# Redefining the kilogram using Planck's constant instead of the density of water among other examples

The kilogram is in the process of being redefined in terms of Planck's constant so as to eliminate its dependence on a physical artefact. Since the length and temperature units are already precisely defined, why not just calculate the density of some substance, say water, at a particular temperature and use that as a standard for mass? Sounds simpler to me.

• How confident are you that the substance is pure? How uniform is the temperature in the substance? How do you account for possible evaporation during the measurement process? Most importantly, how do you calculate density for a particular substance if there is no standard for mass? – probably_someone Jul 13 '17 at 10:31
• Are you proposing to fix that 1 litre of water at 20C and 1atm pressure is the standard? That copies are made (essentially what we have today), or you rely that someone else can replicate the standard anywhere with high accuracy (with all the issues that @probably_someone said, plus variation in gravity, in turn standardization of what us 1atm pressure...)? – Tricky Jul 13 '17 at 11:40
• Water is not a good choice for this sort of thing. Its isotope composition varies, and it tends to dissolve other impurities. Elemental silicon is much better, because it's far easier to prepare it with high purity, due to it being a semiconductor. See en.wikipedia.org/wiki/Zone_melting – PM 2Ring Jul 13 '17 at 17:26
• I don't understand the votes to close as opinion-based. Perhaps the BIPM's rationale for choosing a fundamental constants-based approach is subject to debate, but going back to water is a bad choice. A water-based standard was used initially, with one liter of pure water at standard pressure and at its densest having a mass of exactly one kilogram. This approach was ditched almost immediately for a prototype-based standard because of accuracy issues. – David Hammen Jul 13 '17 at 17:33
• @robertbristow-johnson - Laksh did write "say water" parenthetically. I took the question to be asking why the BIPM isn't using "some substance" in general to redefine the kilogram, rather than "some substance, say water". A carefully measured sphere of $^{28}\text{Si}$, for example, would make for a rather nice choice. But that isn't the route the BIPM is taking. They instead are redefining the base units in terms of fundamental constants. – David Hammen Jul 13 '17 at 18:00

Since the length and temperature units are already precisely defined, why not just calculate the density of some substance, say water, at a particular temperature and use that as a standard for mass?

Water is a lousy choice. The initial proposal for the French metric system used the mass of a cubic decimeter of water. Measurement issues resulted in this being changed to a prototype-based system in just a few years. Issues with those initial prototypes resulted in the current prototype masses. Issues with those newer prototypes are part of what motivated the physics-based redefinition of the International System (SI).

Water is a bad choice, but what about some other substance? The problem with this is that it flies in the face of one of the key goals of the proposed redefinition of the SI, which is to define the base units of time, length, mass, current, and temperature solely in terms of fundamental physical constants. The mole is also being redefined, from the number of atoms in 12 grams of 12C to a specified number.

Other key goals are that the changes should represent improvements and that the redefined base units must be consistent with the past. Those latter two have always been goals. Using a fundamental physics-based approach is new, or almost new. The definitions of the second and meter are fundamentally-based. The improvements that these redefinitions enabled were a strong motivator to continue this process to the remaining three physical units, and to the mole as well.

That said, using a carefully measured quantity of some substance is close to one of the two approaches being used to establish the exact value of Planck's constant. Those two approaches are the Kibble balance (formerly Watt balance), which carefully compares electrical power to mechanical power, and the Avogadro technique, which carefully calculates the number of atoms in a carefully measured sphere of nearly pure 28Si.

The deadline for measurements by multiple groups using these two techniques to estimate Planck's constant passed on July 1. The requirement by the International Bureau of Weights and Measures (BIPM; the acronym is French) was to have at least three experiments with an uncertainty of 50 parts per billion (ppb) or better and at least one with an uncertainty of 20 ppb or better. Previous failures to meet that goal is the key reason the SI redefinition is not yet in place. That goal has now been met. There are multiple experiments with much better than the requisite 50 ppb uncertainty and three with better than 20 ppb.

• Thank you for your answer. Can you throw some light on why the gravitational constant is not being used for this purpose? – Laksh Arora Jul 26 '17 at 13:31
• @LakshArora - Because doing so would represent a huge step backwards. In terms of accuracy, the gravitational constant is one of the least well-known physical constants, with perhaps five places of accuracy at most. That's an uncertainty of hundreds of parts per million. The accuracy of the constant chosen to serve as the basis of the kilogram needs an accuracy of better than 50 parts per billion or better. – David Hammen Jul 26 '17 at 14:44
• – uhoh Nov 17 '18 at 8:23

As mentioned in David Hammen's answer, the core of the answer is accuracy. Defining the kilogram as the mass of some volume $V$ of water at given conditions is fine in principle, but the idea is that the new definition needs to beat the existing one in both accuracy and stability.

Given that the existing drift in the national kilogram standards is about $50\:µ\mathrm g$ over multiple decades, the new definition needs to be able to give stabilities better than 50 parts per billion in order to be an improvement over the existing artefact-based mass standard.

With water, this is extremely hard, because of a number of reasons:

• Most impurities you might find in water are a good deal heavier than the water itself. That means that if you want a density that's accurate to 50 ppb, you're going to want the density of impurities to be at about that level, which isn't easy at all.

• Both oxygen and hydrogen have variable isotopic compositions, with heavier isotopes present at concentrations of about 0.05%; that's fine if you know the isotopic composition, but any variability in that is going to cause problems with the mass standard. As a comparison, when you need water with a well-defined isotopic composition, you turn to standards like VSMOW, which has uncertainties in its composition of the order of 0.1 to 1 ppm. That's already enough to take you out of the running unless you work extremely hard to pin down that composition to about two more significant figures, and that's extremely challenging because isotopes are intrinsically hard to separate.

• The water volumetric expansivity is zero at the stable-density point but the second-order derivative is not, with a ballpark value of about $10^{-4}/\mathrm{K^2}$. If you want a precision superior to $10^{-8}$, you're going to want to find that maximal-density point with millikelvin precision. That's OK for small masses but not so much at the kilogram scale ─ but, more importantly, thermometric metrology is by far the hardest branch of metrology. This kind of scheme would subject huge swathes of metrology to the limitations of thermometry.

Those are just off the top of my head (and not necessarily completely accurate), but they're representative of how effects that look completely negligible at first glance turn out to be extremely hard problems to solve once you want a standard that's stable to eight significant figures.

More practically, though, we can turn to history to see just how hard it is to get water-based mass standards to be accurate: given the original definition based on the density of water, the Kilogramme des Archives moved the kilogram from a constant-based standard to an artefact-based one... in 1799. Now, the eighteenth century was pretty good at making precision masses, and less good at precision metrology than we are, but that's a good indicator of how hard it is to get water to make a precise and stable reference for a mass standard.

• Broadening the question from water, a carefully measured amount of some other substance ($^{28}\text{Si}$, for example) would do quite nicely as a replacement for those kilogram prototypes. But that's not the route the BIPM is taking. There's a reason for this. – David Hammen Jul 13 '17 at 18:13