# Mole (unit)

"Nmol" redirects here. For the mathematical technique, see Method of lines.
Mole
Unit system SI base unit
Unit of Amount of substance
Symbol mol

The mole is a unit of measurement for amount of substance. It is defined as the amount of any chemical substance that contains as many elementary entities, e.g., atoms, molecules, ions, electrons, or photons, as there are atoms in 12 grams of pure carbon-12 (12C), the isotope of carbon with relative atomic mass 12 by definition. This number is expressed by the Avogadro constant, which has a value of 6.02214129(27)×1023. In other words, the mole is the name given to an amount of a substance equal in mass (in grams) to the combined mass (in amu) of the atoms of the constituent molecules of the substance multiplied by Avogadro's number. It is one of the base units in the International System of Units; it has the unit symbol mol.[clarification needed]

Example problem using the mole: 1 hydrogen atom has an atomic mass of 1.008 amu. What is 1 mole of hydrogen? Answer: {1.008 x [6.022 141 29(27) x 10^23]} amu = 1.008 grams.

Another way to look at the mole is to think of it as a mass equal to {[6.022 141 29(27) x 10^23]^2} times the mass of the base-unit of the substance (base-unit as in a molecule or atom of the substance). We know this is true because the number of amu in a gram is Avogadro's number. We can derive the relationship between a mole of a substance and the mass of one atom/molecule of the substance using the previous definition of the mole stated above and the relationship between a gram and an amu (there are Avogadro's-number of amu in a gram) like so:

1 hydrogen atom = 1.008 amu of mass. 1 mole of hydrogen = {1.008 x [6.022 141 29(27) x 10^23]} amu = 1.008 grams. 1 hydrogen atom = {1.008 / [6.022 141 29(27) x 10^23]} grams. We now compare the mass in grams of 1 hydrogen atom with the mass in grams of 1 mole of hydrogen. By dividing the mass of 1 mole of hydrogen by the mass of 1 hydrogen atom we'll obtain the scale-factor between the mass of 1 hydrogen atom and the mass of 1 mole of hydrogen. (We will use the numbers that are 2 lines above this one). Since dividing by a fraction is the same as multiplying by that fraction's reciprocal, we can see that the scale-factor is (1.008 x Avogadro's-number) / 1.008. The 1.008 on top and bottom cancel out and we're left with Avogadro's-number [(6.022 141 29(27) x 10^23]. This scale-factor holds true for all pure substances. For example, a mole of water is Avogadro's-number times the mass of one water molecule (H2O).

The mole is widely used in chemistry as a convenient way to express amounts of reactants and products of chemical reactions and to represent substance in proportions. For example, the chemical equation 2 H2 + O2 → 2 H2O implies that 2 mol of dihydrogen (H2) and 1 mol of dioxygen (O2) react to form 2 mol of water (H2O). The mole may also be used to express the number of atoms, ions, or other elementary entities in a given sample of any substance. The concentration of a solution is commonly expressed by its molarity, defined as the number of moles of the dissolved substance per litre of solution.

The number of molecules in a mole (known as Avogadro's constant) is defined such that the mass of one mole of a substance, expressed in grams, is equal to the mean molecular mass of the substance. For example, the mean molecular mass of natural water is about 18.015, therefore, one mole of water has a mass of about 18.015 grams.

The term gram-molecule was formerly used for essentially the same concept.[1] The term gram-atom has been used for a related but distinct concept, namely a quantity of a substance that contains Avogadro's number of atoms, whether isolated or combined in molecules. Thus, for example, 1 mole of MgB2 is 1 gram-molecule of MgB2 but 3 gram-atoms of MgB2.[2][3]

In honour of the unit, some chemists celebrate October 23 (a reference to the 1023 part of the Avogadro constant) as "Mole Day". Some also do the same for February 6 and June 2.

## Definition and related concepts

As of 2011, the mole is defined by BIPM to be the amount of substance of a system which contains the same number of elementary entities (e.g. atoms, molecules, ions, electrons) as atoms in 0.012 kilograms of carbon-12 (12C), the isotope of carbon with relative atomic mass 12.[1] Thus, by definition, one mole of pure 12C has a mass of exactly 12 g. It also follows from the definition that X moles of any substance will contain the same number of molecules as X moles of any other substance.

The mass per mole of a substance is called its molar mass. Since the standard unit for expressing the mass of molecules or atoms (atomic mass unit or the dalton) is defined as 1/12 of the mass of a 12C atom, it follows that the molar mass of a substance, measured in grams per mole, is exactly equal to its mean molecular or atomic mass, measured in unified atomic mass units or daltons; which is to say, to the substance's mean molecular or relative atomic mass.

The number of elementary entities in a sample of a substance is technically called its (chemical) amount. Therefore, the mole is a convenient unit for that physical quantity. One can determine the chemical amount of a known substance, in moles, by dividing the sample's mass by the substance's molar mass.[4] Other methods include the use of the molar volume or the measurement of electric charge.[4]

The mass of one mole of a substance depends not only on its molecular formula, but also on the proportion of the isotopes of each element present in it. For example, one mole of calcium-40 is 39.96259098 ± 0.00000022 grams, whereas one mole of calcium-42 is 41.95861801 ± 0.00000027 grams, and one mole of calcium with the normal isotopic mix is 40.078 ± 0.004 grams.

Since the definition of the gram is not (as of 2011) mathematically tied to that of the atomic mass unit, the number NA of molecules in a mole (Avogadro's number) must be determined experimentally. The value adopted by CODATA in 2010 is NA = 6.02214129×1023 ± 0.00000027×1023.[5] In 2011 the measurement was refined to 6.02214078×1023 ± 0.00000018×1023.[6]

The number of moles in a sample is simply the sample mass divided by the molar mass of the material.

## History

The history of the mole is intertwined with that of molecular mass, atomic mass unit, Avogadro's number and related concepts.

The first table of relative atomic mass (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 reactions 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, an innovation that 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.

Developments in mass spectrometry led to the adoption of oxygen-16 as the standard substance, in lieu of natural oxygen.[citation needed] The current definition of the mole, based on carbon-12, was approved during the 1960s.[1][7] 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%

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).[8][9][10] However, the related concept of equivalent mass had been in use at least a century earlier.[11]

The mole was made the seventh SI base unit in 1971 by the 14th CGPM.[12]

## The mole as a unit

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

• the number of molecules, etc. in a given lump of material is a fixed dimensionless quantity that can be expressed simply as a number, so does not require its own base unit;[7]
• the SI thermodynamic mole is irrelevant to analytical chemistry and could cause avoidable costs to advanced economies;[13]
• 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;[14]
• the SI defines numbers of entities as quantities of dimension one, and thus ignores the ontological distinction between entities and units of continuous quantities.[15]

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.

## Other units called "mole"

Chemical engineers use the concept extensively, but the unit is rather small for industrial use.[16] For convenience in avoiding conversions in the Imperial (or American 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 mol.[17]

In the metric system, 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), when dealing with laboratory data.[17]

Late 20th century chemical engineering practice came to use the kilomole (kmol), which is numerically identical to the kilogram-mole, but whose name and symbol adopt the SI convention for standard multiples of metric units - thus kmol means 1000 moles. This is analogous 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 the molecular mass not the factor 1000 unless the basic SI unit of mol/s were to be used. Indeed, the appearance of any conversion factors in a model can cause confusion and is to be avoided; possibly a definition of coherence is the absence of conversion factors in sets of equations developed for modelling.

Concentrations expressed as kmol/m3 are numerically the same as those in mol/dm3 i.e. the molarity conventionally used by chemists for bench measurements; this equality can be convenient for scale-up.

## Proposed future definition

Main article: New SI definitions

In 2011, the 24th meeting of the General Conference on Weights and Measures (CGPM) agreed a plan for a possible revision of the SI base unit definitions on an as yet undetermined date. This plan, set forward in the meeting's first resolution, included a proposal to redefine the mole in a way that will fix "the Avogadro constant to be equal to exactly 6.022 14X ×1023 when it is expressed in the SI unit mol−1....[18] .... the symbol X in this Draft Resolution represents one or more additional digits to be added to the numerical values .... using values based on the most recent CODATA adjustment".

## Related units

The SI units for molar concentration are mol/m3. However, most chemical literature traditionally uses mol/dm3, or mol dm−3, which is the same as mol/L. These traditional units are often denoted by a capital letter M (pronounced "molar"), sometimes preceded by an SI prefix, for example, millimoles per litre (mmol/L) or millimolar (mM), micromoles/litre (µmol/L) or micromolar (µM), or nanomoles/L (nmol/L) or nanomolar (nM).

## The unit's holiday

October 23 is called Mole Day.[19] It is an informal holiday in honor of the unit among chemists. The date is derived from Avogadro's constant, 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 or February 6, a reference to the 6.02 part of the constant.[20][21][22]

## Notes and references

1. International Bureau of Weights and Measures (2006), The International System of Units (SI) (PDF) (8th ed.), pp. 114–15, ISBN 92-822-2213-6
2. Wang, Yuxing; Bouquet, Fr d ric; 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. Bibcode:2003JPCM...15..883W. arXiv:. doi:10.1088/0953-8984/15/6/315.
3. 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. Bibcode:2005PhRvB..72b4547L. arXiv:. doi:10.1103/PhysRevB.72.024547.
4. International Bureau of Weights and Measures. "Realising the mole." Retrieved 25 September 2008.
5. [1]
6. Andreas, Birk; et al. (2011). "Determination of the Avogadro Constant by Counting the Atoms in a 28Si Crystal". Physical Review Letters. 106 (3): 30801. Bibcode:2011PhRvL.106c0801A. arXiv:. doi:10.1103/PhysRevLett.106.030801.
7. de Bièvre, P.; Peiser, H.S. (1992). "'Atomic Weight'—The Name, Its History, Definition, and Units" (PDF). Pure Appl. Chem. 64 (10): 1535–43. doi:10.1351/pac199264101535.
8. Helm, Georg (1897). "The Principles of Mathematical Chemistry: The Energetics of Chemical Phenomena". transl. by Livingston, J.; Morgan, R. New York: Wiley: 6.
9. Some sources place the date of first usage in English as 1902. Merriam–Webster proposes an etymology from Molekulärgewicht (molecular weight).
10. Ostwald, Wilhelm (1893). Hand- und Hilfsbuch zur Ausführung Physiko-Chemischer Messungen. Leipzig. p. 119.
11. mole, n.8, Oxford English Dictionary, Draft Revision Dec. 2008
12. 14th CGPM (1971):Resolution 3
13. 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.[2].
14. 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.
15. Cooper, G; Humphry, S (2010). "The ontological distinction between units and entities". Synthese. 187 (2): 393. doi:10.1007/s11229-010-9832-1.
16. In particular, when the mole is used, alongside the SI unit of volume of a cubic metre, in thermodynamic calculations such as the ideal gas law, a factor of 1000 is introduced which engineering practice chooses to simplify by adopting the kilomole.
17. Himmelblau, David (1996). Basic Principles and Calculations in Chemical Engineering (6 ed.). pp. 17–20. ISBN 0-13-305798-4.
18. "RESOLUTIONS ADOPTED BY THE 24TH MEETING OF THE GENERAL CONFERENCE ON WEIGHTS AND MEASURES (CGPM)" (PDF). Paris: BIPM. 17–21 Oct 2011.
19. History of National Mole Day Foundation, Inc
20. Happy Mole Day!, Mary Bigelow. SciLinks blog, National Science Teachers Association. October 17, 2013.
21. What Is Mole Day? - Date and How to Celebrate, Anne Marie Helmenstine. About.com
22. The Perse School (Feb 7, 2013), The Perse School celebrates moles of the chemical variety, Cambridge Network, retrieved Feb 11, 2015, As 6.02 corresponds to 6th February, the School has adopted the date as their ‘Mole Day’.