# Temperature & Thermal Expansion

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.

### Concept of Temperature:

(i) Temperature is a physical quantity which measures the degree of hotness or coldness of a body.

(ii) Temperature determines the direction of flow of heat between two bodies in thermal contact with each other until both acquire same temperature.

(iii) When two bodies are at same temperature then they are said to be in thermal equilibrium with each other.

(iv)In thermal equilibrium the heat in the two bodies may or may not be equal.

### Thermometric Scales

On any Thermometric Scale ,

$\large \frac{Reading – LFP}{UFP-LFP} = Constant$ ;

Where LFP & UFP are Lower Fixed Point & Upper Fixed Point respectively .

Relation Between Temperatures of Different Scales:

$\large \frac{C-0}{100-0} = \frac{F-32}{212-32} = \frac{K-273}{373-273}$

$\large \frac{C}{100} = \frac{F-32}{180} = \frac{K-273}{100}$

$\large \frac{C}{5} = \frac{F-32}{9} = \frac{K-273}{5}$

### 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.

### (1) Linear expansion

Suppose the solid in the form of rod of length Lo 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 − Lo) is

(a)  directly proportional to its original length ( L − Lo) ∝ Lo

(b)  directly proportional to rise in temperature of rod i.e. ( L − Lo) ∝ ΔT

From (a) and (b)

( L − Lo) ∝ Lo ΔT

change in length ΔL = L − Lo = Lo α ΔT

( where α is co-efficient of linear expansion)

L   = Lo + LoαΔT

L   = Lo (1 + αΔT )

( Lo 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 Ao =  initial surface area of solid

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

Change in surface area i.e. (A − Ao)

(a)   will directly proportional to Ao

(b)   will directly proportional to ΔT

From (a) and (b)

A − Ao ∝ Ao ΔT

change in Area ΔA = A − Ao = Aoβ ΔT

(where β is co-efficient of superficial expansion)

A  =  Ao + Aoβ ΔT

A  =  Ao ( 1 + β ΔT )

( Ao 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 Vo =  initial Volume of solid

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

Change in Volume i.e. (V − Vo)

(a)   will directly proportional to Vo

(b)   will directly proportional to ΔT

From (a) and (b)

(V − Vo) ∝ Vo ΔT

change in volume expansion ΔV = V − Vo = Vo γ ΔT

(where γ is co-efficient of volume expansion)

V  =  Vo + Vo γ ΔT

V  =  Vo ( 1 + γ ΔT )

( Vo is initial volume of the rod, where γ is co-efficient of volume expansion)

(1) For isotropic solid , α1 = α2= α3= α (say). So β = 2α and γ = 3α

(2) For anisotropic solids , β = α1 + α2 and γ = α1 + α2 + α3

Here α1 , α2 and α3 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 αc = 1.7 x 10-5 K-1 and αt = 0.4 x 10-5 K-1

Solution : Lo(1 + α1ΔT) = (R + δ/2)φ ….. (1)

Lo(1 + α2ΔT) = (R – δ/2)φ …. (2)

From (1) and (2), we get

RT (α1 – α2) = δ + [δ T/2 ](α1 + α2)

RT (α1 – α2) ≃ δ

$\displaystyle R =\frac{\delta}{(\alpha_1 – \alpha_2)\Delta t}$

= 0.769 m