Illustrated Physics
physics, drawn
Concept

Latent heat — where it appears

The energy a substance absorbs to change phase at fixed temperature, which no thermometer registers and which sets the slope of the boundary curve. It goes into breaking the bonds rather than into raising the kinetic energy, and it falls to zero at a critical point where the two phases have become one.

Named by 18 essays across 2 fields — each of them below, with the objects they name alongside it.

Heating 1 kg of water from -20°C to 130°C. Temperature against heat added for 1 kilogram of water taken from -20 to 130 degrees Celsius. The two flat stretches are the melting and the boiling, where 334 and 2260 kilojoules go in and the temperature does not move. Melting costs as much as warming the water by 80 degrees; boiling costs as much as warming it by 541, which is 73 per cent of the whole journey.

The heat that changes no temperature, and where it actually goes

A kettle reaches a hundred degrees in a minute and takes five more to boil dry. The heat going in during those five minutes changes nothing a thermometer can see, and it is most of the energy in the whole process.

thermodynamics · Phase change Ways to arrange 12 coins. The number of distinct arrangements giving each number of heads, for 12 coins. Every individual arrangement is equally likely; the middle wins because there are more ways to reach it.

What a system actually minimises

A ball falls to the bottom of a bowl and a gas fills a room, and neither of those is the rule. A system in contact with a large reservoir minimises U − TS, and the minus sign is the reservoir's own entropy written in the system's variables — which is why a rubber band pulls harder when it is heated.

thermodynamics · Entropy The phase boundary of water, from one equation. Pressure against temperature for water on a logarithmic pressure axis spanning 6.6 decades. The vaporisation curve is integrated from Clausius–Clapeyron between the triple point at 273.16 kelvin and 611.7 Pa and the critical point at 647.096 kelvin, with a single latent heat of 43.32 kilojoules per mole — the value the two published points on the curve imply. The measured latent heats are 45.05 at the triple point and 40.65 at the reference point, and the fitted value sits between them, because a constant latent heat is an average over the interval. The sublimation curve below the triple point is not measured but predicted, from the two latent heats adding where all three boundaries meet: 51.1 kilojoules per mole, which reaches 103.2 Pa at 253.1 kelvin against a measured 253.15. The melting curve is drawn at the slope Clapeyron gives it, -13.5 megapascals per kelvin, which is a volume ratio and nothing else: water's solid is 917 against 1000 kilograms per cubic metre for its liquid, so melting shrinks it and the line leans backwards. Across the whole 6.6 decades of this axis that line moves 5.2 kelvin, and one atmosphere shifts the melting point by 0.0075 kelvin. At 1 atmosphere the boundary is crossed at 373.1 kelvin, where water boils. The one place the curve fails is its top end: a constant latent heat reaches 37.4 MPa at the critical temperature where the measured critical pressure is 22.1 MPa, 70 per cent high, because the latent heat falls to zero at the critical point and this curve does not know that.

A boiling point is a pressure, not a temperature

Nothing about water names one hundred degrees; the air does. Where a liquid turns to vapour in its bulk is fixed by what pushes on it, which makes the familiar figure a coordinate on a curve — 71 °C on Everest, 119.5 °C in a sealed pot — and one latent heat draws the whole curve.

thermodynamics · Phase change The ceiling, inverted. How many joules of heat a perfect machine can move per joule of work, against the outside temperature, with the inside held at 21 °C. The upper curve is heating — T_h/(T_h − T_c), which is what the Carnot argument becomes when the cycle is run backwards — and the lower one is cooling the outside, T_c/(T_h − T_c). They differ by exactly one everywhere, to 1.8e-15 across the whole range as drawn, because the work put in is delivered as heat along with whatever was moved. The dashed line at one is a resistive heater, which is 100% efficient and is the worst option on the figure. At 7 °C and −7 °C the ideal coefficients are 21.0 and 10.5; a real machine reaching 25% of the ideal gets 5.3 and 2.6, which is still several times what burning the same energy would give.

The engine that pays back more than it takes

Carnot's argument puts a ceiling on how much work a flow of heat can be made to do. Run the same cycle backwards and the ceiling inverts into a floor that is greater than one — so a machine can deliver three or four joules of heat for every joule it consumes, and a perfectly efficient electric heater is the worst way to warm a room.

thermodynamics · Heat engines The part of the curve no fluid follows. Van der Waals' isotherms in reduced units, at 5 temperatures either side of the critical one, so that nothing about any particular substance appears. Above the critical temperature the pressure falls monotonically as the volume grows, which is what a fluid does. Below it the curve develops a loop with a rising section in the middle, and that section says the pressure increases as the substance expands — a material with negative compressibility, which cannot exist, because any fluctuation would run away. What happens instead is drawn as the horizontal line: the substance separates into two phases at one pressure, and the volume moves along the line as the proportions change. Its height, 0.6470 of the critical pressure, is fixed by requiring the two areas the line cuts off to be equal, which is the condition that the two phases have the same Gibbs energy. It meets the curve at volumes 0.603 and 2.349, a ratio of 3.9, and those are the densities of the liquid and its vapour. The two turning points of the loop, at 0.72 and 1.53, bound the part that is not merely unobserved but impossible; between them and the construction the substance can be made to sit, superheated or supercooled, until something nucleates.

The part of the curve no fluid follows

One equation for a real gas produces isotherms with a rising middle section, which says a substance would expand as the pressure on it grows. Nothing does that. What replaces it is a horizontal line whose height is fixed by making two areas equal, and the condition is not a convenience.

thermodynamics · Phase change 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 Melting curves, and the one that leans the wrong way. Melting temperature against pressure for water, benzene, naphthalene, each measured from its own melting point at one atmosphere, with pressure in bars. The slope of every coexistence line is the latent heat divided by the temperature and the change in volume, and the latent heat of melting is positive for everything — so the sign of the slope is the sign of the volume change, and nothing else. Almost everything expands on melting and its line leans forwards. Water's solid is less dense than its liquid, so its line leans backwards at 135 bars a kelvin: pressing on ice at just below zero melts it, and it takes 135 atmospheres to gain a single degree. The anomaly is not in the thermodynamics; it is in the fact that ice floats.

The melting curve that leans the wrong way

The slope of any coexistence line is the latent heat divided by the temperature and the change in volume. Latent heat is always positive, so the sign of the slope is the sign of the volume change — and for water the volume change is negative, which is the whole of why ice floats and why the melting curve leans backwards.

thermodynamics · Phase change The boiling curve, which turns back. Heat flux from a horizontal surface into water boiling at one atmosphere, on a logarithmic axis, against how far the surface is above the boiling point, also logarithmic, from standard correlations. At a few degrees heat leaves by convection; from about 5 K bubbles nucleate and the flux climbs as the cube of the superheat, to the critical heat flux of 1.11 MW/m² at 20 K. Beyond it the flux FALLS as the surface gets hotter — the transition regime, drawn dashed as the interpolation it is — down to 19 kW/m² at 67 K, the Leidenfrost point, where a continuous film of vapour insulates the surface. Past that, film boiling and radiation let the flux rise again, slowly: 310 kW/m² at 1000 K above boiling. Between the crisis and the Leidenfrost point, a surface 3 times hotter passes 58 times less heat.

The boiling curve that turns back

Heat a surface under water and more heat flows from it the hotter it gets — up to a point. Past about twenty degrees above boiling the bubbles crowd together, merge, and begin to blanket the surface with vapour, and from then on a hotter surface passes less heat, not more, until by a few hundred degrees above boiling it is wrapped in an insulating film and passes fifty times less than it did at its best. Shigeo Nukiyama found the turn in 1934 by melting wires. It is why a water drop skates for a minute on a very hot pan but vanishes in two seconds on a merely hot one, and why every boiler, steam generator and water-cooled reactor is designed around a number called the critical heat flux.

thermodynamics · Phase change The temperature through growing lake ice. Temperature against depth below the ice surface in a lake under air 10 K below freezing, after 1, 4 and 16 days, from Stefan's law with the ice's top held at the air's temperature. The ice is 11.1, 22.3, 44.6 cm thick; within it the temperature rises in a straight line from −10 °C at the top to 0 °C at the bottom, and below it the water is at 0 °C throughout. Ice forms only at the bottom, where water meets ice at the freezing point, and the latent heat it releases there has to be carried up through all the ice above by conduction. The straight lines get shallower as the ice thickens: the same temperature difference drives heat across a longer path, so less heat escapes and the ice grows more slowly. Sixteen times as long gives four times the thickness.

The ice that grows more slowly the thicker it gets

A lake under air ten degrees below freezing grows eleven centimetres of ice on the first day and only another eighteen in the following six. The ice forms at its underside, and every gram of it releases latent heat that must be carried up through the ice already there before more can form: the ice is its own insulation. The thickness therefore grows as the square root of time, and the rule, Stefan's, depends on temperature and time only through their product — the freezing degree-days that lake-ice forecasters count. A few centimetres of snow on top slows it more than any amount of cold speeds it up.

thermodynamics · Diffusion How a solute moves both ends of the liquid range. The molar chemical potential of water against temperature near its freezing point (left) and its boiling point (right): liquid (blue), ice (green), vapour (red), each a line whose slope is minus its entropy, drawn relative to the liquid at the transition. A solute at mole fraction 0.05 — about 2.9 mol/kg — lowers the liquid's line by RTx (dashed) and leaves the ice and the vapour unchanged. The crossings move: the freezing point down by RTx divided by the entropy of melting, 5.16 K, the boiling point up by RTx divided by the entropy of boiling, 1.42 K. The same vertical shift moves the freezing crossing 3.6 times as far, because ice's line is nearly parallel to the liquid's — melting changes the entropy by 22.0 J/mol K — while the vapour's line crosses steeply, with 109 J/mol K between them. Lowering the liquid makes it the winner over a wider range of temperature at both ends: the solution is liquid from below 0 °C to above 100 °C.

The degree a spoonful of solute buys

Dissolve anything in water and the water freezes lower and boils higher. The two shifts come from one cause — the solute lowers the liquid water's chemical potential, so the liquid wins against ice and against vapour over a wider range — but they are not the same size. For every mole of particles per kilogram the freezing point falls 1.86 degrees and the boiling point rises only 0.51, three and a half times less, and the reason is in the entropies: ice and water are so alike that a small push moves their crossing a long way. The same arithmetic says salting pasta water does nothing a cook could notice, weighed molecules for a century of chemistry, and in 1887 showed that salts in water are made of ions.

thermodynamics · Chemical potential A parcel that leaves the air behind. A parcel of air lifted from the ground at 30 °C with a dew point of 23 °C, against height, beside the air it rises through, which cools at 6.5 K per kilometre to the tropopause at 15 km. Unsaturated, the parcel cools at the dry rate and is soon colder than its surroundings; at 890 m it reaches saturation, a cloud base, and from there cools along the moist adiabat. From 2080 m it is warmer than the air round it and accelerates upward, until at 13.4 km it is cold again. The shaded area, the buoyancy integrated over that climb, is the convective available potential energy: 2543 J/kg. If all of it went into motion the updraft would reach 71 m/s, the square root of twice the CAPE. The 99 J/kg below the level of free convection is the push it needs to get started.

The cloud that cools more slowly than the air

Dry air carried upward cools at 9.8 degrees per kilometre, and the atmosphere, which cools at about 6.5, is stable against it: a dry parcel nudged upward ends up colder than its surroundings and sinks back. Saturated air is different. As it rises its vapour condenses, the latent heat warms it, and it cools at only four to six degrees per kilometre in the lower atmosphere. The same column of air is then unstable to it. Most of the troposphere, most of the time, is stable until something lifts air to saturation, and then it is not — and that one asymmetry runs from a fair-weather cumulus to a thunderstorm, to the warm wind that comes down the far side of a mountain.

fluids · Stratification 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 The distance at which the air takes over. Evaporation of water at 20 °C into dry air, in kilograms per square metre per second, against the thickness of the layer the vapour must cross before it is carried away, both on logarithmic scales, with the kinetic and diffusive resistances added in series: 1/u + δ/D, where u = √(RT/2πM) is the speed at which the kinetic flux carries vapour off and D the diffusion coefficient. Dashed, the kinetic ceiling. At one atmosphere the two resistances are equal at δ = D/u = 0.17 μm, about two and a half mean free paths of the air; a layer much thinner than that leaves the surface at its ceiling, and a layer much thicker throttles it in proportion to its thickness. At a tenth of an atmosphere diffusion is ten times faster and the crossover moves out to 1.7 μm. Only in vacuum, or across gaps shorter than a mean free path, does the liquid itself set the rate.

The puddle that could dry in a second

A liquid in equilibrium with its vapour loses molecules exactly as fast as the vapour delivers them, and kinetic theory says how fast that is. For water at room temperature it is two and a half kilograms per square metre every second — a puddle two and a half millimetres deep gone in a second. Puddles take hours because almost every molecule that leaves is knocked back by the air within a fraction of a micrometre. The ceiling binds only where the vapour can escape in a single free flight: in vacuum, where water freezes itself, in the smallest droplets, and on the filament of a lamp.

thermodynamics · Kinetic theory 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 How much of supercooled water freezes in an instant. The fraction of a body of supercooled water that freezes at once when it is nucleated, against how far below 0 °C it had been cooled: the ice that forms releases latent heat until the whole of it is back at 0 °C, and freezing stops there until heat is taken away. Solid: with water's measured heat capacity, which rises steeply below about −20 °C; dashed: with the heat capacity it has at 0 °C. At −15 °C, 19.1 per cent freezes at once; at −20 °C, 25.7 per cent; at −38 °C, the coldest liquid water reaches before it freezes by itself, 51 per cent. The rest stays liquid at 0 °C. Water would have to be cooled about 80 degrees for all of it to freeze at once, and no liquid water survives that long.

The supercooled water that only partly freezes

Water can be cooled well below 0 °C and stay liquid, and when a crystal finally starts, the whole bottle seems to turn to ice at once. It does not. Freezing releases heat, and with nowhere for that heat to go in the instant of freezing, it warms everything back to 0 °C — at which point freezing must stop. Water cooled to −15 °C freezes about a fifth at once; to −38 °C, the coldest liquid water gets, about half. The rest has to wait for the surroundings to take the heat away, and a hand warmer is the same arithmetic run above body temperature.

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 Newcomen's dilemma. Two consequences of how cold a Newcomen engine's injection spray leaves its cylinder, against that temperature: the pressure difference the atmosphere then pushes the piston down with, one atmosphere less the vapour pressure of the water left in the cylinder; and the steam the next stroke wastes reheating the cylinder's iron skin, as a fraction of the most it wastes at 30 °C (8.2 cylinder-fulls). A cylinder cooled to 40 °C gives a push of 0.93 atmospheres and wastes 7.0 cylinder-fulls of steam a stroke; one left at 70 °C wastes half as much but pushes with only 0.69. Every choice was bad, because the one vessel had to be both hot, to receive the steam, and cold, to make the vacuum. Newcomen's engineers ran it in between.

The cylinder that must not be cooled

The first steam engines that did useful work, Thomas Newcomen's pumps of 1712, made a vacuum by spraying cold water into a cylinder full of steam. It worked, and on every stroke it cooled the cylinder, so that the next charge of steam condensed on the cold iron until the walls were hot again — several cylinders' worth of steam spent heating metal that the next spray would cool. James Watt saw in 1765 that the vessel was being asked to be hot and cold at once, and moved the cold to a separate condenser. The improvement was a factor of three or four in coal. A century later the reason had a name: heat passing from hot to cold across a large difference, doing nothing on the way, is work thrown away.

thermodynamics · Heat engines 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

Named alongside it

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

EntropyPhase transitionClausius clapeyronNucleationVapour pressureChemical potentialEquilibriumTemperatureFree energyMetastabilityBoiling pointThe Boltzmann factor

All concepts