Avogadro constant

The Avogadro constant (symbols: L, N_A), also called the Avogadro number, is the number of atoms in exactly 12 grams of ¹²C. A mole is defined as this number of "entities" (usually, atoms or molecules) of any material. The currently accepted value for this number is:

<math>N_A = (6.022 \, 141 \, 79\pm 0.000 \, 000 \, 30)\,\times\,10^{23} \mbox{ mol}^{-1} \,</math>

The value of Avogadro's constant was first indicated by Johann Josef Loschmidt who, in 1865, computed the number of particles in one cubic centimetre of gas held at standard conditions. The term Loschmidt constant is thus more correctly applied for this value, which can be said to be proportional to the Avogadro number. However, in the German language literature, "Loschmidt constant" refers to both this quantity as well as the number of entities in a mole.

History and etymology

The Avogadro constant is named after the early nineteenth century Italian scientist Amedeo Avogadro, who is credited (1811) with being the first to realize that the volume of a gas (strictly, of an ideal gas) is proportional to the number of atoms or molecules. The French chemist Jean Baptiste Perrin in 1909 proposed naming the constant in honor of Avogadro. American chemistry textbooks picked it up in the 1930s followed by high school textbooks starting in the 1950s.

Loschmidt: Measuring the Avogadro Number

Avogadro never attempted to measure the constant: the numerical value was first estimated by the Austrian physicist Johann Josef Loschmidt in 1865 using the kinetic theory of gases. In German-speaking countries, the constant is sometimes referred to as the Loschmidt constant or Loschmidt's number, but what Loschmidt actually measured is the number of particles per unit volume of an ideal gas, i.e. the number density of particles in an ideal gas. Thus, the term Loschmidt constant is more appropriate to the number of particles per unit of gas, Symbol: n_o:

<math>n_\circ = \frac{p}{k_B T} \,</math>

equal to (2.686 7774 ± 0.000 0047) × 10²⁵ m⁻³ at 273.15 K and 101.325 kPa with k_B the Boltzmann constant, T the temperature and p the pressure.

This constant is related to the Avogadro constant by the relation:

<math> R = N_Ak_B \,</math>

with k_B the Boltzmann constant, and R the gas constant, hence

<math>N_A = \frac{n_\circ R T}{p} \,</math>

However, in German scientific literature, the term Loschmidt constant is used ambiguously - it often refers to what has come to be known as Avogadro number. However, Boltzmann had first used the term for the sense in which Loschmidt had measured it - the number of particles in a unit volume of gas. Thus, its use in referring to the related Avogadro number leads to confusion and is usually deprecated. For the same reason, sometimes the Avogadro Number is denoted as L as opposed to N_A in German texts.

Standardization of the mole

Before 1960, there were conflicting definitions of the mole, and hence of the Avogadro number (as it was known at the time), based on 16 grams of oxygen: physicists generally used oxygen-16 while chemists generally used the "naturally occurring" isotope ratio.

Switching, in 1960, to 12 grams of carbon-12 as the basis ended this dispute and had other advantages. At this time, the Avogadro number was defined as the number of atoms in 12 grams of carbon-12, that is as a dimensionless quantity, while a mole was defined as one Avogadro number of atoms, molecules or other entities.

When the mole entered the International System of Units (SI), in 1971, the definitions were interchanged.

In 1971, a mole was defined as the number of atoms in 12 grams of carbon-12, with its own dimension, namely "amount of substance". Avogadro's number became a physical constant with the unit of reciprocal moles (mol⁻¹).

While Avogadro’s constant is a dimensionless quantity, it is currently based on a single physical property (the defined gram, hence the golden karat). Since instruments are unable to measure that property accurately beyond 10 places it is not known to an integer value. The Binary Mole, on the other hand, defines a precise integer value for the units that occur in a mole without any such analytical attachment. Thus, the binary mole is an integer of 24 digits:

N_o = 2⁷⁹ = 604 462 909 807 314 587 353 088

or 6.0446 x 10²³ to 5-significant digits for mathematical calculations.

Application

The Avogadro constant 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 g/mol, so N_A iron atoms (i.e. one mole of iron atoms) have a mass of 55.847 g. Conversely, 55.847 g of iron contains N_A iron atoms. The Avogadro constant also enters into the definition of the unified atomic mass unit, u:

<math>1 \ u = \frac{1}{N_A} \ g = (1.660 \, 538\, 86 \pm 0.000\, 000\, 28) 10^{-24} \ g</math>

Additional physical relations

Because of its role as a scaling factor, the Avogadro number provides the link between a number of useful physical constants when moving between the atomic scale and the macroscopic scale. For example, it provides the relationship between:

• the gas constant R and the Boltzmann constant k_B:

<math> R = k_BN_A = 8.314 \, 472 \, \pm \, 0.000 \, 015 \, \mbox{J}\cdot\mbox{mol}^{-1}\mbox{K}^{-1}\,</math>

in J mol⁻¹ K⁻¹

• the Faraday constant F and the elementary charge e:

<math> F = N_Ae = 96 \, 485.3383 \, \pm \,0.0083 \,\, \mbox{C}\cdot\mbox{mol}^{-1} \,</math>

in C mol⁻¹

Measurement of the Avogadro constant

File:Silicon-unit-cell-labelled-3D-balls.png

Ball-and-stick model of the unit cell of silicon. X-ray diffraction experiments can determine the length of the cell, a, which can in turn be used to calculate a value for Avogadro's constant

A number of methods can be used to measure the Avogadro constant. One modern method is to calculate the Avogadro constant from the density (ρ) of a crystal, the relative atomic mass (M), and the unit cell length (a) 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 the Avogadro constant:

<math> N_A = \frac{8M}{a^3\rho} \,</math>.

based on silicon.

See also

• Mole (unit)
• Large numbers

External links

• 1996 definition of the Avogadro constant from the IUPAC Compendium of Chemical Terminology ("Gold Book")
• Some Notes on Avogadro's Number, 6.022Template:E (historical notes)
• An Exact Value for Avogadro's Number -- American Scientist