Calculating Density

Introduction

Knowing what density means is only half the job; chemists also need to measure it. Density is never read directly from a single instrument — it is calculated from two measurements, mass and volume. This page shows how to carry out the calculation reliably and how to measure the density of solids and liquids in the laboratory.

Core explanation

The calculation. Density = mass ÷ volume. Use mass in grams and volume in cm³ to obtain g/cm³; use kilograms and m³ to obtain kg/m³. Never mix a mass in kg with a volume in cm³ without converting.

Density of a regular solid. 1. Weigh the solid on a balance. 2. Measure its length, width and height and calculate the volume (page 38). 3. Divide mass by volume.

Density of an irregular solid. 1. Weigh the solid. 2. Find its volume by displacement (page 39). 3. Divide mass by volume.

Density of a liquid. 1. Place an empty measuring cylinder on the balance and tare it (or record its mass). 2. Pour in a measured volume of the liquid, for example 50.0 cm³, reading the meniscus at eye level. 3. Record the mass of the liquid. 4. Divide mass by volume.

Precision of the answer. A calculated density cannot be more precise than the measurements used. If the volume is known to three significant figures (for example 12.5 cm³), give the density to three significant figures as well. A calculator display of 2.7043478 g/cm³ should be rounded to 2.70 g/cm³.

Checking the answer. Compare your value with data tables. Most metals lie between about 2 and 20 g/cm³, most liquids between 0.7 and 1.5 g/cm³, and gases around 0.001 g/cm³. A metal density of 270 g/cm³ signals a unit error.

Step-by-step reasoning

To calculate the density of a stone:

1. Mass from the balance: 33.8 g. 2. Volume by displacement: water rises from 50.0 cm³ to 63.0 cm³, so V = 13.0 cm³. 3. Density = 33.8 ÷ 13.0 = 2.6 g/cm³ (2.60 to three significant figures). 4. Compare: 2.6 g/cm³ is typical of rocks such as granite, so the answer is reasonable.

Visual explanation

A results table for a liquid density experiment:

Measurement Value --- --- Mass of empty cylinder 72.40 g Mass of cylinder + 50.0 cm³ liquid 111.90 g Mass of liquid 39.50 g Density = 39.50 ÷ 50.0 0.790 g/cm³

The value matches ethanol (0.79 g/cm³).

Real-world analogy

Calculating density is like working out how much a bag of rice costs per kilogram. You need two numbers — the total price and the total mass — and you divide one by the other. The "per kilogram" figure lets you compare bags of different sizes fairly, just as density lets you compare samples of different sizes.

Real-world example

Dairies check milk quality by measuring its density with an instrument called a lactometer. Pure cow's milk has a density of about 1.028–1.034 g/cm³. If water has been added, the density falls towards 1.00 g/cm³, revealing the dilution.

Why?

Why is it better to use a large volume of liquid when measuring density? Every measurement has a fixed uncertainty — about ±0.5 cm³ on a measuring cylinder, for example. With 50 cm³ this is a 1% uncertainty; with 5 cm³ it would be 10%. Larger samples make the fixed uncertainty a smaller fraction of the reading.

Common misconception

Students sometimes divide volume by mass instead of mass by volume. Remember that density is "mass per volume" — the thing after "per" goes on the bottom. A quick check: water's density must come out as about 1 g/cm³, and denser materials must give larger numbers.

Worked example

Question: A rectangular block measures 4.0 cm × 2.5 cm × 2.0 cm and has a mass of 157 g. Calculate its density and suggest the metal.

Reasoning: Volume = 4.0 × 2.5 × 2.0 = 20 cm³. Density = 157 ÷ 20 = 7.85 g/cm³.

Answer: About 7.9 g/cm³ — consistent with iron (7.87 g/cm³) or steel.

Quick check

1. A liquid has a mass of 25.2 g and a volume of 20.0 cm³. Calculate its density. Answer: 25.2 ÷ 20.0 = 1.26 g/cm³.

Exam focus

Write the equation, substitute with units and give the answer with a unit and sensible significant figures. Method questions carry marks for naming the apparatus (balance, measuring cylinder, ruler or displacement) and for stating that mass is divided by volume. Watch for masses given in kg or volumes in dm³.

Advanced insight

A more precise method for liquids uses a density bottle (pycnometer), a small flask with a stopper containing a fine capillary so that it holds exactly the same volume every time. Weighing it empty, full of water and full of the liquid gives densities to four or five significant figures.

Summary

Density is calculated as mass ÷ volume from separate measurements. For regular solids use dimensions, for irregular solids use displacement, and for liquids weigh a measured volume. Keep units consistent, round to match the precision of the data and check the result against typical values.

Practice questions

1. A pebble has a mass of 52.0 g and a volume of 20.0 cm³. Find its density. Answer: 52.0 ÷ 20.0 = 2.60 g/cm³. 2. Describe how to find the density of cooking oil. Answer: Weigh an empty measuring cylinder, add a measured volume of oil (reading at eye level), reweigh to find the oil's mass, then divide mass by volume. 3. A student obtains a density of 0.37 cm³/g for water. What mistake was made? Answer: The student divided volume by mass; density is mass ÷ volume. 4. A block of mass 2.16 kg has a volume of 800 cm³. Find its density in g/cm³. Answer: 2.16 kg = 2160 g; 2160 ÷ 800 = 2.70 g/cm³ (aluminium).