CHAPTER 6
THERMAL
EXPANSION
The use of an equation
THERMAL EXPANSION
• Most substances expand
with increasing
temperature and contract
with decreasing
temperature. This
thermal expansion is
usually quite small, but it
can be an important
effect.
• Suppose the length of a solid rod is Lo at
some reference temperature To.
• If the temperature is changed by an amount
T = T – To
then the length changes by an amount
L = L – Lo
• Experiment shows that under usual
circumstances the change in length is
proportional to the temperature change, at
least for a small temperature change.
• We expect that the change in length
should be proportional to the reference
length Lo.
• That is, if the change in length of a rod 2m
long is 0.4mm, then the change in length
of a 1m rod should be 0.2mm.
• The change in length also depends on the
type of material. For example, copper and
iron rods of equal length at one
temperature have different lengths at other
temperatures.
THE EQUATION
• These features can be put into equation form by
introducing a coefficient that is characteristic of
the material.
• The average coefficient of linear expansion is
denoted by .
• The change in length L for a temperature
change T is given by
L = Lo T
THE EQUATION
• Although  depends on the temperature
interval T and the reference temperature
To, that dependence is usually negligible
for moderate temperature changes. The
coefficient  does not depend on the
length Lo.
• The dimension of  is reciprocal
temperature, and the commonly used unit
is reciprocal degrees Celsius (oC-1).
• Note that this unit is the same as the SI unit,
reciprocal Kelvin (K-1), because we are using
temperature changes.
• The next table lists the values of  for several
common substances.
Linear expansion
Substance (solid)
, 10-5 oC-1
Aluminium
2.4
Copper
1.8
Steel
1.1
Glass
0.1 - 1.3
Concrete
0.7 – 1.4
• Our discussion of thermal expansion has
been based on the change in length of a
rod, but the equation applies to any linear
dimension, such as the diameter of a
cylinder or even the radius of a circular
hole in a plate.
• You can think of thermal expansion as
analogous to a photographic enlargement
in which every linear feature of an isotropic
substance changes proportionally. (An
isotropic substance has the same
properties in all directions.)
• A bimetallic strip bends as its temperature is
increased. The strip is a complete of two strips
of different metals bonded together.
• Why does it bend?
• Ball-and-ring thermal expansion demonstration. The
ball barely fits through the ring when both are at
room temperature. If the temperature of the ball
alone is increased, it will not fit through the ring. If
the temperatures of both the ball and the ring are
increased, the ball again fits through the ring. This
shows that when the ring expands, the size of the
hole increases.
and for LIQUIDS?
• For liquids, as well as for solids, it is convenient
to consider volume changes that correspond to
temperature changes.
• If Vo is the volume of a substance at a
reference temperature To, then the change in
volume V that accompanies a temperature
change T is given by
V =  Vo T
where  is the average coefficient of volume
expansion.
• The following table shows some values of  for
some liquids.
Volume expansion
Substance (liquid)
, 10-5 oC-1
Methanol
113
Glycerin
49
Mercury
18
Turpentine
90
Acetone
132
• Since the product of three linear
dimensions gives a volume, it is not
surprising that linear expansion and
volume expansion are related.
• Experiments shows that
 = 3
for an isotropic substance.
Characteristics of water
• Notable by its absence from the tables is
liquid water.
• The positive values of  and  for the
substances in that table indicate that they
expand with increasing temperature.
• Water also expands (but not linearly) with
a temperature increase in the temperature
range from about 4 to 100 oC.
• However, between 0oC and about 4oC,
water contracts with a temperature
increase.
• This behaviour is shown in the following figure
in which the volume of 1 kg of water is plotted
versus temperature.
• This variation of volume or of density with
temperature is responsible for the stratification
that sometimes occurs in large bodies of fresh
water.
• The anomalous thermal expansion of water is
ultimately due to the interaction of the unusually
shaped water molecules.
Example 1
• Expanding concrete. A concrete slab has length of
12m at -5oC on a winter day. What change in length
occurs from winter to summer, when the temperature
is 35oC?
• Solution. From 1st table,  for concrete is around
1 × 10-5 C-1. Using the first equation, we have
L
= Lo T
= (1 × 10-5 C-1)(12 m)(40 oC)
= 5 mm
• Adjacent slabs in highways and in sidewalks are often
separated by pliable spacers to allow for this kind of
expansion.
Exercise 1
1. A copper rod lengthens by 5mm when its
temperature increases by 40oC. What is
the original length of the rod?
2. A metal rod has a length of 1m. It is
heated through 200oC. If the coefficient
of linear expansion () of the metal is
0.00002 /oC, find the expansion.
3. 50m copper piping is heated through
70oC. What is the expansion?
Example 2
• Volume expansion of a sphere. An aluminium
sphere has a radius R of 3.000 mm at 100.0 oC. What
is its volume at 0.0 oC?
• Solution.
The volume of sphere (4R3/3) at
100oC is V = 113.1 mm3.
From the 1st table
 = 2.4 × 10-5 oC-1
and
 = 3 = 7.2 × 10-5 oC-1
Applying 2nd equation, we obtain
V
= (7.2 × 10-5 oC-1)(113.1 mm3)( -100.0 oC)
= - 0.81 mm3
The volume at 0oC is 113.1 – 0.8 = 112.2 mm3.
• An alternative approach is to evaluate the
radius of the sphere (a linear dimension)
at 0oC and hence calculate the volume
from V = 4R3/3 with the new radius obtained.
• The answer should be the same in any way.
Exercise 2
1. What temperature change would cause
the volume of mercury to change by 0.1
percent?
Questions
1)
a)
b)
c)
A steel rule is calibrated at 22oC against a
standard so that the distance between
numbered divisions is 10.00 mm.
What is the distance between these divisions
when the rule is at -5oC?
If a nominal length of 1m is measured with
the rule at this lower temperature, what
percent error is made?
What absolute error is made for a 100-m
length?
Questions
2)
a)
b)
c)
d)
A copper plate at 0oC has thickness of
5.00mm and a circular hole of radius
75.0mm. Its temperature is raised to 220oC.
Determine the values at this temperature of:
the thickness of the plate,
the radius of the hole,
the circumference of the boundary of the
circular hole,
the area of the hole in the plate.
Questions
3)
a)
b)
A steel shaft has diameter 42.51mm at 28oC.
It is to be fitted to a steel pulley with a
circular hole of diameter 42.50mm at that
temperature.
By how much must the temperature of the
shaft be reduced so that it can fit in the hole?
Suppose the temperature of the entire
structure is reduced to -5oC after the shaft
has been fitted. Will the shaft come loose?
Explain.
Questions
4)
a)
b)
c)
d)
The density of aluminium is 2692 kg/m3 at 20oC.
What is the mass of an aluminium sphere (V =
4R3/3) whose radius R at this temperature is
25.00mm?
What is the mass of the aluminium at 100.0oC?
What is the density of aluminium at 100oC?
What are the answers to parts (b) and (c) if the
aluminium is in the shape of a cube?
Questions
5)
A glass ( = 2.2  10-5 oC-1) bulb is
completely filled with 176.2ml of
mercury ( = 2.2  10-5 oC-1) at
0.0oC. The bulb is fitted, as
illustrated in the following figure,
with a glass tube of inside diameter
2.5mm at 0.0oC. How high does the
mercury rise in the tube if the
temperature of the system is
raised to 50.0oC? The change in
diameter of the glass tube may be
neglected. Why?