Avogadro's Number Calculator: Moles to Particles
Convert between moles and number of particles (atoms, molecules, or ions) using Avogadro's constant, 6.022×10²³ per mole.
Reviewed by Manoj Kumar, Mathematics Educator
Formula
N = n * NA = (m / M) * 6.022e23
The number of particles N equals moles n times Avogadro's number NA (6.022 x 10^23). Moles can be found from mass divided by molar mass (n = m/M). This allows conversion between macroscopic mass and the number of individual atoms or molecules.
Worked Examples
Example 1: Water Molecules
Problem:How many molecules are in 36 g of water (H2O, M = 18.015 g/mol)?
Solution:n = m/M = 36/18.015 = 1.9992 mol\nN = n * NA = 1.9992 * 6.022e23\nN = 1.204e24 molecules
Result:1.204 x 10^24 water molecules
Example 2: Gold Atoms
Problem:How many atoms are in a 10 g gold nugget (Au, M = 196.97 g/mol)?
Solution:n = 10/196.97 = 0.05077 mol\nN = 0.05077 * 6.022e23\nN = 3.057e22 atoms
Result:3.057 x 10^22 gold atoms
Frequently Asked Questions
What is Avogadro's number?
Avogadro's number (NA) is exactly 6.02214076 x 10^23, representing the number of entities (atoms, molecules, ions, or other particles) in one mole of a substance. It was redefined in 2019 by the International Bureau of Weights and Measures as an exact value rather than an experimentally determined one. The number is named after Italian scientist Amedeo Avogadro, who in 1811 proposed that equal volumes of gases at the same temperature and pressure contain equal numbers of molecules, though the actual value was determined much later by Jean Perrin.
How do you convert between grams, moles, and particles?
To convert grams to moles, divide the mass by the molar mass: n = m/M. To convert moles to particles, multiply by Avogadro's number: N = n * NA. To go from particles to grams, first divide by NA to get moles, then multiply by molar mass. For example, 18 g of water (M = 18.015 g/mol) contains 1 mole, which equals 6.022 x 10^23 water molecules or about 1.806 x 10^24 individual atoms (since each water molecule has 3 atoms). These conversions are fundamental to all stoichiometric calculations.
How was Avogadro number determined experimentally?
Several methods have been used to determine Avogadro's number. Jean Perrin earned the 1926 Nobel Prize for measuring it using Brownian motion experiments. Other methods include X-ray crystallography (measuring crystal lattice spacing and density), electrolysis (Faraday's laws relating charge to moles of substance deposited), and the oil drop experiment (Millikan's charge measurement combined with Faraday's constant). The most precise modern method uses silicon sphere counting, where a near-perfect silicon-28 sphere is measured to determine the number of atoms from its macroscopic dimensions and density.
Why is 6.022 x 10^23 such an enormous, hard-to-grasp number?
Avogadro's number is scaled to match the size of individual atoms, which are astonishingly small — it takes roughly 10^23 atoms to make a visible, weighable amount of matter. A common analogy: a mole of standard drinking straws would cover the entire surface of the Earth to a depth of about 6 miles. Another: if you had a mole of unpopped popcorn kernels, they would blanket the entire United States roughly 9 miles deep. The number is large specifically because atoms are that much smaller than the everyday objects we're used to counting.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy