QCC Notes
CLASS 11 · PHYSICSJEE MAIN × NEETहिंदी
§ 10.2NCERT Class 11 · Physics · Chapter 10

Thermal Expansion of Solids, Liquids & Gases – Masterclass

QCC Notes · Class 11 Physics NCERT-aligned Updated 26 Sep 2026 6 min read

Almost all materials expand when heated and contract when cooled. The increase in the dimensions of a body due to an increase in its temperature is called thermal expansion. On the microscopic scale, heating increases atomic vibrational amplitudes within asymmetric interatomic potential energy wells, causing the average interatomic spacing to increase.

1. Thermal Expansion in Solids

Solids have definite shape and volume; hence thermal expansion can occur in one dimension (linear), two dimensions (superficial/areal), or three dimensions (cubical/volume).

A. Linear Expansion (α)

Expansion in length of a slender rod or wire:

ΔL = L0 α ΔT
LT = L0 (1 + α ΔT)

where α is the coefficient of linear expansion. SI unit: K−1 or °C−1. For metals, α is typically ~ 10−5 K−1.

B. Superficial / Areal Expansion (β)

Expansion in the surface area of a sheet or lamina:

ΔA = A0 β ΔT
AT = A0 (1 + β ΔT)

where β is the coefficient of superficial expansion. For isotropic solids, β ≈ 2 α.

C. Cubical / Volume Expansion (γ)

Expansion in the three-dimensional bulk volume of a solid:

ΔV = V0 γ ΔT ⇒ VT = V0 (1 + γ ΔT)

where γ is the coefficient of volume expansion. For isotropic solids, γ ≈ 3 α.

Derivation of α : β : γ = 1 : 2 : 3 (for Isotropic Solids):
Consider a cube of initial side L0. Upon heating by ΔT:
AT = LT2 = [L0(1 + αΔT)]2 = L02(1 + 2αΔT + α2ΔT2). Neglecting α2ΔT2 ≪ 1 gives β = 2α.
VT = LT3 = [L0(1 + αΔT)]3 = L03(1 + 3αΔT + ...). Neglecting higher-order terms gives γ = 3α.
Hence, α / 1 = β / 2 = γ / 3 ⇒ α : β : γ = 1 : 2 : 3.

2. Thermal Stress & Strain in Rigidly Clamped Rods

If a rod is free to expand, it elongates by ΔL = L0 α ΔT without experiencing internal stress. However, if the rod is rigidly clamped at both ends so that its expansion or contraction is completely prevented, severe internal thermal stress develops.

Compressive Thermal Strain: Strain = ΔL / L = α ΔT
Thermal Stress: Stress = Y × Strain = Y α ΔT
Force on Clamps: F = Stress × Area = Y A α ΔT

JEE Insight: If the walls yield slightly by an amount Δx, the actual prevented expansion is (ΔL − Δx), and the modified thermal stress is Stress = Y [(α L ΔT − Δx) / L].

3. The Bimetallic Strip

A bimetallic strip consists of two equal strips of different metals (e.g., Brass with higher α1 and Steel with lower α2) welded or riveted together side-by-side.

  • On Heating (ΔT > 0): The metal with higher α expands more and forms the outer convex side, bending the strip.
  • On Cooling (ΔT < 0): The metal with higher α contracts more and forms the inner concave side.
  • Radius of Curvature (R): For strips of thickness d each, R ≈ d / [(α1 − α2) ΔT].
  • Engineering Uses: Thermostats in irons, refrigerators, bimetallic dial thermometers, and fire alarm circuits.

4. Thermal Expansion of Liquids

Liquids do not have a fixed shape, so only volume expansion (γ) is meaningful. Furthermore, because liquids must be held in a container that also expands, we distinguish between:

  • Real Expansion (γr): The true expansion of the liquid volume itself.
  • Apparent Expansion (γa): The observed expansion of the liquid relative to the expanding vessel.
  • Relationship: γr = γa + γcontainer = γa + 3 αvessel

Liquid Overflow Conditions

A vessel of capacity V0 is completely filled with liquid at T0. When heated by ΔT:

  • If γr > γvessel: Liquid level rises and overflows. Volume overflowed = V0(γr − γvessel)ΔT = V0 γa ΔT.
  • If γr = γvessel: Liquid level remains unchanged.
  • If γr < γvessel: Liquid level falls.

Variation of Density with Temperature

Since mass is conserved while volume increases (VT = V0(1 + γΔT)):

ρT = ρ0 / (1 + γ ΔT) ≈ ρ0 (1 − γ ΔT)

Density decreases almost linearly as temperature increases for normal liquids.

5. Anomalous Expansion of Water

Water exhibits a remarkable exception to normal thermal behaviour in the temperature range between 0 °C and 4 °C:

Anomalous Water Behaviour:
  • When liquid water is heated from 0 °C to 4 °C, its volume contracts rather than expands (γ is negative!).
  • At 4 °C (precisely 3.98 °C), water reaches its minimum volume and maximum density (1000 kg/m3 = 1.000 g/cm3).
  • Above 4 °C, water expands normally like other liquids.

Aquatic Life Preservation: During freezing winters, surface water cools to 4 °C and sinks to the bottom because it is densest. Once the whole lake reaches 4 °C, further surface cooling below 4 °C makes the surface water less dense, so it stays at the top and freezes into ice at 0 °C. Since ice is a poor thermal conductor and floats, it forms a protective blanket while the bottom of the lake remains liquid water at 4 °C, allowing fish and aquatic life to survive.

6. Thermal Expansion of Gases

Gases expand far more than liquids or solids (coefficients are roughly 100 times larger). For an ideal gas at constant pressure (P = constant):

P V = n R T ⇒ P ΔV = n R ΔT ⇒ ΔV / V = ΔT / T
Since ΔV = V γp ΔT ⇒ γp = 1 / T

At 0 °C (T = 273.15 K), γp ≈ 1 / 273.15 ≈ 3.66 × 10−3 K−1, in perfect agreement with Charles's experimental law.

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