Most substances expand when heated and contract when cooled. When a body’s temperature  
changes, its dimensions change as well. The term “thermal expansion” refers to the expansion of  
a body’s size as its temperature rise.  
Terms used  
✓
Expansion is the process whereby object increases its volume due to increase in  
temperature  
✓
Contraction is the process whereby object decreases its volume due to decrease in  
temperature  
Sources of thermal Energy  
The sun (the sun generates its energy by nuclear fusion)  
•
Solar thermal radiation is a process that emits thermal energy from the Sun, in the form of  
electromagnetic waves that can travel freely through space, without any intervening medium  
required for conduction of energy from one point to another  
•
The source of solar thermal energy itself is nuclear hydrogen radioisotopes, which are  
abundant in the Sun. These isotopes are constantly undergoing fusion reactions that lead to  
the formation of helium molecules with release of tremendous amount of radiant heat.  
Fossil fuels  
•
Fossil fuels emit thermal energy through the process of combustion, whereby a fuel is ignited  
and burnt in the presence of oxygen, with release of significant amounts of heat.  
•
Examples of fossil fuels that produce thermal energy are; natural gas, petroleum and coal.  
Nuclear energy (is energy generated from nuclear reactions)  
•
Nuclear energy originates from the splitting of uranium atoms – a process called fission. This  
generates heat to produce steam, which is used by a turbine generator to generate electricity.  
Geothermal energy (the heat energy delivered from the earth core)  
•
The process of geothermal energy production commences with nuclear fission as  
radioisotopes in the Earth's core undergo radioactive decay.  
•
•
Geothermal processes include nuclear fission, thermal conduction and fluid convection.  
The two types of thermal energy transfer that occur in geothermal systems, as indicated  
above, are conduction and convection. These mechanisms help to circulate geothermal heat  
in the subsurface, and are also responsible for the extraction of this heat and its use by  
humans.  
•
Thermal energy from geothermal processes and systems can be utilized to generate  
electricity, as well as for domestic heating and cooling purposes.  
Why Substance expands?  
•
Substance expands when heated because its particles vibrate more rapidly. As a result they  
collide and push each other further apart  
•
All states of matter (solids, liquids and gases) expand when heated.  
Explain what happen when solids (liquids or gases) are heated?  
•
When a solid is heated, its molecules gain kinetic energy and vibrate more vigorously. As the  
vibration become larger, the molecules are pushed further apart and the solid expands  
slightly in all directions.  
Explain what happen when solids (liquids or gases) are cooled?  
•
When solid is cooled, its molecules lose kinetic energy and have less vibration. As the  
vibration become lower, the molecules are pulled closer and the solid contracts slightly in all  
directions  
Thermal Expansion of Solids  
•
•
The expansion of solid substance is so small such that it is difficult to observe its changes  
As the temperature of a solid increases, the atoms vibrate with large amplitudes and the average  
separation between them increases. As a result, the entire solid occupies a large volume as the  
temperature increases.  
This can be seen in the ball and ring experiment as shown in the fig. below  
Observation and Explanations  
•
•
The metal ball can just pass through the ring at room temperature  
On heating, the metal ball expands. There is an increase in volume and the ball cannot pass  
through the ring  
•
On cooling , contraction occurs and the original volume is regained .The ball can now pass  
through the ring  
Linear expansivity (Coefficient of linear expansion)  
•
•
Is the ratio of increase in length to its original length per degree rise in temperature  
OR – Is the increase in length per unit length of the substance when its temperature rises by 1°C  
or 1 K.  
•
•
The SI unit for linear expansivity is K-1  
Mathematically: Linear expansivity =  
풊풏풄풓풆풂풔풆 풊풏 풍풆풏품풕풉 (풆풙풑풂풏풔풊풐풏)  
휟풍  
→
휶 =  
풐풓풊품풊풏풔풍 풍풆풏풕풉 ×풓풊풔풆 풊풏 풕풆풎풑풆풓풂풕풖풓풆  
풍ퟏ × 휟휽  
Where  
:
α = Linear expansivity  
Δθ = (θ2 – θ1) = rise in temperature  
θ2 = initial temperature, θ1 = final temperature  
ΔL = (푳ퟐ  
–
푳ퟏ) = increase in length  
L1 = original length, L2 = new length  
Linear expansivities of different substances.  
Substance  
Linear expansivity Substance  
Linear expansivity  
(K-1 ) x ퟏ−6  
(K-1 ) x ퟏ−6  
Aluminium  
Brass  
Copper  
Iron  
26  
19  
17  
10.2  
1
Steel  
Glass  
Pyrex glass  
Invar  
Silica  
11  
8.5  
3.0  
0.9  
0.42  
Diamond  
•
The knowledge of linear expansivity is used in designing various materials to ensure that  
they are able to operate well under varying thermal conditions. For instance ordinary glass  
has a higher linear expansivity than a pyrex glass. When hot water is put in an ordinary glass,  
it breaks but when a pyrex glass is used it does not crack. The pyrex glass has lower linear  
expansivity and cannot suffer very large forces of expansion while the ordinary glass does as  
it undergoes temperature changes.  
•
In building and construction, concrete is always reinforced using steel because both have the  
same linear expansivity.  
Worked examples  
1. The main span of San Francisco’s Golden Gate Bridge is 1275 m long at its coldest. The bridge  
is exposed to temperatures ranging from –15ºC to 40ºC. What is its change in length between  
these temperatures? Assume that the bridge is made entirely of steel.  
ANS; 푳ퟏ = ퟏퟐퟕퟓ풎, ∝풔풕풆풆풍= ퟏퟏ × ퟏퟎ−ퟔ푲−ퟏ, ∆푻 = ퟒퟎ + ퟏퟓ = ퟓퟓK  
From;  
Linear expansivity =  
휟풍  
풊풏풄풓풆풂풔풆 풊풏 풍풆풏품풕풉  
→
휶 =  
풐풓풊품풊풏풔풍 풍풆풏풕풉 × 풓풊풔풆 풊풏 풕풆풎풑풆풓풂풕풖풓풆  
풍
ퟏ
× 휟푻  
휟풍  
=
∝
→ 휟푳 = ∝풔풕풆풆풍 × 푳ퟏ × 휟푻  
풔풕풆풆풍  
풍
ퟏ
× 휟푻  
∴ 휟푳 = ∝풔풕풆풆풍 × 푳ퟏ × 휟푻 = ퟏퟏ × ퟏퟎ−ퟔ × ퟏퟐퟕퟓ × ퟓퟓ = 0.77m  
2. A rod is heated to 50°C to increase its length from 20 m to 30 m. Calculate the expansion  
coefficient if the room temperature is 20°C.  
ANS;  
휟풍  
풊풏풄풓풆풂풔풆 풊풏 풍풆풏품풕풉  
→
휶 =  
Linear expansivity =  
풐풓풊품풊풏풂풍 풍풆풏풕풉 × 풓풊풔풆 풊풏 풕풆풎풑풆풓풂풕풖풓풆  
풍
ퟏ
× 휟푻  
휟풍  
ퟑퟎ−ퟐퟎ  
ퟏퟎ  
= ퟏ. ퟔퟕ × ퟏퟎ−ퟐ푲−ퟏ  
ퟔퟎퟎ  
=
=
=
Therefore;  
∝
푹풐풅  
ퟐퟎ×(ퟓퟎ−ퟐퟎ)  
풍
ퟏ
× 휟푻  
3. A rod is heated to 30°C to increase its length by 15 m. Calculate the initial length if the  
expansion coefficient is 0.02 K-1 for a room temperature of 10°C.  
ANS; 풍ퟏ =? , 휟풍 = ퟏퟓ풎, 휶 = 0.02K-1, 휟푻 = ퟑퟎ − ퟏퟎ = ퟐퟎ0  
휟풍  
풊풏풄풓풆풂풔풆 풊풏 풍풆풏품풕풉  
→
휶 =  
From; Linear expansivity =  
풐풓풊품풊풏풂풍 풍풆풏풕풉 × 풓풊풔풆 풊풏 풕풆풎풑풆풓풂풕풖풓풆  
풍
ퟏ
× 휟푻  
ퟏퟓ  
ퟏퟒ  
ퟏퟓ  
0.02=  
=
= ퟑퟓ풎  
=
→
∴ 풍ퟏ  
ퟎ.ퟎퟐ×ퟐퟎ  
풍
ퟏ
× ퟐퟎ  
ퟎ.ퟒ  
4.  
An iron rod A and B are of equal length at 00C . If at 1000C they differ by 1mm find their  
lengths at 00C, Given that; ∝퐴= 8 × 10−6, ∝퐵= 12 × 10−6 0C-1  
ANS: L =2.5 m  
At 00;  
풍ퟏ푨 = 풍ퟏ푩 = 풙풎풎  
At 1000C; 풍ퟐ푩 − 풍ퟐ푨 = ퟏ풎풎 -----(i)  
∆푙  
=
---- (ii)  
But;  
∝
→
풍ퟐ푨 = ∝푨 × 풍ퟏ × ∆푻 + 풍ퟏ = (∝푨 × ∆푻 + ퟏ)풍ퟏ  
퐴
푙 ×∆푇  
1
∆푙  
=
∝
→
풍ퟐ푩 = ∝푩× 풍ퟏ × ∆푻 + 풍ퟏ = (∝푩 × ∆푻 + ퟏ)풍ퟏ ---- (iii)  
퐵
푙 ×∆푇  
1
Substitute the values of 풍ퟐ푨 and 풍ퟐ푩 into 풍ퟐ푩 − 풍ퟐ푨 = ퟏ  
(
)
(
)
∝푩× ∆푻 + ퟏ 풍ퟏ − ∝푨× ∆푻 + ퟏ 풍ퟏ = ퟏ  
−6  
−6  
8 × 10 × ퟏퟎퟎ + ퟏ 풍ퟏ = ퟏ  
(
)
(
)
12 × 10  
× ퟏퟎퟎ + ퟏ 풍ퟏ −  
ퟏ. ퟎퟎퟎퟎퟏퟐ풍ퟏ − ퟏ. ퟎퟎퟎퟎퟎퟖ풍ퟏ = ퟏ → ퟎ. ퟎퟎퟎퟎퟒ풍ퟏ = ퟏ  
ퟏ
Therefore; their lengths at 00C is 2500mm (2.5m)  
=
풍ퟏ  
=
ퟐퟓퟎퟎ풎풎  
;
ퟎ.ퟎퟎퟎퟎퟒ  
5. An iron tyre of diameter 50 cm at 150C is to be shrunk on to a wheel of diameter 50.35cm.  
To what temperature must the tyre be heated so that it will slip over the wheel with radial  
gap of 0.5 mm? (Linear expansivity of iron is ퟏ. ퟐ × ퟏퟎ−ퟓ푲−ퟏ  
)
ANS;  
The iron tire has a diameter, D1 of 50 cm at T1=150C, and that diameter must increase  
to 50.35 cm plus  
D2 = 50.35 cm + 2  
2
×
radial clearances of 0.5mm (or 0.05 cm) each, for a new diameter,  
0.05cm = 50.45 cm, see the figure below;  
∆퐷  
∆퐷  
50.45−50  
−ퟓ  
= ퟕퟓퟎ0C  
From; ∝ =  
→ ∆푻 =  
=
50×ퟏ.ퟐ×ퟏퟎ  
퐷 ×∆푇  
1
퐷 ×∝  
1
But; T2= T1 +∆푻 = ퟏퟓ + ퟕퟓퟎ = 7650C  
Therefore, 7650C is the temperature at which the tyre must be heated so that it will slip over the  
wheel of radial gap 0.5mm  
Class Activity – 6:1  
1. A rod is heated to 30°C to increase its length from 10 m to 25 m. Calculate the expansion  
coefficient if the room temperature is 10°C. ANS; 0.075K-1  
2. Calculate the expansion of a 15m copper pipe, when it is heated from 5 0C to 60 0C, if the  
linear expansivity of copper = 0.000012K-1 [ANS; ∆푳 = ퟎ. ퟎퟎퟗퟗ풎  
]
3. A block of concrete 5.0 m long expands to 5.00412 m when heated from 25°C to 100°C.  
Determine the linear expansivity of concrete.( ANS: α = 1.1 x ퟏퟎ−  
℃
= 1.1 x ퟏퟎ− K -1)  
ퟓ
ퟓ
0
4. The length of a wire at a temperature of 30 C is 1.002m. If the temperature of the wire is  
raised to 1050C and the linear expansivity of the wire is 1.89 x 10—5K-1, find the increase in  
the length of the wire. [ANS; ∆푳 = ퟏ. ퟒퟐ × ퟏퟎ−ퟑ풎  
]
5. The difference in length between a brass and an iron rod is 14 cm at 100 C. What must be the  
length of the iron for this difference to remain at 14 cm when both rods are heated to 1000 C?  
Given that the linear expansivity of brass = 19 x 10-6/K and iron = 12 x 10-6/K. (A: L = 38 cm)  
0
6. A metal rod has a length of 99.4cm at 200 C. At what temperature will its length be 100cm, if  
the linear expansivity of the metal is 0.000021K-1 [ANS; 485.60 C]  
7. A part of a steel tape used by a surveyor is 20.00m at 120. What is the overall length measured by using  
this part of the tape one hundred times on a warmer day corresponding to 220?  
ANS; 2000.22m]  
[  
Steel =1.1x10-5]  
8. A metal rod has a length of 100 cm at 2000 C. At what temperature will its length be 99.4 cm  
if the linear expansivity of the material of the rod is 0.00002/K (ANS: T2 = T1+∆푻 = - 1000 C)  
9. A metal pipe which of 1m long at 40°C increases in length by 0.3% when carrying a steam at  
100°C. Find the Coefficient of Linear Expansion (ANS: α = 5 x ퟏퟎ− K )  
ퟓ
10. A brick (30 cm x 18 cm x 10 cm) is at 20°C, If the brick heated to a temperature of 150°C,  
what will be its new dimensions? (The coefficient of linear expansion of concrete is 1.2 x 10-5  
K-1  
(ANS: 30.05 cm x 18.03 cm x 10.02cm)  
11. An iron plate at 20°C has a hole of radius of 8.92 mm in the centre, an iron rivet with radius of  
8.95 mm at 20°C, inserted into the hole. To what temperature the plate heated for the rivet to  
fit into the hole. (Linear expansivity of iron is 1.24 x 10-5K-1). ANS: 291°C  
12. Which is heavier, 1 dm3 of glass at 40 C or 1 dm3 of glass at 100 C? Explain your answer.  
13. A concrete railroad tie has a length 2.45m on a hot sunny day having a temperature of 350C.  
What is the length of the railroad tie in the winter when the temperature dips to –250C.Take  
the coefficient of linear expension of railroad as 12×10−5/ 0C  
∆풍  
[ANS; ∝=  
∆풍 =∝× 풍ퟏ × ∆푻 = −ퟎ. ퟎퟎퟏퟖ풎 → 풍ퟐ = 풍ퟏ + ∆풍 = 2.4482m]  
풍 ×∆푻  
ퟏ
14. Does it really help to run hot water over a tight metal lid on a glass jar before trying to  
open it? Explain your answer  
ANS; Yes. You can try to open a tight metal lid of a glass jar just by pouring hot water over it.  
The thermal expansion helps us to loosen the metal lid. Because it is the process in which  
the metal lid expands in the presence of heat therefore makes it easier for us to open it.  
15. How much gap should be left between two 20m tracks made of steel if it is laid at 220C and is  
to operate up to a temperature of 47 0C?  
ANS; Each track will expand half of the gap on each side. Hence the total expansion will the  
equal to the gap, ie,. ∆풍 =∝× 풍ퟏ × ∆푻 = ퟓ. ퟗ × ퟏퟎ−ퟑ  
풎
16. Eiffel tower is made up of iron and its height is roughly 300 m. During winter season  
0
(January) in France the temperature is 2 C and in hot summer its average temperature  
25°C. Calculate the change in height of Eiffel tower between summer and winter. The linear  
thermal expansion coefficient for iron α = 10 ×10−6 per 0C [ANS; 69cm]  
17. A rod is found to be 0.04 cm longer at 30 0C than it is at 10 0C. Calculate its length at 0 0C if  
coefficient of linear expansion, α = 2 x 10-5 /0C. [ANS; 1m or 100cm]  
18. A brass rod and an iron rod are each 1m in length at 10 0C. Find the difference in their  
lengths at 110 0C.  
∝
for brass is 19 x 10-6 / 0C and  
휶
for iron is 10 x 10-6 / 0C  
[A; 0.9mm or 0.0009m]  
.
Superficial expansion of solids (Areal expansion)  
•
•
is the ratio of increase in area to its original area for every degree increase in temperature  
If the surface area of a solid increases on being heated, it is called Area or superficial  
expansion. If the area of a solid is 푨ퟏ at 00C, then the area of that solid at T0C is given by;  
휟푨 = 휷푨ퟏ휟휽  
•
•
Whereby  
Therefore; 휷 =  
휷
is the coefficient of areal expansion  
푨 −푨  
ퟐ
ퟏ
푨 ×∆휽  
ퟏ
Relationship between coefficient of linear expansion and Areal expansion for a solid  
•
For isotropic materials, and for small expansions, the linear thermal expansion coefficient is  
one half of the area coefficient. To derive the relationship, let’s take a square of steel that has  
sides of length  
L
. The original area will be A=L2,and the new area, after a temperature  
increase, will be  
•
푨 + ∆푨 = (푳 + ∆푳)ퟐ = 푳ퟐ + ퟐ푳∆푳 + ∆푳ퟐ  
ퟐ
= 푳ퟐ + ퟐ푳 ∆푳 + ∆푳ퟐ  
푳
ퟐ
ퟐ푳 ∆푳  
≈ 푳ퟐ +  
,
풂풔 ∆푳ퟐ = (∝ ∆휽푳)ퟐ ≈ ퟎ  
푳
ퟐ
= 푳ퟐ + ퟐ푳 ∆푳 = 푨 + ퟐ푨 ∆푳  
푳
푳
= 푨 + ퟐ푨 ∆푳 = 푨 + ퟐ푨 ∝푳∆휽 = 푨 + ퟐ푨 ∝ ∆휽  
푳
푳
푨 + ∆푨 = 푨 + ퟐ푨 ∝ ∆휽 → ∆푨 = ퟐ푨 ∝ ∆휽  
(
)
But; 휟푨 = 휷푨휟휽  
→ 휷(푨휟휽) = ퟐ ∝ 푨∆휽 → 휷 = ퟐ ∝  
•
Therefore, coefficient of areal expansion is twice the coefficient of linear expansion  
풊풆, . 휷 = ퟐ ∝  
Volume expansion and expansivity of solids  
When a solid increases in volume, on being heated, it is called volume expansion. For  
•
an initial volume  
V
, at 휟휽 change in temperature, volume expansion is given by  
휟푽 = 휸푽ퟏ휟휽  
푽 −푽  
ퟏ
•
ퟐ
Whereby  
휸
is the coefficient of volume expansion  
,
Therefore; 휸 =  
푽 ×∆휽  
ퟏ
Relationship between coefficient of linear expansion and volume expansion for a solid  
•
For isotropic material, and for small expansions, the linear thermal expansion coefficient is  
one third the volumetric coefficient. To derive the relationship, let’s take a cube of steel that  
has sides of length  
L
. The original volume will be V=L3, and the new volume, after a  
temperature increase, will be:  
푽 + ∆푽 = (푳 + ∆푳) = 푳 + ퟑ푳 ∆푳 + ퟑ푳 ∆푳  
ퟐ
ퟑ
ퟑ
ퟑ
ퟐ
•
(
)
(
)
+ ∆푳  
ퟑ
= 푳ퟑ + ퟑ푳 ∆푳 + ퟑ푳 ∆푳  
+ ∆푳  
ퟐ
ퟑ
(
)
(
)
푳
ퟑ
ퟑ푳 ∆푳  
ퟐ
≈ 푳ퟑ +  
≈ ퟎ, ∆푳ퟑ ≈ ퟎ  
(
)
,
풂풔; ퟑ푳 ∆푳  
푳
= 푳ퟑ + ퟑ푳ퟑ ∆푳 = 푽 + ퟑ푽 ∆푳  
푳
푳
= 푽 + ퟑ푽 ∆푳 = 푽 + ퟑ푽 ∝푳∆휽 = 푽 + ퟑ푽 ∝ ∆휽  
푳
푳
푽 + ∆푽 = 푽 + ퟑ푽 ∝ ∆휽 → ∆푽 = ퟑ푽 ∝ ∆휽  
(
)
But; 휟푽 = 휸푽휟휽  
→ 휸(푽휟휽) = ퟑ ∝ 푽∆휽  
→ 휸 = ퟑ ∝  
•
Therefore; the coefficient of cubical expansion is three times the coefficient of linear  
expansion 풊풆, . 휸 = ퟑ ∝  
Class Activity – 6:2  
1. A rod is heated to 40°C to increase its area from 50 sq. m to 100 sq. m. Calculate the  
expansion coefficient if the room temperature is 25°C. [ANS; 0.067K-1]  
2. The coefficient of linear expansivity of a metal is 2.7 x 10-5K-1. If its original area is 600mm2,  
what will be the change in its area if its temperature changes from 600 C to 80 0C?  
ANS; 휷 = ퟐ ∝ = ퟐ × ퟐ. ퟕ × ퟏퟎ−ퟓ = ퟓ. ퟒ × ퟏퟎ−ퟓ, ∆푨 = ퟎ.64mm2  
3. A rod is heated to 30°C to increase its area by 40 sq. m. Calculate the initial area if the  
expansion coefficient is 0.05 K-1 for a room temperature of 10°C. [ANS; 40sq]  
4. An engineer has one piece of metal plate only to be used in fixing a gap on a metal bridge. If the  
steel plate has area of 0.8m2 at 250C, by how much must the engineer raise the temperature of  
the steel plate for it to fit a gap of 0.804m2? use coefficient of linear expansion of steel as  
12x10-5 0  
C
-1 [ANS; ∆푻 =2080C]  
5. A rod is heated to 40°C to increase its volume from 200 cu.m to 300 cu.m. Calculate the  
expansion coefficient if the room temperature is 20°C. [ANS; 0.025K-1]  
0
0
6. By how much will a steel rod 1 m long expand when heated from 25 C to 55 C? The coefficient  
of volume expansion of steel is 3 x 10-5 /0C.  
휟풍  
[A; 휸 = ퟑ ∝ → ∝ = ퟏ × ퟏퟎ−ퟓ푲−ퟏ , →  
휶 =  
∴ ∆풍 = ퟎ. ퟎퟎퟎퟑ풎 (0.3mm)]  
풍
ퟏ
풙 휟휽  
7. The diameter of an alluminium sphere at 200C is 2.5cm. If its diameter increases to 2.51cm  
when heated, what is the final temperature? the coefficient of linear expansion of aluminium is  
2.6x10-5 0C-1  
3
= ퟎ. ퟏcm  
ANS; 휸 = ퟑ ∝ = ퟕ. ퟖ × ퟏퟎ−ퟓ  
ퟒ
)
−ퟏ  
ퟑ
− 푹ퟑ  
(
푲 . ∆푽 = 푽ퟐ − 푽ퟏ = 흅 푹  
ퟐ
ퟏ
∆푽  
ퟑ
ퟒ
3
0
0
흅푹ퟑ  
= ퟖ. ퟏퟖcm  
∆휽 =  
휸푽  
ퟏ
푽ퟏ = ퟑ  
= ퟏퟓퟔ. ퟕ C , ∴ 휽ퟐ = 휽ퟏ + ∆휽 = ퟏퟕퟔ. ퟕ C  
ퟏ
8.  
The coefficient of linear expansion of iron is 0.000012K-1  
(c) Explain the meaning of this statement.  
(d) Calculate the superficial expansion of the iron.  
(e) Determine the cubical expansion of the iron.