Weighing the Kilogram

Proposals to redefine this unit of mass and a related measurement, the mole, may be overlooking a clearer, simpler solution.

Physics Technology

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November-December 2014

Volume 102, Number 6
Page 426

DOI: 10.1511/2014.111.426

Standards of measurement date back to James Clerk Maxwell, nearly one and a half centuries ago. Such standards guarantee uniformity and clarity for comparisons, an especially necessary ability owing to the global nature of the marketplace, technology, and research. Today, in the International System of Units (SI) there are seven base units that define physical quantities. These units include the second, meter, ampere, kelvin, and candela. But the final two have proven particularly problematic over the years: the kilogram (for the physical quantity “mass”), and the mole (for the physical quantity “amount of substance”—the name comes from “gram molecular weight”). Although it is widely accepted that the former needs a modification, there has been slow movement to do so.

Extreme precision in the measurement of all units is more than a semantic point. Improvements in measurement technology, combined with the fundamental properties of nature that can be revealed by nearly infinitesimally small changes in magnitudes, means that more and more precise standards are needed to extract knowledge from experimental determinations or from comparison with theories. The list of Nobel Prize winners is replete with recognition of major discoveries based on precision of the highest degree. Base units, upon which measurements are structured, not surprisingly have been undergoing modernization. But the kilogram and the mole have not been at the top of the list more because of controversy than lack of motivation.

Robert Rathe

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The kilogram, and by extension the mole, are the only base units still tied to a physical object. For now, the 125-year-old kilogram definition is still anchored to a metal artifact: a platinum-iridium alloy cylinder that is kept under lock and key in a suburb of Paris. The mass of this standard kilogram seems to be changing at about the millionth of a percent level over about a century. The cause of this change is uncertain, but likely relates to periodic cleaning and calibration measurements. Small as it may seem, that change is enough to lend urgency to the campaign to redefine the kilogram.

To that end, a symposium on “The Kilogram and the Mole,” sponsored by the president of the American Chemical Society, was held in San Francisco in 2010. There are now a number of formal proposals to redefine the kilogram, but I aim to demonstrate that some of the major options championed are grossly inappropriate. In 2011, Michael Kühne, then the director of the International Bureau of Weights and Measures (BIPM) spoke to the New York Times about the kilogram, saying that “the reason it hasn’t been redefined before now is that nobody has come up with something better.” I think that the BIPM has actually come up with something worse. This is the bureau’s proposed new definition:

The kilogram, kg, is the SI unit of mass; its magnitude is set by fixing the numerical value of the Planck constant to be equal to exactly 6.626069…×10-34 when it is expressed in the unit s-1 m2 kg, which is equal to J s.

I would like to decode that definition, but even its proponents agree that is not a trivial task. What it comes down to is that a device called a watt-balance can be used to realize the connection between Planck’s constant, h, the quantum of action in quantum mechanics, and the kilogram, based on an equation relating electrical and mechanical power that involves both that constant and the velocity of a mass moving under the influence of gravity. Making matters murkier, the suggestion has arisen to further constrain this definition in order to avoid the possibility that Planck’s constant might vary as the universe evolves. To the above would be added the qualifying phrase for the Planck constant “being constant in terrestrial space and in the current era time frame.”

Closely related to the kilogram issue is the one involving the mole. Up to now, the mole has been stipulated to be the number of atoms in exactly 12 grams of carbon-12. (See “An Exact Value for Avogadro’s Number,” March–April 2007. ) Because it is defined in terms of grams, the mole is inextricably entangled with the kilogram. Under the new BIPM proposal, therefore, as the kilogram definition changes the mass of carbon-12 will also change, probably multiple times. Yet most recently, revisions in official atomic mass data by the International Union of Pure and Applied Chemistry still refer to the mass scale’s foothold as embodied in carbon-12 having an atomic mass of exactly 12.

Proponents of the new SI definitions, such as the BIPM, have sought to bring the kilogram and mole in line with the other five standard units, make the complete set of fundamental physical quantities invariant and not based on artifacts. In other words, they aim to rationalize the entire international system of standard units. Quoting Ian Mills, president of the BIPM’s Consultative Committee on Units: “The objective of the proposed changes is to adopt definitions referenced to constants of nature, taken in the widest sense, so that the definitions may be based on what are believed to be true invariants.”

The kilogram, and by extension the mole, are the only base units still tied to a physical object.

Prominent are the phrases “constants of nature” and “true invariants.” The clearest examples of success here are the meter and the second, which are now infinitely precise and permanently defined in terms of integers. For instance, the meter is defined as the length of the path travelled by light in a vacuum during a time interval of 1/299,792,458 of a second. This part of the SI scheme has universal support, which is essential if it is to be extended to the remaining physical quantities. The widely accepted criteria for SI units include the following, according to Mills: the possibility of realization anywhere and at any time, at least in principle; based on commonly accepted laws in physics; and conceptually clear and easy to understand.

These are sensible and laudable goals. But the criteria and the objectives put forward by Mills seriously call into question the wisdom, if not the legitimacy, of the BIPM’s very own proposed changes.

New Kilograms

There are actually two prominent, competing proposals for how to recast the kilogram. One is the “silicon sphere” (the Avogadro constant approach, related to the mole); the other is the “watt balance” (the Planck constant approach). The silicon sphere would use a softball-size orb of a single crystal of silicon-28, which has precise lattice spacing between atoms that, combined with Avogadro’s number, would determine the exact size needed to measure 1 kilogram. In this way, it would maintain the reference kilogram as a physical object. The watt balance uses a moving coil to counteract the gravitational force on an object by electromagnetic induction. A truly ingenious, albeit very expensive device, the watt balance takes advantage of quantum electronic measurements involving g, the acceleration of a mass in the Earth’s gravitational field, linking together the kilogram and Planck’s constant.

Andrew Brookes/NPL/Science Source

The two proposals have been championed to different degrees by the BIPM, proposed for ratification first in 2005 and subsequently in 2011, but postponed because of discordant results between similar balance devices or between independent methods. Several attempts at constructing watt balance devices were abandoned because of the difficulty in achieving the desired precision. Commenting on a definition of the base unit in terms of h, Peter Becker, an expert consultant to the Consultative Committee, reminds us that mass, like the other base units, is a classical concept whereas a definition relating it to the Planck constant will depart from classical physics. Defining a base unit just because its realization invokes modern quantum metrology intimates that the tail is beginning to wag the dog.

In fact, both approaches have serious shortcomings. Rather than advocating either, the BIPM should have resoundingly discarded them together as not satisfying the objective agreed on: “constants of nature” and “true invariants.”

In the case of the silicon sphere with its chemically reactive surface, even if isotopic purity (100 percent silicon-28) is approached, the kilogram would still be based on an artifact. Natural silicon is 92 percent silicon-28. Russian laboratories have provided kilogram quantities of silicon enriched to 99.99 percent at high cost. The surfaces are oxidized. Isotopic analyses by different techniques have produced significantly different results for the composition. Yet the most precise value to date for experimentally determining Avogadro’s number has been extracted from the latest silicon sphere measurement based on precise isotopic masses for silicon compared to carbon-12.

Photograph courtesy of NIST.

The watt balance is predicated on an even more troublesome artifact. Gravitational acceleration is permanently encumbered with a limit of parts-per-billion precision. The proponents of the watt balance approach admit that the lunar cycle and tides affect their measurements, although the corrections are confidently manageable. Local gravitational acceleration ( g, not to be confused with the fundamental gravitational constant, G), of course, is also affected by distribution of mass and by geographical location of the apparatus. Earthquakes, volcanic eruptions, and seasonal hydrology have the largest correctional influence. The 2010 Chilean and 2011 Tohoku earthquakes are reported to have altered the planet’s revolution period, which in turn changes gravitational acceleration at the parts-per-billion level.

Also, landmasses rising and falling and ice caps melting with seasonal temperature changes; polar motions; atmospheric pressure; depth of the local water table; and even human activities affect gravitational acceleration. The crustal lithosphere is constantly stirring around. The planet’s convecting magma below the crust continually and unpredictably (unlike tides and lunar orbits) affect gravitational acceleration in degrees that will continually modify the results of measurements. Global climate changes affect sea levels. Gravitational shifts occur on time scales ranging from minutes to years and spatial extents from meters to global. And that’s just on Earth—on whose mass distribution the watt balance is inescapably based—the largest artifact imaginable! The watt balance does not satisfy the BIPM/SI constraints. The Earth’s gravity is neither a “constant of nature” nor a “true invariant.”

The BIPM should have been aware of their transgression beforehand. James Clerk Maxwell said at the 1870 meeting of the British Association for the Advancement of Science:

If, then we wish to obtain standards of length, time, and mass which shall be absolutely permanent, we must seek them not in the dimensions, or the motion, or the mass of our planet, but in the wavelength, the period of vibration, and the absolute mass of these imperishable and unalterable and perfectly similar molecules.

This cogent quote followed an astute observation by Maxwell that seems to have been overlooked by the metrology community:

Yet, after all, the dimensions of our earth and its time of rotation, though, relatively to our present means of comparison, very permanent, are not so by physical necessity. The Earth might contract by cooling, or it might be enlarged by a layer of meteorites falling on it, or its rate of revolution might slowly slacken, and yet it would continue to be as much a planet as before. But a molecule, say of hydrogen, if either its mass or its time of vibration were to be altered in the least, would no longer be a molecule of hydrogen.

In 2005 and 2006, theoretical physicist Frank Wilczek published a series of brief, thoughtful essays on absolute physical units emphasizing, among other things, relationships among them. Much of his presentation dealt with units of length, time, and mass. Wilczek noted as an illustration that units must be such that one could communicate information, such as body mass, with a correspondent anywhere—on the Moon or even in the Andromeda galaxy—by simply transmitting pure numbers. No such hypothetical exchange of information on mass would be possible under the BIPM proposal, making it a choice that is uniquely and disadvantageously confined in both geography and time. Furthermore, length (the meter), time (the second), and mass (the kilogram) are all comprehensible phenomena to the general public, but the BIPM-proposed kilogram definition is undeniably obtuse, stifling clarity and visualization.

Constant Carbon

An essential insight with respect to the mass fundamental unit is the following dictum: There is, in practice, absolutely no need for parts-per-billion precision in a macroscopic mass such as embodied in the current standard kilogram. But in keeping with the philosophy of the meter and second, there is a need for an equivalently precise fundamental mass standard. Fortunately, such a standard can be quite elegantly and simply instituted microscopically, where it is needed, without the ambiguities and complexities inherent in either of the BIPM proposals. An atomic scale that has the mass of a carbon-12 atom being exactly 12 atomic mass units can be related to the kilogram through Avogadro’s number.

Current measurement techniques have yielded better than parts-per-billion precision masses of atoms, and even the electron and the muon, all relative to carbon-12. Measurements can be on one atom at a time. The mass of carbon-12 is an invariant of nature. Ubiquitous carbon-12 requires no upkeep or security precautions. It would seem, though, that the proposed carbon-12 alternative leaves us with an impractical dilemma: macroscopic mass. But one mole of carbon-12 atoms has a mass of exactly 12 grams, providing a macroscopic connection to the unified atomic mass. Although somewhat arbitrary, this long-standing dictum is nevertheless an intuitive and understandable option.

More problematic, it might initially seem, is the mole. Currently, “the mole is the amount of substance of a system which contains as many elementary entities as there are atoms in 0.012 kilogram of carbon 12.” A proposal considered by the BIPM offered the mole as “the SI unit of amount of substance of a specified elementary entity, which may be an atom, molecule, ion, electron, any other particle or a specified group of such particles; its magnitude is set by fixing the numerical value of the Avogadro constant to be equal to exactly 6.022141079 . . .×1023 when it is expressed in the unit mol-1.”

Illustration by Barbara Aulicino.

I have a simple solution. The mole, also known to many as Avogadro’s number, should be regarded not as a physical constant, but rather as a numerical constant. It is, in the simple scheme of things, the quantitative gauge that links two mass scales, microscopic and macroscopic. The mole does not belong on the list of SI base units for physical quantities (and, in fact, was added to that list only in 1971). To some, such a suggestion is horrifying, chafing at the drilled-in and deeply entrenched notion that the mole is a physical unit, a must for the chemical amount of substance, rather than a numerical quantity. But if we describe three dozen eggs, we symbolically, if not actually, calculate 3×12. When we talk about three moles of methane, we symbolically conjure up 3 × Avogadro’s number of methane molecules. The mole is a number: That number of carbon-12’s defines 12 grams, which in turn gives us the route to defining an invariant kilogram: a constant of nature through the fact that carbon-12 is invariant.

The platinum-iridium metal artifact that currently defines the kilogram, along with all its hundred or so near-clones, in this alternative view become secondary global standards, reference materials, precise to better than 0.1 parts-per-million, sufficient if not well beyond what is necessary in the practical world for now, in the past, and in the foreseeable future. The kilogram as an SI fundamental physical unit becomes defined as the mass of exactly (1,000/12) × Avogadro’s number of carbon-12 atoms. Certainly, this value is an analog to the definition of a meter as related to the speed of light in a vacuum.

Giving “the mole” its due recognition as a number, removing it from the SI list of fundamental physical units, and establishing carbon-12 as the basis of an SI mass system meets the BIPM objective and satisfies the criteria. Moreover, returning to methane to illustrate this point, both “3 moles” of methane and “0.04813 kilograms” of methane conjure up the same thing, an amount of the substance methane.

In the future, instrumentation connecting time and mass is likely to play a role in the kilogram–mole connection. The Compton clock, a device taking advantage of atom interferometry, has recently been constructed and successfully operated. This clock splits a quantum matter wave of a single cesium-133 atom in two, one of which remains stationary. The device uses a laser to compare the phase difference between the beams, and this frequency relates directly to the mass of the atom.

The scheme would provide an absolute measurement of a cesium-133 atom’s mass to an accuracy of 4 parts per billion, which is competitive with other methods and more accurate than in the present SI. The mass determined is an inertial mass, as opposed to the seriously problematic gravitational mass that the watt balance method weighs. Other microscopic masses can be related to cesium, but it remains to be seen how readily and rapidly available such a device might become.

The Wider Community

The redefinitions of the kilogram and mole directly affect chemistry, physics, biology, and the medical sciences, but the BIPM’s deliberations have been largely confined to the community of metrologists. Plainly, it behooves such an international effort, founded on diplomatic agreement, to proactively involve the wider user community.

Indeed, the director of the BIPM, in a February 20, 2011, letter to the New York Times declared “The redefinition is not being held up by the question of whether it should be based on the Planck constant or the Avogadro constant. That issue has been resolved in favor of the Planck constant.” After a significant number of scientific publications aired unhappiness about the proposed revision, two members of the International Union on Pure and Applied Chemistry responded: “Given that the decision to adopt the New SI is in effect a fait accompli, due to the authority vested in the BIPM set up by the Metre Convention of 1875 and amended in 1921, those affected will have to adapt to live with the new system.”

That vexing dismissiveness could be excused if “those affected” had been generally made aware of the new SI proposals and given an opportunity to participate in developments. The good news is that the new standard is not, in fact, a fait accompli. There is time for a wide-reaching discussion to take place. I urge the BIPM to make every effort to ensure that input is received and heard.


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    • Mills, I. M., et al. 2006. Redefinition of the kilogram, ampere, kelvin, and mole: A proposed approach to implementing CIPM recommendation 1 (CI-2005). Metrologia 43:227–246.
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