Matter and the particle model · Quantitative
Density
A kilogram of lead and a kilogram of feathers weigh the same. Everybody knows that, and everybody still feels it is wrong. What they are reaching for is a different quantity altogether.
Start here
A block of iron and a block of oak balance exactly.
On the pan balance in front of you: a small lump of iron on the left, a large block of oak on the right. The beam is dead level. The oak is about twelve times the size.
Which one is heavier?
They weigh the same — that is what a level balance means. What differs is size: it takes about twelve times as much oak to match one lump of iron. Heavy is a property of the object; density is a property of the material, and it is the second one that lets you compare iron with oak at all.
Density is how much mass is packed into a given amount of space. It is worked out by dividing the mass of a sample by the volume that sample takes up, and it is measured in grams per cubic centimetre (g/cm³) or in kilograms per cubic metre (kg/m³). Those are the only two pairings you will meet: grams go with cubic centimetres and kilograms go with cubic metres. There is no such unit as kg/cm³ or g/m³, so if a question gives you a mass in one family and a volume in the other, converting one of them is the first line of the working.
The word people usually reach for is heavy, and it is the wrong word, because heavy is about a particular object. A paving slab is heavy and a chip of the same stone is not, and they are the same material at the same density. Density is a property of the material: every cubic centimetre of iron has a mass of 7.87 g whether it came from a girder or from a nail.
That is why cutting something in half does not change its density. You halve the mass, and you halve the volume at the same time, and the ratio of one to the other is untouched. It is also why density identifies things — measure a mass, measure a volume, divide, and the number that comes out tells you what the material is.
At the bench · six materials, one balance, one measuring cylinder
Same volume, very different masses.
Change a control to begin
Pick a material, choose how big a block of it you want, and read the mass off the balance. The bars are the density league table — and they do not move when you change the volume.
Commit first. You take a 100 cm³ block of aluminium and cut it exactly in half. What happens to its density?
The material on the balance
—
The cylinder says
—
—
The balance says
—
—
So the density is
—
—
Dropped in water it
—
—
Writing it down · the shape of this relationship
Density = mass ÷ volume
The triangle
Cover the one you want
mass = density × volume
density = mass ÷ volume
volume = mass ÷ density
Two things side by side means multiply. One thing over another means divide.
m · mass of the sample · g or kg
V · volume the sample takes up · cm³ or m³
d · density of the material · g/cm³ or kg/m³
g with cm³ gives g/cm³ · kg with m³ gives kg/m³
Worked example · one step at a time
A block has a mass of 54 g and a volume of 20 cm³. What is its density?
Step 0 of 5
Convert
54 g stays 54 g · 20 cm³ stays 20 cm³
The answer is wanted in g/cm³, and the mass is already in grams and the volume already in cubic centimetres, so there is nothing to convert.
Formula
density = mass ÷ volume
Cover d on the triangle: m sits over V, so you divide.
Insert
density = 54 g ÷ 20 cm³
Mass on top, because density is how much mass sits in each cubic centimetre.
Fine-tune
54 ÷ 20 = 2.7
Grams divided by cubic centimetres leaves grams per cubic centimetre.
Answer
density = 2.70 g/cm³
Which identifies it: that is aluminium.
Worked example · one step at a time
A block has a mass of 1.2 kg and a volume of 150 cm³. What is its density in g/cm³?
Step 0 of 5
Convert
1.2 kg × 1000 = 1200 g
A mass in kilograms with a volume in cubic centimetres is a mismatched pair — kg/cm³ is not a unit. Bring the mass into grams and it pairs with cm³.
Formula
density = mass ÷ volume
Cover d on the triangle: m sits over V, so you divide.
Insert
density = 1200 g ÷ 150 cm³
The converted mass goes in. The 1.2 never does.
Fine-tune
1200 ÷ 150 = 8
Grams divided by cubic centimetres leaves grams per cubic centimetre.
Answer
density = 8.00 g/cm³
Insert 1.2 instead of 1200 and the answer comes out 0.008 g/cm³ — light enough to float on a puddle.
Your turn · the same five steps
Your block: 270.0 g of aluminium, taking up 100 cm³.
Write each line out yourself — starting by deciding whether anything needs converting. Then check your working and tick the lines you had.
The five lines · tick what you had
Convert
270.0 g stays 270.0 g · 100 cm³ stays 100 cm³
The balance reads grams and the cylinder reads cubic centimetres, which is what g/cm³ needs, so there is nothing to convert.
Formula
density = mass ÷ volume
Cover d on the triangle: m sits over V, so you divide.
Insert
density = 270.0 g ÷ 100 cm³
Mass on top. Both readings come from the same block.
Fine-tune
270.0 ÷ 100 = 2.70
Grams divided by cubic centimetres leaves grams per cubic centimetre.
Answer
density = 2.70 g/cm³
Change the volume slider and this line does not change. That is the whole point of a density.
The five lines give 2.70 g/cm³ — the bar for aluminium on the chart above.
A steel bolt has a mass of 0.039 kg and a volume of 5.0 cm³. What is its density in g/cm³?
Write each line out yourself — starting by deciding whether anything needs converting. Then check your working and tick the lines you had.
The five lines · tick what you had
Convert
0.039 kg × 1000 = 39 g
Kilograms with cubic centimetres is a mismatched pair. Grams go with cm³, kilograms go with m³ — pick one family and stay in it.
Formula
density = mass ÷ volume
Cover d on the triangle: m sits over V, so you divide.
Insert
density = 39 g ÷ 5.0 cm³
The converted mass goes in. The 0.039 never does.
Fine-tune
39 ÷ 5.0 = 7.8
Grams divided by cubic centimetres leaves grams per cubic centimetre.
Answer
density = 7.80 g/cm³
Insert 0.039 instead of 39 and the bolt comes out at 0.008 g/cm³ — a hundredth the density of water, which would make a steel bolt float.
The five lines give 7.80 g/cm³ — steel, near the iron bar on the chart. The whole question turned on the first one.
Key fact
Density is mass ÷ volume. The units come in matched pairs — grams with cubic centimetres gives g/cm³, kilograms with cubic metres gives kg/m³ — and there is no such unit as kg/cm³ or g/m³, so a mismatched pair must be converted before you divide. Density belongs to the material, not the object: cut a block in half and both halves have the same density. Anything below 1.00 g/cm³, or 1000 kg/m³, floats on water.
Think again
“Heavy things are dense and light things are not.”
Heavy is about the object; dense is about the material. A polystyrene packing block the size of a fridge is awkward to carry and has a density of about 0.02 g/cm³ — a fiftieth that of water. A gold ring is light enough to forget you are wearing and has a density of 19.30. The two words answer different questions, and only one of them survives cutting the object up.
“Things float because they are light and sink because they are heavy.”
An oil tanker floats and a 5p coin sinks. What decides it is not the weight but how that weight compares with the weight of the water pushed out of the way — a comparison of densities, not of masses. Below 1.00 g/cm³ a thing floats on water however large it is; above, it sinks however small. It is also why the same coin sinks in water and floats in mercury, which has a density of 13.5.
Mastery ladder
Not started yet.
Rungs 3 and 4 you mark yourself.
Rung 1 · Calculate
A stone has a mass of 240 g and displaces 80 cm³ of water. What is its density?
Rung 2 · The one that catches people
A student says a 2 kg block of oak must be denser than a 50 g lump of gold, because the oak is forty times heavier. What is right?
Rung 3 · Explain
Describe how you would find the density of a small irregular stone using a balance, a measuring cylinder and water, and say why the cylinder is needed at all.
Rung 4 · Take it somewhere new
A steel ship floats even though steel has a density of about 7.9 g/cm³. A solid steel bar of the same mass sinks. Explain the difference using density, and predict what happens if the hull fills with water.
Key note
Density is mass divided by volume, measured in g/cm³ or in kg/m³ — 1.00 g/cm³ is the same density as 1000 kg/m³. The two units in the pair must match: grams go with cubic centimetres and kilograms with cubic metres, and if the question hands you one of each, converting is the first thing you do. Density is a property of the material rather than of the object, so a block and a chip of the same substance share it, and cutting something in half changes neither. Finding a density means two measurements and one division: a mass from a balance, and a volume from a cylinder or from displacement. Below 1.00 g/cm³ a material floats on water; above it, it sinks.
Going further
The numbers vary so widely because they are set by two things at once: how heavy the individual atoms are, and how tightly the structure packs them. Osmium, the densest element at 22.6 g/cm³, wins on both counts — heavy atoms in a tight hexagonal arrangement. Lithium, at 0.53, has light atoms in a loose one, and would float on water if it did not react with it violently first.
Archimedes is supposed to have solved a density problem in the bath and gone straight out into the streets of Syracuse to announce it. The story is almost certainly invented; the physics underneath it is not. A crown of pure gold and a crown of gold mixed with silver have the same mass and different volumes, so measuring the volume by displacement settles the question without damaging the crown. Every hydrometer in a brewery and every battery tester in a garage is still doing the same trick.
Before this lesson
- Nothing — this is where the unit starts.
Connects to
- Why ice floats
- Temperature, particle motion and internal energy
- Solids, liquids and gases
Where the spacing of the particles in each state comes from — this lesson measures what that spacing does to a density.
At GCSE this becomes
- Density in kg/m³, the required practical for regular and irregular solids and for liquids, and density as the link between particle spacing and state of matter.
Where to next
- Next: Brownian motion
- Previous: How a motor works
Magnetism and electromagnetism
Ask Mr Badmus AI
Got a mass and a volume and want the density?
Eye protection on. Wipe up spilled water straight away — a wet floor is the real hazard here. Measuring cylinders tip easily, so keep them back from the edge of the bench.
The bench is a teaching model. Densities are quoted at room temperature and ordinary pressure to two decimal places: oak 0.65, ice 0.92, water 1.00, aluminium 2.70, iron 7.87 and gold 19.30 g/cm³. Real timber varies widely with species and moisture content, and the figure for oak is a typical seasoned value rather than a measurement of a particular piece. Masses shown are calculated from the quoted density and the chosen volume, and are rounded to one decimal place.
Lesson content © MrBadmusAI.