Thermodynamics

The note that fixed the kelvin

A sphere of argon, struck, rings at a pitch set by the speed of sound in it, and the speed of sound in argon contains only one quantity nobody could compute: the constant that converts a temperature into an energy. Argon's atoms can hold energy in exactly three ways, so its heat capacity is a count rather than a measurement, and its speed of sound is (5/3)kT/m with nothing left over. Ring the sphere at the temperature of melting ice, measure its size with radio waves in the same cavity, and Boltzmann's constant comes out to a part in a million. In 2019 that number stopped being measured and became the definition.

14 min read 5 figures Made into an instrument What stays the same

Assumes: The ball that weighs how a gas stores heat · Half a kT for every way of moving

The ball that weighs how a gas stores heat dropped a steel ball onto a column of gas and timed it bouncing, and read from the period how many ways the gas’s molecules could hold energy. It noted in passing that the speed of sound gives the same number, and that for precision work it is the method of choice. This essay follows the precision to its end, which is unusual: a measurement so good that it changed the definition of a unit. In 2019 the kelvin stopped being defined by the temperature of a particular state of water and was defined instead by fixing the value of Boltzmann’s constant. The value fixed was, more than any other single source, the one read from the pitch of a sphere of argon.

How a sphere of gas rings. The acoustic pressure against distance from the centre of a spherical cavity, for its three lowest radial modes, in which the gas breathes in and out in spherical shells: sin(zr/a)/(zr/a), with z = 4.4934, 7.7253, 10.9041 the roots of tan z = z that make the gas's motion vanish at the wall. Filled with argon at the triple point of water, 273.16 K, a sphere 62 mm in radius rings in these modes at 3551, 6104, 8616 Hz. Radial modes are the ones precision work uses: they have no motion along the wall, so the wall's roughness and the gas's viscosity there barely touch them, and their frequencies are set by the volume rather than the shape — a slightly imperfect sphere shifts them only at second order. — Fig. 1 Acoustic pressure against distance from the centre of a spherical cavity for its three lowest radial modes, sin(zr/a)/(zr/a), with z = 4.4934, 7.7253 and 10.9041, the roots of tan z = z that make the gas’s motion vanish at the wall. Filled with argon at 273.16 K, a 62 mm sphere rings at 3,551, 6,104 and 8,616 Hz.

A constant that only a gas can reveal

Boltzmann’s constant k converts a temperature into an energy: the mean kinetic energy of a molecule in a gas at temperature T is $\tfrac32 kT$ , and half a kT for every way of moving made that the organising rule of how matter stores heat. Before 2019 the kelvin was defined by declaring the triple point of water, the one temperature at which ice, liquid water and vapour coexist, to be 273.16 kelvins exactly, and k was then whatever experiment found it to be. Measuring it meant measuring an energy and a temperature at once, in a system where the relation between them is known exactly.

A gas is such a system, if its molecules can hold energy in a known number of ways. Sound in a gas is a wave of adiabatic compressions, and its speed is $c = \sqrt{\gamma kT/m}$ , where m is the mass of a molecule and γ the ratio of the gas’s heat capacities at constant pressure and at constant volume. The mass is known from the atomic mass of the gas, and γ is set by counting: each way a molecule can hold energy adds $\tfrac12 k$ to its heat capacity at constant volume, and the ratio follows. Measure the speed of sound at a known temperature, and k is the one unknown left.

Why argon

Why the measurement uses a noble gas. The ratio of heat capacities, γ, against temperature, for argon (or any noble gas) and for nitrogen, computed by counting the ways each molecule can hold energy. Argon's atoms can only move, three ways, so γ is 5/3 at every temperature drawn: no internal motion wakes up until its electrons are excited, at tens of thousands of kelvin. Nitrogen also turns, two more ways, giving 7/5 at room temperature, and its vibration wakes gradually: γ is 1.3997 at 300 K, 1.343 at 1000 K and 1.294 at 3000 K. A gas whose heat capacity has to be measured cannot calibrate anything; a monatomic gas's γ is a count, exact, and the speed of sound in it contains no unknown but the constant that relates temperature to energy. — Fig. 2 The ratio of heat capacities against temperature, computed by counting the ways each molecule can hold energy. Argon’s atoms can only move, three ways, so γ = 5/3 at every temperature drawn. Nitrogen also turns, giving 7/5 at room temperature, and its vibration wakes gradually: 1.3997 at 300 K, 1.343 at 1000 K, 1.294 at 3000 K.

The counting must be exact, and only one kind of gas makes it so. A diatomic molecule like nitrogen moves three ways and turns two, so at room temperature its γ is 7/5 — nearly. Its vibration is also beginning to wake, as the heat capacity that goes up and comes down found for every molecule whose vibrational quantum is comparable to kT, and at 300 kelvin it has already lowered γ by three parts in ten thousand, far more than a precision measurement can tolerate. Its γ would have to be measured, and a constant cannot be calibrated against a quantity that itself needs calibrating.

A noble gas has no such problem. Its atoms are spheres, with no axis to turn about that matters and no bond to vibrate; the first way they could hold energy other than moving is by exciting an electron, which needs tens of thousands of kelvin. At any temperature an experiment uses, a monatomic gas’s heat capacity is exactly $\tfrac32 k$ per atom and its γ exactly 5/3. Its speed of sound is then $\sqrt{(5/3)kT/m}$ , 307.8 metres a second in argon at the triple point of water, and the constant is the only thing in it that was not known. The energy that refuses to be shared found quantum mechanics freezing out motions that classical physics would have shared energy with; here the freezing is the whole point, because a frozen motion is one that cannot contaminate the count.

Argon in particular is cheap, available in very pure form, heavy enough that its speed of sound is modest, and its isotopic composition — the share of argon-36, -38 and -40, which sets the mean atomic mass — can be measured. Helium, lighter and simpler, has also been used; its larger speed of sound and its sensitivity to impurities make it harder.

A sphere that rings

The speed of sound is not measured with a stopwatch. It is measured as a pitch. A closed cavity full of gas has resonant frequencies at which sound waves fit exactly inside it, like the notes of a pipe, and each is proportional to the speed of sound divided by the cavity’s size. The cavity of choice is a sphere, and the modes of choice are its radial modes, in which the gas breathes in and out in spherical shells, as the first figure draws. In a radial mode the gas moves only towards and away from the centre. It does not slide along the wall, so the wall’s roughness and the drag of the gas’s viscosity there barely affect it. And a radial mode’s frequency depends on the sphere’s volume rather than its exact shape: a sphere slightly out of round, which every real sphere is, shifts it only at second order in the distortion. The note a bottle sings whatever its shape found the same insensitivity to shape in a Helmholtz resonator, for a different reason.

For a sphere of radius a the radial modes ring at $f = zc/2\pi a$ , with z a root of $\tan z = z$ , the condition that the gas be at rest at the wall: 4.4934, 7.7253, 10.9041. A sphere 62 millimetres in radius, filled with argon at the triple point of water, rings at 3,551 hertz in its lowest radial mode — a high, clear note, which a microphone in the wall can hear and a loudspeaker in the wall can drive. Its frequency is measured to a part in a hundred million by sweeping the drive through the resonance and fitting the response.

The corrections bigger than the answer

The correction that is three hundred times the answer's error. How far each radial mode's frequency is lowered by the thermal boundary layer at the wall, in parts per million, against the argon's pressure. Within a thermal diffusion length of the wall the gas exchanges heat with the metal during each cycle, so its compressions there are isothermal rather than adiabatic, and the gas is a little softer than the 5/3 the speed of sound assumes. At 100 kPa the layer is 40 μm thick at the first mode's frequency and lowers it by 215 parts per million; it thins as the inverse square root of both pressure and frequency, so the correction is smaller for higher modes and at higher pressure. A measurement that aims at a part in a million must compute this correction to better than a part in a hundred, and the agreement of the corrected values from different modes — whose corrections differ — is one of its tests. — Fig. 3 How far each radial mode is lowered by the thermal boundary layer at the wall, in parts per million, against pressure. Within a thermal diffusion length of the wall the gas’s compressions are isothermal rather than adiabatic. At 100 kPa the layer is 40 μm thick at the first mode’s frequency and lowers it by 215 ppm; it thins as the inverse square root of pressure and frequency.

The simple formula assumes the gas is compressed adiabatically everywhere. Near the wall it is not. The metal wall is so much better at holding heat than the gas that the gas touching it stays at the wall’s temperature through each cycle of the sound, and within a thermal diffusion length of the wall, forty micrometres at the first mode’s frequency and atmospheric pressure, its compressions are isothermal. Isothermal gas is softer than adiabatic gas, by the factor γ, and the softening lowers the resonance. The fourth figure computes the shift: 215 parts per million for the first mode at a hundred kilopascals, smaller at higher pressures, where the layer is thinner, and for higher modes.

A correction of two hundred parts per million, in a measurement aiming at less than one, has to be calculated to better than a part in two hundred of itself. It can be, because the physics of the boundary layer is well understood and the gas’s thermal conductivity is measurable — and because the correction differs between modes, which provides a check: if the corrected speed of sound comes out the same from the first, second and third radial modes, whose corrections differ by a factor of two, the correction has been understood. A second correction is for the gas not being ideal. Real argon atoms attract one another slightly, which changes the speed of sound in proportion to the pressure, and the measurements are made at a series of pressures and extrapolated to zero, where the gas is ideal. The ring of the sphere’s own shell, which moves slightly with the pressure inside it, and the holes for the microphone and the gas inlet, are corrected for too.

Measuring the size of the sphere with light

One cavity, two kinds of resonance. The same 62 mm sphere rings with sound in its argon at 3551 Hz and with radio waves in its interior at 2.111 GHz (the lowest electromagnetic mode, TM₁₁) — six decades apart, on a logarithmic axis. Both scale inversely with the radius, so their ratio, 1.682 × 10⁻⁶, depends on the sphere's size not at all: it is the ratio of the speed of sound to the speed of light, times a ratio of two known roots. Since 1983 the speed of light has been exact, so measuring the microwave resonances measures the cavity's size in the units the acoustic measurement needs, in the same apparatus at the same moment — the step that took the acoustic method below a part in a million. — Fig. 4 The same 62 mm sphere rings with sound in its argon at 3,551 Hz and with radio waves in its interior at 2.111 GHz, the lowest electromagnetic mode, six decades apart on a logarithmic axis. Their ratio, 1.682 × 10⁻⁶, does not depend on the sphere’s size: it is the ratio of the speed of sound to the speed of light, times a ratio of known roots.

The pitch gives the speed of sound divided by the sphere’s radius, so the radius must be known as well as the pitch: to half a part in a million for a measurement of k good to one part, since k depends on the square of the speed. Michael Moldover and his colleagues at the US National Bureau of Standards, in the measurement of 1988 that dominated the next decade, found the volume of their sphere by filling it with mercury and weighing the mercury. Later measurements used a cleverer idea, which the fifth figure draws. The same metal cavity that resonates with sound also resonates with microwaves, at frequencies proportional to the speed of light divided by its radius. Since 1983 the speed of light has been exact by definition, so the microwave resonances measure the radius in metres with nothing else assumed, and the ratio of acoustic to microwave frequencies gives the speed of sound as a fraction of the speed of light with the radius cancelling. The cavities used were made deliberately slightly non-spherical, triaxial ellipsoids called quasi-spheres, so that microwave modes that would coincide in a perfect sphere split apart and can be measured separately; the acoustic radial modes, being insensitive to shape, do not care.

From eight parts in a million to none

Boltzmann's constant, as well as it was known. The relative uncertainty of the recommended value of Boltzmann's constant in each international adjustment of the fundamental constants, on a logarithmic scale. It stood at 8.5 parts per million in 1986; an acoustic measurement in argon reported in 1988 brought it to 1.7 in the next adjustment, where it sat for a decade; new acoustic measurements, and others from dielectric-constant and noise thermometry, brought it to 0.37 ppm by 2017. In 2019 the kelvin was redefined by fixing k = 1.380649 × 10⁻²³ J/K exactly (the open circle): the uncertainty moved from the constant to the temperature of the triple point of water, which became a measured quantity. — Fig. 5 The relative uncertainty of the recommended value of Boltzmann’s constant in each international adjustment of the constants: 8.5 ppm in 1986, 1.7 after the acoustic measurement of 1988, 0.37 by 2017 with acoustic, dielectric-constant and noise measurements. In 2019 the value was fixed at 1.380649 × 10⁻²³ J/K exactly (open circle).

The last figure shows what the measurements did. Before 1988 Boltzmann’s constant was known to eight parts in a million, inferred mostly from the gas constant measured in older ways. Moldover’s acoustic measurement in argon brought it to under two, and it stayed there for a decade until a new generation of measurements, around 2010, began to improve it again: acoustic resonators in several national laboratories using argon and helium, measurements of the dielectric constant of helium gas, whose polarisability can be calculated from quantum mechanics, and measurements of the thermal noise voltage in a resistor, the noise half a kT in a piece of wire derived. The best acoustic result, from the UK’s National Physical Laboratory in 2013, reached seven parts in ten million.

When the uncertainties of several independent methods had fallen below a part in a million and agreed with each other, the international bodies that maintain the units fixed the value. Since 20 May 2019 Boltzmann’s constant is exactly 1.380649 × 10⁻²³ joules per kelvin, and the kelvin is defined by it: one kelvin is the temperature change that changes kT by exactly that many joules. The triple point of water, which used to be 273.16 kelvins by definition, is now a measured temperature, equal to 273.16 kelvins within the uncertainty the constant used to have, about a third of a part in a million. Why the triple point is a point explained why water has a single such temperature; it remains the most reproducible temperature there is, and thermometers are still calibrated against it, but it is no longer the foundation.

What fixing a constant changes

Fixing the constant did not change any temperature anyone measures. The value chosen was the best measured value, so the triple point of water stayed at 273.16 kelvins to within a third of a part in a million, and every thermometer calibrated before 2019 still reads correctly. What changed is which quantity carries the uncertainty. Before, the triple point was exact and k was measured relative to it; after, k is exact and the triple point is measured relative to it. The numbers are the same; the foundation moved.

The move matters away from room temperature. A scale anchored to one temperature, the triple point of water, has to be carried to a few millikelvin or to thousands of kelvins through chains of intermediate fixed points and interpolating instruments, and the error grows along the chain. A scale anchored to a constant can be realised directly at any temperature, by any method in which the temperature appears through kT: the pitch of a gas, the noise in a resistor, the radiation from a cavity. The kelvin was redefined on the same day as the kilogram, the ampere and the mole, each by fixing a constant — Planck’s constant, the electron’s charge and Avogadro’s number — for the same reason: an artefact or a particular substance can drift or be lost, and a constant of nature cannot.

The sensitivity of the measurement to its ingredients follows from its formula. Boltzmann’s constant is proportional to the square of the speed of sound, and so to the square of the frequency and the square of the radius, and inversely to the temperature. A measurement aiming at one part in a million therefore needs the frequency and the radius each to half a part in a million and the temperature of the gas to a part in a million, which at 273 kelvins is a quarter of a millikelvin, held across the whole sphere for days.

Where the method holds

The acoustic determination rests on the gas being monatomic and pure, on the atomic mass being known from the gas’s isotopic composition, and on the corrections being right. Each has needed care. Argon from different sources differs in its isotopic make-up enough to shift the mean atomic mass by parts per million, and measuring it was one of the larger contributions to the final uncertainty. Impurities as small as a part per million of a light gas, helium or hydrogen, change the speed of sound measurably. And the extrapolation to zero pressure assumes the virial expansion of the speed of sound holds over the range measured, which is checked by measuring at many pressures. The gas must also be at one temperature throughout, which is why the spheres are machined from copper or aluminium, metals that spread heat quickly, and sit inside nested shields in a bath held at the triple point of water, compared against cells of ice and water by thermometers read to a tenth of a millikelvin.

What the figures cannot show

The figures compute frequencies and corrections; they cannot show what makes a measurement at a part in a million different in kind from one at a part in a thousand, which is that every effect large enough to matter must be named, modelled and checked by a second method. A small leak through a seal, a temperature gradient of a millikelvin across the sphere, the slight compliance of the microphone’s diaphragm: each is a few parts per million, and each has a section in the published analyses. Nor can they show the agreement between methods that persuaded the community to fix the value. An acoustic measurement in argon, a dielectric one in helium and an electrical one in a resistor have almost no systematic errors in common, and their agreement within their uncertainties was the evidence that the number was right.

Still open: how far a gas thermometer can go now

With k fixed, the same apparatus measures temperature instead of the constant: a sphere of argon ringing at a measured frequency, with its size known from microwaves, gives the thermodynamic temperature directly, without reference to any other thermometer. Acoustic gas thermometry has been used since the redefinition to measure the differences between the international practical temperature scale, a set of fixed points and interpolating instruments, and true thermodynamic temperature, which differ by up to tens of millikelvin at some temperatures. How far the method can be extended — to the very low temperatures where helium is the only gas left, and to the high temperatures where the sphere’s own materials begin to change — and whether it can become a practical primary thermometer outside the national laboratories that built it are open questions. The note that fixed the kelvin is now being used to measure it.

The habit worth carrying away is to look for a system whose answer is a count. A monatomic gas holds energy in exactly three ways, so its γ is exactly 5/3 and its speed of sound, $\sqrt{(5/3)kT/m}$ , has no unknown in it but Boltzmann’s constant; a sphere of argon at 273.16 K rings at 3,551 Hz for a 62 mm radius, corrected by 215 ppm for its walls and sized by microwaves at 2.111 GHz, and measurements of such notes took k from 8.5 to 0.37 parts per million before it was fixed at 1.380649 × 10⁻²³ J/K in 2019. Equipartition, a rule about how heat is shared, ended up defining what a degree is.