Density

What is Density?

Density is a measure of how much mass is packed into a given volume. An object with a high density has a lot of mass squeezed into a small space, while an object with a low density has little mass spread over a large volume.

Think of a block of lead versus a block of foam of the same size — the lead feels much heavier because lead atoms are heavier and packed more tightly together, giving lead a much higher density than foam.

Density (ρ) — the mass per unit volume of a substance. It tells us how tightly packed the matter in an object is.

ρ = m ÷ V

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

You can rearrange this equation to find mass or volume:

m = ρ × V V = m ÷ ρ

Symbol	Quantity	SI Unit	Other Common Unit
ρ (rho)	Density	kg/m³	g/cm³
m	Mass	kg	g
V	Volume	m³	cm³

💡 Unit conversions: 1 g/cm³ = 1000 kg/m³. Water has a density of 1.0 g/cm³ or 1000 kg/m³.

Density and States of Matter

The density of a material depends on two things: the mass of individual atoms or molecules, and how closely they are packed together. This is why the same substance can have very different densities in different states.

Solids have particles arranged in a regular, tightly packed lattice structure. The particles vibrate in fixed positions but cannot move past each other. Because particles are so close together, solids generally have the highest densities.

Liquids have particles that are still close together (similar spacing to solids) but arranged randomly and able to flow past each other. Liquids are typically slightly less dense than their solid counterparts — water (1.0 g/cm³) is denser than ice (0.92 g/cm³), which is why ice floats!

Gases have particles that are far apart and move rapidly and randomly. The particles are spread over a very large volume, so gases have very low densities — typically around 1000 times less dense than the liquid or solid form.

State	Particle Arrangement	Typical Density	Example
Solid	Regular, tightly packed	Highest	Iron: 7874 kg/m³
Liquid	Random, close together	Medium	Water: 1000 kg/m³
Gas	Random, far apart	Lowest	Air: ~1.2 kg/m³

🧊 Water is unusual — it is LESS dense as a solid (ice) than as a liquid because ice has a hexagonal open lattice structure. This is why ice floats on water.

Measuring Density of Regular Solids

For a regular solid (one with a predictable geometric shape such as a cube, cuboid, cylinder, or sphere), you can calculate its volume using a ruler and a mathematical formula.

Method:

1. Use a top-pan balance to measure the mass (m) of the object in grams or kilograms.

2. Use a ruler to measure the relevant dimensions:

Cuboid: length (l), width (w), height (h) → V = l × w × h
Cylinder: diameter (d), height (h) → r = d/2 → V = π × r² × h
Sphere: diameter (d) → r = d/2 → V = (4/3) × π × r³

3. Calculate density using ρ = m ÷ V.

📏 To improve accuracy, measure the same dimension in several places and take the mean. This accounts for any irregularity in manufactured objects.

Top-pan balance — an instrument used to measure mass, giving a reading in grams (g) or kilograms (kg).

Measuring Density of Irregular Solids

If the solid has an irregular shape, you cannot use a formula to find its volume. Instead, you use the displacement method, based on Archimedes' principle: when an object is submerged in water, it displaces a volume of water equal to its own volume.

Method using a displacement can (eureka can):

1. Fill the displacement can with water up to the spout. Place an empty measuring cylinder under the spout to collect displaced water.

2. Gently lower the irregular solid into the can using a thread.

3. Collect the water that overflows into the measuring cylinder. The volume of water collected equals the volume of the object.

4. Weigh the object on a mass balance to find its mass.

5. Calculate density: ρ = m ÷ V.

Alternative method (measuring cylinder only):

1. Add a known volume of water (V₁) to a measuring cylinder.

2. Submerge the solid. Read the new water level (V₂).

3. Volume of solid = V₂ − V₁.

💧 Read the bottom of the meniscus (the curved water surface) when taking volume readings from a measuring cylinder.

Measuring Density of Liquids

To find the density of a liquid, you need to measure both its mass and its volume.

Method:

1. Weigh an empty measuring cylinder on a mass balance. Record this as m₁.

2. Pour a known volume of the liquid into the measuring cylinder. Record the volume (V) by reading the scale at the meniscus.

3. Weigh the measuring cylinder with the liquid. Record this as m₂.

4. Mass of liquid: m = m₂ − m₁.

5. Calculate density: ρ = m ÷ V.

Meniscus — the curved surface of a liquid in a container caused by surface tension. Always read from the bottom of the meniscus for accurate volume measurements.

⚖️ Subtracting the mass of the empty container from the total mass gives you just the mass of the liquid — this is a crucial step that students often forget!

Substance	Density (g/cm³)	Density (kg/m³)
Water	1.00	1000
Ethanol	0.79	790
Mercury	13.6	13600
Cooking oil	0.92	920
Glycerol	1.26	1260

❓ A rectangular block of aluminium has a length of 8.0 cm, width of 4.0 cm and height of 2.5 cm. Its mass is 216 g. Calculate the density of aluminium in g/cm³ and kg/m³.

1 Identify the equation: ρ = m ÷ V

2 Calculate the volume of the cuboid:
V = l × w × h = 8.0 × 4.0 × 2.5 = 80 cm³

3 Substitute values:
ρ = 216 ÷ 80 = 2.7 g/cm³

4 Convert to kg/m³:
ρ = 2.7 × 1000 = 2700 kg/m³

ρ = 2.7 g/cm³ = 2700 kg/m³

❓ A student measures the density of an irregular stone using a displacement can. The stone has a mass of 156 g. When lowered into the can, 60 cm³ of water is displaced. Calculate the density of the stone in kg/m³.

1 Identify known values:
m = 156 g = 0.156 kg
V = 60 cm³ = 60 × 10⁻⁶ m³ = 6.0 × 10⁻⁵ m³

2 Write the equation:
ρ = m ÷ V

3 Substitute values (using g and cm³ first):
ρ = 156 ÷ 60 = 2.6 g/cm³

4 Convert to kg/m³:
ρ = 2.6 × 1000 = 2600 kg/m³

ρ = 2600 kg/m³

❓ A measuring cylinder has a mass of 85 g when empty. When 50 cm³ of cooking oil is added, the total mass is 131 g. Calculate the density of the cooking oil in g/cm³.

1 Find the mass of the liquid:
m = m₂ − m₁ = 131 − 85 = 46 g

2 State the volume:
V = 50 cm³

3 Calculate density:
ρ = m ÷ V = 46 ÷ 50 = 0.92 g/cm³

4 Interpret the result:
0.92 g/cm³ < 1.0 g/cm³ (water), so cooking oil floats on water. ✓

ρ = 0.92 g/cm³ (920 kg/m³)

❓ A gold bar has a density of 19 300 kg/m³ and a mass of 12.44 kg. Calculate its volume in cm³.

1 Rearrange the equation for volume:
ρ = m ÷ V → V = m ÷ ρ

2 Substitute values:
V = 12.44 ÷ 19 300 = 6.446 × 10⁻⁴ m³

3 Convert m³ to cm³:
1 m³ = 1 000 000 cm³
V = 6.446 × 10⁻⁴ × 1 000 000 = 644.6 cm³

4 Round to 3 significant figures:
V = 645 cm³

V = 645 cm³

Q1. Which equation correctly defines density?

✅ Correct! Density = mass ÷ volume. The Greek letter ρ (rho) represents density.

Q2. What are the SI units of density?

✅ Correct! The SI unit is kg/m³. Note: g/cm³ is also commonly used but is not the SI unit.

Q3. A cube of iron has sides of 3.0 cm and a mass of 213.6 g. What is its density in g/cm³?

✅ Correct! V = 3³ = 27 cm³. ρ = 213.6 ÷ 27 = 7.9 g/cm³ (this is iron).

Q4. Why do gases have a much lower density than solids?

✅ Correct! In gases, particles are far apart. Large volume + same/similar mass = very low density.

Q5. A liquid has a density of 800 kg/m³. Calculate its mass if its volume is 0.25 m³. Enter your answer in kg.

Challenge Q1 — Multi-step calculation

A solid aluminium cylinder has a diameter of 6.0 cm and a height of 10.0 cm. The density of aluminium is 2700 kg/m³. Calculate the mass of the cylinder in kg. (Use π = 3.14)

Step 1: Find the radius: r = 6.0 ÷ 2 = 3.0 cm = 0.030 m
Step 2: Find the volume: V = π × r² × h = 3.14 × (0.030)² × 0.10 = 3.14 × 0.0009 × 0.10 = 2.826 × 10⁻⁴ m³
Step 3: Use m = ρ × V = 2700 × 2.826 × 10⁻⁴ = 0.763 kg
Answer: m ≈ 0.76 kg (760 g)

Challenge Q2 — Comparing densities and floating

A student has three liquids in a tall glass: water (ρ = 1000 kg/m³), honey (ρ = 1400 kg/m³), and cooking oil (ρ = 920 kg/m³). A rubber duck has density 800 kg/m³ and a marble has density 2500 kg/m³.

(a) State the order of the liquids from bottom to top.

(b) State where the rubber duck floats and explain why.

(a) Bottom to top: honey (1400) → water (1000) → cooking oil (920). Denser liquids sink below less dense liquids.

(b) The rubber duck (800 kg/m³) floats on the cooking oil (920 kg/m³) because the duck's density is less than all three liquids, so it floats on the top layer (cooking oil). It does not sink into the oil because 800 < 920 kg/m³.

(c) The marble (2500 kg/m³) is denser than all three liquids, so it sinks to the very bottom of the glass, resting below the honey layer.

Challenge Q3 — Evaluating experimental method

A student uses a displacement can to find the density of a porous rock. She gets a density of 1800 kg/m³, which is much lower than the accepted value of 2600 kg/m³. Suggest two reasons why her measured density might be too low and explain each.

Reason 1 — Water absorbed into pores: The rock is porous, so water enters the tiny holes (pores) in the rock rather than being displaced over the spout. This means the volume reading is smaller than the true volume of the rock. A lower V gives a lower density from ρ = m ÷ V.

Reason 2 — Overflowing before rock is added: If the can was too full before the rock was added, some water may have already spilled, so the collected water volume (V₂ − V₁) is less than the actual volume of the rock, underestimating the volume and underestimating the density. (Note: underestimating V gives higher density — so this isn't correct here.)

Better Reason 2 — Incomplete displacement: If the rock was not fully submerged, only part of it displaces water. This gives a smaller volume reading, which would actually overestimate density. (A more likely error causing underestimate: parallax error reading the measuring cylinder, giving a volume that is too large.)

Best answer: (1) Water soaks into pores, increasing apparent volume → density too LOW. (2) Parallax error causing volume to be read too high → density too LOW.

Challenge Q4 — Unit conversion and rearrangement

The density of air at room temperature is 1.2 kg/m³. A bedroom measures 5.0 m × 4.0 m × 2.5 m.

(a) Calculate the volume of the bedroom in m³.

(b) Calculate the mass of air in the bedroom in kg.

(a) V = 5.0 × 4.0 × 2.5 = 50 m³

(b) m = ρ × V = 1.2 × 50 = 60 kg

(c) 60 kg = 60 000 g. This is surprisingly heavy — roughly the mass of an adult person! Even though air feels weightless, there is a significant mass of it in a typical room because the volume is so large. This demonstrates that density, even when small, multiplied by a large volume gives a significant mass.

🔬 AQA Required Practical

Measuring Density Using a Mass Balance and Displacement

Aim

To measure the density of a regular solid, an irregular solid, and a liquid using a mass balance and displacement method.

Equipment

Top-pan balance (sensitivity ±0.1 g)
Ruler (sensitivity ±1 mm)
Measuring cylinder (100 cm³, sensitivity ±1 cm³)
Displacement (eureka) can
Thread or string
Regular solid (e.g., aluminium or copper block)
Irregular solid (e.g., a stone or glass stopper)
Liquid sample (e.g., cooking oil or salt solution)
Water
Beaker, paper towels

Method — Part A: Regular Solid

1Place the regular solid on the top-pan balance. Record its mass (m) in grams.

2Use a ruler to measure the length, width and height (for a cuboid) or diameter and height (for a cylinder). Take each measurement twice and calculate the mean.

3Calculate the volume using the appropriate formula (V = l × w × h for a cuboid).

4Calculate density: ρ = m ÷ V. Record in g/cm³ and convert to kg/m³.

Method — Part B: Irregular Solid

1Weigh the irregular solid on the balance. Record mass (m) in grams.

2Fill the displacement can with water until it just drips from the spout. Wait for dripping to stop. Place an empty measuring cylinder under the spout.

3Tie the solid with thread and gently lower it fully into the can. Collect all the displaced water in the measuring cylinder.

4Read the volume of water collected at the bottom of the meniscus. This equals the volume of the solid (V).

5Calculate ρ = m ÷ V.

Method — Part C: Liquid

1Weigh the empty measuring cylinder. Record mass as m₁.

2Pour approximately 50 cm³ of the liquid into the measuring cylinder. Read the volume at the bottom of the meniscus and record as V.

3Weigh the measuring cylinder plus liquid. Record as m₂.

4Mass of liquid: m = m₂ − m₁. Calculate ρ = m ÷ V.

Safety

⚠️ Wipe up spilled water immediately to prevent slipping.
⚠️ Handle glass measuring cylinders carefully — place on a flat surface and do not overfill.
⚠️ If using chemicals other than water, check CLEAPSS guidance and wear safety goggles.
⚠️ Take care with heavy objects — do not drop them into the displacement can as they may crack it.

Results Table

Object	Mass (g)	Volume (cm³)	Density (g/cm³)	Density (kg/m³)
Regular solid (aluminium block)
Irregular solid (stone)
Liquid (cooking oil)

Analysis Questions

1. A student measures a copper block: mass = 89.6 g, dimensions 2.0 cm × 2.0 cm × 5.0 cm. Calculate its density. The accepted value for copper is 8.9 g/cm³. Calculate the percentage error.

V = 2.0 × 2.0 × 5.0 = 20 cm³ | ρ = 89.6 ÷ 20 = 4.48 g/cm³
Hmm — this seems low. If mass = 89.6 g and V = 20 cm³ → ρ = 4.48 g/cm³. For copper the accepted is 8.9 g/cm³, so % error = |8.9 − 4.48| ÷ 8.9 × 100 = 49.7%.
This large error suggests the mass should probably be ~178 g for copper of this size. Demonstrates importance of accurate measurement!

2. Explain why it is important to wait for all water to stop dripping before lowering an object into the displacement can.

If water is still dripping when the object is lowered, some of the water in the measuring cylinder was not displaced by the object — it dripped in naturally. This would make the volume reading too large, giving a density that is too small (ρ = m ÷ V — if V is too large, ρ is too small).

3. State one way to improve the accuracy of the ruler measurements in Part A.

Take measurements at multiple points along each dimension and calculate the mean. This reduces the effect of any slight non-uniformity in the object. Also: ensure the ruler is placed flush against the object to avoid parallax error.

Learning Objectives