1. Energy Equivalents

The atomic mass constant can also be expressed as its energy equivalent, that is m_uc². The CODATA recommended values are:

1.49241808560(45)×10⁻¹⁰ J^[1]

931.49410242(28) MeV^[2]

The megaelectronvolt (MeV) is commonly used as a unit of mass in particle physics, and these values are also important for the practical determination of relative atomic masses.

2. History

2.1. Origin of the Concept

Jean Perrin in 1926. https://handwiki.org/wiki/index.php?curid=1308990

The interpretation of the law of definite proportions in term of the atomic theory of matter implied that the masses of atoms of various elements had definite ratios that depended on the elements. While the actual masses were unknown, the relative masses could be deduced from that law. In 1803 John Dalton proposed to use the (still unknown) atomic mass of the lightest atom, that of hydrogen, as the natural unit of atomic mass. This was the basis of the atomic weight scale.^[3]

For technical reasons, in 1898, chemist Wilhelm Ostwald and others proposed to redefine the unit of atomic mass as 1/16 of the mass of an oxygen atom.^[4] That proposal was formally adopted by the International Committee on Atomic Weights (ICAW) in 1903. That was approximately the mass of one hydrogen atom, but oxygen was more amenable to experimental determination. This suggestion was made before the discovery of the existence of elemental isotopes, which occurred in 1912.^[3] The same definition was adopted in 1909 the physicist Jean Perrin in his extensive experiments to determine the amu.^[5] This definition remained unchanged until 1961.^[6][7] Perrin also defined the "mole" as an amount of a compound that contained as many molecules as 32 grams of oxygen (O₂). He called that number the Avogadro number in honor of physicist Amedeo Avogadro.

2.2. Isotopic Variation

The discovery of isotopes of oxygen in 1929 required a more precise of the unit. Unfortunately, two distinct definitions came into use. Chemists choose to define the amu as 1/16 of the average mass of an oxygen atom as found in nature; that is, the average of the masses of the known isotopes, weighted by their natural abundance. Physicists, on the other hand, defined it as 1/16 of the mass of an atom of the isotope oxygen-16 (¹⁶O).^[4]

2.3. Definition by the IUPAC

The existence of two distinct units with the same name was confusing, and the difference (about 1.000282 in relative terms) was large enough to affect high-precision measurements. Moreover, it was discovered that the isotopes of oxygen had different natural abundances in water and in air. For these and other reasons, in 1961 the International Union of Pure and Applied Chemistry (IUPAC), which had absorbed the ICAW, adopted a new definition of the atomic mass unit for use in both physics and chemistry; namely, 1/12 of the mass of a carbon-12 atom. This new value was intermediate between the two earlier definitions, but closer to the one used by chemists (who would be affected the most by the change).^[3][4]

The new unit was named the "unified atomic mass unit" and given a new symbol "u", to replace the old "amu" that had been used for the oxygen-based units.^[8] However, the old symbol "amu" has often been used, after 1961, to refer to the new unit.

With this new definition, the standard atomic weight of carbon is approximately 12.011 Da, and that of oxygen is approximately 15.999 Da. These values, generally used in chemistry, are based on averages of many samples from Earth's crust, its atmosphere, and organic materials.

2.4. Adoption by the BIPM

The IUPAC 1961 definition of the unified atomic mass unit, with that name and symbol "u", was adopted by the International Bureau for Weights and Measures (BIPM) in 1971 as a "non-SI unit accepted for use with the SI".^[9]

The definition was not affected by the 2019 redefinition of SI base units;^[10][11][12] that is, 1 u in the SI is still 1/12 of the mass of a carbon-12 atom, a quantity that must be determined experimentally. However, the definition of the mole was changed to be "exactly N particles", where N, Avogadro's number, was redefined to be exactly 6.02214076×10²³; and the definition of the kilogram was changed too. As a consequence, the molar mass constant is no longer exactly 1 g/mol, meaning that the mass of one mole of any substance, in grams, is no longer numerically equal to its molecular weight, in unified atomic mass units. In particular, the mass of one mole of atoms of carbon-12 is no longer exactly 12 grams by definition, but exactly 12N u; whose value in grams must be determined experimentally.^[13][14] However, the difference (3.5×10⁻¹⁰ in relative terms, as determined from the 2018 CODATA value^[15]) is insignificant for practical purposes.

As of the redefinition, the relative uncertainty of the atomic mass unit is 3.0×10^−10[16], which is almost entirely due to the uncertainty in the fine-structure constant.

2.5. The Dalton

In 1993, the IUPAC proposed the shorter name "dalton" (with symbol "Da") for the unified atomic mass unit.^[17][18] As with other unit names such as watt and newton, "dalton" is not capitalized in English, but its symbol, "Da", is capitalized. The name was endorsed by the International Union of Pure and Applied Physics (IUPAP) in 2005.^[19]

In 2003 the name was recommended to the BIPM by the Consultative Committee for Units, part of the CIPM, as it "is shorter and works better with [the SI] prefixes".^[20] In 2006, the BIPM included the dalton in its 8th edition of the formal definition of SI.^[21] The name was also listed as an alternative to "unified atomic mass unit" by the International Organization for Standardization in 2009.^[22][23] It is now recommended by several scientific publishers,^[24] and some of them consider "atomic mass unit" and "amu" deprecated.^[25] In 2019, the BIPM retained the dalton in its 9th edition of the formal definition of SI while dropping the unified atomic mass unit from its table of non-SI units accepted for use with the SI.^[12]

2.6. New Proposal

A proposal was made in 2012 to redefine the dalton (and presumably the unified atomic mass unit) as being 1/N grams, thereby breaking the link with ¹²C. This would imply changes in the atomic masses of all elements, when expressed in daltons; but the change would be too small to have practical effects.^[26]

2.7. Legal Implications

The US Supreme Court established a legal precedent of appellate law in a case (Teva Pharmaceuticals v. Sandoz) that hinged upon the interpretation of the term "molecular weight".^[27]

3. Measurement

Although relative atomic masses are defined for neutral atoms, they are measured (by mass spectrometry) for ions: hence, the measured values must be correct for the mass of the electrons that were removed to form the ions, and also for the mass equivalent of the electron binding energy, E_b/m_uc². The total binding energy of the six electrons in a carbon-12 atom is 1030.1089 eV = 1.650 4163×10⁻¹⁶ J: E_b/m_uc² = 1.105 8674×10⁻⁶, or about one part in 10 million of the mass of the atom.^[28]

3.1. Josef Loschmidt

Josef Loschmidt. https://handwiki.org/wiki/index.php?curid=1302124

A reasonably accurate value of the atomic mass unit was first obtained indirectly by Josef Loschmidt in 1865, by estimating the number of particles in a given volume of gas.^[29]

3.2. Jean Perrin

Until May 20, 2019, the Avogadro number N was defined as the number of atoms in k grams of a reference element (oxygen or carbon-12) whose atoms had, by definition, k atomic mass units. Therefore, experiments that aimed to determine of N were essentially determinations of the atomic mass unit. Perrin himself estimated the Avogadro number by a variety of methods, at the turn of the 20th century. He was awarded the 1926 Nobel Prize in Physics, largely for this work.^[30]

3.3. Coulometry

For a while, the most accurate measurements of the atomic mass unit were based on coulometry. The electric charge per mole of electrons is a constant called the Faraday constant, whose value had been essentially known since 1834 when Michael Faraday published his works on electrolysis. In 1910, Robert Millikan obtained the first measurement of the charge on an electron, e. The quotient F/e provided an estimate of Avogadro's number^[31]

The classic experiment is that of Bower and Davis at NIST,^[32] 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_r the atomic weight of silver, then the Faraday constant is given by:

[math]\displaystyle{ F = \frac{A_{\rm r}M_{\rm u}It}{m}. }[/math]

The NIST scientists devised a method to compensate for silver lost from the anode by mechanical causes, and conducted an isotope analysis of the silver used to determine its atomic weight. Their value for the conventional Faraday constant was F₉₀ = 96485.39(13) C/mol, which corresponds to a value for the Avogadro constant of 6.0221449(78)×10²³ mol⁻¹: both values have a relative standard uncertainty of 1.3×10⁻⁶.

3.4. Electron Mass Measurement

In practice, the atomic mass constant is determined from the electron rest mass m_e and the electron relative atomic mass A_r(e) (that is, the mass of the electron on a scale where ¹²C = 12).^[33] The relative atomic mass of the electron can be measured in cyclotron experiments, while the rest mass of the electron can be derived from other physical constants.

[math]\displaystyle{ m_{\rm u} = \frac{m_{\rm e}}{A_{\rm r}({\rm e})} = \frac{2R_\infty h}{A_{\rm r}({\rm e})c\alpha^2} , }[/math]

where c is the speed of light, h is the Planck constant, α is the fine-structure constant, and R_∞ is the Rydberg constant.

As may be observed in the table below, the main limiting factor in the precision of the Avogadro constant is the uncertainty in the value of the Planck constant, as all the other constants that contribute to the calculation are known more precisely.

3.5. X-Ray Crystal Density Methods

Ball-and-stick model of the unit cell of silicon. X-ray diffraction measures the cell parameter, a, which is used to calculate a value for the Avogadro constant. https://handwiki.org/wiki/index.php?curid=1421579

A modern method to determine the Avogadro constant is the use of X-ray crystallography. Silicon single crystals may be produced today in commercial facilities with extremely high purity and with few lattice defects. This method defines the Avogadro constant as the ratio of the molar volume, V_m, to the atomic volume V_atom:

[math]\displaystyle{ N_{\rm A} = \frac{V_{\rm m}}{V_{\rm atom}} }[/math], where [math]\displaystyle{ V_{\rm atom} = \frac{V_{\rm cell}}{n} }[/math] and n is the number of atoms per unit cell of volume V_cell.

The unit cell of silicon has a cubic packing arrangement of 8 atoms, and the unit cell volume may be measured by determining a single unit cell parameter, the length of one of the sides of the cube, a.^[34]

In practice, measurements are carried out on a distance known as d₂₂₀(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 d₂₂₀(Si) is 192.0155762(50) pm, a relative standard uncertainty of 2.8×10⁻⁸, corresponding to a unit cell volume of 1.60193304(13)×10⁻²⁸ m³.

The isotope proportional composition of the sample used must be measured and taken into account. Silicon occurs in three stable isotopes (²⁸Si, ²⁹Si, ³⁰Si), and the natural variation in their proportions is greater than other uncertainties in the measurements. The atomic weight A_r for the sample crystal can be calculated, as the standard atomic weights of the three nuclides are known with great accuracy. This, together with the measured density ρ of the sample, allows the molar volume V_m to be determined:

[math]\displaystyle{ V_{\rm m} = \frac{A_{\rm r}M_{\rm u}}{\rho} }[/math]

where M_u is the molar mass constant. The 2006 CODATA value for the molar volume of silicon is 12.0588349(11) cm³⋅mol⁻¹, with a relative standard uncertainty of 9.1×10⁻⁸.^[35]

As of the 2006 CODATA recommended values, the relative uncertainty in determinations of the Avogadro constant by the X-ray crystal density method is 1.2×10⁻⁷, about two and a half times higher than that of the electron mass method.

International Avogadro Coordination

Achim Leistner at the Australian Centre for Precision Optics (ACPO) holding a one-kilogram single-crystal silicon sphere prepared for the International Avogadro Coordination. https://handwiki.org/wiki/index.php?curid=1858326

The International Avogadro Coordination (IAC), often simply called the "Avogadro project", is a collaboration begun in the early 1990s between various national metrology institutes to measure the Avogadro constant by the X-ray crystal density method to a relative uncertainty of 2×10⁻⁸ or less.^[36] The project was one of the alternative approaches to redefining the kilogram in terms of a universal physical constant, rather than the International Prototype Kilogram, and complemented the measurements of the Planck constant using Kibble balances.^[37][38] Prior to the 2019 redefinition of the SI base units, a measurement of the Avogadro constant was an indirect measurement of the Planck constant:

[math]\displaystyle{ h = \frac{c\alpha^2 A_{\rm r}({\rm e})M_{\rm u}}{2R_{\infty} N_{\rm A}}. }[/math]

The measurements use highly polished spheres of silicon with a mass of one kilogram. Spheres are used to simplify the measurement of the size (and hence the density) and to minimize the effect of the oxide coating that inevitably forms on the surface. The first measurements used spheres of silicon with natural isotopic composition, and had a relative uncertainty of 3.1×10⁻⁷.^[39][40][41] These first results were also inconsistent with values of the Planck constant derived from Kibble balance measurements, although the source of the discrepancy is now believed to be known.^[38]

The main residual uncertainty in the early measurements was in the measurement of the isotopic composition of the silicon to calculate the atomic weight, so in 2007 a 4.8 kg single crystal of isotopically-enriched silicon (99.94% ²⁸Si) was grown,^[42][43][44] and two one-kilogram spheres cut from it. Diameter measurements on the spheres are repeatable to within 0.3 nm, and the uncertainty in the mass is 3 µg. Full results from these determinations were expected in late 2010.^[45] Their paper, published in January 2011, summarized the result of the International Avogadro Coordination and presented a measurement of the Avogadro constant to be 6.02214078(18)×10²³ mol⁻¹.^[46]