Thermal Expansion Formula
Last Updated : 04 Feb, 2024
Thermal expansion is the phenomenon in which a body undergoes a change in its length, area, or volume due to a change in temperature. It happens when an object expands and grows larger owing to a change in its temperature or heat produced. It causes changes in dimension, whether in length, volume, or area. It is directly proportional to the temperature change, it means if temperature increases, then thermal expansion also increases and vice-versa.
Thermal Expansion Formula
The formula of linear expansion states that the ratio of change in dimension to the original dimension equals the product of expansion coefficient and temperature change. The expansion coefficient is denoted by the symbol α. Its unit of measurement is per kelvin (K-1) and the dimensional formula is given by [M0L0T0K-1]. It is categorized into three types: Linear expansion, Area expansion, and Volume expansion. Its value can be positive and negative as well.
Linear expansion
If there is a change in length of an object due to temperature change, it is called linear expansion. The linear expansion coefficient is denoted by the symbol αL. Its formula is given as,
ΔL/L0 = αLΔT
where,
L0 is the original length,
L is the expanded length,
ΔL is the length change,
αL is the length expansion coefficient,
ΔT is the temperature change.
Area expansion
If there is a change in the area of an object due to temperature change, it is called area expansion. The area expansion coefficient is denoted by the symbol αA. Its formula is given as,
ΔA/A0 = αAΔT
where,
A0 is the original area,
A is the expanded area,
ΔA is the area change,
αA is the area expansion coefficient,
ΔT is the temperature change.
Volume expansion
If there is a change in the volume of an object due to temperature change, it is called volume expansion. The volume expansion coefficient is denoted by the symbol αV. Its formula is given as,
ΔV/V0 = αVΔT
where,
V0 is the original volume,
V is the expanded volume,
ΔA is the volume change,
αV is the volume expansion coefficient,
ΔT is the temperature change.
Sample Question
Question 1. A rod is heated to 50°C to increase its length from 20 m to 30 m. Calculate the expansion coefficient if the room temperature is 20°C.
Solution:
We have,
L0 = 20
ΔL = 30 - 20 = 10
ΔT = 50 - 20
= 30°C
= 303.15 K
Using the formula we get,
α = ΔL/(L0 × ΔT)
= 10/(20 × 303.15)
= 10/6063
= 0.0167 K-1
Question 2. A rod is heated to 30°C to increase its length from 10 m to 25 m. Calculate the expansion coefficient if the room temperature is 10°C.
Solution:
We have,
L0 = 10
ΔL = 25 - 10 = 15
ΔT = 30 - 10
= 20°C
= 293.15 K
Using the formula we get,
α = ΔL/(L0 × ΔT)
= 15/(10 × 293.15)
= 15/2931.5
= 0.0051 K-1
Question 3. A rod is heated to 30°C to increase its length by 15 m. Calculate the initial length if the expansion coefficient is 0.02 K-1 for a room temperature of 10°C.
Solution:
We have,
α = 0.02
ΔL = 15
ΔT = 30 - 10
= 20°C
= 293.15 K
Using the formula we get,
L0 = ΔL/(α × ΔT)
= 15/(0.02 × 293.15)
= 15/5.863
= 37.5 m
Question 4. A rod is heated to 40°C to increase its area from 50 sq. m to 100 sq. m. Calculate the expansion coefficient if the room temperature is 25°C.
Solution:
We have,
A0 = 50
ΔA = 100 - 50 = 50
ΔT = 40 - 25
= 15°C
= 288.15 K
Using the formula we get,
α = ΔA/(A0 × ΔT)
= 50/(50 × 288.15)
= 1/288.15
= 0.035 K-1
Question 5. A rod is heated to 30°C to increase its area by 40 sq. m. Calculate the initial area if the expansion coefficient is 0.05 K-1 for a room temperature of 10°C.
Solution:
We have,
α = 0.05
ΔA = 40
ΔT = 30 - 10
= 20°C
= 293.15 K
Using the formula we get,
A0 = ΔA/(α × ΔT)
= 40/(0.05 × 293.15)
= 40/14.64
= 2.73 sq. m
Question 6. A rod is heated to 40°C to increase its volume from 200 cu. m to 300 cu. m. Calculate the expansion coefficient if the room temperature is 20°C.
Solution:
We have,
V0 = 200
ΔV = 300 - 200 = 100
ΔT = 40 - 20
= 20°C
= 293.15 K
Using the formula we get,
α = ΔV/(V0 × ΔT)
= 100/(200 × 293.15)
= 1/586.3
= 0.0017 K-1
Question 7. A rod is heated to 30°C to increase its volume by 70 cu. m. Calculate the initial area if the expansion coefficient is 0.03 K-1 for a room temperature of 10°C.
Solution:
We have,
α = 0.03
ΔV = 70
ΔT = 30 - 10
= 20°C
= 293.15 K
Using the formula we get,
V0 = ΔV/(α × ΔT)
= 70/(0.03 × 293.15)
= 70/8.795
= 116.67 cu. m
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