y, and z directions plus elastic potential energy associ-
ated with the Hooke’s law forces exerted by neighbor-
ing atoms on it in the x, y, and z directions. According
to the theorem of equipartition of energy, assume the
average energy of each atom is 1
T for each degree of
freedom. (a) Prove that the molar specific heat of the
solid is 3R. The Dulong–Petit law states that this result
generally describes pure solids at sufficiently high tem-
peratures. (You may ignore the difference between the
specific heat at constant pressure and the specific heat
at constant volume.) (b) Evaluate the specific heat c of
iron. Explain how it compares with the value listed in
Table 20.1. (c) Repeat the evaluation and comparison
51. A certain ideal gas has a molar specific heat of C
A 2.00-mol sample of the gas always starts at pressure
1.003 105 Pa and temperature 300 K. For each of the
following processes, determine (a) the final pressure,
(b) the final volume, (c) the final temperature, (d) the
change in internal energy of the gas, (e) the energy
added to the gas by heat, and (f) the work done on the
gas. (i) The gas is heated at constant pressure to 400 K.
(ii) The gas is heated at constant volume to 400 K.
(iii) The gas is compressed at constant temperature to
1.20 3 105 Pa. (iv) The gas is compressed adiabatically
to 1.20 3 105 Pa.
52. The compressibility k of a substance is defined as the
fractional change in volume of that substance for a
given change in pressure:
(a) Explain why the negative sign in this expression
ensures k is always positive. (b) Show that if an ideal
gas is compressed isothermally, its compressibility is
given by k
5 1/P. (c) What If? Show that if an ideal gas
is compressed adiabatically, its compressibility is given
5 1/(gP). Determine values for (d) k
and (e) k
for a monatomic ideal gas at a pressure of 2.00 atm.
53. Review. Oxygen at pressures much greater than 1 atm
is toxic to lung cells. Assume a deep-sea diver breathes
a mixture of oxygen (O
) and helium (He). By weight,
what ratio of helium to oxygen must be used if the
diver is at an ocean depth of 50.0 m?
54. Examine the data for polyatomic gases in Table 21.2
and give a reason why sulfur dioxide has a higher spe-
cific heat at constant volume than the other polyatomic
gases at 300 K.
55. Model air as a diatomic ideal gas with M 5 28.9 g/mol.
A cylinder with a piston contains 1.20 kg of air at
25.08C and 2.00 3 105 Pa. Energy is transferred by
heat into the system as it is permitted to expand, with
the pressure rising to 4.00 3 105 Pa. Throughout the
expansion, the relationship between pressure and vol-
ume is given by
P 5 CV1/2
where C is a constant. Find (a) the initial volume, (b) the
final volume, (c) the final temperature, (d) the work
done on the air, and (e) the energy transferred by heat.
this process, what is the ratio of the rms speed of the
molecules remaining in the tank to the rms speed of
those being released at atmospheric pressure?
46. The dimensions of a classroom are 4.20 m 3 3.00m3
2.50 m. (a) Find the number of molecules of air in
the classroom at atmospheric pressure and 20.08C.
(b) Find the mass of this air, assuming the air consists
of diatomic molecules with molar mass 28.9 g/mol.
(c) Find the average kinetic energy of the molecules.
(d) Find the rms molecular speed. (e) What If?
Assume the molar specific heat of the air is inde-
pendent of temperature. Find the change in internal
energy of the air in the room as the temperature is
raised to 25.08C. (f) Explain how you could convince
a fellow student that your answer to part (e) is correct,
even though it sounds surprising.
47. The Earth’s atmosphere consists primarily of oxygen
(21%) and nitrogen (78%). The rms speed of oxygen
) in the atmosphere at a certain loca-
tion is 535m/s. (a) What is the temperature of the
atmosphere at this location? (b) Would the rms speed
of nitrogen molecules (N
) at this location be higher,
equal to, or lower than 535 m/s? Explain. (c) Deter-
mine the rms speed of N
at his location.
48. The mean free path , of a molecule is the average dis-
tance that a molecule travels before colliding with
another molecule. It is given by
where d is the diameter of the molecule and N
number of molecules per unit volume. The number of
collisions that a molecule makes with other molecules
per unit time, or collision frequency f, is given by
(a) If the diameter of an oxygen molecule is 2.00 3
10210m, find the mean free path of the molecules
in a scuba tank that has a volume of 12.0 L and is
filled with oxygen at a gauge pressure of 100 atm at a
temperature of 25.08C. (b)What is the average time
interval between molecular collisions for a molecule
of this gas?
49. An air rifle shoots a lead pellet by allowing high-
pressure air to expand, propelling the pellet down the
rifle barrel. Because this process happens very quickly,
no appreciable thermal conduction occurs and the
expansion is essentially adiabatic. Suppose the rifle
starts with 12.0 cm3 of compressed air, which behaves
as an ideal gas with g 5 1.40. The expanding air
pushes a 1.10-g pellet as a piston with cross-sectional
area 0.030 0 cm2 along the 50.0-cm-long gun barrel.
What initial pressure is required to eject the pellet
with a muzzle speed of 120 m/s? Ignore the effects
of the air in front of the bullet and friction with the
inside walls of the barrel.
50. In a sample of a solid metal, each atom is free to
vibrate about some equilibrium position. The atom’s
energy consists of kinetic energy for motion in the x,
chapter 21 the Kinetic theory of Gases
as that of a molecule in an ideal gas. Consider a spheri-
cal particle of density 1.00 3 103 kg/m3 in water at
20.08C. (a) For a particle of diameter d, evaluate the
rms speed. (b) The particle’s actual motion is a ran-
dom walk, but imagine that it moves with constant
velocity equal in magnitude to its rms speed. In what
time interval would it move by a distance equal to its
own diameter? (c) Evaluate the rms speed and the time
interval for a particle of diameter 3.00 mm. (d) Evalu-
ate the rms speed and the time interval for a sphere of
mass 70.0 kg, modeling your own body.
62. A vessel contains 1.00 3 104 oxygen molecules at 500 K.
(a) Make an accurate graph of the Maxwell speed distri-
bution function versus speed with points at speed inter-
vals of 100 m/s. (b) Determine the most probable speed
from this graph. (c) Calculate the average and rms
speeds for the molecules and label these points on your
graph. (d) From the graph, estimate the fraction of mol-
ecules with speeds in the range 300 m/s to 600 m/s.
63. A pitcher throws a 0.142-kg baseball at 47.2 m/s. As it
travels 16.8 m to home plate, the ball slows down to
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
R 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 tem-
perature of the ball itself does not change. A change
in temperature happens initially only for the air in a
cylinder 16.8 m in length and 3.70 cm in radius. This
air is initially at 20.08C.
64. The latent heat of vaporization for water at room tem-
perature is 2 430 J/g. Consider one particular molecule
at the surface of a glass of liquid water, moving upward
with sufficiently high speed that it will be the next
molecule to join the vapor. (a) Find its translational
kinetic energy. (b)Find its speed. Now consider a thin
gas made only of molecules like that one. (c) What is
its temperature? (d)Why are you not burned by water
evaporating from a vessel at room temperature?
65. A sample of a monatomic ideal gas occupies 5.00 L at
atmospheric pressure and 300 K (point A in Fig. P21.65).
It is warmed 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 origi-
nal state. (a)Find the number of moles in the sample.
56. Review. 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 negligible time
interval or by effective thickness of insulation. The
compressions and rarefactions are adiabatic. (a) Show
that the speed of sound in an ideal gas is
where M is the molar mass. The speed of sound in a
gas is given by Equation 17.8; use that equation and
the definition of the bulk modulus from Section 12.4.
(b)Compute the theoretical speed of sound in air at
20.08C and state how it compares with the value in
Table 17.1. Take M5 28.9 g/mol. (c) Show that the
speed of sound in an ideal gas is
is the mass of one molecule. (d) State how
the result in part (c) compares with the most probable,
average, and rms molecular speeds.
57. Twenty particles, each of mass m
and confined to a
volume V, have various speeds: two have speed v, three
have speed 2v, five have speed 3v, four have speed
4v, three have speed 5v, two have speed 6v, and one
has speed 7v. Find (a)the average speed, (b) the rms
speed, (c) the most probable speed, (d) the average
pressure the particles exert on the walls of the vessel,
and (e) the average kinetic energy per particle.
58. In a cylinder, a sample of an ideal gas with number of
moles n undergoes an adiabatic process. (a) Starting
with the expression W52
P dV and using the condi-
tion PVg 5 constant, show that the work done on the
(b) Starting with the first law of thermodynamics, show
that the work done on the gas is equal to nC
(c)Are these two results consistent with each other?
59. As a 1.00-mol sample of a monatomic ideal gas expands
adiabatically, the work done on it is 22.50 3 103 J. The
initial temperature and pressure of the gas are 500 K
and 3.60atm. Calculate (a) the final temperature and
(b) the final pressure.
60. A sample consists of an amount n in moles of a mona-
tomic ideal gas. The gas expands adiabatically, with
work W done on it. (Work W is a negative number.)
The initial temperature and pressure of the gas are T
. Calculate (a) the final temperature and (b) the
61. When a small particle is suspended in a fluid, bom-
bardment by molecules makes the particle jitter about
at random. Robert Brown discovered this motion in
1827 while studying plant fertilization, and the motion
has become known as Brownian motion. The particle’s
average kinetic energy can be taken as 3
T, the same
70. On the PV diagram for an ideal gas, one isothermal
curve and one adiabatic curve pass through each point
as shown in Figure P21.70. Prove that the slope of the
adiabatic curve is steeper than the slope of the iso-
therm at that point by the factor g.
71. In 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 molar mass of
water is 18.0 g/mol. (a) Find 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 3 1021 kg. How many
of the water molecules originally in the pot are likely to
be present in it again today?
72. Review. (a) If it has enough kinetic energy, a molecule
at the surface of the Earth can “escape the Earth’s grav-
itation” in the sense that it can continue to move away
from the Earth forever as discussed in Section 13.6.
Using the principle of conservation of energy, show
that the minimum kinetic energy needed for “escape”
, where m
is the mass of the molecule, g is
the free-fall acceleration at the surface, and R
radius of the Earth. (b)Calculate the temperature for
which the minimum escape kinetic energy is ten times
the average kinetic energy of an oxygen molecule.
73. 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) Over what approxi-
mate time interval would an atom wander out of the
trap region if there were no trapping action?
74. Equations 21.42 and 21.43 show that v
collection of gas particles, which turns out to be true
whenever the particles have a distribution of speeds.
Let us explore this inequality for a two-particle gas.
Find (b) the temperature at point B, (c) the temperature
at point C, and (d) the volume at point C. (e) Now con-
sider the processes A S B, B S C, and C S A. Describe
how to carry out each process experimentally. (f) Find
Q, W, and DE
for each of the processes. (g) For the
whole cycle A S B S C S A, find Q, W, and DE
66. 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 v.
The force acting on a gas molecule toward the center
of the centrifuge is m
v2r. (a) Discuss how a gas centri-
fuge can be used to separate particles of different mass.
(b) Suppose the centrifuge contains a gas of particles
of identical mass. Show that the density of the particles
as a function of r is
67. For a Maxwellian gas, use a computer or programma-
ble calculator to find the numerical value of the ratio
) for the following values of v: (a) v 5
/10.0), (c) (v
/2.00), (d) v
, and (g) 50.0v
. Give your
results to three significant figures.
68. A triatomic molecule can have a linear configuration,
as does CO
(Fig. P21.68a), or it can be nonlinear, like
O (Fig. P21.68b). Suppose the temperature of a gas
of triatomic molecules is sufficiently low that vibrational
motion is negligible. What is the molar specific heat
at constant volume, expressed as a multiple of the uni-
versal gas constant, (a) if the molecules are linear and
(b) if the molecules are nonlinear? At high tempera-
tures, a triatomic molecule has two modes of vibration,
and each contributes 1
R to the molar specific heat for its
kinetic energy and another
R for its potential energy.
Identify the high-temperature molar specific heat at
constant volume for a triatomic ideal gas of (c) linear
molecules and (d) nonlinear molecules. (e) Explain how
specific heat data can be used to determine whether a
triatomic molecule is linear or nonlinear. Are the data
in Table 21.2 sufficient to make this determination?
69. Using the Maxwell–Boltzmann speed distribution
function, verify Equations 21.42 and 21.43 for (a) the
rms speed and (b) the average speed of the molecules
of a gas at a temperature T. The average value of vn is
Use the table of integrals B.6 in Appendix B.
chapter 21 the Kinetic theory of Gases
parallel to the axis of the cylinder until it comes to
rest at an equilibrium position (Fig. P21.75b). Find the
final temperatures in the two compartments.
= 550 K K T
= 250 K
Let the speed of one particle be v
and the other
particle have speed v
5 (2 2 a)v
. (a) Show that the
average of these two speeds is v
. (b) Show that
(2 2 2a 1 a2)
(c) Argue that the equation in part (b) proves that, in
. (d) Under what special condition
for the two-particle gas?
75. A cylinder is closed at both ends and has insulating
walls. It is divided into two compartments by an insu-
lating piston that is perpendicular to the axis of the
cylinder as shown in Figure P21.75a. Each compart-
ment contains 1.00 mol of oxygen that behaves as an
ideal gas with g 5 1.40. Initially, the two compartments
have equal volumes and their temperatures are 550 K
and 250 K. The piston is then allowed to move slowly
A Stirling engine from the early
nineteenth century. Air is heated in the
lower cylinder using an external source.
As this happens, the air expands and
pushes against a piston, causing it to
move. The air is then cooled, allowing the
cycle to begin again. This is one example
of a heat engine, which we study in this
SSPL/The Image Works)
22.1 Heat Engines and the Second
Law of Thermodynamics
22.2 Heat Pumps and Refrigerators
22.3 Reversible and
22.4 The Carnot Engine
22.5 Gasoline and Diesel Engines
22.7 Changes in Entropy for
22.8 Entropy and the Second Law
c h a p p t t e r
The first law of thermodynamics, which we studied in Chapter 20, is a statement of
conservation of energy and is a special-case reduction of Equation 8.2. This law states
that a change in internal energy in a system can occur as a result of energy transfer by
heat, by work, or by both. Although the first law of thermodynamics is very important,
it makes no distinction between processes that occur spontaneously and those that do
not. Only certain types of energy transformation and energy transfer processes actually
take place in nature, however. The second law of thermodynamics, the major topic in this
chapter, establishes which processes do and do not occur. The following are examples
heat engines, entropy,
and the Second Law of
chapter 22 heat engines, entropy, and the Second Law of thermodynamics
of processes that do not violate the first law of thermodynamics if they proceed in either
direction, but are observed in reality to proceed in only one direction:
• 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 molecules
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.1
22.1 Heat Engines and the Second Law
A heat engine is a device that takes in energy by heat2 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, a fuel such as coal 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
mechanical energy associated with this rotation is used to drive an electric genera-
tor. Another device that can be modeled as a heat engine is the internal combustion
engine in an automobile. This device uses energy from a burning fuel to perform
work on pistons that results in the motion of the automobile.
Let us consider the operation of a heat engine in more detail. A heat engine car-
ries some working substance through a cyclic process during which (1) the working
substance absorbs energy by heat from a high-temperature energy 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 burn-
ing fuel and evaporates to steam, which then does work by expanding against a pis-
ton. 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
| from the hot reservoir. For the mathematical
discussion of heat engines, we use absolute values to make all energy transfers by
heat positive, and the direction of transfer is indicated with an explicit positive or
negative sign. The engine does work W
(so that negative work W 5 2W
on the engine) and then gives up a quantity of energy |Q
| to the cold reservoir.
British physicist and mathematician
Born William Thomson in Belfast, Kel-
vin 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 thermody-
namics led to the idea that energy can-
not pass spontaneously from a colder
object to a hotter object.
1Although a process occurring in the time-reversed sense has never been observed, it is possible for it to occur. As we
shall see later in this chapter, however, the probability of such a process occurring is infinitesimally small. From this
viewpoint, processes occur with a vastly greater probability in one direction than in the opposite direction.
2We use heat as our model for energy transfer into a heat engine. Other methods of energy transfer are 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.
locomotive obtains its energy
by burning wood or coal. The
generated energy vaporizes water
into steam, which powers the
locomotive. 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
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