Avogadro's number

Avogadro's number, also called Avogadro's constant (N_A), named after Amedeo Avogadro, is the number of atoms in a mole of an element or the number of molecules in a mole of a compound. Avogadro's number is formally defined in the SI as the number of carbon-12 atoms in 12 grams (0.012 kg) of unbound carbon-12 in its rest-energy electronic state. The accepted value is 6.0221415 atoms/mole. In the nineteenth century physicists measured the mass of one atom of hydrogen to be about 1/(6.023x10²³) grams; they were trying to evaluate how many molecules of an ideal gas would fit in 1 cubic centimeter ^*. Carbon-12 was chosen as the reference substance over hydrogen because its atomic mass could be measured more accurately.

A mole is defined in the SI as Avogadro's number of particles of any kind of substance (atoms, ions, molecules, or formula units). In the SI, this unit is abbreviated mol. The mole is the basic unit of amount of substance.

History

Avogadro's number is named after the early nineteenth 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 estimated 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: ^*.

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 u, 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

N_{\rm A}

corresponds to the conversion factor between grams (g) and atomic mass units:

1\ \mbox{g}=N_{\rm A} \mbox{u}.

Chemical 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" cylinder of metal in France. This means that the particular value of Avogadro's number is the result of convention; there is no physical reason for it. For this reason, Avogadro's number is not considered a fundamental constant in the strictest sense. However, for practical purposes, Avogadro's number is regarded as a chemical constant.

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" cylinder 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 cylinder 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 1/3 times Avogadro's number of carbon-12 atoms.

Additional physical relations

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

the universal gas constant R and the Boltzmann constant k_B: R = k_B N_A
the Faraday constant F and the elementary charge e: F = e N_A

In the 19th century physicists measured the mass of one atom of hydrogen to be about 1/(6.02214199) grams. The gram was originally defined to be the mass of a cubic centimeter of pure water at standard temperature and pressure 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 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 6.0221415 atoms. The recent history and more details can be found in the document, [http://www.iupac.org/publications/pac/1992/pdf/6410x1535.pdf Atomic Weight: The Name, Its History, Definition and Units.

Numerical value

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

6.022\,1415\times 10^{23}

\pm 0.000\,0010\times 10^{23} \mathrm{mol}^{-1}.

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 as a first approximation, being about 1/1840 of the mass of the proton). 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 it would appear that one mole of carbon-12 atoms, which should consist of 6 moles each of protons, neutrons, and electrons would have a mass greater than 12 g. The discrepancy or mass defect (as it is known) 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 (which means that all energy has mass), the released energy has mass and carries away the loss in the mass of the nucleus relative to that of the separated protons and neutrons (note that mass is conserved in this process just as energy is). Thus, protons and neutrons in the nucleus have masses that are less (about 0.7 percent less) than those of 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 (neucleons) 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, as explained above. For example, iron nucleons will have a significantly lower mass than those in hydrogen or plutonium.