Physics For Scientists And Engineers 6E - part 150

Problems

597

size of the expansion joints that must be built into the
structure.
A copper telephone wire has essentially no sag between

poles 35.0 m apart on a winter day when the temperature
is ! 20.0°C. How much longer is the wire on a summer day
when T

35.0°C?

10. The concrete sections of a certain superhighway are de-

signed to have a length of 25.0 m. The sections are poured
and cured at 10.0°C. What minimum spacing should the
engineer leave between the sections to eliminate buckling
if the concrete is to reach a temperature of 50.0°C?

11. A pair of eyeglass frames is made of epoxy plastic. At room

temperature (20.0°C), the frames have circular lens holes
2.20 cm in radius. To what temperature must the frames
be heated if lenses 2.21 cm in radius are to be inserted in
them? The average coefficient of linear expansion for
epoxy is 1.30 ) 10

(°C)

12. Each year thousands of children are badly burned by hot

tap water. Figure P19.12 shows a cross-sectional view of an
antiscalding  faucet  attachment  designed  to  prevent  such
accidents.  Within  the  device,  a  spring  made  of  material
with  a  high  coefficient  of  thermal  expansion  controls  a
movable plunger. When the water temperature rises above
a preset safe value, the expansion of the spring causes the
plunger to shut off the water flow. If the initial length L of
the unstressed spring is 2.40 cm and its coefficient of lin-
ear  expansion  is  22.0 ) 10

(°C)

, determine the in-

crease in length of the spring when the water temperature
rises by 30.0°C. (You will find the increase in length to be
small. For this reason actual devices have a more compli-
cated mechanical design, to provide a greater variation in
valve opening for the temperature change anticipated.)

floorboard hold stationary the ends of this section of copper
pipe. Find the magnitude and direction of the displacement
of the pipe elbow when the water flow is turned on, raising
the temperature of the pipe from 18.0°C to 46.5°C.

Figure P19.12

The active element of a certain laser is made of a glass

rod 30.0 cm long by 1.50 cm in diameter. If the tempera-
ture of the rod increases by 65.0°C, what is the increase in
(a) its length, (b) its diameter, and (c) its volume? Assume
that the average coefficient of linear expansion of the glass
is 9.00 ) 10

(°C)

14. Review problem. Inside the wall of a house, an L-shaped

section  of  hot-water  pipe  consists  of  a  straight  horizontal
piece  28.0 cm  long,  an  elbow,  and  a  straight  vertical  piece
134 cm  long  (Figure  P19.14).  A  stud  and  a  second-story

13.

Figure P19.14

15.

A brass ring of diameter 10.00 cm at 20.0°C is heated and
slipped over an aluminum rod of diameter 10.01 cm at
20.0°C. Assuming the average coefficients of linear expan-
sion are constant, (a) to what temperature must this com-
bination be cooled to separate them? Is this attainable?
(b) What If? What if the aluminum rod were 10.02 cm in
diameter?

16. A square hole 8.00 cm along each side is cut in a sheet of

copper. (a) Calculate the change in the area of this hole
if the temperature of the sheet is increased by 50.0 K.
(b) Does this change represent an increase or a decrease
in the area enclosed by the hole?

17.

The average coefficient of volume expansion for carbon
tetrachloride is 5.81 ) 10

(°C)

. If a 50.0-gal steel con-

tainer is filled completely with carbon tetrachloride when
the temperature is 10.0°C, how much will spill over when
the temperature rises to 30.0°C?

18.

At  20.0°C,  an  aluminum  ring  has  an  inner  diameter  of
5.000 0 cm and a brass rod has a diameter of 5.050 0 cm.
(a)  If  only  the  ring  is  heated,  what  temperature  must  it
reach so that it will just slip over the rod? (b) What If? If
both  are  heated  together,  what  temperature  must they
both reach so that the ring just slips over the rod? Would
this latter process work?

19.

A volumetric flask made of Pyrex is calibrated at 20.0°C. It
is filled to the 100-mL mark with 35.0°C acetone. (a) What
is the volume of the acetone when it cools to 20.0°C?
(b) How significant is the change in volume of the flask?

598

C H A P T E R 19 • Temperature

20.

A concrete walk is poured on a day when the temperature
is 20.0°C in such a way that the ends are unable to move.
(a) What is the stress in the cement on a hot day of
50.0°C? (b) Does the concrete fracture? Take Young’s
modulus for concrete to be 7.00 ) 10

N/m

and the com-

pressive strength to be 2.00 ) 10

N/m

A hollow aluminum cylinder 20.0 cm deep has an internal

capacity  of  2.000 L  at  20.0°C.  It  is  completely  filled  with
turpentine  and  then  slowly  warmed  to  80.0°C.  (a)  How
much  turpentine  overflows?  (b)  If  the  cylinder  is  then
cooled  back  to  20.0°C,  how  far  below  the  cylinder’s  rim
does the turpentine’s surface recede?

22.

A beaker made of ordinary glass contains a lead sphere of
diameter  4.00 cm  firmly  attached  to  its  bottom.  At  a  uni-
form  temperature  of  ! 10.0°C,  the  beaker  is  filled  to  the
brim  with  118 cm

of mercury, which completely covers

the sphere. How much mercury overflows from the beaker
if the temperature is raised to 30.0°C?

23.

A steel rod undergoes a stretching force of 500 N. Its cross-
sectional area is 2.00 cm

. Find the change in temperature

that would elongate the rod by the same amount as the
500-N force does. Tables 12.1 and 19.1 are available to you.

24.

The Golden Gate Bridge in San Francisco has a main span
of length 1.28 km—one of the longest in the world. Imag-
ine that a taut steel wire with this length and a cross-sec-
tional area of 4.00 ) 10

is laid on the bridge deck

with its ends attached to the towers of the bridge, on a sum-
mer  day  when  the  temperature  of  the  wire  is  35.0°C.
(a) When winter arrives, the towers stay the same distance
apart  and  the  bridge  deck  keeps  the  same  shape  as  its  ex-
pansion  joints  open.  When  the  temperature  drops  to
!

10.0°C, what is the tension in the wire? Take Young’s

modulus for steel to be 20.0 ) 10

N/m

. (b) Permanent

deformation occurs if the stress in the steel exceeds its elas-
tic limit of 3.00 ) 10

N/m

. At what temperature would

this happen? (c) What If? How would your answers to
(a) and (b) change if the Golden Gate Bridge were twice
as long?

25. A certain telescope forms an image of part of a cluster of

stars  on  a  square  silicon  charge-coupled  detector  (CCD)
chip  2.00 cm  on  each  side.  A  star  field  is  focused  on  the
CCD chip when it is first turned on and its temperature is
20.0°C.  The  star  field  contains  5 342  stars  scattered  uni-
formly. To make the detector more sensitive, it is cooled to
!

100°C. How many star images then fit onto the chip?

The average coefficient of linear expansion of silicon is
4.68 ) 10

(°C)

Section 19.5 Macroscopic Description of an Ideal Gas

26. Gas is contained in an 8.00-L vessel at a temperature of

20.0°C and a pressure of 9.00 atm. (a) Determine the

Note: Problem 8 in Chapter 1 can be assigned with this
section.

21.

number of moles of gas in the vessel. (b) How many mole-
cules are there in the vessel?
An automobile tire is inflated with air originally at 10.0°C
and normal atmospheric pressure. During the process, the
air is compressed to 28.0% of its original volume and the
temperature is increased to 40.0°C. (a) What is the tire
pressure? (b) After the car is driven at high speed, the tire
air temperature rises to 85.0°C and the interior volume of
the tire increases by 2.00%. What is the new tire pressure
(absolute) in pascals?

28.

A tank having a volume of 0.100 m

contains helium gas at

150 atm. How many balloons can the tank blow up if each
filled balloon is a sphere 0.300 m in diameter at an ab-
solute pressure of 1.20 atm?
An auditorium has dimensions 10.0 m ) 20.0 m ) 30.0 m.
How many molecules of air fill the auditorium at 20.0°C
and a pressure of 101 kPa?

30. Imagine a baby alien playing with a spherical balloon the

size of the Earth in the outer solar system. Helium gas in-
side the balloon has a uniform temperature of 50.0 K due
to  radiation  from  the  Sun.  The  uniform  pressure  of  the
helium  is  equal  to  normal  atmospheric  pressure  on
Earth. (a)  Find  the  mass  of  the  gas  in  the  balloon.
(b)  The  baby  blows  an  additional  mass  of  8.00 ) 10

of helium into the balloon. At the same time, she wanders
closer to the Sun and the pressure in the balloon doubles.
Find the new temperature inside the balloon, whose vol-
ume remains constant.

31.

Just 9.00 g of water is placed in a 2.00-L pressure cooker
and heated to 500°C. What is the pressure inside the
container?

32. One mole of oxygen gas is at a pressure of 6.00 atm and a

temperature of 27.0°C. (a) If the gas is heated at constant
volume until the pressure triples, what is the final tempera-
ture? (b) If the gas is heated until both the pressure and
volume are doubled, what is the final temperature?

The mass of a hot-air balloon and its cargo (not in-

cluding the air inside) is 200 kg. The air outside is at
10.0°C and 101 kPa. The volume of the balloon is 400 m

To what temperature must the air in the balloon be heated
before the balloon will lift off? (Air density at 10.0°C is
1.25 kg/m

34. Your father and your little brother are confronted with the

same  puzzle.  Your  father’s  garden  sprayer  and  your
brother’s water cannon both have tanks with a capacity of
5.00 L  (Figure  P19.34).  Your  father  inserts  a  negligible
amount  of  concentrated  insecticide  into  his  tank.  They
both  pour  in  4.00 L  of  water  and  seal  up  their  tanks,  so
that  they  also  contain  air  at  atmospheric  pressure.  Next,
each uses a hand-operated piston pump to inject more air,
until  the  absolute  pressure  in  the  tank  reaches  2.40 atm
and  it  becomes  too  difficult  to  move  the  pump  handle.
Now  each  uses  his  device  to  spray  out  water— not  air—
until the stream becomes feeble, as it does when the pres-
sure in the tank reaches 1.20 atm. Then he must pump it
up again, spray again, and so on. In order to spray out all
the water, each finds that he must pump up the tank three

33.

29.

27.

Problems

599

35. (a) Find the number of moles in one cubic meter of an

ideal gas at 20.0°C and atmospheric pressure. (b) For air,
Avogadro’s number of molecules has mass 28.9 g. Calcu-
late the mass of one cubic meter of air. Compare the result
with the tabulated density of air.

36. The void fraction of a porous medium is the ratio of the

void volume to the total volume of the material. The void
is  the  hollow  space  within  the  material;  it  may  be  filled
with  a  fluid.  A  cylindrical  canister  of  diameter  2.54 cm
and height 20.0 cm is filled with activated carbon having
a void fraction of 0.765. Then it is flushed with an ideal
gas at 25.0°C and pressure 12.5 atm. How many moles of
gas  are  contained  in  the  cylinder  at  the  end  of  this
process?

37. A cube 10.0 cm on each edge contains air (with equivalent

molar mass 28.9 g/mol) at atmospheric pressure and tem-
perature 300 K. Find (a) the mass of the gas, (b) its weight,
and (c) the force it exerts on each face of the cube.
(d) Comment on the physical reason why such a small
sample can exert such a great force.

38.

At 25.0 m below the surface of the sea (, # 1 025 kg/m

where the temperature is 5.00°C, a diver exhales an air
bubble having a volume of 1.00 cm

. If the surface temper-

ature of the sea is 20.0°C, what is the volume of the bubble
just before it breaks the surface?
The pressure gauge on a tank registers the gauge pressure,
which is the difference between the interior and exterior
pressure. When the tank is full of oxygen (O

), it contains

12.0 kg of the gas at a gauge pressure of 40.0 atm. Deter-
mine the mass of oxygen that has been withdrawn from
the tank when the pressure reading is 25.0 atm. Assume
that the temperature of the tank remains constant.

39.

40.

Estimate the mass of the air in your bedroom. State the
quantities you take as data and the value you measure or
estimate for each.

41. A popular brand of cola contains 6.50 g of carbon dioxide

dissolved in 1.00 L of soft drink. If the evaporating carbon
dioxide is trapped in a cylinder at 1.00 atm and 20.0°C,
what volume does the gas occupy?

42. In state-of-the-art vacuum systems, pressures as low as

Pa are being attained. Calculate the number of mole-

cules in a 1.00-m

vessel at this pressure if the temperature

is 27.0°C.

43.

A room of volume V contains air having equivalent molar
mass M (in g/mol). If the temperature of the room is
raised from T

to T

, what mass of air will leave the room?

Assume that the air pressure in the room is maintained
at P

44.

A  diving  bell  in  the  shape  of  a  cylinder  with  a  height
of 2.50 m  is  closed  at  the  upper  end  and  open  at  the
lower  end.  The  bell  is  lowered  from  air  into  sea  water
(, # 1.025 g/cm

). The air in the bell is initially at

20.0°C. The bell is lowered to a depth (measured to the
bottom of the bell) of 45.0 fathoms or 82.3 m. At this
depth the water temperature is 4.0°C, and the bell is in
thermal equilibrium with the water. (a) How high does
sea water rise in the bell? (b) To what minimum pressure
must the air in the bell be raised to expel the water that
entered?

Additional Problems

45.

A student measures the length of a brass rod with a steel
tape at 20.0°C. The reading is 95.00 cm. What will the tape
indicate for the length of the rod when the rod and the
tape are at (a) ! 15.0°C and (b) 55.0°C?

46.

The density of gasoline is 730 kg/m

at 0°C. Its average

coefficient of volume expansion is 9.60 ) 10

/°C. If

1.00 gal of gasoline occupies 0.003 80 m

, how many extra

kilograms of gasoline would you get if you bought 10.0 gal
of gasoline at 0°C rather than at 20.0°C from a pump that
is not temperature compensated?
A mercury thermometer is constructed as shown in Figure
P19.47. The capillary tube has a diameter of 0.004 00 cm,

47.

Figure P19.34

∆T

∆h

Figure P19.47 Problems 47 and 48.

times. This is the puzzle: most of the water sprays out as a
result of the second pumping. The first and the third
pumping-up processes seem just as difficult, but result in a
disappointingly small amount of water coming out. Ac-
count for this phenomenon.

600

C H A P T E R 19 • Temperature

and the bulb has a diameter of 0.250 cm. Neglecting the
expansion of the glass, find the change in height of the
mercury column that occurs with a temperature change of
30.0°C.

48.

A liquid with a coefficient of volume expansion ' just fills
a spherical shell of volume V

at a temperature of T

(see

Fig. P19.47). The shell is made of a material that has an av-
erage coefficient of linear expansion &. The liquid is free
to expand into an open capillary of area A projecting from
the top of the sphere. (a) If the temperature increases by
"

T, show that the liquid rises in the capillary by the

amount "h given by "h # (V

/A)(' ! 3&)"T. (b) For a

typical system, such as a mercury thermometer, why is
it a good approximation to neglect the expansion of
the shell?

49.

Review  problem.  An  aluminum  pipe,  0.655 m  long  at
20.0°C  and  open  at  both  ends,  is  used  as  a  flute.  The
pipe  is  cooled  to  a  low  temperature  but  then  is  filled
with  air  at  20.0°C  as  soon  as  you  start  to  play  it.  After
that,  by  how  much  does  its  fundamental  frequency
change as the metal rises in temperature from 5.00°C to
20.0°C?

50.

A cylinder is closed by a piston connected to a spring of
constant 2.00 ) 10

N/m (see Fig. P19.50). With the

spring relaxed, the cylinder is filled with 5.00 L of gas at a
pressure of 1.00 atm and a temperature of 20.0°C. (a) If
the piston has a cross-sectional area of 0.010 0 m

and neg-

ligible mass, how high will it rise when the temperature is
raised to 250°C? (b) What is the pressure of the gas at
250°C?

erated in the respiration recycling of three astronauts
during one week of flight is stored in an originally empty
150-L tank at ! 45.0°C, what is the final pressure in the
tank?

A vertical cylinder of cross-sectional area A is fitted

with a tight-fitting, frictionless piston of mass m (Fig.
P19.53). (a) If n moles of an ideal gas are in the cylinder at
a temperature of T, what is the height h at which the pis-
ton is in equilibrium under its own weight? (b) What is the
value for h if n # 0.200 mol, T # 400 K, A # 0.008 00 m

and m # 20.0 kg?

53.

°C

250

°C

Figure P19.50

Gas

Figure P19.53

A liquid has a density ,. (a) Show that the fractional

change in density for a change in temperature "T is
"

/, # ! ' "T. What does the negative sign signify?

(b) Fresh water has a maximum density of 1.000 0 g/cm

at 4.0°C. At 10.0°C, its density is 0.999 7 g/cm

. What is

for water over this temperature interval?

52. Long-term space missions require reclamation of the oxy-

gen  in  the  carbon  dioxide  exhaled  by  the  crew.  In  one
method of reclamation, 1.00 mol of carbon dioxide pro-
duces  1.00 mol  of  oxygen  and  1.00 mol  of  methane  as  a
byproduct.  The  methane  is  stored  in  a  tank  under  pres-
sure and is available to control the attitude of the space-
craft  by  controlled  venting.  A  single  astronaut  exhales
1.09 kg of carbon dioxide each day. If the methane gen-

51.

54.

A bimetallic strip is made of two ribbons of dissimilar met-
als bonded together. (a) First assume the strip is originally
straight. As they are heated, the metal with the greater aver-
age coefficient of expansion expands more than the other,
forcing the strip into an arc, with the outer radius having a
greater circumference (Fig. P19.54a). Derive an expression
for the angle of bending - as a function of the initial length
of the strips, their average coefficients of linear expansion,
the change in temperature, and the separation of the cen-
ters of the strips ("r # r

). (b) Show that the angle of

bending decreases to zero when "T decreases to zero and
also  when  the  two  average  coefficients  of  expansion  be-
come  equal.  (c)  What  If?  What  happens  if  the  strip  is
cooled? (d) Figure P19.54b shows a compact spiral bimetal-
lic strip in a home thermostat. The equation from part (a)
applies to it as well, if - is interpreted as the angle of addi-
tional bending caused by a change in temperature. The in-
ner end of the spiral strip is fixed, and the outer end is free
to move. Assume the metals are bronze and invar, the thick-
ness of the strip is 2 "r # 0.500 mm, and the overall length
of the spiral strip is 20.0 cm. Find the angle through which
the  free  end  of  the  strip  turns  when  the  temperature
changes  by  one  Celsius  degree.  The  free  end  of  the  strip
supports a capsule partly filled with mercury, visible above
the strip in Figure P19.54b. When the capsule tilts, the mer-
cury shifts from one end to the other, to make or break an
electrical contact switching the furnace on or off.

Content .. 148 149 150 151 ..