Density and the States of Matter - Physics for Pilot Assessments

Density tells you how much matter is packed into a given amount of space, and it is why a lump of lead the size of a bread roll is a struggle to lift while the bread roll is not. By the end of this lesson you should be able to work out a density from a mass and a volume, run that calculation backwards to find either of them, say what physically differs between a solid, a liquid and a gas, and use a density comparison to decide which of two materials will float on the other.

Mass and volume, the two ingredients

Density is built out of two quantities you already have. Mass is the amount of matter in an object, measured in kilograms, and it does not change when you move the object somewhere else — it is not the same thing as weight, which is the pull of gravity on that mass and is measured in newtons. Volume is the amount of space the object occupies, measured in cubic metres: a cubic metre is the volume of a cube one metre along each edge. Smaller volumes are usually quoted in cubic centimetres, the volume of a cube one centimetre along each edge, or in litres — a litre is a thousand cubic centimetres, and a thousand litres make a cubic metre.

Neither quantity on its own tells you anything about how tightly packed the material is, and that is the whole point of the lesson. Two kilograms of anything is two kilograms; what makes lead lead and cork cork is how much space those two kilograms take up. So the quantity you want is a mass divided by the volume it occupies — how many kilograms per cubic metre. That division is density, written with the Greek letter rho, which looks like a p with a descending tail and is pronounced "roe".

Density

\rho = \frac{m}{V}

$\rho$: density, in kilograms per cubic metre (kg/m³)
$m$: mass, in kilograms (kg)
$V$: volume, in cubic metres (m³)

Read it as a rate rather than as an amount: it says nothing about how much material you have, only about what that material is like. A teaspoon of mercury and a bathful of mercury have wildly different masses and wildly different volumes, but exactly the same density of 13 600 kilograms per cubic metre — which is why density is a property of a substance and mass is not.

Rearranging it, both ways

Two of the three quantities in that relationship will be given and the third will be what the item wants, so it is worth deriving the other two arrangements once rather than memorising them separately. Start from $\text{density} = \frac{\text{mass}}{\text{volume}}$ and multiply both sides by the volume: on the right the volume in the numerator and the volume in the denominator cancel, leaving mass, so mass = density × volume. Now take that version and divide both sides by the density: on the right the densities cancel and leave the volume, so $\text{volume} = \frac{\text{mass}}{\text{density}}$ . Three arrangements, one fact, and the same procedure works on every formula in this module.

Numbers worth carrying

Water is the reference everything else gets compared against, and in SI units its density is 1000 kilograms per cubic metre. That is the same statement as "one gram per cubic centimetre", and it is worth seeing why, because the two forms appear interchangeably: a cubic metre contains a million cubic centimetres, since a metre is a hundred centimetres and a hundred multiplied by itself three times is a million; a thousand kilograms is a million grams; so a million grams spread through a million cubic centimetres is one gram in each. Air at sea level is about 1.2 kilograms per cubic metre, which makes it roughly eight hundred times less dense than water.

A few other values are worth having in your head, because items lean on comparisons rather than on precise figures. Ice is about 920, so it is slightly less dense than the water it froze from — which is why ice floats, and why water is a genuinely unusual substance in expanding when it freezes rather than contracting. Petrol and most oils sit between about 700 and 900; dry wood between about 400 and 800 depending on the species; aluminium is 2700; steel about 7800; lead 11 300; mercury 13 600. Notice that every solid metal in that list is denser than water by a factor of several, and that no ordinary solid comes anywhere near being as light as air.

Solids, liquids and gases

Everything in that list is made of particles — individual atoms, or groups of atoms bound together into molecules — and the difference between a solid, a liquid and a gas is not a difference in the particles themselves but in how they are arranged and how freely they move. In a solid the particles sit in fixed positions, usually in a regular repeating pattern, and although they vibrate about those positions they do not swap places. That is what lets a solid hold its own shape without a container, and what makes it very hard to compress: there is almost no spare space to squeeze out.

In a liquid the particles are just as close together — a liquid is typically only a few per cent less dense than the same substance frozen solid — but the regular pattern is gone, and they can slide past one another. That mobility is why a liquid flows, why it takes the shape of whatever you pour it into and settles to a flat surface, and why it nonetheless resists being compressed almost as stubbornly as a solid does. A gas is the outlier. Its particles are far apart compared with their own size, so the container is mostly empty space, and they move fast and in all directions, colliding with each other and with the walls. That is why a gas expands to fill whatever it is put in rather than settling with a surface, and why it can be squeezed into a much smaller volume: there is plenty of empty space to remove.

The particles are the same in all three — only the spacing and the ordering change. A solid keeps its own shape, a liquid takes the container's shape but keeps its own volume, and a gas fills whatever it is put in.

Warm a solid and its particles vibrate harder, until the vibration is violent enough to break the regular pattern apart and they begin to slide over one another: it melts. Warm the liquid further and they break away from each other altogether and spread out: it boils. Cooling reverses both changes. Two things are worth noticing. The first is that the particles themselves are unchanged throughout — melting ice does not alter a single water molecule, only the arrangement of them. The second is that the mass is unchanged too, so if the volume changes then the density must change with it, which is exactly what happens: almost every substance is denser as a solid than as a liquid, and a thousand times or so denser as a liquid than as a gas, because it is the spacing and not the particles that changed.

Why one material floats on another

Comparing two densities tells you which one ends up on top. Pour oil, at perhaps 900 kilograms per cubic metre, onto water at 1000, and the oil sits above the water; drop a steel bolt at 7800 into either and it goes straight to the bottom. The rule is that the less dense material floats on the denser one, and it holds for a solid in a liquid as well as for two liquids that do not mix. Note carefully what the rule does not mention: how big the pieces are, and how heavy either of them is. A single drop of oil floats on a swimming pool of water, and a swimming pool of oil would float on an ocean of water, because the comparison is between properties of the two materials and not between amounts of them.

What actually pushes the lighter material upwards is a pressure difference — a fluid presses harder on the bottom of a submerged object than on its top — and that mechanism needs pressure itself dealt with properly first, so it comes later in this module. The comparison above is not a rough rule standing in for the real explanation; it follows from it exactly. For now it is enough that a density comparison settles the question and a mass comparison does not.

Putting it to work

Example 2 — running it backwards

An aluminium casting has a volume of 0.02 cubic metres, and aluminium has a density of 2700 kilograms per cubic metre. What is the casting's mass? And separately: what volume would 5400 kilograms of aluminium occupy?

For the mass, use the arrangement mass = density × volume: $m = 2700 \times 0.02$
Multiplying by 0.02 is multiplying by 2 and then dividing by 100: $2700 \times 2 = 5400,\ \text{and}\ \frac{5400}{100} = 54$ So the mass is 54 kilograms.
For the volume, use the other arrangement, $\text{volume} = \frac{\text{mass}}{\text{density}}$ : $V = \frac{5400}{2700} = 2\ \text{m}^3$
Check the two answers against each other. 54 kilograms filled 0.02 cubic metres, and 5400 kilograms is a hundred times more mass, so it should fill a hundred times the volume — and 0.02 × 100 is indeed 2.

Example 3 — heavier is not denser

Block A has a mass of 12 kilograms and a volume of 0.004 cubic metres. Block B has a mass of 30 kilograms and a volume of 0.015 cubic metres. Which is heavier, and which is denser?

B is heavier: 30 kilograms against 12. That half of the question needs no calculation at all.
For A's density: $\rho = \frac{12}{0.004}$ Multiply both numbers by 1000 to clear the decimal, giving $\frac{12\,000}{4} = 3000$ kilograms per cubic metre.
For B's density: $\rho = \frac{30}{0.015}$ Multiply both by 1000 again, giving $\frac{30\,000}{15} = 2000$ kilograms per cubic metre.
So A is the denser material even though B is more than twice as heavy, because A's mass is packed into barely a quarter of the space. Both are denser than water, so both sink in it; both are less dense than mercury at 13 600, so both would float on that.

Example 4 — from litres to kilograms

A drum holds 200 litres of oil with a density of 900 kilograms per cubic metre. What is the mass of the oil?

The density is in kilograms per cubic metre, so the volume must be in cubic metres before anything is multiplied. A thousand litres make a cubic metre, so 200 litres is $\frac{200}{1000} = 0.2$ cubic metres.
Now use mass = density × volume: $m = 900 \times 0.2 = 180\ \text{kg}$
Sense-check it against water. Water at 1000 kilograms per cubic metre works out to one kilogram per litre, so 200 litres of water would be 200 kilograms; this oil is a tenth less dense than water, and 180 is a tenth less than 200.
This litres-to-kilograms conversion is exactly the one fuel loading turns on: tanks are filled by the litre, the flight is planned by the kilogram, and the fuel's density is the bridge between the two.

Why it matters in aviation

Air density is the quantity that actually governs how an aircraft performs. Lift, drag and the power an engine can produce all depend on the mass of air being handled each second, not on pressure or temperature taken by themselves, so the figure that matters is kilograms per cubic metre. Warm the air, move to a higher-elevation aerodrome, or add water vapour to it, and in every case the same volume of air contains less mass: the wing generates less lift at a given speed, the engine has less air to work with, the takeoff run lengthens and the climb gradient shallows. That combined effect is what density altitude expresses — it restates the actual conditions as the height in a standard atmosphere at which the air would be equally thin, so that performance figures can be read against a single number.

Fuel density matters for a different reason. Tanks are filled by volume, in litres, while the aircraft is loaded and the flight planned by mass, in kilograms, and the conversion between the two is the density of the fuel — which changes with its temperature. Warmer fuel is less dense, so the same uplift in litres delivers fewer kilograms.

What the test asks

The recurring trap is the word heavier. An item will give you two objects, make one of them plainly more massive, and offer that one as the denser — with no mention of volume anywhere in the option. Density is a ratio, so it cannot be settled by one of its two ingredients: a tonne of expanded polystyrene is far more massive than a marble and thousands of times less dense. Expect also to be asked which substances in a list will float on water, which is a comparison against 1000 kilograms per cubic metre and nothing else, and expect one item on the particle arrangements, where the reliable distractor claims that a liquid's particles are much further apart than a solid's. They are not — that is the gas.