The length of a side of a metallic cube at 20° C is 5.0cm. Given that the linear expansively of the metal is…

Analyze the question: A cube of side 5.0 cm at 20°C is heated to 120°C. The linear expansivity is α = 4.0 × 10⁻⁵ K⁻¹. We need the new volume.

Concept:

• Linear expansion: L = L₀(1 + αΔθ)
• Volume expansion: V = V₀(1 + γΔθ), where γ ≈ 3α

Solution:

1. Initial volume: V₀ = 5.0³ = 125 cm³
2. Temperature change: Δθ = 120 − 20 = 100 K
3. Volume expansivity: γ = 3α = 3 × 4.0 × 10⁻⁵ = 1.2 × 10⁻⁴ K⁻¹
4. Change in volume: ΔV = V₀γΔθ = 125 × 1.2 × 10⁻⁴ × 100 = 1.5 cm³
5. New volume: V = 125 + 1.5 = 126.5 cm³

Exact check: New side = 5(1 + 4×10⁻³) = 5.02 cm. 5.02³ = 126.506 cm³ ≈ 126.5 cm³. Both methods agree.

Tip: For solids, γ = 3α and the area expansivity β = 2α. When Δθ is small, use the approximation V = V₀(1 + 3αΔθ). It saves time in the exam. Also, a temperature change in °C equals the same change in K, so you don't need to convert.