Avogadro's Number: What It Is, the Mole and Molar Mass

Nobody can count atoms one at a time, so chemistry counts them by weight. You put a sample on a scale, and the mass tells you how many particles it holds. The link between the two is the mole, defined by a number that carries the name of Amedeo Avogadro.

What a mole is

In the International System of Units (SI), the mole, symbol mol, is the unit of amount of substance. Since 20 May 2019 its definition has been a count: one mole contains exactly 6.022 140 76 × 10²³ elementary entities. The BIPM, the international bureau that maintains the SI, writes amount of substance as n and uses it to say how many entities of a stated kind a system contains.

What gets counted can be any particle, from an electron to an atom, ion or molecule, or a well-defined group of particles. Since 1971, anyone using the mole has had to say what is being counted, and the BIPM suggests doing so with the chemical formula. A mole of O₂ is 6.022 140 76 × 10²³ oxygen molecules, which means twice that many oxygen atoms.

In 2018 IUPAC, the international union of chemists, stressed that amount of substance is a count of entities, and a sample’s mass is a different quantity. Peter W. Atkins, the first chair of its committee on chemistry education, said he was puzzled that the mole has a reputation as a hard topic. He welcomed the new definition because it goes straight to what a mole means.

Avogadro’s number vs the Avogadro constant

The same value goes by two names. According to the BIPM, the Avogadro constant, symbol N_A, is exactly 6.022 140 76 × 10²³ mol⁻¹, read as “per mole”. Take the number 6.022 140 76 × 10²³ on its own, as a pure number, and you have Avogadro’s number.

The constant is a proportionality factor between two quantities: amount of substance, in moles, and the number of entities, which is a pure number. As a formula, N = n × N_A.

	Mole	Avogadro’s number	Avogadro constant
What it is	unit of amount of substance	pure number	constant with a unit
Symbol and unit	mol	no unit	NA, in mol⁻¹
Value	none of its own, it is the unit	6.022 140 76 × 10²³	6.022 140 76 × 10²³ mol⁻¹
What it does	expresses “how much substance”	says how many entities are in a mole	converts moles into a number of entities

Practical example: how many molecules are there in 0.5 mol of water? Multiply by the constant: 0.5 mol × 6.022 140 76 × 10²³ mol⁻¹ ≈ 3.011 × 10²³ molecules. The “mol” cancels out and a pure number is left.

How to calculate molar mass

Going from grams to particles takes molar mass, M: the mass of one mole of a substance. The BIPM gives the molar mass of an atom or molecule as the Avogadro constant multiplied by the mass of a single entity. That mass can in turn be written as the relative atomic mass A_r, a pure number, times the atomic mass constant m_u, which is one twelfth of the mass of a carbon-12 atom.

The dalton (Da) and the unified atomic mass unit (u) are synonyms, both defined as one twelfth of the mass of an unbound carbon-12 atom, at rest and in its ground state. The CODATA 2022 recommended value is 1.660 539 068 92 × 10⁻²⁷ kg, and since it is measured, it comes with an uncertainty.

Multiply the two and the bridge appears: 6.022 140 76 × 10²³ mol⁻¹ times 1.660 539 068 92 × 10⁻²⁷ kg gives 1 g/mol, give or take digits far to the right of the decimal point. That is why, in classroom calculations, molar mass in g/mol has the same numerical value as relative atomic mass, and a molecule’s molar mass is the sum of its atoms’ values.

Quantity	Symbol	Unit	What it tells you
Relative atomic mass (atomic weight)	Ar	none	how many times heavier an atom is than one twelfth of carbon-12
Mass of one entity	m	Da (or u), or kg	the mass of a single atom or molecule
Molar mass	M	g/mol	the mass of 6.022 140 76 × 10²³ entities

The atomic weights to use are the standard values from CIAAW, the IUPAC commission that maintains atomic weights for the elements of the periodic table. Its abridged table is based on the Atomic Weights 2021 report, and since 2013 the commission has recommended rounding to five figures. The CIAAW page on carbon notes that carbon-12 has anchored the atomic mass unit scale since 1960 and underlies every atomic weight.

Element	Abridged standard atomic weight (CIAAW)
Hydrogen (H)	1.0080 ± 0.0002
Carbon (C)	12.011 ± 0.002
Oxygen (O)	15.999 ± 0.001
Sodium (Na)	22.990 ± 0.001
Chlorine (Cl)	35.45 ± 0.01

Practical example: the molar mass of water, H₂O. There are two hydrogen atoms and one oxygen: 2 × 1.0080 + 15.999 = 18.015. Our calculated molar mass is about 18.015 g/mol.

Practical example: sodium chloride, NaCl. Adding 22.990 and 35.45 gives 58.44, so about 58.44 g/mol. Chlorine is given only to two decimal places, and the result stops there too.

The formula n = m/M, with worked examples

Nearly every mole exercise turns on one relationship: amount of substance equals mass divided by molar mass, n = m/M, with m in grams and M in g/mol. Rearranging it, as you would any first-degree equation, gives m = n × M and M = m/n. One more multiplication, by the Avogadro constant, takes you to the number of particles.

Practical example: how many moles are there in 36 g of water? n = 36 g ÷ 18.015 g/mol ≈ 2.0 mol. The molecules come to about 2.0 × 6.022 × 10²³ ≈ 1.2 × 10²⁴.

Practical example: what is the mass of 0.25 mol of sodium chloride? m = 0.25 mol × 58.44 g/mol ≈ 14.6 g.

NIST, the US national metrology institute, uses water to give a sense of scale. Half a mole of oxygen (O₂) and one mole of hydrogen (H₂) make one mole of water, about 18 grams. Those 18 grams hold roughly 600 billion trillion molecules, a number on the order of 6 × 10²³.

From Avogadro to Perrin: a history of the number

Amedeo Avogadro lived from 1776 to 1856. He trained as a lawyer before turning to chemistry, where his name became attached to a hypothesis about gases: equal volumes at the same temperature and pressure contain the same number of molecules. That is Avogadro’s law, a separate idea from the number. In his day not everyone accepted that atoms existed, and he spoke of “elementary molecules”. NIST says it took about fifty years, by which time he had died, for others to see the value of his ideas. The name “Avogadro’s number” arrived in the early 20th century as a tribute, for the number of molecules in a gram-molecule of oxygen. The word mole dates from the same period: it appeared around 1900 as the German shortening Mol, from Molekül.

Jean Perrin retraced the hunt for that number in his Nobel lecture in physics on 11 December 1926. He defined the gram-molecule against 32 grams of oxygen and called the number of molecules it contains Avogadro’s number, N. Ratios between the weights of atoms and molecules were already known; with N in hand, their absolute masses would follow.

Method cited by Perrin (1926)	Values of Avogadro’s number
Kinetic theory of gases, early estimates	40 × 10²² to 120 × 10²², uncertainty of easily 100% by Perrin’s account
Emulsions of suspended grains	60 × 10²² to 68 × 10²²
Rotation of the grains (rotational Brownian motion)	65 × 10²²
Radioactivity, counting helium atoms	62 × 10²² to 70 × 10²²

His most careful emulsion measurement gave 68 × 10²², or 6.8 × 10²³. The value fixed in 2019 is 6.022 140 76 × 10²³, the same order of magnitude. In the same lecture Perrin argued that the reality of atoms and molecules, still doubted twenty years earlier, could now be treated as a proven principle.

The 2019 definition and the carbon-12 one

The mole joined the SI in 1971 as the seventh base unit, by decision of the 14th General Conference on Weights and Measures (CGPM), and people began to speak of the Avogadro constant. The 1971 definition counted as many entities as there are atoms in 0.012 kg, or 12 grams, of carbon-12, the most common form of carbon. That made the mole depend on the kilogram, which was then the mass of a platinum-iridium cylinder kept near Paris.

IUPAC’s project on a new definition began in 2013, to work out how redefining the kilogram would affect amount of substance. The recommendation, signed by Roberto Marquardt and six co-authors including Jürgen Stohner, appeared in the January 2018 issue of Pure and Applied Chemistry. Its proposed wording is the one later adopted in the SI, and the mole changed along with the kilogram, the ampere and the kelvin.

On 16 November 2018, at Versailles, representatives of more than 50 countries voted for the redefinition, which NIST describes as the outcome of about twenty years of international work. The resolution repealed the 1971 definition, and the new one took effect on 20 May 2019, World Metrology Day, the anniversary of the 1875 Metre Convention.

One of the methods that led there counts the atoms in a silicon sphere using X-rays. Starting in the early 2000s, silicon was enriched in centrifuges from its natural 92% silicon-28 to 99.995%, and Germany’s metrology institute, PTB, made two one-kilogram spheres from it. From the sphere’s mass and volume, plus the spacing between its atoms and the molar mass of silicon, you can work out how many atoms it holds. The method measures the Avogadro constant, and through the Planck constant it also fed into the new kilogram.

	1971 definition	Definition in force since 20 May 2019
Reference	atoms in 0.012 kg of carbon-12	6.022 140 76 × 10²³ entities, a fixed number
Tied to the kilogram	yes	no
Avogadro constant	measured	exact
Molar mass of carbon-12	exactly 0.012 kg/mol	measured

The constant’s value was chosen so that, at the changeover, the molar mass of carbon-12 stayed at 0.012 kg/mol with a relative uncertainty of 4.5 × 10⁻¹⁰.