Density

Section 1 of 11What density means

1Density2 points

• the mass per unit volume of a substance, a measure of how tightly its matter is packed.
• A small piece of lead has more mass than the same volume of wood, so lead is denser.

2Equation2 points

• density = mass ÷ volume, written ρ = m ÷ V.
• The symbol ρ is the Greek letter rho.

3Units2 points

• the SI unit is kg/m³, and g/cm³ is often more convenient for solids and liquids.
• To convert, 1 g/cm³ = 1000 kg/m³.

4Rearranging3 points

• the same equation can be used in three ways.
• Mass m = ρ × V and volume V = m ÷ ρ.
• A formula triangle with m on top and ρ and V below helps to remember the three forms.

5Density does not depend on size2 points

• a large block and a small block of the same pure material have the same density.
• Density is a property of the material, so it can be used to identify it.

Section 2 of 11Typical densities

• Values you should recognise
  • they help you judge whether an answer is sensible.

Material	Density (kg/m3)	Density (g/cm3)
Air	1.2	0.0012
Cork	240	0.24
Ice	920	0.92
Water	1000	1.0
Sea water	1030	1.03
Aluminium	2700	2.7
Iron	7900	7.9
Copper	8900	8.9
Lead	11 300	11.3
Mercury	13 600	13.6

• Gases
  • their particles are far apart, so gases have very low densities.
• Solids and liquids
  • their particles are touching, so their densities are much higher.
  • A solid is usually slightly denser than its liquid, but ice is less dense than water, so it floats.

Section 3 of 11Measuring the density of a regular solid

• Aim
  • find the density of a block with a regular shape.

1. Measure the mass of the block on a top-pan balance.
2. Measure the length, width and height with a ruler or vernier calipers.
3. Calculate the volume V = l × w × h, or πr²h for a cylinder.
4. Calculate the density from ρ = m ÷ V.

• Improving accuracy
  • measure each dimension several times in different places and take a mean.
• Common error
  • if the measurements are in cm, the answer comes out in g/cm³, so check the units before reporting.

Section 4 of 11Measuring the density of an irregular solid

• Aim
  • find the density of an object with no simple shape, such as a stone.

1. Find the mass of the stone on a balance.
2. Half-fill a measuring cylinder with water and record the volume V₁.
3. Tie the stone to a thread and lower it gently into the water until it is completely submerged.
4. Record the new volume V₂.
5. Volume of stone = V₂ − V₁, then calculate ρ = m ÷ V.

• Eureka can method
  • for a larger object, fill the can to the spout and collect the water displaced as the object is lowered in.
  • The volume of displaced water equals the volume of the object.
• Sources of error
  • air bubbles clinging to the stone, parallax when reading the meniscus and water splashing out.

Section 5 of 11Measuring the density of a liquid

• Aim
  • find the density of a liquid such as oil.

1. Place an empty measuring cylinder on a balance and record its mass m₁.
2. Pour in a known volume V of the liquid, such as 50 cm³.
3. Place the cylinder on the balance again and record the total mass m₂.
4. Mass of liquid = m₂ − m₁.
5. Calculate ρ = (m₂ − m₁) ÷ V.

• Using a density bottle
  • a bottle of fixed volume gives a more accurate volume than a measuring cylinder.
• Tare method
  • place the cylinder on the balance and zero it so the balance reads only the liquid's mass.

Section 6 of 11Floating, sinking and flotation

• Rule for floating
  • an object floats in a fluid if its average density is less than the density of the fluid.
  • It sinks if its density is greater than the fluid.
  • If the densities are equal it floats at any depth, as a neutral buoyancy object.
• Examples
  • a cork floats on water because 240 is less than 1000 kg/m³, and iron sinks because 7900 is greater than 1000.
  • A steel ship floats because its hollow shape gives an average density, including the air inside, below that of water.
• Layers of liquids
  • immiscible liquids separate by density, with the least dense on top.
  • Oil floats on water and water floats on mercury.
• Hot air balloon and helium balloons
  • the gas inside is less dense than the surrounding air, so they rise.
• Hydrometer
  • a floating instrument that sinks lower in a less dense liquid and is used to test battery acid and milk.

Section 7 of 11Density calculations step by step

• Method for calculations
  • follow the same routine each time.

1. List the known quantities and convert to consistent units.
2. Write the equation ρ = m ÷ V.
3. Rearrange for the unknown quantity.
4. Substitute the values and calculate.
5. Check that the unit and magnitude are sensible, comparing with the table of typical values.

• Mixed volumes
  • if a hollow object holds air, use the total volume and total mass for its average density.
• Alloys and mixtures
  • the mass of the mixture is the sum of the masses, and the volume is the sum of the volumes if the volumes simply add.

Section 8 of 11Density and the particle model

• Why solids are dense
  • the particles are packed closely together in a regular arrangement, so a lot of mass fits into a small volume.
• Why gases are not dense
  • the particles are far apart with mostly empty space between them, so the mass per unit volume is tiny.
• Effect of temperature
  • heating makes most substances expand, so the volume increases and the density decreases.
  • This is why hot air rises, because warm air is less dense than the cooler air around it.
• Water's anomaly
  • water is densest at about 4 °C, so ice floats and lakes freeze from the top down.
  • Fish can survive below the ice because the water underneath stays liquid.
• Density of mixtures
  • to find the density of a mixture, find the total mass and divide by the total volume.
  • Mixing 100 cm³ of water with 100 cm³ of oil of density 0.80 g/cm³ gives a total mass of 180 g in 200 cm³,
    • so an average density of 0.90 g/cm³ if the volumes add.

Section 9 of 11Worked examples

1. A metal block measures 5.0 cm by 4.0 cm by 2.0 cm and has a mass of 316 g. Calculate its density and suggest the metal. [3]3 marks

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2. A stone of mass 84 g is lowered into a measuring cylinder containing 40 cm³ of water, and the level rises to 70 cm³. Find the density of the stone in g/cm³ and kg/m³. [4]4 marks

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3. An empty measuring cylinder has a mass of 62 g. When 80 cm³ of a liquid is added the total mass is 126 g. Find the density of the liquid and say whether it would float on water. [4]4 marks

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4. A cube of side 0.10 m is made of wood with density 600 kg/m³. Calculate its mass, and its weight (g = 9.8 N/kg). [4]4 marks

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5. Explain why a steel ship floats although a steel bolt sinks. [3]3 marks

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Section 10 of 11Key terms

Term	Meaning
Density	Mass per unit volume of a substance, ρ = m ÷ V.
Volume	The amount of space an object occupies.
Displacement method	Finding the volume of an irregular solid by the rise in water level when it is submerged.
Meniscus	The curved surface of a liquid in a narrow tube, read at the bottom of the curve.
Floating	The state where an object rests on the surface because its average density is below that of the fluid.
Eureka can	A can with a spout used to collect the water displaced by an object.
Hydrometer	A floating device that measures the density of a liquid.

Section 11 of 11Exam technique

1. Using the wrong units for volume — 1 m³ = 10⁶ cm³, so convert before comparing kg/m³ with g/cm³.
2. Forgetting to subtract the starting volume in displacement — the object volume is the change in level.
3. Saying heavy objects sink and light ones float — it depends on density, not total mass.
4. Reading the mass of the cylinder plus liquid as the liquid's mass — subtract the empty cylinder.
5. Confusing density with weight — density is mass per volume, a property of the material.