Reading the Density Graph
Students are given a data table for five unknown solid cubes of equal volume (1.0 cm³ each) with masses of 2.7 g, 7.9 g, 8.9 g, 11.3 g, and 19.3 g. A reference list gives densities: aluminum 2.7 g/cm³, iron 7.9 g/cm³, copper 8.9 g/cm³, lead 11.3 g/cm³, and gold 19.3 g/cm³. A student says, 'The heaviest cube must be the densest, and since all cubes are the same size, the one that feels heaviest is gold.' Another student points out that a cube twice the volume of aluminum could have the same mass as a smaller denser cube. The teacher wants students to connect the mass–volume relationship to the meaning of density and to recognize when a single measurement is insufficient.
1. Explain why, for these equal-volume cubes, ranking by mass is equivalent to ranking by density - and why that equivalence would fail if volumes differed.
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Why it works here: Density = mass ÷ volume. If every cube has volume 1.0 cm³, then density in g/cm³ equals mass in grams numerically. So the mass ranking and density ranking coincide: 2.7 < 7.9 < 8.9 < 11.3 < 19.3.
Why it fails with different volumes: A large low-density block can have more mass than a small high-density block. For example, a 10 cm³ aluminum cube has mass 27 g, far more than a 1 cm³ gold cube at 19.3 g. Mass alone does not identify a material; you need mass and volume to compute density.
Takeaway: The student's shortcut is valid only because the problem fixed volume. The underlying concept is the ratio, not the raw mass.
2. A sixth cube of the same material as one of the listed metals has a mass of 39.5 g and a volume of 5.0 cm³. Identify the likely material and explain your reasoning.
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Compute density: 39.5 g ÷ 5.0 cm³ = 7.9 g/cm³.
Match: 7.9 g/cm³ corresponds to iron on the reference list.
Reasoning: Density is an intensive property - it does not depend on sample size - so a 5 cm³ piece of iron should have the same density as a 1 cm³ piece. The computed ratio matches iron, not just 'a heavy metal.' Note that a single mass reading (39.5 g) would have been misleading; the volume was essential to the identification.
3. A student measures a cube's mass as 8.9 g but is unsure of its volume, guessing 'about 1 cm³.' How should the class treat a density calculated from a guessed volume, and what would improve confidence?
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Treat it cautiously. A density computed from a guessed volume inherits the guess's error. If the true volume were 1.2 cm³, density would be 7.4 g/cm³, not 8.9 - enough to misidentify the metal.
Improvements:
- Measure volume directly - ruler dimensions for a regular cube (length × width × height) or water displacement for irregular shapes.
- Repeat measurements and average to reduce random error.
- Report uncertainty so the class can see whether the value is close enough to a reference density to claim a match.
Reasoning: Identification by density requires both mass and volume to be measured, not assumed. A precise-looking number built on an estimate can be confidently wrong.