Illustrated Physics
physics, drawn
Concept

Adiabatic process — where it appears

A change made without heat crossing the system's boundary, so that any temperature change comes from work alone. A parcel of air lifted a kilometre cools by g/c_p = 9.76 K on this account and by nothing else, which is what makes the number a property of gravity and heat capacity rather than of weather.

Named by 7 essays across 4 fields — each of them below, with the objects they name alongside it.

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 Each stage takes 25.0 per cent of what is left. Entropy against temperature for a spin-½ paramagnet at 0.25 T and 1 T, with the cooling cycle drawn between them: a vertical drop is isothermal magnetisation, a horizontal move is adiabatic demagnetisation. Starting from 1 K the treads are at 1.000 K, 0.250 K, 0.062 K, 0.016 K, 3.91 mK. Each is 0.2500 of the one before — a ratio read back off the drawn treads rather than written into them, and equal to the field ratio 0.25/1 because this refrigerant's entropy depends on the field and the temperature only through their quotient. The steps therefore shrink in proportion to what is left, and no finite number of them arrives.

The staircase that never reaches the floor

Absolute zero is unreachable, and the reason is not that the apparatus is not good enough. Every stage of cooling removes a fixed fraction of what is left rather than a fixed amount, so the steps shrink in proportion to the distance remaining — and the fixed fraction cannot be made one, because the entropy curves at two field strengths are required to meet where the axis is.

thermodynamics · Third law Everything is decided against one line at 9.76 K per kilometre. Temperature against height for five environments, with the dry adiabat drawn heavy. A parcel lifted from the ground cools along the adiabat, at g/c_p = 9.76 K/km — a number with no meteorology in it, only gravity and the heat capacity of air. If the environment cools faster than that, a lifted parcel finds itself warmer than its surroundings and keeps going; if it cools more slowly, the parcel finds itself colder and sinks back. -5 K/km gives N² = 5.02e-4 s⁻², a period of 4.7 min; 0 K/km gives N² = 3.32e-4 s⁻², a period of 5.7 min; 6.5 K/km gives N² = 1.11e-4 s⁻², a period of 9.9 min; 9.8 K/km gives N² = -1.43e-6 s⁻², an e-folding time of 835 s; 12 K/km gives N² = -7.63e-5 s⁻², an e-folding time of 114 s. The classification is a comparison of two slopes and nothing else: no density appears in it, and the same cold air is stable under one profile and unstable under another.

The layer a parcel cannot leave

Whether a column of air overturns is not decided by its density but by a difference of two gradients — the rate the environment cools with height, and the rate a lifted parcel cools on its own. Subtract one from the other and what is left is a restoring force per unit displacement, so a stable atmosphere rings at a period of minutes and an unstable one has no period at all.

fluids · Stratification The deepest water reads warmer than the water above it. A column of seawater 10.9 km deep whose potential temperature — the temperature each parcel would have if brought to the surface without gaining or losing heat — falls from 20 °C at the top to 1 °C in the deep, and the temperature a thermometer lowered into it would read. The two agree near the surface and part company with depth: the thermometer reads a minimum of 1.38 °C near 3.9 km and then rises to 2.35 °C at the bottom, although the water there is the same water, with the same potential temperature, 1.00 °C, as the water two kilometres higher. The warming is compression alone; no heat reaches the deep trench from anywhere.

The water that warms on the way down

Lower a thermometer into the Mariana Trench and it reads colder and colder for four kilometres and then starts to warm, reaching about two and a half degrees at the bottom, eleven kilometres down. No heat is coming up from the sea floor to do it. The water at the bottom is the same water as the water two kilometres higher, carried down and squeezed, and squeezing warms it. In fresh water near four degrees the same squeezing does nothing at all, and below four degrees it cools — so a parcel of cold lake water pushed deep enough keeps sinking on its own.

fluids · Hydrostatics An orbit that flattens without being told to. The path of a classical electron relative to its nucleus, in the laboratory, before and after the nucleus has been accelerated gently — over 200 orbits — from rest to 0.6 of the speed of light along the horizontal, computed from Maxwell's field of a moving charge and the relativistic law of motion alone. Before, the orbit is a circle of radius 1. After, it is an ellipse 0.810 wide along the motion and 1.012 across it, a ratio of 0.8003 against 1/γ = 0.8000, and one turn takes 1.2504 times as long as before, against γ = 1.2500. Nothing in the calculation assumed a contraction or a slowing clock: the forces between the moving charges produced both.

The contraction the forces work out for themselves

Relativity usually introduces length contraction as a consequence of how moving observers disagree about simultaneity, not as something that happens to matter. There is a second way to reach the same number, and it is worth knowing because it is the way the idea was first found. Take an atom, accelerate its nucleus gently, and compute what the electron does using only Maxwell's field of a moving charge and the law of motion. Nobody tells the orbit to contract, and it does, by exactly 1/γ, while each turn takes exactly γ times as long. The forces holding matter together work out the contraction for themselves.

relativity · Length contraction 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

Named alongside it

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

Heat capacityAdiabatic indexDegrees of freedomEfficiencyEquilibriumEquipartitionSimple harmonic motionTemperatureArrhenius lawAtomic structureBulk modulusBuoyancy

All concepts