Thermal Expansion - Relation Between Coefficient of Expansion

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Units and Measurements
Mathematical Methods
Motion in a Plane
Laws of Motion
Gravitation
Mechanical Properties of Solids
Thermal Properties of Matter
- Thermal Properties of Matter
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- Thermal Expansion
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- Relation Between Coefficient of Expansion
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Semiconductors
- Concept of Semiconductors
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- Band Theory of Solids
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- n-type semiconductor
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- Charge neutrality of extrinsic semiconductors
- p-n Junction
- A p-n Junction Diode
- Basics of Semiconductor Devices
- Applications of Semiconductors and P-n Junction Diode
- Thermistor

Estimated time: 16 minutes

Relation between β and α
Relation between γ and α
Example
Real-Life Applications
Key Points: Relation Between Coefficient of Expansion

Maharashtra State Board: Class 11

Relation between β and α

Square plate expanding from l₀ to l_T when heated from 0°C to T°C

Fig: A square plate at 0°C (side l₀) expands to side l_T at T°C. Area goes from A₀ = l₀² to A_T = l_T².

Setup: Consider a thin square metal plate with side length l₀ at 0°C.
When heated to T°C, each side expands to:
l_T = l₀(1 + αT) … (1)
Express area at both temperatures:
Area at 0°C: A₀ = l₀²
Area at T°C: A_T = l_T² = l₀²(1 + αT)²
∴ A_T = A₀(1 + αT)² … (2)
Also, from the definition of areal expansion:
A_T = A₀(1 + βT) … (3)
Compare and simplify: Setting Eq. (2) = Eq. (3):
A₀(1 + αT)² = A₀(1 + βT)
Expanding the left side:
1 + 2αT + α²T² = 1 + βT
After dropping the tiny α²T² term:
2αT = βT
β = 2α

Maharashtra State Board: Class 11

Relation between γ and α

Cube expanding from l₀ to l_T when heated from 0°C to T°C

Fig: A cube at 0°C (side l₀, volume V₀ = l₀³) expands to side l_T at T°C (volume V_T = l_T³).

Setup: Consider a cube with side l₀ at 0°C.
When heated to T°C, each side becomes:
l_T = l₀(1 + αT)
Express volume at both temperatures:
Volume at 0°C: V₀ = l₀³
Volume at T°C: V_T = l_T³ = l₀³(1 + αT)³
∴ V_T = V₀(1 + αT)³ … (1)
Also, from the definition of volume expansion:
V_T = V₀(1 + γT) … (2)
Compare and simplify: Setting Eq. (1) = Eq. (2):
V₀(1 + αT)³ = V₀(1 + γT)
Expanding using the binomial theorem:
1 + 3αT + 3α²T² + α³T³ = 1 + γT
After neglecting higher-order terms:
3αT = γT
γ = 3α

The Master Relation

Combining both derivations gives us the fundamental relationship between all three coefficients:

α = β / 2 = γ / 3

α : β : γ = 1 : 2 : 3

Maharashtra State Board: Class 11

Example

Problem: A sheet of brass is 50 cm long and 8 cm broad at 0°C. If the surface area at 100°C is 401.57 cm², find the coefficient of linear expansion of brass.

Given:

T₁ = 0°C

T₂ = 100°C

A₁ = 50 × 8 = 400 cm²

A₂ = 401.57 cm²

Solution:

Step 1 — Find β using the areal expansion formula:

β = (A₂ - A₁) / [A₁ \times (T₂ - T₁)]

β = (401.57 - 400) / [400 \times (100 - 0)] = 1.57 / 40,000

β = 3.925 \times 10⁻⁵ °C⁻¹

Step 2 — Use β = 2α to find α:

α = β / 2 = 3.925 \times 10⁻⁵ / 2

α = 1.962 × 10⁻⁵ °C⁻¹

Maharashtra State Board: Class 11

Real-Life Applications

Sagging Wires: In summer, metal wires expand and become longer, causing them to sag. Engineers leave slack so they don’t snap in winter when they contract.

Power lines sagging between towers

Railway Expansion Gaps: Metal rails expand in heat. Small gaps (expansion joints) prevent rails from bending or buckling.

Railway track with expansion gap

Shrink-Fitting: A steel wheel is heated to expand, placed on an axle, and then cools and contracts to form a tight fit.

Shrink fitting process using induction heating

Anomalous Expansion of Water: Water is densest at 4°C. Colder water stays on top and freezes, forming ice that insulates aquatic life below.

Diagram showing lake freezing from surface with temperature layers

Maharashtra State Board: Class 11

Key Points: Relation Between Coefficient of Expansion

Expansion Coefficients: Linear (α), areal (β), and volume (γ) expansion follow the relation α: β: γ = 1: 2: 3.
Areal and Volume Relations: β = 2α (surface expansion) and γ = 3α (volume expansion).
Approximation Rule: Higher powers of α are ignored because α is very small (≈10⁻⁵).
Anomalous Expansion of Water: Water has maximum density at 4°C, so lakes freeze from the top, protecting aquatic life below.
Practical Applications: Thermal expansion is used in railway expansion joints, shrink-fitting of wheels, and slack in power lines.