Density and Relative Density: Formulas, Methods, Examples

Why Does Iron Sink and Wood Float?

A tiny iron nail sinks in water, yet a large log of wood floats. A bag of cotton and a bag of cement may be the same size, but you would not want to carry the cement. The property behind all of this is density: how much mass is packed into a given volume. This tutorial covers density and relative density for WAEC, NECO and JAMB physics: the definitions and units, measuring the density of regular solids, irregular solids and liquids, the density bottle, relative density of powders and granules, the hydrometer, mixtures, and why objects float or sink, with worked examples.

What Is Density?

The density of a substance is its mass per unit volume:

density (ρ) = mass (m) ÷ volume (V)

The SI unit is the kilogram per cubic metre (kg/m³). In the laboratory, the gram per cubic centimetre (g/cm³) is often more convenient.

Converting units

1 g/cm³ = 1000 kg/m³, because 1 g = 0.001 kg and 1 cm³ = 0.000001 m³. So to convert from g/cm³ to kg/m³, multiply by 1000.

Some typical densities: water 1000 kg/m³ (1.0 g/cm³), ice 920 kg/m³, kerosene about 800 kg/m³, aluminium 2700 kg/m³, iron about 7900 kg/m³, mercury 13,600 kg/m³ and air about 1.3 kg/m³.

Density depends on the material, not on how much of it there is. A drop of mercury and a bottle of mercury have the same density. Density does change slightly with temperature, because most substances expand when heated, so the same mass occupies a larger volume.

Measuring the Density of a Regular Solid

Measure the mass with a beam balance or electronic balance.
Measure the dimensions with a metre rule, vernier calipers or micrometer screw gauge, and calculate the volume (length × breadth × height for a block, πr²h for a cylinder).
Divide the mass by the volume.

Worked example 1

A rectangular block measures 5 cm × 4 cm × 2 cm and has a mass of 120 g. Find its density in g/cm³ and kg/m³.

V = 5 × 4 × 2 = 40 cm³, so ρ = 120 ÷ 40 = 3 g/cm³ = 3000 kg/m³.

Measuring the Density of an Irregular Solid

For a stone or a key, the volume is found by displacement of water.

Pour water into a measuring cylinder and note the reading.
Tie the solid to a thread and lower it gently until it is completely covered.
The rise in the water level is the volume of the solid.

For a larger object, use a eureka (displacement) can: fill the can to the spout, lower in the object, and collect the overflow in a measuring cylinder.

Worked example 2

The water level in a measuring cylinder rises from 50 cm³ to 68 cm³ when a metal piece of mass 48.6 g is lowered into it. Find the density of the metal.

V = 68 − 50 = 18 cm³, so ρ = 48.6 ÷ 18 = 2.7 g/cm³. This matches aluminium.

Measuring the Density of a Liquid

Weigh an empty beaker, pour in a measured volume of the liquid, and weigh again. The difference in mass, divided by the volume, gives the density. A more accurate method uses a density bottle, described next.

What Is Relative Density?

Relative density (formerly called specific gravity) compares the density of a substance with the density of water:

relative density = density of substance ÷ density of water

Because it is a ratio of two densities, relative density has no units. Since equal volumes are compared, it can also be written as:

relative density = mass of substance ÷ mass of an equal volume of water

For example, aluminium has a relative density of 2.7, so its density is 2.7 × 1000 = 2700 kg/m³. In g/cm³, density and relative density have the same numerical value, because water's density is 1 g/cm³.

The Density Bottle

A density bottle (relative density bottle) has a ground-glass stopper with a fine hole, so it always holds exactly the same volume when filled. Any excess liquid escapes through the hole and is wiped off.

Relative density of a liquid

Weigh the clean, dry, empty bottle: m1.
Fill it with water, insert the stopper, wipe dry and weigh: m2.
Empty and dry it, fill it with the liquid and weigh: m3.

Relative density = (m3 − m1) ÷ (m2 − m1).

Worked example 3

An empty density bottle has a mass of 20 g. Its mass is 70 g when full of water and 60 g when full of kerosene. Find the relative density and the density of kerosene.

Mass of kerosene = 60 − 20 = 40 g, and mass of an equal volume of water = 70 − 20 = 50 g.

Relative density = 40 ÷ 50 = 0.8, so the density = 0.8 × 1000 = 800 kg/m³.

Relative density of sand or granules

Small particles cannot be measured with a ruler, but the density bottle handles them neatly.

Weigh the empty bottle: m1.
Add some sand and weigh: m2.
Fill up the rest of the bottle with water and weigh: m3.
Empty, clean and fill with water only, then weigh: m4.

The mass of sand is m2 − m1. The water displaced by the sand has mass (m4 − m1) − (m3 − m2). So:

relative density = (m2 − m1) ÷ [(m4 − m1) − (m3 − m2)]

Worked example 4

In such an experiment, m1 = 25.0 g, m2 = 45.0 g, m3 = 87.5 g and m4 = 75.0 g. Find the relative density of the sand.

Mass of sand = 45.0 − 25.0 = 20.0 g.

Mass of water displaced = (75.0 − 25.0) − (87.5 − 45.0) = 50.0 − 42.5 = 7.5 g.

Relative density = 20.0 ÷ 7.5 ≈ 2.67.

Relative Density by Archimedes' Principle

A solid that sinks can be weighed in air and then fully immersed in water. The loss in weight equals the weight of water displaced, which is the weight of an equal volume of water. So:

relative density = weight in air ÷ loss of weight in water

For example, a stone that weighs 2.5 N in air and 1.5 N in water has a relative density of 2.5 ÷ (2.5 − 1.5) = 2.5. The tutorial on pressure and Archimedes' principle covers this method in more depth.

Density of Mixtures

When liquids are mixed (assuming no change in total volume), the density of the mixture is the total mass ÷ total volume, not the average of the two densities.

Worked example 5

200 cm³ of a liquid of density 0.8 g/cm³ is mixed with 300 cm³ of water. Find the density of the mixture.

Mass of liquid = 200 × 0.8 = 160 g, and mass of water = 300 × 1.0 = 300 g.

Density = (160 + 300) ÷ (200 + 300) = 460 ÷ 500 = 0.92 g/cm³.

Worked example 6: mass of air in a room

A classroom measures 5 m × 4 m × 3 m. If the density of air is 1.3 kg/m³, find the mass of air in the room.

V = 60 m³, so m = ρV = 1.3 × 60 = 78 kg. Air is light, but there is a surprising amount of it.

Floating, Sinking and the Hydrometer

An object that is less dense than a fluid floats in it, and one that is more dense sinks. Wood (about 600 kg/m³) floats on water, and iron sinks. A steel ship floats because its hollow shape gives it a large volume, so its average density (including the air inside) is less than that of water.

For a floating object, the fraction of its volume below the surface equals the ratio of its density to the fluid's density. Ice of density 920 kg/m³ floats with 920 ÷ 1000 = 92% of its volume submerged, which is why most of an iceberg is hidden.

A hydrometer measures the relative density of a liquid directly. It is a weighted glass float with a graduated stem, and it sinks deeper in a less dense liquid. Hydrometers are used to test the acid in car batteries (to judge the state of charge), the strength of alcoholic drinks, and milk, where a hydrometer called a lactometer can detect added water.

Density and Relative Density: Practice Questions

A cube of side 3 cm has a mass of 216 g. Find its density in kg/m³.
Express 13.6 g/cm³ in kg/m³.
A density bottle weighs 30 g empty, 80 g full of water and 85 g full of a salt solution. Find the relative density of the solution.
A body weighs 6.0 N in air and 4.0 N when fully immersed in water. Find its relative density.
Find the mass of 2.5 litres of a liquid of relative density 0.9.

Answers: 1. V = 27 cm³, ρ = 8 g/cm³ = 8000 kg/m³. 2. 13,600 kg/m³. 3. (85 − 30) ÷ (80 − 30) = 55 ÷ 50 = 1.1. 4. 6.0 ÷ 2.0 = 3.0. 5. 2.5 litres = 2500 cm³, and density = 0.9 g/cm³, so mass = 2250 g = 2.25 kg.

Conclusion

Density is mass per unit volume, measured in kg/m³ or g/cm³, and 1 g/cm³ equals 1000 kg/m³. Relative density compares a substance with water and has no units. Measure volumes of regular solids by calculation, irregular solids by displacement, and liquids and granules with a density bottle. Remember that mixtures use total mass over total volume, and that whether an object floats depends on its average density compared with the fluid.