A 0.06 m³ steel gasoline tank is full of gasoline. If the initial temperature of the tank and the gasoline is 15°C, how much gasoline will have…
A 0.06 m³ steel gasoline tank is full of gasoline. If the initial temperature of the tank and the gasoline is 15°C, how much gasoline will have spilled by the time they warm up to 35°C?
[α_s = 1.17 × 10⁻⁵ K⁻¹, γ_g = 9.5 × 10⁻⁴ K⁻¹]
Explanation
When heated, both the steel tank and the gasoline expand. But gasoline expands much more than steel. The gasoline that cannot fit inside the expanded tank spills over. The amount that spills equals the difference between how much the gasoline expands and how much the tank expands.
Step 1: Find the temperature change.
ΔT = 35°C − 15°C = 20 K
Step 2: Find how much the gasoline expands.
γ_g (volume expansion coefficient of gasoline) = 9.5 × 10⁻⁴ K⁻¹
ΔV_gasoline = V₀ × γ_g × ΔT = 0.06 × 9.5 × 10⁻⁴ × 20
ΔV_gasoline = 0.06 × 0.019 = 1.14 × 10⁻³ m³
Step 3: Find how much the steel tank expands.
The volume expansion coefficient of steel = 3α_s = 3 × 1.17 × 10⁻⁵ = 3.51 × 10⁻⁵ K⁻¹
ΔV_tank = V₀ × 3α_s × ΔT = 0.06 × 3.51 × 10⁻⁵ × 20
ΔV_tank = 0.06 × 7.02 × 10⁻⁴ = 4.21 × 10⁻⁵ m³
Step 4: Find the volume that spills.
Spill = ΔV_gasoline − ΔV_tank
Spill = 1.14 × 10⁻³ − 0.0421 × 10⁻³
Spill ≈ 1.1 × 10⁻³ m³
About 1.1 × 10⁻³ m³ of gasoline spills over. The steel tank barely expands compared to the gasoline, so nearly all the gasoline’s expansion results in spillage.