Theoretical gravity

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Theoretical_gravity

In geodesy and geophysics, theoretical gravity or normal gravity is an approximation of the true gravity on Earth's surface by means of a mathematical model representing Earth. The most common model of a smoothed Earth is a rotating Earth ellipsoid of revolution (i.e., a spheroid).

Principles

The type of gravity model used for the Earth depends upon the degree of fidelity required for a given problem. For many problems such as aircraft simulation, it may be sufficient to consider gravity to be a constant, defined as:

${\displaystyle g=g_{45}=}$ 9.80665 m/s² (32.1740 ft/s²)

based upon data from World Geodetic System 1984 (WGS-84), where ${\displaystyle g}$ is understood to be pointing 'down' in the local frame of reference.

If it is desirable to model an object's weight on Earth as a function of latitude, one could use the following:

${\displaystyle g=g_{45}-\frac{1}{2}(g_{\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{e}\mathrm{s}}-g_{\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{r}})\cos \left(2\phi \cdot \frac{\pi }{180}\right)}$

where

• ${\displaystyle g_{\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{e}\mathrm{s}}}$ = 9.832 m/s² (32.26 ft/s²)
• ${\displaystyle g_{45}}$ = 9.806 m/s² (32.17 ft/s²)
• ${\displaystyle g_{\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{r}}}$ = 9.780 m/s² (32.09 ft/s²)
• ${\displaystyle \phi }$ = latitude, between −90° and +90°

Neither of these accounts for changes in gravity with changes in altitude, but the model with the cosine function does take into account the centrifugal relief that is produced by the rotation of the Earth. For the mass attraction effect by itself, the gravitational acceleration at the equator is about 0.18% less than that at the poles due to being located farther from the mass center. When the rotational component is included (as above), the gravity at the equator is about 0.53% less than that at the poles, with gravity at the poles being unaffected by the rotation. So the rotational component of change due to latitude (0.35%) is about twice as significant as the mass attraction change due to latitude (0.18%), but both reduce strength of gravity at the equator as compared to gravity at the poles.

Note that for satellites, orbits are decoupled from the rotation of the Earth so the orbital period is not necessarily one day, but also that errors can accumulate over multiple orbits so that accuracy is important. For such problems, the rotation of the Earth would be immaterial unless variations with longitude are modeled. Also, the variation in gravity with altitude becomes important, especially for highly elliptical orbits.

The Earth Gravitational Model 1996 (EGM96) contains 130,676 coefficients that refine the model of the Earth's gravitational field. The most significant correction term is about two orders of magnitude more significant than the next largest term. That coefficient is referred to as the ${\displaystyle J_2}$ term, and accounts for the flattening of the poles, or the oblateness, of the Earth. (A shape elongated on its axis of symmetry, like an American football, would be called prolate.) A gravitational potential function can be written for the change in potential energy for a unit mass that is brought from infinity into proximity to the Earth. Taking partial derivatives of that function with respect to a coordinate system will then resolve the directional components of the gravitational acceleration vector, as a function of location. The component due to the Earth's rotation can then be included, if appropriate, based on a sidereal day relative to the stars (≈366.24 days/year) rather than on a solar day (≈365.24 days/year). That component is perpendicular to the axis of rotation rather than to the surface of the Earth.

A similar model adjusted for the geometry and gravitational field for Mars can be found in publication NASA SP-8010.

The barycentric gravitational acceleration at a point in space is given by:

${\displaystyle \mathbf{g}=-\frac{GM}{r^2}\hat{\mathbf{r}}}$

where:

M is the mass of the attracting object, ${\displaystyle \hat{\mathbf{r}}}$ is the unit vector from center-of-mass of the attracting object to the center-of-mass of the object being accelerated, r is the distance between the two objects, and G is the gravitational constant.

When this calculation is done for objects on the surface of the Earth, or aircraft that rotate with the Earth, one has to account for the fact that the Earth is rotating and the centrifugal acceleration has to be subtracted from this. For example, the equation above gives the acceleration at 9.820 m/s², when GM = 3.986 × 10¹⁴ m³/s², and R = 6.371 × 10⁶ m. The centripetal radius is r = R cos(φ), and the centripetal time unit is approximately (day / 2π), reduces this, for r = 5 × 10⁶ metres, to 9.79379 m/s², which is closer to the observed value.

Basic formulas

Various, successively more refined, formulas for computing the theoretical gravity are referred to as the International Gravity Formula, the first of which was proposed in 1930 by the International Association of Geodesy. The general shape of that formula is:

${\displaystyle g(\varphi )=g_e\left(1+A{\sin }^2(\varphi )-B{\sin }^2(2\varphi )\right),}$

in which g(φ) is the gravity as a function of the geographic latitude φ of the position whose gravity is to be determined, ${\displaystyle g_e}$ denotes the gravity at the equator (as determined by measurement), and the coefficients A and B are parameters that must be selected to produce a good global fit to true gravity.

Using the values of the GRS80 reference system, a commonly used specific instantiation of the formula above is given by:

${\displaystyle g(\varphi )=9.780327\left(1+0.0053024{\sin }^2(\varphi )-0.0000058{\sin }^2(2\varphi )\right){\mathrm{m}\mathrm{s}}^{-2}.}$

Using the appropriate double-angle formula in combination with the Pythagorean identity, this can be rewritten in the equivalent forms

${\displaystyle \begin{array}{rl}g(\varphi )& =9.780327\left(1+0.0052792{\sin }^2(\varphi )+0.0000232{\sin }^4(\varphi )\right){\mathrm{m}\mathrm{s}}^{-2},\\ & =9.780327\left(1.0053024-.0053256{\cos }^2(\varphi )+.0000232{\cos }^4(\varphi )\right){\mathrm{m}\mathrm{s}}^{-2},\\ & =9.780327\left(1.0026454-0.0026512\cos (2\varphi )+.0000058{\cos }^2(2\varphi )\right){\mathrm{m}\mathrm{s}}^{-2}.\end{array}}$

Up to the 1960s, formulas based on the Hayford ellipsoid (1924) and of the famous German geodesist Helmert (1906) were often used. The difference between the semi-major axis (equatorial radius) of the Hayford ellipsoid and that of the modern WGS84 ellipsoid is 251 m; for Helmert's ellipsoid it is only 63 m.

Somigliana equation

A more recent theoretical formula for gravity as a function of latitude is the International Gravity Formula 1980 (IGF80), also based on the WGS80 ellipsoid but now using the Somigliana equation (after Carlo Somigliana (1860–1955)):

${\displaystyle g(\varphi )=g_e\left[\frac{1+k{\sin }^2(\varphi )}{\sqrt{1-e^2{\sin }^2(\varphi )}}\right],}$

where,

• ${\displaystyle k=\frac{b{g}_p-a{g}_e}{a{g}_e}}$ (formula constant);
• ${\displaystyle g_e,g_p}$ is the defined gravity at the equator and poles, respectively;
• ${\displaystyle a,b}$ are the equatorial and polar semi-axes, respectively;
• ${\displaystyle e^2=\frac{a^2-b^2}{a^2}}$ is the spheroid's squared eccentricity;

providing,

${\displaystyle g(\varphi )=9.78032677153489285793\left[\frac{1+0.0019318513532606763607{\sin }^2(\varphi )}{\sqrt{1-0.00669438002290341574957{\sin }^2(\varphi )}}\right]{\mathrm{m}\mathrm{s}}^{-2}.}$

A later refinement, based on the WGS84 ellipsoid, is the WGS (World Geodetic System) 1984 Ellipsoidal Gravity Formula:

${\displaystyle g(\varphi )=9.780325335903891718546\left[\frac{1+0.00193185265245827352087{\sin }^2(\varphi )}{\sqrt{1-0.006694379990141316996137{\sin }^2(\varphi )}}\right]{\mathrm{m}\mathrm{s}}^{-2}.}$

(where ${\displaystyle g_p}$ = 9.83218493786340046183 ms⁻², a = exactly 6378137 m and b ${\displaystyle \simeq }$ 6356752.31424517949756 m)

The difference with IGF80 is insignificant when used for geophysical purposes, but may be significant for other uses.

Further details

For the normal gravity ${\displaystyle {\gamma }_0}$ of the sea level ellipsoid, i.e., elevation h = 0, this formula by Somigliana (1929) applies:

${\displaystyle {\gamma }_0(\phi )=\frac{a\cdot {\gamma }_a\cdot {\cos }^2\phi +b\cdot {\gamma }_b\cdot {\sin }^2\phi }{\sqrt{a^2\cdot {\cos }^2\phi +b^2\cdot {\sin }^2\phi }}}$

with

• ${\displaystyle {\gamma }_a}$ = Normal gravity at Equator
• ${\displaystyle {\gamma }_b}$ = Normal gravity at Poles
• a = semi-major axis (Equator radius)
• b = semi-minor axis (Pole radius)
• ${\displaystyle \phi }$ = latitude

Due to numerical issues, the formula is simplified to this:

${\displaystyle {\gamma }_0(\phi )={\gamma }_a\cdot \frac{1+p\cdot {\sin }^2\phi }{\sqrt{1-e^2\cdot {\sin }^2\phi }}}$

with

• ${\displaystyle p=\frac{b\cdot {\gamma }_b}{a\cdot {\gamma }_a}-1}$
• ${\displaystyle e^2=1-\frac{b^2}{a^2};}$(e is the eccentricity)

For the Geodetic Reference System 1980 (GRS 80) the parameters are set to these values:

${\displaystyle a=6378137\mathrm{m}b=6356752.314140347\mathrm{m}}$

${\displaystyle {\gamma }_a=9.7803267715\frac{\mathrm{m}}{{\mathrm{s}}^2}{\gamma }_b=9.8321863685\frac{\mathrm{m}}{{\mathrm{s}}^2}}$

${\displaystyle \Rightarrow p=1.931851353260\cdot {10}^{-3}{e}^2=6.69438002290\cdot {10}^{-3}}$

Approximation formula from series expansions

The Somigliana formula was approximated through different series expansions, following this scheme:

${\displaystyle {\gamma }_0(\phi )={\gamma }_a\cdot (1+\beta \cdot {\sin }^2\phi +{\beta }_1\cdot {\sin }^{2}2\phi +\dots )}$

International gravity formula 1930

The normal gravity formula by Gino Cassinis was determined in 1930 by International Union of Geodesy and Geophysics as international gravity formula along with Hayford ellipsoid. The parameters are:

${\displaystyle {\gamma }_a=9.78049\frac{\mathrm{m}}{{\mathrm{s}}^2}\beta =5.2884\cdot {10}^{-3}{\beta }_1=-5.9\cdot {10}^{-6}}$

In the course of time the values were improved again with newer knowledge and more exact measurement methods.

Harold Jeffreys improved the values in 1948 at:

${\displaystyle {\gamma }_a=9.780373\frac{\mathrm{m}}{{\mathrm{s}}^2}\beta =5.2891\cdot {10}^{-3}{\beta }_1=-5.9\cdot {10}^{-6}}$

International gravity formula 1967

The normal gravity formula of Geodetic Reference System 1967 is defined with the values:

${\displaystyle {\gamma }_a=9.780318\frac{\mathrm{m}}{{\mathrm{s}}^2}\beta =5.3024\cdot {10}^{-3}{\beta }_1=-5.9\cdot {10}^{-6}}$

International gravity formula 1980

From the parameters of GRS 80 comes the classic series expansion:

${\displaystyle {\gamma }_a=9.780327\frac{\mathrm{m}}{{\mathrm{s}}^2}\beta =5.3024\cdot {10}^{-3}{\beta }_1=-5.8\cdot {10}^{-6}}$

The accuracy is about ±10⁻⁶ m/s².

With GRS 80 the following series expansion is also introduced:

${\displaystyle {\gamma }_0(\phi )={\gamma }_a\cdot (1+c_1\cdot {\sin }^2\phi +c_2\cdot {\sin }^4\phi +c_3\cdot {\sin }^6\phi +c_4\cdot {\sin }^8\phi +\dots )}$

As such the parameters are:

• c₁ = 5.279 0414·10⁻³
• c₂ = 2.327 18·10⁻⁵
• c₃ = 1.262·10⁻⁷
• c₄ = 7·10⁻¹⁰

The accuracy is at about ±10⁻⁹ m/s² exact. When the exactness is not required, the terms at further back can be omitted. But it is recommended to use this finalized formula.

Height dependence

Cassinis determined the height dependence, as:

${\displaystyle g(\phi ,h)=g_0(\phi )-\left(3.08\cdot {10}^{-6}\frac{1}{{\mathrm{s}}^2}-4.19\cdot {10}^{-7}\frac{{\mathrm{c}\mathrm{m}}^3}{\mathrm{g}\cdot {\mathrm{s}}^2}\cdot \rho \right)\cdot h}$

The average rock density ρ is no longer considered.

Since GRS 1967 the dependence on the ellipsoidal elevation h is:

${\displaystyle \begin{array}{rl}g(\phi ,h)& =g_0(\phi )-\left(1-1.39\cdot {10}^{-3}\cdot {\sin }^2(\phi )\right)\cdot 3.0877\cdot {10}^{-6}\frac{1}{{\mathrm{s}}^2}\cdot h+7.2\cdot {10}^{-13}\frac{1}{\mathrm{m}\cdot {\mathrm{s}}^2}\cdot h^2\\ & =g_0(\phi )-\left(3.0877\cdot {10}^{-6}-4.3\cdot {10}^{-9}\cdot {\sin }^2(\phi )\right)\frac{1}{{\mathrm{s}}^2}\cdot h+7.2\cdot {10}^{-13}\frac{1}{\mathrm{m}\cdot {\mathrm{s}}^2}\cdot h^2\end{array}}$

Another expression is:

${\displaystyle g(\phi ,h)=g_0(\phi )\cdot (1-(k_1-k_2\cdot {\sin }^2\phi )\cdot h+k_3\cdot h^2)}$

with the parameters derived from GSR80:

• ${\displaystyle k_1=2\cdot (1+f+m)/a=3.15704\cdot {10}^{-7}{\mathrm{m}}^{-1}}$
• ${\displaystyle k_2=4\cdot f/a=2.10269\cdot {10}^{-9}{\mathrm{m}}^{-1}}$
• ${\displaystyle k_3=3/(a^2)=7.37452\cdot {10}^{-14}{\mathrm{m}}^{-2}}$

This adjustment is about right for common heights in Aviation; But for heights up to outer space (over ca. 100 kilometers) it is out of range.

WELMEC formula

In all German standards offices the free-fall acceleration g is calculated in respect to the average latitude φ and the average height above sea level h with the WELMEC–Formel:

${\displaystyle g(\phi ,h)=\left(1+0.0053024\cdot {\sin }^2(\phi )-0.0000058\cdot {\sin }^2(2\phi )\right)\cdot 9.780318\frac{\mathrm{m}}{{\mathrm{s}}^2}-0.000003085\frac{1}{{\mathrm{s}}^2}\cdot h}$

The formula is based on the International gravity formula from 1967.

The scale of free-fall acceleration at a certain place must be determined with precision measurement of several mechanical magnitudes. Weighing scales, the mass of which does measurement because of the weight, relies on the free-fall acceleration, thus for use they must be prepared with different constants in different places of use. Through the concept of so-called gravity zones, which are divided with the use of normal gravity, a weighing scale can be calibrated by the manufacturer before use.

Example

Free-fall acceleration in Schweinfurt:

Data:

• Latitude: 50° 3′ 24″ = 50.0567°
• Height above sea level: 229.7 m
• Density of the rock plates: ca. 2.6 g/cm³
• Measured free-fall acceleration: g = 9.8100 ± 0.0001 m/s²

Free-fall acceleration, calculated through normal gravity formulas:

• Cassinis: g = 9.81038 m/s²
• Jeffreys: g = 9.81027 m/s²
• WELMEC: g = 9.81004 m/s²

Atomic mass

From Wikipedia, the free encyclopedia

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

 

Stylized lithium-7 atom: 3 protons, 4 neutrons, and 3 electrons (total electrons are ~1⁄4300th of the mass of the nucleus). It has a mass of 7.016 Da. Rare lithium-6 (mass of 6.015 Da) has only 3 neutrons, reducing the atomic weight (average) of lithium to 6.941.

The atomic mass (m_a or m) is the mass of an atom. Although the SI unit of mass is the kilogram (symbol: kg), atomic mass is often expressed in the non-SI unit dalton (symbol: Da) – equivalently, unified atomic mass unit (u). 1 Da is defined as 1⁄12 of the mass of a free carbon-12 atom at rest in its ground state. The protons and neutrons of the nucleus account for nearly all of the total mass of atoms, with the electrons and nuclear binding energy making minor contributions. Thus, the numeric value of the atomic mass when expressed in daltons has nearly the same value as the mass number. Conversion between mass in kilograms and mass in daltons can be done using the atomic mass constant ${\displaystyle m_{\mathrm{u}}=\frac{m({}^{12}\mathrm{C})}{12}=1\mathrm{D}\mathrm{a}}$.

The formula used for conversion is:

${\displaystyle 1\mathrm{D}\mathrm{a}=m_{\mathrm{u}}=\frac{M_{\mathrm{u}}}{N_{\mathrm{A}}}=\frac{M{(}^{12}C)}{12N_{\mathrm{A}}}=1.66053906660(50)\times {10}^{-27}\mathrm{k}\mathrm{g},}$

where ${\displaystyle M_{\mathrm{u}}}$ is the molar mass constant, ${\displaystyle N_{\mathrm{A}}}$ is the Avogadro constant, and ${\displaystyle M{(}^{12}\mathrm{C})}$ is the experimentally determined molar mass of carbon-12.

The relative isotopic mass (see section below) can be obtained by dividing the atomic mass m_a of an isotope by the atomic mass constant m_u yielding a dimensionless value. Thus, the atomic mass of a carbon-12 atom is 12 Da by definition, but the relative isotopic mass of a carbon-12 atom is simply 12. The sum of relative isotopic masses of all atoms in a molecule is the relative molecular mass.

The atomic mass of an isotope and the relative isotopic mass refers to a certain specific isotope of an element. Because substances are usually not isotopically pure, it is convenient to use the elemental atomic mass which is the average (mean) atomic mass of an element, weighted by the abundance of the isotopes. The dimensionless (standard) atomic weight is the weighted mean relative isotopic mass of a (typical naturally occurring) mixture of isotopes.

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 (per E = mc²).

Relative isotopic mass

Relative isotopic mass (a property of a single atom) is not to be confused with the averaged quantity atomic weight (see above), that is an average of values for many atoms in a given sample of a chemical element.

While atomic mass is an absolute mass, relative isotopic mass is a dimensionless number with no units. This loss of units results from the use of a scaling ratio with respect to a carbon-12 standard, and the word "relative" in the term "relative isotopic mass" refers to this scaling relative to carbon-12.

The relative isotopic mass, then, is the mass of a given isotope (specifically, any single nuclide), when this value is scaled by the mass of carbon-12, where the latter has to be determined experimentally. Equivalently, the relative isotopic mass of an isotope or nuclide is the mass of the isotope relative to 1/12 of the mass of a carbon-12 atom.

For example, the relative isotopic mass of a carbon-12 atom is exactly 12. For comparison, the atomic mass of a carbon-12 atom is exactly 12 daltons. Alternately, the atomic mass of a carbon-12 atom may be expressed in any other mass units: for example, the atomic mass of a carbon-12 atom is 1.99264687992(60)×10⁻²⁶ kg.

As is the case for the related atomic mass when expressed in daltons, the relative isotopic mass numbers of nuclides other than carbon-12 are not whole numbers, but are always close to whole numbers. This is discussed fully below.

Similar terms for different quantities

The atomic mass or relative isotopic mass are sometimes confused, or incorrectly used, as synonyms of relative atomic mass (also known as atomic weight) or the standard atomic weight (a particular variety of atomic weight, in the sense that it is standardized). However, as noted in the introduction, atomic mass is an absolute mass while all other terms are dimensionless. Relative atomic mass and standard atomic weight represent terms for (abundance-weighted) averages of relative atomic masses in elemental samples, not for single nuclides. As such, relative atomic mass and standard atomic weight often differ numerically from the relative isotopic mass.

The atomic mass (relative isotopic mass) is defined as the mass of a single atom, which can only be one isotope (nuclide) at a time, and is not an abundance-weighted average, as in the case of relative atomic mass/atomic weight. The atomic mass or relative isotopic mass of each isotope and nuclide of a chemical element is, therefore, a number that can in principle be measured to high precision, since every specimen of such a nuclide is expected to be exactly identical to every other specimen, as all atoms of a given type in the same energy state, and every specimen of a particular nuclide, are expected to be exactly identical in mass to every other specimen of that nuclide. For example, every atom of oxygen-16 is expected to have exactly the same atomic mass (relative isotopic mass) as every other atom of oxygen-16.

In the case of many elements that have one naturally occurring isotope (mononuclidic elements) or one dominant isotope, the difference between the atomic mass of the most common isotope, and the (standard) relative atomic mass or (standard) atomic weight can be small or even nil, and does not affect most bulk calculations. However, such an error can exist and even be important when considering individual atoms for elements that are not mononuclidic.

For non-mononuclidic elements that have more than one common isotope, the numerical difference in relative atomic mass (atomic weight) from even the most common relative isotopic mass, can be half a mass unit or more (e.g. see the case of chlorine where atomic weight and standard atomic weight are about 35.45). The atomic mass (relative isotopic mass) of an uncommon isotope can differ from the relative atomic mass, atomic weight, or standard atomic weight, by several mass units.

Relative isotopic masses are always close to whole-number values, but never (except in the case of carbon-12) exactly a whole number, for two reasons:

• protons and neutrons have different masses, and different nuclides have different ratios of protons and neutrons.
• atomic masses are reduced, to different extents, by their binding energies.

The ratio of atomic mass to mass number (number of nucleons) varies from 0.9988381346(51) for ⁵⁶Fe to 1.007825031898(14) for ¹H.

Any mass defect due to nuclear binding energy is experimentally a small fraction (less than 1%) of the mass of an equal number of free nucleons. When compared to the average mass per nucleon in carbon-12, which is moderately strongly-bound compared with other atoms, the mass defect of binding for most atoms is an even smaller fraction of a dalton (unified atomic mass unit, based on carbon-12). Since free protons and neutrons differ from each other in mass by a small fraction of a dalton (1.38844933(49)×10⁻³ Da), rounding the relative isotopic mass, or the atomic mass of any given nuclide given in daltons to the nearest whole number, always gives the nucleon count, or mass number. Additionally, the neutron count (neutron number) may then be derived by subtracting the number of protons (atomic number) from the mass number (nucleon count).

Mass defects in atomic masses

Binding energy per nucleon of common isotopes. A graph of the ratio of mass number to atomic mass would be similar.

The amount that the ratio of atomic masses to mass number deviates from 1 is as follows: the deviation starts positive at hydrogen-1, then decreases until it reaches a local minimum at helium-4. Isotopes of lithium, beryllium, and boron are less strongly bound than helium, as shown by their increasing mass-to-mass number ratios.

At carbon, the ratio of mass (in daltons) to mass number is defined as 1, and after carbon it becomes less than one until a minimum is reached at iron-56 (with only slightly higher values for iron-58 and nickel-62), then increases to positive values in the heavy isotopes, with increasing atomic number. This corresponds to the fact that nuclear fission in an element heavier than zirconium produces energy, and fission in any element lighter than niobium requires energy. On the other hand, nuclear fusion of two atoms of an element lighter than scandium (except for helium) produces energy, whereas fusion in elements heavier than calcium requires energy. The fusion of two atoms of ⁴He yielding beryllium-8 would require energy, and the beryllium would quickly fall apart again. ⁴He can fuse with tritium (³H) or with ³He; these processes occurred during Big Bang nucleosynthesis. The formation of elements with more than seven nucleons requires the fusion of three atoms of ⁴He in the triple alpha process, skipping over lithium, beryllium, and boron to produce carbon-12.

Here are some values of the ratio of atomic mass to mass number:

Nuclide	Ratio of atomic mass to mass number
1H	1.007825031898(14)
2H	1.0070508889220(75)
3H	1.005349760440(27)
3He	1.005343107322(20)
4He	1.000650813533(40)
6Li	1.00252048124(26)
12C	1
14N	1.000219571732(17)
16O	0.999682163704(20)
56Fe	0.9988381346(51)
210Po	0.9999184461(59)
232Th	1.0001640242(66)
238U	1.0002133905(67)

Measurement of atomic masses

Direct comparison and measurement of the masses of atoms is achieved with mass spectrometry.

Relationship between atomic and molecular masses

Similar definitions apply to molecules. One can calculate the molecular mass of a compound by adding the atomic masses (not the standard atomic weights) of its constituent atoms. Conversely, the molar mass is usually computed from the standard atomic weights (not the atomic or nuclide masses). Thus, molecular mass and molar mass differ slightly in numerical value and represent different concepts. Molecular mass is the mass of a molecule, which is the sum of its constituent atomic masses. Molar mass is an average of the masses of the constituent molecules in a chemically pure but isotopically heterogeneous ensemble. In both cases, the multiplicity of the atoms (the number of times it occurs) must be taken into account, usually by multiplication of each unique mass by its multiplicity.

Molar mass of CH4
	Standard atomic weight	Number	Total molar mass (g/mol) or molecular weight (Da or g/mol)
C	12.011	1	12.011
H	1.008	4	4.032
CH4			16.043
Molecular mass of 12C1H4
	Nuclide mass	Number	Total molecular mass (Da or u)
12C	12.00	1	12.00
1H	1.007825	4	4.0313
CH4			16.0313

History

Main articles: History of chemistry and Unified atomic mass unit

The first scientists to determine relative atomic masses were John Dalton and Thomas Thomson between 1803 and 1805 and Jöns Jakob Berzelius between 1808 and 1826. Relative atomic mass (Atomic weight) was originally defined relative to that of the lightest element, hydrogen, which was taken as 1.00, and in the 1820s, Prout's hypothesis stated that atomic masses of all elements would prove to be exact multiples of that of hydrogen. Berzelius, however, soon proved that this was not even approximately true, and for some elements, such as chlorine, relative atomic mass, at about 35.5, falls almost exactly halfway between two integral multiples of that of hydrogen. Still later, this was shown to be largely due to a mix of isotopes, and that the atomic masses of pure isotopes, or nuclides, are multiples of the hydrogen mass, to within about 1%.

In the 1860s, Stanislao Cannizzaro refined relative atomic masses by applying Avogadro's law (notably at the Karlsruhe Congress of 1860). He formulated a law to determine relative atomic masses of elements: the different quantities of the same element contained in different molecules are all whole multiples of the atomic weight and determined relative atomic masses and molecular masses by comparing the vapor density of a collection of gases with molecules containing one or more of the chemical element in question.

In the 20th century, until the 1960s, chemists and physicists used two different atomic-mass scales. The chemists used a "atomic mass unit" (amu) scale such that the natural mixture of oxygen isotopes had an atomic mass 16, while the physicists assigned the same number 16 to only the atomic mass of the most common oxygen isotope (¹⁶O, containing eight protons and eight neutrons). However, because oxygen-17 and oxygen-18 are also present in natural oxygen this led to two different tables of atomic mass. The unified scale based on carbon-12, ¹²C, met the physicists' need to base the scale on a pure isotope, while being numerically close to the chemists' scale. This was adopted as the 'unified atomic mass unit'. The current International System of Units (SI) primary recommendation for the name of this unit is the dalton and symbol 'Da'. The name 'unified atomic mass unit' and symbol 'u' are recognized names and symbols for the same unit.

The term atomic weight is being phased out slowly and being replaced by relative atomic mass, in most current usage. This shift in nomenclature reaches back to the 1960s and has been the source of much debate in the scientific community, which was triggered by the adoption of the unified atomic mass unit and the realization that weight was in some ways an inappropriate term. The argument for keeping the term "atomic weight" was primarily that it was a well understood term to those in the field, that the term "atomic mass" was already in use (as it is currently defined) and that the term "relative atomic mass" might be easily confused with relative isotopic mass (the mass of a single atom of a given nuclide, expressed dimensionlessly relative to 1/12 of the mass of carbon-12; see section above).

In 1979, as a compromise, the term "relative atomic mass" was introduced as a secondary synonym for atomic weight. Twenty years later the primacy of these synonyms was reversed, and the term "relative atomic mass" is now the preferred term.

However, the term "standard atomic weights" (referring to the standardized expectation atomic weights of differing samples) has not been changed, because simple replacement of "atomic weight" with "relative atomic mass" would have resulted in the term "standard relative atomic mass."