Physics and Mathematics
Thermal Expansion
1. Concept Overview
Thermal Expansion is the phenomenon in which the size of a body (length, area, or volume) increases with the increase in temperature.
When a body is heated, the average kinetic energy of its molecules increases, causing them to vibrate more vigorously and occupy a greater mean separation. As a result, the body expands.
This expansion occurs in all states of matter, but it is most prominent in solids (especially metals) due to their regular structure and measurable linear expansion.
2. Explanation and Mathematical Derivation
The expansion of solids can occur in three ways:
(a) Linear Expansion
When a solid rod of initial length [L₀] is heated through a temperature rise [ΔT], its increase in length (ΔL) is proportional to [L₀] and [ΔT].
[\Delta L ∝ L₀ \Delta T]
[\Delta L = \alpha L₀ \Delta T]
Where α is the Coefficient of Linear Expansion.
Rearranging,
[\alpha = \dfrac{\Delta L}{L₀ \Delta T}]
(b) Areal Expansion
For a thin plate of initial area [A₀], the increase in area (ΔA) due to temperature rise [ΔT] is given by:
[\Delta A = \beta A₀ \Delta T]
Where β is the Coefficient of Areal (or Superficial) Expansion.
Mathematically,
[\beta = \dfrac{\Delta A}{A₀ \Delta T}]
And it can be shown that approximately,
[\beta = 2\alpha]
(c) Volumetric Expansion
For a solid or liquid of initial volume [V₀], the increase in volume (ΔV) due to temperature rise [ΔT] is given by:
[\Delta V = \gamma V₀ \Delta T]
Where γ is the Coefficient of Cubical (or Volumetric) Expansion.
Also,
[\gamma = \dfrac{\Delta V}{V₀ \Delta T}]
and
[\gamma = 3\alpha]
3. Dimensions and Units
| Quantity | Symbol | SI Unit | Dimensions |
|---|---|---|---|
| Coefficient of Linear Expansion | [α] | K⁻¹ | [K⁻¹] |
| Coefficient of Areal Expansion | [β] | K⁻¹ | [K⁻¹] |
| Coefficient of Cubical Expansion | [γ] | K⁻¹ | [K⁻¹] |
4. Key Features
- Expansion occurs because of increased molecular vibration on heating.
- The extent of expansion depends on the nature of the material and temperature rise.
- α, β, and γ are constants for a given substance at a given temperature range.
- Liquids generally exhibit volumetric expansion, not linear or areal.
- Gases show large expansion and are governed by gas laws.
- For small temperature changes, expansion is linear with respect to temperature.
- Thermal expansion has many applications such as in thermometers, bimetallic strips, and railway track gaps.
5. Important Formulas to Remember
| Type of Expansion | Formula | Coefficient Relation |
|---|---|---|
| Linear Expansion | [\Delta L = \alpha L₀ \Delta T] | — |
| Areal Expansion | [\Delta A = \beta A₀ \Delta T] | [\beta = 2\alpha] |
| Volumetric Expansion | [\Delta V = \gamma V₀ \Delta T] | [\gamma = 3\alpha] |
| Final Length | [L = L₀ (1 + \alpha \Delta T)] | — |
| Final Volume | [V = V₀ (1 + \gamma \Delta T)] | — |
6. Conceptual Questions with Solutions
1. Why do solids expand on heating?
Because the average kinetic energy of their molecules increases, leading to greater average separation between them.
2. What does coefficient of linear expansion signify?
It represents the fractional change in length per unit original length per unit rise in temperature.
3. Why is expansion in gases much greater than in solids?
Because intermolecular forces in gases are negligible, allowing larger changes in separation for the same temperature rise.
4. Why do bridges have small gaps between their sections?
To allow expansion of materials during hot weather and prevent structural damage.
5. Why is mercury used in thermometers?
Because it expands uniformly with temperature and does not wet the glass.
6. What happens if α, β, and γ are large?
The material expands more for the same temperature rise.
7. What is the relation between β and α?
Approximately, [\beta = 2\alpha].
8. What is the relation between γ and α?
Approximately, [\gamma = 3\alpha].
9. Why do glass tumblers sometimes crack when hot water is poured into them?
Because the inner surface expands more rapidly than the outer surface, producing thermal stress.
10. Why are overhead electric wires left loose in summer?
Because they expand when heated and contract in winter; if too tight, they could snap when cold.
11. Does expansion depend on the initial length or volume of the material?
Yes, expansion is directly proportional to the initial dimension of the body.
12. Can the coefficient of expansion vary with temperature?
Yes, for large temperature ranges, α, β, and γ vary slightly with temperature.
13. What is meant by “anomalous expansion of water”?
Between 0°C and 4°C, water contracts on heating instead of expanding.
14. What is the physical meaning of α = 2 × 10⁻⁵ K⁻¹?
It means the length of the material increases by 2 × 10⁻⁵ of its original length for every 1 K rise in temperature.
15. Why are bimetallic strips used in thermostats?
Because different metals have different α values, causing bending that can switch circuits on/off automatically.
7. FAQ / Common Misconceptions
1. Does thermal expansion depend on shape?
No, it depends only on the material and temperature change, not on shape.
2. Is the coefficient of expansion the same for all materials?
No, different materials have different coefficients.
3. Do liquids have linear expansion?
No, liquids only show volumetric expansion.
4. Is expansion always linear?
Only for small temperature ranges. For large temperature changes, expansion becomes non-linear.
5. Can solids contract on heating?
Only under exceptional conditions like constrained systems or specific temperature ranges (e.g., water anomaly).
6. Why is β ≈ 2α?
Because area expansion occurs in two perpendicular directions, doubling the linear effect.
7. Why is γ ≈ 3α?
Because volume expansion occurs in three perpendicular directions.
8. Is thermal expansion reversible?
Yes, on cooling, the body contracts to its original dimensions.
9. Does α depend on pressure?
Slightly — high pressure tends to reduce expansion.
10. Can gases have constant expansion coefficients?
No, gas expansion depends on pressure and temperature (see Ideal Gas Laws).
8. Practice Questions with Step-by-Step Solutions
Q1. A metal rod of length [1 m] expands by [1 mm] when heated from [0°C] to [100°C]. Find its coefficient of linear expansion.
Solution:
[\alpha] [= \dfrac{\Delta L}{L₀ \Delta T}] [= \dfrac{1 \times 10^{-3}}{1 \times 100}] [= 1 \times 10^{-5} K^{-1}]
Q2. A brass plate of area [0.5 m²] is heated through [100°C]. Find the increase in area if [\beta = 2.1 \times 10^{-5} K^{-1}].
Solution:
[\Delta A] [= \beta A₀ \Delta T = 2.1 \times 10^{-5} \times 0.5 \times 100] [= 1.05 \times 10^{-3} m²]
Q3. The volume of a copper cube is [1000 cm³] at 0°C. Find its volume at 100°C if [\gamma = 5.1 \times 10^{-5} K^{-1}].
Solution:
[\Delta V] [= \gamma V₀ \Delta T] [= 5.1 \times 10^{-5} \times 1000 \times 100] [= 5.1 cm³]
Hence, [V = 1005.1 cm³].
Q4. If [\alpha = 2 \times 10^{-5} K^{-1}], find [\beta] and [\gamma].
Solution:
[\beta] [= 2\alpha] [= 4 \times 10^{-5} K^{-1}]
[\gamma] [= 3\alpha] [= 6 \times 10^{-5} K^{-1}]
Q5. An aluminum rod 2 m long expands by 0.6 cm when heated from 20°C to 120°C. Find α.
Solution:
[\Delta L = 0.6 cm = 6 \times 10^{-3} m]
[\alpha] [= \dfrac{\Delta L}{L₀ \Delta T}] [= \dfrac{6 \times 10^{-3}}{2 \times 100}] [= 3 \times 10^{-5} K^{-1}]
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