A glass bottle of initial volume 2 x 10 ^4 cm ^3 is heated from 20ºC to 50ºC. If the linear expansivity of…

Analyze the question

We are given the initial volume of a glass bottle, a temperature rise, and the linear expansivity of glass. We need the final volume.

Concept

Volume (cubic) expansivity γ is about 3 times the linear expansivity α:

γ = 3α

Final volume: V₂ = V₁(1 + γΔθ)

Solution

• V₁ = 2 × 10⁴ cm³ = 20000 cm³
• Δθ = 50 − 20 = 30 K
• α = 9 × 10⁻⁶ K⁻¹, so γ = 3 × 9 × 10⁻⁶ = 2.7 × 10⁻⁵ K⁻¹

Increase in volume: ΔV = V₁γΔθ = 20000 × 2.7 × 10⁻⁵ × 30 = 20000 × 8.1 × 10⁻⁴ = 16.2 cm³

Final volume: V₂ = 20000 + 16.2 = 20016.2 cm³

Common mistakes

• Using α instead of γ gives ΔV = 5.4 cm³, i.e. 20005.4 cm³ (option B). That is the trap: the question gives linear expansivity but asks about volume.
• Option C (8.1) comes from using 2α or a similar slip.

Exam tip

Remember the ratio α : β : γ = 1 : 2 : 3 (linear : area : volume). Whenever the question asks about volume and gives linear expansivity, multiply by 3 first.