The mole (symbol: mol) is the base unit of amount of substance ("number of substance") in the International System of Units or System International (SI), defined as exactly 6.02214076×1023 particles, e.g., atoms, molecules, ions or electrons.^[1]

The current definition was adopted in November 2018,^[1] revising its old definition based on the number of atoms in 12 grams of carbon-12 (12C) (the isotope of carbon with relative atomic mass 12 Da by definition).

The number 6.02214076×1023 (the Avogadro number) was chosen so that the mass of one mole of a chemical compound, in grams, is numerically equal (for all practical purposes) to the average mass of one molecule of the compound, in daltons. Thus, for example, one mole of water contains 6.02214076×1023 molecules, whose total mass is about 18.015 grams – and the mean mass of one molecule of water is about 18.015 daltons.

The mole is widely used in chemistry as a convenient way to express amounts of reactants and products of chemical reactions. For example, the chemical equation 2H2 + O2 → 2H2O can be interpreted to mean that 2 mol dihydrogen (H2) and 1 mol dioxygen (O2) react to form 2 mol water (H2O). The mole may also be used to represent the number of atoms, ions, or other entities in a given sample of a substance. The concentration of a solution is commonly expressed by its molarity, defined as the amount of dissolved substance per unit volume of solution, for which the unit typically used is moles per litre (mol/l), commonly abbreviated M.

The term gram-molecule was formerly used for "mole of molecules",^[2] and gram-atom for "mole of atoms". Thus, for example, 1 mole of MgBr2 is 1 gram-molecule of MgBr2 but 3 gram-atoms of MgBr2.^[3][4]

In principle, the word "mole" should always be qualified by the nature of the particles one is referring to (molecules, atoms, ions, electrons, etc.). Usually, however, the latter is implied by context, or by the nature of the substances.

For chemical substances with covalently bound neutral separate molecules, like water, "one mole" generally means "of molecules". For metals and elements with extended bonding networks, like carbon, it usually means "of atoms".

For ionic solids and polymeric substances, it means "of the minimal elemental formula". Thus, for example, one mole of calcium chloride CaCl2 means an amount of the salt that contains one mole of Ca2+ ions and two moles of Cl− ions; and one mole of silica SiO2 has one mole of silicon atoms and two of oxygen atoms, even through silica does not have discrete SiO2 molecules.

For solids with water of hydration (or other solvation molecules), the latter is normally included in the "molecule" of the substance. For example, one mole of anhydrous copper sulfate CuSO4 is about 159.609 g, whereas one mole of the pentrahydrate CuSO4·5H2O is usually understood to be about 159.609 + 5 × 18.015 = 249.684 g. Both contain one mole of Cu2+ ions and one mole of SO2−4 ions.

One must be aware that, depending on the context, "one mole of nitrogen" may mean "one mole of N2 molecules" (that is, about 28.014 grams) or "one mole of N atoms" (that is, about 14.007 grams).

A mole of a substance comprises of an amount of that substance equal numerically to the molecular mass in grams of that substance. For example, 1 mole of Aluminum is 26.982 g, 1 mole of lithium is 6.941 g etc.

This can be understood through the justification as follows:

Let the mass of one atom of hydrogen be equal to mH.

Then mass of x atoms of hydrogen will be x mH.

We take 1 gram of hydrogen. 1 gram of hydrogen will contain some number of atoms. Assume by some means we count this number and that the number turns out to be 6.02214076×1023 , the Avogadro’s number.

Now, we show that 12 g of carbon will contain 6.02214076×1023 atoms.

One atom of C is 12 times as massive as H. That is mC = 12 mH.

So q atoms of C will weigh q 12 mH. In other words, q atoms of C will weigh 12 q mH.

Now, if we take q = 6.02214076×1023 , then q mH will be equivalent to a mass of 1 g.

So, 6.02214076×1023 atoms of C will weigh 12 q mH = 12 x 1 g = 12 g.

The same argument goes to any other element, and hence a mole of a substance comprises of an amount of that substance equal numerically to the molecular mass in grams of that substance.

The molar mass of a substance is the mass of a sample, in grams (or kilograms), divided by the number of moles of the substance in that sample, and expressed in units of grams (or kilograms) per mole. This is a characteristic property of the substance; numerically, it is equal (for all practical purposes) to the mean mass of one molecule, expressed in atomic mass units. Thus, for example, the molar mass of water is 18.015 g/mol.^[5] Other methods include the use of the molar volume or the measurement of electric charge.^[5]

Conversely, the number moles of a substance in a sample can be obtained by dividing the mass of the sample by the molar mass of the compound. For example, 100 g of water is about (100 g)/(18.015 g/mol) = 5.551 mol of water.^[5]

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

The molar concentration of a solution of some substance is the number of moles per unit of volume of the final solution (which may be slightly different from the amount of solvent in it).

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 demal (D) is an obsolete unit for expressing the concentration of a solution. It is equal to molar concentration at 0 °C, i.e., 1 D represents 1 mol of the solute present in one cubic decimetre of the solution at 0 °C.^[6] It was first proposed in 1924 as a unit of concentration based on the decimetre rather than the litre; at the time there was a factor of 1.000028 difference between the litre and the cubic decimetre.^[7] The demal was used as a unit of concentration in electrolytic conductivity primary standards.^[8] These standards were later redefined in terms of molar concentration.^[9]

When expressing the molar concentration of solutions of a substance that may undergo partial dissociation or hydration, "one mole" usually means "one mole of the original substance before dissolution". Thus, for example, a "0.5 molar solution of hydrogen chloride" is obtained by dissolving 0.5 moles of HCl in enough water to make 1 litre of solution; even though most HCl molecules will dissociate into H3O+ and Cl− ions in the liquid. Likewise, one litre of a "0.5 molar solution of formaldehyde" will actually contain some methanediol, some formaldehyhde, metaformaldehyde, other oligomers and polymers.

The molar fraction or mole fraction of a substance in a mixture (such as a solution) is the number of moles of the compound in one sample of the mixture, divided by the total number of moles of all components. For example, if 20 g of NaCl is dissolved in 100 g of water, the amounts of the two substances in the solution will be (20 g)/(58.443 g/mol) = 0.34221 mol and (100 g)/(18.015 g/mol) = 5.5509 mol, respectively; and the molar fraction of NaCl will be 0.34221/(0.34221 + 5.5509) = 0.05807.

In a mixture of gases, the partial pressure of each component will be proportional to its molar ratio.

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

The first table of standard atomic weight (atomic mass) 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).^[10][11][12] However, the related concept of equivalent mass had been in use at least a century earlier.^[13]

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

The oxygen-16 definition was replaced with one based on carbon-12 during the 1960s. The mole was defined by International Bureau of Weights and Measures as "the amount of substance of a system which contains as many elementary entities as there are atoms in 0.012 kilogram of carbon-12." Thus, by that definition, one mole of pure 12C had a mass of exactly 12 g.^[2][14] The four different definitions were equivalent to within 1%.

Since the definition of the gram was not mathematically tied to that of the atomic mass unit, 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±0.00000027)×1023 mol−1.^[15] In 2011 the measurement was refined to (6.02214078±0.00000018)×1023 mol−1.^[16]

The mole was made the seventh SI base unit in 1971 by the 14th CGPM.^[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 constants that are, in their nature, exact.^[1]

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

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:

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.

Chemical engineers use the unit extensively, and decimal multiples may be more suitable for industrial use.^[23] 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,^[24] which value is the same as the number of grams in an international avoirdupois pound.

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.^[24]

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 mol. 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 without the factor 1000 unless the basic SI unit of mol/s were to be used.

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 when scaling up.

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.^[25]

October 23, denoted 10/23 in the US, is recognized by some as Mole Day.^[26] 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 or February 6, a reference to the 6.02 part of the constant.^[27][28][29]