Definitions

gram-atom

Mole (unit)

The mole (symbol: mol) is a unit of amount of substance: it is an SI base unit, and almost the only unit to be used to measure this physical quantity. The name "mole" was coined in German (as Mol) by Wilhelm Ostwald in 1893, although the related concept of equivalent mass had been in use at least a century earlier. The name is assumed to be derived from the word Molekül (molecule). The first usage in English dates from 1897, in a work translated from German. The names gram-atom and gram-molecule have also been used in the same sense as "mole", but these names are now obsolete.

The mole is defined as the amount of substance of a system which contains as many "elemental entities" (eg, atoms, molecules, ions, electrons) as there are atoms in 12 g of carbon-12. Hence:

  • one mole of iron contains the same number of atoms as one mole of gold;
  • one mole of benzene contains the same number of molecules as one mole of water;
  • the number of atoms in one mole of iron is equal to the number of molecules in one mole of water.

It is a common misconception that the mole is defined in terms of the Avogadro constant (also, anachronistically, known as "Avogadro's number"). It is not necessary to know the number of atoms or molecules which are present in order to use the mole as a unit of measurement, and indeed the first measurements of amount of substance predate modern atomic theory and any measurements of atomic weight. The current definition of the mole was approved during the 1960s: Prior to that, there had been definitions based on the atomic weight of hydrogen, the atomic weight of oxygen and the relative atomic mass of oxygen-16: the four different definitions are equivalent to within 1%.

The most common method of measuring an amount of substance is to measure its mass and then to divide by the molar mass of the substance. Molar masses may be easily calculated from tabulated values of atomic weights and the molar mass constant (which has a convenient defined value of 1 g/mol). Other methods include the use of the molar volume or the measurement of electric charge.

Definition

The legal definition of the mole in the International System of Units (SI) was approved by the 14th General Conference on Weights and Measures (CGPM) in 1971, based on recommendations from the International Union of Pure and Applied Physics (IUPAP), the International Union of Pure and Applied Chemistry (IUPAC) and the International Committee for Weights and Measures (CIPM).

1. The mole is the amount of substance of a system which contains as many elementary entities as there are atoms in 0.012 kilogram of carbon 12; its symbol is “mol”.
2. When the mole is used, the elementary entities must be specified and may be atoms, molecules, ions, electrons, other particles, or specified groups of such particles.

The CIPM approved a clarification to the definition in 1980, stating:

In this definition, it is understood that unbound atoms of carbon 12, at rest and in their ground state, are referred to.

Amount of substance

Amount of substance is a physical quantity which is proportional to the number of "elementary entities" (eg, atoms, molecules) which are present. The ratio between the measured amount of substance and the number of atoms or molecules need not be known so long as it is a true constant, ie the same for all measurements.

The constant of proportionality is called the Avogadro constant (N), named after Amadeo Avogadro (1776–1856) who was the first to propose that such a fixed ratio could exist (Avogadro's law, 1811). The current best estimate of the value of the Avogadro constant in SI units is 6.022 141 79(30)×10 mol: its measurement (using several different methods) by Jean Baptiste Perrin earned him the 1926 Nobel Prize for Physics.

Amount of substance, like electric charge, is quantized at the microscopic scale but continuous at the macroscopic scale. Hence it is appropriate to talk about numbers of atoms or electrons at the microscopic scale, but to use macroscopic physical quantities and units (such as the mole and the coulomb) when the number of quanta becomes so large that it is effectively uncountable.

Elementary entities

The term "elementary entities" is defined in the second part of the CGPM definition of the mole: "atoms, molecules, ions, electrons, other particles, or specified groups of such particles." These are "elementary" in the sense that they cannot be broken down into anything simpler without completely changing their nature: half a benzene molecule has no chemical ressemblance to a benzene molecule, and half a uranium atom is simply not uranium (it would presumably be a nuclide of niobium).

Several popular science texts contain "examples" of the mole which ignore this part of the definition, for example:

  • 1 mole of marshmallows would be enough marshmallows to make a 12 mile thick layer of marshmallows covering the entire face of the Earth.
  • A mole of popcorn kernels could be spread uniformly over the USA if the thickness of the layer was about 9 miles.
  • A mole of donut holes would cover the earth and be 5 miles (8 km) deep.
  • A mole of blood cells would be more than the total number of blood cells found in every human on earth.

These (and similar) are examples of the fact that atoms and molecules are very small compared to everyday objects. Marshmallows, popcorn kernels, donut holes and blood cells are not "elementary entities", nor are they "substances" whose amount is measured in moles.

In some cases, there could be confusion as to which elementary entity the measurement refers. Hence, "one mole of hydrogen" could refer to hydrogen atoms or H molecules. In these cases, it is essential to specify which entities are being referred to.

Unbound atoms at rest and in their ground state

The 1980 clarification from the CIPM refers to "unbound atoms of carbon 12, at rest and in their ground state" as the basis for the definition of the mole. The distinction is important, as accurate measurements of relative atomic mass, which form the basis for measurements of amount of substance, are not carried out on atoms at rest but rather on ions in a Penning trap. Hence three corrections must be made to the results before they can be used in more normal laboratory situations:

  • a correction for the mass of the electrons lost, which corresponds to approximately 1 part in 3646 of the mass of the carbon-12 atom;
  • a correction for the mass equivalent of the electron binding energy, which is about 1 part in 10 of the mass of the carbon-12 atom;
  • a relativistic correction for the speed of the ions in the Penning trap, which depends on the exact details of the experiment.

The small correction for the mass equivalent of the bond energy in solid carbon, about 5 parts in 10 of the mass of a carbon atom, is not of practical importance: only a very few relative atomic masses are known to this level of precision, and none of these values were obtained from bound carbon atoms.

The mole as a unit

Since its adoption into the International System of Units, there have been a number of criticisms of the concept of the mole being a unit in the same way as the metre or the second. These criticisms may be briefly summarised as:

  • amount of substance is not a true physical quantity (or dimension), and is redundant to mass, so should not have its own base unit;
  • the mole is simply a shorthand way of referring to a large number.

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 which 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, most notably 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.

The second misconception, that the mole is simply a counting aid, has found its way even into elementary chemistry textbooks. It contends that the mole is defined in terms of the Avogadro constant, rather than the other way round, and so is equal to 6.022 141 79×10 elementary entities (or marshmallows, etc.).

Consider the measurement of one mole of silicon. As silicon is a solid at room temperature, the convenient method is by weighing. By consulting published tables, it can easily be found that the atomic weight of silicon is 28.0855. Multiplying by the molar mass constant M gives the molar mass in any desired mass units: assuming the measurement is to be made in grams, M = 1 g/mol, and so the molar mass of silicon is 28.0855 g/mol. Hence, 28.0855 g of silicon is equivalent to one mole of silicon, without the Avogadro constant ever having come into play.

Counting (or calculating) the number of atoms in 28.0855 g of silicon is one way of determining the Avogadro constant, N, and a way which is currently receiving a lot of attention (see below) although, as of the 2006 CODATA values of the physical constants, it is not the most accurate. It is only a method of determining N because it is known by other means that 28.0855 g of silicon is equivalent to one mole. Those other means are:

  • the very accurate determination of the ratios of the masses of each of the three silicon nuclides to the mass of an atom of carbon-12, in such a way that it is known that a silicon-28 atom is [27.976 926 5327(20)/12] times as massive as a carbon-12 atom;
  • the determination of the isotopic abundance of silicon in the samples used to make the measurements, allowing the calculation of the atomic weight of silicon in each individual sample;
  • the definition of 12 g of carbon-12 atoms to be equivalent to one mole.

History

The first table of atomic weights was published by John Dalton (1766–1844) in 1805, based on a system in which the atomic weight of hydrogen was defined as 1. These atomic weights were based on the stoichiometric proportions of chemical reactions and compounds, a fact which 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 weights (promoted by proponants of atomic theory) and equivalent weights (promoted by its opponants and which sometimes differed from atomic weights by an integer factor), which would last throught much of the nineteenth century.

Jöns Jacob Berzelius (1779–1848) was instrumental in the determination of atomic weights to ever inceasing accuracy. He was also the first chemist to use oxygen as the standard to which other weights 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 weight of oxygen as 100, an innovation which did not catch on.

Charles Frédéric Gerhardt (1816–56), Henri Victor Regnault (1810–78) and Stanislao Cannizzaro (1826–1910) expanded on Berzelius' work, resolving many of the problems of unknown stoichiometry of compounds, and the use of atomic weights attracted a large consensus by the time of the Karlsruhe Congress (1860). The convention had reverted to defining the atomic weight 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 weight standard became ever more evident with advances in analytical chemistry and the need for ever more accurate atomic weight determinations.

With the discovery of stable isotopes in 1913 and the development of mass spectrometry came a divergence between physicists and chemists as to the definition of the atomic weight scale. Chemists continued to measure atomic weights by chemical analysis, and naturally chose to fix their values against the atomic weight of natural oxygen (which contains three different isotopes). Physicists were measuring the relative atomic masses of discrete particles, and so naturally chose to fix their values against the nuclide of oxygen with a mass number of 16. The numerical difference between the two scales is minute, but became significant as measurements became ever more accurate. The dispute was finally resolved in 1959/1960, and a unified atomic weight scale based on C = 12 was adopted by IUPAP and IUPAP. This is the scale that forms the basis for the SI definition of the mole, adopted in 1972.

Scale basis Scale basis
relative to C = 12
Relative deviation
from the C = 12 scale
Atomic weight of hydrogen = 1 1.007 94(7) –0.788%
Atomic weight of oxygen = 16 15.9994(3) +37.5 ppm
Relative atomic mass of O = 16 15.994 914 6221(15) +318 ppm

Proposed future definition

Kilogram

As with other SI base units, there have been proposals to redefine the kilogram in such a way as to define some presently measured physical constants to fixed values. One proposed definition of the kilogram is:
The kilogram is the mass of exactly (6.0221415/0.012) unbound carbon-12 atoms at rest and in their ground state.
This would have the effect of defining the Avogadro constant to be precisely 6.0221415 elementary entities per mole.

See also

References

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