Density

Density is one of the most fundamental properties of matter. It tells you how much mass is packed into a given volume and is essential for understanding why some objects float while others sink, why materials are chosen for specific engineering purposes, and how to identify unknown substances.

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Defining Density

Density is defined as mass per unit volume:

$\rho = \frac{m}{V}$ρ=Vm​

where:

• ρ (Greek letter rho) is density, measured in kg m⁻³ (SI unit) or g cm⁻³
• m is mass in kg (or g)
• V is volume in m³ (or cm³)

To convert between the two common unit systems: 1000 kg m⁻³ = 1 g cm⁻³. Water has a density of approximately 1000 kg m⁻³ or 1.0 g cm⁻³.

Unit conversion trap: Many students lose marks by forgetting to convert cm³ to m³. Remember: 1 cm = 0.01 m, so 1 cm³ = (0.01)³ m³ = 1 × 10⁻⁶ m³. This means 1 m³ = 10⁶ cm³, not 100 cm³.

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Reference Table: Densities of Common Materials

Material	Density / kg m⁻³	Density / g cm⁻³	State at room temp
Air (at sea level)	1.2	0.0012	Gas
Helium	0.164	0.000164	Gas
Cork	120	0.12	Solid
Softwood (pine)	500	0.50	Solid
Ice	917	0.917	Solid
Water	1000	1.00	Liquid
Seawater	1025	1.025	Liquid
Mercury	13 600	13.6	Liquid
Aluminium	2700	2.70	Solid
Glass	2500	2.50	Solid
Titanium	4500	4.50	Solid
Iron/Steel	7800	7.80	Solid
Copper	8900	8.90	Solid
Lead	11 300	11.3	Solid
Gold	19 300	19.3	Solid
Osmium (densest element)	22 590	22.6	Solid

These values explain material selection in engineering. Aircraft use aluminium alloys (low density, reasonable strength) rather than steel. Electrical wiring uses copper (excellent conductor) despite its relatively high density because conductivity matters more than weight in that application.

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Measuring Density of Regular Solids

For objects with regular geometric shapes (cubes, cuboids, cylinders, spheres), you can calculate volume directly from measured dimensions.

Method:

1. Measure the mass using a digital balance (record in kg or g).
2. Measure the relevant dimensions using a ruler, vernier callipers, or micrometer.
3. Calculate volume using the appropriate formula:
   • Cuboid: V = l × w × h
   • Cylinder: V = πr²h
   • Sphere: V = (4/3)πr³
4. Calculate density: ρ = m/V.

Worked example 1: A metal cylinder has mass 0.350 kg, diameter 2.40 cm, and height 5.00 cm.

• Radius = 2.40/2 = 1.20 cm = 0.0120 m
• Volume = π × (0.0120)² × 0.0500 = π × 1.44 × 10⁻⁴ × 0.0500 = 2.26 × 10⁻⁵ m³
• Density = 0.350 / 2.26 × 10⁻⁵ = 15 500 kg m⁻³

This is close to the density of tungsten (19 300 kg m⁻³) or lead (11 300 kg m⁻³), suggesting it could be a lead alloy.

Worked example 2: A solid sphere has a mass of 4.50 kg and a diameter of 12.0 cm. Identify the material.

• Radius = 0.060 m
• Volume = (4/3)π(0.060)³ = (4/3)π × 2.16 × 10⁻⁴ = 9.047 × 10⁻⁴ m³
• Density = 4.50 / 9.047 × 10⁻⁴ = 4973 kg m⁻³ ≈ 5000 kg m⁻³

Checking the table: this sits between titanium (4500) and iron (7800). It could be a titanium alloy or a composite material.

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Measuring Density of Irregular Solids

When the shape is irregular and you cannot calculate volume from dimensions, you use displacement.

Method:

1. Measure the mass of the object on a balance.
2. Fill a measuring cylinder with water and record the initial volume V₁.
3. Carefully lower the object into the water (ensuring it is fully submerged and no air bubbles are trapped).
4. Record the new volume V₂.
5. Volume of object = V₂ − V₁.
6. Calculate density: ρ = m / (V₂ − V₁).

For larger objects, a eureka can (displacement can) can be used. Water overflows into a measuring cylinder through a spout when the object is submerged, and the collected water volume equals the object’s volume.

Key source of error: Surface tension can cause water to cling to the spout, so you should wait until dripping stops completely before reading the volume.

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Uncertainty Analysis in Density Measurements

Understanding uncertainty is essential for practical work and exam questions.

Worked example 3 (with uncertainty): A student measures a cylinder: mass = 45.2 ± 0.1 g, diameter = 1.20 ± 0.02 cm, height = 8.50 ± 0.05 cm.

Step 1 — Calculate the density:

• A = π(0.60)² = 1.131 cm²; V = 1.131 × 8.50 = 9.613 cm³
• ρ = 45.2 / 9.613 = 4.70 g cm⁻³

Step 2 — Percentage uncertainties:

• Mass: (0.1/45.2) × 100 = 0.22%
• Height: (0.05/8.50) × 100 = 0.59%
• Diameter: (0.02/1.20) × 100 = 1.67% → but area uses d², so % uncertainty in area = 2 × 1.67% = 3.33%

Step 3 — Total % uncertainty in density = 0.22% + 0.59% + 3.33% = 4.14%

Step 4 — Absolute uncertainty = 4.14% × 4.70 = ±0.19 g cm⁻³

Final answer: ρ = 4.70 ± 0.19 g cm⁻³ (or 4700 ± 190 kg m⁻³)

Exam tip: The diameter has the largest percentage uncertainty because it is the smallest measurement AND it is squared in the area calculation. This is a very common exam question.

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Measuring Density of Liquids

To find the density of a liquid:

1. Place an empty measuring cylinder on a balance and record its mass m₁.
2. Pour a known volume V of liquid into the measuring cylinder.
3. Record the new mass m₂.
4. Mass of liquid = m₂ − m₁.
5. Density = (m₂ − m₁) / V.

Read the meniscus at eye level to avoid parallax error. For water and most liquids, read from the bottom of the meniscus. For mercury, read from the top.

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Floating and Sinking

An object floats if its average density is less than the density of the fluid it is placed in. An object sinks if its average density is greater than the fluid density.

This explains why:

• A steel ship floats — the hull encloses a large volume of air, making the ship’s average density much less than water.
• Ice floats on water — ice has a density of about 917 kg m⁻³, which is less than water’s 1000 kg m⁻³.
• A helium balloon rises — helium (0.164 kg m⁻³) is less dense than air (1.225 kg m⁻³).

When an object floats, it displaces a weight of fluid equal to its own weight. This is a direct consequence of Archimedes’ principle, which you will study in the next lesson.

Worked example 4: An iceberg of volume 500 m³ and density 917 kg m⁻³ floats in seawater (ρ = 1025 kg m⁻³). What volume is above the surface?

• Fraction submerged = ρ_ice / ρ_seawater = 917 / 1025 = 0.8946
• Volume submerged = 0.8946 × 500 = 447.3 m³
• Volume above surface = 500 − 447.3 = 52.7 m³ (about 10.5%)

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Density and the Particle Model

At the microscopic level, density depends on:

• The mass of individual atoms or molecules — heavier atoms mean higher density.
• How closely packed the particles are — tightly packed structures (like metals) have higher density.
• The arrangement of particles — diamond (3500 kg m⁻³) and graphite (2200 kg m⁻³) are both pure carbon, but their different crystal structures give different densities.

In solids, particles are closely packed in fixed positions, giving high density. In gases, particles are widely spaced, giving very low density. Liquids fall in between, with densities typically similar to (but slightly less than) the corresponding solid — water being a notable exception where the solid (ice) is less dense than the liquid.

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Alloy Density Calculation

When two materials are mixed to form an alloy, the resulting density can be estimated using volume fractions:

$\rho_{\text{alloy}} = f_1 \rho_1 + f_2 \rho_2$ρalloy​=f1​ρ1​+f2​ρ2​

where f₁ and f₂ are the volume fractions of each component.

Worked example 5: An alloy is 70% aluminium (ρ = 2700 kg m⁻³) and 30% copper (ρ = 8900 kg m⁻³) by volume.

ρ_alloy = 0.70 × 2700 + 0.30 × 8900 = 1890 + 2670 = 4560 kg m⁻³

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Common Exam Mistakes

1. Forgetting volume unit conversion: 1 cm³ = 1 × 10⁻⁶ m³, not 1 × 10⁻² m³.
2. Using diameter instead of radius when calculating volumes of cylinders or spheres.
3. Confusing mass and weight in density calculations — density uses mass (kg), not weight (N).
4. Ignoring air bubbles when using displacement — trapped air reduces the apparent volume of the object, giving a density that is too high.
5. Assuming all objects of a material float or sink — it is the average density of the whole object (including air spaces) that matters, not just the density of the material itself.