Physics For Scientists And Engineers 6E - part 148

equal length are raised by the same amount from some common initial value, the brass
rod expands more than the steel rod does because brass has a greater average coefficient
of expansion than steel does. A simple mechanism called a bimetallic strip utilizes this
principle and is found in practical devices such as thermostats. It consists of two thin
strips of dissimilar metals bonded together. As the temperature of the strip increases, the
two metals expand by different amounts and the strip bends, as shown in Figure 19.9.

S E C T I O N 19 . 4 • Thermal Expansion of Solids and Liquids

589

Quick Quiz 19.3

If you are asked to make a very sensitive glass thermome-

ter, which of the following working liquids would you choose? (a) mercury (b) alcohol
(c) gasoline (d) glycerin

Quick Quiz 19.4

Two spheres are made of the same metal and have the

same radius, but one is hollow and the other is solid. The spheres are taken through
the same temperature increase. Which sphere expands more? (a) solid sphere (b) hol-
low sphere (c) They expand by the same amount. (d) not enough information to say

(b)

(a)

Steel

Brass

Room temperature

Higher temperature

Bimetallic strip

Off

°C

Figure 19.9 (a) A bimetallic strip bends as the temperature changes because the two

metals have different expansion coefficients. (b) A bimetallic strip used in a thermostat

to break or make electrical contact.

Example 19.3 Expansion of a Railroad Track

(B)

Suppose that the ends of the rail are rigidly clamped at

0.0°C so that expansion is prevented. What is the thermal
stress set up in the rail if its temperature is raised to 40.0°C?

Solution The thermal stress will be the same as that in the
situation in which we allow the rail to expand freely and
then compress it with a mechanical force F back to its origi-
nal length. From the definition of Young’s modulus for a
solid (see Eq. 12.6), we have

Because Y for steel is 20 ) 10

N/m

(see Table 12.1), we

have

Tensile stress #

A segment of steel railroad track has a length of 30.000 m
when the temperature is 0.0°C.

(A)

What is its length when the temperature is 40.0°C?

Solution Making use of Table 19.1 and noting that the
change in temperature is 40.0°C, we find that the increase
in length is

If the track is 30.000 m long at 0.0°C, its length at 40.0°C is

30.013 m.

# 0.013 m

L # &L

T # [11 ) 10

(%C)

](30.000 m)(40.0%C)

The Unusual Behavior of Water

Liquids generally increase in volume with increasing temperature and have average coeffi-
cients of volume expansion about ten times greater than those of solids. Cold water is an
exception to this rule, as we can see from its density-versus-temperature curve, shown in
Figure 19.11. As the temperature increases from 0°C to 4°C, water contracts and thus its
density increases. Above 4°C, water expands with increasing temperature, and so its den-
sity decreases. Thus, the density of water reaches a maximum value of 1.000 g/cm

at 4°C.

We can use this unusual thermal-expansion behavior of water to explain why a

pond begins freezing at the surface rather than at the bottom. When the atmospheric
temperature drops from, for example, 7°C to 6°C, the surface water also cools and con-
sequently decreases in volume. This means that the surface water is denser than the
water below it, which has not cooled and decreased in volume. As a result, the surface
water sinks, and warmer water from below is forced to the surface to be cooled. When
the atmospheric temperature is between 4°C and 0°C, however, the surface water ex-
pands as it cools, becoming less dense than the water below it. The mixing process

590

C H A P T E R 19 • Temperature

Solving for "T, we find

Thus, the temperature at which the bolts touch is 27°C $

7.4°C #

To finalize this problem, note that this

temperature is possible if the air conditioning in the building
housing the device fails for a long period on a very hot sum-
mer day.

34°C

# 7.4%C

5.0 ) 10

(19 ) 10

)(0.030 m) $ (11 ) 10

)(0.010 m)

T #

5.0 ) 10

i, br

i, st

$ "

T $ &

T # 5.0 ) 10

An electronic device has been poorly designed so that two
bolts attached to different parts of the device almost touch
each other in its interior, as in Figure 19.10. The steel and
brass bolts are at different electric potentials and if they
touch, a short circuit will develop, damaging the device. (We
will study electric potential in Chapter 25.) If the initial gap
between the ends of the bolts is 5.0 *m at 27°C, at what tem-
perature will the bolts touch?

Solution We can conceptualize the situation by imagining
that the ends of both bolts expand into the gap between
them as the temperature rises. We categorize this as a ther-
mal expansion problem, in which the sum of the changes in
length of the two bolts must equal the length of the initial
gap between the ends. To analyze the problem, we write this
condition mathematically:

Example 19.4 The Thermal Electrical Short

creases.  Thus,  if  there  is  an  increase  in  length  of  0.013 m
when  the  temperature  increases  by  40°C,  then  there  is  a
decrease  in  length  of  0.013 m  when  the  temperature
decreases  by  40°C.  (We  assume  that & is  constant  over
the entire  range  of  temperatures.)  The  new  length  at  the
colder temperature is 30.000 m ! 0.013 m # 29.987 m.

What If?

What if the temperature drops to ! 40.0°C? What

is the length of the unclamped segment?

The expression for the change in length in Equation 19.4
is the same whether the temperature increases or de-

8.7 ) 10

N/m

(20 ) 10

N/m

)

0.013 m

30.000 m

0.010 m

0.030 m

5.0 m

Steel

Brass

Figure 19.10 (Example 19.4) Two bolts attached to different parts of an electrical de-

vice are almost touching when the temperature is 27°C. As the temperature increases,

the ends of the bolts move toward each other.

stops, and eventually the surface water freezes. As the water freezes, the ice remains on
the surface because ice is less dense than water. The ice continues to build up at the
surface, while water near the bottom remains at 4°C. If this were not the case, then fish
and other forms of marine life would not survive.

19.5 Macroscopic Description of an Ideal Gas

The volume expansion equation "V # 'V

T is based on the assumption that the ma-

terial has an initial volume V

before the temperature change occurs. This is the case

for solids and liquids because they have a fixed volume at a given temperature.

The case for gases is completely different. The interatomic forces within gases are

very weak, and, in many cases, we can imagine these forces to be nonexistent and still
make very good approximations. Note that there is no equilibrium separation for the atoms
and, thus, no “standard” volume at a given temperature. As a result, we cannot express
changes in volume "V in a process on a gas with Equation 19.6 because we have no de-
fined volume V

at the beginning of the process. For a gas, the volume is entirely deter-

mined by the container holding the gas. Thus, equations involving gases will contain
the volume V as a variable, rather than focusing on a change in the volume from an ini-
tial value.

For a gas, it is useful to know how the quantities volume V, pressure P, and temper-

ature T are related for a sample of gas of mass m . In general, the equation that interre-
lates these quantities, called the equation of state, is very complicated. However, if the gas
is maintained at a very low pressure (or low density), the equation of state is quite sim-
ple and can be found experimentally. Such a low-density gas is commonly referred to
as an ideal gas.

It is convenient to express the amount of gas in a given volume in terms of the

number of moles n. One

mole of any substance is that amount of the substance that

S E C T I O N 19 . 5 • Macroscopic Description of an Ideal Gas

591

1.00

0.99
0.98
0.97

0.96
0.95

100

Temperature (

°C)

(g/cm

)

0.9999

1.0000

0.9998
0.9997
0.9996
0.9995

2 4 6 8 10 12

Temperature (

°C)

(g/cm

)

Figure 19.11 The variation in the density of water at atmospheric pressure with

temperature. The inset at the right shows that the maximum density of water occurs

at 4°C.

To be more specific, the assumption here is that the temperature of the gas must not be too low

(the gas must not condense into a liquid) or too high, and that the pressure must be low. The concept
of an ideal gas implies that the gas molecules do not interact except upon collision, and that the molec-
ular volume is negligible compared with the volume of the container. In reality, an ideal gas does not
exist. However, the concept of an ideal gas is very useful because real gases at low pressures behave as
ideal gases do.

contains

Avogadro’s number N

6.022 ) 10

of constituent particles (atoms or

molecules). The number of moles n of a substance is related to its mass m through the
expression

(19.7)

where M is the molar mass of the substance. The molar mass of each chemical element
is the atomic mass (from the periodic table, Appendix C) expressed in g/mol. For ex-
ample, the mass of one He atom is 4.00 u (atomic mass units), so the molar mass of
He is 4.00 g/mol. For a molecular substance or a chemical compound, you can add up
the molar mass from its molecular formula. The molar mass of stable diatomic oxygen
(O

) is 32.0 g/mol.

Now suppose that an ideal gas is confined to a cylindrical container whose volume

can be varied by means of a movable piston, as in Figure 19.12. If we assume that the
cylinder does not leak, the mass (or the number of moles) of the gas remains constant.
For such a system, experiments provide the following information. First, when the gas
is kept at a constant temperature, its pressure is inversely proportional to its volume
(Boyle’s law). Second, when the pressure of the gas is kept constant, its volume is di-
rectly proportional to its temperature (the law of Charles and Gay-Lussac). These
observations are summarized by the

equation of state for an ideal gas:

(19.8)

In this expression, known as the

ideal gas law, R is a constant and n is the number of

moles of gas in the sample. Experiments on numerous gases show that as the pressure
approaches zero, the quantity PV/nT approaches the same value R for all gases. For
this reason, R is called the

universal gas constant. In SI units, in which pressure is ex-

pressed in pascals (1 Pa # 1 N/m

) and volume in cubic meters, the product PV has

units of newton+ meters, or joules, and R has the value

(19.9)

If the pressure is expressed in atmospheres and the volume in liters (1 L # 10

), then R has the value

Using this value of R and Equation 19.8, we find that the volume occupied by 1 mol of
any gas at atmospheric pressure and at 0°C (273 K) is 22.4 L.

The ideal gas law states that if the volume and temperature of a fixed amount of

gas do not change, then the pressure also remains constant. Consider a bottle of cham-
pagne that is shaken and then spews liquid when opened, as shown in Figure 19.13.

R # 0.082

L+atm/mol+K

R # 8.314

J/mol+K

PV # nRT

n #

592

C H A P T E R 19 • Temperature

Gas

Active Figure 19.12 An ideal gas

confined to a cylinder whose volume

can be varied by means of a movable

piston.

Equation of state for an ideal

gas

At the Active Figures link

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

can choose to keep either the

temperature or the pressure

constant and verify Boyle’s law

and the law of Charles and

Gay–Lussac.

Figure 19.13 A bottle of champagne is

shaken and opened. Liquid spews out of the

opening. A common misconception is that

the pressure inside the bottle is increased

due to the shaking.

Steve Niedorf/Getty Images

Content .. 146 147 148 149 ..