The mole and Avogadro constant

The mole solves a practical problem: reactions occur between particles, while laboratories measure grams and millilitres. The mole provides the bridge between those scales.

The mole is the intermediate quantity. Convert mass to n first, then n to particle count.

Definition and amount of substance

The mole, symbol mol, is the SI unit of amount of substance. One mole contains exactly 6.02214076 × 10²³ specified elementary entities. This is the Avogadro constant N_A, in mol⁻¹.

Amount of substance is neither mass nor volume. One mole of iron atoms and one mole of water molecules contain the same number of chosen entities but have very different masses and volumes.

Entities → moles

n = N/N_A

Convert a microscopic count into amount of substance.

Moles → entities

N = nN_A

Convert a laboratory amount into a particle count.

Since the 2019 SI redefinition, N_A is fixed exactly. The mole is no longer defined through 12 g of carbon-12.

Specify the entity

One mole of O atoms contains N_A oxygen atoms. One mole of O₂ contains N_A O₂ molecules, hence 2 mol O atoms.

For ionic solids such as NaCl, count formula units rather than discrete molecules. 1 mol NaCl corresponds to 1 mol Na⁺ and 1 mol Cl⁻. For CaCl₂, 1 mol formula units corresponds to 1 mol Ca²⁺ and 2 mol Cl⁻.

Writing the entity explicitly—mol H₂O molecules, mol H atoms, mol Cl⁻ ions—prevents many conceptual errors before any arithmetic begins.

Molar mass and the periodic table

Molar mass M connects amount to mass: M = m/n, so n = m/M and m = nM.

H₂O

M ≈ 18.015 g mol⁻¹

1 mol H₂O molecules has a mass of about 18.015 g.

Fe

M ≈ 55.845 g mol⁻¹

1 mol Fe atoms has a mass of about 55.845 g.

NaCl

M ≈ 58.44 g mol⁻¹

1 mol NaCl formula units has a mass of about 58.44 g.

Particle mass in u and molar mass in g mol⁻¹ are different quantities even though their numerical values are nearly identical. That numerical correspondence is what makes periodic-table masses so useful in the laboratory.

Worked example: 9.00 g of water

Step 1 — mass → moles
n = 9.00 g / 18.015 g mol⁻¹ ≈ 0.4996 mol H₂O.

Step 2 — moles → molecules
N = 0.4996 × 6.02214076 × 10²³ ≈ 3.01 × 10²³ H₂O molecules.

Step 3 — read the formula
Each H₂O contains two H atoms, so the sample contains about 6.02 × 10²³ H atoms, almost exactly 1 mol H atoms.

The calculation becomes easier to audit when the entity is written at every stage. “0.4996 mol H₂O molecules” is more informative than “0.4996 mol”.

Stoichiometry: use mole ratios

In 2 H₂ + O₂ → 2 H₂O, coefficients describe both molecule ratios and mole ratios. Two moles H₂ react with one mole O₂ to form two moles H₂O.

With 3.00 mol H₂ and excess O₂, the H₂:H₂O ratio is 1:1, so 3.00 mol H₂O forms, corresponding to 3.00 × 18.015 ≈ 54.0 g.

La mole est l’étape centrale. Pour passer d’une mass de réactif à une mass de produit, on convertit d’abord en moles, on applique le ratio stœchiométrique, puis on revient aux grammes.

For a mass-to-mass problem, going directly from reactant grams to product grams hides the chemical ratio. The reliable path is mass → moles → stoichiometric ratio → moles → mass.

Solutions, gases and electrons

Solution

Using c = n/V, 250.0 mL of 0.200 mol L⁻¹ NaCl contains 0.0500 mol NaCl.

Gas

22.4 L mol⁻¹ applies only under particular conditions. Molar volume depends on T and P; the general relation is PV = nRT.

Electrons

1 mol electrons corresponds to about 96 485 C. This connects measured charge to amount of substance in electrochemistry.

The same change of scale appears in R = N_Ak_B: Boltzmann constant works per particle, while R works per mole.

A small laboratory amount is still a huge particle population. 1.00 mmol = 1.00 × 10⁻³ mol, which corresponds to about 6.02 × 10²⁰ entities. This is why millimoles and micromoles are convenient laboratory units: they are small macroscopic amounts but still contain enormous numbers of atoms, molecules or ions.

Exact constants do not make experimental data exact. If a mass is measured to three significant figures, multiplying by the exact value of N_A does not create extra physical precision. Keep exact constants unrounded during the calculation, then round the final result to the precision justified by the measured quantities.

Common reasoning errors

Error	Correction
“A mole is a mass.”	No: mass depends on M.
“A mole always means molecules.”	No: the entity must be specified.
“One mole of gas always occupies 22.4 L.”	No: volume depends on T and P.
“Exact N_A makes the result exact.”	No: measured mass, volume and concentration retain uncertainty.
“1 mol H₂O contains 1 mol H atoms.”	No: it contains 2 mol H atoms.

Exercises

Solid to entity count

A sample contains 11.7 g NaCl. Take M(NaCl) = 58.44 g mol⁻¹. How many formula units are present?

Solution

n = 11.7/58.44 ≈ 0.200 mol. N ≈ 0.200 × 6.022 × 10²³ = 1.20 × 10²³ formula units.

Solution to ions

How many moles of Na⁺ are present in 100.0 mL of 0.150 mol L⁻¹ NaCl, assuming complete dissociation?

Solution

n(NaCl) = cV = 0.150 × 0.1000 = 0.0150 mol. The NaCl:Na⁺ ratio is 1:1, so n(Na⁺) = 0.0150 mol.

Reaction

You have 0.50 mol O₂ and excess H₂. How much H₂O can form?

Solution

From 2 H₂ + O₂ → 2 H₂O, 1 mol O₂ gives 2 mol H₂O. Therefore 0.50 mol O₂ gives 1.00 mol H₂O.