Dalton (unit)

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https://en.wikipedia.org/wiki/Dalton_(unit) 

 

dalton (unified atomic mass unit)
Unit of	mass
Symbol	Da or u
Named after	John 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 unit of mass widely used in physics and chemistry. It is 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. The atomic mass constant, denoted m_u, is defined identically, giving m_u = m(¹²C)/12 = 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
₉H
₈O
₄, has an average mass of approximately 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 kilodaltons (kDa), megadaltons (MDa), etc. 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, widely used in chemistry and physics, which was originally defined so that the mass of one mole of a substance, measured in grams, would be numerically equal to the average mass of one of its constituent particles, measured 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 way mole was redefined on 20 May 2019.

In general, the mass in daltons of an atom is numerically close but not exactly equal to the number of nucleons A contained 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 one unbound atom of the common hydrogen isotope (hydrogen-1, protium) is 1.007825032241(94) Da, the mass of the proton is 1.007276466621(53) Da, the mass of one free neutron is 1.00866491595(49) Da, and the mass of one 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 atomic units systems, which is the electron rest mass (m_e).

Energy equivalents

The atomic mass constant can also be expressed as its energy-equivalent, m_uc². The 2018 CODATA recommended values are:

m_uc² = 1.49241808560(45)×10⁻¹⁰ J = 931.49410242(28) MeV

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

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, that of 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 of 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 the existence of elemental isotopes, which occurred in 1912. The 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. 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).

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).

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. 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 approximately 12.011 Da, and that of oxygen is approximately 15.999 Da. These values, generally used in chemistry, are based on averages of many samples from Earth's crust, its atmosphere, and organic materials.

Adoption by the BIPM

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

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 [the SI] prefixes". In 2006, the BIPM included the dalton in its 8th edition of the formal definition of 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 formal definition of SI 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 redefinition of the SI base units

The definition of the dalton was not affected by the 2019 redefinition of SI base units, 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 is no longer exactly 1 g/mol, meaning that the number of grams in the mass of one mole of any substance is no longer exactly equal to the number of daltons in its average molecular mass.

Measurement

Although 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_uc². 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_uc² = 1.1058674×10⁻⁶, or about one part in 10 million of the mass of the atom.

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

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

Main article: 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:

${\displaystyle F=\frac{A_{\mathrm{r}}{M}_{\mathrm{u}}It}{m}.}$

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 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.

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

${\displaystyle m_{\mathrm{u}}=\frac{M_{\mathrm{u}}}{N_{\mathrm{A}}},}$

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

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	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

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_m, to the atomic volume V_atom:

${\displaystyle N_{\mathrm{A}}=\frac{V_{\mathrm{m}}}{V_{\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{m}}}}$,

where

• ${\displaystyle V_{\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{m}}=\frac{V_{\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}}{n}}$, 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 2018 CODATA value of a for silicon is 5.431020511(89)×10⁻¹⁰ m.

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 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:

${\displaystyle V_{\mathrm{m}}=\frac{A_{\mathrm{r}}{M}_{\mathrm{u}}}{\rho }}$

where M_u 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.

Rydberg constant

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Rydberg_constant 

In spectroscopy, the Rydberg constant, symbol ${\displaystyle R_{\mathrm{\infty }}}$ for heavy atoms or ${\displaystyle R_{\text{H}}}$ for hydrogen, named after the Swedish physicist Johannes Rydberg, is a physical constant relating to the electromagnetic spectra of an atom. The constant first arose as an empirical fitting parameter in the Rydberg formula for the hydrogen spectral series, but Niels Bohr later showed that its value could be calculated from more fundamental constants via his Bohr model. As of 2018, ${\displaystyle R_{\mathrm{\infty }}}$ and electron spin g-factor are the most accurately measured physical constants.

The constant is expressed for either hydrogen as ${\displaystyle R_{\text{H}}}$, or at the limit of infinite nuclear mass as ${\displaystyle R_{\mathrm{\infty }}}$. In either case, the constant is used to express the limiting value of the highest wavenumber (inverse wavelength) of any photon that can be emitted from an atom, or, alternatively, the wavenumber of the lowest-energy photon capable of ionizing an atom from its ground state. The hydrogen spectral series can be expressed simply in terms of the Rydberg constant for hydrogen ${\displaystyle R_{\text{H}}}$ and the Rydberg formula.

In atomic physics, Rydberg unit of energy, symbol Ry, corresponds to the energy of the photon whose wavenumber is the Rydberg constant, i.e. the ionization energy of the hydrogen atom in a simplified Bohr model.

Value

Rydberg constant

The CODATA value is

${\displaystyle R_{\mathrm{\infty }}=\frac{m_{\text{e}}{e}^4}{8{\epsilon }_0^2{h}^{3}c}}$ = 10973731.568160(21) m⁻¹,

where

• ${\displaystyle m_{\text{e}}}$ is the rest mass of the electron,
• ${\displaystyle e}$ is the elementary charge,
• ${\displaystyle {\epsilon }_0}$ is the permittivity of free space,
• ${\displaystyle h}$ is the Planck constant, and
• ${\displaystyle c}$ is the speed of light in vacuum.

The Rydberg constant for hydrogen may be calculated from the reduced mass of the electron:

${\displaystyle R_{\text{H}}=R_{\mathrm{\infty }}\frac{m_{\text{p}}}{m_{\text{e}}+m_{\text{p}}}\approx 1.09678\times {10}^7{\text{ m}}^{-1},}$

where

• ${\displaystyle m_{\text{e}}}$ is the mass of the electron,
• ${\displaystyle m_{\text{p}}}$ is the mass of the nucleus (a proton).

Rydberg unit of energy

The Rydberg unit of energy is equivalent to joules and electronvolts in the following manner:

${\displaystyle 1\text{Ry}\equiv hc{R}_{\mathrm{\infty }}=\frac{m_{\text{e}}{e}^4}{8{\epsilon }_0^2{h}^2}=\frac{e^2}{8\pi {\epsilon }_0{a}_0}=2.1798723611035(42)\times {10}^{-18}\text{J}=13.605693122994(26)\text{eV}.}$

Rydberg frequency

${\displaystyle c{R}_{\mathrm{\infty }}=3.2898419602508(64)\times {10}^{15}\text{Hz}.}$

Rydberg wavelength

${\displaystyle \frac{1}{R_{\mathrm{\infty }}}=9.112670505824(17)\times {10}^{-8}\text{m}}$.

The angular wavelength is

${\displaystyle \frac{1}{2\pi R_{\mathrm{\infty }}}=1.4503265557696(28)\times {10}^{-8}\text{m}}$.

Occurrence in Bohr model

Main article: Bohr model

The Bohr model explains the atomic spectrum of hydrogen (see hydrogen spectral series) as well as various other atoms and ions. It is not perfectly accurate, but is a remarkably good approximation in many cases, and historically played an important role in the development of quantum mechanics. The Bohr model posits that electrons revolve around the atomic nucleus in a manner analogous to planets revolving around the sun.

In the simplest version of the Bohr model, the mass of the atomic nucleus is considered to be infinite compared to the mass of the electron, so that the center of mass of the system, the barycenter, lies at the center of the nucleus. This infinite mass approximation is what is alluded to with the ${\displaystyle \mathrm{\infty }}$ subscript. The Bohr model then predicts that the wavelengths of hydrogen atomic transitions are (see Rydberg formula):

${\displaystyle \frac{1}{\lambda }=\mathrm{R}\mathrm{y}\cdot \frac{1}{hc}\left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right)=\frac{m_{\text{e}}{e}^4}{8{\epsilon }_0^2{h}^{3}c}\left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right)}$

where n₁ and n₂ are any two different positive integers (1, 2, 3, ...), and ${\displaystyle \lambda }$ is the wavelength (in vacuum) of the emitted or absorbed light.

${\displaystyle \frac{1}{\lambda }=R_M\left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right)}$

where ${\displaystyle R_M=\frac{R_{\mathrm{\infty }}}{1+\frac{m_{\text{e}}}{M}},}$ and M is the total mass of the nucleus. This formula comes from substituting the reduced mass of the electron.

Precision measurement

See also: Precision tests of QED

The Rydberg constant is one of the most precisely determined physical constants, with a relative standard uncertainty of under 2 parts in 10¹². This precision constrains the values of the other physical constants that define it.

Since the Bohr model is not perfectly accurate, due to fine structure, hyperfine splitting, and other such effects, the Rydberg constant ${\displaystyle R_{\mathrm{\infty }}}$ cannot be directly measured at very high accuracy from the atomic transition frequencies of hydrogen alone. Instead, the Rydberg constant is inferred from measurements of atomic transition frequencies in three different atoms (hydrogen, deuterium, and antiprotonic helium). Detailed theoretical calculations in the framework of quantum electrodynamics are used to account for the effects of finite nuclear mass, fine structure, hyperfine splitting, and so on. Finally, the value of ${\displaystyle R_{\mathrm{\infty }}}$ is determined from the best fit of the measurements to the theory.

Alternative expressions

The Rydberg constant can also be expressed as in the following equations.

${\displaystyle R_{\mathrm{\infty }}=\frac{{\alpha }^2{m}_{\text{e}}c}{4\pi \hslash }=\frac{{\alpha }^2}{2{\lambda }_{\text{e}}}=\frac{\alpha }{4\pi a_0}}$

and in energy units

${\displaystyle \text{Ry}=hc{R}_{\mathrm{\infty }}=\frac{1}{2}{m}_{\text{e}}{c}^2{\alpha }^2=\frac{1}{2}\frac{e^4{m}_{\text{e}}}{(4\pi {\epsilon }_0{)}^2{\hslash }^2}=\frac{1}{2}\frac{m_{\text{e}}{c}^2{r}_{\text{e}}}{a_0}=\frac{1}{2}\frac{hc{\alpha }^2}{{\lambda }_{\text{e}}}=\frac{1}{2}h{f}_{\text{C}}{\alpha }^2=\frac{1}{2}\hslash {\omega }_{\text{C}}{\alpha }^2=\frac{1}{2m_{\text{e}}}{\left({\displaystyle \frac{\hslash }{a_0}}\right)}^2=\frac{1}{2}\frac{e^2}{(4\pi {\epsilon }_0)a_0}.}$

where

• ${\displaystyle m_{\text{e}}}$ is the electron rest mass,
• ${\displaystyle e}$ is the electric charge of the electron,
• ${\displaystyle h}$ is the Planck constant,
• ${\displaystyle \hslash =h/2\pi }$ is the reduced Planck constant,
• ${\displaystyle c}$ is the speed of light in a vacuum,
• ${\displaystyle {\epsilon }_0}$ is the electrical field constant (permittivity) of free space,
• ${\displaystyle \alpha =\frac{1}{4\pi {\epsilon }_0}\frac{e^2}{\hslash c}}$ is the fine-structure constant,
• ${\displaystyle {\lambda }_{\text{e}}=h/m_{\text{e}}c}$ is the Compton wavelength of the electron,
• ${\displaystyle f_{\text{C}}=m_{\text{e}}{c}^2/h}$ is the Compton frequency of the electron,
• ${\displaystyle {\omega }_{\text{C}}=2\pi f_{\text{C}}}$ is the Compton angular frequency of the electron,
• ${\displaystyle a_0=\frac{4\pi {\epsilon }_0{\hslash }^2}{e^2{m}_{\text{e}}}}$ is the Bohr radius,
• ${\displaystyle r_{\mathrm{e}}=\frac{1}{4\pi {\epsilon }_0}\frac{e^2}{m_{\mathrm{e}}{c}^2}}$ is the classical electron radius.

The last expression in the first equation shows that the wavelength of light needed to ionize a hydrogen atom is 4π/α times the Bohr radius of the atom.

The second equation is relevant because its value is the coefficient for the energy of the atomic orbitals of a hydrogen atom: ${\displaystyle E_n=-hc{R}_{\mathrm{\infty }}/n^2}$.