Avogadro's number, also called Avogadro's constant (NA) is a large constant used in chemistry and physics.

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## Definition

Avogadro's number is formally defined as the number of carbon-12 atoms in 0.012 kg of carbon-12. It is approximately 6.022 × 1023 particles/mole. Historically, carbon-12 was chosen as the reference substance because its atomic mass could be measured particularly accurately.

A mole is defined as Avogadro's number of particles of any kind of substance (atoms, ions, molecules, or formula units).

## History

Avogadro's number is named after the early 19th century Italian scientist Amedeo Avogadro. It appears that Jean Baptiste Perrin was the first to name it. Perrin called it "Avogadro's constant" and it is still sometimes known by that name. The numerical value was first calculated by Johann Josef Loschmidt in 1865 using the kinetic gas theory. In German-speaking countries, the number may still be referred to as Loschmidt's number. Unfortunately, in a few cases (mainly in the older literature) Loschmidt's number refers to the number of atoms (or molecules) in a cubic centimeter, a usage now disparaged, viz:  (http://gemini.tntech.edu/~tfurtsch/scihist/loschmid.html)

## Application

Avogadro's number can be applied to any substance. It corresponds to the number of atoms or molecules needed to make up a mass equal to the substance's atomic or molecular mass, in grams. For example, the atomic mass of iron is 55.847 amu, so Avogadro's number of iron atoms (i.e. one mole of iron atoms) have a mass of 55.847 g. Conversely, 55.847 g of iron contains Avogadro's number of iron atoms. Thus Avogadro's number corresponds to the conversion factor between grams (g) and atomic mass units:

[itex]1\ \mbox{g}=N_{\rm A}\ \mbox{amu}.[itex]

## Physical significance of Avogadro's number

The value of Avogadro's number depends on the definition of the mole, which depends on the definition of the kilogram. Both definitions, especially that of the kilogram, are arbitrary: the kilogram system is currently based on the mass of a particular "standard" bar of metal in France. Clearly, this means that the value of Avogadro's number is less fundamental than other physical constants in the sense that there is no physical reason for its particular value. However, Avogadro's number is still a fundamental constant: all constants depend on the units used and on the definition of the units, and therefore, such a dependence does not exclude that a constant can be called fundamental.

Avogadro's number can be regarded as a conversion factor between the microscopic mass system (atomic mass units or Daltons) and the kilogram system. The microscopic mass system is based on the mass of carbon-12, while the kilogram system is currently based on the mass of a particular "standard" bar of metal in France. So naturally there's no simple conversion factor between the two. However, if a method were developed to count atoms, it would be possible to redefine the kilogram in a way that did not depend on an arbitrary bar of metal. The number of atoms picked would presumably be equal or close to the latest accepted value of Avogadro's number. In that case, the kilogram would be redefined as the mass of 1/0.012=83.333 times Avogadro's number of Carbon atoms.

Because of its role as a scaling factor, Avogadro's number provides the link between a number of useful physical constants when we move between an atomic mass scale and a kilogram (SI) scale. For example, it provides the relationship between:

In the 19th century physicists measured the mass of one atom of hydrogen to be about 1/(6.023x1023) grams. The gram was originally defined to be the mass of a cubic centimeter of pure water at standard temperature and pressure  (http://www.factmonster.com/ce6/sci/A0832931.html). As experiments became more accurate, it was found that water was contaminated with variable amounts of heavy water, which made it undesirable to maintain a standard with hydrogen having one a.m.u. (atomic mass unit). Carbon was found to have a more constant isotopic composition, and it was also possible to separate pure carbon-12. Therefore, the atomic mass unit was changed to 1/12 the mass of an atom of carbon-12 . Hence 12 grams of carbon-12 has about 6.02214x1023 atoms. The recent history and more details can be found in the document, Atomic Weight: The Name, Its History, Definition and Units (http://www.iupac.org/publications/pac/1992/pdf/6410x1535.pdf).

## Numerical value

At present it is not technologically feasible to count the exact number of atoms in .012 kg of carbon-12, so the precise value of Avogadro's number is unknown. The 2002 CODATA recommended value for Avogadro's number is

[itex]6.0221415(10)\times 10^{23}\hbox{ mol}^{-1},[itex]

where the number in parentheses represents the one standard deviation uncertainty in the last digits of the value.

A number of methods can be used to measure Avogadro's number. One modern method is to calculate Avogadro's number from the density of a crystal, the relative atomic mass, and the unit cell length determined from x-ray crystallography. Very accurate values of these quantities for silicon have been measured at the National Institute of Standards and Technology (NIST) and used to obtain the value of Avogadro's number.

## Connection to masses of protons and neutrons

A carbon-12 atom consists of 6 protons and 6 neutrons (which have approximately the same mass) and 6 electrons (whose mass is negligible in comparison). One could therefore think that NA is the number of protons or neutrons that have a mass of 1 gram. While this is approximately correct, the mass of a free proton is 1.00727 amu, so a mole of protons would actually have a mass of 1.00727 g. Similarly, a mole of neutrons has a mass of 1.00866 g. Clearly, 6 moles of protons combined with six moles of neutrons would have a mass greater than 12 g. So, you might ask how one mole of carbon-12 atoms, which should consist of 6 moles each of protons, neutrons, and electrons could possibly have a mass of only 12 g? What happened to the excess mass? The answer is related to the equivalence of matter and energy discovered by Albert Einstein as part of the theory of special relativity. When an atom is formed, the protons and neutrons in the nucleus are bound together by the strong nuclear force. This binding results in the formation of a low energy state and is accompanied by a large release of energy. Since energy is equivalent to mass, the released energy corresponds to a loss in the mass of the nucleus relative to that of the separated protons and neutrons. Thus, protons and neutrons in the nucleus have masses that are less (about 0.7 percent less) than free protons and neutrons. The precise amount of mass loss is related to the binding energy of the nucleus and varies depending on the type of atom.

One may therefore say that NA is approximately the number of nuclear neutrons or protons that have a mass of 1 gram. This is approximate because the precise mass of a nuclear proton or neutron depends on the composition of the nucleus. For example, iron nucleons will have a significantly lower mass than those in hydrogen or plutonium.

Avogadro's number often yields practical reasonings in real life. For example, the fact that a known number of atoms are in a given amount of a substance is one reason for scientific criticism of homeopathy, in which medicinal substances are often diluted to the extent that a simple calculation involving Avogadro's number would imply that less than a single molecule remains.

• Journal of Physical and Chemical Reference Data, 28 (1999) 1713.

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