Illustrated Physics
physics, drawn
Concept

Thermal expansion — where it appears

The change in a body's dimensions with temperature, of order ten parts per million per kelvin for most solids. Prevented rather than allowed, it produces a stress of the modulus times the coefficient times the temperature change, which contains no length and no area — so a short member reaches the same stress as a long one.

Named by 6 essays across 3 fields — each of them below, with the objects they name alongside it.

The average position of something that is only shaking. The mean displacement of an oscillator against temperature, taken as a Boltzmann average over each well rather than from any expansion of it, with the temperature measured against each well's own depth so that unlike bonds can share an axis. A symmetric well gives exactly zero at every temperature — heating a harmonic solid makes it vibrate harder and does not make it larger. The others drift outward, because the outward side is the shallower one, and the measured slopes are a pendulum 0.000, a chemical bond 0.766, a pair of atoms 0.150. Thermal expansion is not a property a spring has; it is one a spring lacks.

Why heating a perfect spring changes nothing

A harmonic solid vibrates harder when heated and does not get any longer. Thermal expansion lives entirely in the term that the harmonic approximation throws away — and so does the fact that a solid conducts heat at a finite rate, which is the same discarded term doing a second job nobody would have connected to the first.

thermodynamics · Harmonic approximation The stress a temperature change puts into a bar that cannot move. The stress in a member held between supports that will not let it change length, against how much its temperature changes, for four materials. Each line is EαΔT, computed here from a free expansion and the force needed to undo it, and checked by evaluating it for a bar half a metre long and one thirty-seven metres long: the two agree to every figure carried, because neither the length nor the cross-section appears in the answer. steel develops 2.40 MPa for every kelvin and reaches yield at 104 K; aluminium develops 1.59 MPa for every kelvin and reaches yield at 151 K; concrete develops 0.30 MPa for every kelvin and reaches cracking at 10 K; invar develops 0.17 MPa for every kelvin and reaches yield at 1655 K. Concrete reaches its cracking stress after ten kelvin, which is less than a sunny afternoon, and is why every slab has movement joints in it. Invar is in the comparison because it was made to have a small product: it is as stiff as steel and develops a fourteenth of the stress, which is a statement about the expansion coefficient and nothing else.

The load nobody applied

A redundant structure develops forces with nothing on it. Change its temperature and the same extra constraint that made statics unanswerable also refuses the expansion — and a restrained steel member reaches its yield stress after a hundred and four kelvin, a figure that contains no length, no area and no load.

mechanics · Free-body 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 The gas that has to stop expanding. The thermal expansion coefficient at constant pressure times the temperature, αT, against temperature over the Fermi temperature, on logarithmic axes, for an ideal classical gas and an ideal gas of spin-half fermions at the same density. For the classical gas αT is exactly one at every temperature, so α = 1/T grows without limit as the gas is cooled, which the third law forbids: by a Maxwell relation α is minus the pressure derivative of the entropy per unit volume, and that must vanish at absolute zero. The Fermi gas agrees with the classical one when hot and departs from it near the Fermi temperature; at a tenth of it αT is 0.048, and at a hundredth 0.002, falling as π²/2 times the square of the temperature over the Fermi temperature. Its expansion vanishes as the law requires.

The two laws absolute zero will not allow

The ideal-gas law and Curie's law are the first two equations of state most people meet. The third law of thermodynamics says both must be wrong at low temperature, and it says so without any model of what replaces them. A Maxwell relation turns every derivative of the entropy into something measurable with a ruler or a magnetometer: the thermal expansion, the slope of the magnetisation with temperature. If the entropy stops depending on pressure and field at absolute zero, those must vanish. The classical gas expands as 1/T and Curie's magnetisation climbs as 1/T, and both slopes grow without limit instead.

thermodynamics · Third law The rise a watt-year buys at each depth. The rise in sea level, in millimetres, from one watt-year per square metre of heat (3.16 × 10⁷ joules) stored in a thin layer at each depth, for two water columns: a subtropical one, 24 °C at the surface cooling to 1.5 °C in the deep, and a polar one, 1.5 °C at the surface and 0.5 °C below. In the warm column the yield is 2.36 mm at the surface, falls to 1.02 mm near 2.0 km where the water is cold but the pressure still modest, and recovers to 1.49 mm at 6 km as pressure takes over. In the polar column the same heat buys 0.63 mm at the surface and 1.13 mm at 4 km. A joule stored in cold, deep water raises the sea about twice as much as one stored in the cold water above it.

The rise a joule of heat buys

Warm a column of seawater and it stands taller without weighing any more, so the sea rises while the pressure on the sea floor stays exactly the same. How much it rises depends on where the heat goes. Seawater's expansion coefficient grows with temperature and with pressure, so a joule stored in warm tropical water lifts the sea more than six times as far as the same joule in Arctic surface water. Cold water in the deep, squeezed by four kilometres above it, expands about twice as much per joule as the same cold water at the surface. That is why the budget of sea-level rise turns on thousandths of a degree in the least-measured part of the ocean.

fluids · Hydrostatics A gridiron of steel and brass drawn to the length that holds. The rods of a gridiron pendulum, to scale, for a seconds pendulum of 0.994 m: steel rods (dark) hang from the suspension and carry cross-bars from which brass rods (light) push upward, and from their tops the next steel rods hang down again, until the last carries the bob. Steel expands downward, lowering the bob; brass expands upward, raising it. They cancel when the steel's total length times its expansion equals the brass's: 2.52 m of steel against 1.52 m of brass, a ratio of 0.605. That is why gridirons are built of so many rods — the steel alone is 2.53 times the pendulum's length, folded into 3 stages. With zinc instead of brass, which expands 1.58 times as much, 1.61 m of steel would do. Real gridirons double every rod, side by side, so that the bob hangs symmetrically.

The pendulum that must not grow when it warms

A pendulum clock keeps time by the length of its pendulum, and every material grows when it is warmed. A steel rod a metre long lengthens by a hundredth of a millimetre for each degree, and the clock loses half a second a day — a minute a fortnight between a cold winter and a warm summer, far worse than anything the clockmaker's escapement could cause. The cure, found in the 1720s, was to let the pendulum grow in two directions at once: one part downward, another upward, in exactly the proportion that leaves the bob where it was. The same trick, with different metals, kept the best clocks to a hundredth of a second a day for two centuries, and its limits are set by heat that does not spread evenly or quickly.

mechanics · Pendulum

Named alongside it

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

CompressibilityPotential temperatureAdiabatic processAnharmonicityApproximationThe Boltzmann factorBucklingCentre of oscillationClockConstraint countingConvectionCurie law

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