Relative atomic mass

Relative atomic mass (symbol: Ar) or atomic weight is a dimensionless physical quantity defined as the ratio of the average mass of atoms of a chemical element in a given sample to one unified atomic mass unit. The unified atomic mass unit (symbol: u or Da) is defined as being ​112 of the atomic mass of a carbon-12 atom.[1][2] Since both values in the ratio are expressed in the same unit (u), the resulting value is dimensionless; hence the value is said to be relative.

For a single given sample, the relative atomic mass of a given element is the weighted arithmetic mean of the masses of the individual atoms (including their isotopes) that are present in the sample. This quantity can vary substantially between samples because the sample's origin (and therefore its radioactive history or diffusion history) may have produced unique combinations of isotopic abundances. For example, due to a different mixture of stable carbon-12 and carbon-13 isotopes, a sample of elemental carbon from volcanic methane will have a different relative atomic mass than one collected from plant or animal tissues.

The more common, and more specific quantity known as standard atomic weight (Ar, standard) is an application of the relative atomic mass values obtained from multiple different samples. It is sometimes interpreted as the expected range of the relative atomic mass values for the atoms of a given element from all terrestrial sources, with the various sources being taken from Earth.[3] "Atomic weight" is often loosely and incorrectly used as a synonym for standard atomic weight (incorrectly because standard atomic weights are not from a single sample). Standard atomic weight is nevertheless the most widely published variant of relative atomic mass.

Additionally, the continued use of the term "atomic weight" (for any element) as opposed to "relative atomic mass" has attracted considerable controversy since at least the 1960s, mainly due to the technical difference between weight and mass in physics.[4] Still, both terms are officially sanctioned by the IUPAC. The term "relative atomic mass" now seems to be replacing "atomic weight" as the preferred term, although the term "standard atomic weight" (as opposed to the more correct "standard relative atomic mass") continues to be used.

Definition

Relative atomic mass is determined by the average atomic mass, or the weighted mean of the atomic masses of all the atoms of a particular chemical element found in a particular sample, which is then compared to the atomic mass of carbon-12.[5] This comparison is the quotient of the two weights, which makes the value dimensionless (no unit appended). This quotient also explains the word relative: the sample mass value is considered relative to that of carbon-12.

It is a synonym for atomic weight, though it is not to be confused with relative isotopic mass. Relative atomic mass is also frequently used as a synonym for standard atomic weight and these quantities may have overlapping values if the relative atomic mass used is that for an element from Earth under defined conditions. However, relative atomic mass (atomic weight) is still technically distinct from standard atomic weight because of its application only to the atoms obtained from a single sample; it is also not restricted to terrestrial samples, whereas standard atomic weight averages multiple samples but only from terrestrial sources. Relative atomic mass is therefore a more general term that can more broadly refer to samples taken from non-terrestrial environments or highly specific terrestrial environments which may differ substantially from Earth-average or reflect different degrees of certainty (e.g. in number of significant figures) than those reflected in standard atomic weights.

Current definition

The prevailing IUPAC definitions (as taken from the "Gold Book") are:

atomic weight — See: relative atomic mass[6]

and

relative atomic mass (atomic weight) — The ratio of the average mass of the atom to the unified atomic mass unit.[7]

Here the "unified atomic mass unit" refers to ​112 of the mass of an atom of 12C in its ground state.[8]

The IUPAC definition[1] of relative atomic mass is:

An atomic weight (relative atomic mass) of an element from a specified source is the ratio of the average mass per atom of the element to 1/12 of the mass of an atom of 12C.

The definition deliberately specifies "An atomic weight…", as an element will have different relative atomic masses depending on the source. For example, boron from Turkey has a lower relative atomic mass than boron from California, because of its different isotopic composition.[9][10] Nevertheless, given the cost and difficulty of isotope analysis, it is common practice to instead substitute the tabulated values of standard atomic weights, which are ubiquitous in chemical laboratories and which are revised biennially by the IUPAC's Commission on Isotopic Abundances and Atomic Weights (CIAAW).[11]

Historical usage

Older (pre-1961) historical relative scales based on the atomic mass unit (symbol: a.m.u. or amu) used either the oxygen-16 relative isotopic mass or else the oxygen relative atomic mass (i.e., atomic weight) for reference. See the article on the history of the modern unified atomic mass unit for the resolution of these problems.

Standard atomic weight

The IUPAC commission CIAAW maintains an expectation-interval value for relative atomic mass (or atomic weight) on Earth named standard atomic weight. Standard atomic weight requires the sources be terrestrial, natural, and stable with regard to radioactivity. Also, there are requirements for the research process. For 84 stable elements, CIAAW has determined this standard atomic weight. These values are widely published and referred to loosely as 'the' atomic weight of elements for real-life substances like pharmaceuticals and commercial trade.

Also, CIAAW has published abridged (rounded) values and simplified values (for when the Earthly sources vary systematically).

Other measures of the mass of atoms

Atomic mass (ma) is the mass of a single atom, with unit Da or u (the unified atomic mass unit). It defines the mass of a specific isotope, which is an input value for the determination of the relative atomic mass. An example for three silicon isotopes is given below.

The relative isotopic mass is specifically the ratio of the mass of a single atom to the mass of a unified atomic mass unit. This value, too, is relative, and therefore dimensionless.

Determination of relative atomic mass

Modern relative atomic masses (a term specific to a given element sample) are calculated from measured values of atomic mass (for each nuclide) and isotopic composition of a sample. Highly accurate atomic masses are available[12][13] for virtually all non-radioactive nuclides, but isotopic compositions are both harder to measure to high precision and more subject to variation between samples.[14][15] For this reason, the relative atomic masses of the 22 mononuclidic elements (which are the same as the isotopic masses for each of the single naturally occurring nuclides of these elements) are known to especially high accuracy. For example, there is an uncertainty of only one part in 38 million for the relative atomic mass of fluorine, a precision which is greater than the current best value for the Avogadro constant (one part in 20 million).

Isotope Atomic mass[13] Abundance[14]
Standard Range
28Si 27.97692653246(194) 92.2297(7)% 92.21–92.25%
29Si 28.976494700(22) 4.6832(5)% 4.67–4.69%
30Si 29.973770171(32) 3.0872(5)% 3.08–3.10%

The calculation is exemplified for silicon, whose relative atomic mass is especially important in metrology. Silicon exists in nature as a mixture of three isotopes: 28Si, 29Si and 30Si. The atomic masses of these nuclides are known to a precision of one part in 14 billion for 28Si and about one part in one billion for the others. However, the range of natural abundance for the isotopes is such that the standard abundance can only be given to about ±0.001% (see table).

The calculation is as follows:

Ar(Si) = (27.97693 × 0.922297) + (28.97649 × 0.046832) + (29.97377 × 0.030872) = 28.0854

The estimation of the uncertainty is complicated,[16] especially as the sample distribution is not necessarily symmetrical: the IUPAC standard relative atomic masses are quoted with estimated symmetrical uncertainties,[17] and the value for silicon is 28.0855(3). The relative standard uncertainty in this value is 1×10–5 or 10 ppm.

Apart from this uncertainty by measurement, some elements have variation over sources. That is, different sources (ocean water, rocks) have a different radioactive history and so different isotopic composition. To reflect this natural variability, the IUPAC made the decision in 2010 to list the standard relative atomic masses of 12 elements as an interval rather than a fixed number.[18]

See also

  • Possolo, Antonio; van der Veen, Adriaan M.H.; Meija, Juris; Brynn Hibbert, D. (2018-01-04). "Interpreting and propagating the uncertainty of the standard atomic weights (IUPAC Technical Report)". Retrieved 2019-02-08.

References

  1. ^ a b International Union of Pure and Applied Chemistry (1980). "Atomic Weights of the Elements 1979". Pure Appl. Chem. 52 (10): 2349–84. doi:10.1351/pac198052102349.
  2. ^ International Union of Pure and Applied Chemistry (1993). Quantities, Units and Symbols in Physical Chemistry, 2nd edition, Oxford: Blackwell Science. ISBN 0-632-03583-8. p. 41. Electronic version.
  3. ^ Definition of element sample
  4. ^ de Bièvre, P.; Peiser, H. S. (1992). "'Atomic Weight'—The Name, Its History, Definition, and Units". Pure Appl. Chem. 64 (10): 1535–43. doi:10.1351/pac199264101535.
  5. ^ IUPAC, Compendium of Chemical Terminology, 2nd ed. (the "Gold Book") (1997). Online corrected version:  (2006–) "relative atomic mass". doi:10.1351/goldbook.R05258
  6. ^ IUPAC Gold Book - atomic weight
  7. ^ IUPAC Gold Book - relative atomic mass (atomic weight), A r
  8. ^ IUPAC Gold Book - unified atomic mass unit
  9. ^ Greenwood, Norman N.; Earnshaw, Alan (1984). Chemistry of the Elements. Oxford: Pergamon Press. pp. 21, 160. ISBN 0-08-022057-6.
  10. ^ International Union of Pure and Applied Chemistry (2003). "Atomic Weights of the Elements: Review 2000". Pure Appl. Chem. 75 (6): 683–800. doi:10.1351/pac200375060683.
  11. ^ IUPAC Gold Book - standard atomic weights
  12. ^ National Institute of Standards and Technology. Atomic Weights and Isotopic Compositions for All Elements.
  13. ^ a b Wapstra, A.H.; Audi, G.; Thibault, C. (2003), The AME2003 Atomic Mass Evaluation (Online ed.), National Nuclear Data Center. Based on:
  14. ^ a b Rosman, K. J. R.; Taylor, P. D. P. (1998), "Isotopic Compositions of the Elements 1997" (PDF), Pure and Applied Chemistry, 70 (1): 217–35, doi:10.1351/pac199870010217
  15. ^ Coplen, T. B.; et al. (2002), "Isotopic Abundance Variations of Selected Elements" (PDF), Pure and Applied Chemistry, 74 (10): 1987–2017, doi:10.1351/pac200274101987
  16. ^ Meija, Juris; Mester, Zoltán (2008). "Uncertainty propagation of atomic weight measurement results". Metrologia. 45: 53–62. Bibcode:2008Metro..45...53M. doi:10.1088/0026-1394/45/1/008.
  17. ^ Holden, Norman E. (2004). "Atomic Weights and the International Committee—A Historical Review". Chemistry International. 26 (1): 4–7.
  18. ^ IUPAC - International Union of Pure and Applied Chemistry: Atomic Weights of Ten Chemical Elements About to Change

External links

Atomic mass

The atomic mass (ma) is the mass of an atom. Its unit is the unified atomic mass units (abbr. u) where 1 unified atomic mass unit is defined as ​1⁄12 of the mass of a single carbon-12 atom, at rest. For atoms, the protons and neutrons of the nucleus account for nearly all of the total mass, and the atomic mass measured in u has nearly the same value as the mass number.

When divided by unified atomic mass units, or daltons (abbr. Da), to form a pure numeric ratio, the atomic mass of an atom becomes a dimensionless value called the relative isotopic mass (see section below). Thus, the atomic mass of a carbon-12 atom is 12 u (or 12 Da), but the relative isotopic mass of a carbon-12 atom is simply 12.

The atomic mass or relative isotopic mass refers to the mass of a single particle, and therefore is tied to a certain specific isotope of an element. The dimensionless standard atomic weight instead refers to the average (mathematical mean) of atomic mass values of a typical naturally-occurring mixture of isotopes for a sample of an element. Atomic mass values are thus commonly reported to many more significant figures than atomic weights. Standard atomic weight is related to atomic mass by the abundance ranking of isotopes for each element. It is usually about the same value as the atomic mass of the most abundant isotope, other than what looks like (but is not actually) a rounding difference.

The atomic mass of atoms, ions, or atomic nuclei is slightly less than the sum of the masses of their constituent protons, neutrons, and electrons, due to binding energy mass loss (as per E = mc2).

Atomic mass constant

In physics and chemistry, the atomic mass constant, mu, is one twelfth of the mass of an unbound atom of carbon-12 at rest and in its ground state. It serves to define the atomic mass unit and is, by definition, equal to 1 u. It is inverse of Avogadro constant (1/NA) when expressed in grams (instead of SI unit kilogram). The CODATA recommended value is 1.660539040(20)×10−27 kg.

In practice, the atomic mass constant is determined as the ratio of the electron rest mass me to the electron relative atomic mass Ar(e) (that is, the mass of the electron on a scale where 12C = 12). 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.

The current uncertainty in the value of the atomic mass constant – one part in 20 million – is almost entirely due to the uncertainty in the value of the Planck constant.

Atomic mass unit

The unified atomic mass unit or dalton (symbol: u, or Da or AMU) is a standard unit of mass that quantifies mass on an atomic or molecular scale (atomic mass). One unified atomic mass unit is approximately the mass of one nucleon (either a single proton or neutron) and is numerically equivalent to 1 g/mol. It is defined as one twelfth of the mass of an unbound neutral atom of carbon-12 in its nuclear and electronic ground state and at rest, and has a value of 1.660539040(20)×10−27 kg, or approximately 1.66 yoctograms. The CIPM has categorised it as a non-SI unit accepted for use with the SI, and whose value in SI units must be obtained experimentally.The atomic mass unit (amu) without the "unified" prefix is technically an obsolete unit based on oxygen, which was replaced in 1961. However, many sources still use the term amu but now define it in the same way as u (i.e., based on carbon-12). In this sense, most uses of the terms atomic mass units and amu, today, actually refer to unified atomic mass unit. For standardization, a specific atomic nucleus (carbon-12 vs. oxygen-16) had to be chosen because the average mass of a nucleon depends on the count of the nucleons in the atomic nucleus due to mass defect. This is also why the mass of a proton or neutron by itself is more than (and not equal to) 1 u.

The atomic mass unit is not the unit of mass in the atomic units system, which is rather the electron rest mass (me).

Until the 2019 redefinition of SI base units, the number of daltons in a gram is exactly the Avogadro number by definition, or equivalently, a dalton is exactly equivalent to 1 gram/mol. Thereafter, these relationships will no longer be exact, but they will still be extremely accurate approximations.

Atomic number

The atomic number or proton number (symbol Z) of a chemical element is the number of protons found in the nucleus of an atom. It is identical to the charge number of the nucleus. The atomic number uniquely identifies a chemical element. In an uncharged atom, the atomic number is also equal to the number of electrons.

The sum of the atomic number Z and the number of neutrons, N, gives the mass number A of an atom. Since protons and neutrons have approximately the same mass (and the mass of the electrons is negligible for many purposes) and the mass defect of nucleon binding is always small compared to the nucleon mass, the atomic mass of any atom, when expressed in unified atomic mass units (making a quantity called the "relative isotopic mass"), is within 1% of the whole number A.

Atoms with the same atomic number Z but different neutron numbers N, and hence different atomic masses, are known as isotopes. A little more than three-quarters of naturally occurring elements exist as a mixture of isotopes (see monoisotopic elements), and the average isotopic mass of an isotopic mixture for an element (called the relative atomic mass) in a defined environment on Earth, determines the element's standard atomic weight. Historically, it was these atomic weights of elements (in comparison to hydrogen) that were the quantities measurable by chemists in the 19th century.

The conventional symbol Z comes from the German word Zahl meaning number, which, before the modern synthesis of ideas from chemistry and physics, merely denoted an element's numerical place in the periodic table, whose order is approximately, but not completely, consistent with the order of the elements by atomic weights. Only after 1915, with the suggestion and evidence that this Z number was also the nuclear charge and a physical characteristic of atoms, did the word Atomzahl (and its English equivalent atomic number) come into common use in this context.

Banana equivalent dose

Banana equivalent dose (BED) is an informal measurement of ionizing radiation exposure, intended as a general educational example to compare a dose of radioactivity to the dose one is exposed to by eating one average-sized banana. Bananas contain naturally occurring radioactive isotopes, particularly potassium-40 (40K), one of several naturally-occurring isotopes of potassium. One BED is often correlated to 10-7 Sievert (0.1 µSv); however, in practice, this dose is not cumulative, as the principal radioactive component is excreted to maintain metabolic equilibrium. The BED is only meant to inform the public about the existence of very low levels of natural radioactivity within a natural food and is not a formally adopted dose measurement.

Dimensionless quantity

In dimensional analysis, a dimensionless quantity is a quantity to which no physical dimension is assigned. It is also known as a bare number or pure number or a quantity of dimension one and the corresponding unit of measurement in the SI is one (or 1) unit and it is not explicitly shown. Dimensionless quantities are widely used in many fields, such as mathematics, physics, chemistry, engineering, and economics. Examples of quantities to which dimensions are regularly assigned are length, time, and speed, which are measured in dimensional units, such as metre, second and metre per second. This is considered to aid intuitive understanding. However, especially in mathematical physics, it is often more convenient to drop the assignment of explicit dimensions and express the quantities without dimensions, e.g., addressing the speed of light simply by the dimensionless number 1.

Electron rest mass

The electron rest mass (symbol: me) is the mass of a stationary electron, also known as the invariant mass of the electron. It is one of the fundamental constants of physics. It has a value of about 9.109×10−31 kilograms or about 5.486×10−4 atomic mass units, equivalent to an energy of about 8.187×10−14 joules or about 0.5110 MeV.

Gas constant

The gas constant is also known as the molar, universal, or ideal gas constant, denoted by the symbol R or R and is equivalent to the Boltzmann constant, but expressed in units of energy per temperature increment per mole, i.e. the pressure–volume product, rather than energy per temperature increment per particle. The constant is also a combination of the constants from Boyle's law, Charles's law, Avogadro's law, and Gay-Lussac's law. It is a physical constant that is featured in many fundamental equations in the physical sciences, such as the ideal gas law and the Nernst equation.

Physically, the gas constant is the constant of proportionality that happens to relate the energy scale in physics to the temperature scale, when a mole of particles at the stated temperature is being considered. Thus, the value of the gas constant ultimately derives from historical decisions and accidents in the setting of the energy and temperature scales, plus similar historical setting of the value of the molar scale used for the counting of particles. The last factor is not a consideration in the value of the Boltzmann constant, which does a similar job of equating linear energy and temperature scales.

The gas constant value is

8.3144598(48) J⋅mol−1⋅K−1

The two digits in parentheses are the uncertainty (standard deviation) in the last two digits of the value. The relative uncertainty is 5.7×10−7. Some have suggested that it might be appropriate to name the symbol R the Regnault constant in honour of the French chemist Henri Victor Regnault, whose accurate experimental data were used to calculate the early value of the constant; however, the exact reason for the original representation of the constant by the letter R is elusive.

The gas constant occurs in the ideal gas law, as follows:

where P is the absolute pressure (SI unit pascals), V is the volume of gas (SI unit cubic metres), n is the amount of gas (SI unit moles), m is the mass (SI unit kilograms) contained in V, and T is the thermodynamic temperature (SI unit kelvins). Rspecific is the molar-weight-specific gas constant, discussed below. The gas constant is expressed in the same physical units as molar entropy and molar heat capacity.

ISO 31-8

ISO 31-8 is the part of international standard ISO 31 that defines names and symbols for quantities and units related to physical chemistry and molecular physics.

Mass number

The mass number (symbol A, from the German word Atomgewicht (atomic weight), also called atomic mass number or nucleon number, is the total number of protons and neutrons (together known as nucleons) in an atomic nucleus. It determines the atomic mass of atoms. Because protons and neutrons both are baryons, the mass number A is identical with the baryon number B as of the nucleus as of the whole atom or ion. The mass number is different for each different isotope of a chemical element. This is not the same as the atomic number (Z) which denotes the number of protons in a nucleus, and thus uniquely identifies an element. Hence, the difference between the mass number and the atomic number gives the number of neutrons (N) in a given nucleus: .

The mass number is written either after the element name or as a superscript to the left of an element's symbol. For example, the most common isotope of carbon is carbon-12, or 12
C
, which has 6 protons and 6 neutrons. The full isotope symbol would also have the atomic number (Z) as a subscript to the left of the element symbol directly below the mass number: 12
6
C
. This is technically redundant, as each element is defined by its atomic number, so it is often omitted.

Molar mass

In chemistry, the molar mass M is a physical property defined as the mass of a given substance (chemical element or chemical compound) divided by the amount of substance. The base SI unit for molar mass is kg/mol. However, for historical reasons, molar masses are almost always expressed in g/mol.

In simple terms, molar mass of a substance is the total weight of that substance (in either kilogram or gram) for one mole of that substance. That is, the weight of a substance for 6.02214076×10^23 molecules or atoms of that substance.

As an example, the molar mass of water: M(H2O) ≈ 18.015 g/mol.

Molar mass constant

The molar mass constant, symbol Mu, is a physical constant which relates relative atomic mass and molar mass. Its value is defined to be 1 g/mol in SI units.

The molar mass constant is important in writing dimensionally correct equations. It is common to see phrases such as

The molar mass of an element is the atomic weight in grams per mole.

However, atomic weight, i.e., relative atomic mass, is a dimensionless quantity, and cannot take the units of grams per mole. Formally, the operation is the multiplication by a constant which has the value 1 g/mol, that is the molar mass constant.

The molar mass constant is unusual (but not unique) among physical constants by having an exactly defined value rather than being measured experimentally. It is fixed by the definitions of the mole and of relative atomic mass. From the definition of the mole, the molar mass of carbon 12 is exactly 12 g/mol. From the definition of relative atomic mass, the relative atomic mass of carbon 12, that is the atomic weight of a sample of pure carbon 12, is exactly 12. The molar mass constant is given by

The speed of light, the electric constant and the magnetic constant are other examples of physical constants whose values are fixed by the definitions of the International System of Units (SI), in these cases by the definitions of the metre and the ampere.

The molar mass constant is also related to the mass of a carbon-12 atom in grams:

Hence the uncertainty in the value of the mass of a carbon-12 atom in SI units is governed by the uncertainty in the Avogadro constant: the CODATA 2006 recommended value is 1.992 646 54(10)×10−26 kg (ur = 5×10−8).

The relatively simple value of the molar mass constant in SI units is also a consequence of the way in which the International System of Units is defined. It is possible to quote the value of the molar mass constant in other units: for example, it is equal to (1/453.592 37) lb/mol ~ 2.204 623 262 × 10−3 lb/mol.

Mole (unit)

The mole is the base unit of amount of substance in the International System of Units (SI). Effective 20 May 2019, the mole is defined as the amount of a chemical substance that contains exactly 6.02214076×1023 (Avogadro constant) constitutive particles, e.g., atoms, molecules, ions or electrons.This definition was adopted in November 2018, revising its old definition based on the number of atoms in 12 grams of carbon-12 (12C) (the isotope of carbon with relative atomic mass 12 Da by definition). The mole is an SI base unit, with the unit symbol mol.

The mole is widely used in chemistry as a convenient way to express amounts of reactants and products of chemical reactions. For example, the chemical equation 2H2 + O2 → 2H2O can be interpreted to mean that 2 mol dihydrogen (H2) and 1 mol dioxygen (O2) react to form 2 mol water (H2O). The mole may also be used to represent the number of atoms, ions, or other entities in a given sample of a substance. The concentration of a solution is commonly expressed by its molarity, defined as the amount of dissolved substance per unit volume of solution, for which the unit typically used is moles per litre (mol/l).

The term gram-molecule was formerly used for essentially the same concept. The term gram-atom has been used for a related but distinct concept, namely a quantity of a substance that contains Avogadro number of atoms, whether isolated or combined in molecules. Thus, for example, 1 mole of MgBr2 is 1 gram-molecule of MgBr2 but 3 gram-atoms of MgBr2.

Natural abundance

In physics, natural abundance (NA) refers to the abundance of isotopes of a chemical element as naturally found on a planet. The relative atomic mass (a weighted average, weighted by mole-fraction abundance figures) of these isotopes is the atomic weight listed for the element in the periodic table. The abundance of an isotope varies from planet to planet, and even from place to place on the Earth, but remains relatively constant in time (on a short-term scale).

As an example, uranium has three naturally occurring isotopes: 238U, 235U and 234U. Their respective natural mole-fraction abundances are 99.2739–99.2752%, 0.7198–0.7202%, and 0.0050–0.0059%. For example, if 100,000 uranium atoms were analyzed, one would expect to find approximately 99,274 238U atoms, approximately 720 235U atoms, and very few (most likely 5 or 6) 234U atoms. This is because 238U is much more stable than 235U or 234U, as the half-life of each isotope reveals: 4.468 × 109 years for 238U compared with 7.038 × 108 years for 235U and 245,500 years for 234U.

Exactly because the different uranium isotopes have different half-lives, when the Earth was younger, the isotopic composition of uranium was different. As an example, 1.7×109 years ago the NA of 235U was 3.1% compared with today's 0.7%, and for that reason a natural nuclear fission reactor was able to form, something that cannot happen today.

However, the natural abundance of a given isotope is also affected by the probability of its creation in nucleosynthesis (as in the case of samarium; radioactive 147Sm and 148Sm are much more abundant than stable 144Sm) and by production of a given isotope as a daughter of natural radioactive isotopes (as in the case of radiogenic isotopes of lead).

Planck constant

The Planck constant (denoted h, also called Planck's constant) is a physical constant that is the quantum of electromagnetic action, which relates the energy carried by a photon to its frequency. A photon's energy is equal to its frequency multiplied by the Planck constant. The Planck constant is of fundamental importance in quantum mechanics, and in metrology it is the basis for the definition of the kilogram.

At the end of the 19th century, physicists were unable to explain why the observed spectrum of black body radiation, which by then had been accurately measured, diverged significantly at higher frequencies from that predicted by existing theories. In 1900, Max Planck empirically derived a formula for the observed spectrum by assuming that a hypothetical electrically charged oscillator in a cavity that contained black body radiation could only change its energy in a minimal increment, E, that was proportional to the frequency of its associated electromagnetic wave. He was able to calculate the proportionality constant, h, from the experimental measurements, and that constant is named in his honor. In 1905, the value E was associated by Albert Einstein with a "quantum" or minimal element of the energy of the electromagnetic wave itself. The light quantum behaved in some respects as an electrically neutral particle, as opposed to an electromagnetic wave. It was eventually called a photon.

Since energy and mass are equivalent, the Planck constant also relates mass to frequency. By 2017, the Planck constant had been measured with sufficient accuracy in terms of the SI base units, that it was central to replacing the metal cylinder, called the International Prototype of the Kilogram (IPK), that had defined the kilogram since 1889. The new definition was unanimously approved at the General Conference on Weights and Measures (CGPM) on 16 November 2018 as part of the 2019 redefinition of SI base units. For this new definition of the kilogram, the Planck constant, as defined by the ISO standard, was set to 6.62607015×10−34 J⋅s exactly. The kilogram was the last SI base unit to be re-defined by a fundamental physical property to replace a physical artefact.

Standard atomic weight

The standard atomic weight (Ar, standard, a relative atomic mass) is the atomic weight (Ar) of a chemical element, as appearing and met in the earthly environment. It reflects the variance of natural isotopes (and so weight differences) of an element. Values are defined by (restricted to) the IUPAC (CIAAW) definition of natural, stable, terrestrial sources. It is the most common and practical atomic weight used, for example to determine molar mass.

The specified definition is to use many representative sources (samples) from the Earth, so that the value can widely be used as 'the' atomic weight for real life substances—for example, in pharmaceuticals and scientific research. Atomic weights are specific to single sources and samples of an element, such as the atomic weight of carbon in a particular bone from a particular archeological site. Standard atomic weight generalizes such values to the range of atomic weights which a chemist might expect to derive from many random samples from Earth. This range is the cause of the interval notation in some standard atomic weight values.

Out of the 118 known chemical elements, 84 are stable and have this Earth-environment based value. Typically, such a value is, for example helium: Ar, standard(He) = 4.002602(2). The "(2)" indicates the uncertainty in the last digit shown, to read 4.002602 ±0.000002. IUPAC also publishes abridged values, rounded to five significant figures. For helium, Ar, abridged(He) = 4.0026.

For twelve elements the samples diverge on this value, because their sample sources have had a different decay history. For example, thallium (Tl) in sedimentary rocks has a different isotopic composition than in igneous rocks and volcanic gases. For these elements, the standard atomic weight is noted as an interval: Ar, standard(Tl) = [204.38, 204.39]. With such an interval, for less demanding situations, IUPAC also publishes an conventional value. For thallium, Ar, conventional(Tl) = 204.38.

Stefan–Boltzmann constant

The Stefan–Boltzmann constant (also Stefan's constant), a physical constant denoted by the Greek letter σ (sigma), is the constant of proportionality in the Stefan–Boltzmann law: "the total intensity radiated over all wavelengths increases as the temperature increases", of a black body which is proportional to the fourth power of the thermodynamic temperature. The theory of thermal radiation lays down the theory of quantum mechanics, by using physics to relate to molecular, atomic and sub-atomic levels. Slovenian physicist Josef Stefan formulated the constant in 1879, and it was later derived in 1884 by Austrian physicist Ludwig Boltzmann. The equation can also be derived from Planck's law, by integrating over all wavelengths at a given temperature, which will represent a small flat black body box. "The amount of thermal radiation emitted increases rapidly and the principal frequency of the radiation becomes higher with increasing temperatures". The Stefan–Boltzmann constant can be used to measure the amount of heat that is emitted by a blackbody, which absorbs all of the radiant energy that hits it, and will emit all the radiant energy. Furthermore, the Stefan–Boltzmann constant allows for temperature (K) to be converted to units for intensity (W⋅m−2), which is power per unit area.

The value of the Stefan–Boltzmann constant is given in SI units by

σ = 5.670367(13)×10−8 W⋅m−2⋅K−4.

In cgs units the Stefan–Boltzmann constant is:

σ5.6704×10−5 erg⋅cm−2⋅s−1⋅K−4.

In thermochemistry the Stefan–Boltzmann constant is often expressed in cal⋅cm−2⋅day−1⋅K−4:

σ11.7×10−8 cal cm−2⋅day−1⋅K−4.

In US customary units the Stefan–Boltzmann constant is:

σ1.714×10−9 BTU⋅hr−1⋅ft−2⋅°R−4.

The value of the Stefan–Boltzmann constant is derivable as well as experimentally determinable; see Stefan–Boltzmann law for details. It can be defined in terms of the Boltzmann constant as:

where:

The CODATA recommended value is calculated from the measured value of the gas constant:

where:

Dimensional formula: M1T−3Θ−4

A related constant is the radiation constant (or radiation density constant) a which is given by:

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