Illustrated Physics
physics, drawn
Concept

Thermal energy — where it appears

The energy a body holds in the disordered motion of its molecules, of order kT for each way any one of them can move. It is not a property a body possesses in a definite amount but the disordered part of its internal energy, and the distinction is what separates heat from work.

Named by 5 essays across 2 fields — each of them below, with the objects they name alongside it.

A ratio in the exponent. The Boltzmann factor against the energy of a state measured in units of kT, with the logarithm up the axis so that the straight line is the whole content. Every kT of energy costs a factor of e, so 10, 20, 30 times kT are factors of 10^-4.3, 10^-8.7, 10^-13.0. At 300 K, kT is 25.9 meV, so a barrier of 0.35 eV is 13.5 kT and a factor of 1.3e-6. That is the sense in which a third of an electronvolt is not a small energy: it is small compared with a chemical bond and enormous compared with kT, and it is the second comparison that decides whether anything happens.

The exponential that decides everything

Maximising the number of ways a reservoir can arrange what is left after taking E out of it gives one factor, e to the minus E over kT. Its exponent is a ratio, which is why a barrier of a third of an electronvolt — nothing at all by chemical standards — is the difference between instantly and never.

thermodynamics · Entropy Newton's answer, Laplace's, and the measurement. The speed of sound in 4 gases at 273.15 K. The short bar is Newton's √(RT/M), which assumes the compressions stay at one temperature; the long one is the same multiplied by √γ, which is what they come to if no heat crosses between a compression and the rarefaction beside it; the upright mark is the measured value. Newton's is 15.5% low for air, 22.5% low for helium, 22.5% low for argon, 12.0% low for carbon dioxide, and the corrected one is right to 0.05% for every gas here. Turning it round: γ read off each pair of bars is 1.400 for air, 1.665 for helium, 1.666 for argon, 1.290 for carbon dioxide, which is 1 + 2/f with f = 5.0, 3.0, 3.0, 6.9 ways of holding energy — three for the monatomic gases, five for the diatomic ones, and nearly seven for carbon dioxide, whose bending modes have begun to take a share at this temperature and its stretch has not. A speed measured with a stopwatch counts the ways a molecule can move.

The correction that took a century

Newton derived the speed of sound in 1687 and got 290 metres a second against a measured 340. The arithmetic was right; the assumption was not. Heat cannot cross a wavelength in a period, so the compressions are adiabatic — and the factor that repairs the answer turns out to be a count of the ways a molecule can move.

waves · Wave motion An f² law that is right in shape and out by 30× in size. Two absorption curves for air against frequency, both logarithmic, in decibels per kilometre. The lower one is the classical Stokes–Kirchhoff result computed from air's viscosity and thermal conductivity alone, and it goes as f^2.000 — exactly two, because the loss per cycle is fixed and the number of cycles per metre is proportional to the frequency. The upper one is the measured atmospheric absorption at 20 °C and 50 per cent humidity, which fits f^1.42 and is 30 times larger at 1 kHz and 211 times at 125 Hz. The excess is not a correction to viscosity: it is nitrogen and oxygen storing energy in vibration and giving it back late, at a rate the water vapour sets, and it is the mechanism that actually removes the treble from a distant sound.

The distance that takes the treble out

Spreading treats every frequency alike; absorption does not. The loss per cycle is roughly fixed and the number of cycles per metre goes as the frequency, so absorption climbs as f² and a sound gets duller with distance as well as quieter — which is the whole account of why a nearby thunderclap cracks and a distant one rumbles.

waves · Attenuation A ratio in the exponent. The Boltzmann factor against the energy of a state measured in units of kT, with the logarithm up the axis so that the straight line is the whole content. Every kT of energy costs a factor of e, so 5, 12, 20 times kT are factors of 10^-2.2, 10^-5.2, 10^-8.7. At 295 K, kT is 25.4 meV, so a barrier of 0.4 eV is 15.7 kT and a factor of 1.5e-7. That is the sense in which a third of an electronvolt is not a small energy: it is small compared with a chemical bond and enormous compared with kT, and it is the second comparison that decides whether anything happens.

The temperature a molecule does not have

Temperature fixes a system's average energy and nothing more. The actual energy wanders, by an amount tied to the heat capacity, and the relative size of the wandering falls as one over the square root of the number of degrees of freedom — so a mole has a temperature and a molecule does not.

thermodynamics · Equipartition A beam of sound has a weight. The force a fully absorbed acoustic beam exerts, against its power, for 3 media. It is the power divided by the speed of sound and nothing else — a watt in water gives 675 micronewtons, which is the weight of 68 milligrams, and a watt in air gives 2.9 millinewtons because the sound is slower there. This is not an analogy with light: it is the same statement, that a wave carrying energy carries momentum, with a much smaller speed in the denominator. The consequence is that acoustic power is measured by weighing. A radiation-force balance — an absorbing target on a laboratory balance, with the transducer beneath it — is the primary standard for ultrasonic output, and every therapeutic and diagnostic transducer is calibrated against one.

Where the loudness goes

An absorption coefficient removes energy from a wave, and energy removed has to appear somewhere. It appears twice, from the same coefficient: as heat, and as momentum. So a beam of sound has a weight — a watt absorbed in water weighs sixty-eight milligrams — and acoustic power is measured by putting an absorber on a balance. The ratio of the force to the heating contains no intensity at all.

waves · Attenuation

Named alongside it

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

AbsorptionAttenuationThe Boltzmann factorDegrees of freedomDissipationEquipartitionHeat capacityIntensityTransportViscosityWave speedAcoustic streaming

All concepts