# Expansion of Solids

 Science >  Expansion of Solids > You are Here
• Whenever there is an increase in the dimensions of a body due to heating, then the body is said to be expanded and the phenomenon is known as expansion.
• Solids undergo three types of expansions a) Linear (Longitudinal) expansions, b) Superficial expansions (Arial) and  c) Cubical expansions (Volumetric)

### Linear Expansion of Solid:

• Whenever there is an increase in the length of a body due to heating then the expansion is called as linear 0r longitudinal expansion.

#### Expression for the Coefficient of Linear Expansion of a Solid:

• Consider a metal rod of length ‘l0’ at temperature 0 °C. Let the rod be heated to some higher temperature say t °C. Let ‘l’  be the length of the rod at temperature t °C.

∴  Change in temperature = t2 – t1 = t – 0 = t

and Change in length = l l0

Experimentally it is found that the change is length ( l0) is

Directly proportional to the original length (l0)

l0   ∝  l0   ………………. (1)

Directly proportional to the change in temperature (t)

l0   ∝  t   ………………. (1)

Dependent upon the material of the rod.

From the equation (1) and (2)

l0   ∝  lt

∴   l l0   =  α lt    …………… (3)

Where ‘α’ is a constant called a coefficient of linear expansion This is an expression for the coefficient of linear expansion of a solid.

• The coefficient of linear-expansion is defined as the increase in length per unit original length at 00c per unit rise in temperature.

From equation (3) we get

∴   l   = l0  +   α lt

∴   l   = l(1 +    α t) ………….. (4)

This is an expression for length of rod at t °C

#### Note:

• The magnitude of the coefficient of linear expansion is so small that it is not necessary to take initial temperature as 0 °C.
• Consider a metal rod of length ‘l1’ at temperature t10 °C. Let the rod be heated to some higher temperature say t °C. Let ‘l2’  be the length of the rod at temperature t2 °C. Let l0’ be the length of the rod at the temperature 0 °C. Let α be the coefficient of linear expansion, then we have

l1  = l(1 +    α t1) ………….. (2)

l2  = l(1 +    α t2) ………….. (2)

Dividing equation (2) by (1) we get • The coefficient of linear expansion is different for different material

Superficial Expansion of Solid:

• Whenever there is an increase in the area of a solid body due to heating then the expansion is called as superficial or arial expansion.

#### Expression for the Coefficient of Superficial Expansion of a Solid:

• Consider a thin metal plate of area ‘A0’ at temperature 0 °C. Let the plate be heated to some higher temperature say t °C. Let ‘A’  be the area of the plate at temperature t °C.

∴  Change in temperature = t2 – t1 = t – 0 = t

and Change in area = A – A0

Experimentally it is found that the change is area (A – A0) is

Directly proportional to the original area (A0)

A – A0   ∝  A0   ………………. (1)

Directly proportional to the change in temperature (t)

A A0   ∝  t   ………………. (1)

Dependent upon the material of the plate

From the equation (1) and (2)

A – A0   ∝  At

∴   A – A0   =  β At    …………… (3)

Where ‘β’ is a constant called a coefficient of superficial expansion This is an expression for the coefficient of superficial expansion of a solid.

• The coefficient of superficial expansion is defined as the increase in area per unit original area at 00c per unit rise in temperature.

From equation (3) we get

∴  A   = A0  +  β At

∴  A   = A(1 +    Βt) ………….. (4)

This is an expression for the area of the plate at t °C

#### Note:

• The magnitude of the coefficient of superficial expansion is so small that it is not necessary to take initial temperature as 0 °C.
• Consider a thin metal plate of area ‘A1’ at temperature t10 °C. Let the plate be heated to some higher temperature say t °C. Let ‘A2’  be the area of the plate at temperature t2 °C. Let ‘A0’ be the area of the plate at the temperature 0 °C. Let β be the coefficient of superficial expansion, then we have

A1  = A(1 +    β t1) ………….. (2)

A2  = A(1 +    β t2) ………….. (2)

Dividing equation (2) by (1) we get • The coefficient of superficial expansion is different for different material

Cubical Expansion of Solid:

• Whenever there is an increase in the volume of the body due to heating the expansion is called as cubical or volumetric expansion.

#### Expression for the Coefficient of Cubical Expansion of a Solid:

• Consider a solid body of volume ‘V0’ at temperature 0 °C. Let the body be heated to some higher temperature say t °C. Let ‘V’  be the volume of the body at temperature t °C.

∴  Change in temperature = t2 – t1 = t – 0 = t

and Change in volume = V – V0

Experimentally it is found that the change is volume ( V – V0) is

Directly proportional to the original volume (V0)

V – V0  ∝  V0   ………………. (1)

Directly proportional to the change in temperature (t)

V – V0  ∝  t   ………………. (1)

Dependent upon the material of the body.

From the equation (1) and (2)

V – V0  ∝  Vt

∴  V – V0   =  γ Vt    …………… (3)

Where ‘γ’ is a constant called a coefficient of cubical expansion This is an expression for the coefficient of cubicalexpansion of a solid.

• The coefficient cubical expansion is defined as an increase in volume per unit original volume at 00c per unit rise in temperature.

From equation (3) we get

∴   V   = V0  +  γ Vt

∴   V   = V(1 +    γ t) ………….. (4)

This is an expression for volume of the body at t °C

#### Note:

• The magnitude of the coefficient of cubical expansion is so small that it is not necessary to take initial temperature as 0 °C.
• Consider a solid body of volume ‘V1’ at temperature t10 °C. Let the body be heated to some higher temperature say t °C. Let ‘V2’  be the volume of the body at temperature t2 °C. Let ‘V0’ be the volume of the body at the temperature 0 °C. Let γ be the coefficient of cubical-expansion, then we have

V1  = V(1 +   γ t1) ………….. (2)

V2  = V(1 +  γ t2) ………….. (2)

Dividing equation (2) by (1) we get #### Relation Between α and β:

• Consider a thin metal plate of length, breadth, and area l0, b0, and Aat temperature 0 °C. Let the plate be heated to some higher temperature say t °C. Let l, b and A  be the length, breadth, and area of the plate at temperature t °C.

Then original area   =   A0 =   l0 b0 ,,,,,,,,,,,,,,,,  (1)

Consider linear expansion

Length,   l =   l0 (1+ αt)

Breadth, b = b0 (1 + αt)

where α = coefficient of linear expansion

Final area   =    A    =  l  b  =  l0 (1+ αt) × b0 (1 + αt)

∴   A    =  l0 b0  (1+ 2 αt + α²t²)

Now α is very small hence α2 is still small, hence quantity α²t² can be neglected

∴   A    =  A(1+ 2 αt)  ,,,,,,,,,,,,,,,,,,  (2)

Consider superficial expansion of the plate area.

A = A0( 1+ βt) ,,,,,,,,,,,,,,,,,  (3)

From (2) and (3)

β = 2α

Thus the coefficient of superficial expansion is twice coefficient of linear expansion.

#### Relation Between α and γ:

• Consider a thin rectangular parallelopiped solid of length, breadth, height, and volume l0, b0, h0, and Vat temperature 0 °C. Let the solid be heated to some higher temperature say t °C. Let l, b, h and V  be the length, breadth, height, and volume of the solid at temperature t °C.

Then original volume    =   V0 =   l0 b0 h0   ,,,,,,,,,,,,,,,,  (1)

Consider linear expansion

Length,   l =   l0 (1+ αt)

Breadth, b = b0 (1 + αt)

Height   h  = h0 (1 + αt)

where α = coefficient of linear expansion

Final volume    =    V    =  l  b h  =  l0 (1+ αt) × b0 (1 + αt)× h0 (1 + αt)

∴   V    =  l0 b0  h(1+ 3 αt + 3 α²t² + α³t³ )

Now α is very small hence α2 is still small, hence quantity α²t², α³t³ can be neglected

∴   V    =  V(1+ 3 αt)  ,,,,,,,,,,,,,,,,,,  (2)

Consider cubical expansion of the solid.

V = V0( 1+ γt) ,,,,,,,,,,,,,,,,,  (3)

From (2) and (3)

γ = 3α

Thus the coefficient of cubical expansion is thrice coefficient of linear expansion.

#### Notes:

We have β = 2α   hence α = β/2   ………………… (1)

We have γ = 3α   hence α = γ/3   ………………… (2)

From relations (1) and (2) we get

α = β/2  = γ/3

Hence 6 α = 3 β   = 2γ

 Science >  Expansion of Solids > You are Here

1. Shreyas Mohan Padghan
• Hemant More