Definitions

The Avogadro constant (symbols: L, NA), also called Avogadro's number, is the number of "elementary entities" (usually atoms or molecules) in one mole, that is (from the definition of the mole) the number of atoms in exactly 12 grams of carbon-12. The 2006 CODATA recommended value is entities per mole.

The Avogadro constant is named after the early nineteenth century Italian scientist Amedeo Avogadro, who, in 1811, first proposed that the volume of a gas (at a given pressure and temperature) is proportional to the number of atoms or molecules regardless of the nature of the gas. The French physicist Jean Perrin in 1909 proposed naming the constant in honour of Avogadro. Perrin would win the 1926 Nobel Prize in Physics, in a large part for his work in determining the Avogadro constant by several different methods.

The value of the Avogadro constant was first indicated by Johann Josef Loschmidt who, in 1865, estimated the average diameter of the molecules in air by a method that is equivalent to calculating the number of particles in a given volume of gas. This latter value, the number density of particles in an ideal gas, is now called the Loschmidt constant in his honour, and is approximately proportional to the Avogadro constant. The connection with Loschmidt is the root of the symbol L sometimes used for the Avogadro constant, and German language literature may refer to both constants by the same name, distinguished only by the units of measurement.

## In other units

While it is rare to use units of amount of substance other than the mole, the Avogadro constant can also be defined in units such as the pound mole (lb-mol.) and the ounce mole (oz-mol.).
N = 2.731 597 57(14) lb-mol.–1 = 1.707 248 479(85) oz-mol.–1

## 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 NA iron atoms (i.e. one mole of iron atoms) have a mass of 55.847 g. Conversely, 55.847 g of iron contains NA iron atoms. The Avogadro constant also enters into the definition of the unified atomic mass unit, u:
$1 mathrm\left\{u\right\} = frac\left\{1\right\}\left\{N_A\right\} mathrm\left\{g\right\} = \left(1.660 , 538, 86 pm 0.000, 000, 28\right) 10^\left\{-24\right\} mathrm\left\{g\right\}$

Because of its role as a scaling factor, the Avogadro constant 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:

$R = k_BN_A = 8.314 , 472 , pm , 0.000 , 015 , mbox\left\{J\right\}cdotmbox\left\{mol\right\}^\left\{-1\right\}mbox\left\{K\right\}^\left\{-1\right\},$

in J mol−1 K−1

$F = N_Ae = 96 , 485.3383 , pm ,0.0083 ,, mbox\left\{C\right\}cdotmbox\left\{mol\right\}^\left\{-1\right\} ,$

in C mol−1

## Measurement

### Coulometry

The earliest accurate method to measure the value of the Avogadro constant was based on coulometry. The principle is to measure the Faraday constant F, which is the electric charge carried by one mole of electrons, and to divide by the elementary charge e to obtain the Avogadro constant.
$N_\left\{rm A\right\} = frac\left\{F\right\}\left\{e\right\}$
The classic experiment is that of Bowers and Davis at NIST, and relies on dissolving silver metal away from the anode of an electrolysis cell, while passing a constant electric current I for a known time t. If m is the mass of silver lost from the anode and A the atomic weight of silver, then the Faraday constant is given by:
$F = frac\left\{A_\left\{rm r\right\}M_\left\{rm u\right\}It\right\}\left\{m\right\}$
The NIST workers devised an ingenious method to compensate for silver that was lost from the anode for mechanical reasons, and conducted an isotope analysis of their silver to determine the appropriate atomic weight. Their value for the conventional Faraday constant is F = 96 485.39(13) C/mol, which corresponds to a value for the Avogadro constant of 6.022 1449(78) mol–1: both values have a relative standard uncertainty of 1.3.

### Electron mass method (CODATA)

The CODATA value for the Avogadro constant is determined from the ratio of the molar mass of the electron A(e)M to the rest mass of the electron m:
$N_\left\{rm A\right\} = frac\left\{A_\left\{rm r\right\}\left(\left\{rm e\right\}\right)M_\left\{rm u\right\}\right\}\left\{m_\left\{rm e\right\}\right\}$
The "relative atomic mass" of the electron, A(e), is a directly-measured quantity, and the molar mass constant, M, is a defined constant in the SI system. The electron rest mass, however, is calculated from other measured constants:
$m_\left\{rm e\right\} = frac\left\{2R_\left\{infty\right\}h\right\}\left\{calpha^2\right\}$
As can be seen from the table of 2006 CODATA values below, the main limiting factor in the accuracy to which the value of the Avogadro constant is known is the uncertainty in the value of the Planck constant, as all the other constants which contribute to the calculation are known much more accurately.

Constant Symbol 2006 CODATA value Relative standard uncertainty Correlation coefficient
with N
Electron relative atomic mass A(e) 5.485 799 0943(23) 4.2 0.0082
Molar mass constant M 0.001 kg/mol defined
Rydberg constant R 10 973 731.568 527(73) m–1 6.6 0.0000
Planck constant h 6.626 068 96(33) Js 5.0 –0.9996
Speed of light c 299 792 458 m/s defined
Fine structure constant α 7.297 352 5376(50) 6.8 0.0269
Avogadro constant N 6.022 141 79(30) mol–1 5.0

### X-ray crystal density method

One modern method to calculate the Avogadro constant is to use ratio of the molar volume V to the unit cell volume V for a single crystal of silicon:

$N_\left\{rm A\right\} = frac\left\{8V_\left\{rm m\right\}\left(\left\{rm Si\right\}\right)\right\}\left\{V_\left\{rm cell\right\}\right\}$
The factor of eight arises because there are eight silicon atoms in each unit cell.

The unit cell volume can be obtained by X-ray crystallography: as the unit cell is cubic, the volume is the cube of the length of one side (known as the unit cell parameter, a. In practice, measurements are carried out on a distance known as d(Si), which is the distance between the planes denoted by the Miller indices {220} and is equal to a/√8. The 2006 CODATA value for d(Si) is 192.015 5762(50) pm, a relative uncertainty of 2.8, corresponding to a unit cell volume of 3.128 775 48(27) m3.

The molar volume requires a series of measurements to be determined. Silicon occurs with three stable isotopes28Si, 29Si, 30Si – and the natural variation in the proportions of these isotopes is greater than the other uncertainties in the other measurements, so the proportions must be determined for each crystal which is used. With these values, the atomic weight A for that crystal can be calculated, as the relative atomic masses of the three nuclides are known with great accuracy. The crystal must also be weighed and measured to determine its density ρ. Once all these quantities are known, the molar volume V is given by:

$V_\left\{rm m\right\} = frac\left\{A_\left\{rm r\right\}M_\left\{rm u\right\}\right\}\left\{rho\right\}$
where M is the molar mass constant. The 2006 CODATA value for the molar volume of silicon is 12.058 8349(11) cm3mol–1, with a relative standard uncertainty of 9.1.

As of the 2006 CODATA recommended values, the relative uncertainty in determinations of the Avogadro constant by the X-ray crystal density method is 1.2, about two and a half times higher than that of the electron mass method.