Converting Between Moles and Particles

To convert between moles and particles, use Avogadro’s number as the conversion factor: number of particles = moles × 6.022 × 10²³, and moles = number of particles ÷ 6.022 × 10²³. In brief: 0.250 mol of water contains 0.250 × 6.022 × 10²³ = 1.51 × 10²³ molecules, and since each molecule has 3 atoms, that is 4.52 × 10²³ atoms; to go from a mass to particles, first convert grams to moles and then moles to particles.

This conversion links the visible world (grams) with the microscopic one (atoms, molecules and ions). It is needed to count particles, to find the number of atoms of a particular element in a sample and to estimate sizes of atoms. This article gives the method, the three-step route from mass to particles and back, shows how to count atoms and ions in formulas, includes a map of conversions in a table, works through examples with large and small numbers in scientific notation and explains common problems.

What is the basic conversion?

One mole contains N_A = 6.022 × 10²³ particles. The conversion factors are (6.022 × 10²³ particles ÷ 1 mol) and (1 mol ÷ 6.022 × 10²³ particles). Since the numbers are very large or very small, use scientific notation.

Moles to particles: N = n × 6.022 × 10²³.
Particles to moles: n = N ÷ 6.022 × 10²³.

How do you go from mass to particles?

A balance measures mass, not particles, so a conversion in two steps is needed. Mass is converted to moles with the molar mass, and moles are converted to particles with Avogadro’s number:

mass (g) → ÷ M → moles → × N_A → particles

And the reverse: particles → ÷ N_A → moles → × M → mass.

How do you count atoms and ions in a formula?

The subscripts tell the number of atoms of each element per formula unit or molecule. Multiply the number of molecules by the subscript to find the number of atoms of one element.

Particles in 1 mole of some substances
Substance	Particles per mole	Atoms or ions per particle	Total atoms or ions in 1 mol
He	6.022 × 10²³ atoms	1	6.022 × 10²³
O₂	6.022 × 10²³ molecules	2	1.204 × 10²⁴ atoms
H₂O	6.022 × 10²³ molecules	3	1.807 × 10²⁴ atoms
NaCl	6.022 × 10²³ formula units	2 ions	1.204 × 10²⁴ ions
CaCl₂	6.022 × 10²³ formula units	3 ions	1.807 × 10²⁴ ions
C₆H₁₂O₆	6.022 × 10²³ molecules	24 atoms	1.445 × 10²⁵ atoms

Moles are the middle step between the mass you weigh and the number of particles. Going right, divide by the molar mass and then multiply by Avogadro’s number; going left, reverse each step. The row of numbers follows 10.0 g of CO₂ (44.009 g/mol), which is 0.227 mol or 1.37 × 10²³ molecules.

Conversion map

Conversion between mass, moles, particles and gas volume
From	To	Multiply by
Mass (g)	Moles	1 mol ÷ M
Moles	Particles	6.022 × 10²³ ÷ 1 mol
Particles	Moles	1 mol ÷ 6.022 × 10²³
Moles	Mass (g)	M ÷ 1 mol
Moles of gas	Volume at STP (L)	22.4 L ÷ 1 mol

Worked examples

Moles to molecules. How many molecules are in 0.250 mol of H₂O? N = 0.250 × 6.022 × 10²³ = 1.51 × 10²³ molecules.
Moles to atoms. How many atoms of hydrogen are in 0.250 mol of water? Each molecule has 2 H atoms: 1.51 × 10²³ × 2 = 3.01 × 10²³ H atoms. The total number of atoms is 3 × 1.51 × 10²³ = 4.52 × 10²³.
Particles to moles. A sample contains 2.40 × 10²⁴ atoms of silver. n = 2.40 × 10²⁴ ÷ 6.022 × 10²³ = 3.99 mol. The mass is 3.99 × 107.87 = 430 g.
Mass to particles. How many molecules are in 10.0 g of CO₂ (44.009 g/mol)? n = 10.0 ÷ 44.009 = 0.2272 mol. N = 0.2272 × 6.022 × 10²³ = 1.37 × 10²³ molecules. The number of oxygen atoms is twice as large: 2.74 × 10²³.
Particles to mass. What is the mass of 1.00 × 10²² atoms of copper? n = 1.00 × 10²² ÷ 6.022 × 10²³ = 0.01661 mol. m = 0.01661 × 63.546 = 1.06 g.
Ions in a solution. A solution has 0.100 mol of CaCl₂ dissolved. It contains 0.100 mol of Ca²⁺ and 0.200 mol of Cl⁻, which is 0.300 mol of ions, or 0.300 × 6.022 × 10²³ = 1.81 × 10²³ ions.
A drop of water. A drop of 0.050 g contains 0.050 ÷ 18.015 = 2.78 × 10⁻³ mol, which is 2.78 × 10⁻³ × 6.022 × 10²³ = 1.67 × 10²¹ molecules.
Mass of one molecule. For water, 18.015 g/mol ÷ 6.022 × 10²³ mol⁻¹ = 2.99 × 10⁻²³ g.

What do the numbers mean?

The numbers of particles in everyday samples are huge. A teaspoon of water (about 5 g) has 1.7 × 10²³ molecules. In contrast, the mass of a single atom is tiny: a single atom of iron has a mass of about 9.3 × 10⁻²³ g. Because the number of particles in a visible sample is so large and the mass of each particle so small, chemists count in moles.

How was the number of particles measured?

Josef Loschmidt gave the first estimate of the size of air molecules and the number of molecules in a given volume in 1865. Jean Perrin made precise measurements around 1908 using several methods, such as the distribution of small particles in a liquid under gravity and Brownian motion, and obtained values that agreed with each other, which strongly supported the existence of atoms. Robert Millikan’s measurement of the electron charge, together with the Faraday constant (the charge of a mole of electrons, 96,485 C/mol), gave another value: N_A = F ÷ e = 96,485 ÷ 1.602 × 10⁻¹⁹ = 6.02 × 10²³. In 2019 the value was fixed exactly by definition.

Which mistakes are common?

Mistake 1: Skipping the mole step

You cannot convert grams directly to particles. Always go through moles.

Mistake 2: Mixing up molecules and atoms

Count the atoms using the subscripts. 1 mol of H₂O has 1 mol of molecules but 3 mol of atoms.

Mistake 3: Entering scientific notation incorrectly on a calculator

Use the EXP key. Typing 6.022 × 10 ^ 23 can give a different result if the brackets are missing.

Mistake 4: Dividing when you should multiply

More moles mean more particles: multiply moles by Avogadro’s number. Check that the unit cancels.