Avogadro's number

Avogadro's number, N_A, is defined as the number of atoms in 12 gram of carbon-12 atoms in their ground state at rest. By definition it is related to the atomic mass constant m_u by the relation

$N_{\mathrm {A} }m_{\mathrm {u} }=1/1000\quad {\hbox{exact}}.$

The exact factor 1/1000 appears here by the historic facts that the kilogram is the unit of mass and that in chemistry the mole is preferred over the Kmole. Recall that the atomic mass constant has the mass 1 u exactly (u is the unified atomic mass unit). Avogadro's number is indeed defined as number, a dimensionless quantity. Its latest numeric value is N_A = 6.022 141 79 10²³.

The SI definition of Avogadro's constant (also designated by N_A) is: the number of entities (such as atoms, ions, or molecules) per mole. (This definition requires, of course, a definition of mole that does not rely on N_A, but one that is in terms of ¹²C atoms). In this definition N_A has dimension mol⁻¹. The numeric value of Avogadro's constant is N_A = 6.022 141 79 10²³ mol⁻¹.

Because the mole and Avogadro's number are defined in terms of the atomic mass constant (one twelfth of the mass of a ¹²C atom), Avogadro's constant and Avogadro's number have by definition the same numerical value. In practice the two terms are used interchangeably.

History of Avogadro's number

Since 1811, when Amedeo Avogadro put forward his law stating that equal volumes of gas (we now know ideal gas) contain equal number of particles, increasingly sophisticated methods of determining Avogadro’s constant have been developed. These include the kinetic theory of gases, Brownian motion, measurement of the electron charge, black-body radiation, alpha particle emission, and X-ray measurements of crystals.

Without the belief that a macroscopic substance consists of minute particles (initially called atoms, later also molecules), it does not make sense to speak of Avogadro's number. This belief—called atomism— was born in antiquity and grew further in importance with the developments of chemistry early in the 19th century. An important milestone was John Dalton’s law of multiple proportions published in 1804 that gave rise to the first table of the relative weights of atoms. In 1808 Joseph-Louis Gay-Lussac published his law for the combining volumes of gases, namely that gases combine among themselves in very simple proportions of their volumes, and if the products are gases, their volumes are also in simple proportions. Especially this latter law was of great influence on Avogadro's historical publication of 1811, in which he introduced the term "molecule" and enunciated his law.

In his 1811 paper Avogadro discusses Dalton’s atomic theory and calculates from gas densities that the molecular weight of nitrogen is nearly fourteen times the molecular weight of hydrogen. Avogadro was the first to propose that the gaseous elements, hydrogen, oxygen, and nitrogen, were diatomic molecules. He deduced that the molecule of water contains half a molecule of oxygen and one molecule (or two half molecules) of hydrogen. Dalton, who had assumed earlier that water is formed from a molecule each of oxygen and hydrogen, rejected Avogadro's and Gay-Lussac's laws. There are no testimonials that Avogadro ever speculated on the number of molecules in a given gas volume.

Avogadro's law went for a long time largely unnoticed, not in the least because it was not recognized that the law holds strictly only for ideal gases, which many dissociating and associating organic compounds are not. Four years after his death, at the historic (1860) chemistry conference in Karlsruhe, his countryman Stanislao Cannizaro explained why the exceptions to Avogadro's law happen and that it can determine molar masses.

Kinetic theory estimates

The first estimate of Avogadro’s constant is attributed to Johann Josef Loschmidt (1865). He gave a value for L, the number of molecules in 1 cm³ at standard temperature and pressure. The number L is called Loschmidt’s number; the Avogadro equivalent of this is

N_A = 0.410×10²³.

Loschmidt estimated his number by applying the kinetic gas theory of James Clerk Maxwell and Rudolph Clausius, together with experimental data on gas viscosities and atomic volumes. Loschmidt’s work was the first to show that Avogadro’s constant very large. In 1873 Maxwell used his kinetic theory of the diffusion coefficient of a gas to obtain a ten times larger value: N_A = 4.2×10²³.

A simple method for getting the actual volume of molecules is to use the 1873 Van der Waals equation that contains a parameter b, which is the volume of a single molecule. From this and the volume of the total gas, an estimate of the number of molecules in the gas can be obtained. Much later (1923) Perrin measured b for mercury vapor, and combining this with results from viscosity measurements, he calculated Avogadro's number to be 6.2×10²³. which is a very good value.

Estimates from Brownian motion

Before Einstein wrote his famous 1905 paper on Brownian motion, he had already worked on the size of molecules and the closely related problem of the magnitude of N_A. In his 1905 Ph.D. thesis, that dealt with the size of molecules, Einstein gave the estimate 2.1×10²³. In a paper based on his doctorate work that appeared in 1906 Einstein gave the estimate N_A = 4.15×10²³, close to Maxwell's value of 1873. Later it was discovered that Einstein made an algebraic error in his thesis and the paper based on it. When this was corrected the very same experimental data gave N_A = 6.6×10²³.

The phenomenon of Brownian motion was first described by Robert Brown in 1828 as the "tremulous motion" of pollen grains observed as suspensions in liquids. A theory of Brownian motion was developed by Einstein in a paper from 1905.