Thermal Expansion - Areal Expansion

The increase ΔA in the surface area of a solid, on heating, is called areal expansion or superficial expansion.

The coefficient of areal expansion of a solid is defined as the increase in the area per unit original area at 0°C for one degree rise in temperature.

Unit: °C⁻¹ or K⁻¹
Dimensions: [L⁰ M⁰ T⁰ K⁻¹]

\[\beta=\frac{\Delta A}{A\cdot\Delta T}=\frac{A_T-A_0}{A_0\left(T-T_0\right)}\]

This is one of the most important results in thermal expansion — and a frequent exam question. Here's how β relates to the linear expansion coefficient α.

A thin aluminium plate has an area of 286 cm² at 20 °C. Find its area when it is heated to 180 °C. (β for aluminium = 4.9×10-5 /°C)

Given:

T₁ = 20°C,
T₂ = 180°C,
A₁ = 286 cm²,
β = 4.9 × 10⁻⁵/°C

Solution: Using Eq.:

The area increased by only ~2.24 cm² — illustrating why thermal expansion in solids is hard to notice with the naked eye.

• Loosening Jar Lids: Running hot water over a metal lid causes it to undergo a small amount of superficial expansion — its area increases slightly, loosening the grip on the glass jar (glass expands much less than metal).
•  Bridge Expansion Joints: Metal deck plates on bridges have small gaps. On hot days, the plates expand in area; without these gaps, they would buckle and crack.
• Aircraft Panels: At cruising altitude, the outer skin of an aircraft experiences temperature changes. Engineers account for areal expansion when designing panel joints to prevent structural stress.
• Holes in Metal Plates: A hole drilled in a metal sheet expands when heated (not shrinks!). The hole behaves as if it were filled with the same metal.

• Areal expansion = increase in surface area of a solid due to heating.
• Working formula: A₂ = A₁[1 + β(T₂ − T₁)]
• Unit of β: °C⁻¹ or K⁻¹
• Key derivation: β = 2α (from expanding a square plate and neglecting α² term)
• Master ratio: α : β : γ = 1 : 2 : 3
• A hole in a metal plate expands on heating — it does NOT shrink.
• Exam shortcut: If given α, multiply by 2 for β. If given γ, divide by 3 for α, then multiply by 2.