Density of materials

What you'll learn

This revision guide covers everything you need to know about density for your AQA GCSE Physics exam. You'll learn how to define and calculate density, understand why different materials have different densities, and master the required practical techniques for measuring density of regular solids, irregular solids, and liquids. These concepts appear frequently in Paper 1 and are essential for achieving top grades.

Key terms and definitions

Density — the mass per unit volume of a substance, measured in kg/m³ or g/cm³

Mass — the quantity of matter in an object, measured in kilograms (kg) or grams (g)

Volume — the amount of three-dimensional space occupied by an object, measured in m³ or cm³

Displacement — a method used to find the volume of an irregular object by measuring how much fluid it pushes aside

Eureka can — a displacement vessel with a spout used to collect displaced water when finding the volume of irregular objects

Regular solid — an object with a uniform geometric shape whose volume can be calculated using a mathematical formula

Compression — when particles in a substance are forced closer together, increasing density

Particle arrangement — how closely packed the particles are in a substance, determining its density

Core concepts

The density equation

The relationship between density, mass, and volume is expressed by the equation:

ρ = m/V

Where:

• ρ (rho) = density in kg/m³ or g/cm³
• m = mass in kg or g
• V = volume in m³ or cm³

This equation can be rearranged using the formula triangle:

• Density = mass ÷ volume
• Mass = density × volume
• Volume = mass ÷ density

Unit conversions are critical when working with density:

• 1 g/cm³ = 1000 kg/m³
• To convert cm³ to m³, divide by 1,000,000 (or multiply by 10⁻⁶)
• To convert g to kg, divide by 1000

Typical density values for different states of matter

Understanding the typical densities of solids, liquids, and gases helps explain their properties:

Solids have the highest densities:

• Steel: 7800 kg/m³
• Aluminium: 2700 kg/m³
• Ice: 920 kg/m³
• Particles are very close together in a fixed, regular arrangement
• Strong forces hold particles in position
• Very little empty space between particles

Liquids have medium densities:

• Water: 1000 kg/m³ (or 1 g/cm³)
• Oil: approximately 900 kg/m³
• Particles are close together but can move around
• Less regular arrangement than solids
• Slightly more space between particles than in solids

Gases have very low densities:

• Air: approximately 1.2 kg/m³
• Oxygen: 1.3 kg/m³
• Carbon dioxide: 1.8 kg/m³
• Particles are far apart and move randomly at high speeds
• Very large spaces between particles
• Weak forces between particles
• Densities are typically 1000 times less than solids or liquids

Measuring the density of a regular solid

The required practical for measuring density of regular solids involves these steps:

Equipment needed:

• Balance (to measure mass)
• Ruler or callipers (to measure dimensions)
• Regular solid object (cube, cuboid, cylinder, or sphere)

Method:

1. Measure the mass of the object using a balance and record in grams or kilograms
2. Measure the dimensions needed to calculate volume:
   • For a cuboid: length, width, and height (V = l × w × h)
   • For a cube: length of one side (V = l³)
   • For a cylinder: diameter and height (V = πr²h)
   • For a sphere: diameter (V = 4/3πr³)
3. Calculate the volume using the appropriate formula
4. Use the equation ρ = m/V to calculate density
5. State the answer with appropriate units

Accuracy improvements:

• Use digital callipers instead of a ruler for more precise measurements
• Measure each dimension multiple times and calculate a mean
• Use a sensitive balance reading to 0.01 g
• Ensure the object is dry before measuring mass

Measuring the density of an irregular solid

For objects without regular geometric shapes, the displacement method must be used:

Method 1: Using a Eureka can (displacement can)

1. Fill a Eureka can with water until it begins to drip from the spout
2. Allow the water to stop dripping, then place an empty measuring cylinder under the spout
3. Carefully lower the irregular object into the water on a string
4. Collect the displaced water in the measuring cylinder
5. The volume of water displaced equals the volume of the object
6. Measure the mass of the object using a balance
7. Calculate density using ρ = m/V

Method 2: Using a measuring cylinder

1. Partly fill a measuring cylinder with water and record the initial volume (V₁)
2. Carefully tilt the measuring cylinder and slide the object in to avoid splashing
3. Record the new volume reading (V₂)
4. Calculate the volume of the object: V = V₂ - V₁
5. Measure the mass of the object using a balance
6. Calculate density using ρ = m/V

Key considerations:

• The object must be insoluble (doesn't dissolve in water)
• The object must be more dense than water (sinks rather than floats)
• Dry the object before measuring its mass
• Avoid trapping air bubbles on the object's surface
• Read the meniscus at eye level for accurate volume measurements

Measuring the density of a liquid

Equipment needed:

• Balance
• Measuring cylinder
• The liquid sample

Method:

1. Place an empty measuring cylinder on a balance and measure its mass (m₁)
2. Pour a known volume of liquid into the measuring cylinder and record the volume (V)
3. Measure the total mass of the measuring cylinder and liquid (m₂)
4. Calculate the mass of the liquid: m = m₂ - m₁
5. Calculate density using ρ = m/V

Alternative method:

1. Measure the mass of an empty measuring cylinder
2. Add the liquid and record both the new mass and the volume
3. Subtract to find the liquid mass
4. Calculate density

Accuracy tips:

• Read the measuring cylinder at eye level
• Read from the bottom of the meniscus for water and aqueous solutions
• Use a larger volume of liquid for more accurate results
• Ensure the measuring cylinder is on a level surface
• Avoid parallax error when reading the scale

Explaining density differences using particle theory

The particle model explains why different materials have different densities:

Factors affecting density:

1. Mass of particles — materials made of heavier atoms/molecules have higher densities

   • Lead atoms are heavier than aluminium atoms, so lead is denser

2. Spacing between particles — more closely packed particles mean higher density

   • Solid copper has particles close together (high density)
   • Gaseous copper has particles far apart (low density)

3. Particle arrangement — how efficiently particles are packed

   • Regular crystalline structures can pack more tightly
   • Irregular arrangements leave more gaps

Changes in state and density:

When substances change state, their density changes:

• Freezing (liquid → solid): usually increases density as particles pack closer
• Exception: Water becomes less dense when it freezes to ice (920 kg/m³ vs 1000 kg/m³)
• Melting (solid → liquid): usually decreases density slightly
• Boiling (liquid → gas): massively decreases density (particles move far apart)
• Condensing (gas → liquid): massively increases density

Compression of gases:

Gases can be compressed because particles are far apart with large spaces:

• Compressing a gas pushes particles closer together
• Volume decreases significantly
• Mass remains the same
• Density increases (ρ = m/V, so smaller V means larger ρ)

Liquids and solids cannot be compressed significantly because particles are already close together.

Worked examples

Example 1: Calculating density of a regular solid

Question: A student has a metal cube with sides of length 2.0 cm. The mass of the cube is 62.4 g. Calculate the density of the metal in g/cm³ and identify whether it could be aluminium (density 2.7 g/cm³) or copper (density 8.9 g/cm³). [4 marks]

Solution:

Step 1: Calculate the volume of the cube

• V = l³ = 2.0³ = 8.0 cm³ [1 mark]

Step 2: Use the density equation

• ρ = m/V [1 mark]
• ρ = 62.4/8.0 = 7.8 g/cm³ [1 mark]

Step 3: Compare with given values

• The density is closer to copper than aluminium, but lower than 8.9 g/cm³
• The cube could be made of a copper alloy or could be a different metal such as steel [1 mark]

Example 2: Finding mass using density

Question: A storage tank has a volume of 2.5 m³ and is completely filled with oil. The density of the oil is 900 kg/m³. Calculate the mass of oil in the tank. [3 marks]

Solution:

Step 1: Identify the equation needed

• Rearrange ρ = m/V to give m = ρ × V [1 mark]

Step 2: Substitute values

• m = 900 × 2.5 [1 mark]

Step 3: Calculate and state unit

• m = 2250 kg [1 mark]

Example 3: Measuring density of an irregular object

Question: A student uses a displacement can to find the volume of a small stone. The stone has a mass of 45 g. When placed in the displacement can, it causes 15 cm³ of water to overflow into the measuring cylinder. Calculate the density of the stone in g/cm³. [3 marks]

Solution:

Step 1: Identify that volume of stone = volume of displaced water

• V = 15 cm³ [1 mark]

Step 2: Apply the density equation

• ρ = m/V = 45/15 [1 mark]

Step 3: Calculate answer with unit

• ρ = 3.0 g/cm³ [1 mark]

Common mistakes and how to avoid them

• Mixing units — Always ensure mass and volume are in compatible units. If volume is in cm³, mass should be in g. If volume is in m³, mass should be in kg. Convert before calculating, not after.

• Formula rearrangement errors — Use a formula triangle or practise rearranging ρ = m/V correctly. Remember: to find mass, multiply density by volume (m = ρV); to find volume, divide mass by density (V = m/ρ).

• Reading scales incorrectly — When using measuring cylinders, always read at eye level from the bottom of the meniscus. For mass balances, ensure the balance reads zero before adding the object (tare function).

• Forgetting to subtract the initial mass or volume — When measuring liquid density, subtract the mass of the empty container. When using displacement with a measuring cylinder, subtract the initial water volume from the final volume.

• Not showing working — Even if you can do the calculation mentally, always show each step. In AQA exams, marks are awarded for method, not just the final answer. A correct method with a calculation error still earns marks.

• Unit conversion mistakes — Remember that 1 g/cm³ = 1000 kg/m³ (not 100). To convert cm³ to m³, divide by 1,000,000 because you're converting three dimensions (100 × 100 × 100).

Exam technique for "Density of materials"

• Command words matter: "Calculate" means show working and include units. "State" requires just the answer. "Explain" needs a because/therefore statement linking cause and effect. "Describe the method" requires ordered steps without explanation.

• Required practical questions: When describing the method to measure density, include specific equipment names (Eureka can, measuring cylinder, balance), actual measurements taken (mass in g, volume in cm³), and the calculation performed. Questions worth 6 marks typically require 6-8 distinct points.

• Significant figures: Give answers to the same number of significant figures as the data provided, or to 2-3 significant figures if not specified. For density calculations, 2 or 3 significant figures is usually appropriate.

• Show units throughout: Include units at every stage of the calculation, not just the final answer. This demonstrates understanding and helps you spot unit conversion errors. State whether density is in g/cm³ or kg/m³.

Quick revision summary

Density is mass per unit volume (ρ = m/V), measured in kg/m³ or g/cm³. Solids have the highest densities (particles close together), liquids have medium densities, and gases have very low densities (particles far apart). Measure density of regular solids by calculating volume from dimensions; use displacement methods (Eureka can or measuring cylinder) for irregular solids; measure liquid density by finding mass of a known volume. Always ensure consistent units and show full working in calculations.