Atomic weights: not so constant after all
The IUPAC Commission on Isotopic Abundances and Atomic Weights has changed the way certain atomic weights are reported [1, 2, 3]. In particular, the fixed ‘standard atomic weight’ has been replaced by an atomic weight ‘interval’ for ten predominantly light elements (H, Li, B, C, N, O, Si, S, Cl and Tl), reflecting the variability of their isotopic composition. Although the respective span (the ‘range’) may not be large, the precise atomic weight of an element in a particular batch or reservoir has to be estimated from other material properties, or has to measured with higher precision, depending on experimental requirements. In the future, the atomic weight intervals are expected to increase with new findings, such as the discovery of new materials or isotopically exotic sources of materials, rather than becoming more tightly constrained by new measurements.
A century ago the discovery of the stable neon isotopes, 20Ne and 22Ne, by Thomson  and the subsequent systematic quest for further occurrences of isotopes by Aston  came to the rescue of the periodic table of the elements when it became obvious that the periodicity in the chemical properties of elements was not governed by atomic weight  alone.1 The flip side of the coin, however, namely that atomic weights are in principle not necessarily constants of nature, has not precipitated into the day-to-day thinking of chemists, possibly mainly for practical reasons (it does not matter much in daily tasks in stoichiometry). Moreover, the exploration of small variations of isotope ratios in nature has been a specialist discipline mainly in geology and related disciplines, confined in addition to those light elements which could be converted into the form of simple gases for (gas) isotope ratio mass spectrometry. For the heavier elements, the precision of isotope ratio measurements to clearly identify isotope variations outside the corresponding measurement uncertainty was not sufficient for a long time. With modern high-precision isotope ratio mass spectrometers, such as multicollector inductively coupled plasma mass spectrometry systems, it is now possible to reach relative standard deviations of a few parts in 105 in isotope ratio measurements for most heavy elements. Hence, differences in the isotope ratios can now be identified at the level of few parts in 105 .
We have learned to consider various phenomena as true constants of nature. These are either constants so fundamental that they form the framework of the SI system, such as the speed of light (c), Planck’s constant (h), and elementary charge (e), or they can be measured with high precision, often better than a few parts in 108, limited only by our actual capabilities of making such measurements.
The mass of the proton, the neutron, the electron and the masses of individual atoms or isotopes belong to the latter category: they are true invariants in nature and their masses can be compared with each other with very high precision. With new experimental abilities, we can expect to improve our precision even further and thus our knowledge of the accurate masses of isotopes. On the other hand, we do not expect to suddenly find a finer structure in any of these masses and they will retain a single value accompanied by a (measurement) uncertainty. The same cannot be said for atomic weights.
Atomic weights as well as nuclide masses are reported in atomic mass units, also named dalton (Da). Until 1961 , 1 Da was defined as 1/16 of the atomic weight of oxygen: 1 Da = Ar(O)/16. Because of the recognition that different sources of oxygen, such as air, rocks, ocean water and precipitation, have significantly different atomic weights, in 1961 it was decided to redefine the atomic mass unit as 1/12 of the invariant mass of 12C. The selection of 12C was made so that the change in the magnitude of the atomic mass unit (dalton), i.e. Ar(O)/16 – Ar(12C)/12, would be as small as possible.
The dalton is the elementary mass unit on the atomic and subatomic scale. All other masses at the microscopic level are commonly expressed in terms of the dalton and not the kilogram. The microscopic world and the macroscopic world in practice employ different units for mass, the dalton for the atomic realm, and the kilogram, which is defined by a man-made Pt–Ir cylinder kept at the International Bureau of Weights and Measures (BIPM) in Paris, for the macroscopic world (although the kilogram is currently subject to a redefinition). The relation of the microscopic and the macroscopic mass scales is made through the Avogadro number: 12 g of pure 12C (ma = 12 Da) contains 6.022 14X × 1023 atoms of 12C.
Isotopic abundance variations in nature
Atomic weights can be constants of nature, but only in a few special cases. In general, chemical elements are made of a number of stable isotopes which differ only in the neutron count in their nuclei. The special cases where atomic weights of an element are in fact constants of nature refer to 21 elements2 with only one stable isotope. Beryllium (9Be), fluorine (19F), sodium (23Na), phosphorous (31P) and gold (197Au) are such examples (admittedly, definition of ‘stable’ is a subjective matter). The atomic weights of these elements are constants of nature because the (rest) masses of the isotopes are invariant quantities. In the other 62 non-radioactive elements with more than one stable isotope, the atomic weight further depends on the relative abundances of the isotopes in the particular material. This relative abundance in turn depends on the origin and history of the material.
The primordial nucleosynthesis established the initial isotopic composition of all elements. Consequently, isotopic abundances and thus atomic weights differ for different planetary systems and, to a lesser extent, from planet to planet. Often, the origin of meteorites can be traced by studying their isotopic composition.
The atomic weight can be altered by radiogenic production of isotopes. As an example, 206Pb is formed during the radioactive decay of 238U. Argon, for instance, was present in the early solar system mainly as a mixture of two isotopes: 36Ar and 38Ar. With time, however, 40Ar has been produced from the radioactive decay of 40K (t1/2 ~ 1.25 × 109 years) and the abundance of 40Ar has now reached 99.6 %. This value dominates the atomic weight of argon that we find in the atmosphere today. Such processes of radioactive decay often serve as natural clocks for investigating the age of rocks, meteorites or even the Moon.
Intervals, ranges and confusion
When quantity values such as the mass of an isotope are reported with an uncertainty margin, this is usually understood as a symmetric (Gaussian) error around the mean; hence, it is related to the implicit, yet mostly not proven assumption that the reported values originate from repeated measurements of the same (invariant) quantity and the error is governed by statistics. The isotopes fully represent such cases. Their mass (expressed in daltons) can be determined with high precision, usually by employing high-resolution mass spectrometry. This is not the case for atomic weights of all elements.
In the traditional tables3 of the IUPAC Commission on Isotopic Abundances and Atomic Weights, standard atomic weights have so far been listed in the form xx.xxx(yy), with x denoting the numerical value and (yy) denoting the uncertainty applicable to the last digits . Another common way of expressing the atomic weight is xx.xxx ± 0.0yy. As an example, the atomic weight of (monoisotopic) fluorine is listed as 18.998 4032(5) Da, with the last number (5) in parentheses denoting the uncertainty of the mass assignment of 19F. Likewise, the standard atomic weight of mercury, with its seven stable isotopes, is listed as 200.59(2) Da, with the number (2) in parentheses also incorporating the uncertainty arising from the variability of the isotopic abundances of mercury on Earth. Note the difference in relative uncertainty of almost three orders of magnitude (2.6 × 10-8 vs. 1 × 10-5) for fluorine and mercury, respectively, representing a true isotope mass measurement precision for fluorine and a natural mixing variability for mercury.
The atomic weights of polyisotopic elements such as mercury depend on their history. Instead of a symmetric Gaussian distribution, the atomic weight function of each of these elements could be depicted in a diagram with the atomic weight on the x-axis and the frequency of occurrence on the y-axis, i.e. as a frequency diagram. However, there are multiple options for the choice of compartment: we may look at whole-Earth distributions, or just the accessible Earth’s crust, or even at commercially available specimens on the laboratory shelf. We could refine the diagram and consider only a special reservoir characteristic of a particular isotopic distribution or mean value. However, presently we do not have any precise information on any of the major reservoirs providing reliable isotopic distributions of the elements in question.
As a consequence, the concept of a constant atomic weight of an element, however handy and historically relevant, must be abandoned. It cannot be replaced by a better number for the element, unless the specific isotopic value of a given material has been determined. If this has not been done or cannot be done for whatever reason, the atomic weight has to be determined by means of a well-founded (educated) guess.
How can this educated guess of a specific atomic weight be made in practice, and how can it be used in international trade and commerce?
The examples given and the size of the ranges show that the deviations appear small. However, size does not always coincide with relevance. The mass fraction of CO2 in air used to be between 220 and 280 μmol/mol (or 220–280 ppm) , at least for more than the last 800,000 years, but probably much longer. Although this value has been different during certain time periods, we can assume that it has been within this interval most of the time (today, of course, owing to burning of fossil fuel and change in land use, it is much higher, amounting to approximately 400 ppm at the time of writing, and is still rising). Although 220–280 μmol/mol, or 0.022–0.028 %, does not seem much, this is the level of CO2 at which life existed for a long time.
Where does precise knowledge of atomic weights matter and, perhaps more importantly, where does it not matter?
For the everyday chemist, who needs to weigh materials in stoichiometric proportions, the consequences are negligible. For this purpose, the atomic weight of carbon can often simply be taken as 12 Da (instead of [12.0096; 12.0116] Da); the approximately 1.1 % contribution of 13C can be safely neglected, and even more so its variability. High precision is not an issue.
For legal purposes such as in trade and commerce, it might matter that a single number has been replaced by an interval of values. Such intervals do not fit well into forms or spreadsheets. Recognizing that error margins are usually not taken into account here, nor is the fine-structure distribution of atomic weights, the IUPAC Commission on Isotopic Abundances and Atomic Weights has issued a legal table of standard atomic weights, called the conventional atomic weights, for those elements whose atomic weights are expressed in the interval notation (Table 6 in Wieser and Coplen ). The conventional atomic weight of carbon is listed there as 12.011 Da with no further uncertainty. These values are coarse enough and no variations in the atomic weights are visible at this level (for most elements).
Tracer studies are common in many research areas, including medicine, agronomy, and ecology. In all cases, precise knowledge of the molecular weight and isotopic composition of the substrate is required. The accuracy determines the analysed mixing ratios as well as the time for which the respective signal can be studied and distinguished from the background.
The triple point of water defines the thermodynamic temperature scale. The eighth edition of The International System of Units  states on page 114: “The kelvin, unit of thermodynamic temperature, is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water.” The definition of the triple point of water implies a certain molecular weight of water. In this case the water in question is VSMOW, the international isotopic reference material, which closely represents the isotopic composition (and thus molecular weight) of ocean water. Note the difference from the previous standard atomic weight (see Fig. 2).
For science as a whole, i.e. to be able to find answers to questions we will ask in the future, the atomic weights should be provided with no compromise. They should be given at the highest level of knowledge we have reached at any given time when the tables are revised. To neglect the fine structure, which is already well described and therefore has become part of common knowledge is not an option.
Here is an example. The abundance of CO2 in Earth’s atmosphere has been increasing steadily since the onset of industrialization. Measurements of the CO2 mass fraction (currently close to 400 ppm) are usually performed with a precision of ±0.1 ppm, and better precision is urgently desired. In fact, the current precision limits are set not by the measurement techniques, but rather by the scale consistency, which is provided by the Central Calibration Laboratory at the National Oceanic and Atmospheric Administration/Climate Monitoring and Diagnostics Laboratory in Boulder, Colorado. This laboratory guards the commonly accepted scale, which is partly accomplished using a set of primary cylinders filled with compressed air and covering a CO2 abundance interval of 300–600 ppm. This interval was originally generated by adding various amounts of extraneous CO2 to the freshly sampled clean atmospheric air. Unfortunately, the CO2 did not have the same isotopic composition as the air—it had less 13C than the atmospheric CO2 because it had been obtained by burning fossil fuel. Hence, the primary cylinders contain CO2 with a varying isotopic composition. On the other hand, different measurement techniques look at different properties. When gas chromatography is used to determine the CO2 content of an air sample, 13CO2 and 12CO2 contribute equally to the analytical signal. This is not true for photoabsorption techniques, such as nondispersive infrared spectroscopy and laser spectroscopy. These optical absorption techniques often sample only part of the spectrum, sometimes discriminating only slightly against the minor isotopologues, sometimes ignoring them altogether. This implies that the different techniques will measure different abundances for the same sample, which can be corrected only when the isotopic composition (and thus the molecular weight of the respective CO2) is known and taken into account when calculating the true mixing ratios from the initial experimental results.
What would have happened if the IUPAC Commission on Isotopic Abundances and Atomic Weights had not decided to introduce atomic weight intervals?
If the IUPAC Commission on Isotopic Abundances and Atomic Weights had not decided to introduced atomic weight intervals, there would have been no distinction between measurement precision of isotopic masses, on the one hand, and isotope mixing ratios in the elements, on the other hand: The old notation suggests that the standard atomic weight has an uncertainty. This, however, is not the case. Rather, the following statement holds: in any material likely to appear on a laboratory shelf the atomic weight of the element in question is likely to be within a certain interval. The width of this interval covers the range of atomic weights (and thus isotopic abundances) occurring on Earth.
The implicitly suggested finite precision for the atomic weights could be interpreted as the measurement uncertainty. This serious misinterpretation of the situation could have prevailed without the introduction of the interval notation for the atomic weight. The isotopic values can have a rather narrow window or exhibit a larger distribution. Ocean water, for instance, is well mixed across the globe. It has a narrow distribution of its hydrogen and oxygen isotopes close (but not identical) to that of the VSMOW isotopic reference material (see Fig. 2). In contrast, meteoric water, including the polar ice sheets, in general has less deuterium than water in the world oceans owing to evapotranspiration processes, and the amount can vary substantially depending on the location and time. Antarctic ice from the South Pole has about 43 % less deuterium than VSMOW. It also has about 5 % less 18O. Air oxygen in comparison has 2.4 % more 18O than VSMOW, resulting mainly from fractionation during respiration processes. A similar situation can be found for carbon. CO2 dissolved in ocean water and that in the atmosphere have almost identical 13C abundances (about 0.8 % less than the international carbonate reference VPDB). During photosynthesis, a major discrimination against the heavier isotope occurs, leaving the carbon in living matter depleted by about 2.5 % versus VPDB. The lightest carbon can be found in CH4, where the 13C deficit can reach values up to 10 %. The average δ13CVPDB value for methane in the atmosphere is close to –5 %. The carbon isotope signature can even vary considerably within a molecule, in particular after biosynthesis. This 13C ‘anatomy’ hitherto is difficult to measure, but constitutes a robust property of bioorganic compounds that can be used in food authenticity or doping control studies.
There are many more larger or subtler variations in isotopic composition and, thus, atomic weight. These variations can no longer be neglected, and they certainly cannot be correctly encompassed by a simplifying uncertainty statement in the IUPAC tables of standard atomic weights. As a consequence, the ‘uncertainty notation’ of atomic weights had to be abandoned and replaced by a specified interval of values in cases where we know that measurement precision is not limiting our knowledge of the atomic weights. The probability of finding a specific atomic weight in a particular material cannot be assessed by statistical means. Rather, the origin and history of the material can be used to estimate a more precise atomic weight. If even more accurate values are required, they must be measured using specialized equipment such as an isotope ratio mass spectrometer.
Atomic weight intervals have not been constant in the past. With new compounds found on Earth that stretch the existing limits of the isotopic compositions further to a new limit, an extension of the interval will be necessary to encompass the new range. In contrast, we can narrow the error margins on the isotopic base mass by new, more accurate and precise measurements.
Future investigations with further improved or even newly developed instrumentation (including multicollector inductively coupled plasma mass spectrometry , double-focusing gas isotope ratio mass spectrometry  or laser absorption  techniques) will reveal new details and fine structures in the atomic weights of those elements which have two or more stable isotopes, and are presently still listed using the traditional notation. These elements (including specifically a number of metals) will eventually be expressed using the interval notation, with new or refined information on typical atomic weights for particular compartments or reservoirs.
The term ‘atomic weight’ is kept for historical reasons (see page 96 in ).
A new illustrative periodic table of the isotopes has been issued by the IUPAC Commission on Isotopic Abundances and Atomic Weights (see http://www.ciaaw.org).
A complete list of references to these tables can be found at http://www.ciaaw.org.
VSMOW is an acronym of ‘Vienna Standard Mean Ocean Water’. However, it designates a particular larger sample of ocean water, kept at the IAEA in Vienna, from where aliquots in glass ampoules can be obtained.
Following agreement within the IUPAC Commission on Isotopic Abundances and Atomic Weights, we apply the stable-isotope terminology guidelines compiled by Coplen .
VPDB stands for ‘Vienna Pee Dee Belemnite’. Corresponding reference materials are available from the IAEA in Vienna.
This article has benefited from discussions with Michael Rothe and Peter Sperlich, both at the Max Planck Institute for Biogeochemistry in Jena. I am indebted to Klaus Heumann for a number of valuable suggestions and for his patience and encouragement. Two anonymous reviewers made valuable suggestions which helped to improve the readability and language.
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