Mole to Atoms Converter | Avogadro's Number Calculator

What is a mole converter?

A mole converter is a tool that switches between moles and the number of particles (atoms, molecules, or ions) in a sample. It uses Avogadro's number, about 6.022 × 10²³, the count of particles in one mole of any substance.

Atoms and molecules are far too small and too numerous to count one by one. Chemists solve this by counting in moles instead of individual particles, the same way a shopper buys eggs by the dozen instead of one at a time.

What is Avogadro's number?

Avogadro's number is the number of elementary particles in one mole of a substance: about 6.022 × 10²³. It is named after the Italian scientist Amedeo Avogadro, though he never estimated the value himself.

Since 2019, the number has been fixed by international agreement at exactly 6.022 140 76 × 10²³ per mole, as part of a redefinition of the SI system of units. The value has no measurement uncertainty. This calculator uses the exact value and shows results to seven significant figures. For rough hand calculations, 6.022 × 10²³ is close enough.

Mole conversion formulas

Two formulas cover every conversion this calculator performs.

Moles to particles

$\text{Particles} = \text{Moles} \times 6.02214076 \times 10^{23}$

Multiply the number of moles by Avogadro's number to get the number of atoms or molecules.

Particles to moles

$\text{Moles} = \frac{\text{Particles}}{6.02214076 \times 10^{23}}$

Divide the number of particles by Avogadro's number to get the number of moles.

How to use the mole converter calculator

To convert moles to atoms or molecules:

1. Choose a particle type: atoms or molecules.
2. Enter the number of moles.
3. The particle count appears automatically.

To convert atoms or molecules to moles:

1. Choose a particle type: atoms or molecules.
2. Enter the number of particles. Scientific notation such as 3.011e23 is accepted.
3. The number of moles appears automatically.

The calculator switches to scientific notation for very large or very small results, so long strings of zeros do not appear on screen.

Worked examples

The results below are the values the calculator displays, using the exact constant.

Example 1: moles to molecules. How many water molecules are in 0.05 moles of water?

$0.05 \times 6.02214076 \times 10^{23} = 3.011070 \times 10^{22} \text{ molecules}$

Example 2: atoms to moles. How many moles are in 1.204428 × 10²⁴ carbon atoms?

$\frac{1.204428 \times 10^{24}}{6.02214076 \times 10^{23}} = 2.0 \text{ mol}$

Example 3: moles to atoms. How many sodium atoms are in 0.25 moles of sodium chloride (NaCl)?

$0.25 \times 6.02214076 \times 10^{23} = 1.505535 \times 10^{23} \text{ atoms}$

This gives the number of sodium atoms only. An equal number of chlorine atoms is also present, since each NaCl unit has one atom of each.

Common mistakes in mole conversions

Confusing atoms with molecules. One mole of water (H₂O) contains 6.022141 × 10²³ molecules, but each molecule has 3 atoms, so it contains 1.806642 × 10²⁴ atoms in total.

Ignoring equation coefficients. In 2H₂ + O₂ → 2H₂O, the coefficient 2 in front of H₂ means 2 moles of hydrogen gas, or 2 × 6.02214076 × 10²³ = 1.204428 × 10²⁴ molecules, not one mole's worth.

Forgetting dissociation in solution. Dissolving 1 mole of NaCl in water produces 1 mole of Na⁺ ions and 1 mole of Cl⁻ ions, for 2 moles of dissolved particles in total.

Rounding Avogadro's number too early. Using 6 × 10²³ instead of 6.022 × 10²³ introduces error that grows through a multi-step calculation.

Double-counting scientific notation. A value written as 3.011e23 already includes the × 10²³. It should not be multiplied by Avogadro's number a second time.

History of the mole and Avogadro's number

In 1811, Amedeo Avogadro proposed that equal volumes of gas at the same temperature and pressure contain equal numbers of molecules, an idea that helped separate the concepts of atoms and molecules. He did not know the actual particle count.

In 1865, Johann Josef Loschmidt made the first estimate of the number of molecules in a cubic centimeter of gas. The number of molecules in a mole was later worked out from it. In the early 1900s, French physicist Jean Perrin confirmed by experiment that molecules are real, partly through his study of Brownian motion. He received the 1926 Nobel Prize in Physics for this work on "the discontinuous structure of matter".

The chemist Wilhelm Ostwald introduced the term "mole". The mole became an SI base unit in 1971, defined at the time as the amount of substance containing as many elementary entities as there are atoms in 12 grams of carbon-12. In 2019, the SI system was redefined so that Avogadro's number itself became the fixed reference point. One mole now contains exactly 6.022 140 76 × 10²³ elementary entities.

Frequently asked questions

What is a mole in chemistry? A mole is the SI unit for the amount of a substance. One mole always contains 6.022 140 76 × 10²³ elementary entities, whether they are atoms, molecules, or ions. Before 2019 the mole was defined so that one mole of carbon-12 atoms had a mass of exactly 12 grams. Since 2019 that mass is very close to 12 grams but no longer exact. Either way, chemists can count particles by weighing them.

How is the number of atoms calculated from moles? Multiply the number of moles by Avogadro's number. For example, 2 moles of carbon contains 2 × 6.02214076 × 10²³ = 1.204428 × 10²⁴ atoms.

How is the number of moles calculated from a particle count? Divide the particle count by Avogadro's number. For example, 3.011070 × 10²³ water molecules is 3.011070 × 10²³ ÷ 6.02214076 × 10²³ = 0.5 moles of water.

Is Avogadro's number the same for every substance? Yes. One mole of any substance contains 6.022 140 76 × 10²³ particles. What differs between substances is molar mass, the mass of one mole, measured in grams per mole.

What is the difference between counting atoms and counting molecules? The arithmetic is the same, but the particle type matters. One mole of water is 6.022141 × 10²³ molecules. Because each water molecule contains 3 atoms, the same sample contains 1.806642 × 10²⁴ atoms.

Why is Avogadro's number so large? Individual atoms have extremely little mass, so an enormous number of them is needed before the total mass becomes large enough to measure on a laboratory scale. Avogadro's number is the specific value that makes atomic mass units correspond to grams per mole.

References

1. National Institute of Standards and Technology (NIST). CODATA Value: Avogadro constant. Avogadro constant — N_A = 6.022 140 76 × 10²³ mol⁻¹, exact.
2. International Bureau of Weights and Measures (BIPM). SI base unit: mole. Mole — one mole contains exactly 6.022 140 76 × 10²³ elementary entities.
3. National Institute of Standards and Technology (NIST). Redefining the Mole. Redefining the mole — Avogadro's equal-volumes hypothesis; mole added to the SI in 1971; former carbon-12 definition.
4. Masaryk University, Loschmidt Laboratories. Physics Work of Josef Loschmidt. Physics work of Josef Loschmidt — Avogadro's 1811 hypothesis, his lack of an estimate, and Loschmidt's 1865 estimate.
5. Nobel Prize Outreach. Jean Baptiste Perrin – Facts. Perrin facts — 1926 Nobel Prize in Physics; Brownian motion experiments.