# Dalton (unit)

The dalton or unified atomic mass unit (symbols: Da or u, respectively) 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. It is a non-SI unit accepted for use with SI. The atomic mass constant, denoted m$_u$, is defined identically, giving m$_u$ = ⁠1/12⁠ m($^{12}$C) = 1 Da.
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 unit kilodalton (kDa) and megadalton (MDa). Titin, one of the largest known proteins, has a molecular mass of between 3 and 3.7 megadaltons. 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.
The 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 grams 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 Da, and the mass of one mole of water is about 18.0153 g. 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.
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, the mass of a proton is 1.0072764665789(83) Da, the mass of a free neutron is 1.00866491606(40) Da, and the mass of a hydrogen-2 (deuterium) atom is 2.014101778114(122) Da. 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$_e$).

## Energy equivalents

The atomic mass constant can also be expressed as its energy-equivalent, m$_u$c$^2$. The CODATA recommended values are:
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

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.
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. 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. Physicist Jean Perrin had adopted the same definition in 1909 during his experiments to determine the atomic masses and the Avogadro constant. This definition remained unchanged until 1961. 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 ($^{16}$O).

### 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).
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. 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.

### Unit name

In 1993, the IUPAC proposed the shorter name "dalton" (with symbol "Da") for the unified atomic mass unit. 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.
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". 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. The name was also listed as an alternative to "unified atomic mass unit" by the International Organization for Standardization in 2009. It is now recommended by several scientific publishers, and some of them consider "atomic mass unit" and "amu" deprecated. 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.

### 2019 revision of the SI

The definition of the dalton was not affected by the 2019 revision of the SI, 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 6.02214076×10$^{23}$ 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, although the relative standard uncertainty of 4.5×10$^{−10}$ at the time of the redefinition is insignificant for all practical purposes. 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$_0$ ent. This means that the appropriate atomic-scale unit for molar mass is dalton per entity, Da/ent = M$_u$, 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$_0$ 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$_b$/m$_u$c$^2$. The total binding energy of the six electrons in a carbon-12 atom is 1030.1089 eV = 1.6504163×10$^{−16}$ J: E$_b$/m$_u$c$^2$ = 1.1058674×10$^{−6}$, or about one part in 10 million of the mass of the atom.
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

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.

### 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.

### 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.
The classic experiment is that of Bower and Davis at NIST, 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:
$F={\frac {A_{\rm {r}}M_{\rm {u}}It}{m}}.$
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 F$_{90}$ = 96485.39(13) C/mol, which corresponds to a value for the Avogadro constant of 6.0221449(78)×10$^{23}$ 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 m$_e$ and the electron relative atomic mass A$_r$(e) (that is, the mass of electron divided by the atomic mass constant). 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.
$m_{\rm {u}}={\frac {m_{\rm {e}}}{A_{\rm {r}}({\rm {e}})}}={\frac {2R_{\infty }h}{A_{\rm {r}}({\rm {e}})c\alpha ^{2}}},$
$m_{\rm {u}}={\frac {M_{\rm {u}}}{N_{\rm {A}}}},$
$N_{\rm {A}}={\frac {M_{\rm {u}}A_{\rm {r}}({\rm {e}})}{m_{\rm {e}}}}={\frac {M_{\rm {u}}A_{\rm {r}}({\rm {e}})c\alpha ^{2}}{2R_{\infty }h}},$
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 N$_A$ |
| Proton–electron mass ratio | m$_p$/m$_e$ | 1836.15267389(17) | 9.5×10$^{−11}$ | −0.0003 |
| Molar mass constant | M$_u$ | 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 | N$_A$ | 6.022140857(74)×10$^{23}$ mol$^{−1}$ | 1.2×10$^{−8}$ | 1 |
The power of having defined values of universal constants as is presently the case can be understood from the table below (2018 CODATA).
| Constant | Symbol | 2018 CODATA values | Relative standard uncertainty | Correlation coefficient with N$_A$ |
| Proton–electron mass ratio | m$_p$/m$_e$ | 1836.15267343(11) | 6.0×10$^{−11}$ | — |
| Molar mass constant | M$_u$ | 0.99999999965(30) g/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 | N$_A$ | 6.02214076×10$^{23}$ 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, V$_m$, to the atomic volume V$_{atom}$: $N_{\rm {A}}={\frac {V_{\rm {m}}}{V_{\rm {atom}}}},$ where V$_{atom}$ = ⁠V$_{cell}$/n⁠ 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 a of one of the sides of the cube. The CODATA value of a for silicon is 5.431020511(89)×10$^{−10}$ m.
In practice, measurements are carried out on a distance known as d$_{220}$(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 ($^{28}$Si, $^{29}$Si, $^{30}$Si), 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: $V_{\rm {m}}={\frac {A_{\rm {r}}M_{\rm {u}}}{\rho }},$ where M$_u$ is the molar mass constant. The CODATA value for the molar volume of silicon is 1.205883199(60)×10$^{−5}$ m$^3$⋅mol$^{−1}$, with a relative standard uncertainty of 4.9×10$^{−8}$.