Illustrated Physics
physics, drawn
Concept

Adiabatic index — where it appears

The exponent relating pressure to volume when a substance is compressed without heat entering or leaving it. It is 5/3 for a monatomic gas, 7/5 for a diatomic one and exactly 4/3 for radiation, and that last value is the one at which a self-gravitating body stops being stable against collapse.

Named by 7 essays across 2 fields — each of them below, with the objects they name alongside it.

What a coordinate is worth, by the shape of its energy. The mean energy stored in one coordinate, in units of kT, against the power with which that coordinate enters the energy. The curve is 1/n and the points are the same average obtained by integrating the Boltzmann weight numerically, agreeing to 3.3e-7 per cent at worst. an energy going as |x|^1 holds 1.0000 kT; an energy going as |x|^1.5 holds 0.6667 kT; an energy going as |x|^2 holds 0.5000 kT; an energy going as |x|^3 holds 0.3333 kT; an energy going as |x|^4 holds 0.2500 kT; an energy going as |x|^6 holds 0.1667 kT. The familiar half a kT is the case n = 2 and nothing more general than that: a coordinate whose energy is linear in it — the momentum of an ultrarelativistic particle, or a field with no restoring force but a constant tension — carries a whole kT, and one confined by a very steep wall carries almost nothing. Equipartition is a theorem about quadratic terms, and calling it a theorem about degrees of freedom is the substitution that makes it fail.

The share that is not half a kT

Equipartition is quoted as half a kT for every degree of freedom, and it is nothing of the kind. It is half a kT for every *quadratic* term. A coordinate whose energy is linear in it carries a whole kT, and a gas hot enough that its particles' energy is pc rather than p²/2m therefore holds twice what the counting says — which drops its ratio of specific heats to four thirds and puts a star on the edge of being able to hold itself up.

thermodynamics · Equipartition Energy, pressure and entropy of a gas nobody counted. The energy density, pressure and entropy density of blackbody radiation against temperature, on logarithmic axes, together with the pressure a monatomic gas of the same energy density would have. Every curve is a power of the temperature — the fourth for energy and pressure, the third for entropy — because the only length in the problem is the thermal wavelength and the only energy is kT. The pressure is exactly a third of the energy density, where an ordinary gas's is two thirds, a factor of 2: a photon carries momentum E/c and a slow molecule carries √(2mE), and that difference is the whole of it. Some values: at room temperature the radiation pressure is 1.86e-6 pascals, which is a ten thousand millionth of an atmosphere; at 1e+7 kelvin it is 2.52e+12, which is where radiation rather than matter holds a star up.

The gas that nobody counted

A box of gas holds however many molecules were put in it. A box of radiation holds however many photons the temperature says, because the walls make and destroy them until the free energy is least — and one dropped assumption changes every result. The pressure becomes a third of the energy density instead of two thirds, the entropy goes as the cube of the temperature, and the adiabatic index comes out at exactly four thirds.

thermodynamics · Blackbody The stiffness that falls below four-thirds. The first adiabatic exponent Γ₁ — how steeply pressure rises when a gas is squeezed without heat escaping — of hot oxygen with its radiation and its equilibrium electron–positron pairs, against temperature, at densities of 10³, 10⁴, 10⁵, 10⁶ g/cm³. At 10³ g/cm³ its least value is 1.222, near 1.2 billion kelvin; at 10⁴ g/cm³ its least value is 1.235, near 1.3 billion kelvin; at 10⁵ g/cm³ its least value is 1.285, near 1.8 billion kelvin; at 10⁶ g/cm³ its least value is 1.339, near 3.1 billion kelvin. A star can hold itself up only if its core's Γ₁ averages above 4/3, the value for a gas of light alone; a core that dips below it has no stable size, and the pair production that pulls it below is strongest where the gas is hot and thin — the cores of the most massive stars.

The heat that turns into matter

Thermodynamics counts particles and relativity lets them be made. Heat a gas to a billion kelvin and its light carries enough energy to create electrons and positrons, which the gas then holds in equilibrium like any other ingredient. The heat spent making them raises neither the temperature nor the pressure, so the gas grows soft — softer, in the cores of the most massive stars, than a star can survive. Those stars collapse, ignite their oxygen all at once and blow themselves entirely apart; and the same pairs, vanishing a few seconds after the Big Bang, left the universe's light hotter than its neutrinos by a factor that is the cube root of eleven over four.

relativity · Relativistic thermodynamics What the heat shares out, and what it does not. Two averages per particle in the Maxwell–Jüttner gas, in units of kT, against kT/mc² on a logarithmic scale: the momentum times the velocity, p·v, whose average over the three directions gives the pressure, and the kinetic energy. The first is 3 at every temperature, computed here by averaging over the distribution and equal to 3 to four decimal places across the whole figure — so the pressure is exactly nkT, the ideal gas law, from a gas colder than a star to one hotter than the early universe. The kinetic energy is not shared out the same way. It is 1.506 kT when the gas is cold, Boltzmann's 3/2, and rises to 3.00 kT when it is ultra-relativistic, because for a fast particle the energy is p·c and p·v, which is what equipartition hands out, is the whole of it rather than twice it.

The gas law relativity leaves exact

Heat a gas until its particles move at nearly the speed of light and almost everything Maxwell and Boltzmann said about it changes. The speeds pile up just below c instead of spreading out. The energy per particle climbs from three halves of kT to three. The heat capacity doubles, and the ratio of heat capacities falls from five-thirds to four-thirds. One law survives untouched: the pressure is still exactly nkT, at a thousand degrees or a hundred thousand million. The reason it survives is the reason the others change. What heat hands out equally to every way of moving is not energy but momentum times velocity, and for slow particles those happen to differ only by a factor of two.

relativity · Relativistic thermodynamics The period of a ball bouncing on a gas. The period of a 16 mm steel ball (16.8 g) oscillating in a vertical tube that fits it on top of a flask, against the flask's volume on a logarithmic axis, for helium, air and carbon dioxide at room temperature, if every compression is adiabatic: T = 2π√(mV/γPA²). On a 10 litre flask the ball swings with a period of 0.983 s on helium, 1.072 s on air and 1.117 s on carbon dioxide. The gas is the spring, and its stiffness is γ times the pressure: the gas whose molecules have fewest ways to store energy is the stiffest, because less of the work done in compressing it goes into motions that do not push on the walls.

The ball that weighs how a gas stores heat

Drop a steel ball into a glass tube that fits it closely, on top of a flask of gas, and it does not fall to the bottom. It bounces, up and down, on the gas trapped beneath it, with a period of about a second. The gas is a spring, and its stiffness is the pressure multiplied by a number that counts how many ways the gas's molecules can hold energy. So a stopwatch on a bouncing ball measures something that otherwise needs a calorimeter: whether the molecules can rotate, whether they can vibrate, and how far a vibration has woken up at the temperature of the room.

thermodynamics · Equipartition A race between the flame and the end gas. The Livengood–Wu integral of the end gas — the fraction of its ignition delay used up — against time from the spark, for 95-octane fuel at compression ratios 9, 11 and 13; the dotted line marks when nine-tenths of the charge has burned. If the integral reaches one before then, the end gas ignites by itself while a sizeable part of the charge is still unburned, and knocks. At a ratio of 9 the flame wins easily: the integral ends at 0.90. At 11 it reaches one when 92 per cent of the charge has burned, just short of knock; at 13, when only 78 per cent has, and the rest detonates. The integral climbs fastest at the end, where the end gas is hottest and most compressed, which is why knock is heard late in the burn.

The squeeze a petrol engine is not allowed

Every textbook gives a petrol engine's ideal efficiency as one minus the compression ratio raised to a small negative power, and every reader can see that the efficiency keeps climbing as the ratio rises. Real petrol engines stop near ten or eleven. Nothing in the thermodynamics stops them. The fuel does: the charge being compressed is already fuel and air, and the burning gas behind the flame squeezes the rest until it ignites on its own, in a hammer-blow that wrecks pistons. An octane number buys nothing but time against that, and a diesel escapes it by compressing air and adding the fuel only when it wants ignition.

thermodynamics · Heat engines 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.

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.

thermodynamics · Equipartition

Named alongside it

The objects these essays reach for when they reach for this one.

EquipartitionHeat capacityRelativistic gasAdiabatic processChemical potentialDegrees of freedomEquation of stateArrhenius lawBlackbodyBoltzmann constantThe Boltzmann factorCosmic microwave background

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