Module 8 · thermodynamics
Module 4 built a gas out of bouncing balls. This module asks the harder question: why does heat only ever flow one way? Every other law of physics works equally well backwards. This one looks as though it doesn't — and the reason turns out to have nothing to do with forces and everything to do with counting.
Temperature, heat and internal energy are three different things, and almost every confusion in this subject comes from mixing them up. Get them straight now and the rest of the module is easy.
| Word | What it actually means | The test |
|---|---|---|
| Internal energy | the total energy of all the jiggling molecules in an object | depends on how much stuff there is. |
| Temperature | the average kinetic energy per molecule (Module 4: ⟨½mv²⟩ = ³⁄₂kT) | does not depend on how much stuff there is. A cup of boiling water and a boiling swimming pool are both at 100 °C. |
| Heat | energy on the move between two bodies because there is a temperature difference between them. Left alone it always crosses from the hotter body to the colder one; the only way to send it the other way is to do work on the system, which is exactly what a fridge does (Bench 4). | a verb pretending to be a noun. An object does not "contain heat"; it contains internal energy. Heat is what crosses the boundary. |
Which contains more internal energy: a cup of boiling water at 100 °C, or an iceberg at −10 °C?
Two things happened there, and only one of them is obvious.
The obvious one: energy flowed from hot to cold until both sides sat at the same temperature. That final shared temperature is not the average of the two starting temperatures unless the blocks are the same size — it is the weighted average, weighted by how much energy each block needs per degree. That quantity has a name: heat capacity, and per kilogram it is the specific heat capacity c.
Water's absurd heat capacity is why life is possible and why Chennai is not Rajasthan. It takes five times as much energy to warm a kilogram of water by one degree as a kilogram of sand, so the sea warms slowly all morning and cools slowly all night, and the coast stays within a few degrees of itself while the inland desert swings by twenty. It is why your body — mostly water — does not cook when you run, why a hot-water bottle beats a hot-sand bottle, and why car engines are cooled with water rather than anything cleverer.
The unobvious thing that happened: the energy went one way. Nothing in Newton's laws forbids the cold block spontaneously giving up energy to the hot one and getting colder still. Run a film of two colliding molecules backwards and you see two colliding molecules — perfectly legal physics. Yet a film of a cold block warming a hot one would be instantly recognisable as running backwards. Something is going on that is not in the force laws at all, and Bench 3 is where we find out what.
Two objects are in thermal equilibrium when they have been in contact long enough that no net energy passes between them any more. The zeroth law says that if A is in thermal equilibrium with B, and B with C, then A is in equilibrium with C. It sounds too obvious to be worth saying. It is in fact the statement that makes thermometers possible: every object carries a single number — call it temperature — such that two objects with the same value exchange no net heat, whatever they are made of. Without it, "this mercury column reads 37" would tell you about the mercury and nothing about the patient. It was named the zeroth law because it was recognised as necessary after the first and second laws were already famous, and physicists refused to renumber them.
Put a pan of ice on a steady flame and plot the temperature as the energy goes in. You would expect a straight climb. What you get has two flat shelves in it. Very little else about the physical world matters as much as those two shelves do.
During a shelf, energy pours in and the temperature does not move. The energy is not being wasted, and it is not being stored as motion. It is being spent breaking the bonds that hold the molecules in place. In ice, each water molecule is locked into a crystal by its neighbours; to melt it you must pay the bill to release every one of them, and until the last one is free the temperature cannot rise. That bill is the latent heat — latent meaning hidden, because a thermometer cannot see it. Not gone, though: freezing the water back hands the same energy out again, which is why the frost-protection row below works at all.
| Consequence | Why |
|---|---|
| Sweating cools you | every gram of sweat that evaporates takes about 2,400 J with it (the bill is a little higher at skin temperature than at 100 °C), and it takes that energy from your skin. This is the single most effective cooling mechanism the human body has, and it stops working when the air is already saturated — already carrying as much water vapour as it can hold — which is why humid heat is dangerous in a way that dry heat is not. |
| A steam burn is far worse than a boiling-water burn | both arrive at 100 °C, but the steam must first condense on your skin, dumping 2,260 J per gram before the cooling even starts. |
| Ice in a drink works so well | a gram of ice at 0 °C absorbs 334 J becoming water at 0 °C — as much as cooling 80 g of water by one degree. Ice-cold water without the ice is a much poorer coolant. |
| Frost protection in orchards | farmers spray water on the trees before a freeze. As it freezes it releases 334 J/g, holding the buds at exactly 0 °C — cold, but not colder. |
| A pressure cooker | raising the pressure raises the boiling point above 100 °C, so the boiling shelf sits higher and the food cooks faster. Module 4's gas laws, doing kitchen work. |
Three mechanisms, and almost every everyday thermal question is about telling them apart.
Put a bowl of water in the freezer with a thermometer in it, and read the temperature every ten minutes. It drops steadily to 0 °C — and then stops, for a long time, while the ice forms, before dropping again. You are watching a latent-heat shelf in your own kitchen. (If it drops a little below zero and then jumps back up to zero, congratulations: you have supercooled the water. Very still, clean water can be cooled below 0 °C without freezing, because the first ice crystal needs a speck of something to grow on. When freezing finally starts, the latent heat it releases warms the bowl straight back up to zero — which is the jump you just saw.)
Here is the puzzle, stated as sharply as it can be. Every fundamental law of motion we have met works identically forwards and backwards in time. Newton's laws, gravity, the bouncing collisions of Module 2 — reverse every velocity and the whole film runs backwards and every law is still obeyed. And yet:
Where does the direction come from, if not from the laws? The answer, and it is one of the deepest ideas in science, is that it comes from counting.
Ten gas molecules bounce around in a box. At any instant each one is equally likely to be in either half, and takes no notice of where the others are — so one snapshot of the box is like ten independent coin tosses. What is the chance that all ten are on the left at the same instant?
Run it with 6 molecules and watch the counter. Every so often — every minute or so — all six are on the left again, and the "all on the left" light comes on. Nothing has been violated; it is simply a 1-in-64 event, and if you wait, 1-in-64 events happen.
Now drag the slider to 200 and try again. The light never comes on. Not because it is forbidden, but because the chance is (½)²⁰⁰ ≈ 1 in 10⁶⁰, and you would wait far longer than the age of the universe. And 200 is a laughably small number of molecules: a real thimble of air holds about 10¹⁹ molecules, and a lungful about 10²². For that many, the probability is not merely small; it is a number with no meaning left in it.
So the second law of thermodynamics is not a law like Newton's F = ma, where F is a force. It is a statement about overwhelming odds, and it is worth writing out in full. In a system that is isolated — nothing going in, nothing coming out, nobody doing work on it — entropy is overwhelmingly likely to rise and overwhelmingly unlikely to fall. You have just watched it fall: with six molecules the light came on about once a minute, and nothing was broken. With the 10²² molecules of anything real, unlikely becomes so strong a word that never decreases is the honest way to say it. Both sentences are the same sentence, counted for different numbers of molecules. Things go from arrangements there are few of to arrangements there are many of, because there are more of the latter. That is the whole mechanism. Heat flows from hot to cold for exactly the same reason: there are vastly more ways to share energy evenly among all the molecules than to keep it concentrated in half of them.
And this is where the arrow of time comes from. Not from the laws of motion, which have no preferred direction — and not from the counting either, since a system in an unlikely state now was probably in a likelier one a minute ago just as it will probably be in a likelier one a minute from now, so the counting alone points both ways in time. It comes from the fact that the universe began in an extraordinarily improbable, low-entropy state, and has been wandering toward more probable ones ever since. Every time you remember something, break something, or grow a day older, you are riding that wander.
No, and the word that saves us is isolated. A seed becoming a tree is a spectacular local decrease in entropy — and it is paid for many times over by the entropy the tree exports: warm air, water vapour, waste, and above all a swap of light for light. Sunlight arrives as a few very energetic photons from one small hot patch of sky — few ways to arrange that, so low entropy. The tree returns the same energy as many feeble infrared photons in every direction — enormously more ways, so high entropy. By the counting rule of Bench 3 that swap alone is a large entropy increase, and the tree's tidiness is small change beside it. The Earth is not an isolated system; it sits in a stream of order flowing from a very hot Sun into very cold space, and life is a small eddy in that stream. Add up the entropy of the Earth and its surroundings and the total goes up, as always. You are permitted to tidy your room. You are not permitted to do it without getting warm.
An engine takes in heat from something hot, does useful work, and dumps the remainder into something cold. Petrol engine, steam turbine, jet, your body: all the same shape — and it is Module 5's first law, ΔU = Q + W, run as a cycle: after one full cycle the engine's own internal energy is back where it started (ΔU = 0), so whatever heat went in has to come back out, either as work delivered or as heat dumped in the cold sink. Nothing else is allowed to happen to it. The obvious question is how much of that heat you can turn into work. The answer is one of the most surprising results in physics. It was worked out in 1824 by a 28-year-old French engineer named Sadi Carnot, before anyone even knew what heat was.
One more distinction before you predict. Heat is energy shared out randomly among countless molecules, each going its own way. Work is energy marching in step — a piston moving one way, a shaft turning one way. Bench 3's counting applies to that difference too: there are astronomically more ways to arrange energy in the first form than in the second.
Why can't an engine convert all the heat it takes in into useful work — not even a perfect one, with no friction anywhere?
Bench 3 counted arrangements; Bench 4 needs a way to turn those counts into joules and kelvins. One equation does that:
Three readings of that formula are worth having.
A modern coal power station burns at about 830 K and exhausts at about 300 K, so its Carnot limit is 64%; it achieves around 40%, and the remaining 60% of the coal's energy leaves as warm water and warm air. That is not incompetence — it is the second law, and it is why power stations are built next to rivers. A car engine manages around 25–30%. Your own body converts food energy to mechanical work at about 20%, which is why hard exercise makes you hot rather than efficient. And a heat pump, which is a fridge run to warm your house, can deliver three or four joules of heat for every joule of electricity, because it is not creating the heat — only carrying it in from outside. That looks like magic and is merely the Carnot formula run in reverse: pumping heat from 273 K into a 293 K room needs at best (293−273)/293 of the delivered heat as work — about one joule for every fifteen delivered — and real heat pumps do somewhat worse.
1. You take a metal spoon and a wooden spoon out of the same drawer. The metal one feels colder. Why?
2. It takes about 335 kJ to melt 1 kg of ice at 0 °C into water at 0 °C. During that process, the temperature of the mixture:
3. An inventor shows you an engine that takes in heat at 400 K, exhausts at 300 K, and converts 40% of the input heat into work. Your response: