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# Dalton (unit)

dalton
(unified atomic mass unit)
Unit ofmass
SymbolDa or u
Named afterJohn Dalton
Conversions
1 Da or u in ...... is equal to ...
kg   1.66053906660(50)×10−27
mu   1
me   1822.888486209(53)
MeV/c2   931.49410242(28)

The dalton or unified atomic mass unit (symbols: Da or u) is a non-SI 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] The atomic mass constant, denoted mu, is defined identically, giving mu = 1/12 m(12C) = 1 Da.[3]

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
9
H
8
O
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 units kilodalton (kDa) and megadalton (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, widely used in chemistry and physics, which was originally defined so that the mass of one mole of a substance, in grams, would be numerically equal to the average mass of one of its constituent particles, in daltons. That is, the molar mass of a chemical compound was 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 almost all practical purposes, it is now 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 1.007276466621(53) Da,[7], the mass of a free neutron is 1.00866491595(49) 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 atomic units systems, which is the electron rest mass (me).

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## Energy equivalents

The atomic mass constant can also be expressed as its energy-equivalent, muc2. The 2018 CODATA recommended values are:

muc2 = 1.49241808560(45)×10−10 J[10] = 931.49410242(28) MeV[11]

The megaelectronvolt mass-equivalent (MeV/c2) 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.

## History

### Origin of the concept

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. 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).[13]

### 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).[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 units.[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.

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 redefinition of the SI base units

The definition of the dalton was not affected by the 2019 redefinition of SI base units,[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 redefinition of SI units, experiments were aimed to determine the value of the Avogadro constant for finding the value of the unified atomic mass unit.

### 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:

${\displaystyle F={\frac {A_{\rm {r}}M_{\rm {u}}It}{m}}.}$

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.

${\displaystyle m_{\rm {u}}={\frac {m_{\rm {e}}}{A_{\rm {r}}({\rm {e}})}}={\frac {2R_{\infty }h}{A_{\rm {r}}({\rm {e}})c\alpha ^{2}}},}$
${\displaystyle m_{\rm {u}}={\frac {M_{\rm {u}}}{N_{\rm {A}}}},}$
${\displaystyle N_{\rm {A}}={\frac {M_{\rm {u}}A_{\rm {r}}({\rm {e}})}{m_{\rm {e}}}}={\frac {M_{\rm {u}}A_{\rm {r}}({\rm {e}})c\alpha ^{2}}{2R_{\infty }h}}}$

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.

Constant Symbol 2014 CODATA values Relative standard uncertainty Correlation coefficient with NA
Proton–electron mass ratio mp/me 1836.15267389(17) 9.5×10−11 −0.0003
Molar mass constant Mu 0.001 kg/mol = 1 g/mol 0 (defined)  —
Rydberg constant R 10973731.568508(65) m−1 5.9×10−12 −0.0002
Planck constant h 6.626070040(81)×10−34 J⋅s 1.2×10−8 −0.9993
Speed of light c 299792458 m/s 0 (defined)  —
Fine structure constant α 7.2973525664(17)×10−3 2.3×10−10 0.0193
Avogadro constant NA 6.022140857(74)×1023 mol−1 1.2×10−8 1

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

Constant Symbol 2018 CODATA values[37] Relative standard uncertainty Correlation coefficient with NA
Proton–electron mass ratio mp/me 1836.15267343(11) 6.0×10−11  —
Molar mass constant Mu 0.99999999965(30)×10−3 kg/mol 3.0×10−10  —
Rydberg constant R 10973731.568160(21) m−1 1.9×10−12  —
Planck constant h 6.62607015×10−34 J⋅s 0 (defined)  —
Speed of light c 299792458 m/s 0 (defined)  —
Fine structure constant α 7.2973525693(11)×10−3 1.5×10−10  —
Avogadro constant NA 6.02214076×1023 mol−1 0 (defined)  —

### X-ray crystal density methods

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:

${\displaystyle N_{\rm {A}}={\frac {V_{\rm {m}}}{V_{\rm {atom}}}},}$

where ${\displaystyle V_{\rm {atom}}={\frac {V_{\rm {cell}}}{n}}}$ 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 2018 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:

${\displaystyle V_{\rm {m}}={\frac {A_{\rm {r}}M_{\rm {u}}}{\rho }},}$

where Mu is the molar mass constant. The 2018 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]

## Notes

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

## References

1. Bureau International des Poids et Mesures (2019): The International System of Units (SI), 9th edition, English version, page 146. Available at the BIPM website.
2. ^ IUPAC, Compendium of Chemical Terminology, 2nd ed. (the "Gold Book") (1997). Online corrected version: (2006–) "atomic mass constant". doi:10.1351/goldbook.A00497
3. ^ Taylor, Barry N. (2009). "Molar mass and related quantities in the new SI". Metrologia. 46 (3): L16–L19. doi:10.1088/0026-1394/46/3/L01. S2CID 115540416.
4. ^ a b Berg, Jeremy M.; Tymoczko, John L.; Stryer, Lubert (2007). "2". Biochemistry (6th ed.). Macmillan. p. 35. ISBN 978-0-7167-8724-2.
5. ^ Opitz CA, Kulke M, Leake MC, Neagoe C, Hinssen H, Hajjar RJ, Linke WA (October 2003). "Damped elastic recoil of the titin spring in myofibrils of human myocardium". Proc. Natl. Acad. Sci. U.S.A. 100 (22): 12688–93. Bibcode:2003PNAS..10012688O. doi:10.1073/pnas.2133733100. PMC 240679. PMID 14563922.
6. ^ Integrated DNA Technologies (2011): "Molecular Facts and Figures Archived 2020-04-18 at the Wayback Machine". Article on the IDT website, Support & Education section Archived 2021-01-19 at the Wayback Machine, accessed on 2019-07-08.
7. ^ "2018 CODATA Value: proton mass in u". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2022-09-11.
8. ^ "2018 CODATA Value: neutron mass in u". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2020-06-24.
9. ^ Meng Wang, G. Audi, F.G. Kondev, W.J. Huang, S. Naimi, and Xing Xu (2017): "The Ame2016 atomic mass evaluation (II). Tables, graphs and references". Chinese Physics C, volume 41, issue 3, article 030003, pages 1-441. doi:10.1088/1674-1137/41/3/030003
10. ^ "2018 CODATA Value: atomic mass constant energy equivalent". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-07-21.
11. ^ "2018 CODATA Value: atomic mass constant energy equivalent in MeV". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-07-21.
12. ^ a b c Petley, B. W. (1989). "The atomic mass unit". IEEE Trans. Instrum. Meas. 38 (2): 175–179. Bibcode:1989ITIM...38..175P. doi:10.1109/19.192268.
13. ^ a b c Holden, Norman E. (2004). "Atomic Weights and the International Committee—A Historical Review". Chemistry International. 26 (1): 4–7.
14. ^ Perrin, Jean (1909). "Mouvement brownien et réalité moléculaire". Annales de Chimie et de Physique. 8e Série. 18: 1–114. Extract in English, translation by Frederick Soddy.
15. ^ Chang, Raymond (2005). Physical Chemistry for the Biosciences. University Science Books. p. 5. ISBN 978-1-891389-33-7.
16. ^ Kelter, Paul B.; Mosher, Michael D.; Scott, Andrew (2008). Chemistry: The Practical Science. Vol. 10. Cengage Learning. p. 60. ISBN 978-0-547-05393-6.
17. ^ IUPAC, Compendium of Chemical Terminology, 2nd ed. (the "Gold Book") (1997). Online corrected version: (2006–) "unified atomic mass unit". doi:10.1351/goldbook.U06554
18. ^ Bureau International des Poids et Mesures (1971): 14th Conference Générale des Poids et Mesures Archived 2020-09-23 at the Wayback Machine Available at the BIPM website.
19. ^ Mills, Ian; Cvitaš, Tomislav; Homann, Klaus; Kallay, Nikola; Kuchitsu, Kozo (1993). Quantities, Units and Symbols in Physical Chemistry International Union of Pure and Applied Chemistry; Physical Chemistry Division (2nd ed.). International Union of Pure and Applied Chemistry and published for them by Blackwell Science Ltd. ISBN 978-0-632-03583-0.
20. ^ IUPAC, Compendium of Chemical Terminology, 2nd ed. (the "Gold Book") (1997). Online corrected version: (2006–) "dalton". doi:10.1351/goldbook.D01514
21. ^ "IUPAP: C2: Report 2005". Retrieved 2018-07-15.
22. ^
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". doi:10.1515/iupac.68.2930. {{cite journal}}: Cite journal requires |journal= (help)
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. ^ "2018 CODATA Value: Lattice parameter of silicon". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-08-23.
40. ^ "2018 CODATA Value: molar volume of silicon". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-08-23.
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