Thermal Expansion - Volume Expansion

When a solid, liquid, or gas is heated, it expands in all three dimensions: length, breadth, and height, resulting in an increase in its overall volume. This phenomenon is called volume expansion (also known as cubical expansion).

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

\[\gamma=\frac{\Delta V}{V\Delta T}=\frac{V_T-V_0}{V_0(T-T_0)}\]

where,
V₀ = volume at 0 °C
V_T = volume when heated to T °C
T₀ = 0 °C is initial temperature
T = final temperature
∆V = V_T - V₀ = change in volume
∆T = T - T₀ = rise in temperature.

\[\gamma_1=\frac{V_2-V_1}{V_1(T_2-T_1)}\]

Typical average values in the temperature range 0°C to 100°C:

The coefficient of volume expansion depends on temperature. At low temperatures, γ rises steeply; at high temperatures, it levels off and becomes approximately constant.

Fig.: Coefficient of volume expansion (γ) of copper as a function of temperature — becomes constant only at high temperatures.

Problem: A liquid at 0°C is poured in a glass beaker of volume 600 cm³ to fill it completely. The beaker is then heated to 90°C. How much liquid will overflow?

(γ_liquid = 1.75 × 10⁻⁴ /°C,   γ_glass = 2.75 × 10⁻⁵ /°C)

Given
V₁600 cm³
T₁ → T₂0°C → 90°C
ΔT90°C
γ_liquid1.75 × 10⁻⁴ /°C
γ_glass2.75 × 10⁻⁵ /°C

• Volume expansion = increase in volume due to heating; relevant for solids, liquids, and gases.
• The formula is ΔV/V = γ · ΔT, where γ is the coefficient of volume expansion (unit: K⁻¹).
• γ = 3α for isotropic solids (α = coefficient of linear expansion).
• Liquids expand much more than solids (γ_liquid ≫ γ_solid); this is why thermometers work.
• When heating a liquid in a container, account for both expansions: γ_real = γ_apparent + γ_container.
• Water is anomalous: it expands when cooled from 4°C to 0°C — crucial for the survival of aquatic life.
• Heating increases volume → decreases density: ρ_T ≈ ρ₀(1 − γ · ΔT).
• γ varies with temperature but is treated as constant for most problems.