The Mole

The single most useful thing anyone can tell you about the mole is that it is a counting word. It belongs to the same family as pair, dozen, score, gross and ream. A dozen eggs is twelve eggs; a ream of paper is five hundred sheets; a mole of anything is 602,214,076,000,000,000,000,000 of them.

That is the whole idea. Everything else on this page is consequences of it.

Most of the difficulty students have with moles comes from being introduced to the concept through the formula n = m / M before anyone has said what the thing being counted is. The formula then looks like an arbitrary rule, and arbitrary rules are impossible to apply to an unfamiliar question. If you know that "mole" means "this many things", every mole calculation becomes a unit conversion of the sort you have been doing since primary school.

Why chemists invented a counting word this awkward

Reactions happen between individual particles, in whole-number ratios. Two hydrogen molecules react with one oxygen molecule. Not two grams with one gram — two molecules with one molecule.

But nobody can count out two molecules. What a chemist can do is weigh things. So there is a permanent mismatch between the currency reactions are conducted in (numbers of particles) and the currency laboratories are equipped for (mass). The mole exists to convert between the two, and it was chosen at a size that makes that conversion painless: it is defined so that a mole of particles weighs, in grams, the same number as one particle weighs in atomic mass units.

A single carbon-12 atom weighs 12 u. A mole of carbon-12 atoms weighs almost exactly 12 g. That correspondence is not a natural law — it is a design decision, and it is the reason the periodic table doubles as a table of masses per mole. The molar mass guide works through why the correspondence holds and where it is only approximate.

Worked: turning grams into a particle count

Question. How many sodium atoms are in 4.60 g of sodium?

Sodium's standard atomic weight is 22.990, so one mole of sodium atoms has a mass of 22.990 g. Work in two hops — grams to moles, then moles to atoms — and never try to jump straight from grams to atoms.

Step 1 — grams to moles.

n = 4.60 g ÷ 22.990 g/mol = 0.2001 mol

Look at how the units behave: grams divided by grams-per-mole leaves moles. If your units do not cancel to what the question asked for, you have divided the wrong way round, and this check catches that before the arithmetic does.

Step 2 — moles to atoms.

N = 0.2001 mol × 6.02214076 × 10²³ mol⁻¹ = 1.205 × 10²³ atoms

So a small pinch of sodium, less than a fifth of a mole, still contains more than a hundred thousand billion billion atoms. That gap between the size of the sample and the size of the count is the reason the unit has to be so enormous.

Worked: running it backwards

Question. A sample contains 1.00 × 10²⁴ molecules of carbon dioxide. What is its mass?

Step 1 — particles to moles. Divide by the number of particles per mole:

n = (1.00 × 10²⁴) ÷ (6.02214076 × 10²³) = 1.661 mol

Step 2 — moles to grams. CO₂ has a molar mass of 44.009 g/mol, so:

m = 1.661 mol × 44.009 g/mol = 73.1 g

Notice that 10²⁴ molecules is only a little more than one and a half moles. Powers of ten compress differences that feel enormous written out, and a sense of "one mole is about 6 × 10²³, so 10²⁴ must be a bit under two moles" is worth more in an exam than the calculator is.

The mole conversion calculator does both directions for any formula if you want to check working you have already done by hand.

"A mole of hydrogen" is an incomplete sentence

This is the ambiguity that quietly ruins answers. The definition of the mole says elementary entities, and it insists that the entity be specified. It can be an atom, a molecule, an ion, an electron, a formula unit, or a specified group of such particles.

A mole of hydrogen atoms has a mass of about 1.008 g. A mole of hydrogen molecules, H₂, has a mass of about 2.016 g and contains two moles of atoms. Both are "a mole of hydrogen" in loose speech, and they differ by a factor of two — which in a marked calculation is the difference between full credit and none.

The habit that fixes it: whenever you write "mol" in your working, write what of. 0.5 mol H₂, not 0.5 mol hydrogen. Ionic compounds get the same treatment — you count formula units of NaCl, because no NaCl molecule exists in a salt crystal, only a repeating one-to-one arrangement of ions.

One mole of electrons is a quantity with a name of its own: it carries 96,485 coulombs of charge, the Faraday constant, which is simply the charge on one electron multiplied out by the mole.

What changed on 20 May 2019

Until 2019 the mole was defined as the number of atoms in exactly 12 g of carbon-12. Nobody knew what that number was to better than about eight significant figures, because it had to be measured, and the measurement kept improving.

The revised SI turned the relationship inside out. The number is now fixed by decree — one mole contains exactly 6.02214076 × 10²³ elementary entities — and the mass of 12 g of carbon-12 has become the measured quantity instead.

For your homework this changes nothing you can detect. The chosen value was picked to match the old definition as closely as the best measurements allowed, so the mass of a mole of carbon-12 is still 12 g to within about one part in a billion. What it changes is the logic: the mole is no longer parasitic on the kilogram, and a definition that once depended on a lump of metal in France now depends on nothing but a counted number.

The value itself, its history and how it came to be measured that precisely belong to Avogadro's number, which is a different question from what the unit is for.

The mistakes that actually cost marks

• Multiplying when you should divide. Moles are almost always the smaller number when starting from grams of an everyday sample, and always the smaller number when starting from a particle count. If your answer is 10²⁵ moles, you have gone the wrong way.
• Using the atomic number instead of the atomic weight. Carbon is 12.011, not 6. This one is invisible in your working and catastrophic in your answer.
• Treating "mole" as a mass. A mole is not 6 g, or 12 g, or any fixed mass. A mole of lead weighs about 207 g and a mole of helium about 4 g; they contain the same number of atoms.
• Losing the entity halfway through. Starting a problem in moles of O₂ and finishing in moles of O without saying so is the most common single error in stoichiometry, and it is a factor of two every time.
• Rounding Avogadro's number to 6 × 10²³ and then quoting four significant figures. The shortened constant is about 0.4% low, which is a bigger error than four figures claims to have.

A word on where the name came from

The German Mol was in use before the modern concept settled, and credit for it is genuinely contested. Wilhelm Ostwald is usually named, having used and popularised Mol in his 1890s textbooks as a clipping of Molekül; August Horstmann had used the same word in the early 1880s for a related quantity, and some historians of chemistry give him priority. What is not disputed is that the term arrived as laboratory shorthand and was only later promoted to an SI base unit, in 1971 — which is late enough that many chemists working today were taught it as a convenience rather than as a fundamental unit of measurement.