# Avogadro constant

The Avogadro constant, commonly denoted N$_A$ or L, is an SI defining constant with an exact value of 6.02214076×10$^{23}$ mol$^{−1}$ (reciprocal moles). It defines the number of constituent particles in one mole, where the particles in question can be either molecules, atoms, ions, ion pairs, or any other elementary entities. The number 6.02214076×10$^{23}$ is a dimensionless physical constant known as the Avogadro number, commonly denoted N$_0$ and often confused with the constant. Both are named after the Italian physicist and chemist Amedeo Avogadro.
The Avogadro constant is used as a normalization factor in relating the amount of substance n(X){\textstyle n(\mathrm {X} )}, in a sample of a substance $\mathrm {X}$, to the corresponding number of elementary entities $N(\mathrm {X} )$:
$n(\mathrm {X} )={\frac {N(\mathrm {X} )}{N_{\mathrm {A} }}}=N(\mathrm {X} )\,\left({\frac {1\;\mathrm {mol} }{N_{0}}}\right)$
The Avogadro constant N$_A$ is also the factor that converts the average mass ($m$) of one particle, in grams, to the molar mass ($M$) of the substance, in grams per mole (g/mol). That is, $M=m\cdot N_{\text{A}}$. Historically, this was precisely true, but since the 2019 revision of the SI, the relation is now merely approximate.
The constant N$_A$ also relates the molar volume (the volume per mole) of a substance to the average volume nominally occupied by one of its particles, when both are expressed in the same units of volume. For example, since the molar volume of water in ordinary conditions is about 18 mL/mol, the volume occupied by one molecule of water is about 18/(6.022×10$^{23}$) mL, or about 0.030 nm$^3$ (cubic nanometres). For a crystalline substance, N$_0$ relates the volume of a crystal with one mole worth of repeating unit cells, to the volume of a single cell (both in the same units).

## Definition

The Avogadro constant was historically derived from the old definition of the mole as the amount of substance in 12 grams of carbon-12 ($^{12}$C); or, equivalently, the number of daltons in a gram, where the dalton is defined as ⁠1/12⁠ of the mass of a $^{12}$C atom. By this old definition, the numerical value of the Avogadro constant in mol$^{−1}$ (the Avogadro number) was a physical constant that had to be determined experimentally.
The redefinition of the mole in 2019, as being the amount of substance containing exactly 6.02214076×10$^{23}$ particles, meant that the mass of 1 mole of a substance is now exactly the product of the Avogadro number and the average mass of its particles. The dalton, however, is still defined as ⁠1/12⁠ of the mass of a $^{12}$C atom, which must be determined experimentally and is known only with finite accuracy. The prior experiments that aimed to determine the Avogadro constant are now re-interpreted as measurements of the value in grams of the dalton.
By the old definition of mole, the numerical value of one mole of a substance, expressed in grams, was precisely equal to the average mass of one particle in daltons. With the new definition, this numerical equivalence is no longer exact, as it is affected by the uncertainty of the value of the dalton in SI units. However, it is still applicable for all practical purposes. For example, the average mass of one molecule of water is about 18.0153 daltons, and of one mole of water is about 18.0153 grams. Also, the Avogadro number is the approximate number of nucleons (protons and neutrons) in one gram of ordinary matter.
In older literature, the Avogadro number was also denoted N, although that conflicts with the symbol for number of particles in statistical mechanics.

## History
### Origin of the concept

The Avogadro constant is named after the Italian scientist Amedeo Avogadro (1776–1856), who, in 1811, first proposed that the volume of a gas (at a given pressure and temperature) is proportional to the number of atoms or molecules regardless of the nature of the gas.
Avogadro's hypothesis was popularized four years after his death by Stanislao Cannizzaro, who advocated Avogadro's work at the Karlsruhe Congress in 1860.
The name Avogadro's number was coined in 1909 by the physicist Jean Perrin, who defined it as the number of molecules in exactly 32 grams of oxygen gas. The goal of this definition was to make the mass of a mole of a substance, in grams, be numerically equal to the mass of one molecule relative to the mass of the hydrogen atom; which, because of the law of definite proportions, was the natural unit of atomic mass, and was assumed to be ⁠1/16⁠ of the atomic mass of oxygen.

### First measurements

The value of Avogadro's number (not yet known by that name) was first obtained indirectly by Josef Loschmidt in 1865, by estimating the number of particles in a given volume of gas. This value, the number density n$_0$ of particles in an ideal gas, is now called the Loschmidt constant in his honor, and is related to the Avogadro constant, N$_A$, by
    $n_{0}={\frac {p_{0}N_{\rm {A}}}{R\,T_{0}}},$
where p$_0$ is the pressure, R is the gas constant, and T$_0$ is the absolute temperature. Because of this work, the symbol L is sometimes used for the Avogadro constant, and, in German literature, that name may be used for both constants, distinguished only by the units of measurement. (However, N$_A$ should not be confused with the entirely different Loschmidt constant in English-language literature.)
Perrin himself determined the Avogadro number, which he called "Avogadro's constant" (constante d'Avogadro), by several different experimental methods. He was awarded the 1926 Nobel Prize in Physics, largely for this work.
The electric charge per mole of electrons is a constant called the Faraday constant and has been known since 1834, when Michael Faraday published his works on electrolysis. In 1910, Robert Millikan with the help of Harvey Fletcher obtained the first measurement of the charge on an electron. Dividing the charge on a mole of electrons by the charge on a single electron provided a more accurate estimate of the Avogadro number.

### SI definition of 1971

In 1971, in its 14th conference, the International Bureau of Weights and Measures (BIPM) decided to regard the amount of substance as an independent dimension of measurement, with the mole as its base unit in the International System of Units (SI). Specifically, the mole was defined as the amount of a substance that contains as many elementary entities as there are atoms in 12 grams (0.012 kilograms) of carbon-12 ($^{12}$C). Thus, in particular, an amount of one mole of carbon 12 had a corresponding mass that was exactly 12 grams of that element.
By this definition, one mole of any substance contained exactly as many elementary entities as one mole of any other substance. However, this number N$_0$ was a physical constant that had to be experimentally determined since it depended on the mass (in grams) of one atom of $^{12}$C, and therefore, it was known only to a limited number of decimal digits. The common rule of thumb that "one gram of matter contains N$_0$ nucleons" was exact for carbon-12, but slightly inexact for other elements and isotopes.
In the same conference, the BIPM also named N$_A$ (the factor that related the amount of a substance to the corresponding number of particles) the "Avogadro constant". However, the term "Avogadro number" continued to be used, especially in introductory works. As a consequence of this definition, N$_A$ was not a pure number, but had the metric dimension of reciprocal of amount of substance (mol$^{−1}$).

### SI redefinition of 2019

In its 26th Conference, the BIPM adopted a different approach: effective 20 May 2019, it defined the Avogadro constant N$_A$ as the exact value 6.02214076×10$^{23}$ mol$^{−1}$, thus redefining the mole as exactly 6.02214076×10$^{23}$ constituent particles of the substance under consideration. One consequence of this change is that the mass of a mole of $^{12}$C atoms is no longer exactly 0.012 kg. On the other hand, the dalton (a.k.a. universal atomic mass unit) remains unchanged as ⁠1/12⁠ of the mass of $^{12}$C. Thus, the molar mass constant remains very close to but no longer exactly equal to 1 g/mol, although the difference (4.5×10$^{−10}$ in relative terms, as of March 2019) is insignificant for all practical purposes.

## Connection to other constants

The Avogadro constant N$_A$ is related to other physical constants and properties.
* It relates the molar gas constant R and the Boltzmann constant k$_B$, which in the SI is defined to be exactly 1.380649×10$^{−23}$ J/K:
    R = k$_B$ N$_A$ = 8.314462618... J⋅mol$^{−1}$⋅K$^{−1}$
* It relates the Faraday constant F and the elementary charge e, which in the SI is defined as exactly 1.602176634×10$^{−19}$ coulombs:
    F = e N$_A$ = 9.648533212...×10$^4$ C⋅mol$^{−1}$
* It relates the molar mass constant M$_u$ and the atomic mass constant m$_u$ currently 1.66053906892(52)×10$^{−27}$ kg
    M$_u$ = m$_u$ N$_A$ = 1.00000000105(31)×10$^{−3}$ kg⋅mol$^{−1}$