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[[File:Amadeo Avogadro.png|right|thumb|[[Amedeo Avogadro]]]]

The '''Avogadro number''', sometimes denoted ''N''<ref name=pauling/><ref name=mcgraw/> or ''N''<sub>0</sub>,<ref name=feynman/><ref name=born/> is the number of [[particle number|constituent particle]]s (usually [[molecule]]s, [[atom]]s or [[ion]]s) that are contained in one [[mole (unit)|mole]], the [[International System of Units|international (SI)]] unit of [[amount of substance]]: by definition, exactly {{physconst|NA|unit=no|ref=no}}, and it is dimensionless.<ref name=cgpm26/><ref name=bipm9th/>  It is named after the scientist [[Amedeo Avogadro]] (1776–1856).<ref name=perrin1909/>

The '''Avogadro constant''', usually denoted by ''N''<sub>A</sub><ref name=bipm9th/> or ''L''<ref name=iupac1996/> is the factor that, multiplied by the amount of substance in a sample, measured in moles, gives the number of constituent particles in that sample.  Its numerical value is defined as being the Avogadro number, and its unit is the reciprocal of mole; that is, {{physconst|NA|symbol=yes|after=.}}<ref name=bipm9th/><ref name=cgpm26/><ref name=goldbook/><ref name=ciaaw/>

The value of the Avogadro constant was chosen so that the mass of one mole of a [[chemical compound]], in [[gram]]s, is numerically equal (for all practical purposes) to the average mass of one molecule of the compound, in [[atomic mass unit|daltons (universal atomic mass units)]]; one dalton being {{sfrac|1|12}} of the mass of one [[carbon-12]] atom, which is approximately the mass of one [[nucleon]] ([[proton]] or [[neutron]]).  For example, the average mass of one molecule of [[water]] is about 18.0153 daltons, and one mole of water (''N'' molecules) is about 18.0153 grams. Thus, the Avogadro constant ''N''<sub>A</sub> is the [[proportionality factor]] that relates the [[molar mass]] of a substance to the average [[mass]] of one molecule; and the Avogadro number is also the approximate number of nucleons in one gram of ordinary [[matter]].<ref name=okun/>

The Avogadro constant also relates the [[molar volume]] of a substance to the average volume nominally occupied by one of its particle, when both are expressed in the same units of volume.  For example, since the molar volume of water in ordinary conditions is about 18 [[millilitre|mL]]/mol, the volume occupied by one molecule of water is about {{sfrac|18|6.022}}{{e|-23}}&nbsp;mL, or about 30&nbsp;[[angstrom (unit)|Å]]<sup>3</sup> (cubic [[angstrom (unit)|angstroms]]).  For a [[crystal]]ine substance, it similarly relates its molar volume (in mL/mol), the volume of the repeating [[crystal structure|unit cell]] of the crystals (in mL), and the number of molecules in that cell.

The Avogadro number (or constant) has been defined in many different ways through its long history. Its approximate value was first determined, indirectly, by [[Johann Josef Loschmidt|Josef Loschmidt]] in 1865.<ref name="losch1865"/> (Avogadro's number is closely related to the [[Loschmidt constant]], and the two concepts are sometimes confused.) It was initially defined by [[Jean Baptiste Perrin|Jean Perrin]] as the number of atoms in 16 grams of [[oxygen]].<ref name=perrin1909/>  It was later redefined in the 14th conference of the [[International Bureau of Weights and Measures]] (BIPM) as the number of atoms in 12 grams of the isotope [[carbon-12]] (<sup>12</sup>C).<ref name=bipm8th/>  In each case, the mole was defined as the quantity of a substance that contained the same number of atoms as those reference samples.  In particular, when carbon-12 was the reference, one mole of carbon-12 was exactly 12 grams of the element.

These definitions meant that the value of the Avogadro number depended on the experimentally determined value of the mass (in grams) of one atom of those elements, and therefore it was known only to a limited number of decimal digits.  However, in its 26th Conference, the BIPM adopted a different approach: effective 20 May 2019, it defined the Avogadro number as the exact integer {{nowrap|1=''N'' = {{physconst|NA|unit=no|ref=no}}}}, and redefined the mole as ''N'' constituent particles of the substance in consideration.  Under the new definition, the mass of one mole of any substance (including hydrogen, carbon-12, and oxygen-16) is ''N'' times the average mass of one of its constituent particles—a physical quantity whose precise value has to be determined experimentally for each substance.

==History==

===Origin of the concept===
[[File:Jean Perrin 1926.jpg|right|thumb|Jean Perrin in 1926]]
The Avogadro constant is named after the Italian scientist [[Amedeo Avogadro]], who, in 1811, first proposed that the volume of a gas (at a given pressure and temperature) is proportional to the number of [[atom]]s or [[molecule]]s regardless of the nature of the gas.<ref name=avog1811/>

The name ''Avogadro's number'' was coined in 1909 by the physicist [[Jean Baptiste Perrin|Jean Perrin]], who defined it as the number of molecules in exactly 32 grams of [[oxygen]].<ref name=perrin1909/>  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===
[[File:Johann Josef Loschmidt portrait plaque.jpg|right|thumb|Josef Loschmidt]]
The value of Avogadro's number (still not known by that name) was first obtained indirectly by [[Johann Josef Loschmidt|Josef Loschmidt]] in 1865, by estimating the number of particles in a given volume of gas.<ref name=losch1865/> This value, the [[number density]] ''n''<sub>0</sub> of particles in an [[ideal gas]], is now called the [[Loschmidt constant]] in his honor, and is related to the Avogadro constant, ''N''<sub>A</sub>, by
:<math>n_0 = \frac{p_0N_{\rm A}}{R\,T_0} ,</math>
where ''p''<sub>0</sub> is the [[pressure]], ''R'' is the [[gas constant]], and ''T''<sub>0</sub> is the [[absolute temperature]].  Because of this work, the symbol ''L'' is sometimes used for the Avogadro constant,<ref name=bipm1971/> and, in [[German language|German]] literature, that name may be used for both constants, distinguished only by the [[units of measurement]].<ref name=virgo1933/> (However, ''N''<sub>A</sub> should not to be confused with the entirely different [[Loschmidt constant]] in the English language literature.)

Perrin himself determined Avogadro's number by several different experimental methods.  He was awarded the 1926 [[Nobel Prize in Physics]], largely for this work.<ref name=oseen1926/>

The electric charge per [[Mole (unit)|mole]] of electrons is a constant called the [[Faraday constant]] and had been known since 1834 when [[Michael Faraday]] published [[Faraday's laws of electrolysis|his works on electrolysis]]. In 1910, [[Robert Millikan]] obtained the first measurement of the [[elementary charge|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.<ref name=ebrit1974/>

===SI definition of 1971===
In 1971 the International Bureau of Weights and Measures (IBPM) decided to regard the [[amount of substance]] as an independent [[dimensional analysis|dimension of measurement]], with the mole as its [[SI base unit|base unit]] in the [[International System of Units]] (SI).<ref name=bipm1971/>  Specifically, the mole was defined as an amount of a substance that contains as many elementary entities as there are atoms in 0.012 kilogram of [[carbon-12]].

By this definition, the common rule of thumb "one gram of matter contains ''N''<sub>0</sub> nucleons" was exact for carbon-12, but slightly inexact for other elements and isotopes.  On the other hand, one mole of any substance contained exactly as many molecules as one mole of any other substance.

As a consequence of this definition, in the SI system the Avogadro constant ''N''<sub>A</sub> had the [[dimensional analysis|dimensionality]] of reciprocal of amount of substance rather than of a pure number, and had the approximate value {{val|6.02|e=23}} with units of mol<sup>−1</sup>.<ref name=bipm1971/>  By this definition, the value of ''N''<sub>A</sub> inherently had to be determined experimentally.

The IBPM also named ''N''<sub>A</sub> the "Avogadro ''constant''", but the term "Avogadro number" continued to be used especially in introductory works.<ref name=kotz2008/>

===SI redefinition of 2019===
{{main|2019 redefinition of SI base units}}
In 2017, the IBPM decided to change the definitions of mole and amount of substance.<ref name=cipm106/><ref name=cgpm26/>  The mole was redefined as being the amount of substance containing exactly {{val|6.02214076|e=23}} elementary entities. One consequence of this change is that the mass of a mole of <sup>12</sup>C atoms is no longer exactly 0.012&nbsp;kg. On the other hand, the dalton ({{aka}} universal atomic mass unit) remains unchanged as 1/12 of the mass of <sup>12</sup>C.<ref name=pave2018/><ref name=lehm2016/>  Thus, the [[molar mass constant]] is no longer exactly 1&nbsp;g/mol, although the difference ({{val|3.5|e=-10}} in relative terms, as of March 2019) is insignificant for practical purposes.<ref name=cgpm26/>

==The Avogadro constant in other unit systems==

In other measurement systems where the unit of [[amount of substance]] is not the SI mole, the Avogadro constant ''N''<sub>A</sub> is taken to mean the number of particles in said unit, and will have a different value.

For example, if quantities of a substance are measured in [[pound-mole]] (lb-mol), then the Avogadro constant ''N''<sub>A</sub> is {{val|2.73159734|(12)|e=26}} lb-mol<sup>−1</sup>.  If the [[ounce-mole]] (oz-mol) is used instead, ''N''<sub>A</sub> is {{val|1.707248434|(77)|e=25}} oz-mol<sup>−1</sup>.{{citation needed|date=May 2019}}  However, these units are hardly used nowadays.

==Connection to other constants==

The Avogadro constant, ''N''<sub>A</sub> is related to other physical constants and properties.

* It relates the [[molar gas constant]] ''R'' and the [[Boltzmann constant]] ''k''<sub>B</sub>, which in the SI (since 20 May 2019) is defined to be exactly {{val|1.380649|e=−23|u=J/K}}:<ref name=cgpm26/>
*:<math>R = k_{\text{B}} N_{\text{A}}</math> = {{val|8.31446261815324|u=J⋅K<sup>−1</sup>⋅mol<sup>−1</sup>}}

* It relates the [[Faraday constant]] ''F'' and the [[elementary charge]] ''e'', which in the SI (since 20 May 2019) is defined as exactly {{val|1.602176634|e=−19|u=[[coulomb]]s}}:<ref name=cgpm26/>
*:<math>F = e N_{\text{A}}</math> = {{val|96485.3321233100184|u=C/mol}}

* It relates the [[molar mass constant]], ''M''<sub>u</sub> and the [[atomic mass constant]] ''m''<sub>u</sub>, currently {{physconst|mu|after=:}}
*:<math>M_{\text{u}} = m_{\text{u}} N_{\text{A}}</math> = {{physconst|Mu}}.

==See also==
* [[Mole Day]]
* [[CODATA 2018]]

==References==
<references>

<ref name="perrin1909">{{cite journal|first = Jean|last = Perrin|authorlink = Jean Baptiste Perrin|title = Mouvement brownien et réalité moléculaire|journal = [[Annales de Chimie et de Physique]] |series=8<sup>e</sup> Série|volume = 18|pages = 1–114|year = 1909}} [http://web.lemoyne.edu/~giunta/perrin.html Extract in English, translation by Frederick Soddy].</ref>

<ref name="losch1865">{{cite journal|first = J.|last = Loschmidt|authorlink = Johann Josef Loschmidt|title = Zur Grösse der Luftmoleküle|journal = Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften Wien|volume = 52|issue = 2|pages = 395–413|year =1865}} [https://web.archive.org/web/20060207130125/http://dbhs.wvusd.k12.ca.us/webdocs/Chem-History/Loschmidt-1865.html English translation].</ref>

<ref name=bipm8th>{{SIbrochure8th|pages=114–15}}</ref>

<ref name=bipm9th>Bureau International des Poids et Mesures (2019): ''[https://www.bipm.org/utils/common/pdf/si-brochure/SI-Brochure-9-EN.pdf The International System of Units (SI)]'', 9th edition, English version, page 134. Available at the [https://www.bipm.org/en/publications/si-brochure/ BIPM website].</ref>

<ref name=feynman>Richard P. Feynman (1963): ''The Feynman Lectures on Physics'', Volume II, 2nd edition; 512 pages. {{isbn|9780805390476}}</ref>

<ref name=mcgraw>Marvin Yelles (1971): ''McGraw-Hill Encyclopedia of Science and Technology'', Volume 9, 3rd edition; 707 pages. {{isbn|9780070797987}}</ref>

<ref name=born>Max Born (1969): ''[https://books.google.com/books?id=zv3DAgAAQBAJ&lpg=PA3&pg=PA3#v=onepage&f=false Atomic Physics]'', 8th Edition. Dover edition, reprinted by Courier in 2013; 544 pages. {{isbn|9780486318585}}</ref>

<ref name=bipm1971>Bureau International des Poids et Mesures (1971): ''[https://www.bipm.org/jsp/en/ListCGPMResolution.jsp?CGPM=14 14th Conference Générale des Poids et Mesures]'' Available at the [https://www.bipm.org/ BIPM website].</ref>

<ref name=iupac1996>H. P. Lehmann, X. Fuentes-Arderiu, and L. F. Bertello (1996): "Glossary of terms in quantities and units in Clinical Chemistry (IUPAC-IFCC Recommendations 1996)"; page 963, item "[http://goldbook.iupac.org/terms/view/A00543 Avogadro constant]". ''Pure and Applied Chemistry'', volume 68, issue 4, pages 957–1000. {{doi|10.1351/pac199668040957}}</ref>

<ref name=goldbook>{{GoldBookRef |title=Avogadro constant |file=A00543 |accessdate=2019-02-08 }}</ref>

<ref name=ciaaw>{{cite journal|title=Atomic Weight: The Name, Its History, Definition and Units|last1=de Bievre|first1=P.|last2=Peiser|first2=H. S.|journal=[[Pure and Applied Chemistry]]|year=1992|volume=64|issue=10|pages=1535–1543|doi=10.1351/pac199264101535}}</ref>

<ref name=okun>{{cite book|last = Okun|first = Lev B.|author2= Lee, A. G.|title = Particle Physics: The Quest for the Substance of Substance|url = https://books.google.com/?id=UBidG9OKrQ8C&printsec=frontcover#v=onepage|pages= 86|year = 1985|publisher = OPA Ltd.|isbn = 978-3-7186-0228-5}}</ref>

<ref name=pauling>Linus Pauling (1970), ''[https://books.google.com/books?id=FjKlBQAAQBAJ&lpg=PA96&pg=PA96#v=onepage&f=false General Chemistry]'', page 96. Dover Edition, reprinted by Courier in 2014; 992 pages. {{isbn|9780486134659}}</ref>

<ref name=avog1811>{{cite journal|first = Amedeo|last = Avogadro|authorlink = Amedeo Avogadro|title = Essai d'une maniere de determiner les masses relatives des molecules elementaires des corps, et les proportions selon lesquelles elles entrent dans ces combinaisons|journal = Journal de Physique|year = 1811|volume = 73|pages = 58–76}} [http://web.lemoyne.edu/~giunta/avogadro.html English translation].</ref>

<ref name=oseen1926>[[Carl Wilhelm Oseen|Oseen, C.W.]] (December 10, 1926). ''[http://nobelprize.org/nobel_prizes/physics/laureates/1926/press.html Presentation Speech for the 1926 Nobel Prize in Physics]''.</ref>

<ref name=virgo1933>{{cite journal|last=Virgo |first=S.E. |url=http://gemini.tntech.edu/~tfurtsch/scihist/loschmid.html |title=Loschmidt's Number |journal=Science Progress |volume=27 |year=1933 |pages=634–649 |url-status=dead |archiveurl=https://web.archive.org/web/20050404003919/http://gemini.tntech.edu/~tfurtsch/scihist/loschmid.html |archivedate=2005-04-04 }}</ref>

<ref name=ebrit1974>(1974): ''[https://physics.nist.gov/cuu/Constants/historical1.html Introduction to the constants for nonexperts, 1900–1920]'' From the ''Encyclopaedia Britannica'', 15th edition; reproduced by [[NIST]]. Accessed on 2019-07-03.</ref>

<ref name=kotz2008>{{cite book|last = Kotz|first = John C.|author2 = Treichel, Paul M.|author3 = Townsend, John R.|title = Chemistry and Chemical Reactivity|edition = 7th|url = http://cengagesites.com/academic/kotz.cfm?site=2719&section=home|archive-url = https://web.archive.org/web/20081016082922/http://cengagesites.com/academic/kotz.cfm?site=2719&section=home|url-status = dead|archive-date = 2008-10-16|year = 2008|publisher = Brooks/Cole|isbn = 978-0-495-38703-9}}</ref>

<ref name=cipm106>International Bureau for Weights and Measures (2017): ''[https://www.bipm.org/utils/en/pdf/CIPM/CIPM2017-EN.pdf Proceedings of the 106th meeting of the International Committee for Weights and Measures (CIPM), 16-17 and 20 October 2017]'', page 23. Available at the [https://www.bipm.org/en/committees/cipm/meeting/106.html BIPM website].</ref>

<ref name=cgpm26>International Bureau for Weights and Measures (2018): ''[https://www.bipm.org/utils/common/pdf/CGPM-2018/26th-CGPM-Resolutions.pdf Resolutions Adopted - 26th Confernce Générale des Poids et Mesures]''. Available at the [https://www.bipm.org/en/cgpm-2018/ BIPM website].</ref>

<ref name=pave2018>{{Cite journal|last=Pavese|first=Franco|date=January 2018|title=A possible draft of the CGPM Resolution for the revised SI, compared with the CCU last draft of the 9th SI Brochure|journal=Measurement|volume=114|pages=478–483|doi=10.1016/j.measurement.2017.08.020|issn=0263-2241}}</ref>

<ref name=lehm2016>{{Cite journal|title=Unified Atomic Mass Unit|last=Lehmann|first=H. P.|last2=Fuentes-Arderiu|first2=X.|date=2016-02-29|last3=Bertello|first3=L. F.|doi = 10.1515/iupac.68.2930}}</ref>

</references>

==External links==
* [http://goldbook.iupac.org/A00543.html 1996 definition of the Avogadro constant] from the [[IUPAC]] ''[[Compendium of Chemical Terminology]]'' ("''Gold Book''")
* [https://web.archive.org/web/20140428004944/http://iweb.tntech.edu/tfurtsch/scihist//avogadro.htm Some Notes on Avogadro's Number, {{val|6.022|e=23}}] ''(historical notes)''
* [http://www.americanscientist.org/issues/pub/2007/2/an-exact-value-for-avogadros-number An Exact Value for Avogadro's Number] – [[American Scientist]]
* [https://web.archive.org/web/20110717124427/http://www.inrim.it/Nah/Web_Nah/home.htm Avogadro and molar Planck constants for the redefinition of the kilogram]
* [http://onlinelibrary.wiley.com/doi/10.1002/1522-2675%2820010613%2984:6%3C1314::AID-HLCA1314%3E3.0.CO;2-Q/abstract;jsessionid=2BF5A65667EEDDCD40A985289CFDDC0B.f01t01?deniedAccessCustomisedMessage=&userIsAuthenticated=false Murrell, J. 2001 ''Avogadro and his Constant'', Helvitica Chemica Acta, '''84''', 6, p.1314-1327]
* Scanned version of "Two hypothesis of Avogadro", 1811 Avogadro's article, on ''[https://translate.google.com/translate?&us=auto&tl=en&u=https%3A%2F%2Fwww.bibnum.education.fr%2Fchimie%2Ftheorie-chimique%2Fles-deux-hypotheses-d-avogadro-en-1811 BibNum]''

{{Mole concepts}}
{{Scientists whose names are used in physical constants}}

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