Metric prefix

From Wikipedia, the free encyclopedia

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

A metric prefix is a unit prefix that precedes a basic unit of measure to indicate a multiple or submultiple of the unit. All metric prefixes used today are decadic. Each prefix has a unique symbol that is prepended to any unit symbol. The prefix kilo-, for example, may be added to gram to indicate multiplication by one thousand: one kilogram is equal to one thousand grams. The prefix milli-, likewise, may be added to metre to indicate division by one thousand; one millimetre is equal to one thousandth of a metre.

A metric power is an integer unit affix, written in superscript in formal typography, that follows the basic unit of measure to indicate a multiplicity of the basic unit. In electronic plain text where superscript is not available, the subscript is often omitted, or where confusion is possible, indicated by placing the caret symbol ^ between the base unit and the integer power, thus km², km2, and km^2 are variously encountered. When no integer affix is supplied, the implied power is 1. When a unit is not mentioned at all, the implied power is 0. Negative powers imply division. With extreme formality, the unit m/s² can also be rendered m¹s^-2, but the literal present of the implied integer 1 is considered unconventional in common usage. Often all the units with positive prefixes will be listed first (in some natural order), followed by all the units with negative prefixes (in some natural order); this semi-canonical form is most easily mapped by the mind onto division notation, and makes switching between the two conventions less mentally onerous.

Decimal multiplicative prefixes have been a feature of all forms of the metric system, with six of these dating back to the system's introduction in the 1790s. Metric prefixes have also been used with some non-metric units. The SI prefixes are metric prefixes that were standardized for use in the International System of Units (SI) by the International Bureau of Weights and Measures (BIPM) in resolutions dating from 1960 to 1991. Since 2009, they have formed part of the International System of Quantities. They are also used in the Unified Code for Units of Measure (UCUM)

List of SI prefixes

The BIPM specifies twenty prefixes for the International System of Units (SI).

SI prefixes

Prefix		Base 10	Decimal	English word		Adoption
Name	Symbol			Short scale	Long scale
yotta	Y	1024	1000000000000000000000000	septillion	quadrillion	1991
zetta	Z	1021	1000000000000000000000	sextillion	trilliard	1991
exa	E	1018	1000000000000000000	quintillion	trillion	1975
peta	P	1015	1000000000000000	quadrillion	billiard	1975
tera	T	1012	1000000000000	trillion	billion	1960
giga	G	109	1000000000	billion	milliard	1960
mega	M	106	1000000	million		1873
kilo	k	103	1000	thousand		1795
hecto	h	102	100	hundred		1795
deca	da	101	10	ten		1795
		100	1	one		–
deci	d	10−1	0.1	tenth		1795
centi	c	10−2	0.01	hundredth		1795
milli	m	10−3	0.001	thousandth		1795
micro	μ	10−6	0.000001	millionth		1873
nano	n	10−9	0.000000001	billionth	milliardth	1960
pico	p	10−12	0.000000000001	trillionth	billionth	1960
femto	f	10−15	0.000000000000001	quadrillionth	billiardth	1964
atto	a	10−18	0.000000000000000001	quintillionth	trillionth	1964
zepto	z	10−21	0.000000000000000000001	sextillionth	trilliardth	1991
yocto	y	10−24	0.000000000000000000000001	septillionth	quadrillionth	1991

Prefixes adopted before 1960 already existed before SI. The introduction of the CGS system was in 1873.

Each prefix name has a symbol that is used in combination with the symbols for units of measure. For example, the symbol for kilo- is k, and is used to produce km, kg, and kW, which are the SI symbols for kilometre, kilogram, and kilowatt, respectively. Except for the early prefixes of kilo-, hecto-, and deca-, the symbols for the multiplicative prefixes are uppercase letters, and those for the fractional prefixes are lowercase letters. There is a Unicode symbol for micro µ for use when the Greek letter μ is unavailable. When both are unavailable, the visually similar lowercase Latin letter u is commonly used instead. SI unit symbols are never italicised.

Prefixes corresponding to an integer power of one thousand are generally preferred. Hence 100 m is preferred over 1 hm (hectometre) or 10 dam (decametres). The prefixes deci-, and centi-, and less frequently hecto- and deca-, are commonly used for everyday purposes, and the centimetre (cm) is especially common. Some modern building codes require that the millimetre be used in preference to the centimetre, because "use of centimetres leads to extensive usage of decimal points and confusion".

Prefixes may not be used in combination. This also applies to mass, for which the SI base unit (kilogram) already contains a prefix. For example, milligram (mg) is used instead of microkilogram (μkg).

In the arithmetic of measurements having units, the units are treated as multiplicative factors to values. If they have prefixes, all but one of the prefixes must be expanded to their numeric multiplier, except when combining values with identical units. Hence:

• 5 mV × 5 mA = 5×10⁻³ V × 5×10⁻³ A = 25×10⁻⁶ V⋅A = 25 μW.
• 5.00 mV + 10 μV = 5.00 mV + 0.01 mV = 5.01 mV.

Metric powers

When powers of units occur, for example, squared or cubed, the multiplicative prefix must be considered part of the unit, and thus included in the exponentiation:

• 1 km² means one square kilometre, or the area of a square of 1000 m by 1000 m. In other words, an area of 1000000 square metres and not 1000 square metres.
• 2 Mm³ means two cubic megametres, or the volume of two cubes of 1000000 m by 1000000 m by 1000000 m or 2×10¹⁸ m³, and not 2000000 cubic metres (2×10⁶ m³).

Examples with prefixes and powers

• 5 cm = 5×10⁻² m = 5 × 0.01 m = 0.05 m.
• 9 km² = 9 × (10³ m)² = 9 × (10³)² × m² = 9×10⁶ m² = 9 × 1000000 m² = 9000000 m².
• 3 MW = 3×10⁶ W = 3 × 1000000 W = 3000000 W.

Application to units of measurement

The use of prefixes can be traced back to the introduction of the metric system in the 1790s, long before the 1960 introduction of the SI. The prefixes, including those introduced after 1960, are used with any metric unit, whether officially included in the SI or not (e.g., millidynes and milligauss). Metric prefixes may also be used with non-metric units.

The choice of prefixes with a given unit is usually dictated by convenience of use. Unit prefixes for amounts that are much larger or smaller than those actually encountered are seldom used.

Metric units

Mass

The units kilogram, gram, milligram, microgram, and smaller are commonly used for measurement of mass. However, megagram, gigagram, and larger are rarely used; tonnes (and kilotonnes, megatonnes, etc.) or scientific notation are used instead. Megagram and teragram are occasionally used to disambiguate the tonne from other units with the name "ton".

The kilogram is the only base unit of the International System of Units that includes a metric prefix.

Volume

The litre (equal to a cubic decimetre), millilitre (equal to a cubic centimetre), microlitre, and smaller are common. In Europe, the centilitre is often used for liquids, and the decilitre is used less frequently. Bulk agricultural products, such as grain, beer and wine, are often measured in hectolitres (each 100 litres in size).

Larger volumes are usually denoted in kilolitres, megalitres or gigalitres, or else in cubic metres (1 cubic metre = 1 kilolitre) or cubic kilometres (1 cubic kilometre = 1 teralitre). For scientific purposes, the cubic metre is usually used.

Length

The kilometre, metre, centimetre, millimetre, and smaller units are common. The decimetre is rarely used. The micrometre is often referred to by the older non-SI name micron. In some fields, such as chemistry, the ångström (0.1 nm) has been used commonly instead of the nanometre. The femtometre, used mainly in particle physics, is sometimes called a fermi. For large scales, megametre, gigametre, and larger are rarely used. Instead, ad hoc non-metric units are used, such as the solar radius, astronomical units, light years, and parsecs; the astronomical unit is mentioned in the SI standards as an accepted non-SI unit.

Time

See also: Metric time and Orders of magnitude (time)

Prefixes for the SI standard unit second are most commonly encountered for quantities less than one second. For larger quantities, the system of minutes (60 seconds), hours (60 minutes) and days (24 hours) is accepted for use with the SI and more commonly used. When speaking of spans of time, the length of the day is usually standardized to 86400 seconds so as not to create issues with the irregular leap second.

Larger multiples of the second such as kiloseconds and megaseconds are occasionally encountered in scientific contexts, but are seldom used in common parlance. For long-scale scientific work, particularly in astronomy, the Julian year or annum is a standardized variant of the year, equal to exactly 31557600 SI seconds (365 days, 6 hours). The unit is so named because it was the average length of a year in the Julian calendar. Long time periods are then expressed by using metric prefixes with the annum, such as megaannum or gigaannum.

Angle

The SI unit of angle is the radian, but degrees, as well as arc-minutes and arc-seconds, see some scientific use.

Temperature

Official policy also varies from common practice for the degree Celsius (°C). NIST states: "Prefix symbols may be used with the unit symbol °C and prefix names may be used with the unit name degree Celsius. For example, 12 m°C (12 millidegrees Celsius) is acceptable." In practice, it is more common for prefixes to be used with the kelvin when it is desirable to denote extremely large or small absolute temperatures or temperature differences. Thus, temperatures of star interiors may be given in units of MK (megakelvins), and molecular cooling may be described in mK (millikelvins).

Energy

In use the joule and kilojoule are common, with larger multiples seen in limited contexts. In addition, the kilowatt-hour, a composite unit formed from the kilowatt and hour, is often used for electrical energy; other multiples can be formed by modifying the prefix of watt (e.g. terawatt-hour).

There exist a number of definitions for the non-SI unit, the calorie. There are gram calories and kilogram calories. One kilogram calorie, which equals one thousand gram calories, often appears capitalized and without a prefix (i.e. Cal) when referring to "dietary calories" in food. It is common to apply metric prefixes to the gram calorie, but not to the kilogram calorie: thus, 1 kcal = 1000 cal = 1 Cal.

Non-metric units

Metric prefixes are widely used outside the metric SI system. Common examples include the megabyte and the decibel. Metric prefixes rarely appear with imperial or US units except in some special cases (e.g., microinch, kilofoot, kilopound). They are also used with other specialized units used in particular fields (e.g., megaelectronvolt, gigaparsec, millibarn). They are also occasionally used with currency units (e.g., gigadollar), mainly by people who are familiar with the prefixes from scientific usage. In astronomy, geology, and paleontology, the year, with symbol a (from the Latin annus), is commonly used with metric prefixes: ka, Ma, and Ga.

Official policies about the use of SI prefixes with non-SI units vary slightly between the International Bureau of Weights and Measures (BIPM) and the American National Institute of Standards and Technology (NIST). For instance, the NIST advises that 'to avoid confusion, prefix symbols (and prefix names) are not used with the time-related unit symbols (names) min (minute), h (hour), d (day); nor with the angle-related symbols (names) ° (degree), ′ (minute), and ″ (second), whereas the BIPM adds information about the use of prefixes with the symbol as for arcsecond when they state: "However astronomers use milliarcsecond, which they denote mas, and microarcsecond, μas, which they use as units for measuring very small angles."

An advantage of the SI system decimal prefixes is that they make for simplicity of calculation and conversion involving units of different sizes; consider for example the simplicity of buying 13 items of 390 g weight at €12.34 per kilogram, compared with items of 13+3⁄4 oz at $4.79 per pound (or, worse, with old non-decimalized currency: £4/15/9+1⁄2). In the units used in the US, combining of units that are not decimal multiples of each other is often avoided by not mixing the units used, e.g., using inches, feet or miles only: 89  inches rather than 7 feet 5 inches (or 2 yards, 1 foot 5 inches).

Presentation

Pronunciation

When a metric prefix is affixed to a root word, the prefix carries the stress, while the root drops its stress but retains a full vowel in the syllable that is stressed when the root word stands alone. For example, kilobyte is /ˈkɪlɒbaɪt/, with stress on the first syllable. However, units in common use outside the scientific community may be stressed idiosyncratically. In English-speaking countries, kilometre is the most conspicuous example. It is often pronounced /kɪˈlɒmɪtər/, with reduced vowels on both syllables of metre. This stress is not applied to other multiples or sub-multiples of metre, or to other units prefixed with kilo-.

The prefix giga is usually pronounced in English as /ˈɡɪɡə/, with hard ⟨g⟩ as in get, but sometimes /ˈdʒɪɡə/, with soft ⟨g⟩ as in gin.

Typesetting

The LaTeX typesetting system features an SIunitx package in which the units of measurement are spelled out, for example, \SI{3}{\tera\hertz} formats as "3 THz".

Non-standard prefixes

See also: Unit prefix § Unofficial prefixes

 

Distance marker on the Rhine: 36 (XXXVI) myriametres from Basel. The stated distance is 360 km; the decimal mark in Germany is a comma.

Obsolete metric prefixes

Some of the prefixes formerly used in the metric system have fallen into disuse and were not adopted into the SI. The decimal prefix for ten thousand, myria- (sometimes spelled myrio-), and the prefixes double- (2×) and demi- (1/2×) were parts of the original metric system adopted by France in 1795, but were not retained when the SI prefixes were internationally adopted by the 11th CGPM conference in 1960.

Other metric prefixes used historically include hebdo- (10⁷) and micri- (10⁻¹⁴).

Double prefixes

Double prefixes have been used in the past, such as micromillimetres or millimicrons (now nanometres), micromicrofarads (μμF; now picofarads, pF), kilomegatons (now gigatons), hectokilometres (now 100 kilometres) and the derived adjective hectokilometric (typically used for qualifying the fuel consumption measures). These are not compatible with the SI.

Other obsolete double prefixes included "decimilli-" (10⁻⁴), which was contracted to "dimi-" and standardized in France up to 1961.

Proposed prefixes

A proposal made in 2019 to the BIPM is ronna (R) for 10²⁷, quecca (Q) for 10³⁰, ronto (r) for 10⁻²⁷ and quecto (q) for 10⁻³⁰. A 2022 draft resolution by the BIPM revises this with quetta in place of quecca.

Similar symbols and abbreviations

In written English, the symbol K is often used informally to indicate a multiple of thousand in many contexts. For example, one may talk of a 40K salary (40000), or call the Year 2000 problem the Y2K problem. In these cases, an uppercase K is often used with an implied unit (although it could then be confused with the symbol for the kelvin temperature unit if the context is unclear). This informal postfix is read or spoken as "thousand" or "grand", or just "k".

The financial and general news media mostly use m or M, b or B, and t or T as abbreviations for million, billion (10⁹) and trillion (10¹²), respectively, for large quantities, typically currency and population.

The medical and automotive fields in the United States use the abbreviations cc or ccm for cubic centimetres. One cubic centimetre is equal to one millilitre.

For nearly a century, engineers used the abbreviation MCM to designate a "thousand circular mils" in specifying the cross-sectional area of large electrical cables. Since the mid-1990s, kcmil has been adopted as the official designation of a thousand circular mils, but the designation MCM still remains in wide use. A similar system is used in natural gas sales in the United States: m (or M) for thousands and mm (or MM) for millions of British thermal units or therms, and in the oil industry, where MMbbl is the symbol for "millions of barrels". This usage of the capital letter M for "thousand" is from Roman numerals, in which M means 1000.

Binary prefixes

In some fields of information technology, it has been common to designate non-decimal multiples based on powers of 1024, rather than 1000, for some SI prefixes (kilo-, mega-, giga-), contrary to the definitions in the International System of Units (SI). This practice was once sanctioned by some industry associations, including JEDEC. The International Electrotechnical Commission (IEC) standardized the system of binary prefixes (kibi-, mebi-, gibi-, etc.) for this purpose.

Dalton (unit)

From Wikipedia, the free encyclopedia

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.