Relative atomic mass (symbol: Ar), or atomic weight, is a dimensionless (number only) physical quantity. In its modern definition, it is the ratio of the average mass of atoms of an element (in a given sample) to one unified atomic mass unit.
The unified atomic mass unit, symbol u, is defined being of the mass of a carbon-12 atom. Since both values in the ratio are expressed in the same unit (u), the resulting value is dimensionless; hence the value is relative.
For one sample, its relative atomic mass (atomic weight) is a straight average over the individual atom weights (isotopes) present. Between sources, the atomic weight can vary when the source's origin (its radioactive history or diffusion history) caused a different mixture of isotopic concentrations. For example, due to a different mix of stable carbon-12 and carbon-13, a sample of elemental carbon from volcanic methane will have a different relative atomic mass than one collected from plant or animal tissues.
The well-known standard atomic weight is an application of this relative atomic mass values from different samples. It is expected range of the relative atomic mass values, with the various sources being terrestrial (taken from Earth). These standard atomic weights are what chemists loosely and so incorrectly call "atomic weights" (incorrectly, because they are not from a single sample). They are the most published form of the relative atomic mass, because they are not sample-specific, but cover a broad range of expected Earth samples.
The continued use of the term "atomic weight" (of 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. Both terms are officially sanctioned by IUPAC. The term "relative atomic mass" now seems to be gaining as the preferred term over "atomic weight", although in the case of "standard atomic weight", this shorter term (as opposed to the more correct "standard relative atomic mass") continues to be used.
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. 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 made relative to carbon-12.
It is a synonym for atomic weight (and is not to be confused with relative isotopic mass). Relative atomic mass is frequently used as a synonym for the standard atomic weight and these will 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) covers more than standard atomic weights, and is more general term that may more broadly refer to non-terrestrial environments and highly specific terrestrial environments that deviate from Earth-average or have different certainties (number of significant figures) than do the standard atomic weights.
Prevailing IUPAC definitions taken from the "Gold Book" are
Here the "unified atomic mass unit" refers to of the mass of an atom of 12C in its ground state.
The IUPAC definition 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. Nevertheless, given the cost and difficulty of isotope analysis, it is usual to use the tabulated values of standard atomic weights which are ubiquitous in chemical laboratories.
Older (pre-1961) historical relative scales (based on the atomic mass unit, or a.m.u., or amu) used either the oxygen-16 relative isotopic mass for reference, 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 in 1961.
The IUPAC commission CIAAW maintains a 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).
Atomic mass (ma) is the mass of a single atom, with unit Da or u (the unified atomic mass unit). It defines the weight 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 in determination of relative atomic mass
From this mass, 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 so dimensionless.
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 for virtually all non-radioactive nuclides, but isotopic compositions are both harder to measure to high precision and more subject to variation between samples. 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).
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
The estimation of the uncertainty is complicated, especially as the sample distribution is not necessarily symmetrical: the IUPAC standard relative atomic masses are quoted with estimated symmetrical uncertainties, 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, in 2010 IUPAC made the decision to list the standard relative atomic masses of 12 elements as an interval rather than a fixed number.