Avogadro's Number Calculator - Mole to Molecule Converter

What is Avogadro's number?

Avogadro's number is the count of particles in one mole of a substance: exactly 6.02214076 × 10²³. A mole is a unit chemists use to count atoms, molecules, or ions, the same way "dozen" is a unit for counting twelve of something. This calculator converts between moles and the number of molecules using that constant.

Avogadro's number formula

The number of molecules equals the number of moles times Avogadro's constant:

$N = n \times N_A$

• $N$ is the number of molecules (or atoms, ions, or other particles)
• $n$ is the number of moles
• $N_A$ is Avogadro's constant, 6.02214076 × 10²³ per mole

To go the other way, divide the number of molecules by the constant:

$n = N \div N_A$

Since 2019, Avogadro's constant has had this exact value by definition. It is one of the seven defining constants of the International System of Units (SI), so it carries no measurement uncertainty.

How to calculate molecules from moles

1. Take the number of moles.
2. Multiply it by 6.02214076 × 10²³.
3. The result is the number of molecules.

Worked example

Start with 2 moles of a substance.

$N = 2 \times 6.02214076 \times 10^{23} = 1.204428152 \times 10^{24}$

So 2 moles contain about 1,204,428,152,000,000,000,000,000 molecules, usually written as 1.204428 × 10²⁴.

How to calculate moles from molecules

1. Take the number of molecules.
2. Divide it by 6.02214076 × 10²³.
3. The result is the number of moles.

Worked example

Start with 3.01107038 × 10²³ molecules.

$n = 3.01107038 \times 10^{23} \div 6.02214076 \times 10^{23} = 0.5$

So 3.01107038 × 10²³ molecules make up half a mole.

Using the calculator

Enter a value in the moles field to get the number of molecules, or enter a value in the molecules field to get the number of moles. If both fields have a value, the calculator uses the moles field. A substance name can be added for reference; it is included in the copied result but does not affect the calculation. Negative numbers are not accepted, because a mole count cannot be less than zero. Zero is a valid input and produces a result of zero.

Where Avogadro's number is used

Chemists use Avogadro's number to move between two scales: the moles used in a balanced chemical equation and the actual number of atoms or molecules reacting. It appears across several areas of science.

• Stoichiometry: converting between moles of reactants and products in a chemical equation.
• Gas laws: relating the number of gas molecules to volume, pressure, and temperature.
• Solution chemistry: finding how many molecules are dissolved in a solution of known molarity.
• Molar mass: converting between the mass of a sample and its number of moles, which then gives a molecule count.
• Biochemistry: estimating how many protein, DNA, or enzyme molecules are present in a sample.

History of Avogadro's number

The constant is named after the Italian scientist Amedeo Avogadro (1776–1856), though he never measured it. In 1811 he proposed that equal volumes of gas, at the same temperature and pressure, contain equal numbers of molecules. This idea is now called Avogadro's law.

The Austrian physicist Johann Josef Loschmidt made the first estimate of the number of molecules in a given volume of gas in 1865. The name "Avogadro's number" came later, coined by the French physicist Jean Perrin in 1909 while he was studying Brownian motion, the random jittering of small particles suspended in a fluid. Perrin's measurements of the constant contributed to his 1926 Nobel Prize in Physics.

In 2019, an international redefinition of the SI base units fixed Avogadro's constant at exactly 6.02214076 × 10²³ per mole, ending centuries of increasingly precise experimental measurement.

Frequently asked questions

What is Avogadro's number? It is the number of particles, such as atoms or molecules, in one mole of a substance: 6.02214076 × 10²³.

How do you convert moles to molecules? Multiply the number of moles by 6.02214076 × 10²³. For example, 2 moles gives 2 × 6.02214076 × 10²³ = 1.204428 × 10²⁴ molecules.

How do you convert molecules to moles? Divide the number of molecules by 6.02214076 × 10²³. For example, 3.01107038 × 10²³ molecules divided by the constant gives 0.5 mole.

Does Avogadro's number apply only to molecules? No. It applies to any kind of particle, including atoms, ions, and formula units. One mole of any of these contains 6.02214076 × 10²³ particles.

What is the difference between a mole and Avogadro's number? A mole is a unit, similar to a dozen. Avogadro's number is the count of particles that make up one mole, similar to how twelve makes up one dozen.

Why is Avogadro's number so large? Atoms and molecules are extremely small. A single water molecule is about 0.3 nanometers across. It takes a huge number of them to add up to an amount of matter that can be measured on a lab scale, such as a gram.

References

1. International Bureau of Weights and Measures (BIPM), "The International System of Units (SI)," 9th edition, 2019.
2. IUPAC, Compendium of Chemical Terminology (the "Gold Book"), 2nd ed., 1997.
3. Perrin, J. (1909). "Mouvement brownien et réalité moléculaire." Annales de Chimie et de Physique, 8th series, 18: 1–114.