What is a mole — Avogadro's number — Chemistry

Before this

This leads to

Too many atoms to count

Pick up a single grain of sand, about half a millimetre across. Sand is mostly quartz, silicon dioxide. A grain that size has a mass of roughly 0.17 mg, and it contains about $5 \times 1 0^{18}$ atoms. Suppose you counted them, one atom every second, without stopping. You would need about 165 billion years. That is around twelve times the age of the Universe.

So chemists cannot count atoms. But chemistry is about counting atoms. The equation $2 H_{2} + O_{2} \to 2 H_{2} O$ says: two hydrogen molecules react with one oxygen molecule. It is a statement about numbers of particles. In a laboratory, though, nobody can pick out two molecules. What a chemist can do is weigh grams on a balance.

This is the gap that this lecture closes. On one side are particles, which we cannot count. On the other side are grams, which we can measure. The bridge between them is the mole. Earlier lectures already used it, in phrases such as "the energy to remove one mole of electrons" or "786 kJ for every mole of sodium chloride". Here is what it means.

Counting by weighing

You already know a way to count things without counting them. A bank does not count a large bag of identical coins one by one. It weighs the bag. If one coin has a mass of 5.00 g and the bag of coins has a mass of 500 g, the bag holds $500 \div 5.00 = 100$ coins. Weighing replaces counting, as long as you know the mass of one item.

Now suppose the bank receives two bags: one of small coins with a mass of 5 g each, and one of large coins with a mass of 10 g each. How could it make two piles with the same number of coins, using only the balance? Weigh out masses in the same ratio as the coins: 50 g of small coins and 100 g of large coins. Both piles hold 10 coins. You do not even need to know the number. You only need the ratio of the masses of one coin of each kind.

That is exactly the situation in chemistry. We do know the ratios of atomic masses, very precisely. In the lecture on protons, neutrons and electrons, you met the relative atomic mass, $A_{r}$ : the average mass of an element's atoms, compared with one-twelfth of the mass of a carbon-12 atom. Carbon has $A_{r} = 12$ and magnesium has $A_{r} = 24$ . So a magnesium atom is, on average, twice as heavy as a carbon atom.

Follow the coin logic. Weigh out 12 g of carbon and 24 g of magnesium. The masses are in the ratio 1 : 2, and the atoms are in the ratio 1 : 2. So the two piles contain the same number of atoms. The same is true of 16 g of oxygen atoms, 40 g of calcium, or 197 g of gold. Each is the relative atomic mass, written in grams. Each contains the same number of atoms.

That number is so useful that it has a name. It is the number of particles in one mole.

The mole

One mole (symbol mol) of any substance contains exactly $6.022 14076 \times 1 0^{23}$ particles. This number per mole is the Avogadro constant, $N_{A} = 6.022 \times 1 0^{23} mol^{- 1}$ . In calculations at this level we use $6.02 \times 1 0^{23}$ .

The mass of one mole is the molar mass, $M$ , in g/mol. It has the same number as the relative atomic mass $A_{r}$ or relative formula mass $M_{r}$ .

The "particles" can be anything, but you must say what they are: atoms, molecules, ions, electrons. A mole of carbon atoms, a mole of water molecules and a mole of sodium ions each contain $6.02 \times 1 0^{23}$ of that particle. The mole is a counting unit, like a dozen. A dozen eggs is 12 eggs. A mole of eggs would be $6.02 \times 1 0^{23}$ eggs. It is just a much larger count, chosen to suit things as small as atoms.

The same multiplier links every row. One particle has a mass in atomic mass units (u); one mole of the same particle has the same number of grams. Masses are rounded to whole numbers.

Why that strange number?

Why $6.02 \times 1 0^{23}$ , and not a round number such as $1 0^{24}$ ? Because the number is set by two sizes that nobody chose to match: the gram, a unit made for everyday objects, and the mass of an atom.

Atomic masses are measured in the atomic mass unit, u, which is one-twelfth of the mass of a carbon-12 atom. One u is about $1.66 \times 1 0^{- 24}$ g. Ask how many of these tiny units fit into one gram, and you get about $6.02 \times 1 0^{23}$ . So a particle with a mass of 24 u, multiplied by $6.02 \times 1 0^{23}$ , gives 24 g. The Avogadro constant is the exchange rate between atomic mass units and grams. That is why the molar mass in grams has the same number as the relative mass. It is not a coincidence. The mole was built to make it true.

For most of the twentieth century, the definition said this directly: one mole was the number of atoms in exactly 12 g of carbon-12. Then, from 20 May 2019, the world's units were redefined, and the Avogadro constant was fixed at exactly $6.022 14076 \times 1 0^{23} mol^{- 1}$ (International Bureau of Weights and Measures, BIPM). Now the number is exact, and the mass of a mole of carbon-12 is measured instead. It still comes out at 12 g, to within about one part in a billion. For every calculation you will ever do at this level, "molar mass in grams equals the relative mass" is still true.

Equal masses, different numbers

The key relationship is the coin-bag rule, written in chemistry's symbols:

n = \frac{m}{M}

Here $n$ is the amount of substance in moles, $m$ is the mass in grams, and $M$ is the molar mass in g/mol. To turn moles into a number of particles, multiply by the Avogadro constant: $N = n \times N_{A}$ .

Try it below. Choose a mass, and compare how many atoms that same mass of different elements contains.

Explore — the same mass, different numbers of atoms

Choose a mass. Every bar shows how many atoms that mass of an element contains. Tap an element to see the working. Change the mass and watch whether the order of the bars, or their ratios, ever change.

Mass of each sample: 24 g

Try:

The table shows 24 g of seven elements. Each amount is $n = m \div M$ , and each number of atoms is $n \times 6.02 \times 1 0^{23}$ .

Element	M (g/mol)	Moles in 24 g	Atoms in 24 g
carbon	12	2	≈ 1.20 × 10²⁴
sodium	23	≈ 1.04	≈ 6.28 × 10²³
magnesium	24	1	6.02 × 10²³
sulfur	32	0.75	≈ 4.52 × 10²³
calcium	40	0.6	≈ 3.61 × 10²³
iron	56	≈ 0.429	≈ 2.58 × 10²³
gold	197	≈ 0.122	≈ 7.33 × 10²²

What did you see? The bars never change their order or their ratios, whatever mass you choose. Carbon always has the most atoms, and gold the fewest. With 24 g, magnesium has exactly one mole; with 12 g, carbon does; with 197 g, gold does. The mole is simply the mass at which each element reaches the same count.

Stop and think

Which contains more atoms: 1 g of hydrogen atoms or 1 g of gold?

Show the answer

Hydrogen, by far. With $A_{r} = 1$ , 1 g of hydrogen atoms is 1 mol, which is $6.02 \times 1 0^{23}$ atoms. Gold has $A_{r} = 197$ , so 1 g of gold is only $1 \div 197 \approx 0.005 08$ mol, about $3.06 \times 1 0^{21}$ atoms. Hydrogen has 197 times as many. A gram of a light element holds far more atoms than a gram of a heavy one. Equal masses almost never mean equal numbers.

Molecules, formula units and atoms

For a compound, add up the relative atomic masses in the formula to get the relative formula mass, $M_{r}$ . Water, H₂O, has $M_{r} = 2 \times 1 + 16 = 18$ , so its molar mass is 18 g/mol. One mole of water is 18 g: about a tablespoon.

Ionic compounds such as sodium chloride do not contain molecules. As the lecture on lattice structures showed, they are giant lattices of ions. So for NaCl, a "particle" means one formula unit: one Na⁺ and one Cl⁻ together. One mole of NaCl is 58.5 g and contains $6.02 \times 1 0^{23}$ formula units, which is $6.02 \times 1 0^{23}$ sodium ions and the same number of chloride ions.

This is where most mistakes in this topic happen. One mole of a compound contains more than one mole of atoms. Commit to an answer below before you read on.

Predict, then reveal — which sample has the most atoms?

Four samples: 18 g of water (H₂O), 12 g of carbon (C), 32 g of oxygen gas (O₂) and 23 g of sodium (Na). Which contains the most atoms? Choose one.

Each sample is exactly one mole of its own particle: water molecules, carbon atoms, O₂ molecules or sodium atoms. But the number of atoms differs. 18 g of water is 1 mol of H₂O, which is 3 mol of atoms, about $1.81 \times 1 0^{24}$ . 32 g of O₂ is 2 mol of atoms, about $1.20 \times 1 0^{24}$ . Carbon and sodium are each 1 mol of atoms, $6.02 \times 1 0^{23}$ . So the water has the most atoms.

The lesson of that question is a habit. Before you multiply by $N_{A}$ , ask: a mole of what? Then count how many of the particles you want are inside each particle you have.

Worked examples

Worked Example 1 — Carbon and magnesium

How many atoms are there in 12 g of carbon, and in 24 g of magnesium?

Carbon: $n = 12 \div 12 = 1$ mol, so $N = 1 \times 6.02 \times 1 0^{23} = 6.02 \times 1 0^{23}$ atoms.

Magnesium: $n = 24 \div 24 = 1$ mol, so again $6.02 \times 1 0^{23}$ atoms.

The magnesium sample is twice as heavy, yet the numbers are equal. This is because each magnesium atom is twice as heavy as each carbon atom.

Worked Example 2 — Sodium

How many moles, and how many atoms, are in 46 g of sodium?

$M (Na) = 23$ g/mol, so $n = 46 \div 23 = 2$ mol.

$N = 2 \times 6.02 \times 1 0^{23} = 12.04 \times 1 0^{23} \approx 1.20 \times 1 0^{24}$ atoms.

Check: the mass is twice the molar mass, so the answer should be two moles' worth of atoms, about twice $6 \times 1 0^{23}$ . ✓ Notice the last step: $12.04 \times 1 0^{23}$ is not in standard form, because 12.04 is bigger than 10. Moving the decimal point one place left and adding 1 to the power gives $1.204 \times 1 0^{24}$ .

Worked Example 3 — Molecules or atoms?

For 36 g of water, find the number of molecules, the number of atoms, and the number of hydrogen atoms.

$M (H_{2} O) = 2 \times 1 + 16 = 18$ g/mol, so $n = 36 \div 18 = 2$ mol of molecules.

Molecules: $2 \times 6.02 \times 1 0^{23} \approx 1.20 \times 1 0^{24}$ .

Atoms: each molecule has 3 atoms, so there are $3 \times 2 = 6$ mol of atoms, $6 \times 6.02 \times 1 0^{23} \approx 3.61 \times 1 0^{24}$ atoms.

Hydrogen atoms: each molecule has 2, so $2 \times 2 = 4$ mol, about $2.41 \times 1 0^{24}$ hydrogen atoms.

Work in moles for as long as you can, and multiply by $N_{A}$ only at the end. It keeps the arithmetic simple and avoids rounding errors.

Worked Example 4 — Calcium carbonate, both directions

(a) How many moles are in 25 g of calcium carbonate, CaCO₃? How many oxygen atoms does it contain? (b) What is the mass of $3.01 \times 1 0^{23}$ calcium atoms?

(a) $M (CaCO_{3}) = 40 + 12 + 3 \times 16 = 100$ g/mol. So $n = 25 \div 100 = 0.25$ mol. Each formula unit has 3 oxygen atoms, so there are $3 \times 0.25 = 0.75$ mol of oxygen atoms, about $0.75 \times 6.02 \times 1 0^{23} \approx 4.52 \times 1 0^{23}$ .

(b) Go backwards. $n = N \div N_{A} = 3.01 \times 1 0^{23} \div 6.02 \times 1 0^{23} = 0.5$ mol. Then $m = n \times M = 0.5 \times 40 = 20$ g.

The three quantities, mass, moles and number of particles, form a chain. Moles are always the middle link: divide by $M$ to get from grams to moles, and multiply by $N_{A}$ to get from moles to particles.

How big is a mole?

A mole is an enormous count. Counting $6.02 \times 1 0^{23}$ at one per second would take about $1.9 \times 1 0^{16}$ years. Even a mole of sand grains is huge. Take grains half a millimetre across, packed with no gaps. A mole of them would cover the whole surface of the Earth, land and sea, in a layer about 8 cm deep. (You may read that it would be kilometres deep. The arithmetic does not support that for ordinary sand.)

Yet a mole of water is about a tablespoon. A mole of carbon is a lump of charcoal you could close your hand around. That is it. That is the whole idea. A mole is an astronomically large number of particles, but it is an ordinary, everyday mass of stuff. It was designed to be. It lets a chemist hold the invisible world on a laboratory balance.

Three classic mistakes

Trap 1 — Forgetting the atoms inside a molecule

"36 g of water contains $1.20 \times 1 0^{24}$ atoms" is wrong: that is the number of molecules. Read the question again. If it asks for atoms, multiply by the number of atoms in each molecule.

Trap 2 — Using A_r for a diatomic gas

Oxygen gas is O₂, not O. Its molar mass is 32 g/mol, not 16 g/mol. So 32 g of oxygen gas is 1 mol of O₂ molecules, not 2 mol. The same applies to H₂, N₂, Cl₂ and the other gases that exist as pairs of atoms.

Trap 3 — Multiplying when you should divide

From grams to moles you divide by $M$ . From moles to particles you multiply by $N_{A}$ . Use a size check every time: a sample you can see always contains a huge number of atoms, roughly $1 0^{20}$ to $1 0^{25}$ . An answer such as $3.6 \times 1 0^{- 24}$ atoms, or $1.5 \times 1 0^{25}$ moles, means a step went the wrong way.

Try it yourself

Work each one on paper first. Open a hint only when you are stuck. Use $N_{A} = 6.02 \times 1 0^{23} mol^{- 1}$ and the relative masses: H 1, C 12, O 16, Na 23, Mg 24, S 32, Ca 40, Fe 56, Au 197.

How many moles, and how many atoms, are in 64 g of sulfur?

Hint 1
Moles first: divide the mass by the molar mass.

Hint 2
$n = 64 \div 32$ . Then multiply by $6.02 \times 1 0^{23}$ .

Full answer
$n = 64 \div 32 = 2$ mol. $N = 2 \times 6.02 \times 1 0^{23} \approx 1.20 \times 1 0^{24}$ atoms. Check: twice the molar mass gives twice a mole's worth of atoms ✓.
Which contains more atoms, 10 g of calcium or 10 g of magnesium? How many times more?

Hint 1
Find the moles of each. More moles means more atoms.

Hint 2
The ratio of the numbers of atoms is the ratio of the moles, $(10 \div 24) \div (10 \div 40)$ .

Full answer
Magnesium: $10 \div 24 \approx 0.417$ mol, about $2.51 \times 1 0^{23}$ atoms. Calcium: $10 \div 40 = 0.25$ mol, about $1.51 \times 1 0^{23}$ atoms. Magnesium has more, by a factor of $40 \div 24 \approx 1.67$ . Notice that the masses cancel: the ratio depends only on the molar masses, which is why the bars in the explorer kept the same ratios at every mass.
Find the mass of one gold atom, in grams. A thin gold ring has a mass of 1.0 g. About how many gold atoms does it contain?

Hint 1
One mole of gold atoms has a mass of 197 g and contains $6.02 \times 1 0^{23}$ atoms.

Hint 2
Mass of one atom = $197 \div (6.02 \times 1 0^{23})$ . For the ring, find the moles first.

Full answer
One atom: $197 \div (6.02 \times 1 0^{23}) \approx 3.27 \times 1 0^{- 22}$ g. The ring: $n = 1.0 \div 197 \approx 0.005 08$ mol, so $N \approx 0.005 08 \times 6.02 \times 1 0^{23} \approx 3.06 \times 1 0^{21}$ atoms. Check another way: $1.0 \div (3.27 \times 1 0^{- 22}) \approx 3.06 \times 1 0^{21}$ ✓.
One mole of liquid water has a mass of 18 g and a volume of about 18 cm³, since 1 cm³ of water has a mass of about 1 g. Use this to estimate the volume taken up by one water molecule, and the size of a cube with that volume, in nanometres (1 nm = $1 0^{- 7}$ cm).

Hint 1
18 cm³ is shared among $6.02 \times 1 0^{23}$ molecules.

Hint 2
Volume per molecule = $18 \div (6.02 \times 1 0^{23})$ cm³. The side of a cube is the cube root of its volume.

Full answer
Volume per molecule: $18 \div (6.02 \times 1 0^{23}) \approx 2.99 \times 1 0^{- 23}$ cm³. The side of a cube with this volume is $3 2.99 \times 1 0^{- 23} \approx 3.1 \times 1 0^{- 8}$ cm, which is about 0.31 nm. So each water molecule, with the space around it in the liquid, takes up a box about 0.3 nm across. This simple estimate uses only a kitchen fact and the mole, and it gives the right size: molecules are a few tenths of a nanometre across. The mole lets you measure something invisible with a spoon.

The bigger idea

The German chemist Wilhelm Ostwald coined the unit "Mol" in 1894, from the German word for molecule. At that time, some scientists still doubted that atoms were real. The doubt ended largely through the number in this lecture. In 1909, the French physicist Jean Perrin, after experiments on the jittering of tiny particles in water (Brownian motion, explained by Einstein in 1905), proposed calling it "Avogadro's constant". The value he found agreed with values found in completely different ways, from gases, from electric charge and later from crystals. Many independent routes giving one number was very strong evidence that atoms exist.

The mole became one of the seven base units of the international system in 1971. Since 2019 its number has been fixed exactly. It now joins the second, the metre and the kilogram as a unit defined by a constant of nature. Every calculation in the rest of this unit, from formulae to reacting masses to yields, is a calculation in moles. The next lecture adds two more ways to find moles: from the volume of a gas and from the concentration of a solution.

Key concepts in this lesson

A mole is a counting unit: one mole of any particle contains exactly $6.022 14076 \times 1 0^{23}$ of that particle, and we use $N_{A} = 6.02 \times 1 0^{23} mol^{- 1}$ in calculations.
The molar mass in g/mol has the same number as the relative mass, because the Avogadro constant is the exchange rate between atomic mass units and grams.
Moles are the middle link in every calculation: $n = m \div M$ turns grams into moles, and $N = n \times N_{A}$ turns moles into particles.
Equal masses of different substances contain different numbers of particles, because their particles have different masses.
Always ask "a mole of what?", since one mole of a compound or a diatomic gas contains more than one mole of atoms.