Counting atoms by weighing: the mole, hands on

How to use this simulation

1. The pan starts with 18 g of water. Look at the boxes on the right: 1.00 mol, 6.02 × 10²³ molecules. That is exactly one mole of water.
2. Pick another substance in “Choose a substance”, then drag the mass slider. The dots pile up on the pan, and the moles and the particle count follow. The label at the top right says how much substance one dot stands for.
3. Switch on “Weigh exactly 1 mole” and step through all the substances. The pan reading changes every time, but the number of particles never does. For oxygen or carbon dioxide the STP volume appears too.
4. Open “How big is Avogadro’s number” to count paper sheets, sand grains or seconds. Raise the count as 10ⁿ, then switch on “Count exactly 1 mole” and watch the red dot jump along the ruler.
5. Open “Percentage composition” and choose glucose. A coloured bar shows how the mass splits between carbon, hydrogen and oxygen, with the arithmetic printed underneath.

A dozen eggs, a ream of paper, a mole of water

Ask a shopkeeper for a dozen eggs and you get 12. Why not just say twelve? Because a dozen is a counting unit: it wraps a bigger number in one short word. Paper works the same way. A ream is 500 sheets, and the whole pile is only about 5 cm tall.

Chemists have a bigger problem. The atoms and molecules they handle are so small that one grain of sugar holds trillions upon trillions of them. Yet a reaction only makes sense if you know how many molecules meet how many others. You cannot count them one by one.

A bank teller shows the way out. A sack of coins is not counted coin by coin; it is put on a scale, because every coin weighs the same. Chemistry uses the same trick. The mass of each atom is known, so weighing a sample tells you how many atoms it holds. The bridge between weighing and counting is the mole.

Think of a mole as a dozen with a much larger number: a dozen is 12 things, a mole is 6.022 × 10²³. And the mass of one mole is neat, too. 18 g of water, a little more than a tablespoon, holds exactly one mole of molecules, which is 6.02 × 10²³.

The mole from scratch

The mole (symbol mol) is the SI unit for amount of substance. One mole contains exactly 6.022 × 10²³ elementary entities, and the entity can be an atom, a molecule, an ion or an electron. So always say what you are counting: a mole of oxygen atoms is not a mole of oxygen molecules.

The number 6.022 × 10²³ is Avogadro’s number, written Nₐ. Amedeo Avogadro never measured it; later scientists named it after his idea that equal volumes of gases hold equal numbers of molecules. It has this value because the mole was originally defined as the number of atoms in 12 g of carbon-12.

That choice gives a lovely shortcut. A carbon atom has atomic mass 12, so 12 g of carbon holds 6.02 × 10²³ atoms. Hydrogen has atomic mass 1, so 1 g of hydrogen holds the same number of atoms. The mass of one mole in grams equals the atomic or molecular mass as a plain number.

Water, H₂O, has molecular mass 18: two hydrogens (2 × 1) plus one oxygen (16). So a mole of water has mass 18 g. Keep two terms apart. Molecular mass is the mass of one molecule, in amu. Molar mass is the mass of one mole, in g/mol. Same number, different unit.

See how tiny a molecule is: one water molecule has mass 18 ÷ 6.02 × 10²³, about 2.99 × 10⁻²³ g. Nobody can weigh that, but 6.02 × 10²³ of them can be weighed. That is what the mole buys you.

The words you need

Exams reward exact definitions, so here are the key terms side by side.

The conversion triangle: mass, moles, number

Three quantities keep turning up in mole problems: mass in grams, amount in moles and the number of particles. Moles are the bridge. You cannot jump from mass straight to particles; go through moles first.

For gases there is a fourth quantity, volume. At 0 °C and 1 atm (STP) one mole of any gas takes up about 22.4 L, so a gas volume converts to moles as well. The triangle on the right of the simulation lays out exactly these four quantities and how they connect.

How big is Avogadro’s number, really?

6.02 × 10²³. It is easy to write and hard to picture, so let us build it out of things you know. Take paper 0.1 mm thick. A mole of sheets, piled up, would be 6.02 × 10¹⁹ m tall. That is about 6,365 light years, or 4.0 × 10⁸ times the distance from Earth to the Sun.

Try sand. A mole of grains of 1 mm³ each, spread evenly over all of Bangladesh, would bury the country about 4.1 km deep. And counting one per second, a mole of seconds takes 1.91 × 10¹⁶ years, which is 1.4 × 10⁶ times the age of the universe.

So the mole is a tool for the very small. Nothing in everyday life needs a number this large.

Percentage composition: how much of each element?

The percentage composition of a compound tells you how many grams of each element are in 100 g of it. You do not need a lab for that; the formula and the atomic masses are enough.

Think of a cake recipe. Flour, sugar and eggs add up to the total weight, and each ingredient is some percentage of it. In a compound the ingredients are elements and the total is the molar mass.

Try it in the simulation

Run these experiments one after the other and the idea will click.

• Experiment 1: take water at 18 g and read 1.00 mol. Halve the mass and the moles halve too.
• Experiment 2: choose oxygen and set 16 g. You get 0.50 mol and 11.2 L at STP. Double the mass and watch the volume double.
• Experiment 3: keep “Weigh exactly 1 mole” on and step through the substances. The pan reading changes, the particle count does not. Compare with the table below.
• Experiment 4: open “Percentage composition”, choose glucose and set 90 g. It contains 36.0 g of carbon, 6.0 g of hydrogen and 48.0 g of oxygen.
• Experiment 5: in “How big is Avogadro’s number” count paper with n = 12. The stack is 1.00 × 10⁸ m tall. Now switch on exactly 1 mole and see how far the dot jumps.

Solved problems

Five classic question types, worked step by step. Every figure is computed.

Mistakes everyone makes

Spot these in advance and you keep your marks.

• Mixing up atoms and molecules: oxygen gas is O₂, so its molar mass is 32, not 16.
• Forgetting the atom count in a formula: in H₂O hydrogen contributes 2 × 1, not 1.
• Multiplying where you should divide: mass is divided by molar mass to get moles.
• Using 22.4 L for everything: that value is for gases at STP only, never water or salt.
• Treating molecular mass and molar mass as the same unit: one is amu, the other g/mol.
• Calling NaCl a molecule: it is an ionic compound, so say formula unit.

The mole in real life

The mole is not just an exam topic; factories and labs run on it every day.

• Medicine: a dose is planned from the moles of active ingredient in each tablet.
• Fertiliser plants: the moles of ammonia and carbon dioxide in a urea batch decide how much of each raw material to buy.
• Body chemistry: blood glucose is reported in millimoles per litre, not grams.
• Environmental science: the mass of carbon dioxide released converts to moles and then to gas volume and pollution.
• Lab solutions: a 1 molar solution holds one mole of solute in every litre.

Exam corner (CBSE and NCERT, class 9 and 11)

Typical NCERT-style questions on this chapter. Each one uses only the formulas above.

• Short answer: define the mole. State the value of Avogadro’s number. What is STP?
• Numerical: find the mass of 0.5 mol of water. Answer: 9 g.
• Numerical: how many moles are in 11 g of CO₂? Answer: 0.25 mol.
• Numerical: how many molecules are in 4.4 g of CO₂? Answer: 6.022 × 10²².
• Numerical: find the percentage of sodium in NaCl. Answer: 39.32%.

Revision summary

Read this list the night before the exam.

• One mole holds 6.022 × 10²³ particles.
• Molar mass in g/mol equals the atomic or molecular mass as a number.
• n = m ÷ M converts mass to moles.
• N = n × Nₐ converts moles to particles.
• V = n × 22.4 L gives the volume of a gas at STP.
• Percentage of an element = (its total mass ÷ molar mass) × 100.

Frequently asked questions