The mass of an empty density bottle is 30g. When filled with a liquid, the mass of the bottle and the liquid…

Analyze the question

A density bottle is filled with liquid. When heated, the liquid expands more than the bottle, so some liquid spills out. The mass lost is the mass of liquid that overflowed. We want the apparent cubic expansivity, which ignores the expansion of the bottle itself.

Concept

Apparent cubic expansivity: γ_app = (apparent increase in volume) / (original volume × temperature rise)

Since the volume is proportional to the mass of liquid at the same density (original density), this becomes:

γ_app = (mass of liquid expelled) / (mass of liquid remaining × Δθ)

The "remaining" liquid is the mass left in the bottle after heating, which is the usual textbook form. It is the mass of liquid that would occupy the bottle's volume at the higher temperature.

Solution

• Mass of liquid originally = 40 − 30 = 10 g
• Mass of liquid lost = 40 − 38 = 2 g
• Mass of liquid remaining = 38 − 30 = 8 g
• Δθ = 40 K

Using the remaining mass:

γ_app = 2 / (8 × 40) = 2 / 320 = 6.25 × 10⁻³ K⁻¹

Using the original mass instead gives 2 / (10 × 40) = 5 × 10⁻³ K⁻¹, which is not an option. So the intended method uses the remaining liquid, which is the standard approach for the weight-loss method: γ_app = (M₂ − M₃) / ((M₃ − M₁) × Δθ), where M₁ is the empty bottle, M₂ is the bottle plus liquid, and M₃ is the bottle plus the remaining liquid after heating.

Answer: B (6.25 × 10⁻³ K⁻¹)

Tip: For the apparent expansivity from a density bottle, remember the formula: mass expelled ÷ (mass remaining × temperature rise). Always subtract the empty bottle's mass first. Options A and B differ only in the power of ten, so check your exponent carefully.