# Thermal Expansion

## What is called Thermal Expansion?

Upon heating, all matters which are either solid, liquid or gas expand in length, area or volume. This is called thermal expansion.

All matters are composed of atoms and molecules. Upon heating, increase in the of vibrations of these atoms and molecules cause thermal expansion. The cause of thermal expansion can be explained as follows –

• All matters are made of small particles such as atoms and molecules.
• These particles remain in vibration in random fashion.
• Upon heating, internal energy of body increases which increases the amplitude of vibrations of atoms and molecules.
• Hence, the volume of body increases causing thermal expansion.

### Linear Thermal Expansion

Increase in length of a body upon heating, is called Linear Thermal Expansion. It is associated with the thermal expansion of solids.

Suppose a solid rod of length ( l )  is heated through a temperature raise ( \Delta T ) . Due to thermal expansion, let its final length becomes ( l' ) .

Then, law of linear thermal expansion states that –

1. Increase in length is proportional to the rise in temperature i.e. ( l' - l ) \propto \Delta T
2. Increase in length is proportional to the original length i.e. ( l' - l ) \propto l

Combining above two relations, we will get –

( l' - l ) \propto l \Delta T

Or, \quad ( l' - l ) = \alpha l \Delta T .

So, \quad l' = ( l + \alpha l \Delta T )

= l ( 1 + \alpha \Delta T )

Here ( \alpha ) is a constant of proportionality. It is called coefficient of linear expansion.

Therefore, coefficient of linear expansion is given by –

\alpha = \left [ \frac { \left ( l' - l \right ) }{ l \Delta T } \right ]

= \left ( \frac { \Delta l }{ l } \right ) \left ( \frac { 1 }{ \Delta T } \right )

Hence, coefficient of linear expansion of a material is defined as the increase in length per unit original length per degree rise in temperature.

Unit for coefficient of linear expansion is ( \degree C^{-1} ) or ( \degree K^{-1} )

### Superficial Thermal Expansion

Increase in surface area of a surface upon heating, is called Superficial Thermal Expansion. It is associated with thermal expansion of solids and liquids.

Suppose, a metal sheet of surface area ( S )  is heated through a temperature of ( \Delta T ) . Due to thermal expansion, let its final surface area becomes ( S' ) .

Then, law of superficial expansion states that –

1. Increase in surface area is proportional to the rise in temperature i.e. \left ( S' - S \right ) \propto \Delta T
2. Increase in surface area is proportional to original surface area i.e. \left ( S' - S \right ) \propto S

Combining above two relations, we will get –

( S' - S ) \propto S \Delta T

Or, \quad ( S' - S ) = \beta S \Delta T

So, \quad S' = ( S + \beta S \Delta T )

= S ( 1 + \beta \Delta T )

Here \left ( \beta \right ) is constant of proportionality. It is called coefficient of superficial expansion.

Therefore, coefficient of superficial expansion is given by –

\beta = \left [ \frac { \left ( S' - S \right ) }{ S \Delta T } \right ]

= \left ( \frac { \Delta S }{ S } \right ) \left ( \frac { 1 }{ \Delta T } \right )

Hence, coefficient of superficial expansion of a material is defined as the increase in surface area per unit original surface area per degree rise in temperature.

Unit for coefficient of superficial expansion is ( \degree C^{-1} ) or ( \degree K^{-1} )

### Volumetric Thermal Expansion

Increase in volume of a body upon heating, is called Volumetric Thermal Expansion. It is associated with thermal expansion of solids, liquids and gases.

Suppose, a solid block of volume ( V )  is heated through a temperature of ( \Delta T ) . Due to thermal expansion, let its final volume becomes ( V' ) .

Then, law of volumetric expansion states that –

1. Increase in volume is proportional to the rise in temperature i.e. \left ( V' - V \right ) \propto \Delta T
2. Increase in volume is proportional to original volume i.e. \left ( V' - V \right ) \propto V

Combining above two relations, we will get –

( V' - V ) \propto V \Delta T

= \gamma V \Delta T .

Or, \quad V' = V ( 1 + \gamma \Delta T )

Here ( \gamma ) is the constant of proportionality. It is called coefficient of volumetric expansion.

Therefore, coefficient of volumetric expansion is given by –

\gamma = \left [ \frac { \left ( V' - V \right ) }{ V \Delta T } \right ]

= \left ( \frac { \Delta V }{ V } \right ) \left ( \frac { 1 }{ \Delta T } \right )

Hence, coefficient of volumetric expansion of a material is defined as the increase in volume per unit original volume per degree rise in temperature.

Unit for coefficient of volumetric expansion is ( \degree C^{-1} ) or ( \degree K^{-1} )

Volumetric expansion is also called as cubical expansion.

## Relation between Coefficients of Expansions

Consider about a cube of sides ( l ) . Its original volume is ( V = l^3 ) . Suppose the cube is heated so that its temperature rises by ( \Delta T ) . Due to linear thermal expansion, let its sides will become ( l' ) .

From linear expansion, we get the relation –

l' = l ( 1 + \alpha \Delta T )

Therefore, final volume of the cube will be –

V' = ( l' )^3 = l^3 ( 1 + \alpha \Delta T )^3

Or, \quad V' = V ( 1 + 3 \alpha \Delta T + 3 \alpha ^2 \Delta T^2 + \alpha ^3 \Delta T^3 )

Because, ( \alpha ) is small. So its higher terms containing ( \alpha ^2 ) \ \text {and} \ ( \alpha ^3 ) can be neglected.

Then, \quad V' = V ( 1 + 3 \alpha \Delta T )

Or, \quad ( V' - V ) = 3 V \alpha \Delta T

Therefore, change in volume of cube is given by –

\Delta V = 3 V \alpha \Delta T ….. (1)

By volumetric expansion, we get the relation –

\gamma = \left ( \frac { \Delta V }{ V } \right ) \left ( \frac { 1 }{ \Delta T } \right )

Or, \quad \Delta V = \gamma V \Delta T …… (2)

By comparing equations (1) and (2), we get –

\gamma V \Delta T = 3 V \alpha \Delta T

Or, \quad \gamma = 3 \alpha

Similarly, it can be proved that –

\beta = 2 \alpha

Hence, \quad ( \gamma = 3 \alpha ) and ( \beta = 2 \alpha )

## Differential Expansion

Coefficient for linear expansion ( \alpha ) depends upon the nature of material. It is different for different materials.

Therefore, upon heating by the same extent, the elongation is different for different materials. This is called differential expansion of solids.

Consider about two rods of different materials of lengths ( l_1 ) and ( l_2 ) .

Let, ( l_1 > l_2 ) and  ( l_1 - l_2 ) = S

Both the rods are initially at temperature ( t_1 \degree C ) and let they are heated to increase their temperature to ( t_2 \degree C )

Let, the new lengths of the rods are ( l'_1 ) and ( l'_2 )

From linear expansion of rods –

l'_1 = l_1 [ 1 + \alpha_1 ( t_2 - t_1 ) ]

And \quad l'_2 = l_2 [ 1 + \alpha_2 ( t_2 - t_1 ) ]

New difference in their lengths will be –

S' = ( l'_2 - l'_1 ) = l_2 [ 1 + \alpha_2 ( t_2 - t_1 ) ] - l_1 [ 1 + \alpha_1 ( t_2 - t_1 ) ]

Or, \quad S' = ( l_2 - l_1 ) + ( l_2 \alpha_2 - l_1 \alpha_1 ) ( t_2 - t_1 )

= S + ( l_2 \alpha_2 - l_1 \alpha_1 ) ( t_2 - t_1 )

Therefore, \quad ( S' - S ) = ( l_2 \alpha_2 - l_1 \alpha_1 ) ( t_2 - t_1 )

Sometimes, two rods of same length but of different materials are bolted together, so that the difference in their lengths remain fixed. This is done to get some desired effects of thermal expansion. This is also called bimetallic strip such as thermostats.

For this type of combination \quad ( S' - S ) = 0

Or, \quad ( l_2 \alpha_2 - l_1 \alpha_1 ) ( t_2 - t_1 ) = 0

Or, \quad ( \alpha_2 l_2 - \alpha_1 l_1 ) = 0

Therefore, \quad \alpha_2 l_2 = \alpha_1 l_1

## Bimetallic Strip

Different materials have different coefficient of thermal expansion. Extent of elongation and contraction upon heating or cooling by same amount is also different.

• When strips of different materials are welded together, it is called a bimetallic strip.
• When it is heated, it bends into an arc due to difference in expansion extents.
• Metal with higher coefficient of linear expansion lies on convex side of arc because it expands more as compared to the other metal.

The radius of arc thus formed is given by –

R = \left [ \frac {d}{(\alpha_2 - \alpha_1)( t_2 - t_1 )} \right ]

Here, ( d )  is the thickness of strip.

Or, \quad R = \left [ \frac {d}{(\alpha_2 - \alpha_1) \Delta t} \right ]

This property is used in thermal switches of electrical appliances called thermostats.

## Thermal Stress

When a rod is heated or cooled, it expands or contracts freely. In this case, stress is not developed in the rod material. But when ends of rods are clamped together before heating or cooling, it is prevented in expansion or contraction. This produces stress in the rod upon heating or cooling. This stress is called thermal stress.

Let, ( l_1 ) and ( l_2 ) are the free lengths of the rod at temperatures ( t_1 \degree C ) and ( t_2 \degree C ) respectively.

Then, \quad l_2 = l_1 [ 1 + \alpha ( t_2 - t_1 )]

Or, \quad ( \frac { l_2 - l_1 }{ l_1 } ) = \alpha ( t_2 - t_1 )

The above relationship indicates that the thermal strain developed in the rod will be –

\left ( \frac {l_2 - l_1}{l_1} \right ) = \alpha ( t_2 - t_1 )

If E is the Young’s modulus of elasticity of the material, then thermal stress is given by –

E \alpha ( t_2 - t_1 )