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

Put a kilogram of ice at $-20°\mathrm{C}$ on a hotplate delivering a steady kilowatt and watch a thermometer in it. For forty seconds the reading climbs. Then it stops at zero and sits there for five and a half minutes while the hotplate keeps delivering. Then it climbs again to a hundred, takes seven minutes to get there — and stops again, for thirty-eight minutes, while the water boils away.

The thermometer is not broken and no energy has been lost. It has gone somewhere a thermometer cannot see, and identifying where is the argument of this rung.

Temperature is not energy #

The plateaus are impossible if temperature means “how much energy is in there”, and the resolution is that it does not.

Temperature is a measure of the average kinetic energy per degree of freedom — of how fast the molecules are moving, not of how much energy the substance contains. Energy delivered to a substance can go into two quite different places: into making the molecules move faster, which raises the temperature, or into pulling them apart against the forces holding them together, which does not. A heat pump moves the second kind about, which is how it pays back more than it takes.

Heat that raises the temperature makes a molecule rattle faster inside the well it sits in; heat that melts or boils lifts it out of the well altogether. The depth of the well is the latent heat, and the distinction is the whole essay — temperature reports how fast things are moving within an arrangement, and a phase change is a change of arrangement, so nothing about the temperature need alter while it happens.

During melting, the energy goes into breaking the ordered arrangement of the solid: the molecules are not moving faster, they are further apart and less constrained. During boiling, it goes into removing them from each other’s attraction entirely, and against the atmosphere pushing back as the vapour expands. The temperature is fixed throughout because the molecules that remain in the liquid are moving no faster than before.

That is why the quantity is called latent — Black’s word, from the 1760s, meaning hidden. He identified it by exactly the experiment above, noticing that ice in a warm room takes hours to melt while the water it produces warms in minutes, and that this is unreasonable unless the melting is consuming something.

The size of the plateaus #

The numbers are worth carrying because they are much larger than intuition suggests.

Melting a kilogram of ice takes 334 kJ. Raising a kilogram of water by one degree takes 4.18 kJ. So melting the ice costs the same as heating the resulting water by 80 degrees — from freezing to nearly boiling.

Boiling costs 2,260 kJ, which is the same as heating the water by 541 degrees, a temperature interval that does not exist for liquid water at atmospheric pressure. Boiling a pan dry takes five and a half times as long as bringing it from cold to the boil, and every cook knows this as a fact about kettles without its being a fact about kettles.

The ratio between them is worth its own line: vaporisation costs 6.8 times as much as fusion. Melting has to defeat the arrangement of the molecules while leaving them touching; boiling has to separate them altogether, against the full depth of the attraction and against an atmosphere that has to be pushed aside to make room for a vapour a thousand times less dense. The larger number is the one that does the work in the world, and the expansion term is a real part of it: of the 2,260 kilojoules, about 170 are spent purely on shoving the surrounding air out of the way, which is why the latent heat of vaporisation falls as the ambient pressure falls and why water at the top of a mountain both boils cooler and takes slightly less energy to boil.

What the plateaus do for the planet #

Water’s latent heats are large — anomalously so, because of hydrogen bonding — and several familiar features of the world are consequences.

Sweat. Evaporating a litre of sweat removes 2,260 kJ from a body. That is the whole of human thermoregulation above about 35 °C ambient, at which point radiation and convection stop working because the surroundings are no cooler than the skin. The mechanism is a phase change and nothing else, which is why humidity matters more than temperature for how bearable a hot day is: the heat is not carried by the air, it is carried by the transition.

Storms. A hurricane is a heat engine whose fuel is latent heat: water evaporates from a warm ocean, is carried upward, condenses, and releases 2,260 kJ per kilogram at altitude. The energy released by a large storm in a day is comparable to the world’s annual electricity generation, and none of it would exist if condensing were a small effect.

Mild coasts. The sea holds an enormous amount of energy per degree, and the phase changes at its surface buffer it further, which is why maritime climates have small annual temperature ranges and continental ones do not.

Refrigeration. Every refrigerator and air conditioner is a device for moving latent heat: a fluid is boiled where heat is to be removed and condensed where it is to be dumped. The Carnot ceiling governs how much work that costs, and the latent heat governs how much heat is moved per kilogram circulated — which is why the choice of working fluid is a choice about the size of a plateau.

The boundaries in the pressure–temperature plane #

Melting at 0 °C and boiling at 100 °C are not properties of water. They are properties of water at one atmosphere, and the full statement is a map.

Three features of that map are worth reading.

The vaporisation curve ends. It stops at the critical point, 647 K and 22.1 MPa, beyond which there is no distinction between liquid and vapour at all — a substance can be taken continuously from one to the other by going round the end of the line, without ever crossing a boundary or absorbing a latent heat. The line has an end, which no boundary between genuinely different things could have.

The fusion curve leans backwards. For nearly every substance the solid is denser than the liquid and the melting curve leans forwards; for water it leans back, at $-13.5$ MPa per kelvin as Clapeyron’s relation gives, because ice floats. Pressure therefore lowers the melting point of ice, by about 0.0075 K per atmosphere. This is a genuinely small effect and it is routinely over-claimed: the pressure of a skate blade lowers the melting point by a fraction of a degree, not by the several degrees skating on ice at $-10°\mathrm{C}$ would require, and the modern account of why ice is slippery involves a surface layer that is liquid regardless of pressure.

The triple point is one point. Solid, liquid and vapour coexist at exactly one pressure and one temperature, 611.657 Pa and 273.16 K. That sharpness made it a defining fixed point of the kelvin until the 2019 redefinition, because it is a temperature reproducible in any laboratory with no calibration against anything else.

Why the boundary is sharp at all #

A plateau implies a sharp transition — one temperature at which the substance is changing state and no range around it — and that sharpness is a stronger claim than it looks.

Nothing about individual molecules is sharp. In ice at $-5°\mathrm{C}$, some molecules are momentarily energetic enough to escape the lattice and some in water at $+5$ are momentarily slow enough to be captured. The distribution of speeds is continuous and has tails at every temperature.

There is no speed above which molecules exist and below which they do not, and no temperature at which the tail of the distribution abruptly changes character. So the sharpness of a phase boundary does not come from anything sharp in the molecular motion — the distribution is smooth on both sides of the melting point and smooth through it.

What decides the boundary is a competition of counts. Below the melting point the ordered arrangement is cheaper in energy; above it the disordered one is overwhelming in number; and the crossover between “cheaper” and “more numerous” is abrupt because both quantities are exponential in the size of the system. Sharpness at ordinary scales comes from largeness, and a sufficiently small cluster has no melting point at all.

The sharpness is collective. A phase transition happens when the free energy of one arrangement drops below the free energy of another, and free energy is a property of the whole system: the balance between the energy cost of an arrangement and the number of ways it can be realised. Below the melting point the ordered arrangement wins on energy; above it the disordered one wins on entropy, because $T\Delta S$ has grown large enough — which is the quantity a system actually minimises; and at the melting point the two are exactly equal, so both can coexist in any proportion at no cost. That equality is what makes the plateau flat: adding heat converts some solid to liquid rather than raising the temperature, because the two phases have the same free energy and the system is indifferent between them.

That also gives the latent heat a second definition, which is the more fundamental one: $L = T\Delta S$. The heat absorbed at a transition is the temperature times the jump in entropy, so melting is the amount of disorder gained multiplied by the temperature at which it was gained.

Black’s measurement, and the engine that followed it #

Joseph Black established latent heat around 1761 by an experiment that is still the cleanest way to see it, and the sequel is one of the better examples of a measurement changing an industry.

His observation was that ice in a warm room melts slowly. If melting required no energy beyond that needed to bring the ice to zero, a block would collapse into water almost the moment it reached freezing point, and it does not — it sits there for hours absorbing heat at whatever rate the room can supply. Black measured the rate, measured how long the melting took, and got a number for the heat consumed. He repeated it for boiling and got a much larger one.

Two things follow that are worth separating. The first is that heat and temperature had to be recognised as distinct quantities before either could be measured properly, and Black’s experiment is what forced the distinction: before it, the two words were used interchangeably, and “quantity of heat” meant nothing precise.

The second is Watt. He was employed to repair a model Newcomen engine and found it consumed absurd quantities of steam, and — having attended Black’s lectures — he could say why. The Newcomen cycle admitted steam into a cylinder and then cooled the whole cylinder with a water spray to condense it, so every stroke had to reheat the cylinder walls back through the boiling plateau. The latent heat, being 6.8 times the heat needed to raise the water from freezing to boiling, was being paid twice per stroke and thrown away.

Watt’s separate condenser — condensing the steam in a second vessel kept permanently cold, so the working cylinder never cools — is a direct consequence of knowing the size of that plateau. The improvement was a factor of four in fuel, and it is a case of a quantity being measured in a laboratory and an engineering decision falling out of the number rather than out of trial.

The plateau that makes a thunderstorm #

The atmosphere’s most violent behaviour is a latent heat released halfway up, and the mechanism is worth following because it turns a stable arrangement into an unstable one without anything else changing.

A parcel of dry air pushed upward expands and cools, at very nearly ten degrees per kilometre. If the air around it cools faster than that with height, the parcel finds itself warmer than its surroundings and keeps rising; if slower, it sinks back. That comparison is what decides whether the atmosphere overturns.

Now let the parcel be moist. It cools as it rises until it reaches saturation, and from then on every further metre of ascent condenses a little water — which releases latent heat into the parcel and slows its cooling to something like five degrees per kilometre.

The consequence is that the same environment can be stable for a dry parcel and unstable for a saturated one. An atmosphere cooling at seven degrees per kilometre holds down anything dry and cannot hold down anything that has reached its condensation level, so a parcel lifted just far enough to start condensing accelerates away and keeps accelerating for as long as there is water in it. That is conditional instability, and every thunderstorm is one.

The energy involved is what the plateau’s size implies. A storm that condenses its water over ten kilometres of ascent has of order a thousand joules per kilogram available, which corresponds to updraughts of tens of metres a second — and it is all latent heat, delivered where the condensation happens rather than where the sun fell.

The same asymmetry has a gentler face. Air forced over a mountain range rises at the moist rate, drops its water as rain on the windward side, and descends the far side dry, warming at the full ten degrees per kilometre. It arrives at the bottom hotter than it set out, because it left its latent heat on the other side of the mountain.

Storing heat in a flat line #

The plateau’s flatness is a nuisance to a cook and a specification to an engineer.

A material that absorbs heat at constant temperature is a thermal store with two useful properties. It holds far more energy per kilogram than the same material merely warmed — melting ice absorbs the same as heating water through eighty degrees — and it does so at a fixed temperature, so whatever it is protecting stays where it is rather than drifting.

Both are exploited. Large buildings are cooled by freezing tanks of water overnight, when electricity is cheap and the condenser has cold night air to reject heat into, and melting them through the working day; the store is eight times more compact than a tank of chilled water would be for the same duty. Vaccine and food shipments travel with panels of a material chosen to melt at the temperature that must be held, so that the contents sit on a plateau rather than warming gradually. Waxes and salt hydrates with melting points near room temperature are built into wallboard, absorbing the afternoon’s heat and releasing it overnight.

The engineering difficulties are the ones this essay’s last section names. A material that supercools does not release its heat where it should, so nucleating agents are added; a salt hydrate that separates into its components on melting does not remelt properly and its capacity fades over cycles. The physics of the plateau is settled and the problem is making a substance sit on it a thousand times without changing.

What the curve cannot show #

The heating curve is the argument of this page and it hides three things by construction.

It has no time axis. The horizontal axis is heat added, not time, which is what makes the plateaus a statement about energy rather than about a hotplate. Redrawing it against time at constant power gives the same shape, and at variable power it does not — a fact worth stating because the domestic experience of the plateaus is entirely temporal.

Nothing is happening below the plateau. The flat stretch draws the substance as though it were a single object at 0 °C throughout. What is actually present is a mixture, whose proportions shift continuously from all-ice to all-water while the temperature does not move. The figure’s flat line is the whole of a process that has an internal state the drawing does not represent, and the fraction melted is the missing coordinate.

Evaporation is left out entirely. Water does not wait for 100 °C to become vapour; it evaporates at every temperature, because the speed distribution always has molecules in its tail with enough energy to escape the surface. Boiling is the special condition where vapour can form in the bulk rather than only at the surface, which requires the vapour pressure to reach the ambient pressure. The plateau at 100 °C is therefore not the point at which water starts turning to vapour but the point at which it can do so everywhere at once, and a curve of temperature against heat has no way to say so.

Where the model stops #

Sharpness needs an infinite system. Strictly, a phase transition is discontinuous only in the limit of infinitely many particles. A small enough droplet melts over a range rather than at a point, and nanoparticles have melting points that depend on their size — gold that melts at 1,064 °C in bulk melts a few hundred degrees lower at 2 nm across.

Superheating and supercooling. A liquid can be taken above its boiling point if there is nowhere for a bubble to start, because a very small bubble has so much surface that it is unstable and collapses. Clean water in a smooth container in a microwave routinely reaches several degrees above 100 °C and then boils explosively when disturbed. Water can equally be cooled to $-40°\mathrm{C}$ without freezing if it contains no nucleation sites. The phase diagram gives the equilibrium boundaries, and the kinetics of getting there is a separate subject.

Not every transition has a latent heat. Those on this page are first-order, with a discontinuous entropy and a plateau. Others — the ferromagnetic transition at the Curie point, the superfluid transition in helium — are continuous: no latent heat, no plateau, and a heat capacity that diverges instead. The whole subject of critical phenomena lives there, and none of it is visible in the figures here.

The constant latent heat is an approximation, and the figure prices it. The phase diagram’s boundaries are integrated with $L$ held fixed, which puts the boiling point 2.5 per cent high. $L$ actually falls with temperature and reaches zero at the critical point, which is the honest statement of why the curve has to end there and why the approximation gets worse as it approaches it.

Where the ladder goes next #

The rungs from here: the Clausius–Clapeyron relation derived rather than quoted; vapour pressure, and why a liquid evaporates below its boiling point at all; nucleation, and why the kinetics and the thermodynamics give different answers; the critical point and the disappearance of the distinction; second-order transitions and critical exponents; the Gibbs phase rule, which counts how many phases can coexist and gets the triple point’s uniqueness from arithmetic; and the free-energy argument in full, which is where the plateau’s flatness comes from.

The claim to carry forward is the first one. Heat and temperature are different quantities, and the clearest demonstration is a substance absorbing two thousand kilojoules while a thermometer in it reports nothing at all.