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Mole (unit)

## Summary

The mole (symbol mol) is a unit of measurement, the base unit in the International System of Units (SI) for amount of substance, a quantity proportional to the number of elementary entities of a substance. One mole contains exactly 6.02214076×1023 elementary entities (approximately 602 sextillion or 602 billion times a trillion), which can be atoms, molecules, ions, ion pairs, or other particles. The number of particles in a mole is the Avogadro number (symbol N0) and the numerical value of the Avogadro constant (symbol NA) expressed in mol-1.[1] The value was chosen on the basis of the historical definition of the mole as the amount of substance that corresponds to the number of atoms in 12 grams of 12C,[1] which made the mass of a mole of a compound expressed in grams, numerically equal to the average molecular mass or formula mass of the compound expressed in daltons. With the 2019 revision of the SI, the numerical equivalence is now only approximate but may be assumed for all practical purposes.

mole
One mole is exactly 6.02214076×1023 elementary entities, approximately equivalent to the number of atoms in 12 grams of carbon-12 in the historical definition
General information
Unit systemSI
Unit ofamount of substance
Symbolmol

The mole is widely used in chemistry as a convenient way to express amounts of reactants and amounts of products of chemical reactions. For example, the chemical equation 2 H2 + O2 → 2 H2O can be interpreted to mean that for each 2 mol molecular hydrogen (H2) and 1 mol molecular oxygen (O2) that react, 2 mol of water (H2O) form. The concentration of a solution is commonly expressed by its molar concentration, defined as the amount of dissolved substance per unit volume of solution, for which the unit typically used is mole per litre (mol/L).

## Concepts

### Relation to the Avogadro constant

The number of entities (symbol N) in a one-mole sample equals the Avogadro number (symbol N0), a dimensionless quantity.[1] Historically, N0 approximates the number of nucleons (protons or neutrons) in one gram of ordinary matter. The Avogadro constant (symbol NA = N0/mol) has numerical multiplier given by the Avogadro number with the unit reciprocal mole (mol−1).[2] The ratio n = N/NA is a measure of the amount of substance (with the unit mole).[2][3][4]

### Nature of the entities

Depending on the nature of the substance, an elementary entity may be an atom, a molecule, an ion, an ion pair, or a subatomic particle such as a proton. For example, 10 moles of water (a chemical compound) and 10 moles of mercury (a chemical element) contain equal numbers of substance, with one atom of mercury for each molecule of water, despite the two quantities having different volumes and different masses.

The mole corresponds to a given count of entities.[5] Usually, the entities counted are chemically identical and individually distinct. For example, a solution may contain a certain number of dissolved molecules that are more or less independent of each other. However, the constituent entities in a solid are fixed and bound in a lattice arrangement, yet they may be separable without losing their chemical identity. Thus, the solid is composed of a certain number of moles of such entities. In yet other cases, such as diamond, where the entire crystal is essentially a single molecule, the mole is still used to express the number of atoms bound together, rather than a count of molecules. Thus, common chemical conventions apply to the definition of the constituent entities of a substance, in other cases exact definitions may be specified. The mass of a substance is equal to its relative atomic (or molecular) mass multiplied by the molar mass constant, which is almost exactly 1 g/mol.

## History

The history of the mole is intertwined with that of units of molecular mass, and the Avogadro constant.

The first table of standard atomic weight was published by John Dalton (1766–1844) in 1805, based on a system in which the relative atomic mass of hydrogen was defined as 1. These relative atomic masses were based on the stoichiometric proportions of chemical reaction and compounds, a fact that greatly aided their acceptance: It was not necessary for a chemist to subscribe to atomic theory (an unproven hypothesis at the time) to make practical use of the tables. This would lead to some confusion between atomic masses (promoted by proponents of atomic theory) and equivalent weights (promoted by its opponents and which sometimes differed from relative atomic masses by an integer factor), which would last throughout much of the nineteenth century.

Jöns Jacob Berzelius (1779–1848) was instrumental in the determination of relative atomic masses to ever-increasing accuracy. He was also the first chemist to use oxygen as the standard to which other masses were referred. Oxygen is a useful standard, as, unlike hydrogen, it forms compounds with most other elements, especially metals. However, he chose to fix the atomic mass of oxygen as 100, which did not catch on.

Charles Frédéric Gerhardt (1816–56), Henri Victor Regnault (1810–78) and Stanislao Cannizzaro (1826–1910) expanded on Berzelius' works, resolving many of the problems of unknown stoichiometry of compounds, and the use of atomic masses attracted a large consensus by the time of the Karlsruhe Congress (1860). The convention had reverted to defining the atomic mass of hydrogen as 1, although at the level of precision of measurements at that time – relative uncertainties of around 1% – this was numerically equivalent to the later standard of oxygen = 16. However the chemical convenience of having oxygen as the primary atomic mass standard became ever more evident with advances in analytical chemistry and the need for ever more accurate atomic mass determinations.

The name mole is an 1897 translation of the German unit Mol, coined by the chemist Wilhelm Ostwald in 1894 from the German word Molekül (molecule).[6][7][8] The related concept of equivalent mass had been in use at least a century earlier.[9]

### Standardization

Developments in mass spectrometry led to the adoption of oxygen-16 as the standard substance, in lieu of natural oxygen.[10]

The oxygen-16 definition was replaced with one based on carbon-12 during the 1960s. The International Bureau of Weights and Measures defined the mole as "the amount of substance of a system which contains as many elementary entities as there are atoms in 0.012 kilograms of carbon-12." Thus, by that definition, one mole of pure 12C had a mass of exactly 12 g.[11][5] The four different definitions were equivalent to within 1%.

Scale basis Scale basis
relative to 12C = 12
Relative deviation
from the 12C = 12 scale
Atomic mass of hydrogen = 1 1.00794(7) −0.788%
Atomic mass of oxygen = 16 15.9994(3) +0.00375%
Relative atomic mass of 16O = 16 15.9949146221(15) +0.0318%

Because a dalton, a unit commonly used to measure atomic mass, is exactly 1/12 of the mass of a carbon-12 atom, this definition of the mole entailed that the mass of one mole of a compound or element in grams was numerically equal to the average mass of one molecule or atom of the substance in daltons, and that the number of daltons in a gram was equal to the number of elementary entities in a mole. Because the mass of a nucleon (i.e. a proton or neutron) is approximately 1 dalton and the nucleons in an atom's nucleus make up the overwhelming majority of its mass, this definition also entailed that the mass of one mole of a substance was roughly equivalent to the number of nucleons in one atom or molecule of that substance.

Since the definition of the gram was not mathematically tied to that of the dalton, the number of molecules per mole NA (the Avogadro constant) had to be determined experimentally. The experimental value adopted by CODATA in 2010 is NA = 6.02214129(27)×1023 mol−1.[12] In 2011 the measurement was refined to 6.02214078(18)×1023 mol−1.[13]

The mole was made the seventh SI base unit in 1971 by the 14th CGPM.[14]

### 2019 revision of the SI

Before the 2019 revision of the SI, the mole was defined as the amount of substance of a system that contains as many elementary entities as there are atoms in 12 grams of carbon-12 (the most common isotope of carbon).[15] The term gram-molecule was formerly used to mean one mole of molecules, and gram-atom for one mole of atoms.[11] For example, 1 mole of MgBr2 is 1 gram-molecule of MgBr2 but 3 gram-atoms of MgBr2.[16][17]

In 2011, the 24th meeting of the General Conference on Weights and Measures (CGPM) agreed to a plan for a possible revision of the SI base unit definitions at an undetermined date.

On 16 November 2018, after a meeting of scientists from more than 60 countries at the CGPM in Versailles, France, all SI base units were defined in terms of physical constants. This meant that each SI unit, including the mole, would not be defined in terms of any physical objects but rather they would be defined by physical constants that are, in their nature, exact.[3]

Such changes officially came into effect on 20 May 2019. Following such changes, "one mole" of a substance was redefined as containing "exactly 6.02214076×1023 elementary entities" of that substance.[18][19]

## Criticism

Since its adoption into the International System of Units in 1971, numerous criticisms of the concept of the mole as a unit like the metre or the second have arisen:

• the number of molecules, etc. in a given amount of material is a fixed dimensionless quantity that can be expressed simply as a number, not requiring a distinct base unit;[5][20]
• The SI thermodynamic mole is irrelevant to analytical chemistry and could cause avoidable costs to advanced economies[21]
• The mole is not a true metric (i.e. measuring) unit, rather it is a parametric unit, and amount of substance is a parametric base quantity[22]
• the SI defines numbers of entities as quantities of dimension one, and thus ignores the ontological distinction between entities and units of continuous quantities[23]

In chemistry, it has been known since Proust's law of definite proportions (1794) that knowledge of the mass of each of the components in a chemical system is not sufficient to define the system. Amount of substance can be described as mass divided by Proust's "definite proportions", and contains information that is missing from the measurement of mass alone. As demonstrated by Dalton's law of partial pressures (1803), a measurement of mass is not even necessary to measure the amount of substance (although in practice it is usual). There are many physical relationships between amount of substance and other physical quantities, the most notable one being the ideal gas law (where the relationship was first demonstrated in 1857). The term "mole" was first used in a textbook describing these colligative properties.[24]

## Similar units

Like chemists, chemical engineers use the unit mole extensively, but different unit multiples may be more suitable for industrial use. For example, the SI unit for volume is the cubic metre, a much larger unit than the commonly used litre in the chemical laboratory. When amount of substance is also expressed in kmol (1000 mol) in industrial-scaled processes, the numerical value of molarity remains the same, as ${\textstyle {\frac {\text{kmol}}{{\text{m}}^{3}}}={\frac {1000{\text{ mol}}}{1000{\text{ L}}}}={\frac {\text{mol}}{\text{L}}}}$ . Chemical engineers once used the kilogram-mole (notation kg-mol), which is defined as the number of entities in 12 kg of 12C, and often referred to the mole as the gram-mole (notation g-mol), then defined as the number of entities in 12 g of 12C, when dealing with laboratory data.[25]

Late 20th-century chemical engineering practice came to use the kilomole (kmol), which was numerically identical to the kilogram-mole (until the 2019 revision of the SI, which redefined the mole by fixing the value of the Avogadro constant, making it very nearly equivalent to but no longer exactly equal to the gram-mole), but whose name and symbol adopt the SI convention for standard multiples of metric units – thus, kmol means 1000 mol. This is equivalent to the use of kg instead of g. The use of kmol is not only for "magnitude convenience" but also makes the equations used for modelling chemical engineering systems coherent. For example, the conversion of a flowrate of kg/s to kmol/s only requires dividing by the molar mass in g/mol (as ${\textstyle {\frac {\text{kg}}{\text{kmol}}}={\frac {1000{\text{ g}}}{1000{\text{ mol}}}}={\frac {\text{g}}{\text{mol}}}}$ ) without multiplying by 1000 unless the basic SI unit of mol/s were to be used, which would otherwise require the molar mass to be converted to kg/mol.

For convenience in avoiding conversions in the imperial (or US customary units), some engineers adopted the pound-mole (notation lb-mol or lbmol), which is defined as the number of entities in 12 lb of 12C. One lb-mol is equal to 453.59237 g‑mol,[25] which is the same numerical value as the number of grams in an international avoirdupois pound.

Greenhouse and growth chamber lighting for plants is sometimes expressed in micromoles per square metre per second, where 1 mol photons ≈ 6.02×1023 photons.[26] The obsolete unit einstein is variously defined as the energy in one mole of photons and also as simply one mole of photons.

## Derived units and SI multiples

The only SI derived unit with a special name derived from the mole is the katal, defined as one mole per second of catalytic activity. Like other SI units, the mole can also be modified by adding a metric prefix that multiplies it by a power of 10:

SI multiples of mole (mol)
Submultiples Multiples
Value SI symbol Name Value SI symbol Name
10−1 mol dmol decimole 101 mol damol decamole
10−2 mol cmol centimole 102 mol hmol hectomole
10−3 mol mmol millimole 103 mol kmol kilomole
10−6 mol μmol micromole 106 mol Mmol megamole
10−9 mol nmol nanomole 109 mol Gmol gigamole
10−12 mol pmol picomole 1012 mol Tmol teramole
10−15 mol fmol femtomole 1015 mol Pmol petamole
10−18 mol amol attomole 1018 mol Emol examole
10−21 mol zmol zeptomole 1021 mol Zmol zettamole
10−24 mol ymol yoctomole 1024 mol Ymol yottamole
10−27 mol rmol rontomole 1027 mol Rmol ronnamole
10−30 mol qmol quectomole 1030 mol Qmol quettamole

One femtomole is exactly 602,214,076 molecules; attomole and smaller quantities cannot be exactly realized. The yoctomole, equal to around 0.6 of an individual molecule, did make appearances in scientific journals in the year the yocto- prefix was officially implemented.[27]

## Mole Day

October 23, denoted 10/23 in the US, is recognized by some as Mole Day.[28] It is an informal holiday in honor of the unit among chemists. The date is derived from the Avogadro number, which is approximately 6.022×1023. It starts at 6:02 a.m. and ends at 6:02 p.m. Alternatively, some chemists celebrate June 2 (06/02), June 22 (6/22), or 6 February (06.02), a reference to the 6.02 or 6.022 part of the constant.[29][30][31]

## References

1. ^ a b c The International System of Units (PDF) (9th ed.), International Bureau of Weights and Measures, Dec 2022, ISBN 978-92-822-2272-0
2. ^ a b IUPAC Gold Book. IUPAC – mole (M03980). International Union of Pure and Applied Chemistry. doi:10.1351/goldbook.M03980. S2CID 241546445.
3. ^ a b "On the revision of the International System of Units – International Union of Pure and Applied Chemistry". International Union of Pure and Applied Chemistry. 16 November 2018. Retrieved 1 March 2021.
4. ^ BIPM (20 May 2019). "Mise en pratique for the definition of the mole in the SI". BIPM.org. Retrieved 18 February 2022.
5. ^ a b c de Bièvre, Paul; Peiser, H. Steffen (1992). "'Atomic Weight' — The Name, Its History, Definition, and Units" (PDF). Pure and Applied Chemistry. 64 (10): 1535–43. doi:10.1351/pac199264101535.
6. ^ Helm, Georg (1897). The Principles of Mathematical Chemistry: The Energetics of Chemical Phenomena. transl. by Livingston, J.; Morgan, R. New York: Wiley. p. 6.
7. ^ Some sources place the date of first usage in English as 1902. Merriam–Webster proposes Archived 2011-11-02 at the Wayback Machine an etymology from Molekulärgewicht (molecular weight).
8. ^ Ostwald, Wilhelm (1893). Hand- und Hilfsbuch zur Ausführung Physiko-Chemischer Messungen [Handbook and Auxiliary Book for Conducting Physico-Chemical Measurements]. Leipzig, Germany: Wilhelm Engelmann. p. 119. From p. 119: "Nennen wir allgemein das Gewicht in Grammen, welches dem Molekulargewicht eines gegebenen Stoffes numerisch gleich ist, ein Mol, so ... " (If we call in general the weight in grams, which is numerically equal to the molecular weight of a given substance, a "mol", then ... )
9. ^ mole, n.8, Oxford English Dictionary, Draft Revision Dec. 2008
10. ^ Busch, Kenneth (May 2, 2003). "Units in Mass Spectrometry" (PDF). Current Trends in Mass Spectrometry. 18 (5S): S32-S34 [S33]. Retrieved 29 April 2023.
11. ^ a b International Bureau of Weights and Measures (2006), The International System of Units (SI) (PDF) (8th ed.), pp. 114–15, ISBN 92-822-2213-6, archived (PDF) from the original on 2021-06-04, retrieved 2021-12-16
12. ^ physics.nist.gov/ Archived 2015-06-29 at the Wayback Machine Fundamental Physical Constants: Avogadro Constant
13. ^ Andreas, Birk; et al. (2011). "Determination of the Avogadro Constant by Counting the Atoms in a 28Si Crystal". Physical Review Letters. 106 (3): 30801. arXiv:1010.2317. Bibcode:2011PhRvL.106c0801A. doi:10.1103/PhysRevLett.106.030801. PMID 21405263. S2CID 18291648.
14. ^ "BIPM – Resolution 3 of the 14th CGPM". www.bipm.org. Archived from the original on 9 October 2017. Retrieved 1 May 2018.
15. ^ International Bureau of Weights and Measures (2006), The International System of Units (SI) (PDF) (8th ed.), ISBN 92-822-2213-6, archived (PDF) from the original on 2021-06-04, retrieved 2021-12-16
16. ^ Wang, Yuxing; Bouquet, Frédéric; Sheikin, Ilya; Toulemonde, Pierre; Revaz, Bernard; Eisterer, Michael; Weber, Harald W.; Hinderer, Joerg; Junod, Alain; et al. (2003). "Specific heat of MgB2 after irradiation". Journal of Physics: Condensed Matter. 15 (6): 883–893. arXiv:cond-mat/0208169. Bibcode:2003JPCM...15..883W. doi:10.1088/0953-8984/15/6/315. S2CID 16981008.
17. ^ Lortz, R.; Wang, Y.; Abe, S.; Meingast, C.; Paderno, Yu.; Filippov, V.; Junod, A.; et al. (2005). "Specific heat, magnetic susceptibility, resistivity and thermal expansion of the superconductor ZrB12". Phys. Rev. B. 72 (2): 024547. arXiv:cond-mat/0502193. Bibcode:2005PhRvB..72b4547L. doi:10.1103/PhysRevB.72.024547. S2CID 38571250.
18. ^ CIPM Report of 106th Meeting Archived 2018-01-27 at the Wayback Machine Retrieved 7 April 2018
19. ^ "Redefining the Mole". NIST. 2018-10-23. Retrieved 24 October 2018.
20. ^ Barański, Andrzej (2012). "The Atomic Mass Unit, the Avogadro Constant, and the Mole: A Way to Understanding". Journal of Chemical Education. 89 (1): 97–102. Bibcode:2012JChEd..89...97B. doi:10.1021/ed2001957.
21. ^ Price, Gary (2010). "Failures of the global measurement system. Part 1: the case of chemistry". Accreditation and Quality Assurance. 15 (7): 421–427. doi:10.1007/s00769-010-0655-z. S2CID 95388009.
22. ^ Johansson, Ingvar (2010). "Metrological thinking needs the notions of parametric quantities, units, and dimensions". Metrologia. 47 (3): 219–230. Bibcode:2010Metro..47..219J. doi:10.1088/0026-1394/47/3/012. S2CID 122242959.
23. ^ Cooper, G.; Humphry, S. (2010). "The ontological distinction between units and entities". Synthese. 187 (2): 393–401. doi:10.1007/s11229-010-9832-1. S2CID 46532636.
24. ^ The scientific foundations of analytical chemistry: Treated in an elementary manner. Macmillan and co., limited. 1900. OL 7204743M.
25. ^ a b Himmelblau, David (1996). Basic Principles and Calculations in Chemical Engineering (6 ed.). Prentice Hall PTR. pp. 17–20. ISBN 978-0-13-305798-0.
26. ^ "Lighting Radiation Conversion". Archived from the original on March 11, 2016. Retrieved March 10, 2016.
27. ^ Chen, Da Yong; et al. (1991). "Low-cost, high-sensitivity laser-induced fluorescence detection for DNA sequencing by capillary gel electrophoresis". Journal of Chromatography. 559 (1–2): 237–246. doi:10.1016/0021-9673(91)80074-Q. PMID 1761625.
28. ^ History of National Mole Day Foundation, Inc. Archived 2010-10-23 at the Wayback Machine.
29. ^ Happy Mole Day! Archived 2014-07-29 at the Wayback Machine, Mary Bigelow. SciLinks blog, National Science Teachers Association. October 17, 2013.
30. ^ What Is Mole Day? – Date and How to Celebrate. Archived 2014-07-30 at Wikiwix, Anne Marie Helmenstine. About.com.
31. ^ The Perse School (Feb 7, 2013), The Perse School celebrates moles of the chemical variety, Cambridge Network, archived from the original on 2015-02-11, retrieved Feb 11, 2015, As 6.02 corresponds to 6th February, the School has adopted the date as their 'Mole Day'.