Relative Atomic Mass · aiwiki.page

Relative atomic mass, symbol $A_{\mathrm r}$ , is the ratio of the average mass of an atom of a chemical element in a specified sample to the unified atomic mass unit. Also called atomic weight, it accounts for the masses and relative abundances of the element’s isotopes. Because it is a ratio of masses, it is dimensionless: it has no mass unit attached. (goldbook.iupac.org)

Definition and reference scale

The reference quantity is the atomic mass constant, $m_{\mathrm u}$ , defined by

m_{\mathrm u}=\frac{m_{\mathrm a}({}^{12}\mathrm C)}{12},

where $m_{\mathrm a}({}^{12}\mathrm C)$ is the mass of a free, neutral carbon-12 atom at rest in its nuclear and electronic ground state. The dalton, symbol Da, and the unified atomic mass unit, symbol u, are alternative names for the corresponding mass unit:

1\ \mathrm{Da}=1\ \mathrm u=m_{\mathrm u}.

For an element $E$ in a sample $P$ ,

A_{\mathrm r}(E,P) =\frac{\overline{m_{\mathrm a}}(E,P)}{m_{\mathrm u}},

where the bar denotes the mean mass per atom of that element. The sample designation is often omitted when its isotopic composition is understood. (goldbook.iupac.org)

Pure carbon-12 therefore has a relative atomic mass of exactly 12. Ordinary carbon, however, also contains carbon-13, so its relative atomic mass is greater than 12 and varies slightly with isotopic composition. The exact reference value applies to carbon-12, not to every sample of the element carbon. (ciaaw.org)

Calculation from isotopic composition

Relative atomic mass is an abundance-weighted mean. If isotope $i$ has atomic mass $m_i$ and number fraction $x_i$ , then

A_{\mathrm r}(E,P) =\sum_i x_i\frac{m_i}{m_{\mathrm u}}, \qquad \sum_i x_i=1.

The fractions describe the proportions of atoms, not the proportions of the sample’s mass. For isotopes of one element, number fractions are also amount-of-substance fractions. Consequently, an isotope present at high abundance contributes more strongly to the average than a rare isotope. (ciaaw.org)

For example, chlorine’s two principal naturally occurring isotopes have atomic masses of approximately 34.96885 Da for chlorine-35 and 36.96590 Da for chlorine-37. For an illustrative sample containing 75.8% chlorine-35 and 24.2% chlorine-37 by number,

A_{\mathrm r}(\mathrm{Cl}) \approx 0.758(34.96885)+0.242(36.96590) \approx35.452.

This calculated value describes that assumed composition; it is not an exact constant for all chlorine. Neither isotope individually has the average mass represented by the result. (ciaaw.org)

Distinction from related quantities

Several quantities associated with atomic mass must be distinguished:

Quantity	Meaning	Units or character
[[atomic-mass	Atomic mass]]	Rest mass of an individual atom in its ground state
Relative isotopic mass	Mass of an atom of a specified isotope divided by $m_{\mathrm u}$	Dimensionless
Relative atomic mass	Mean atomic mass for an element in a specified sample divided by $m_{\mathrm u}$	Dimensionless
[[mass-number	Mass number]]	Number of protons and neutrons in an atomic nucleus
[[molar-mass	Molar mass]]	Mass divided by amount of substance

Atomic mass and relative atomic mass can have the same numerical value when the former is expressed in daltons, but they are different kinds of quantity. For example, a mean atomic mass of 35.452 Da corresponds to a relative atomic mass of 35.452, without units. Mass number, by contrast, counts protons and neutrons in the atomic nucleus; it does not specify an experimentally measured atomic mass. (goldbook.iupac.org)

The expression atomic weight remains an accepted synonym for relative atomic mass. In this context, “weight” does not mean the gravitational force on an atom. (goldbook.iupac.org)

Standard atomic weights and natural variation

A sample-specific relative atomic mass is not necessarily identical to the standard atomic weight printed in a periodic table. Standard atomic weights are recommended by the Commission on Isotopic Abundances and Atomic Weights, associated with the International Union of Pure and Applied Chemistry. They are intended to apply to normal materials rather than to one uniquely specified sample. Materials whose isotopic composition has been substantially modified, such as deliberately isotope-enriched products, need not fall within these recommendations. (goldbook.iupac.org)

For some elements, natural differences in isotope abundances are large enough that a single value with a small uncertainty would be misleading. Their standard atomic weights are therefore expressed as intervals. For example, the recommended interval for carbon is

[12.0096,\;12.0116].

This interval represents variation among normal materials, not merely uncertainty in measuring one sample. Nor does it imply that all values within the interval are equally probable. A sample with a well-characterized isotopic composition may have a relative atomic mass determined much more precisely than the interval’s width suggests. (ciaaw.org)

Abridged tables provide convenient rounded values, such as approximately 12.011 for carbon and 35.45 for chlorine. Such entries are useful for routine calculations, but they do not eliminate real variation between samples. Elements lacking a characteristic isotopic composition generally have no standard atomic weight; an isotope’s mass number, where shown instead, is not an abundance-weighted atomic weight. (ciaaw.org)

Measurement and uncertainty

Determining relative atomic mass requires both isotope masses and isotope abundances. Modern mass spectrometry supplies isotope-ratio measurements, while precision mass measurements establish the masses of individual isotopes. Historically, chemical measurements of mass ratios in compounds also played an important role in determining atomic weights. (ciaaw.org)

Measurement uncertainty in a sample’s relative atomic mass must be distinguished from differences between samples. More precise measurements can reduce uncertainty about a particular composition, but they cannot make genuinely different isotopic compositions identical. Chlorine provides an example: isotope-ratio measurements revealed variability that eventually led to interval notation for its standard atomic weight in 2009. (ciaaw.org)

Relation to molar mass

Relative atomic mass connects atomic-scale mass ratios with the mole through the Avogadro constant, $N_{\mathrm A}$ . For an element with a specified isotopic composition,

M(E)=A_{\mathrm r}(E)M_{\mathrm u}, \qquad M_{\mathrm u}=N_{\mathrm A}m_{\mathrm u},

where $M_{\mathrm u}$ is the molar mass constant. Its value is extremely close to $1\ \mathrm{g\,mol^{-1}}$ , so relative atomic mass and molar mass expressed in g mol $^{-1}$ are numerically almost identical. (bipm.org)

Before the revised International System of Units took effect on 20 May 2019, the molar mass constant was exactly $1\ \mathrm{g\,mol^{-1}}$ . Under the revised SI, the Avogadro constant has an exact stipulated value, whereas the molar mass constant is experimentally determined. The carbon-12 definition of the dalton, and the exact relative atomic mass of pure carbon-12, remain unchanged. (bipm.org)

Historical development

John Dalton published an early table of atomic weights in the early nineteenth century. Subsequent work established increasingly reliable relative mass scales, and international tables helped standardize the values used in chemistry. The commission’s first international atomic-weight table appeared in 1902. (ciaaw.org)

The earlier oxygen-based scale was replaced by the carbon-12 scale through decisions adopted by the International Union of Pure and Applied Physics in 1960 and the International Union of Pure and Applied Chemistry in 1961. Assigning carbon-12 the exact value 12 provided a common reference for atomic weights and isotope masses. Later developments increasingly distinguished uncertainty in measurement from genuine natural variation in isotope abundances, including the introduction of standard atomic-weight intervals for selected elements in 2009. (ciaaw.org)

References

Frequently Asked Questions — CIAAWciaaw.org
Atomic Weight of Carbon — CIAAWciaaw.org
Atomic Weight of Chlorine — CIAAWciaaw.org
Standard Atomic Weights — CIAAWciaaw.org
Abridged Standard Atomic Weights — CIAAWciaaw.org
Abridged Standard Atomic Weights — CIAAWciaaw.org
Standard Atomic Weights of the Elements 2011ciaaw.org