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, properties of liquid solutions, 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 gas 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 is 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 liquid solutions

Einstein wrote in his 1905 Ph.D. thesis about the size of molecules and the closely related problem of the magnitude of N_A. He derived equations for diffusion coefficients and viscosities in which Avogadro's number appears. From experimental values of the diffusion coefficients and viscosities of sugar solutions in water Einstein gave the estimate N_A = 2.1×10²³. In a later paper derived from his doctorate work he gave a better estimate from improved experimental data: N_A = 4.15×10²³, close to Maxwell's value of 1873. Later (1911) 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 (small solid particles of diameter on the order of a micrometer) suspended in water. Einstein's famous 1905 paper on the theory of Brownian motion gives a method for determining N_A, but not yet a value.

The first to give a value to N_A from Brownian motion was Perrin in 1908. He considered the distribution of Brownian particles in a vertical column in the earths's gravitational field, and he used a similar mathematical approach to that which leads to the distribution of gas molecules in a vertical column of the atmosphere. This is a Boltzmann distribution that contains among the mass of the particle and the gravitational acceleration g the Boltzmann constant k. For suspended Brownian particles one has to make a correction for the buoyancy of the particles in the liquid (Archimedes principle), by using the expressions that contain the densities of the particles and the liquid. Measuring the numbers of particles at two different heights allows one to determine Boltzmann’s constant, k, and Avogadro’s constant through k = N_A/R (the molar gas constant R already being known with high precision). Perrin in his first experiments prepared a monodisperse colloid of a gum called gamboge. The particle masses were determined by direct weighing of a specified number, and their radii (hence their volumes and densities) by using the Stokes-Einstein law for diffusion. Perrin’s first value for Avogadro’s number was N_A = 7.05×10²³. In 1909 Perrin coined the name Avogadro's constant when he wrote:

This invariant number N is a universal constant, which may, with justification, be called Avogadro’s constant

Estimates from the electron charge

Robert A. Millikan and his student H. Fletcher gave in 1910 and 1911 the first reasonably accurate values for the charge e of the electron. In 1917 Millikan gave the improved value e = 1.591×10⁻¹⁹ C. The current accepted value is 1.6022×10⁻¹⁹ C.

The charge carried by a mole of singly charged ions in an electrochemical cell, which is known as Faraday's constant, F, was already known for quite some time when the electron charge was determined. It was 9.6489×10⁴ C/mol. As F = e ⋅N_A, the 1917 value of the electron charge gave Avogadro’s constant as N_A = 6.064×10²³.

Estimates from black-body radiation

In 1900 Planck gave birth to quantum theory by showing that the distribution of black body radiation as a function of temperature could be explained by assuming that oscillators in the body of frequency n could only take up or release energy in integer packets of hn, where the proportionality constant h is now known as Planck's constant.

Planck pointed out that a comparison of his theoretical distribution with the experimental curve allowed the determination of h and Boltzmann's constant k. From the ratio of k and the gas constant R Avogadro’s constant could be determined. Planck's estimate was N_A= 6.175×10²³.

Estimate from counting alpha particles

In 1908, Rutherford and Geiger concluded that their scintillation technique for detecting α particles recorded 100% of the particles (which are He nuclei) emitted during the radioactive decay of radium. They found that a gram of radium emitted 3.4×10¹⁰ particles per second. Counting atoms clearly provides a method for determining Avogadro’s constant. Counting gives the number of a particles produced per second and one only has to measure the volume of helium gas produced per unit of time to know the number atoms per volume, i.e., Avogadro's constant.

In 1911 Boltwood and Rutherford measured the amount of helium produced by two radium samples after 83 days and after 132 days, respectively. The first experiment gave 6.58 mm³ of helium gas at 0°C and 760 mm pressure, while the second gave gave 10.38 mm³ of gas. From this the helium production was found to be 2.09x10⁻² mm³/day, and 2.03×10⁻² mm³/day, which are satisfactorily consistent results. Boltwood and Rutherford did not state the value of Avogadro’s constant, which can be deduced from their experiments and the rate of production of a particles. But, knowing the amount of radium in the sample, and the amount of helium emitted per gram of radium, one can easily deduce that N_A= 6.1×10²³.

(To be continued)