Illustrated Physics
physics, drawn
Concept

Arrhenius law — where it appears

The rule that a rate of crossing a barrier is an attempt frequency multiplied by the exponential of minus the barrier over the thermal energy. Because the exponent is a ratio, a barrier of a few tens of thermal energies turns a nanosecond attempt into a lifetime of years, and a ten per cent change in it moves the answer by decades.

Named by 4 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 The barrier a reverse field takes away. The energy of a single-domain particle against the direction its moment points, in units of its anisotropy energy, for four strengths of reverse field along the easy axis. With no field the two directions are equally good and the barrier between them is exactly the anisotropy energy. A reverse field tilts the landscape and lowers the barrier out of the forward well as the square of the field: at 0.0 of the anisotropy field the barrier is 1.000 KV, at 0.1 of the anisotropy field the barrier is 0.810 KV, at 0.3 of the anisotropy field the barrier is 0.490 KV, at 0.6 of the anisotropy field the barrier is 0.160 KV. Each barrier is located by scanning two hundred thousand directions for the stationary points rather than by substituting the formula. The whole of the argument follows from the barrier being finite: a particle does not need the field that removes the barrier, only time enough to be shaken over what is left of it.

Nothing keeps a magnetisation for ever

A magnetised particle sits in a well with a barrier between it and the other direction, and a barrier of finite height is crossed eventually. So remanence has a lifetime, coercivity is a different number depending on how fast it is measured, and a grain below about twenty nanometres of iron forgets within a second at room temperature.

electromagnetism · Magnetisation The fold beyond which no heap is steady. The centre's steady temperature rise, in units of RTₐ²/E, against the Frank-Kamenetskii parameter δ, for a slab, a long cylinder and a sphere, found by integrating the steady heat equation outward from the centre: each shape has two steady states for every δ below a critical value — a cool, stable one and a hot, unstable one — and none above it. The fold is at δ = 0.878, 2.000, 3.322 for the slab, cylinder and sphere: a sphere tolerates nearly four times a slab's δ at the same half-width, because its surface is larger in proportion to its volume and heat from its centre has more ways out. A real heap, which is neither, has a critical δ between, and fixing it for a given shape is the first step in calculating the size at which a store becomes unsafe.

The heap that sets itself alight

A haystack, a coal stockpile, a bin of oily rags or a silo of damp grain can catch fire with no spark at all. A slow reaction inside makes a little heat; the heap holds it in; the warmer middle reacts faster and makes more. Whether that ends in a warm heap or a burning one is decided by a single number built from the heap's size, the air's temperature and the reaction's speed — and past one value of it there is no steady temperature at all. The critical size shrinks steeply as the weather warms, and the heap gives almost no warning before it goes.

thermodynamics · Diffusion 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

Named alongside it

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

Activation energyAdiabatic indexAdiabatic processBarrierBifurcationThe Boltzmann factorCoercivityEfficiencyEntropyHeat conductionHeat engineHysteresis

All concepts