Moles, Molar Mass, and Avogadro's Number

Counting particles by weighing them

Atoms are far too small to count one at a time, so chemists count them in fixed batches, the way a store counts eggs by the dozen. The batch is the mole. Since the 2019 revision of the SI, one mole contains exactly 6.02214076 × 10²³ specified entities, and the Avogadro constant is defined as exactly 6.02214076 × 10²³ mol⁻¹. For calculations, 6.022 × 10²³ mol⁻¹ is usually enough.

The entity must always be stated. A mole of oxygen atoms (O) and a mole of oxygen molecules (O₂) contain the same number of entities, but the second contains twice as many atoms and has twice the mass. Writing the formula next to every amount, as in 0.50 mol O₂, prevents most of the confusion students meet with moles.

Why grams per mole matches the formula weight

The standard atomic weight of an element is a dimensionless average of its isotopes' masses, weighted by their natural abundance and expressed relative to one twelfth of the mass of a carbon-12 atom. The mole was originally defined so that one mole of a substance has a mass in grams numerically equal to its atomic or formula weight. Since the 2019 redefinition that match is no longer exact by definition, but it still holds to about one part in a billion. Chlorine's atomic weight is about 35.45, so its molar mass is 35.45 g/mol, even though no single chlorine atom has that mass; real chlorine is a mixture of chlorine-35 and chlorine-37.

This guide uses IUPAC standard atomic weights rounded to two decimal places: H 1.01, C 12.01, N 14.01, O 16.00, Al 26.98, and S 32.06. To find the molar mass of a compound, add the atomic weights of every atom in the formula, multiplying each by its subscript and by any multiplier outside parentheses.

For aluminum sulfate, Al₂(SO₄)₃, the parentheses mean three sulfate groups. The formula unit therefore contains 2 Al, 3 S, and 12 O: 2(26.98) + 3(32.06) + 12(16.00) = 53.96 + 96.18 + 192.00 = 342.14 g/mol.

The grams, moles, and particles map

Every problem in this topic is a path through three boxes: mass in grams, amount in moles, and number of particles. Molar mass connects the first two; the Avogadro constant connects the last two. Moles are always the middle stop. There is no direct conversion from grams to particles that skips the mole.

Inside the particles box there is one more step. A formula tells you how many atoms or ions of each kind are in one molecule or formula unit, and those subscripts are exact counting factors.

• Grams to moles: divide by molar mass (multiply by 1 mol / M g).
• Moles to grams: multiply by molar mass.
• Moles to particles: multiply by 6.022 × 10²³ per mole.
• Particles to moles: divide by 6.022 × 10²³ per mole.
• Molecules or formula units to atoms or ions: multiply by the subscript for that atom or ion.

Worked example: from milligrams of caffeine to nitrogen atoms

A sample of caffeine, C₈H₁₀N₄O₂, has a mass of 95.0 mg. How many caffeine molecules and how many nitrogen atoms does it contain?

Step 1, molar mass: 8(12.01) + 10(1.01) + 4(14.01) + 2(16.00) = 96.08 + 10.10 + 56.04 + 32.00 = 194.22 g/mol. Step 2, convert the mass to grams and then to moles: 95.0 mg × (10⁻³ g / 1 mg) = 0.0950 g, and 0.0950 g × (1 mol / 194.22 g) = 4.8914 × 10⁻⁴ mol. Keep the extra digits for now.

Step 3, moles to molecules: 4.8914 × 10⁻⁴ mol × 6.022 × 10²³ molecules/mol = 2.9456 × 10²⁰, reported as 2.95 × 10²⁰ molecules. Step 4, each molecule has four nitrogen atoms, an exact count: 2.9456 × 10²⁰ × 4 = 1.18 × 10²¹ nitrogen atoms. The mass 95.0 mg has three significant figures, so both answers do too.

Check the size: a tenth of a gram is a small fraction of a mole of a substance whose molar mass is about 200 g/mol, so the amount should be roughly 0.1 ÷ 200 = 5 × 10⁻⁴ mol, and the molecule count should be far below 10²³. Both checks pass.

Worked example: from a count of formula units back to grams

A sample of aluminum sulfate contains 3.15 × 10²² formula units. What is its mass, and how many sulfate ions and oxygen atoms does it contain?

Step 1, particles to moles: 3.15 × 10²² ÷ 6.022 × 10²³ mol⁻¹ = 0.052308 mol. Step 2, moles to grams using the molar mass found earlier: 0.052308 mol × 342.14 g/mol = 17.897 g, reported as 17.9 g.

Step 3, use the formula as a counting tool. Each formula unit Al₂(SO₄)₃ contains three sulfate ions, SO₄²⁻, so the sample holds 3 × 3.15 × 10²² = 9.45 × 10²² sulfate ions. Each formula unit has 12 oxygen atoms, so the sample holds 12 × 3.15 × 10²² = 3.78 × 10²³ oxygen atoms, a little more than half a mole of oxygen atoms. Ionic compounds are made of formula units rather than molecules, but the counting logic is identical.

Mole errors that come from losing track of the entity

Many mole mistakes are not arithmetic errors. They come from forgetting what is being counted or from applying a factor in the wrong direction. A quick estimate before you calculate catches most of them.

Write every quantity with both its unit and its entity, such as 0.0523 mol Al₂(SO₄)₃ or 9.45 × 10²² SO₄²⁻ ions, and name the arrow on the grams, moles, and particles map that each step follows. A step that matches no arrow is usually the error.

• Mixing up atoms and molecules: 1.0 mol of O₂ has a mass of 32.00 g, not 16.00 g, and contains 2.0 mol of O atoms.
• Ignoring the multiplier outside parentheses: in Al₂(SO₄)₃ there are 12 oxygen atoms per formula unit, not 4.
• Multiplying by the Avogadro constant when converting particles to moles: an amount in moles for an ordinary lab sample should be a modest number, not something near 10²³ or above.
• Using a mass number instead of an atomic weight: chlorine is 35.45 g/mol, not 35 or 37.
• Skipping the mole: grams cannot be converted directly to particles or to grams of a different substance.
• Rounding molar mass or intermediate amounts too early, which can change the last reported digit.