The ball that weighs how a gas stores heat

Half a kT for every way of moving found that a molecule in a gas holds, on average, half of $kT$ of energy in every quadratic term of its energy — every direction it can move in, every axis it can spin about, every vibration it can make. Count the terms and the heat capacity follows. The share that is not half a kT then found the qualification that made the theory right: a motion whose quantum of energy is much larger than $kT$ is frozen out and holds nothing, which is why diatomic molecules at room temperature rotate but do not vibrate.

Both results are claims about heat capacities, and heat capacities of gases are hard to measure directly. A gas has little heat capacity per litre, and any container holding it has far more. In 1929 Eduard Rüchardt found a way round that needs a flask, a glass tube, a ball and a stopwatch, and that measures the count of active motions in the time it takes the ball to bounce.

A gas spring #

The apparatus is a large flask with a long, vertical precision-bore tube fitted to its neck, and a steel ball that slides in the tube with very little clearance. Dropped into the tube, the ball does not fall into the flask; it compresses the gas below it, which pushes it back up, and it oscillates up and down a few centimetres for many cycles before friction and leakage bring it to rest.

The gas is a spring. If the ball moves down by $x$, the gas’s volume falls by $Ax$, where $A$ is the tube’s cross-section, and its pressure rises. How much it rises depends on what happens to the heat the compression produces. If the compression is fast enough that no heat leaves the gas — an adiabatic compression — then $PV^\gamma$ stays constant, and a small change of volume $dV$ changes the pressure by $-\gamma P\,dV/V$. The restoring force is $A$ times that, $\gamma PA^2x/V$, which makes the ball a simple harmonic oscillator, as every minimum is a parabola would have it, with angular frequency

$$\omega^2 = \frac{\gamma P A^2}{m V},$$

so that one period, measured, gives

$$\gamma = \frac{4\pi^2 m V}{A^2 P T^2}.$$

Here $P$ is the gas’s pressure with the ball resting on it — the atmosphere’s plus the ball’s weight over the bore — and every other quantity is a length or a mass that can be measured with ordinary instruments.

The periods are about a second for a flask of a few litres, easy to time over ten or twenty swings. And they differ between gases at the same pressure, because $\gamma$ does: the ball swings fastest on helium and slowest on carbon dioxide. The stiffness of a gas spring is not just its pressure. It is its pressure multiplied by a number that reflects what the molecules do with the energy the compression gives them.

The precision needed is modest. $\gamma$ depends on the square of the period, so timing twenty swings to a hundredth of a second each determines it to about two parts in a thousand, far better than the difference between air and carbon dioxide, whose periods differ by four per cent. The hard parts are the bore of the tube, which enters as its square, and the volume of the flask, which must include the tube below the ball.

The arrangement is, in fact, an old acoustic instrument in disguise. The note a bottle sings whatever its shape found that a bottle’s tone is set by the air in its neck bouncing on the springy air in its body — a Helmholtz resonator, with $\omega^2 = \gamma PA/\rho LV$ for a neck of length $L$. Rüchardt replaced the plug of air in the neck with a steel ball a thousand times heavier, which lowers the note from a few hundred hertz to one cycle a second, slow enough to time with a watch. The formula is the same, and so is the $\gamma$ in it.

Why the stiffness counts motions #

Compressing a gas does work on it. If the gas cannot lose heat, all that work stays in the gas as internal energy. Part of the energy goes into the molecules’ motion towards and away from the walls — their translational motion — which is what raises the pressure and pushes back on the ball. The speeds in a still room found pressure to be exactly two-thirds of the translational kinetic energy per unit volume: only that share of the energy pushes. The rest goes into motions that do not push on the walls: rotation, vibration. A gas with many such motions shares the work among more of them, its translational energy and its temperature rise less for a given compression, and its pressure rises less. It is a softer spring.

The arithmetic is equipartition. A molecule with $f$ active quadratic degrees of freedom has a heat capacity at constant volume of $\tfrac f2 k$ and at constant pressure of $(\tfrac f2 + 1)k$, so

$$\gamma = \frac{c_p}{c_v} = 1 + \frac{2}{f}.$$

Three degrees of freedom — translation only — give $5/3$. Five — translation and two rotations — give $7/5$. More give less.

The noble gases sit exactly at three, the common diatomic gases at five. Those two numbers were among the earliest evidence that something was wrong with classical physics: a diatomic molecule made of two atoms joined by a bond has, classically, a vibration as well as its rotations, which should add two more terms and bring $\gamma$ to $9/7$. It does not, at room temperature, because the vibration’s quantum of energy is more than ten times $kT$ and the vibration is frozen. Rüchardt’s ball reads the frozen vibration directly: a period that would be longer if the bond could vibrate, and is not.

Carbon dioxide is linear, with two rotations, and its $\gamma$ of 1.289 corresponds not to five degrees of freedom or to the classical thirteen but to 6.9. Its two bending vibrations, at a quantum of about 960 kelvin, are partly awake at room temperature — each holding a little under half of the classical share of two quadratic terms — while its stretching vibrations are still asleep. Sulphur hexafluoride, a heavy molecule with fifteen vibrational modes, many of them soft, has so many partly active motions that its $\gamma$ is barely above one and its ball swings slowly. The ball’s period reports, in one number, how many of a molecule’s motions have woken up.

When the compression is not adiabatic #

The derivation assumed that no heat leaves the gas during a swing. That is not automatic. The gas next to the flask’s walls exchanges heat with the glass quickly, and the heat spreads through the rest of the gas by conduction, taking a time $\tau$ that grows with the flask’s size. If the ball swings slowly compared with $\tau$, the gas stays at the walls’ temperature throughout, the compression is isothermal, and the spring’s stiffness is $PA^2/V$ — with $\gamma$ replaced by one.

Between the two limits the gas is part-way, and something else happens: heat flows into the walls as the gas is compressed and back out as it expands, a little later, so that the pressure lags the volume and the gas spring absorbs energy each cycle. The loss is largest when the oscillation and the heat leak have similar time scales, and it vanishes at both extremes. The note too high for thin air found the same structure in sound: a wave whose period is comparable to the time the gas takes to relax is absorbed most strongly. The same lag, between a stress and a response that needs time to catch up, gives every relaxation its peak of loss.

Rüchardt’s measurement works because a flask of a few litres has a heat-leak time of seconds to tens of seconds, while the ball swings in about one. The leak is not uniform, though. Heat diffuses through air with a diffusivity of about $2\times10^{-5}$ square metres per second, so in the second of one swing it spreads only $\sqrt{D/\omega}$, a couple of millimetres. The gas within that distance of the glass is effectively isothermal and the rest adiabatic. In a ten-litre flask the isothermal skin holds a few per cent of the gas, and it lowers the measured $\gamma$ by a few parts in a hundred of $\gamma - 1$ — small, but larger than the timing error, which is why the most careful versions of the experiment correct for it from the flask’s surface area. The compression is close to adiabatic and the error small. A small flask, a slow ball or a gas that conducts heat well, such as helium, pushes the measurement towards the isothermal end, lengthens the period and damps the swing, and a careful experiment corrects for it — either by measuring the damping, which tells how far from adiabatic the gas is, or by using flasks of different sizes and extrapolating. The other source of damping, friction and leakage between the ball and the tube, is a nuisance that only careful machining reduces; it is why a precision-bore tube and a carefully matched ball are the whole of the apparatus’s cost.

Other ways to the same number #

Rüchardt’s was not the first method, and it was quickly varied. Nicolas Clément and Charles Desormes in 1819 let a little gas escape from a slightly pressurised vessel, quickly enough that the expansion was adiabatic, closed it again, and watched the pressure recover as the gas warmed back to the room’s temperature; the two pressure changes, adiabatic and then isothermal, give $\gamma$ from their ratio. Rinkel, also in 1929, released Rüchardt’s ball from rest and measured how far it fell before turning, which gives $\gamma$ from a length rather than a time. The speed of sound gives $\gamma$ too, through $\sqrt{\gamma P/\rho}$, and for precision work it is the method of choice. What the ball adds is directness: its spring is the same gas, compressed by the same few per cent, and its period is a single reading that anyone can take.

A period that tracks a vibration waking up #

The degrees of freedom a gas has are not fixed numbers but functions of temperature, and the ball’s period should follow them.

Each vibration’s contribution follows Einstein’s formula for a quantum oscillator: a mode whose quantum is $k\theta$ adds $k\,x^2e^x/(e^x-1)^2$ to the heat capacity, with $x = \theta/T$. The function is nearly zero when $T$ is a tenth of $\theta$, reaches half of the classical value near $T = \theta/3$, and is within about a tenth of it by $T = \theta$. A vibration wakes over a factor of five or so in temperature, not suddenly, and a gas whose modes have quanta spread across that range has a $\gamma$ that drifts continuously over hundreds of kelvin — the drift the ball’s period would follow, a few per cent over the range drawn.

Nitrogen’s single vibration has a quantum of about 3,400 kelvin, and at room temperature it is completely frozen: $\gamma$ is 1.400 to three decimal places. It begins to wake only well above 1,000 kelvin, and by 2,000 its $\gamma$ has fallen to 1.30. Carbon dioxide is already in the middle of its transition at room temperature. Its bending vibrations wake between about 200 and 1,000 kelvin, and its $\gamma$ falls steadily across that range — so the speed of sound in carbon dioxide, and the period of a Rüchardt ball on it, change with temperature in a way they would not if the gas were classical or completely frozen. The heat capacity that goes up and comes down followed a heat capacity with a peak from two levels; the vibrations here give smooth steps, one for each mode, at temperatures set by their quanta.

Why it matters which motions are awake #

The ratio $\gamma$ is not only a number in thermodynamics textbooks. It sets the speed of sound in a gas, $\sqrt{\gamma P/\rho}$, which is how Laplace corrected Newton’s estimate of the speed of sound by fifteen per cent in 1816: Newton had assumed sound was isothermal, and it is adiabatic. It sets how hot a gas gets when compressed, which is why diesel engines can ignite their fuel by compression alone, and how much work an engine extracts from a given expansion, which the ceiling on every engine began from. It sets how a planet’s atmosphere cools with height, and whether a layer of air will convect or stay stratified. In each case the number of active degrees of freedom in the gas enters through $\gamma$, and carbon dioxide’s half-awake bends, which make its $\gamma$ depend on temperature, are part of why its behaviour in a planetary atmosphere is more complicated than a textbook polyatomic gas’s.

Rüchardt’s ball is a demonstration that one of the central results of statistical physics — that energy is shared equally among the active motions, and that quantum mechanics decides which are active — can be seen in a measurement a schoolroom can make. The ball does not know about molecules. It feels a spring, times its swing, and the spring’s stiffness carries the count.

What the drawings leave out #

The figures treat the gas as ideal and the ball as frictionless, with no leakage of gas past it, and model the heat exchange with the walls as a single relaxation time — a simplification of what is really a diffusion of heat from the walls inward, with a spread of time scales. Real flasks have necks and tubes whose own volume adds to the gas spring, and the ball’s motion is not exactly sinusoidal when its amplitude is large, because the gas’s pressure–volume relation is not linear; the simulation in the first figure’s check uses the full adiabatic law and confirms that the small-amplitude formula is accurate to a part in a thousand for swings of a tenth of a millimetre, and real swings are larger. The vibrational heat capacities are those of independent Einstein oscillators with the molecules’ measured frequencies, which neglects the coupling between rotation and vibration and the anharmonicity of real vibrations; both are small at the temperatures drawn. The domain of the drawings is ideal gases at atmospheric pressure, flasks of half a litre to twenty litres, and temperatures from 200 to 2,000 kelvin.

Still open: how small a gas spring can be made #

Rüchardt’s measurement relies on a flask large enough for the compression to be adiabatic, and the trend in measurement is towards smaller samples. A gas in a micro-machined cavity, or a gas confined in the pores of a material, exchanges heat with its walls almost instantly, and any oscillation of it is isothermal; but the same cavities are now made as resonators whose frequencies, quality factors and losses can be measured very precisely, and from the dependence of the loss on frequency the heat-exchange time and the gas’s heat capacity can in principle be extracted together. How far such methods can be pushed — to gases at very low pressure, where the molecules hit the walls more often than each other, or to gases whose vibrational relaxation is itself slow — is a question for acoustic thermometry and for measurements of the properties of gases in confinement, where the bulk value of $\gamma$ is not guaranteed to hold.

The physics of the ball is complete in one line. A ball of mass m on a gas of volume V in a tube of bore A oscillates with $\omega^2 = \gamma P A^2/mV$, where $\gamma = 1 + 2/f$ counts the gas’s active degrees of freedom: on a 10 litre flask a 16 mm ball swings in 0.983 s on helium, 1.072 s on air and 1.117 s on carbon dioxide, whose half-awake bends give it 6.9 degrees of freedom and a γ that falls from 1.314 at 250 K to 1.182 at 1000 K — provided the swing is fast compared with heat leaking to the walls, or the gas turns isothermal and the swing dies. A stopwatch on a bouncing ball is a census of what molecules are doing with their energy.