Two Students, Two Densities for the Same Metal
Two lab partners determine the density of an unknown metal cylinder. Student A measures mass on a balance (24.87 g) and volume by water displacement in a 50 mL graduated cylinder (initial 20.0 mL, final 23.1 mL). Student B measures the same cylinder's dimensions with calipers (diameter 1.20 cm, height 2.10 cm) and computes volume geometrically. Student A reports 8.0 g/mL; Student B reports 8.4 g/mL. The known densities of common metals include iron 7.87, copper 8.96, and zinc 7.14 g/mL. The instructor asks them to reconcile the discrepancy before accepting either value.
1. Identify the most likely source of the discrepancy and explain which measurement is more uncertain, with quantitative support.
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The discrepancy almost certainly comes from volume measurement uncertainty, and Student A's displacement method is the weaker one.
- Student A's volume: 23.1 − 20.0 = 3.1 mL. A typical 50 mL graduated cylinder reads to ±0.5 mL (often ±1 mL), so the volume has roughly ±0.5–1.0 mL uncertainty - that is 16–32% of a 3.1 mL displacement. Small displacements amplify relative error badly.
- Student B's volume from calipers: V = π(0.60)²(2.10) ≈ 2.37 cm³. Calipers read to ±0.01 cm; propagated relative uncertainty is only a few percent.
So Student A's 8.0 g/mL carries large volume-driven uncertainty, while Student B's 8.4 g/mL is more precise. Neither matches iron (7.87), copper (8.96), or zinc (7.14) cleanly, so identification is premature. The lesson: choose a method whose resolution is appropriate to the size of the quantity being measured.
2. What concrete procedural change would most improve the reliability of the displacement measurement, and how would it change the reported uncertainty?
Read the worked answer
Use a smaller cylinder or more sample. The core problem is that 3.1 mL is small relative to the cylinder's ±0.5–1.0 mL resolution.
Options, in order of impact:
- Use a 10 mL graduated cylinder (reads to ±0.1 mL) - reduces volume uncertainty to ~±0.2 mL, i.e., ~6% of 3.1 mL instead of 16–32%.
- Use several cylinders or a larger piece so displacement is, say, 15–20 mL, making the same absolute resolution a much smaller fraction.
- Ensure the cylinder is fully submerged with no trapped air bubbles and tap the cylinder to dislodge them - a classic systematic error that inflates apparent volume and lowers computed density.
With a 10 mL cylinder, the propagated density uncertainty drops to a few percent, likely reconciling the two students' values within combined uncertainty and permitting a defensible identification.