Choosing a Measuring Instrument for a Density Practical
A class must determine the density of an irregular small stone and of a clear liquid. Available equipment: a 100 mL measuring cylinder with 2 mL graduations, a 10 mL measuring cylinder with 0.2 mL graduations, a top-pan balance reading to 0.01 g, a 50 mL beaker, and a ruler. The stone has a mass of about 12 g. The liquid sample is only 8 mL. A student plans to find the stone's volume by displacement in the 100 mL cylinder and the liquid's density by weighing 8 mL in the 50 mL beaker. A second student argues the results will be too imprecise and proposes different choices.
1. Calculate the likely percentage uncertainty in the stone's volume if measured in the 100 mL cylinder, and explain why the second student is concerned.
Read the worked answer
The 100 mL cylinder has graduations of 2 mL. Reading uncertainty is typically about half a division, so roughly ±1 mL for each reading.
Displacement volume = final reading − initial reading. The uncertainties add: about ±1.4 mL (or, conservatively, ±2 mL).
If the stone displaces about 4–5 mL (typical for a 12 g stone of density around 2.5–3 g/cm³), then:
Percentage uncertainty ≈ (1.4 ÷ 4.5) × 100 ≈ 31% (or about 40% using ±2 mL).
This is far too large. The density would be quoted with an uncertainty of roughly a third of its value, making comparisons meaningless. The concern is justified: the instrument's resolution is coarse relative to the small volume being measured, so random reading error dominates the result.
2. Recommend the best instrument choices for each measurement and justify them in terms of resolution and percentage uncertainty.
Read the worked answer
Stone volume: the 10 mL cylinder (0.2 mL graduations) gives a reading uncertainty of about ±0.1 mL per reading, so ±0.14 mL on a 4.5 mL displacement - about 3%. This is ten times better than the 100 mL cylinder. The stone (about 12 g, a few mL) fits easily.
Liquid mass: weigh the empty 50 mL beaker, then add the 8 mL and reweigh. The balance reads to 0.01 g, so uncertainty is about ±0.02 g on a mass of roughly 8 g (for water-like liquid) - about 0.25%. This is excellent.
Liquid volume: the 10 mL cylinder (0.2 mL graduations) gives about ±0.14 mL on 8 mL - about 1.8%. The 50 mL beaker has no fine volume scale and should not be used to measure volume; it is only a container.
Overall: density uncertainty is the sum of the mass and volume percentage uncertainties. Using the 10 mL cylinder for both volume measurements and the balance for mass keeps total uncertainty to a few percent, which is fit for purpose. The 100 mL cylinder should be reserved for volumes of 40 mL or more.
3. The stone is porous and absorbs a little water. Explain how this would bias the result and suggest a modification.
Read the worked answer
If the stone absorbs water, the cylinder reading after immersion includes water taken up into the pores as well as water displaced by the solid. The measured 'volume' is therefore larger than the true external volume of the stone.
Density = mass ÷ volume, so an over-large volume gives a density that is too low - a systematic error, not random scatter. Repeating the measurement will not fix it.
Modifications:
- Seal the stone with a thin coat of wax or varnish before immersion, then account for the coating (or use a coating of negligible and known volume).
- Alternatively, soak the stone in water until it is saturated, blot the surface, and then measure displacement - the pores are already full so no further uptake occurs.
- Use a displacement liquid that does not enter the pores, such as dry sand or a fine powder, though this is harder to control.
Any of these converts the systematic error into a much smaller one and gives a density closer to the true value.