**Definition of Heat :** Heat is energy in transit which is transferred from one body to the other, due to difference in temperature, without any mechanical work involved.

__THERMAL EXPANSION:__

Expansion due to increase in temperature.

**Important facts :-**

– In every type of expansion, the increase in dimension is observed to be proportional to the original dimension and the rise in temperature.

– Solids are made up of atoms and molecules. At a given temperature, the atoms and molecules are located at some equilibrium distance. When heat is added to a solid. The amplitude of vibrations of atoms and molecules increase. Due to this, inter-atomic separation increases, which results in the expansion of solids.

**Cause of thermal expansion :** Molecules are held together by elastic forces and they vibrates with some constant mean distance between them.

As temperature increases, vibration energy of the constituent particles increases which results in increase in separation between the particles and hence there is thermal expansion.

__TYPE OF THERMAL EXPANSION:__

__(1) Linear expansion__

Suppose the solid in the form of rod of length L_{o} is heated, till its temperature rise to ΔT. If the length of the rod becomes L (say), then it is found that increase in length (* L − L _{o}*) is

(a) directly proportional to its original length (* L − L _{o}*) ∝ L

_{o}

(b) directly proportional to rise in temperature of rod i.e. (* L − L _{o}*) ∝ ΔT

From (a) and (b)

(* L − L _{o}*) ∝ L

_{o}ΔT

change in length ΔL = L − L_{o} = L_{o }α ΔT

( where α is co-efficient of linear expansion)

L = L_{o} + L_{o}αΔT

**L = L _{o} (1 + αΔT )**

( L_{o} is initial length of the rod , where α is co-efficient of linear expansion)

__Definition of Coefficient of linear expansion__

It is defined as the change in length per unit original length per unit change in its temperature.

** Note :** It may be pointed out that value of coefficient of liner expansion of solid rod does not depend upon the shape of the cross-section of the rod.

__(2) Superficial expansion__

__Superficial Expansion__ : When a solid is heated its surface area and volume as well as its linear dimensions will increase. Increase of area is spoken of as superficial expansion .

Let A_{o} = initial surface area of solid

A = surface area of solid when temperature changed by ΔT , then

Change in surface area i.e. (A − A_{o})

(a) will directly proportional to A_{o}

(b) will directly proportional to ΔT

From (a) and (b)

A − A_{o} ∝ A_{o }ΔT

change in Area ΔA = A − A_{o} = A_{o}β ΔT

(where β is co-efficient of superficial expansion)

A = A_{o} + A_{o}β ΔT

A = A_{o} ( 1 + A_{o }β ΔT )

( A_{o} is initial area of the rod, where β is co-efficient of superficial expansion)

__Definition of Coefficient of superficial expansion:__

It is defined as the change in surface area per unit surface area per unit change in its temperature.

__(3) Volume expansion__

Let V_{o} = initial Volume of solid

V = Volume of solid when temperature changed by ΔT , then

Change in Volume i.e. (V − V_{o})

(a) will directly proportional to V_{o}

(b) will directly proportional to ΔT

From (a) and (b)

(V − V_{o}) ∝ V_{o }ΔT

change in volume expansion ΔV = V − V_{o} = V_{o} γ ΔT

(where γ is co-efficient of volume expansion)

V = V_{o} + V_{o} γ ΔT

V = V_{o} ( 1 + V_{o} γ ΔT )

( V_{o} is initial volume of the rod, where γ is co-efficient of volume expansion)

(1) **For isotropic solid** , α_{1} = α_{2}= α_{3}= α (say). So β = 2α and γ = 3α

(2) **For anisotropic solids** , β = α_{1} + α_{2} and γ = α_{1} + α_{2} + α_{3}

Here α_{1} , α_{2} and α_{3} are coefficient of linear expansion in X , Y and Z directions respectively.

**Example :** A copper and a tungsten plate having a thickness δ = 2mm each are riveted together so that at 0°C they form a flat bimetallic plate. Find the average radius of curvature of this plate at T = 200°C. The coefficients of linear expansion for copper and tungsten are and α_{c} = 1.7 x 10^{-5 }K^{-1 }and α_{t} = 0.4 x 10^{-5 }K^{-1 }

**Solution :**

L_{o}(1 + α_{1}ΔT) = (R + δ/2)φ ….. (1)

L_{o}(1 + α_{2}ΔT) = (R – δ/2)φ …. (2)

From (1) and (2), we get

RT (α_{1} – α_{2}) = δ + [δ T/2 ](α_{1} + α_{2})

RT (α_{1} – α_{2}) ≃ δ