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. The atomic mass constant m_u has the mass 1 u exactly (u is the unified atomic mass unit). Avogadro's number is defined here as a number, i.e., a dimensionless quantity. Its latest numeric value is N_A = 6.022 141 79 10²³.

The SI definition of Avogadro's constant (also designated N_A) is: the number of entities (such as atoms, ions, or molecules) per mole. (This definition requires 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 number have by definition the same numerical value. In practice the two terms are used interchangeably.

The symbol L (after Loschmidt) is also a recognized SI symbol with the same value as N_A.

History of Avogadro's number

Since 1811, when Amedeo Avogadro put forward his law stating that equal volumes of gas (strictly 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 Gay-Lussac's gas law and Dalton’s atomic theory. He calculates from gas densities that the molecular weight of nitrogen is 13.238 times the molecular weight of hydrogen. Avogadro was the first to propose that the gaseous elements, hydrogen, oxygen, and nitrogen, are diatomic molecules. He deduced that the molecule of water contains a molecule of oxygen and two 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 and his 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 Avogadro's 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.

In the beginning of the twentieth century some scientists (the most notable ones being Friedrich Wilhelm Ostwald and Ernst Mach) still denied the existence of molecules. As discussed by Pais the large number of different experiments that had led to basically the same values of Avogadro's constant, finally convinced Ostwald of the reality of molecules. Mach died in 1916 as disbeliever. The different experiments for determining N_A will be briefly reviewed.

Estimates from kinetic gas theory

The first estimate of Avogadro’s constant is attributed to Loschmidt (1865). He gave the value 0.969 nm for the diameter of "air molecules" and an equation that relates this diameter to the number L of molecules in one cm³ at standard temperature and pressure. The number L is called Loschmidt’s number; the Avogadro equivalent of Loschmidt's estimate is: N_A = 0.4×10²³. (Note that at present the SI definitions of L and N_A are the same, i.e., the different symbols stand for the same physical quantity). Loschmidt obtained this number by applying the kinetic gas theory of James Clerk Maxwell and Rudolph Clausius, together with experimental data on mean lengths of free paths of molecules in gases and molecular volumes of nitrogen-oxygen compounds. His value 0.969 nm was a factor three too high, and since his equation for L had an inverse-square dependence on the diameter, his estimate for L was an order of magnitude too low. In any case, Loschmidt’s work was the first to show that Avogadro’s constant is very large. In 1873 Maxwell used his own 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. In the early twentieth century 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 was already 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 Andrews Millikan and his student Harvey 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 the value of the elementary charge 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.

Estimates from counting alpha particles

In 1908, Ernest Rutherford and Hans Geiger concluded that their scintillation technique for detecting α particles (He nuclei) recorded 100% of the particles which are emitted during the radioactive decay of radium. They found that a gram of radium emitted 6.2×10¹⁰ 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 Rutherford and his friend Boltwood, who spent a year's leave at Rutherford's laboratory in Manchester, 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.

Estimates from crystal lattice spacings

Although X-rays have been used since 1912 to determine the lattice spacing of a crystal, it was not until 1930 that the technique was used to determine Avogadro’s constant. Before the 1930s, X-ray wavelengths were not known with enough accuracy. However, at present lattice spacings of the silicon crystal form the most reliable source of Avogadro's constant. The silicon crystal is cubic and has n = 8 atoms per face-centered unit cell of edge length $\scriptstyle a=543$ pm. The three naturally occurring isotopes of Si are ²⁸Si, ²⁹Si, and ³⁰Si, and the amount-of-substance percentages of natural silicon are approximately 92%, 5%, and 3%, respectively. An extensive international effort has been under way since the early 1990s to reduce the relative standard uncertainty of the measured value of the Avogadro constant, so that serious consideration can be given to replacing the current SI unit of mass—the international prototype of the kilogram—by a definition based on an invariant of nature. We sketch the sources of error in this determination.

First, in order to obtain an exact mole of silicon crystal (or a fraction there of), it is necessary to know the exact molar mass and hence the exact isotopic composition and the amount of impurities of the silicon sample. The determination of the molar mass is at present the limiting factor in the accuracy of Avogadro's constant. The number density ρ of a crystal is defined as N_A/V_m, where the molar volume V_m (the volume of one mole) is the second unknown that requires careful measurement. We make the reasonable assumption that the number density ρ of the crystal is the same as the number density n/V_u of the unit cell. The number of atoms n in the unit cell is known, and the volume V_u of the unit cell can be obtained from the lattice spacing of the crystal, provided the geometry of the unit cell is known. Measurement of the lattice spacing is a third source of error. Obviously, the wavelength of the X-rays has to be known with great accuracy, which is obtained from combined optical and X-ray interferometry. In summary, N_A follows from

$\rho =N_{\mathrm {A} }/V_{\mathrm {m} }=n/V_{\mathrm {u} }\Longrightarrow N_{A}=n{\frac {V_{\mathrm {m} }}{V_{\mathrm {u} }}}=n{\frac {V_{\mathrm {m} }}{a^{3}}},$

where $\scriptstyle V_{\mathrm {u} }=a^{3}$ in the case of a cubic unit cell with $\scriptstyle a$ the edge length of the unit cell. This is equation (143) of Ref.

External links

• An important part of the present article is based on the extensive article by: J. Murrell, Avogadro and his constant
• English translation of Avogadro's 1811 article Essay on a manner ...

References

Further references:

• A. Bader and L. Parker, Joseph Loschmidt, Physicist and Chemist, Physics Today, vol. 54(3), March pp. 45-50 (2001).
• J. Wisniak, Amedeo Avogadro The Man, the Hypothesis, and the Number, Chem. Educator, vol. 5, pp. 263–-268 (2000) (This paper gives the value for Avogadro's constant obtained by Loschmidt in 1865 a factor ten too high).