avogadro's number

Avogadro's number, also called Avogadro's constant (N_A) is a large constant used in chemistry and physics, formally defined as the number of carbon-12 atoms in 0.012 kg of carbon-12. Avogadro's number 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.

Definition

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:

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

Physical significance of Avogadro's number

Some school chemistry courses introduce Avogadro's number as if it was some kind of fundamental constant. This is not the case. Avogadro's number is an artifact of the definition of the kilogram. Avogadro's number is essentially just 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" lump of metal in France. So naturally there's no simple conversion factor between the two. However, it wouldn't be totally outlandish to redefine the kilogram in terms of some particular number of atoms, rather than an arbitrary lump. If the atoms picked were carbon-12, then you would also end up redefining Avogadro's number to be an exact numerical value, rather than one obtained by measurement.

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 we move 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: R = k_B N_A
and the Faraday constant F and the elementary charge e: F = e N_A

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

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

where the number in parenthesis 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 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. For example, iron nucleons will have a significantly lower mass than those in hydrogen or plutonium.

Avogadro's number in life

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.