Dalton (unit)

Last updated
dalton
(unified atomic mass unit)
Unit of mass
SymbolDa or u
Named after John Dalton
Conversions
1 Da or u in ...... is equal to ...
   kg   1.66053906892(52)×10−27
   mu   1
   MeV/c2   931.49410372(29)

The dalton or unified atomic mass unit (symbols: Da or u) is a unit of mass defined as ⁠1/12⁠ of the mass of an unbound neutral atom of carbon-12 in its nuclear and electronic ground state and at rest. [1] [2] It is a non-SI unit accepted for use with SI. The atomic mass constant, denoted mu, is defined identically, giving mu = ⁠1/12⁠m(12C) = 1 Da. [3]

Contents

This unit is commonly used in physics and chemistry to express the mass of atomic-scale objects, such as atoms, molecules, and elementary particles, both for discrete instances and multiple types of ensemble averages. For example, an atom of helium-4 has a mass of 4.0026 Da. This is an intrinsic property of the isotope and all helium-4 atoms have the same mass. Acetylsalicylic acid (aspirin), C
9H
8O
4, has an average mass of about 180.157 Da. However, there are no acetylsalicylic acid molecules with this mass. The two most common masses of individual acetylsalicylic acid molecules are 180.0423 Da, having the most common isotopes, and 181.0456 Da, in which one carbon is carbon-13.

The molecular masses of proteins, nucleic acids, and other large polymers are often expressed with the unit kilo dalton (kDa) and mega dalton (MDa). [4] Titin, one of the largest known proteins, has a molecular mass of between 3 and 3.7 megadaltons. [5] The DNA of chromosome 1 in the human genome has about 249 million base pairs, each with an average mass of about 650 Da, or 156 GDa total. [6]

The mole is a unit of amount of substance used in chemistry and physics, which defines the mass of one mole of a substance in grams as numerically equal to the average mass of one of its particles in daltons. That is, the molar mass of a chemical compound is meant to be numerically equal to its average molecular mass. For example, the average mass of one molecule of water is about 18.0153 daltons, and one mole of water is about 18.0153 grams. A protein whose molecule has an average mass of 64 kDa would have a molar mass of 64 kg/mol. However, while this equality can be assumed for practical purposes, it is only approximate, because of the 2019 redefinition of the mole. [4] [1]

In general, the mass in daltons of an atom is numerically close but not exactly equal to the number of nucleons in its nucleus. It follows that the molar mass of a compound (grams per mole) is numerically close to the average number of nucleons contained in each molecule. By definition, the mass of an atom of carbon-12 is 12 daltons, which corresponds with the number of nucleons that it has (6  protons and 6  neutrons). However, the mass of an atomic-scale object is affected by the binding energy of the nucleons in its atomic nuclei, as well as the mass and binding energy of its electrons. Therefore, this equality holds only for the carbon-12 atom in the stated conditions, and will vary for other substances. For example, the mass of an unbound atom of the common hydrogen isotope (hydrogen-1, protium) is 1.007825032241(94) Da, [a] the mass of a proton is 1.0072764665789(83) Da, [7] the mass of a free neutron is 1.00866491606(40) Da, [8] and the mass of a hydrogen-2 (deuterium) atom is 2.014101778114(122) Da. [9] In general, the difference (absolute mass excess) is less than 0.1%; exceptions include hydrogen-1 (about 0.8%), helium-3 (0.5%), lithium-6 (0.25%) and beryllium (0.14%).

The dalton differs from the unit of mass in the system of atomic units, which is the electron rest mass (me).

Energy equivalents

The atomic mass constant can also be expressed as its energy-equivalent, muc2. The CODATA recommended values are:

muc2 = 1.49241808768(46)×10−10 J‍ [10] = 931.49410372(29) MeV‍ [11]

The mass-equivalent is commonly used in place of a unit of mass in particle physics, and these values are also important for the practical determination of relative atomic masses.

History

Origin of the concept

Jean Perrin in 1926 Jean Perrin 1926.jpg
Jean Perrin in 1926

The interpretation of the law of definite proportions in terms 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, hydrogen, as the natural unit of atomic mass. This was the basis of the atomic weight scale. [12]

For technical reasons, in 1898, chemist Wilhelm Ostwald and others proposed to redefine the unit of atomic mass as ⁠1/16⁠ the mass of an oxygen atom. [13] 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 isotopes in 1912. [12] Physicist Jean Perrin had adopted the same definition in 1909 during his experiments to determine the atomic masses and the Avogadro constant. [14] This definition remained unchanged until 1961. [15] [16] Perrin also defined the "mole" as an amount of a compound that contained as many molecules as 32 grams of oxygen (O
2). He called that number the Avogadro number in honor of physicist Amedeo Avogadro.

Isotopic variation

The discovery of isotopes of oxygen in 1929 required a more precise definition of the unit. 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). [13]

Definition by 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). [12] [13]

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 unit. [17] However, the old symbol "amu" has sometimes been used, after 1961, to refer to the new unit, particularly in lay and preparatory contexts.

With this new definition, the standard atomic weight of carbon is about 12.011 Da, and that of oxygen is about 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.

Adoption by 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. [18]

Unit name

In 1993, the IUPAC proposed the shorter name "dalton" (with symbol "Da") for the unified atomic mass unit. [19] [20] 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. [21]

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 [SI] prefixes". [22] In 2006, the BIPM included the dalton in its 8th edition of the SI brochure of formal definitions as a non-SI unit accepted for use with the SI. [23] The name was also listed as an alternative to "unified atomic mass unit" by the International Organization for Standardization in 2009. [24] [25] It is now recommended by several scientific publishers, [26] and some of them consider "atomic mass unit" and "amu" deprecated. [27] In 2019, the BIPM retained the dalton in its 9th edition of the SI brochure, while dropping the unified atomic mass unit from its table of non-SI units accepted for use with the SI, but secondarily notes that the dalton (Da) and the unified atomic mass unit (u) are alternative names (and symbols) for the same unit. [1]

2019 revision of the SI

The definition of the dalton was not affected by the 2019 revision of the SI, [28] [29] [1] that is, 1 Da in the SI is still ⁠1/12⁠ of the mass of a carbon-12 atom, a quantity that must be determined experimentally in terms of SI units. However, the definition of a mole was changed to be the amount of substance consisting of exactly 6.02214076×1023 entities and the definition of the kilogram was changed as well. As a consequence, the molar mass constant remains close to but no longer exactly 1 g/mol, meaning that the mass in grams of one mole of any substance remains nearly but no longer exactly numerically equal to its average molecular mass in daltons, [30] although the relative standard uncertainty of 4.5×10−10 at the time of the redefinition is insignificant for all practical purposes. [1]

Measurement

Though relative atomic masses are defined for neutral atoms, they are measured (by mass spectrometry) for ions: hence, the measured values must be corrected for the mass of the electrons that were removed to form the ions, and also for the mass equivalent of the electron binding energy, Eb/muc2. The total binding energy of the six electrons in a carbon-12 atom is 1030.1089 eV = 1.6504163×10−16 J: Eb/muc2 = 1.1058674×10−6, or about one part in 10 million of the mass of the atom. [31]

Before the 2019 revision of the SI, experiments were aimed to determine the value of the Avogadro constant for finding the value of the unified atomic mass unit.

Josef Loschmidt

Josef Loschmidt Johann Josef Loschmidt portrait plaque.jpg
Josef Loschmidt

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. [32]

Jean Perrin

Perrin 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. [33]

Coulometry

The electric charge per mole of elementary charges is a constant called the Faraday constant, F, 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 the Avogadro constant. [34]

The classic experiment is that of Bower and Davis at NIST, [35] 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 Ar the atomic weight of silver, then the Faraday constant is given by:

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.

Electron mass measurement

In practice, the atomic mass constant is determined from the electron rest mass me and the electron relative atomic mass Ar(e) (that is, the mass of electron divided by the atomic mass constant). [36] 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.

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 from the old values (2014 CODATA) in the table below, the main limiting factor in the precision of the Avogadro constant was the uncertainty in the value of the Planck constant, as all the other constants that contribute to the calculation were known more precisely.

ConstantSymbol2014 CODATA valuesRelative
standard
uncertainty		Correlation
coefficient
with NA
Proton–electron mass ratio mp/me1836.15267389(17)9.5×10−11−0.0003
Molar mass constant Mu1 g/mol0 (defined)—
Rydberg constant R∞10973731.568508(65) m−15.9×10−12−0.0002
Planck constant h6.626070040(81)×10−34 J⋅s1.2×10−8−0.9993
Speed of light c299792458 m/s0 (defined)—
Fine structure constant α7.2973525664(17)×10−32.3×10−100.0193
Avogadro constant NA6.022140857(74)×1023 mol−11.2×10−81

The power of having defined values of universal constants as is presently the case can be understood from the table below (2018 CODATA).

ConstantSymbol2018 CODATA values [37] Relative
standard
uncertainty		Correlation
coefficient
with NA
Proton–electron mass ratio mp/me1836.15267343(11)6.0×10−11—
Molar mass constant Mu0.99999999965(30) g/mol3.0×10−10—
Rydberg constant R∞10973731.568160(21) m−11.9×10−12—
Planck constant h6.62607015×10−34 J⋅s0 (defined)—
Speed of light c299792458 m/s0 (defined)—
Fine structure constant α7.2973525693(11)×10−31.5×10−10—
Avogadro constant NA6.02214076×1023 mol−10 (defined)—

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. Silicon-unit-cell-labelled-3D-balls.png
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.

Silicon single crystals may be produced today in commercial facilities with extremely high purity and with few lattice defects. This method defined the Avogadro constant as the ratio of the molar volume, Vm, to the atomic volume Vatom: where Vatom = ⁠Vcell/n⁠ and n is the number of atoms per unit cell of volume Vcell.

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 a of one of the sides of the cube. [38] The CODATA value of a for silicon is 5.431020511(89)×10−10 m. [39]

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 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 Ar 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 Vm to be determined: where Mu is the molar mass constant. The CODATA value for the molar volume of silicon is 1.205883199(60)×10−5 m3⋅mol−1, with a relative standard uncertainty of 4.9×10−8. [40]

See also

Notes

  1. ↑ The digits in parentheses indicate the uncertainty; see Uncertainty notation.

Related Research Articles

The molecular mass is the mass of a given molecule. Units of daltons (Da) are often used. Different molecules of the same compound may have different molecular masses because they contain different isotopes of an element. The derived quantity relative molecular mass is the unitless ratio of the mass of a molecule to the atomic mass constant.

<span class="mw-page-title-main">Mole (unit)</span> SI unit of amount of substance

The mole (symbol mol) is a unit of measurement, the base unit in the International System of Units (SI) for amount of substance, a quantity proportional to the number of elementary entities of a substance. One mole contains exactly 6.02214076×1023 elementary entities (approximately 602 sextillion or 602 billion times a trillion), which can be atoms, molecules, ions, ion pairs, or other particles. The number of particles in a mole is the Avogadro number (symbol N0) and the numerical value of the Avogadro constant (symbol NA) expressed in mol-1. The value was chosen on the basis of the historical definition of the mole as the amount of substance that corresponds to the number of atoms in 12 grams of 12C, which made the mass of a mole of a compound expressed in grams, numerically equal to the average molecular mass or formula mass of the compound expressed in daltons. With the 2019 revision of the SI, the numerical equivalence is now only approximate but may be assumed for all practical purposes.

<span class="mw-page-title-main">Avogadro constant</span> Fundamental metric system constant defined as the number of particles per mole

The Avogadro constant, commonly denoted NA or L, is an SI defining constant with an exact value of 6.02214076×1023 mol−1 (reciprocal moles). It is this defined number of constituent particles (usually molecules, atoms, ions, or ion pairs—in general, entities) per mole (SI unit) and used as a normalization factor in relating the amount of substance, n(X), in a sample of a substance X to the corresponding number of entities, N(X): n(X) = N(X)(1/NA), an aggregate of N(X) reciprocal Avogadro constants. By setting N(X) = 1, a reciprocal Avogadro constant is seen to be equal to one entity, which means that n(X) is more easily interpreted as an aggregate of N(X) entities. In the SI dimensional analysis of measurement units, the dimension of the Avogadro constant is the reciprocal of amount of substance, denoted N−1. The Avogadro number, sometimes denoted N0, is the numeric value of the Avogadro constant (i.e., without a unit), namely the dimensionless number 6.02214076×1023; the value chosen based on the number of atoms in 12 grams of carbon-12 in alignment with the historical definition of a mole. The constant is named after the Italian physicist and chemist Amedeo Avogadro (1776–1856).

<span class="mw-page-title-main">Coulomb</span> SI derived unit of electric charge

The coulomb (symbol: C) is the unit of electric charge in the International System of Units (SI). It is equal to the electric charge delivered by a 1 ampere current in 1 second and is defined in terms of the elementary charge e, at about 6.241509×1018 e.

<span class="mw-page-title-main">Boltzmann constant</span> Physical constant relating particle kinetic energy with temperature

The Boltzmann constant is the proportionality factor that relates the average relative thermal energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin (K) and the gas constant, in Planck's law of black-body radiation and Boltzmann's entropy formula, and is used in calculating thermal noise in resistors. The Boltzmann constant has dimensions of energy divided by temperature, the same as entropy and heat capacity. It is named after the Austrian scientist Ludwig Boltzmann.

<span class="mw-page-title-main">Gas constant</span> Physical constant equivalent to the Boltzmann constant, but in different units

The molar gas constant is denoted by the symbol R or R. It is the molar equivalent to the Boltzmann constant, expressed in units of energy per temperature increment per amount of substance, rather than energy per temperature increment per particle. The constant is also a combination of the constants from Boyle's law, Charles's law, Avogadro's law, and Gay-Lussac's law. It is a physical constant that is featured in many fundamental equations in the physical sciences, such as the ideal gas law, the Arrhenius equation, and the Nernst equation.

In chemistry and related fields, the molar volume, symbol Vm, or of a substance is the ratio of the volume (V) occupied by a substance to the amount of substance (n), usually at a given temperature and pressure. It is also equal to the molar mass (M) divided by the mass density (ρ):

<span class="mw-page-title-main">Molar mass</span> Mass per amount of substance

In chemistry, the molar mass of a chemical compound is defined as the ratio between the mass and the amount of substance of any sample of the compound. The molar mass is a bulk, not molecular, property of a substance. The molar mass is an average of many instances of the compound, which often vary in mass due to the presence of isotopes. Most commonly, the molar mass is computed from the standard atomic weights and is thus a terrestrial average and a function of the relative abundance of the isotopes of the constituent atoms on Earth. The molar mass is appropriate for converting between the mass of a substance and the amount of a substance for bulk quantities.

The atomic units are a system of natural units of measurement that is especially convenient for calculations in atomic physics and related scientific fields, such as computational chemistry and atomic spectroscopy. They were originally suggested and named by the physicist Douglas Hartree. Atomic units are often abbreviated "a.u." or "au", not to be confused with similar abbreviations used for astronomical units, arbitrary units, and absorbance units in other contexts.

The elementary charge, usually denoted by e, is a fundamental physical constant, defined as the electric charge carried by a single proton or, equivalently, the magnitude of the negative electric charge carried by a single electron, which has charge −1 e.

Relative atomic mass, also known by the deprecated synonym atomic weight, is a dimensionless physical quantity defined as the ratio of the average mass of atoms of a chemical element in a given sample to the atomic mass constant. The atomic mass constant is defined as being ⁠1/12⁠ of the mass of a carbon-12 atom. Since both quantities in the ratio are masses, the resulting value is dimensionless. These definitions remain valid even after the 2019 revision of the SI.

<span class="mw-page-title-main">Amount of substance</span> Ratio of the number of particles in a sample to Avogadros constant

In chemistry, the amount of substance (symbol n) in a given sample of matter is defined as a ratio (n = N/NA) between the number of elementary entities (N) and the Avogadro constant (NA). The entities are usually molecules, atoms, ions, or ion pairs of a specified kind. The particular substance sampled may be specified using a subscript, e.g., the amount of sodium chloride (NaCl) would be denoted as nNaCl. The unit of amount of substance in the International System of Units is the mole (symbol: mol), a base unit. Since 2019, the value of the Avogadro constant NA is defined to be exactly 6.02214076×1023 mol−1. Sometimes, the amount of substance is referred to as the chemical amount or, informally, as the "number of moles" in a given sample of matter.

The molar mass constant, usually denoted by Mu, is a physical constant defined as one twelfth of the molar mass of carbon-12: Mu = M(12C)/12. The molar mass of an element or compound is its relative atomic mass or relative molecular mass multiplied by the molar mass constant.

A conventional electrical unit is a unit of measurement in the field of electricity which is based on the so-called "conventional values" of the Josephson constant, the von Klitzing constant agreed by the International Committee for Weights and Measures (CIPM) in 1988, as well as ΔνCs used to define the second. These units are very similar in scale to their corresponding SI units, but are not identical because of the different values used for the constants. They are distinguished from the corresponding SI units by setting the symbol in italic typeface and adding a subscript "90" – e.g., the conventional volt has the symbol V90 – as they came into international use on 1 January 1990.

<span class="mw-page-title-main">Atomic mass</span> Rest mass of an atom in its ground state

The atomic mass (ma or m) is the mass of an atom. Although the SI unit of mass is the kilogram (symbol: kg), atomic mass is often expressed in the non-SI unit dalton (symbol: Da) – equivalently, unified atomic mass unit (u). 1 Da is defined as 1⁄12 of the mass of a free carbon-12 atom at rest in its ground state. The protons and neutrons of the nucleus account for nearly all of the total mass of atoms, with the electrons and nuclear binding energy making minor contributions. Thus, the numeric value of the atomic mass when expressed in daltons has nearly the same value as the mass number. Conversion between mass in kilograms and mass in daltons can be done using the atomic mass constant .

In particle physics, the electron mass is the mass of a stationary electron, also known as the invariant mass of the electron. It is one of the fundamental constants of physics. It has a value of about 9.109×10−31 kilograms or about 5.486×10−4 daltons, which has an energy-equivalent of about 8.187×10−14 joules or about 0.5110 MeV.

<span class="mw-page-title-main">2019 revision of the SI</span> Definition of the units kg, A, K and mol

In 2019, four of the seven SI base units specified in the International System of Quantities were redefined in terms of natural physical constants, rather than human artefacts such as the standard kilogram. Effective 20 May 2019, the 144th anniversary of the Metre Convention, the kilogram, ampere, kelvin, and mole are now defined by setting exact numerical values, when expressed in SI units, for the Planck constant, the elementary electric charge, the Boltzmann constant, and the Avogadro constant, respectively. The second, metre, and candela had previously been redefined using physical constants. The four new definitions aimed to improve the SI without changing the value of any units, ensuring continuity with existing measurements. In November 2018, the 26th General Conference on Weights and Measures (CGPM) unanimously approved these changes, which the International Committee for Weights and Measures (CIPM) had proposed earlier that year after determining that previously agreed conditions for the change had been met. These conditions were satisfied by a series of experiments that measured the constants to high accuracy relative to the old SI definitions, and were the culmination of decades of research.

In physics, natural unit systems are measurement systems for which selected physical constants have been set to 1 through nondimensionalization of physical units. For example, the speed of light c may be set to 1, and it may then be omitted, equating mass and energy directly E = m rather than using c as a conversion factor in the typical mass–energy equivalence equation E = mc2. A purely natural system of units has all of its dimensions collapsed, such that the physical constants completely define the system of units and the relevant physical laws contain no conversion constants.

<span class="mw-page-title-main">Alternative approaches to redefining the kilogram</span>

The scientific community examined several approaches to redefining the kilogram before deciding on a revision of the SI in November 2018. Each approach had advantages and disadvantages.

References

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  21. ↑ "IUPAP: C2: Report 2005" . Retrieved 2018-07-15.
  22. ↑ "Consultative Committee for Units (CCU); Report of the 15th meeting (17–18 April 2003) to the International Committee for Weights and Measures" (PDF). Retrieved 14 Aug 2010.
  23. ↑ International Bureau of Weights and Measures (2006), The International System of Units (SI) (PDF) (8th ed.), pp. 114–15, ISBN   92-822-2213-6, archived (PDF) from the original on 2021-06-04, retrieved 2021-12-16
  24. ↑ International Standard ISO 80000-1:2009 – Quantities and Units – Part 1: General. International Organization for Standardization. 2009.
  25. ↑ International Standard ISO 80000-10:2009 – Quantities and units – Part 10: Atomic and nuclear physics, International Organization for Standardization, 2009
  26. ↑ "Instructions to Authors". AoB Plants. Oxford journals; Oxford University Press. Archived from the original on 2011-11-03. Retrieved 2010-08-22.
  27. ↑ "Author guidelines". Rapid Communications in Mass Spectrometry. Wiley-Blackwell. 2010.
  28. ↑ International Bureau for Weights and Measures (2017): Proceedings of the 106th meeting of the International Committee for Weights and Measures (CIPM), 16-17 and 20 October 2017 , page 23. Available at the BIPM website Archived 2021-02-21 at the Wayback Machine .
  29. ↑ International Bureau for Weights and Measures (2018): Resolutions Adopted - 26th Conference Générale des Poids et Mesures Archived 2018-11-19 at the Wayback Machine . Available at the BIPM website.
  30. ↑ Lehmann, H. P.; Fuentes-Arderiu, X.; Bertello, L. F. (2016-02-29). "Unified Atomic Mass Unit". Glossary of Terms in Quantities and Units in Clinical Chemistry. doi: 10.1515/iupac.68.2930 .
  31. ↑ Mohr, Peter J.; Taylor, Barry N. (2005). "CODATA recommended values of the fundamental physical constants: 2002" (PDF). Reviews of Modern Physics . 77 (1): 1–107. Bibcode:2005RvMP...77....1M. doi:10.1103/RevModPhys.77.1. Archived from the original (PDF) on 2017-10-01.
  32. ↑ Loschmidt, J. (1865). "Zur Grösse der Luftmoleküle". Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften Wien. 52 (2): 395–413. English translation.
  33. ↑ Oseen, C.W. (December 10, 1926). Presentation Speech for the 1926 Nobel Prize in Physics .
  34. ↑ (1974): Introduction to the constants for nonexperts, 1900–1920 From the Encyclopaedia Britannica, 15th edition; reproduced by NIST. Accessed on 2019-07-03.
  35. ↑ This account is based on the review in Mohr, Peter J.; Taylor, Barry N. (1999). "CODATA recommended values of the fundamental physical constants: 1998" (PDF). Journal of Physical and Chemical Reference Data . 28 (6): 1713–1852. Bibcode:1999JPCRD..28.1713M. doi:10.1063/1.556049. Archived from the original (PDF) on 2017-10-01.
  36. ↑ Mohr, Peter J.; Taylor, Barry N. (1999). "CODATA recommended values of the fundamental physical constants: 1998" (PDF). Journal of Physical and Chemical Reference Data . 28 (6): 1713–1852. Bibcode:1999JPCRD..28.1713M. doi:10.1063/1.556049. Archived from the original (PDF) on 2017-10-01.
  37. ↑ "Constants bibliography, source of the CODATA internationally recommended values". The NIST Reference on Constants, Units, and Uncertainty. Retrieved 4 August 2021.
  38. ↑ "Unit Cell Formula". Mineralogy Database. 2000–2005. Retrieved 2007-12-09.
  39. ↑ "2022 CODATA Value: lattice parameter of silicon". The NIST Reference on Constants, Units, and Uncertainty. NIST. May 2024. Retrieved 2024-05-18.
  40. ↑ "2022 CODATA Value: molar volume of silicon". The NIST Reference on Constants, Units, and Uncertainty. NIST. May 2024. Retrieved 2024-05-18.
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