Density Data, Air Bubbles, and an Unknown Solid
Students receive an unknown metallic solid, a 100 mL graduated cylinder with 2 mL graduations, a balance reading to 0.01 g, and water. They are told to identify the metal from its density. Trial 1 has mass 24.36 g, initial water volume 50.0 mL, and final volume 58.5 mL. Trial 2 with the same sample has initial volume 50.0 mL and final volume 59.0 mL. Trial 3 has final volume 58.0 mL. The sample has sharp edges and is small enough to slide into the cylinder. A student wants to report the average density and identify the metal as aluminum (2.70 g/cm³). Another student notices that the sample sometimes traps an air bubble and that the cylinder's meniscus is hard to read. The teacher asks the group to analyze the data, decide whether the identification is defensible, and propose a safer, more reliable procedure. The unknown metal is not labeled, and the lab safety sheet says to treat unknown solids as potentially irritating.
1. Calculate the density from each trial and the average. Estimate the uncertainty from the graduated cylinder and decide whether the aluminum identification is defensible.
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Volumes by displacement: trial 1 V = 58.5 − 50.0 = 8.5 mL, so ρ = 24.36/8.5 = 2.87 g/cm³; trial 2 V = 9.0 mL, so ρ = 2.71 g/cm³; trial 3 V = 8.0 mL, so ρ = 3.05 g/cm³. The mean is about 2.87 g/cm³. A 2 mL graduated cylinder is read to about ±0.5 mL at each meniscus, so a difference has uncertainty sqrt(0.5² + 0.5²) ≈ 0.7 mL. On an 8.5 mL displacement that is about 8%, or ±0.24 g/cm³ in density. The trial spread from 2.71 to 3.05 is consistent with that. Aluminum's 2.70 g/cm³ lies at the low edge of the range, so the data are consistent with aluminum but do not uniquely identify it; another material or an air bubble could produce the same result.
2. How would an air bubble affect the result? Propose a safer and more reliable procedure, including measurement and safety steps.
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An adhering air bubble increases the final volume reading, so the apparent displaced volume is too large and the calculated density is too low. The 2.71 trial, with 9.0 mL, is the most suspicious. Parallax in reading the meniscus can shift both initial and final readings. Improvements: use a narrower cylinder, such as 10 mL with 0.1 mL divisions, or a buret or overflow can; use a larger sample so the displacement is much larger than the reading error; dry the sample and tap the cylinder to dislodge bubbles; lower the sample with a thin wire; read at eye level; repeat and report standard deviation. Safety: wear goggles and gloves, use tongs, treat the unknown solid as potentially irritating, check with the instructor before adding it to water in case it reacts, and dispose of it as directed. Do not identify a metal by taste or touch.
3. Suppose the sample mass were 124.36 g and the displacement were 46.0 mL. Calculate the density and uncertainty. Does this improve the identification, and what conclusion should be reported?
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With 124.36 g and 46.0 mL, ρ = 124.36/46.0 = 2.70 g/cm³. The volume uncertainty is still about 0.7 mL, now only 1.5% of 46.0 mL, so the density uncertainty is about ±0.04 g/cm³. This is a much stronger match to aluminum (2.70) and the air-bubble effect is relatively smaller. However, a single density match still does not prove identity: alloys or other materials can have similar densities, and a systematic volume offset would shift the result. The group should report 2.70 ± 0.04 g/cm³, state that it is consistent with aluminum, and confirm with a second property or a known-sample comparison. The larger sample is the key improvement because it makes the displacement large compared with the cylinder's resolution.