Half a kT for every way of moving

Warming something up means putting energy into it, and how much energy a kelvin costs is a measurement anyone can make with a kettle and a thermometer. Classical physics has an answer to that question which requires no knowledge of what the substance is made of, what holds it together, or how its parts interact. It requires only a count.

The classical prediction is the top plateau, at every temperature including four kelvin. What the measurement shows instead is a staircase, and the height of each step is a count while the position of each step is a quantum constant.

The theorem, and how little it assumes #

Equipartition says: in thermal equilibrium at temperature $T$, every term in the energy that is quadratic in some coordinate or momentum carries an average of $\tfrac12 kT$.

The proof is a one-line Gaussian integral. If the energy contains a term $ax^2$, the Boltzmann weight for that variable is $e^{-ax^2/kT}$, and the average of $ax^2$ under that weight is $\tfrac12kT$ regardless of what $a$ is. The stiffness cancels. That cancellation is the whole content of the theorem and it is why the answer contains no property of the substance.

The distribution the average is taken over matters less than one might expect, which is the theorem’s whole point. Each of the three velocity components contributes a term $\tfrac12mv^2$ to the energy, each carries $\tfrac12kT$, and the total translational energy is $\tfrac32kT$ per molecule whatever the molecule is made of. The most immediately checkable consequence is that a helium atom and a xenon atom at the same temperature have the same kinetic energy and speeds differing by a factor of six.

The strength of the argument is what it does not need. It does not need the terms to be kinetic — a spring’s $\tfrac12kx^2$ counts, which is why a solid has $3kT$ per atom rather than $\tfrac32kT$. It does not need the particles to collide. It does not need them to be identical, or independent, or dilute. It needs the energy to be a sum of quadratic terms and the system to be in equilibrium, and nothing else at all.

Counting, against measuring #

The theorem turns a heat capacity into a piece of arithmetic: count the quadratic terms, halve, multiply by $R$.

Two features of that table were a scandal for fifty years. The first is that a diatomic molecule appears to have five degrees of freedom rather than seven — the vibration is missing. The second is subtler and was noticed by Maxwell in 1875 as the outstanding difficulty of the theory: why do the rotations count only twice? A dumbbell has three rotational axes; rotation about its own long axis is missing too, and there is nothing in classical mechanics that says a molecule cannot spin about that axis.

Maxwell’s own verdict, in a lecture, was that this was the one thing about the molecular theory that could not be reconciled with experiment, and that the discrepancy was of a kind no adjustment of the model would remove. He was right, and the reason is that the trouble is not in the model of the molecule.

Why a term can be missing #

A quadratic term carries $\tfrac12kT$ only if the variable it belongs to can take a continuous range of values. If instead the energy in that mode comes in steps, and the first step is large compared with $kT$, then the mode is almost certainly in its ground state and contributes nothing.

That resolves both of Maxwell’s difficulties at once. The vibration is frozen because its steps are enormous. The third rotation is frozen because the moment of inertia about the molecular axis is tiny — the mass is in the nuclei, which sit on that axis — and the rotational level spacing goes as one over the moment of inertia, so the first step for that axis is tens of thousands of kelvin.

The two rotational modes that do count have a characteristic temperature of 85 K for hydrogen, which is the lowest of any molecule because hydrogen has the smallest moment of inertia of any molecule. That is why hydrogen is the one gas in which the rotational step can be watched happening: for nitrogen the characteristic temperature is 2.9 K, so the rotations are fully on everywhere the gas is a gas.

The same failure happens on a very much larger stage. Applying equipartition to the modes of the electromagnetic field in a box gives $\tfrac12kT$ to each of them, of which there are infinitely many at short wavelength, so the predicted energy density diverges. The measured curve turns over instead. The mechanism is identical to the frozen vibration, and the size of the quantum that does the freezing is $h\nu$.

The constant a kettle can measure #

Reading the argument backwards makes a heat capacity into an instrument for something quite unexpected.

The position of a step on the temperature axis is the characteristic temperature of the mode, which is $\hbar\omega/k$ for a vibration and $\hbar^2/2Ik$ for a rotation. Both contain $\hbar$. So measuring where a heat capacity changes — a purely thermal measurement, made with a calorimeter, involving no spectroscopy and no light — determines a quantum constant.

For hydrogen the rotational step at 85.4 K gives the moment of inertia and thus the bond length: 74 picometres, from a heat-capacity curve. That is the same number spectroscopy gives, obtained by an entirely different route, and the agreement is one of the better pieces of evidence that the quantum treatment of a molecule is describing the actual object rather than fitting a curve.

The same reasoning applied to solids gave Einstein his 1907 paper — the first application of quantisation to anything other than radiation — and then Debye’s improvement in 1912. Both are attempts to explain why the heat capacity of a solid, which equipartition puts at $3R$ per mole and which is indeed $3R$ at room temperature for most elements, falls to zero at low temperature. Debye’s answer, that it falls as $T^3$, is exact in the limit and is the standard way of extracting a material’s Debye temperature to this day. What that temperature amounts to is the point at which the stiffest vibration of the lattice has a quantum comparable with $kT$ — so it is a measurement of the curvature of the well each atom sits in, read from a calorimeter rather than from a spectrometer.

The counting behind that constant is worth a sentence. The number of arrangements available to a collection of molecules is overwhelmingly concentrated on the arrangements that share the energy out evenly, and “evenly” here means evenly per quadratic term rather than per molecule. That is the whole of the theorem: it is a statement about how a large number of ways of arranging things is distributed, and it says nothing about mechanism.

The ratio that gets measured instead #

A heat capacity at constant volume is awkward to measure on a gas, because holding the volume fixed while adding heat means fighting the pressure. What is easy to measure is the ratio of the two heat capacities, $\gamma = C_p/C_v$, and it is easy because it can be got from a speed rather than from a calorimeter.

Sound in a gas is a compression fast enough to be adiabatic, so its speed is $\sqrt{\gamma P/\rho}$ — and measuring the speed of sound in a gas of known density and pressure gives γ directly. For a monatomic gas equipartition predicts 5/3, for a diatomic one 7/5, and both are confirmed to three figures by a stopwatch and a tube. Newton’s own calculation of the speed of sound assumed the compression was isothermal, missed the γ entirely, and came out 18 per cent low; the discrepancy stood for over a century until Laplace identified the missing factor in 1816.

The mechanism underneath both quantities is molecules striking a wall and delivering momentum. Pressure is that tally per unit area per unit time, and it involves only the translational motion — which is why the difference between the two heat capacities is $R$ for every ideal gas whatever its internal structure. The extra energy needed at constant pressure is the work of pushing the wall back, and rotations and vibrations play no part in it. The ratio depends on the internal structure; the difference does not.

That split is worth keeping. $C_p - C_v = R$ is a statement about the equation of state and holds for a frozen mode and an active one alike. $C_p/C_v$ contains the count, and is therefore the quantity that measures how many ways a molecule can hold energy — from the speed of a sound.

Read the way a kettle presents it, heat capacity is the slope of temperature against energy supplied — or rather its reciprocal. The flat stretches on such a curve are phase changes, where the energy goes into breaking bonds rather than into any of the modes counted above, and the equipartition argument has nothing whatever to say about them. It counts the ways of moving within a phase and is silent about leaving one.

The law that weighed the atoms #

The solid case deserves more than the clause it got above, because for sixty years it was the most useful thing equipartition did — and what it was used for was not thermodynamics.

An atom in a solid sits in a well and oscillates about the bottom of it in three directions. Each direction supplies two quadratic terms, kinetic and potential, so each atom carries $3kT$ and a mole of any solid element should have a heat capacity of $3R$, which is 24.9 joules per kelvin per mole. That number contains nothing about the element.

Dulong and Petit noticed the regularity in 1819, before any of the theory existed, by measuring thirteen metals and observing that the product of the specific heat and the atomic weight came out nearly the same for all of them. They had, without knowing it, measured $3R$ thirteen times.

The use follows from reading it backwards. A specific heat is easy to measure — grams and degrees — and an atomic weight in 1819 was not: chemistry gave equivalent weights, which are atomic weights divided by an unknown small integer, and deciding that integer was the central difficulty of the period. Dulong and Petit’s law breaks the deadlock, because it fixes the heat capacity per mole rather than per gram, so dividing the measured specific heat into 25 gives the atomic weight directly.

That is what it was used for. Mendeleev applied it when placing elements in the periodic table and used it to correct the accepted weights of indium and uranium, both of which had been assigned wrong multiples and both of which then fitted the table. A thermal measurement of no interest to a chemist settled a chemical question no chemical measurement could.

And the exceptions were the clue to everything that came later. Three elements refused the law at room temperature — diamond, boron and beryllium — with diamond only about a quarter of the predicted value. Weber measured diamond over a wide range in 1875 and found the heat capacity climbing steadily toward $3R$ as the temperature rose, so it was not that diamond had fewer modes but that its modes were not yet awake.

Diamond is the stiffest and lightest common solid, so its vibrational quanta are the largest of any material — its characteristic temperature is above 2,000 K. The three elements that broke Dulong and Petit’s law are exactly the three whose atoms are light and whose bonds are stiff, which is to say the three where $\hbar\omega$ is largest. That pattern was visible in 1875 and unexplained until 1907.

Above the last step #

The staircase has a top, at $7/2$ for a diatomic gas, and above it the classical count is exhausted. What happens above it is that the heat capacity climbs again — steeply — and the reason has nothing to do with counting ways of moving.

Heat a diatomic gas past a few thousand kelvin and the molecules start coming apart. Breaking the bond in oxygen costs 5.1 electronvolts and in nitrogen 9.8, which are enormous compared with $kT$ at any of these temperatures, so only the extreme tail of the distribution dissociates at first — and the fraction climbs steeply with temperature in exactly the way an exponential in $E/kT$ climbs.

Energy going into dissociation is energy that does not raise the temperature, so the effective heat capacity becomes very large: adding heat to air at 4,000 K mostly separates oxygen molecules rather than speeding anything up. Push further and the same happens with nitrogen, and further still and the atoms begin to ionise, each ionisation absorbing another fourteen electronvolts.

None of that is equipartition and none of it is a degree of freedom. It is a chemical composition that depends on temperature, and the heat capacity of such a mixture includes a term for the heat of reaction times how fast the composition is changing. The result is a curve with broad humps rather than steps, each hump sitting where one process is halfway done.

The engineering consequence is large enough to be worth stating. A vehicle entering an atmosphere at eight kilometres a second puts an enormous energy flux through the shock in front of it, and if that energy all went into raising the gas temperature the temperature would be far higher than it is. Instead most of it goes into dissociating and ionising the air, which is why the shock layer is a plasma and why the temperature behind it is thousands rather than tens of thousands of kelvin.

And the energy is not gone, only stored. Atoms reaching the vehicle’s surface can recombine there, releasing the bond energy exactly where it is least wanted — so the heat load depends on how catalytic the surface is to recombination, and a coating chosen to be a poor catalyst can reduce the heating substantially. A property with no place in equipartition at all becomes, at high enough temperature, the thing the whole calculation turns on.

What it costs, and where the model stops #

It fails wherever the levels are coarse. That is the whole of the argument above, and it is not a small exception: it excludes every vibration at room temperature, every electronic excitation at any ordinary temperature, and the entire electromagnetic field. Equipartition is a high-temperature limit, and “high” means large compared with the spacing of whatever is being counted.

It fails for a degenerate gas. The conduction electrons in a metal have three translational degrees of freedom each and should contribute $\tfrac32R$ per mole to the heat capacity. They contribute about a hundredth of that at room temperature, because the exclusion principle prevents almost all of them from changing state at all — only those within $kT$ of the Fermi level can absorb anything. That was another standing embarrassment of the classical theory, and the resolution has nothing to do with level spacing and everything to do with what identical particles are allowed to do. It is the same freezing by a different mechanism: in one case a mode cannot be excited because the step is too large, and in the other because the state it would move into is occupied.

It fails for a non-quadratic energy. A relativistic gas has $E=pc$ rather than $p^2/2m$, and the theorem for a term linear in the variable gives $kT$ rather than $\tfrac12kT$. The general statement is that a term going as $x^n$ carries $kT/n$, and the familiar half is the case $n=2$.

Equipartition says nothing about how long it takes. A system with a mode that is weakly coupled to the others can sit for a very long time with that mode out of equilibrium — which is why a shock-heated gas has a translational temperature and a vibrational temperature that differ for microseconds, and why the phrase “the temperature of the gas” needs qualifying in any fast process.

Why it matters that the answer contains no chemistry #

The most useful property of equipartition is one that is easy to walk past: the prediction for a monatomic gas is $\tfrac32R$ per mole for helium, neon, argon, krypton, xenon and mercury vapour alike, and it is right for all of them to three figures. Six substances with different masses, different sizes, different chemistries and boiling points spanning six hundred kelvin have the same heat capacity per mole.

That is a strong statement about what a temperature is. It says the energy per molecule is set by the temperature and nothing else — not by how heavy the molecule is, not by how it interacts, not by what phase the substance was in an hour ago. A hot gas of heavy atoms and a hot gas of light ones differ in their speeds and not in their energies, which is why a mixture reaches a common temperature rather than a common speed, and why an atmosphere sorts by mass with height rather than by temperature.

It is also what makes the anomalies worth anything. A prediction that contains a fitted parameter can absorb a discrepancy; a prediction with no parameters cannot, so when hydrogen came out at 5/2 instead of 7/2, there was nowhere to put the difference. The classical theory’s greatest strength as a piece of physics was that it was too rigid to be repaired, and its failure was therefore informative rather than merely inconvenient.

What the picture cannot show #

The staircase at the top of this page is drawn for a rigid rotor and a harmonic oscillator, which are the two idealisations that make the calculation closed-form. A real molecule is neither: it stretches as it rotates, its vibration is anharmonic, and the two motions are coupled. The corrections are a per cent or two at moderate temperature and are the entire subject of molecular spectroscopy at high resolution.

The curve is also drawn for hydrogen without the complication that makes hydrogen unique. Because the two nuclei are identical fermions, the molecule’s rotational states are tied to its nuclear spin state, and ordinary hydrogen is a mixture of two species — ortho and para — which interconvert so slowly that a cooled sample does not follow the equilibrium curve at all but a weighted average of two curves. The measured heat capacity of hydrogen below 200 K therefore depends on how long the sample has been cold, which is not a fact any equilibrium theory can contain. Commercial liquid hydrogen has to be catalysed into the equilibrium mixture before storage for exactly this reason: the slow conversion releases enough heat to boil away a large fraction of the tank.

The domain of validity is: equilibrium, a Hamiltonian that is a sum of quadratic terms, and a temperature high compared with every level spacing that is not being deliberately excluded. Inside it the theorem is exact and requires nothing else. Outside it, it gives an upper bound, and the gap between that bound and the measurement is where all the interesting physics of the twentieth century came from.

The ladder from here #

Later rungs on this anchor: the theorem proved properly from the partition function, with the $x^n$ generalisation; the Einstein and Debye models of a solid and the $T^3$ law; the electronic heat capacity of a metal and the Sommerfeld expansion; the equipartition theorem in the presence of constraints, where a rigid bond is a frozen mode taken to its limit; and the fluctuation-dissipation connection, in which the same $\tfrac12kT$ per mode becomes the noise voltage across a resistor.

The neighbouring ladders are kinetic theory, where the three translational terms are counted for the first time, the blackbody spectrum, which is equipartition applied to the field and failing spectacularly, and entropy, whose counting is what the averages here are taken over.