Thermal expansion is the increase in the size of a substance (length, area, or volume) when its temperature rises.
- It occurs because particles gain kinetic energy on heating and move farther apart.
- Amount of expansion depends on the material and temperature change and is observed in solids, liquids, and gases.
Three types of expansions can take place in solids:
1. Linear expansion (Change in Length)
Linear expansion refers to the increase in the length of a solid when its temperature is raised. If a substance is in the form of a long rod, a small change in temperature (ΔT) causes a change in its length (Δl). The fractional change in length (Δl / l) is directly proportional to the change in temperature.
The expression for the linear expansion is,
where αlis the coefficient of linear expansion of the given solid.
The unit of α is per degree Celsius (°C⁻¹) in the CGS and per Kelvin (K⁻¹) in the SI system.
The values of the coefficient of linear expansion for some material are shown in the below table.
| Materials | αl (10–5 K–1) |
|---|---|
| Aluminum | 2.5 |
| Brass | 1.8 |
| Iron | 1.2 |
| Copper | 1.7 |
| Silver | 1.9 |
| Gold | 1.4 |
| Glass (pyrex) | 0.32 |
| Lead | 0.29 |
2. Superficial or areal expansion (Change in area)
Superficial or area expansion refers to the increase in the surface area of a solid when its temperature is raised. A small change in temperature (ΔT) causes a change in its surface area (ΔA). The fractional change in area (ΔA / A) is directly proportional to the change in temperature.
The expression for the Area expansion is
\dfrac{\Delta A}{A} = \alpha _{A}\cdot\Delta T
where αAis the coefficient of area expansion of the given solid.
3. Volume expansion (Change in volume)
Volume expansion refers to the increase in the volume of a substance when its temperature is raised. A small change in temperature (ΔT) causes a change in volume (ΔV). The fractional change in volume (ΔV / V) is directly proportional to the change in temperature.
The expression for volume expansion is:
\dfrac{\Delta V}{V} = \alpha _{V}\cdot\Delta T
where αᵥ is the coefficient of volume (cubical) expansion of the substance.
The coefficient of volume expansion is a characteristic property of the material and may vary slightly with temperature, but it is generally considered constant over a small temperature range. Liquids usually have a higher coefficient of volume expansion than solids. Example: Ethyl alcohol expands more than mercury for the same rise in temperature.
The graph plot below shows the Coefficient of volume expansion of copper as a function of temperature.
Relation Between Coefficients
For isotropic solids, the coefficients of linear, area, and volume expansion are related to each other. The coefficient of area expansion is twice the coefficient of linear expansion, while the coefficient of volume expansion is three times the coefficient of linear expansion.
- αA = 2αl
- αV = 3αl
These relations are valid for small temperature changes and are widely used in numerical problems.
Thermal Stress
Thermal stress is the stress developed in a body when its expansion or contraction is prevented. When a material is heated or cooled but is not allowed to change its dimensions, internal forces develop within it, resulting in stress.
The expression for thermal stress is:
where Y is Young’s modulus of the material, α is the coefficient of linear expansion, and ΔT is the change in temperature.
Anomalous Expansion of Water
Anomalous expansion of water is the unusual property of water in which it expands when cooled from 4°C to 0°C instead of contracting, resulting in a decrease in density.
Water behaves normally when cooled from higher temperatures to 4°C, as its volume decreases and density increases. At 4°C, water attains maximum density. However, when the temperature falls below 4°C, water begins to expand, and its density decreases.
This property has important natural effects. In lakes and ponds, water at 4°C sinks to the bottom, as it is the densest, while colder water (below 4°C) remains at the surface. As a result, the surface water freezes first, forming ice, while the lower layers remain liquid, allowing aquatic life to survive.
Sample Problems
Question 1: A rod of length 2 m is heated by 50°C. If the coefficient of linear expansion is 2 × 10⁻⁵ K⁻¹, find the increase in length.
Solution: Given
l = 2 m, ΔT = 50°C, α = 2 × 1-5 K-1
Formula:
\Delta l = l \alpha \Delta T Δl = 2 × (2 × 10-5 ) × 50
Δl = 2 × 10-3 m
Δl = 0.002 m (2 mm)
Question 2: A square plate of area 1 m²2 is heated through 100°C. If the coefficient of area expansion is 4 × 10⁻⁵ K⁻¹, find the increase in area.
Solution: Given
A = 1 m², ΔT = 100°C, αₐ = 4 × 10-5
\Delta A = A \alpha_A \Delta T ΔA = 1 × (4 × 10-5 ) × 100
ΔA = 4 × 10-3 m2
ΔA = 0.004 m²
Question 3: A liquid has an initial volume of 0.5 m³ and is heated by 80°C. If the coefficient of volume expansion is 6 × 10⁻⁵ K⁻¹, find the increase in volume.
Solution: Given
V = 0.5 m³, ΔT = 80°C, αᵥ = 6 × 10-5
Formula:
\Delta V = V \alpha_V \Delta T ΔV = 0.5 × (6 × 10-5 ) × 80
ΔV = 2.4 × 10-3 m3
ΔV = 0.0024 m3
Question 4: A steel rod is heated by 100°C but not allowed to expand. If Young’s modulus is 2 × 10 N/m2 and α = 1 × 10⁻⁵ K⁻¹, find the stress developed.
Solution: Formula:
Stress =YαΔT
Stress = (2 × 1011 ) × (1 × 10-5 ) × 100
Stress = 2 × 108 Pa
Unsolved Problems
Question 1: A copper wire of length 2 m is heated through 70°C. If the coefficient of linear expansion is 1.7 × 10⁻⁵ K⁻¹, find the increase in length.
Question 2: A rectangular metal sheet has an initial area of 3 m2. It is heated to 50°C. If the coefficient of area expansion is 2.5 × 10⁻⁵ K⁻¹, calculate the increase in area.
Question 3: A liquid occupies a volume of 1.2 m3 at a certain temperature. It is heated to 90°C. If the coefficient of volume expansion is 4 × 10⁻⁵ K⁻¹, find the increase in volume.
Question 4: A steel rod is heated by 150°C, but its expansion is completely prevented. If Young’s modulus is 2 × 10 N/m2 and the coefficient of linear expansion is 1.2 × 10⁻⁵ K⁻¹, calculate the stress developed.
Question 5: A square metal plate has a side of 1.5 m. If the coefficient of linear expansion is 2 × 10⁻⁵ K⁻¹, find the coefficient of area expansion.