Editing Dalton (unit)

{{Short description|Standard unit of mass for atomic-scale chemical species}}
{{distinguish|atomic units}}
{{Infobox unit
| bgcolour =
| name = dalton{{br}}(unified atomic mass unit)
| image =
| standard =
| quantity = [[mass]]
| symbol = Da or u
| namedafter = [[John Dalton]]
| units1 = kg
| inunits1 = {{physconst|mu|unit=no|ref=no}}
| units2 = ''m''<sub>u</sub>
| inunits2 = {{val|1}}
| units3 = MeV/''c''<sup>2</sup>
| inunits3 = {{physconst|muc2_MeV|unit=no|ref=no}}
}}

The '''dalton''' or '''unified atomic mass unit''' (symbols: '''Da''' or '''u''', respectively) is a unit of [[mass]] defined as {{sfrac|1|12}} of the mass of an [[Bound state|unbound]] neutral atom of [[carbon-12]] in its nuclear and electronic [[ground state]] and [[invariant mass|at rest]].<ref name=bipm9th/><ref name=goldbAtMaCo/><ref name=ISO10/> It is a [[Non-SI units mentioned in the SI|non-SI unit accepted for use with SI]]. The word "unified" emphasizes that the definition was accepted by both [[IUPAP]] and [[IUPAC]].
<ref name=hold2004></ref>
The [[Molar mass constant|atomic mass constant]], denoted ''m''<sub>u</sub>, is defined identically, giving {{nowrap|1= ''m''<sub>u</sub> = {{sfrac|1|12}} ''m''({{sup|12}}C) = 1 Da}}.<ref>{{cite journal |first=Barry N. |last=Taylor |year=2009 |title=Molar mass and related quantities in the new SI |journal=Metrologia |volume=46 |issue=3 |pages=L16–L19 |doi=10.1088/0026-1394/46/3/L01 |s2cid=115540416 |url=https://tsapps.nist.gov/publication/get_pdf.cfm?pub_id=901156}}</ref>

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

The [[molecular mass]]es of [[protein]]s, [[nucleic acid]]s, and other large [[polymer]]s are often expressed with the unit [[Kilo-|kilo]]<nowiki/>dalton (kDa) and [[Mega-|mega]]<nowiki/>dalton (MDa).<ref name=berg2007p35/> [[Titin]], one of the largest known proteins, has a molecular mass of between 3 and 3.7 megadaltons.<ref name="opitz2003" /> The DNA of [[chromosome 1]] in the [[human genome]] has about 249 million [[base pair]]s, each with an average mass of about {{val|650|u=Da}}, or {{val|156|u=GDa}} total.<ref name=molfat/>

The [[Mole (unit)|mole]] is a unit of [[amount of substance]] used in chemistry and physics, such that the mass of one mole of a substance expressed in [[gram]]s is numerically equal to the average mass of one of its particles expressed in daltons. That is, the [[molar mass]] of a chemical compound expressed in g/mol or kg/kmol is numerically equal to its average molecular mass expressed in Da. For example, the average mass of one molecule of [[water]] is about 18.0153&nbsp;Da, and the mass of one mole of water is about 18.0153&nbsp;g. A protein whose molecule has an average mass of {{val|64|u=kDa}} would have a molar mass of {{val|64|u=kg/mol}}. However, while this equality can be assumed for practical purposes, it is only approximate, because of the [[2019 revision of the SI|2019 redefinition of the mole]].<ref name=berg2007p35/><ref name=bipm9th/>

In general, the mass in daltons of an atom is numerically close but not exactly equal to the [[atomic mass number|number of nucleons]] in its [[atomic nucleus|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&nbsp;daltons, which corresponds with the number of nucleons that it has (6&nbsp;[[proton]]s and 6&nbsp;[[neutron]]s). 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 [[electron]]s.  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 {{val|1.007825032241|(94)|u=Da}},{{efn|The digits in parentheses indicate the uncertainty; see ''[[Uncertainty notation]]''.}}
the mass of a proton is {{physconst|mp_Da|after=,}} the mass of a free neutron is {{physconst|mn_Da|after=,}} and the mass of a [[hydrogen-2]] (deuterium) atom is {{val|2.014101778114|(122)|u=Da}}.<ref name=wang2017ii/>  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]] (''m''{{sub|e}}).

== Energy equivalents ==
The atomic mass constant can also be expressed as its [[mass–energy equivalence|energy-equivalent]], ''m''{{sub|u}}''c''{{sup|2}}. The CODATA recommended values are:
{{block indent|{{physconst|muc2|symbol=yes}} {{=}} {{physconst|muc2_MeV}}}}
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 ===

[[File:Jean Perrin 1926.jpg|right|thumb|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 [[standard atomic weight|atomic weight scale]].<ref name=petley1989/>

For technical reasons, in 1898, chemist [[Wilhelm Ostwald]] and others proposed to redefine the unit of atomic mass as {{sfrac|1|16}} the mass of an oxygen atom.<ref name=hold2004/> That proposal was formally adopted by the [[Commission on Isotopic Abundances and Atomic Weights|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.<ref name=petley1989/> Physicist [[Jean Baptiste Perrin|Jean Perrin]] had adopted the same definition in 1909 during his experiments to determine the atomic masses and the [[Avogadro constant]].<ref name=perrin1909/> This definition remained unchanged until 1961.<ref name=chang2005/><ref name=kelt2008/> Perrin also defined the "mole" as an amount of a compound that contained as many molecules as 32 grams of oxygen ({{chem|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 {{sfrac|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 {{sfrac|1|16}} of the mass of an atom of the isotope oxygen-16 (<sup>16</sup>O).<ref name=hold2004/>

=== Joint definition by IUPAP and IUPAC ===

The existence of two distinct units with the same name was confusing, and the difference (about {{val|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. 

In April 1957 [[Alfred O. C. Nier]] suggested to [[Josef Mattauch]] that the [[carbon-12]] be adopted as mass scale because of carbon's use as a secondary standard in [[mass spectrometry]]. Also, carbon-12 implied acceptable relative changes in the atomic weight scale, i.e., 42 parts-per-million (ppm) compared to 275 ppm for [[oxygen-16]] which would not acceptable to chemists. 

Following the approval of the [[International Union of Pure and Applied Physics]] (IUPAP) General Assembly at Ottawa, Canada in 1960 and the [[International Union of Pure and Applied Chemistry]] (IUPAC) General Assembly at Montreal, Canada in 1961, the atomic weights were officially given on the carbon-12 scale for the first time.<ref name=petley1989/><ref name=hold2004/>

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.<ref name=goldbUnAtMaUn/> 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 {{val|12.011|u=Da}}, and that of oxygen is about {{val|15.999|u=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]].<ref name=bipm1971/>

=== Unit name ===
In 1993, the IUPAC proposed the shorter name "dalton" (with symbol "Da") for the unified atomic mass unit.<ref name=mills1993/><ref name=goldbDa/> 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.<ref name=iupap2005/>

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".<ref name=bipm-ccu15/> In 2006, the BIPM included the dalton in its 8th edition of the [[SI]] brochure of formal definitions as a [[non-SI units accepted for use with the SI|non-SI unit accepted for use with the SI]].<ref name=bipm8th/> The name was also listed as an alternative to "unified atomic mass unit" by the [[International Organization for Standardization]] in 2009.<ref name=ISO1/><ref name=ISO10/> It is now recommended by several scientific publishers,<ref name=oxsty/> and some of them consider "atomic mass unit" and "amu" deprecated.<ref name=rapsty/> 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&nbsp;(Da) and the unified atomic mass unit&nbsp;(u) are alternative names (and symbols) for the same unit.<ref name=bipm9th/>

=== 2019 revision of the SI ===
The definition of the dalton was not affected by the [[2019 revision of the SI]],<ref name=cipm106/><ref name=cgpm26/><ref name=bipm9th/> that is, 1&nbsp;Da in the SI is still {{sfrac|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 {{physconst|NA|unit=no|ref=no}} 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&nbsp;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,<ref name=lehm2016/> although the relative standard uncertainty of {{val|4.5|e=-10}} at the time of the redefinition is insignificant for all practical purposes.<ref name="bipm9th" />  One entity, symbol ent, is the smallest amount of any substance (retaining its chemical properties). One mole is an aggregate of an Avogadro number of entities, 1 mol = ''N''<sub>0</sub> ent. This means that the appropriate atomic-scale unit for molar mass is dalton per entity, Da/ent = ''M''<sub>u</sub>, very nearly equal to 1 g/mol. For Da/ent to be exactly equal to g/mol, the dalton would need to be redefined exactly in terms of the (fixed-''h'') kilogram. Then, in addition to the identity g = (g/Da) Da, we would have the parallel relationship mol = (g/Da) ent = ''N''<sub>0</sub> ent, conforming to the original mole concept—that the Avogadro number is the gram-to-dalton mass unit ratio. 

== 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]], ''E''{{sub|b}}/''m''{{sub|u}}''c''{{sup|2}}. The total binding energy of the six electrons in a carbon-12 atom is {{val|1030.1089|u=eV}}&nbsp;= {{val|1.6504163|e=−16|u=J}}: ''E''<sub>b</sub>/''m''<sub>u</sub>''c''<sup>2</sup>&nbsp;= {{val|1.1058674|e=−6}}, or about one part in 10&nbsp;million of the mass of the atom.<ref>{{cite journal |author-first1=Peter J. |author-last1=Mohr |author-first2=Barry N. |author-last2=Taylor |title=CODATA recommended values of the fundamental physical constants: 2002 |url=https://www.nist.gov/pml/div684/fcdc/upload/rmp2002-2.pdf |journal=[[Reviews of Modern Physics]] |volume=77 |issue=1 |pages=1–107 |bibcode=2005RvMP...77....1M |doi=10.1103/RevModPhys.77.1 |year=2005 |archive-url=https://web.archive.org/web/20171001121924/https://www.nist.gov/sites/default/files/documents/pml/div684/fcdc/rmp2002-2.pdf|archive-date=2017-10-01}}</ref>

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 ===
[[File:Johann Josef Loschmidt portrait plaque.jpg|right|thumb|Josef Loschmidt]]
A reasonably accurate value of the atomic mass unit was first obtained indirectly by [[Johann Josef Loschmidt|Josef Loschmidt]] in 1865, by estimating the number of particles in a given volume of gas.<ref name=losch1865/>

=== 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.<ref name=oseen1926/>

=== Coulometry ===
{{main|Coulometry}}
The electric charge per [[mole (unit)|mole]] of [[elementary charge]]s is a constant called the [[Faraday constant]], ''F'', whose value had been essentially known since 1834 when [[Michael Faraday]] published [[Faraday's laws of electrolysis|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.<ref name=ebrit1974/>

The classic experiment is that of Bower and Davis at [[NIST]],<ref>This account is based on the review in {{cite journal |author-first1=Peter J. |author-last1=Mohr |author-first2=Barry N. |author-last2=Taylor |title=CODATA recommended values of the fundamental physical constants: 1998 |journal=[[Journal of Physical and Chemical Reference Data]] |volume=28 |issue=6 |pages=1713–1852 |bibcode=1999JPCRD..28.1713M |doi=10.1063/1.556049 |year=1999 |url=https://www.nist.gov/pml/div684/fcdc/upload/rmp1998-2.pdf |archive-url=https://web.archive.org/web/20171001122752/https://www.nist.gov/sites/default/files/documents/pml/div684/fcdc/rmp1998-2.pdf|archive-date=2017-10-01}}</ref> 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''{{sub|r}} the atomic weight of silver, then the Faraday constant is given by:
{{block indent|<math>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 ''F''{{sub|90}}&nbsp;= {{val|96485.39|(13)|u=C|up=mol}}, which corresponds to a value for the Avogadro constant of {{val|6.0221449|(78)|e=23|u=mol-1}}: both values have a relative standard uncertainty of {{val|1.3|e=-6}}.

=== Electron mass measurement ===
In practice, the atomic mass constant is determined from the [[electron rest mass]] ''m''{{sub|e}} and the [[electron relative atomic mass]] ''A''{{sub|r}}(e) (that is, the mass of electron divided by the atomic mass constant).<ref>{{cite journal |author-first1=Peter J. |author-last1=Mohr |author-first2=Barry N. |author-last2=Taylor |title=CODATA recommended values of the fundamental physical constants: 1998 |journal=[[Journal of Physical and Chemical Reference Data]] |volume=28 |issue=6 |pages=1713–1852 |bibcode=1999JPCRD..28.1713M |doi=10.1063/1.556049 |year=1999 |url=https://www.nist.gov/pml/div684/fcdc/upload/rmp1998-2.pdf |archive-url=https://web.archive.org/web/20171001122752/https://www.nist.gov/sites/default/files/documents/pml/div684/fcdc/rmp1998-2.pdf|archive-date=2017-10-01}}</ref> 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.
{{block indent|<math>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>}}
{{block indent|<math>m_{\rm u} = \frac{M_{\rm u}}{N_{\rm A}} ,</math>}}
{{block indent|<math>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} ,</math>}}
where ''c'' is the [[speed of light]], ''h'' is the [[Planck constant]], ''α'' is the [[fine-structure constant]], and ''R''{{sub|∞}} 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.
{| class="wikitable"
|- style="line-height:133%"
! Constant
! Symbol
! 2014 [[Committee on Data for Science and Technology|CODATA]] values
! Relative{{br}}standard{{br}}uncertainty
! Correlation{{br}}coefficient{{br}}with ''N''{{sub|A}}
|-
| [[Proton–electron mass ratio]]
| align="center" |''m''{{sub|p}}/''m''{{sub|e}}
| {{val|1836.15267389|(17)}}
| align="center" |{{val|9.5|e=-11}}
| {{val|−0.0003}}
|-
| [[Molar mass constant]]
| align="center" |''M''{{sub|u}}
| 1 g/mol
| align="center" |'''0''' (defined)
| —
|-
| [[Rydberg constant]]
| align="center" |''R''{{sub|∞}}
| {{val|10973731.568508|(65)|u=m-1}}
| align="center" |{{val|5.9|e=-12}}
| −0.0002
|-
| [[Planck constant]]
| align="center" |''h''
| {{val|6.626070040|(81)|e=-34|u=J.s}}
| align="center" |{{val|1.2|e=-8}} 
| −0.9993
|-
| [[Speed of light]]
| align="center" |''c''
| {{val|299792458|u=m|up=s}}
| align="center" |'''0''' (defined)
| —
|-
| [[Fine structure constant]]
| align="center" |''α''
| {{val|7.2973525664|(17)|e=-3}}
| align="center" |{{val|2.3|e=-10}}
| 0.0193
|-
| [[Avogadro constant]]
| align="center" |''N''{{sub|A}}
| {{val|6.022140857|(74)|e=23|u=mol-1}}
| align="center" |{{val|1.2|e=-8}} 
| 1
|-
|}

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

{| class="wikitable"
|- style="line-height:133%"
! Constant
! Symbol
! 2018 [[Committee on Data for Science and Technology|CODATA]] values<ref>{{Cite web|url=https://physics.nist.gov/cuu/Constants/bibliography.html|title=Constants bibliography, source of the CODATA internationally recommended values|website=The NIST Reference on Constants, Units, and Uncertainty|access-date=4 August 2021}}</ref>
! Relative{{br}}standard{{br}}uncertainty
! Correlation{{br}}coefficient{{br}}with ''N''{{sub|A}}
|-
| [[Proton–electron mass ratio]]
| align="center" |''m''{{sub|p}}/''m''{{sub|e}}
| {{val|1836.15267343|(11)}}
| align="center" |{{val|6.0|e=-11}}
| —
|-
| [[Molar mass constant]]
| align="center" |''M''{{sub|u}}
| {{val|0.99999999965|(30)|u=g|up=mol}}
| align="center" |{{val|3.0|e=-10}}
| —
|-
| [[Rydberg constant]]
| align="center" |''R''{{sub|∞}}
| {{val|10973731.568160|(21)|u=m-1}}
| align="center" |{{val|1.9|e=-12}}
| —
|-
| [[Planck constant]]
| align="center" |''h''
| {{val|6.62607015|e=-34|u=J.s}}
| align="center" |'''0''' (defined)
| —
|-
| [[Speed of light]]
| align="center" |''c''
| {{val|299792458|u=m|up=s}}
| align="center" |'''0''' (defined)
| —
|-
| [[Fine structure constant]]
| align="center" |''α''
| {{val|7.2973525693|(11)|e=-3}}
| align="center" |{{val|1.5|e=-10}}
| —
|-
| [[Avogadro constant]]
| align="center" |''N''{{sub|A}}
| {{val|6.02214076|e=23|u=mol-1}}
| align="center" |'''0''' (defined)
| —
|-
|}

=== X-ray crystal density methods ===
[[Image:Silicon-unit-cell-labelled-3D-balls.png|thumb|right|200px|[[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]], ''V''{{sub|m}}, to the atomic volume ''V''{{sub|atom}}:
<math display=block>N_{\rm A}  =  \frac{V_{\rm m}}{V_{\rm atom}},</math>
where {{nowrap|1= ''V''{{sub|atom}} = {{sfrac|''V''{{sub|cell}}|''n''}}}} and ''n'' is the number of atoms per unit cell of volume ''V''<sub>cell</sub>.

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.<ref>{{cite web|url=https://webmineral.com/help/CellDimensions.shtml|title=Unit Cell Formula|work=Mineralogy Database|date=2000–2005|access-date=2007-12-09}}</ref> The CODATA value of ''a'' for silicon is {{physconst|asil|after=.}}

In practice, measurements are carried out on a distance known as ''d''{{sub|220}}(Si), which is the distance between the planes denoted by the [[Miller index|Miller indices]] {220}, and is equal to {{nowrap|''a''/{{radic|8}}}}.

The [[isotope]] proportional composition of the sample used must be measured and taken into account. Silicon occurs in three stable isotopes ({{sup|28}}Si, {{sup|29}}Si, {{sup|30}}Si), and the natural variation in their proportions is greater than other uncertainties in the measurements. The [[atomic weight]] ''A''{{sub|r}} for the sample crystal can be calculated, as the [[standard atomic weight]]s of the three [[nuclide]]s are known with great accuracy. This, together with the measured [[density]] ''ρ'' of the sample, allows the molar volume ''V''{{sub|m}} to be determined:
<math display=block>V_{\rm m} = \frac{A_{\rm r}M_{\rm u}}{\rho},</math>
where ''M''{{sub|u}} is the molar mass constant. The CODATA value for the molar volume of silicon is {{physconst|VmSi|ref=no}}, with a relative standard uncertainty of {{physconst|VmSi|runc=yes|after=.}}

== See also ==
{{Portal|Physics}}
* [[Mass (mass spectrometry)]]
** [[Kendrick mass]]
** [[Monoisotopic mass]]
* [[Mass-to-charge ratio]]

== Notes ==
{{notelist}}

== References ==
{{reflist|25em|refs=

<ref name=goldbAtMaCo>{{GoldBookRef | title = atomic mass constant | file = A00497}}</ref>

<ref name=goldbUnAtMaUn>{{GoldBookRef | title = unified atomic mass unit | file = U06554 | accessdate = 2010-07-16}}</ref>

<ref name=goldbDa>{{GoldBookRef | title = dalton | file = D01514 | accessdate = 2015-04-28}}</ref>

<ref name=wang2017ii>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}}</ref>

<ref name=chang2005>{{cite book | last = Chang | first = Raymond | date = 2005 | title = Physical Chemistry for the Biosciences | isbn = 978-1-891389-33-7 | page = 5 | publisher = University Science Books | url = https://books.google.com/books?id=PNH1fHj5Tw0C&pg=PA5
}}</ref>

<ref name=mills1993>{{cite book | title = Quantities, Units and Symbols in Physical Chemistry International Union of Pure and Applied Chemistry; Physical Chemistry Division | url = https://archive.org/details/quantitiesunitss0000unse | publisher = International Union of Pure and Applied Chemistry and published for them by Blackwell Science Ltd | edition = 2nd | year = 1993 | first1 = Ian | last1 = Mills | first2 = Tomislav | last2 = Cvitaš | first3 = Klaus | last3 = Homann | first4 = Nikola | last4 = Kallay | first5 = Kozo | last5 = Kuchitsu | isbn = 978-0-632-03583-0 | url-access = registration }}</ref>

<ref name=bipm-ccu15>{{cite web | url = https://www.bipm.org/utils/common/pdf/CC/CCU/CCU15.pdf | title = Consultative Committee for Units (CCU); Report of the 15th meeting (17–18 April 2003) to the International Committee for Weights and Measures | access-date = 14 Aug 2010}}</ref>

<ref name=ISO1>{{cite book | title = International Standard ISO 80000-1:2009 – Quantities and Units – Part 1: General | publisher = International Organization for Standardization | year = 2009}}</ref>

<ref name=ISO10>{{citation | title = International Standard ISO 80000-10:2019 – Quantities and units – Part 10: Atomic and nuclear physics | publisher = International Organization for Standardization | year = 2019}}</ref>

<ref name=oxsty>{{cite web | url = https://www.oxfordjournals.org/our_journals/aobpla/for_authors/ | archive-url = https://web.archive.org/web/20111103195158/https://www.oxfordjournals.org/our_journals/aobpla/for_authors/ | url-status = dead | archive-date = 2011-11-03 | title = Instructions to Authors | work = AoB Plants | publisher = Oxford journals; Oxford University Press | access-date = 2010-08-22}}</ref>

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<ref name=molfat>Integrated DNA Technologies (2011): "[https://sfvideo.blob.core.windows.net/sitefinity/docs/default-source/biotech-basics/molecular-facts-and-figures.pdf?sfvrsn=4563407_4 Molecular Facts and Figures] {{Webarchive|url=https://web.archive.org/web/20200418202241/https://sfvideo.blob.core.windows.net/sitefinity/docs/default-source/biotech-basics/molecular-facts-and-figures.pdf?sfvrsn=4563407_4 |date=2020-04-18 }}". Article on the [https://www.idtdna.com/pages/education/biotech-basics IDT website, Support & Education section] {{Webarchive|url=https://web.archive.org/web/20210119024825/https://www.idtdna.com/pages/education/biotech-basics |date=2021-01-19 }}, accessed on 2019-07-08.</ref>

<ref name=opitz2003>{{cite journal |vauthors=Opitz CA, [[Matthew Kulke|Kulke M]], Leake MC, Neagoe C, Hinssen H, Hajjar RJ, Linke WA | title = Damped elastic recoil of the titin spring in myofibrils of human myocardium | journal = Proc. Natl. Acad. Sci. U.S.A. | volume = 100 | issue = 22 | pages = 12688–93 |date=October 2003 | pmid = 14563922 | pmc = 240679 | doi = 10.1073/pnas.2133733100 |bibcode = 2003PNAS..10012688O | doi-access = free }}</ref>

<ref name=kelt2008>{{cite book|last1=Kelter|first1=Paul B.|last2=Mosher|first2=Michael D.|last3=Scott|first3=Andrew | date = 2008 | title = Chemistry: The Practical Science | isbn = 978-0-547-05393-6 | volume = 10 | page = 60 |publisher=Cengage Learning | url = https://books.google.com/books?id=VfcKIManfkUC&pg=PA60}}</ref>

<ref name=perrin1909>{{cite journal | first = Jean | last = Perrin | author-link = Jean Baptiste Perrin | title = Mouvement brownien et réalité moléculaire | journal = [[Annales de Chimie et de Physique]] | series = 8<sup>e</sup> Série | volume = 18 | pages = 1–114 | year = 1909}} [https://web.lemoyne.edu/~giunta/perrin.html Extract in English, translation by Frederick Soddy].</ref>

<ref name=bipm8th>{{SIbrochure8th | pages = 114–15}}</ref>

<ref name=bipm9th>Bureau International des Poids et Mesures (2019): ''[https://www.bipm.org/utils/common/pdf/si-brochure/SI-Brochure-9-EN.pdf The International System of Units (SI)]'', 9th edition, English version, page 146. Available at the [https://www.bipm.org/en/publications/si-brochure/ BIPM website].</ref>

<ref name=petley1989>{{cite journal | last = Petley | first = B. W. | title = The atomic mass unit | journal = IEEE Trans. Instrum. Meas. | volume = 38 | issue = 2 | pages = 175–179 | doi = 10.1109/19.192268 | year = 1989 | bibcode = 1989ITIM...38..175P | url = https://zenodo.org/record/1260639}}</ref>

<ref name=hold2004>{{cite journal | last = Holden | first = Norman E. | year = 2004 | title = Atomic Weights and the International Committee—A Historical Review | journal = Chemistry International | volume = 26 | issue = 1 | pages = 4–7 | url = https://www.iupac.org/publications/ci/2004/2601/1_holden.html}}</ref>

<ref name=cipm106>International Bureau for Weights and Measures (2017): ''[https://www.bipm.org/utils/en/pdf/CIPM/CIPM2017-EN.pdf 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 [https://www.bipm.org/en/committees/cipm/meeting/106.html BIPM website] {{Webarchive|url=https://web.archive.org/web/20210221105820/https://www.bipm.org/en/committees/cipm/meeting/106.html |date=2021-02-21 }}.</ref>

<ref name=cgpm26>International Bureau for Weights and Measures (2018): ''[https://www.bipm.org/utils/common/pdf/CGPM-2018/26th-CGPM-Resolutions.pdf Resolutions Adopted – 26th Conference Générale des Poids et Mesures] {{Webarchive|url=https://web.archive.org/web/20181119214326/https://www.bipm.org/utils/common/pdf/CGPM-2018/26th-CGPM-Resolutions.pdf |date=2018-11-19 }}''. Available at the [https://www.bipm.org/en/cgpm-2018/ BIPM website].</ref>

<ref name=iupap2005>{{cite web | url = https://archive.iupap.org/commissions/c2/reports/ga-05.html | title = IUPAP: C2: Report 2005 | access-date = 2018-07-15}}</ref>

<ref name=oseen1926>[[Carl Wilhelm Oseen|Oseen, C.W.]] (December 10, 1926). ''[https://nobelprize.org/nobel_prizes/physics/laureates/1926/press.html Presentation Speech for the 1926 Nobel Prize in Physics]''.</ref>

<ref name=losch1865>{{cite journal | first = J. | last = Loschmidt | author-link = Johann Josef Loschmidt | title = Zur Grösse der Luftmoleküle | journal = Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften Wien | volume = 52 | issue = 2 | pages = 395–413 | year = 1865}} [https://web.archive.org/web/20060207130125/https://dbhs.wvusd.k12.ca.us/webdocs/Chem-History/Loschmidt-1865.html English translation].</ref>

<ref name=ebrit1974>(1974): ''[https://physics.nist.gov/cuu/Constants/historical1.html Introduction to the constants for nonexperts, 1900–1920]'' From the ''Encyclopaedia Britannica'', 15th edition; reproduced by [[NIST]]. Accessed on 2019-07-03.</ref>

<ref name=lehm2016>{{Cite journal | title = Unified Atomic Mass Unit | last1 = Lehmann | first1 = H. P.|last2=Fuentes-Arderiu|first2=X. | date = 2016-02-29|last3=Bertello|first3=L. F. | doi = 10.1515/iupac.68.2930 |periodical=Glossary of Terms in Quantities and Units in Clinical Chemistry |doi-access=free }}</ref>

<ref name=bipm1971>Bureau International des Poids et Mesures (1971): ''[https://www.bipm.org/jsp/en/ListCGPMResolution.jsp?CGPM=14 14th Conference Générale des Poids et Mesures] {{Webarchive|url=https://web.archive.org/web/20200923114009/https://www.bipm.org/jsp/en/ListCGPMResolution.jsp?CGPM=14 |date=2020-09-23 }}'' Available at the [https://www.bipm.org/ BIPM website].</ref>

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<ref name=feynman>Richard P. Feynman (1963): ''The Feynman Lectures on Physics'', Volume II, 2nd edition; 512 pages. {{isbn|9780805390476}}</ref>

<ref name=mcgraw>Marvin Yelles (1971): ''McGraw-Hill Encyclopedia of Science and Technology'', Volume 9, 3rd edition; 707 pages. {{isbn|9780070797987}}</ref>

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<ref name=virgo1933>{{cite journal | last = Virgo | first = S.E. | url = https://gemini.tntech.edu/~tfurtsch/scihist/loschmid.html | title = Loschmidt's Number | journal = Science Progress | volume = 27 | year = 1933 | pages = 634–649 | url-status = dead | archive-url = https://web.archive.org/web/20050404003919/https://gemini.tntech.edu/~tfurtsch/scihist/loschmid.html | archive-date = 2005-04-04 }}</ref>

<ref name=kotz2008>{{cite book | last = Kotz | first = John C.|author2 = Treichel, Paul M.|author3 = Townsend, John R. | title = Chemistry and Chemical Reactivity | edition = 7th | url = https://cengagesites.com/academic/kotz.cfm?site=2719&section=home|archive-url = https://web.archive.org/web/20081016082922/https://cengagesites.com/academic/kotz.cfm?site=2719&section=home|url-status = dead|archive-date = 2008-10-16 | year = 2008 | publisher = Brooks/Cole | isbn = 978-0-495-38703-9}}</ref>

-->

}} <!-- end "refs=" -->

== External links ==
* {{cite web |title=Atomic weights and isotopic compositions |series=Physical Reference Data |website=physics.nist.gov |date=23 August 2009 |publisher=[[National Institute for Standards and Technology]] |url=https://physics.nist.gov/PhysRefData/Compositions/ }}
* {{cite web |title=Atomic mass unit |website=sizes.com |url=https://www.sizes.com/units/atomic_mass_unit.htm |url-status=dead <!-- presumed --> |archive-url=https://web.archive.org/web/20080115211624/https://www.sizes.com/units/atomic_mass_unit.htm |archive-date=2008-01-15 }}

{{SI units|state=collapsed}}

[[Category:Metrology]]
[[Category:Nuclear chemistry]]
[[Category:Units of chemical measurement]]
[[Category:Units of mass]]