What is Thermal Expansion

 

What is Thermal Expansion :- Most materials expand when their temperatures increase. This phenomenon is known as thermal expansion and it has an important role in numerous engineering applications. For example, thermal-expansion joints, are included in buildings, concrete highways, railroad tracks, brick walls, and bridges to compensate for dimensional changes that occur as the temperature changes.

Reason of Thermal expansion

(What is Thermal Expansion)

Thermal expansion is a consequence of the change in the average separation between the atoms in an object. At ordinary temperatures, the atoms in a solid oscillate about their equilibrium positions with a frequency of approximately 10¹³ Hz. The average spacing between the atoms is about 10^-10 m. As the temperature of the solid increases, the atoms oscillate with greater amplitudes; as a result, the average separation between them increases. Consequently the object expands.

 

Types of thermal expansion

(What is Thermal Expansion)

Thermal Expansion is of three types :-

1. Linear Expansion
2. Area / areal Expansion
3. Volume / volumetric expansion

(1) Linear Expansion (What is Thermal Expansion):-

Suppose a rod of some material has a length L₀ at some initial temperature T₀. Now if the temperature of the rod increases by ΔT, the length of the rod changes by ΔL, then experiments show that [if ΔT is not too large (say, less than 100ºC or so)], change in length ΔL is directly proportional to change in temperature ΔT, i.e.,

(i) ΔL is directly proportional to ΔT

i.e.      ΔL ∝ ΔT     ……….(1)

 

(ii) ΔL must be directly proportional to L₀

If two rods are made of the same material and have the same temperature change, but one is twice as long as the other, then the change in its length is also twice as great. Therefore

i.e.      ΔL ∝ L₀     ……….(2)

Now using equations (1) and (2) we get:-

ΔL ∝ L₀ΔT

introducing a proportionality constant α (which is different for different materials), we may express above relation in an equation:-

ΔL = αL₀ΔT     ……….(3)

If a rod has length L₀ at temperature T₀ then its length L at a temperature  T = T₀+ΔT  is

L = L₀ + ΔL = L₀ + αL₀ΔT

or

 L = L₀ (1 + αΔT )     ……….(4)

The constant α which describes the thermal expansion properties of a particular material, is called the coefficient of linear expansion. The units of α are K^-1 or ºC^-1.

This is because no matter what unit of temperature you are taking ( K or ºC) , the temperature interval(change in temperature) is the same in the Kelvin and Celsius scales.

From ΔL = αL₀ΔT , we get

α = ΔL / L₀ΔT

From above equation….

Definition of α :- The coefficient of linear expansion α may be defined as change in length(ΔL), per unit original length(L₀), per degree Celsius (or per Kelvin) rise in temperature(ΔT).

Note :- It may be helpful to think of thermal expansion as an effective magnification of an object. For example, when a metal washer is heated all dimensions, including the radius of the hole, increase according to
Equation (4).

Notice that this is equivalent to saying that a cavity in a piece of material expands in the same way as if the cavity were filled with the material.

Because the linear dimensions of an object change with temperature, it follows that surface area and volume change as well :-

(2) Area / areal Expansion(What is Thermal Expansion):-

Suppose a sheet of some material has an area A₀ at some initial temperature T₀. Now if the temperature of the sheet increases by ΔT, the area of the sheet changes by ΔA.

Experiments show that [if ΔT is not too large (say, less than 100ºC or so)] :-

(i) ΔA is directly proportional to ΔT

i.e.      ΔA ∝ ΔT     ……….(5)

and

(ii) ΔA is also directly proportional to A₀

i.e.      ΔA ∝ A₀     ……….(6)

From (5) and (6)

ΔA ∝ A₀ΔT

removing proportionality sign….

ΔA =βA₀ΔT     ……….(7)

If the sheet has area A₀ at temperature T₀ then its area A at a temperature  T = T₀+ΔT  is

A = A₀ + ΔA = A₀ + β A₀ΔT

or

 A = A₀ (1 + βΔT )……….(8)

The constant β which describes the thermal expansion properties of a particular material, is called the coefficient of areal/area expansion. The units of β are K^-1 or ºC^-1.

From ΔA = βA₀ΔT , we get

β = ΔA / A₀ΔT

From above equation….

Definition of β :- The coefficient of areal expansion β may be defined as change in area(ΔA), per unit original area(A₀), per degree Celsius (or per Kelvin) rise in temperature(ΔT).

(3) Volumetric / Volume Expansion(What is Thermal Expansion):-

Suppose a cube of some material has a volume V₀ at some initial temperature T₀. Now if the temperature of the cube increases by ΔT, the volume of the cube changes by ΔV.

Experiments show that [if ΔT is not too large (say, less than 100ºC or so)] :-

(i) ΔV is directly proportional to ΔT

i.e.      ΔV ∝ ΔT     ……….(9)

and

(ii) ΔV must be directly proportional to V₀

i.e.      ΔV ∝ V₀     ……….(10)

From (9) and (10)

ΔV ∝ V₀ΔT

removing proportionality sign….

ΔV =γV₀ΔT     ……….(11)

If the cube has volume V₀ at temperature T₀ then its volume V at a temperature  T = T₀+ΔT  is

V = V₀ + ΔV = V₀ + γV₀ΔT

or

 V = V₀(1 + γΔT )……….(12)

The constant γ which describes the thermal expansion properties of a particular material, is called the coefficient of volumetric/volume expansion. The units of γ are K^-1 or ºC^-1.

From ΔV = γV₀ΔT , we get

γ = ΔV / V₀ΔT

From above equation….

Definition of γ :- The coefficient of volumetric expansion γ may be defined as change in volume(ΔV), per unit original volume(V₀), per degree Celsius (or per Kelvin) rise in temperature(ΔT).

 

Note :-

1. Actually thermal expansion is always three dimensional. When other two dimensions of object are negligible with respect to one, then observations are significant only in one dimension and it is known as linear expansion.
2. If expansion coefficients α, β or γ varies with distance, then we have to integrate change in length(dl) to find total expansion. For example let α = ax + b. Then total expansion .
3. If α varies with temperature, let α = f(T),  Then total expansion .

 

Example 1. In the given figure, when temperature is increased then which of the following increases

(A) R₁           (B) R₂           (C) R₂ – R₁

Solution :- As the intermolecular distance between atoms increases on heating hence the inner and outer perimeter increases. Also if the atomic arrangement in radial direction is observed then we can say that it also increases hence all A,B,C are true.

Example 2. A rectangular plate has a circular cavity as shown in the figure. If we increase its temperature then which dimension will increase in following figure.

Solution :- Distance between any two point on an object increases with increase in temperature. So, all dimension a, b, c and d will increase.

Example 3. What is the percentage change in length of 1m iron rod if its temperature changes by 100ºC. α for iron is 2 × 10^–5/ºC.

Solution :- Percentage change in length of rod due to change in temperature is given by

Percentage change in length(%l) =

%l = 2 × 10^–5 × 100 × 100 = 0.2%

Example 4. The coefficient of linear expansion ′α′ of a rod of length 2m varies with the distance x from the end of the rod as α=α₀​+α₁​x where α₀ ​= 1.76×10⁻⁵ ºC^-1 and α₁​ = 1.2×10⁻⁶m⁻¹ºC⁻¹. The increase in the length of the rod, when heated through $100ºC$ is :-

(A) 2 cm  (B) 3.76 mm   (C) 1.2 mm  (D) None of these

Solution :- Here coefficient of linear expansion α varies with position x along the rod, so using the relation given in Note 2, i.e.,

Here α=α₀​+α₁​x, so

 

 

 

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