Avogadro Constant

The Avogadro constant, symbol $N_{\mathrm A}$, relates the number of specified entities in a sample to its amount of substance, measured in moles. Its exact value is $6.02214076\times10^{23}\ \mathrm{mol^{-1}}$. It provides the connection between counting individual particles and describing laboratory-scale quantities of matter. Since May 20, 2019, it has been one of the seven defining constants of the International System of Units (SI). (goldbook.iupac.org)

Definition and terminology

For a system containing $N$ specified entities and having amount of substance $n$,

The entity count $N$ is dimensionless, whereas $n$ has the SI unit mol. Consequently, the Avogadro constant has the unit reciprocal mole, $\mathrm{mol^{-1}}$; it is not itself dimensionless. The term Avogadro number strictly denotes its numerical value when expressed in that unit: the dimensionless number $6.02214076\times10^{23}$. Although the two expressions are often used interchangeably, they distinguish a physical quantity from its numerical value. (goldbook.iupac.org)

The entities must be specified. They may be atoms, molecules, ions, electrons, other particles, or defined groups of particles. “Elementary entity” in this context does not mean an indivisible elementary particle: a molecule qualifies even though it contains atoms. One mole contains exactly the same number of specified entities regardless of their identity, but the entities need not have the same mass. (nist.gov)

Historical development

The constant is named after Amedeo Avogadro, whose 1811 paper proposed that equal volumes of gases at the same temperature and pressure contain equal numbers of molecules. This proposition, now called Avogadro’s law, concerned the relative numbers of molecules in gases, rather than a determination of the modern numerical constant. Its name therefore commemorates the molecular hypothesis underlying the concept. (nist.gov)

Experimental determinations later connected molecular counts with measurable bulk properties. Jean Perrin obtained estimates through the equilibrium distribution and Brownian motion of particles suspended in liquids. His measurements tested the theory developed by Albert Einstein and Marian Smoluchowski and supported the physical existence of molecules. Perrin described several independent determinations in his 1926 Nobel lecture; that year he received the Nobel Prize in Physics for research on the discontinuous structure of matter, particularly sedimentation equilibrium. (nobelprize.org)

The 2019 SI redefinition

Before 2019, the mole was defined through the number of atoms in 0.012 kilograms of carbon-12. This made the molar mass of carbon-12 exactly $0.012\ \mathrm{kg\,mol^{-1}}$, while the Avogadro constant had to be determined experimentally. (nist.gov)

The General Conference on Weights and Measures adopted the revised SI in November 2018, with the new definitions taking effect on May 20, 2019. The logical relationship was reversed: the Avogadro constant became exact, and the mole became the amount containing exactly $6.02214076\times10^{23}$ specified entities. The chosen value preserved continuity with the previous definition within the uncertainty of the measurements available at adoption. (bipm.org)

Carbon-12 remains important in atomic-mass relationships, but its molar mass is no longer exact in SI units. Thus, stating that precisely 12 grams of carbon-12 contains exactly one mole describes the former definition, not the present one. Fixing $N_{\mathrm A}$ also does not make practical measurements exact: preparing or characterizing a sample still involves uncertainties in mass, purity, composition, and measurement procedures. (nist.gov)

Experimental determination

A major route to high-precision determination before the redefinition was the International Avogadro Project. Researchers used nearly spherical single crystals highly enriched in the isotope silicon-28. A regular crystal structure allows the number of atoms to be inferred from bulk dimensions and atomic spacing, rather than counted individually. (nist.gov)

The method combined measurements of sphere mass and volume, crystal-lattice dimensions, and molar mass. Optical interferometry measured sphere dimensions, while X-ray methods determined lattice spacing. Mass spectrometry established isotopic composition, an essential input to the molar mass. Surface contamination and imperfections also required characterization. These coordinated measurements supplied evidence for selecting the exact value used in the revised SI. After redefinition, such experiments test measurement consistency and realize units rather than refine the defined value of $N_{\mathrm A}$. (nist.gov)

Chemical calculations and related constants

The relationship $N=N_{\mathrm A}n$ converts amount of substance into an entity count. As an illustrative calculation, $0.500\ \mathrm{mol}$ corresponds to $3.01107038\times10^{23}$ specified entities before rounding to reflect measurement precision. The constant introduces no uncertainty into this conversion; any uncertainty comes from the measured amount. (goldbook.iupac.org)

For identical entities of mass $m_{\text{entity}}$, their molar mass satisfies

This connects microscopic particle masses with macroscopic sample masses. Molar mass is mass divided by amount of substance, not the number of entities in a mole. (nist.gov)

The Avogadro constant also connects particle-scale and molar constants:

where $R$ is the molar gas constant, $k_{\mathrm B}$ the Boltzmann constant, $F$ the Faraday constant, and $e$ the elementary charge. These relations translate thermal quantities and electric charge from individual entities to molar quantities. Because all three defining constants on the right-hand sides are exact in the revised SI, both products are exact as well. (goldbook.iupac.org)