Around us there are various things including air, water, soil, stone, light and wood. Do all of them occupy space? Do all of them have mass? Air can be collected in a balloon or a football. Could light or sound be collected in the same way?
Things that have mass and volume are matter. Shadow, heat and light are not matter, because they have neither volume nor mass.
Some matter occupies little space and yet has a great deal of mass, while some matter occupies a great deal of space and yet has little mass. The mass of a metal such as iron or gold is far greater than that of paper or plastic of the very same size. In things that occupy little space and yet have a lot of mass, the particles are packed closely together.
Density of matter
Between a piece of iron and a piece of wood of equal volume, which will have the greater mass, and why? And between one litre of water and one litre of kerosene, which will have the greater mass, and what might be the reason for it?
At ordinary temperature some matter is found in the solid state, some in the liquid state and some in the gas state. In liquid matter the molecules are in a looser arrangement than in solid matter, and in gaseous matter they are looser still than in liquid matter. Objects of equal volume may not have equal mass. The reason for this is that their compactness is different from one another. The compactness of matter is what we call the density of that matter.
The exercise at the end of this unit gives you three boxes of particles labelled substance A, substance B and substance C, and asks you to name the state of each one from how closely the particles are packed. You can answer it before you have read another word, because the whole rule is in the paragraph above. Count the empty space. In substance A the particles are few and far apart with a great deal of space between them, so it is a gas. In substance B they are close to one another but still able to slide about, with small gaps here and there, so it is a liquid. In substance C they are packed tight in neat rows with almost no space at all, so it is a solid. Notice that the particles themselves are drawn exactly the same size in all three boxes, and that is the honest part of the picture: changing state never changes the particles, it only changes how far apart they sit.
The book explains density by how tightly the particles are packed, and that is half of the answer. There is a second half worth knowing, because together they explain everything. Imagine two sacks of exactly the same size. Fill one with tennis balls and the other with the same number of iron balls of the same size. The packing is identical, yet the second sack is far heavier, and the only difference is that each iron ball weighs more than each tennis ball. So a substance can be dense for two separate reasons: its particles may be heavy, or its particles may be closely packed, and usually both are at work at once. Iron is dense because iron atoms are heavy and they sit close together. Air has very little density because its particles are light and there are enormous gaps between them. That is why the same substance changes density when it changes state, and why a gas is always far less dense than the liquid or solid it came from: the particles have not changed at all, only the spacing has.
The formula
The mass of matter contained in unit volume is called the density of that matter. The relationship between the density, the mass and the volume of an object can be shown by the formula given below.
| \( D = \dfrac{M}{V} \) |
| If you measure | The unit of density is | Which system |
|---|---|---|
| Mass in kilogram (kg) and volume in cubic metre (m³) | kg/m³ | The SI unit of density |
| Mass in gram (g) and volume in cubic centimetre (cm³) | g/cm³ | The CGS unit of density |
Look at any density table and you will notice something suspicious. Ice is 920 kg/m³ and 0.92 g/cm³. Iron is 7800 kg/m³ and 7.8 g/cm³. The number in kg/m³ is always exactly a thousand times the number in g/cm³, and that is not a coincidence. Work it out. One kilogram is a thousand grams, so switching from kilograms to grams multiplies the top of the fraction by a thousand. One cubic metre is a hundred centimetres by a hundred by a hundred, which is a million cubic centimetres, so switching from cubic metres to cubic centimetres divides the bottom by a million. A thousand on top and a million on the bottom leaves the number a thousand times smaller. So to change kg/m³ into g/cm³, divide by a thousand, and to go back, multiply by a thousand. Knowing this saves a great deal of trouble, because a question may give you a density in one unit and a mass in the other.
Density is a property of the substance, not of the lump
Here is the idea that makes density genuinely useful, and it follows from the fact that density is a ratio. Take a bar of gold and cut it in half. The mass halves, and the volume halves too, so the density, which is one divided by the other, does not change at all. Cut it again and again and the answer is still the same. A gold ring, a gold chain and a gold bar all have exactly the same density, because density does not care how much of the substance you happen to have.
| Quantity | What it tells you | Does it change if you take a smaller piece? |
|---|---|---|
| Mass | How much matter this particular object contains | Yes. Half the object has half the mass |
| Volume | How much space this particular object takes up | Yes. Half the object takes half the space |
| Density | How much matter is packed into every unit of space | No. It is the same for every piece of that substance |
That last row is why density is worth learning. Since it stays the same for every piece, density behaves like a fingerprint for a substance. Measure the mass and the volume of an unknown lump, work out the density, and compare it with a table, and you have a good idea of what the lump is made of. This is how somebody can check whether a piece of jewellery is really gold without cutting it open, and it is how the same trick catches a fake.
Worked example 1: finding a density
If the mass of a stone whose volume is 2 cubic metres is 5000 kg, what will the density of the stone be?
| Step | Working |
|---|---|
| Given | V = 2 m³, m = 5000 kg, D = ? |
| Formula | \( D = \dfrac{m}{V} \) |
| Substitute | \( D = \dfrac{5000}{2} \) |
| Answer | D = 2500 kg/m³. The density of the stone is 2500 kg/m³ |
Worked example 2: the same formula, rearranged
How many kilograms of water would be needed to fill a drum of volume 6 cubic metres with water whose density is 1000 kg per cubic metre?
| Step | Working |
|---|---|
| Given | V = 6 m³, D = 1000 kg/m³, m = ? |
| Formula | \( D = \dfrac{m}{V} \) |
| Substitute | \( 1000 = \dfrac{m}{6} \) |
| Rearrange | \( m = 1000 \times 6 \) |
| Answer | m = 6000 kg. The mass of the water is 6000 kg |
Notice that the two worked examples used the same formula and looked completely different, which is the whole trick with this topic. There is only one relationship between the three quantities, and a question can hide any one of the three and ask you to find it. If the density is missing, divide the mass by the volume. If the mass is missing, multiply the density by the volume. And if the volume is missing, divide the mass by the density. Rather than memorising three formulas, write down \( D = \dfrac{m}{V} \) every single time, put in the two numbers you have been given, and rearrange. Then check the units before you write the answer down, because they will catch most mistakes on their own: an answer in kilograms cannot be a density, and an answer in kg/m³ cannot be a mass.
Worked example 3: an aluminium block
What will the mass of a piece of aluminium of volume 2 cubic metres be, if the density of aluminium is 2700 kg per cubic metre?
| Step | Working |
|---|---|
| Given | V = 2 m³, D = 2700 kg/m³, m = ? |
| Rearrange the formula | \( m = D \times V \) |
| Substitute | \( m = 2700 \times 2 = 5400 \) |
| Answer | m = 5400 kg |
Before you accept an answer like that, hold it against something you know. A cubic metre of water has a mass of 1000 kg, and aluminium is a metal, so it ought to come out heavier than water for the same volume. It does: 2700 kg for every cubic metre against 1000 kg for water. The answer is sensible. If your arithmetic had given something like 54 kg you should have been suspicious at once, because two cubic metres of any metal is far too big a lump to weigh less than a person.
The full lesson is waiting for you
Create a free account to unlock activities, practice tests, presentations and videos - and to track your progress and confidence score for every subject.
The full lesson is waiting for you
Create a free account to unlock activities, practice tests, presentations and videos - and to track your progress and confidence score for every subject.