Illustrated Physics
physics, drawn
Concept

Clausius clapeyron — where it appears

The relation giving the slope of a phase boundary in pressure and temperature as the entropy change of the transition divided by its volume change. It explains why boiling points rise with pressure, why ice melts under pressure, and why helium-3 freezes when heated.

Named by 8 essays across one field — each of them below, with the objects they name alongside it.

Three curves that can only meet at a point. Water's phase diagram within a kelvin of its triple point, with each boundary drawn at the slope Clausius and Clapeyron give it from measured latent heats and densities: 44.4 Pa/K for boiling, 50.3 for sublimation, and -135 bar per kelvin for melting — negative, and so steep that it is drawn vertical here, because ice is the less dense phase and pressure therefore melts it. Two things follow that no amount of measurement could adjust. The three slopes are not independent: the sublimation and vaporisation curves differ in slope by 5.92 Pa/K, which is exactly what the fusion latent heat predicts, because going solid → gas directly and solid → liquid → gas must cost the same enthalpy. And the sublimation curve is the steeper of the two, so the two cross rather than touch — which is why the triple point is a crossing and not a tangency, and why ice sublimes below it and melts above it.

Why the triple point is a point

Three phases of one substance coexist at one temperature and one pressure and nowhere else, and the reason is arithmetic rather than chemistry: count the numbers that describe the state, count the conditions equilibrium imposes, and subtract. The same subtraction says four phases of one substance are impossible, and it says so without knowing what the substance is.

thermodynamics · Phase change The melting curve that turns back, then lies flat. The pressure at which liquid and solid helium-3 coexist, against temperature from 0.9 millikelvin to 1 kelvin on a logarithmic axis, from the international provisional low-temperature scale PLTS-2000, which is defined by this curve. Above 315.2 mK the curve behaves like any other: a higher temperature needs a higher pressure to freeze the liquid. Below it the curve turns back — at its minimum, 2.93113 MPa, it takes a higher pressure to freeze colder helium — and it rises to 3.4391 MPa at 1 mK, where it lies almost flat, as the third law requires of every coexistence line. The kinks that mark the liquid becoming superfluid and the solid's spins ordering sit on that flat end, at 2.444, 1.896 and 0.902 mK.

The melting curve that has to arrive flat

The slope of every line on which two phases coexist is their difference in entropy divided by their difference in volume. The third law says that entropy differences vanish at absolute zero, so every such line — melting, boiling, a superconductor's critical field — must reach the bottom of the temperature scale lying flat. Helium-3 reaches it in the strangest way available: below a third of a kelvin its solid is more disordered than its liquid, the melting curve runs backwards, and squeezing the liquid into solid absorbs heat. That backwards curve was the refrigerator in which superfluid helium-3 was found, and it is now the definition of temperature below one kelvin.

thermodynamics · Third law The melting line of water, through five ices. The pressure at which water freezes, in megapascals on a logarithmic axis, against temperature, along each of the five solid phases that meet the liquid, from the international standard equations. Along ordinary ice, Ih, the melting pressure rises as the temperature falls, from 611 Pa at 273.16 K to 209 MPa at 251.165 K — the backwards slope of the familiar melting line. There the liquid meets ice III, and from then on every ice melts at a higher pressure as the temperature rises: ice V, ice VI, and ice VII, which reaches 20.6 GPa at 715 K. The dots are the four triple points where two ices and the liquid coexist. The lowest temperature at which liquid water can exist in equilibrium, 251.165 K, is at the first of them.

The melting line that turns round

Squeeze water and it freezes at a lower temperature — the backwards slope of its melting line, and the reason ice floats. Keep squeezing and the slope reverses. Past two thousand atmospheres ordinary ice gives way to a denser ice, then another, and from there on every ice melts at a higher temperature under more pressure, exactly as a normal substance's does. At twenty thousand atmospheres water freezes into an ice that is still solid at a hundred degrees Celsius; at two hundred thousand, one still solid at four hundred. The famous anomaly belongs to one ice out of five, and the same relation that explained it explains its end.

thermodynamics · Phase change Why silt heaves and sand and clay do not. The relative rate at which water is drawn to a growing ice lens, against grain diameter on a logarithmic scale, for three loads on the soil above it, in a capillary model: the rate is the soil's permeability, which grows as the grain diameter squared, times the suction the lens can sustain less the load. The suction is capped both by the supercooling at the lens, here 2 K, and by the Gibbs–Thomson pressure at which ice would break into the pores. Coarse soil has open pores that ice simply invades, so the suction never exceeds the load: under 10 kPa, half a metre of soil, nothing heaves above 58 μm. Fine clay can sustain the full suction but passes water too slowly to feed it. Between lies a fastest grain size, 29 μm under 10 kPa, 10 μm under 30 kPa and 2.9 μm under 100 kPa: silt, the soil engineers have long called frost-susceptible. Real heave pressures exceed this model's, and the fit is to the shape, not the scale.

The frost that lifts by drinking

Frozen ground rises in winter, sometimes by tens of centimetres, and the usual explanation — water expands by nine per cent when it freezes — is wrong: soil soaked in benzene, which shrinks as it freezes, heaves too. The lift comes from layers of clear ice that grow by pulling liquid water up through the soil from below. Ice and water touching each other need not be at the same pressure, and when the water is free to flow, ice a degree below its melting point can push with 1.12 megapascals while it drinks — twelve times less than freezing water in a sealed pipe, and enough to lift a house at a tenth of a degree.

thermodynamics · Chemical potential Latent heat of boiling against boiling point. The heat needed to boil one mole of a liquid at its normal boiling point, against that boiling point, both on logarithmic axes, for twenty-one substances from helium to sodium. The line is ΔH = 87.2 J/(mol K) × Tb, the mean entropy of boiling of the ordinary liquids boiling between 250 and 700 K, which scatter about it by 3.3 per cent; the ordinary liquids as a whole (blue), from neon at 27 K to sodium at 1,156 K, by 10.0 per cent, the coldest lying low. Water, ammonia and the alcohols (red) need more heat than the line predicts, acetic acid (green) less; the quantum liquids helium and hydrogen (violet) fall far below.

The entropy every liquid gains on boiling

Divide the heat it takes to boil a liquid by the temperature at which it boils, and for benzene, chloroform, acetone, hexane, ether and carbon tetrachloride the answer comes out within a few per cent of the same number — about ten and a half times the gas constant. Frederick Trouton noticed it in 1884. It is a statement about entropy, and once it is read that way the liquids it fails for become the interesting ones: water and the alcohols, whose hydrogen bonds hold the liquid in order; acetic acid, whose vapour is made of pairs; and helium, whose liquid is already as disordered as quantum mechanics allows.

thermodynamics · Phase change The boiling point in a column of water, and the depth heating reaches it first. The temperature at which water boils against depth in a geyser's conduit 2240 m above sea level, where the air's pressure is 77.6 kPa (solid): 92.7 °C at the surface, 103.6 °C at 4 m, 115.5 °C at 10 m and 131.8 °C at 22 m — rising steeply near the top and more gently below, because each extra metre adds the same pressure to water whose boiling point responds less as it gets hotter. Dashed: the water as it heats from beneath, cooled to 80 °C at the top by the pool and below that warming by only 0.8 °C per metre, mixed by convection; 6 °C and 3 °C short of the curve at 4 m, and then touching it. It touches at 4 m, not at the bottom: there the water is still 13.7 °C below boiling.

The column that boils from the middle

Water at the bottom of a geyser's tube is hotter than water anywhere else in it, and it is not the water that boils first. The weight of the column raises the boiling point with depth faster than the heat from below raises the temperature, so the column reaches its boiling curve partway down, under a cooler surface layer. A little boiling there lifts the water above. Everything beneath is then at a lower pressure than the one it was heated under, so it is superheated and flashes, and the eruption feeds itself: past about fourteen centimetres of water thrown out, each slug of steam makes more room than it was given.

thermodynamics · Phase change The pipe that out-conducts any solid. The temperature difference needed to carry 30 W along a rod 300 mm long and 6 mm across, on a logarithmic scale: for solid aluminium 1343 K, for copper 796 K, for diamond, the best conductor of heat at room temperature, 159 K; and for a water heat pipe of the same size 0.8 K, almost all of it across the thin wet wicks where the heat goes in and comes out. Solid metal could not do it at all: the hot end of the copper rod would have to be hundreds of degrees above the cold. The heat pipe carries the heat as latent heat in moving vapour, and the vapour needs almost no temperature difference to flow. As an effective conductivity that is 1000 times copper's — which is why every laptop and many satellites move their heat in pipes rather than through metal.

The pipe that carries heat as vapour

A copper rod six millimetres across and thirty centimetres long, asked to carry thirty watts from a laptop's processor to its fan, would need its hot end eight hundred degrees above its cold end. The flattened copper tube that actually does the job manages it with less than a degree. It is a heat pipe: sealed, emptied of air, and holding a few drops of water and their vapour. The water boils at the hot end, the vapour rushes to the cold end and condenses, and a wick of sintered copper powder drags the liquid back by capillarity. The heat goes as latent heat, and vapour needs almost no temperature difference to flow — which makes the pipe a thousand times better than copper, as long as its wick can keep up.

thermodynamics · Phase change The heat runs round in a circle through the wire. A section through a wire moving down through ice, not to scale. Under the wire pressure is high and the melting point is slightly below zero, so ice melts there, and that needs heat. Above it, water at low pressure re-freezes at slightly higher temperature, which gives heat out. The heat has to get from above the wire to below it, and it goes through the wire, so the rate of cutting is set by how fast that one short circuit conducts. The water squeezes round in a thin film. Ice closes behind the wire and it leaves the block whole.

The wire that goes through the ice and leaves it whole

Hang a weight on a wire over a block of ice, and the wire sinks slowly through it, leaving the block in one piece behind. The usual explanation is that pressure melts the ice under the wire. That is true and misses the point, because the pressure changes the melting point by a few thousandths of a degree. What limits the speed is how fast the wire can carry heat, and a copper wire goes through 160 times faster than a nylon line.

thermodynamics · Phase change

Named alongside it

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

Latent heatChemical potentialVapour pressureBoiling pointEntropyMeltingPhase diagramPhase transitionThermal conductivityTriple pointCapillary pressureCoexistence

All concepts