Last class you were asked how many atoms were in a paperclip, and the room stalled in an interesting way. Nobody doubted the number existed. The problem was that atoms cannot be counted the way marbles can, and the balance in front of you reports grams, which is the wrong quantity entirely. The mole is how chemists get from the quantity they can measure to the quantity the reaction actually cares about.
The reaction counts, your balance weighs
A balanced equation is a statement about numbers of particles. When you write
the 2 does not mean two grams of hydrochloric acid or two millilitres of it. It means that for every one zinc atom, two molecules of are consumed. The equation is a recipe written in particles.
Your equipment cannot count particles. So you need a bridge — a way of saying “this mass corresponds to that many particles” — and the bridge has to be built once, carefully, and then used everywhere.
A counting unit, chosen to be convenient
You already accept counting units. A dozen is twelve of anything; a ream is five hundred sheets. A mole is the same idea, sized for atoms:
That number — Avogadro’s number — looks arbitrary and is not. It was chosen so that one mole of an element has a mass in grams equal to the number printed under its symbol on the periodic table. One mole of carbon atoms is 12.01 g. One mole of zinc atoms is 65.38 g. The whole point of the number is to make the periodic table double as a conversion chart.1
"Mole" names a number, not a substance
A mole of water and a mole of lead are the same count and wildly different masses. A mole of is 18.02 g; a mole of lead is 207.2 g. If a sentence you have written would still make sense with “mole” replaced by “gram”, something has gone wrong.
Moving between mass, moles, and particles
Two relationships do all the work. With for amount in moles, for mass in grams, for molar mass in grams per mole, and for the number of particles:
Which means every problem of this kind is the same problem wearing different clothes:
| You are given | You want | Route |
|---|---|---|
| Mass | Moles | Divide by molar mass |
| Moles | Mass | Multiply by molar mass |
| Moles | Particles | Multiply by Avogadro’s number |
| Mass | Particles | Both steps, in that order |
There is no fourth trick. If a question looks unfamiliar, find where it sits in that table before you write anything down.
Worked: the paperclip
A steel paperclip has a mass of about 1.0 g and is mostly iron, so treat it as iron. The molar mass of iron is 55.85 g/mol.
atoms
Eleven thousand billion billion atoms, from a measurement you took in four seconds. Note the answer carries two significant figures, because the mass did — see Significant Figures and Units. Writing would be claiming a precision the balance never gave you.
Why this is the hinge of the course
Everything quantitative from here runs through the mole. Percentage composition, empirical formulas, the amount of precipitate you should have recovered, the volume of gas a reaction will produce, the concentration of a solution — each one is a mass or a volume converted into a count, put through a balanced equation, and converted back.
Get comfortable with the conversions in Mole Conversions Practice until they stop being a procedure you look up. Then Molar Mass and Composition handles compounds rather than elements, and Stoichiometry puts the count through the equation, which is where this is all going.
Curriculum connection
D3.1
explain the law of definite proportions
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D3.2
describe the relationships between Avogadro’s number, the mole concept, and the molar mass of any given substance
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Footnotes
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Since 2019 the mole is defined as exactly elementary entities, rather than being derived from a mass of carbon-12 as it was before. The number stayed the same to more decimal places than any school measurement could detect; what changed is which quantity is treated as fixed and which is measured. It is a good example of a definition being tidied up long after everyone had agreed on the value. ↩