Avogadro constant

The Avogadro constant, symbol N_A, is the number of constituent particles in one mole of a substance. Since the revision of the International System of Units that took effect on May 20, 2019, its value is exact: 6.022 140 76 × 10²³ per mole.^[1] It is named for Amedeo Avogadro, who proposed in 1811 that equal volumes of gas at the same temperature and pressure hold equal numbers of molecules, and who never tried to count them. The name was given in 1909 by Jean Perrin, who in 1913 set side by side the values that thirteen different phenomena gave.^[2]

The constant converts between the two scales on which chemistry is done. A balance weighs grams of copper, and a mole of copper is the amount that holds 6.022 140 76 × 10²³ atoms. Dividing the molar mass by the Avogadro constant gives the mass of one atom.

Avogadro's hypothesis of 1811 was accepted slowly. Chemists did not generally take up the distinction between atoms and molecules on which it depends for decades, and Stanislao Cannizzaro defended it at the Karlsruhe congress of 1860.^[3][4] The hypothesis implies that a gram-molecule of any substance contains the same number of molecules, but it does not say what the number is. In his memoir of 1909, in a section headed "The atoms. Avogadro's constant", Perrin defined the gram-molecule as the mass of a substance that occupies as a gas the same volume as 2 grams of hydrogen, and wrote: "This invariable number N is a universal constant, which may appropriately be designated Avogadro's Constant."^[2] In Atoms (1913) the sentence has become "what is called Avogadro's Constant or Avogadro's Number".^[5] Both Avogadro and Josef Loschmidt, the first to estimate the number, were dead by then.

Loschmidt in 1865 used the kinetic theory of gases to estimate how many particles fill a given volume of gas.^[6] The number density of an ideal gas at standard conditions is still called the Loschmidt constant, and it is now exact, 2.651 645 8 × 10²⁵ per cubic meter to eight digits at 273.15 K and 100 kPa.^[7] Perrin recalled in his Nobel lecture that such calculations rested on simplifying assumptions and gave values of the Avogadro constant uncertain by "largely 100 per cent".^[8] The thirteen determinations in Les Atomes, by Brownian motion and by other phenomena, fell between 6.0 and 7.5 × 10²³ per mole, and Perrin adopted 6.8 × 10²³ for his own calculations of molecular magnitudes.^[5]

A second route went through electricity. Electrolysis fixes the charge carried by a mole of a monovalent ion, the faraday, and Perrin pointed out that dividing it by the Avogadro constant gives the charge of one "atom of electricity".^[5] Robert Millikan reversed the division. His oil-drop measurements, published in 1913 as "On the Elementary Electrical Charge and the Avogadro Constant", gave an electron charge of 4.774 × 10⁻¹⁰ electrostatic units, and from the faraday a value for the constant of 6.062 ± 0.012 × 10²³.^[9] The modern value is 0.7 percent lower.

The best measurements of the 21st century counted atoms. In the X-ray crystal-density method a crystal of silicon enriched in the isotope silicon-28 is machined into a sphere of about a kilogram, its volume is measured by interferometry, and X-ray interferometry gives the lattice parameter a of the cubic unit cell, which holds 8 atoms. The number of atoms is $N=8V_s/a^3$, corrected for impurities in the crystal. A collaboration called the International Avogadro Coordination reported a relative standard uncertainty of 3 parts in 10⁸ in 2011, and by 2015 reconditioned spheres had reached 2 parts in 10⁸.^[10][11][12]

From 1971 the mole was "the amount of substance of a system which contains as many elementary entities as there are atoms in 0.012 kilogram of carbon 12", which left the Avogadro constant to be measured. The 26th General Conference on Weights and Measures voted in 2018 to fix the number instead, and the new definition began on May 20, 2019: "One mole contains exactly 6.022 140 76 × 10²³ elementary entities." The 2019 text adds that the numerical value is "fixed with no uncertainty".^[12]

The change moved the uncertainty to the other side. The molar mass of carbon-12 is no longer exactly 12 grams per mole, and the molar mass constant now differs from 1 gram per mole by about one part in 10⁹.^[13] The molar Planck constant N_Ah, the product of the two fixed constants, is exact, 3.990 312 71 × 10⁻¹⁰ J Hz⁻¹ mol⁻¹ to nine digits, and so is the Faraday constant, 96,485.332 12 C/mol to ten digits.^[14][15]

In a nanopore measurement the Avogadro constant appears as a count of charges. A current of 1 pA is 6.24 million elementary charges per second, and the faraday divided by the current gives the time that a mole of charge would take to pass.

The first current is from the 1996 experiment in which single strands of RNA passed through alpha-hemolysin.^[16] The last comes from a Signal & Noise article of July 2026 on the company's live stream, which works out the current of such a pore by hand and puts it at about 19 billion ions per second.^[17][a]

At the small end the constant gives counts too. A zeptoliter (Zeptoliter), 10⁻²¹ L, of a 1 M solution holds about 602 ions of each sign.

• Les Atomes
• Jean Perrin
• Brownian motion
• Ludwig Boltzmann
• Molecular reality (history of science)