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Avogadro's number

Avogadro's number, named in honor of Amedeo Avogadro and denoted by N_A, is the number of items in a mole. It is formally defined as the number of Carbon-12 atoms in 0.012 kg of Carbon-12.

Although defined in terms of Carbon-12, Avogadro's Number can be applied to any material. It corresponds to the number of atoms or molecules needed to make up a mass equal to the material's atomic or molecular weight (in grams). For example, the atomic weight of iron is 55.847, 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 also corresponds to the conversion factor between grams (g) and atomic mass units (amu):

Numerical value

At present it is not technologically feasible to count the exact number of atoms in 0.012 kg of Carbon-12, so the precise value of Avogadro's Number is unknown. The 1998 CODATA Recommended Value for Avogadro's Number is

where the number in parenthesis represents the one-standard-deviation uncertainty in the last digits of the value [1]. For the purposes of simplicity, Avogadro's Number is sometimes rounded to merely:

which is sufficiently accurate for many applications.

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 mass 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 N_A 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 N_A 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.