Amount of substance

In chemistry, the amount of substance (symbol Template:Mvar) in a given sample of matter is defined as a ratio (Template:Math) between the number of elementary entities (Template:Mvar) and the Avogadro constant (Template:Math). The unit of amount of substance in the International System of Units is the mole (symbol: mol), a base unit.<ref name="bipm9th">Template:SIbrochure9th p. 134</ref> Since 2019, the mole has been defined such that the value of the Avogadro constant Template:Math is exactly Template:Physconst, defining a macroscopic unit convenient for use in laboratory-scale chemistry. The elementary entities are usually molecules, atoms, ions, or ion pairs of a specified kind. The particular substance sampled may be specified using a subscript or in parentheses, e.g., the amount of sodium chloride (NaCl) could be denoted as Template:Math or Template:Math. Sometimes, the amount of substance is referred to as the chemical amount or, informally, as the "number of moles" in a given sample of matter. The latter term is deprecated by the IUPAC because, for a substance Template:Math, the correct meaning of "number of moles" is Template:Math/mol. The amount of substance in a sample can be calculated from measured quantities, such as mass or volume, given the molar mass of the substance or the molar volume of an ideal gas at a given temperature and pressure.

Usage

Because of the way the mole and the dalton are defined, the mass in grams of one mole of a chemical compound is numerically very nearly equal to the mass of one molecule or formula unit of the compound in daltons. For example, a single molecule of water has a mass of about 18.0153 daltons on average, whereas a mole of water (which contains Template:Physconst water molecules) has a mass of about 18.0153 grams on average. The molar mass of an isotope in grams per mole is approximately equal to the mass number. Before the mole was redefined in 2019, this equality was exact by definition for carbon-12.

In chemistry, because of the law of multiple proportions, it is often more convenient to work with amounts of substances denominated in moles, than with masses (grams) or volumes (liters). For example, the chemical fact "1 molecule of oxygen (Template:Chem) will react with 2 molecules of hydrogen (Template:Chem) to make 2 molecules of water (Template:Chem2)" can also be stated as "1 mole of Template:Chem2 will react with 2 moles of Template:Chem2 to form 2 moles of water". The same chemical fact, expressed in terms of masses, would be "32.0 g of oxygen (1 mole of Template:Chem) will react with approximately 4.0 g hydrogen (2 moles of Template:Chem) to make approximately 36.0 g of water (2 moles of Template:Chem2)" (and the numbers would depend on the isotopic composition of the reagents). In terms of volume, the numbers would depend on the pressure and temperature of the reagents and products, although the volume of an ideal gas is proportional to the amount in moles or number of molecules at constant temperature and pressure. For the same reasons, the concentrations of reagents and products in solution are often specified in moles per liter, rather than grams per liter.

The amount of substance is also a convenient concept in thermodynamics. For example, the pressure of a certain quantity of a noble gas in a recipient of a given volume, at a given temperature, is directly related to the number of molecules in the gas (through the ideal gas law), not to its mass.

This technical sense of the term "amount of substance" should not be confused with the general sense of "amount" in the English language. The latter may refer to other measurements such as mass or volume,<ref name="Giunta">Template:Cite journal</ref> rather than the number of particles. There are proposals to replace "amount of substance" with more easily-distinguishable terms, such as enplethy<ref name="Cohen">Template:GreenBook3rd</ref> and stoichiometric amount.<ref name="Giunta"/>

The IUPAC recommends that "amount of substance" should be used instead of "number of moles", just as the quantity mass should not be called "number of (kilo)grams".<ref name="Cohen"/>

Nature of the particles

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Template:See also Template:Sister project To avoid ambiguity, the nature of the particles should be specified in any measurement of the amount of substance: thus, a sample of 1 mol of molecules of oxygen (Template:Chem) has a mass of about 32.00 g, whereas a sample of 1 mol of atoms of oxygen (Template:Chem) has a mass of about 16.00 g.<ref name="GoldBook">Template:GoldBookRef</ref><ref>Template:GreenBook3rd</ref>

Derived quantities

Molar quantities (per mole)

File:Mass versus moles of iron vs gold.svg

A diagram comparing moles and molar masses of iron and gold samples that have equal masses

The quotient of some extensive physical quantity of a homogeneous sample by its amount of substance is an intensive property of the substance, usually named by the prefix "molar" or the suffix "per mole".<ref name="molar">Template:GreenBook3rd</ref><ref name="ISO80000-1">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

For example, the quotient of the mass of a sample by its amount of substance is its molar mass, for which the SI unit kilogram per mole or gram per mole may be used. This is about 18.015 g/mol for water, and 55.845 g/mol for iron. Similarly for volume, one gets the molar volume, which is about 18.069 millilitres per mole for liquid water and 7.092 mL/mol for iron at room temperature. From the heat capacity, one gets the molar heat capacity, which is about 75.385 J/(K⋅mol) for water and about 25.10 J/(K⋅mol) for iron.

Molar mass and molar volume

The molar mass (<math>M</math>) of a substance is the ratio of the mass (<math>m</math>) of a sample of that substance to its amount of substance (<math>n</math>): <math>M=m/n</math>. The amount of substance is given as the number of moles in the sample. For most practical purposes, the numerical value of the molar mass in grams per mole is the same as that of the mean mass of one molecule or formula unit of the substance in daltons, as the mole was historically defined such that the molar mass constant was exactly 1 g/mol. Thus, given the molecular mass or formula mass in daltons, the same number in grams gives an amount very close to one mole of the substance. For example, the average molecular mass of water is about 18.015 Da and the molar mass of water is about 18.015 g/mol. This allows for accurate determination of the amount in moles of a substance by measuring its mass and dividing by the molar mass of the compound: <math>n=m/M</math>.<ref name="BIPMrealise">International Bureau of Weights and Measures. Realising the mole Template:Webarchive. Retrieved 25 September 2008.</ref> For example, 100 g of water is about 5.5509 mol of water.

The molar mass of a substance depends not only on its molecular formula, but also on the distribution of isotopes of each chemical element present in it. For example, the molar mass of calcium-40 is Template:Val, whereas the molar mass of calcium-42 is Template:Val, and of calcium with the normal isotopic mix is Template:Val.

Other methods of determining the amount of substance include the use of the molar volume (<math>V_m</math>) for ideal gases at a given temperature <math>T</math> and pressure <math>P</math> (through the relationship <math>V_m = V/n = RT/P</math>) or the measurement of electric charge (using Faraday's laws of electrolysis).<ref name="BIPMrealise" /> For example, the molar volume of an ideal gas under standard conditions of 0 °C (273.15 K) and 1 atm (101.325 kPa) is about 22.414 L/mol, and 1 m³ of an ideal gas under the same conditions is about 44.615 mol of gas.

Amount (molar) concentration (moles per liter)

Another important derived quantity is the molar concentration (<math>c</math>) (also called amount of substance concentration,<ref>Template:GoldBookRef</ref> amount concentration, or substance concentration,<ref>Template:Cite journal</ref> especially in clinical chemistry), defined as the amount in moles (<math>n</math>) of a specific substance (solute in a solution or component of a mixture), divided by the volume (<math>V</math>) of the solution or mixture: <math>c=n/V</math>.

The standard SI unit of this quantity is mol/m³, although more practical units are commonly used, such as mole per liter (mol/L, equivalent to mol/dm³). For example, the amount concentration of sodium chloride in ocean water is typically about 0.599 mol/L.

The denominator is the volume of the solution, not of the solvent. Thus, for example, one liter of standard vodka contains about 0.40 L of ethanol (315 g, 6.85 mol) and 0.60 L of water. The amount concentration of ethanol is therefore (6.85 mol of ethanol)/(1 L of vodka) = 6.85 mol/L, not (6.85 mol of ethanol)/(0.60 L of water), which would be 11.4 mol/L.

In chemistry, it is customary to read the unit "mol/L" as molar, and denote it by the symbol "M" (both following the numeric value). Thus, for example, each liter of a "0.5 molar" or "0.5 M" solution of urea (Template:Chem) in water contains 0.5 moles of that molecule. By extension, the amount concentration is also commonly called the molarity of the substance of interest in the solution. However, as of May 2007, these terms and symbols are not condoned by IUPAC.<ref>Template:GreenBook3rd</ref>

This quantity should not be confused with the mass concentration, which is the mass of the substance of interest divided by the volume of the solution (about 35 g/L for sodium chloride in ocean water).

Amount (molar) fraction (moles per mole)

Confusingly, the amount (molar) concentration should also be distinguished from the molar fraction (also called mole fraction or amount fraction) of a substance in a mixture (such as a solution), which 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 Template:Chem 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 Template:Chem will be Template:Nowrap.

In a mixture of gases, the partial pressure of each component is proportional to its molar fraction.

History

The alchemists, and especially the early metallurgists, probably had some notion of amount of substance, but there are no surviving records of any generalization of the idea beyond a set of recipes. In 1758, Mikhail Lomonosov questioned the idea that mass was the only measure of the quantity of matter,<ref>Template:Cite book</ref> but he did so only in relation to his theories on gravitation. The development of the concept of amount of substance was coincidental with, and vital to, the birth of modern chemistry.

• 1777: Wenzel publishes Lessons on Affinity, in which he demonstrates that the proportions of the "base component" and the "acid component" (cation and anion in modern terminology) remain the same during reactions between two neutral salts.<ref name="Larousse">Template:Cite journal. Template:In lang</ref>
• 1789: Lavoisier publishes Treatise of Elementary Chemistry, introducing the concept of a chemical element and clarifying the law of conservation of mass for chemical reactions.<ref>Template:Cite book. Template:In lang</ref>
• 1792: Richter publishes the first volume of Stoichiometry or the Art of Measuring the Chemical Elements (publication of subsequent volumes continues until 1802). The term "stoichiometry" is used for the first time. The first tables of equivalent weights are published for acid–base reactions. Richter also notes that, for a given acid, the equivalent mass of the acid is proportional to the mass of oxygen in the base.<ref name="Larousse" />
• 1794: Proust's law of definite proportions generalizes the concept of equivalent weights to all types of chemical reaction, not simply acid–base reactions.<ref name="Larousse" />
• 1805: Dalton publishes his first paper on modern atomic theory, including a "Table of the relative weights of the ultimate particles of gaseous and other bodies".<ref>Template:Cite journal</ref>

  The concept of atoms raised the question of their weight. While many were skeptical about the reality of atoms, chemists quickly found atomic weights to be an invaluable tool in expressing stoichiometric relationships.

• 1808: Publication of Dalton's A New System of Chemical Philosophy, containing the first table of atomic weights (based on H = 1).<ref>Template:Cite book</ref>
• 1809: Gay-Lussac's law of combining volumes, stating an integer relationship between the volumes of reactants and products in the chemical reactions of gases.<ref>Template:Cite journal English translation.</ref>
• 1811: Avogadro hypothesizes that equal volumes of different gases (at same temperature and pressure) contain equal numbers of particles, now known as Avogadro's law.<ref>Template:Cite journal English translation.</ref>
• 1813/1814: Berzelius publishes the first of several tables of atomic weights based on the scale of m(O) = 100.<ref name="Larousse" /><ref>Excerpts from Berzelius' essay: Part II; Part III.</ref><ref>Berzelius' first atomic weight measurements were published in Swedish in 1810: Template:Cite journal</ref>
• 1815: Prout publishes his hypothesis that all atomic weights are integer multiple of the atomic weight of hydrogen.<ref>Template:Cite journal</ref> The hypothesis is later abandoned given the observed atomic weight of chlorine (approx. 35.5 relative to hydrogen).
• 1819: Dulong–Petit law relating the atomic weight of a solid element to its specific heat capacity.<ref>Template:Cite journal English translation</ref>
• 1819: Mitscherlich's work on crystal isomorphism allows many chemical formulae to be clarified, resolving several ambiguities in the calculation of atomic weights.<ref name="Larousse" />
• 1834: Clapeyron states the ideal gas law.<ref>Template:Cite journal</ref>

  The ideal gas law was the first to be discovered of many relationships between the number of atoms or molecules in a system and other physical properties of the system, apart from its mass. However, this was not sufficient to convince all scientists of the existence of atoms and molecules, many considered it simply being a useful tool for calculation.

• 1834: Faraday states his laws of electrolysis, in particular that "the chemical decomposing action of a current is constant for a constant quantity of electricity".<ref>Template:Cite journal</ref>
• 1856: Krönig derives the ideal gas law from kinetic theory.<ref>Template:Cite journal</ref> Clausius publishes an independent derivation the following year.<ref>Template:Cite journal</ref>
• 1860: The Karlsruhe Congress debates the relation between "physical molecules", "chemical molecules" and atoms, without reaching consensus.<ref>Wurtz's Account of the Sessions of the International Congress of Chemists in Karlsruhe, on 3, 4, and 5 September 1860.</ref>
• 1865: Loschmidt makes the first estimate of the size of gas molecules and hence of number of molecules in a given volume of gas, now known as the Loschmidt constant.<ref>Template:Cite journal English translation Template:Webarchive.</ref>
• 1886: van't Hoff demonstrates the similarities in behaviour between dilute solutions and ideal gases.
• 1886: Eugen Goldstein observes discrete particle rays in gas discharges, laying the foundation of mass spectrometry, a tool subsequently used to establish the masses of atoms and molecules.
• 1887: Arrhenius describes the dissociation of electrolyte in solution, resolving one of the problems in the study of colligative properties.<ref>Template:Cite journal English translation Template:Webarchive.</ref>
• 1893: First recorded use of the term mole to describe a unit of amount of substance by Ostwald in a university textbook.<ref>Template:Cite book</ref>
• 1897: First recorded use of the term mole in English.<ref>Template:Cite book</ref>
• By the turn of the twentieth century, the concept of atomic and molecular entities was generally accepted, but many questions remained, not least the size of atoms and their number in a given sample. The concurrent development of mass spectrometry, starting in 1886, supported the concept of atomic and molecular mass and provided a tool of direct relative measurement.
• 1905: Einstein's paper on Brownian motion dispels any last doubts on the physical reality of atoms, and opens the way for an accurate determination of their mass.<ref>Template:Cite journal</ref>
• 1909: Perrin coins the name Avogadro constant and estimates its value.<ref>Template:Cite journal Extract in English, translation by Frederick Soddy.</ref>
• 1913: Discovery of isotopes of non-radioactive elements by Soddy<ref>Template:Cite journal</ref> and Thomson.<ref>Template:Cite journal</ref>
• 1914: Richards receives the Nobel Prize in Chemistry for "his determinations of the atomic weight of a large number of elements".<ref>Söderbaum, H.G. (November 11, 1915). Statement regarding the 1914 Nobel Prize in Chemistry.</ref>
• 1920: Aston proposes the whole number rule, an updated version of Prout's hypothesis.<ref>Template:Cite journal</ref>
• 1921: Soddy receives the Nobel Prize in Chemistry "for his work on the chemistry of radioactive substances and investigations into isotopes".<ref>Söderbaum, H.G. (December 10, 1921). Presentation Speech for the 1921 Nobel Prize in Chemistry.</ref>
• 1922: Aston receives the Nobel Prize in Chemistry "for his discovery of isotopes in a large number of non-radioactive elements, and for his whole-number rule".<ref>Söderbaum, H.G. (December 10, 1922). Presentation Speech for the 1922 Nobel Prize in Chemistry.</ref>
• 1926: Perrin receives the Nobel Prize in Physics, in part for his work in measuring the Avogadro constant.<ref>Oseen, C.W. (December 10, 1926). Presentation Speech for the 1926 Nobel Prize in Physics.</ref>
• 1959/1960: Unified atomic mass unit scale based on m(¹²C) = 12 u adopted by IUPAP and IUPAC.<ref name="Holden">Template:Cite journal</ref>
• 1968: The mole is recommended for inclusion in the International System of Units (SI) by the International Committee for Weights and Measures (CIPM).<ref name="SI">Template:SIbrochure8th</ref>
• 1972: The mole is approved as the SI base unit of amount of substance.<ref name="SI" />
• 2019: The mole is redefined in the SI as "the amount of substance of a system that contains Template:Val specified elementary entities".<ref name=bipm9th/>

See also

• International System of Quantities
• Quantity of matter