Density

This lesson covers the concept of density, its definition, the equation linking density to mass and volume, and the practical methods used to measure the density of regular and irregular solids. This is a key topic in the Edexcel GCSE Combined Science specification (1SC0) Particle Model unit.

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What Is Density?

Density is a measure of how much mass is contained in a given volume. It tells us how tightly packed the particles are in a substance.

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

Symbol	Quantity	Unit
(\rho) (rho)	Density	kilograms per cubic metre (kg/m³) or grams per cubic centimetre (g/cm³)
(m)	Mass	kilograms (kg) or grams (g)
(V)	Volume	cubic metres (m³) or cubic centimetres (cm³)

Exam Tip: The Greek letter (\rho) (rho) is used for density — do not confuse it with the letter p. Make sure your handwriting clearly distinguishes the two.

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Rearranging the Density Equation

You must be able to rearrange the equation for any of the three quantities:

To find	Rearrangement
Density	(\rho = \frac{m}{V})
Mass	(m = \rho \times V)
Volume	(V = \frac{m}{\rho})

The Formula Triangle

A useful memory aid is the formula triangle:

graph TD
    A["m"] --- B["ρ × V"]

Cover the quantity you want to find:

• Cover m → (\rho \times V)
• Cover (\rho) → (m \div V)
• Cover V → (m \div \rho)

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Typical Densities

Material	Density (kg/m³)	Density (g/cm³)
Air	1.2	0.0012
Water	1000	1.0
Ice	917	0.917
Aluminium	2700	2.7
Iron/Steel	7800	7.8
Gold	19 300	19.3

Exam Tip: You should know that the density of water is 1000 kg/m³ (or 1.0 g/cm³). This is a very commonly used value in calculations.

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Converting Units

To convert between the two common density units:

• g/cm³ → kg/m³: multiply by 1000
• kg/m³ → g/cm³: divide by 1000

Worked Example 1

A block of aluminium has a mass of 5.4 kg and a volume of 0.002 m³. Calculate its density.

$$\rho = \frac{m}{V} = \frac{5.4}{0.002} = 2700 \text{ kg/m³}$$

Worked Example 2

A gold ring has a mass of 19.3 g and a volume of 1.0 cm³. Calculate its density.

$$\rho = \frac{m}{V} = \frac{19.3}{1.0} = 19.3 \text{ g/cm³}$$

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Measuring Density of a Regular Solid

For a regularly shaped solid (cuboid, cylinder, sphere):

1. Measure the mass using a balance (in grams).
2. Measure the dimensions using a ruler or vernier calliper.
3. Calculate the volume using the appropriate formula:
   • Cuboid: (V = l \times w \times h)
   • Cylinder: (V = \pi r^2 h)
   • Sphere: (V = \frac{4}{3} \pi r^3)
4. Calculate density using (\rho = m / V).

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Measuring Density of an Irregular Solid

For an irregularly shaped solid (e.g. a stone, a key), the volume cannot be calculated from dimensions. Instead, use the displacement method:

1. Measure the mass of the object using a balance.
2. Fill a measuring cylinder with water and record the initial volume (V_1).
3. Carefully lower the object into the water.
4. Record the new volume (V_2).
5. Volume of object = (V_2 - V_1).
6. Calculate density using (\rho = m / V).

Alternatively, use a eureka can (displacement can):

1. Fill the can until water reaches the spout.
2. Place a measuring cylinder under the spout.
3. Lower the object into the can.
4. The displaced water collects in the measuring cylinder — this volume equals the volume of the object.

graph LR
    A["Object placed in eureka can"] --> B["Water displaced through spout"]
    B --> C["Volume measured in cylinder"]
    C --> D["ρ = m / V"]

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Floating and Sinking

Whether an object floats or sinks depends on its density compared with the density of the liquid:

Condition	Result
Object density < liquid density	Object floats
Object density > liquid density	Object sinks
Object density = liquid density	Object is neutrally buoyant (stays at any depth)

Why Does Ice Float?

• Ice has a density of about 917 kg/m³, which is less than water at 1000 kg/m³.
• The water molecules in ice form a rigid crystalline structure with gaps, making ice less dense than liquid water.
• This is unusual — most substances are denser as a solid than as a liquid.

Exam Tip: A common exam question asks you to explain why ice floats on water. Always state that ice is less dense than water, and link this to the particle arrangement.

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Worked Example 3

A student measures the mass of a stone as 150 g. She places it in a measuring cylinder containing 40 cm³ of water. The water level rises to 98 cm³. Calculate the density of the stone.

Step 1 — Find the volume:

$$V = V_2 - V_1 = 98 - 40 = 58 \text{ cm}^3$$

Step 2 — Calculate density:

$$\rho = \frac{m}{V} = \frac{150}{58} = 2.59 \text{ g/cm}^3 \text{ (to 3 s.f.)}$$

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Common Misconceptions

Misconception	Correction
Heavy objects always sink	It depends on density, not just mass — a large ship floats because it has a low average density
Density and mass are the same thing	Density is mass per unit volume — two objects can have the same mass but very different densities
All metals sink in water	Some metals (e.g. lithium, sodium, potassium) are less dense than water and float

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Summary

• Density = mass / volume: (\rho = m / V).
• Units: kg/m³ or g/cm³.
• Regular solids: measure dimensions and calculate volume.
• Irregular solids: use the displacement method (measuring cylinder or eureka can).
• An object floats if its density is less than the liquid it is placed in; it sinks if its density is greater.
• Ice floats because it is less dense than liquid water due to its open crystalline structure.