Density of Materials

Density tells you how much mass is squeezed into a given volume of a material. A block of lead and a block of expanded polystyrene can be exactly the same size, yet the lead is far heavier because its mass is packed into that volume much more tightly.

Density is the mass per unit volume of a material.

$\rho =\frac{m}{V}$

Here $\rho$ (the Greek letter "rho") is density in kilograms per cubic metre (kg/m³), $m$ is mass in kilograms (kg), and $V$ is volume in cubic metres (m³).

You must recall and apply this equation. It is not given on the Physics equation sheet.

Density is a property of the material itself, not of how much of it you have. A gold ring and a gold bar have the same density even though the bar has far more mass, because mass and volume both scale up together.

The particle model represents matter as tiny particles. The way those particles are arranged and how they move explains why solids, liquids and gases behave so differently, including why they have different densities.

State	Arrangement	Spacing	Movement
Solid	Regular, fixed pattern	Very close	Vibrate about fixed positions
Liquid	Random, close together	Close	Move around, slide past each other
Gas	Random, spread out	Far apart	Move quickly in all directions

Simple diagrams model these states as circles: solids as a neat close-packed grid, liquids as touching but disordered circles, and gases as a few scattered circles with wide gaps between them.

In a solid the particles cannot move from place to place; they only vibrate. This is why a solid keeps a fixed shape while a gas fills its container.

Density depends on how much mass fits into a volume, and the particle model explains why that differs between states.

In a solid, particles are packed closely in a regular arrangement, so a small volume contains a large number of particles and therefore a large mass. Solids are usually the densest state.

In a liquid, particles are still close together but arranged randomly, so the density is slightly lower than the solid, though often similar.

In a gas, the particles are spread far apart with large empty gaps between them. The same volume contains very few particles and very little mass, so gases are far less dense than solids or liquids, typically about a thousand times less.

Water is an unusual exception: solid ice is slightly less dense than liquid water, which is why ice floats. This is extra context and not required by the spec, but it shows density is not always highest in the solid.

The density equation can be rearranged to find mass or volume, depending on what a question gives you.

$\rho =\frac{m}{V}m=\rho VV=\frac{m}{\rho }$

Worked example — a metal block has a mass of 2.7 kg and a volume of 0.001 m³. Find its density.

$\rho =\frac{m}{V}=\frac{2.7}{0.001}=2700{\text{kg/m}}^3$

Worked example — a sample of oil has a density of 920 kg/m³ and a volume of 0.5 m³. Find its mass.

$m=\rho V=920\times 0.5=460\text{kg}$

Watch your units. Densities are often quoted in g/cm³ (water is 1 g/cm³) but the SI unit is kg/m³ (water is 1000 kg/m³). To convert g/cm³ to kg/m³, multiply by 1000.

Required Practical 5 measures the densities of regular solids, irregular solids and liquids. For a regular solid such as a cube or cylinder, the volume is calculated from its measured dimensions.

Apparatus: the regular object, a balance, and a ruler (or Vernier callipers or a micrometer for greater precision on small objects).

Method:

1. Place the object on the balance and record its mass in grams.
2. Measure the length, width and height (or the diameter and length of a cylinder) with the ruler or callipers.
3. Calculate the volume from the dimensions, for example $V=\text{length}\times \text{width}\times \text{height}$ for a cuboid.
4. Calculate density using $\rho =m/V$.

Callipers and a micrometer are used for small objects because a ruler cannot measure a few millimetres precisely; a measurement error there causes a large percentage error in the volume.

Worked example — a cuboid measures 4 cm × 3 cm × 2 cm and has a mass of 216 g.

$V=4\times 3\times 2=24{\text{cm}}^3\rho =\frac{216}{24}=9{\text{g/cm}}^3$

For an irregular solid the volume cannot be found from dimensions, so it is found by displacement: the object pushes aside a volume of water equal to its own volume.

Apparatus: the irregular object, a balance, a eureka (displacement) can with a spout, and a measuring cylinder.

Method:

1. Find the mass of the object on the balance.
2. Fill the eureka can until water just runs out of the spout, then wait for it to stop dripping.
3. Place an empty measuring cylinder under the spout.
4. Gently lower the object fully into the water and collect the water that overflows.
5. The volume of water collected equals the volume of the object.
6. Calculate density using $\rho =m/V$.

For a liquid, place a measuring cylinder on the balance and record its mass empty, add a known volume of liquid read from the scale, record the new mass, then divide the mass of liquid by its volume.

Lower the object slowly so no water splashes out and no drops are lost. Any water that escapes without being collected makes the measured volume too small and the density too high.

1. Using the wrong volume formula

Read whether the object is regular or irregular. Regular objects use dimensions; irregular objects use displacement. Do not try to measure an irregular pebble with a ruler.

2. Forgetting to convert units

If mass is in grams and volume in cm³, the density is in g/cm³. To give an answer in kg/m³ you must convert. Mixing grams with m³ gives a nonsense answer.

3. Assuming density means the same as mass

A large object is not automatically denser than a small one. Density compares mass to volume, so a huge low-density object can be lighter than a tiny dense one.

4. Thinking changing state changes the mass

When a substance melts, freezes or boils, mass is conserved. Only the volume changes as particles rearrange, so any change in density comes entirely from the change in volume.

5. Losing water in the displacement method

Fill the eureka can only to the point where it stops dripping before you start, and collect every drop that overflows. Spilled or uncollected water makes the volume reading too small.