, T is constant by definition. There-

fore, the internal energy of the gas does not change.

21.6 (d). The value of 29.1 J/mol & K is 7R/2. According to

Figure 21.7, this suggests that all three types of motion are
occurring.

21.7 (c). The highest possible value of C

for a diatomic gas is

7R/2, so the gas must be polyatomic.

21.8 (a). Because the hydrogen atoms are lighter than the ni-

trogen molecules, they move with a higher average speed
and the distribution curve is stretched out more along the
horizontal axis. See Equation 21.26 for a mathematical
statement of the dependence of N

on m.

P(atm)

V(L)

Figure P21.67

By permission of John Hart and Creators Syndicate, Inc.

Heat Engines, Entropy, and the

Second Law of Thermodynamics

C H A P T E R O U T L I N E

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

Chapter 22

▲

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)

667

668

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:

• 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.

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.

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.

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.

Lord Kelvin

British physicist and
mathematician (1824–1907)

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.)

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