Thermodynamics
Why Heat Flows and Engines Run
Energy Never Runs Out
A cup of coffee left on a desk cools to the temperature of the room. Its energy has not vanished. It has passed to the air, the desk and the walls, warming them by a tiny amount, and the total is just what it was. Energy is never made or destroyed: that is the first law of thermodynamics. It rules out the oldest dream of perpetual motion, a machine that makes work out of nothing.
Yet the cold coffee can no longer warm your hands, and the energy now spread through the room can do almost nothing at all. What was used up was not the energy. It was the difference: hot coffee in a cooler room. Thermodynamics is about what energy can do as it moves, and why it can do less and less.
Its laws do not care what things are made of. They were worked out before anyone could see an atom, and they have never been found to fail for anything large enough to see. Energy is kept in every single collision. The other laws follow from counting the ways that countless atoms can share it.
Heat and Work
Energy passes from one thing to another in two ways. Work is a push that moves something: gas shoving a piston, a spoon stirring a pot. Heat is energy that passes because one thing is hotter than another.
Heat itself travels three ways. It moves by contact, as a spoon in hot coffee warms all the way to its handle. It is carried along by moving fluid, as warm air rises from a radiator. And it travels as light, which needs nothing in between. Warm things around us shine infrared, light whose waves are too long for our eyes to see. The far hotter Sun sends about two fifths of its energy as visible light, and its light warms Earth across 150 million kilometers of empty space.
Touch is a poor thermometer. A metal railing and a wooden bench in the same winter air are at the same temperature, but the metal feels colder, because it carries heat away from your hand faster. Leave two things touching long enough, and heat flows until they are equally warm, and then stops.
Two things that each match a third must match each other. That is all a thermometer needs: it settles to the temperature of your tongue, and then shows its own. The rule sounds too plain to need saying, but every other law of heat leans on it, so it came to be called the zeroth law.
Work can warm a thing just as heat can, and the two are counted in the same units of energy. Let water fall 427 meters and turn all the energy of its fall into the jiggle of its molecules, and it comes out one degree warmer, whether it is a drop or a lake. Water falling the height of Niagara Falls warms by about a tenth of a degree.
Turning one into the other is easy only one way. Work can go wholly into warmth, with no effort: rub your hands, or brake a car. In an engine that keeps running, heat becomes work only in part, and only on one condition. That one-sidedness is what most of thermodynamics is about.
An Engine Runs on a Difference
A small engine of a kind sold as a desk toy runs on a cup of hot coffee. Its base takes heat from the cup and its top gives heat to the air. Inside, a loose-fitting disc, driven by the wheel, moves up and down and shuttles the sealed air between the hot base and the cool top. So the air is warmed and cooled by turns: it swells and shrinks, and pushes a small piston that turns the wheel.
Set the same engine on a table, and it stands still. The table holds far more energy than the coffee, but it is as warm as the air, so no heat flows through the engine. Put ice on its top, and it runs again, on heat that flows up from the table into the ice, until the ice has melted and warmed. Set it on a bowl of ice instead, and it turns the other way.
Every heat engine works this way. Heat flows through it from a hot side to a cold side, and the engine turns part of that flow into work. The rest goes out into the cold side, and part of it always must. That part is not waste that better engineering could save. An engine that kept it, turning heat wholly into work and changing nothing else, would break the second law of thermodynamics. It would be perpetual motion of a subtler kind: a ship could sail on the warmth of the sea and leave a wake of ice.
The reason is counting. Heat arrives through the jumbled jiggling of countless atoms, and work leaves as motion in step. Physicists call the number of ways the atoms can share their energy, or more exactly the number of digits in it, entropy. In anything large, entropy never shrinks overall, so the jumble cannot simply vanish: some heat must carry it out into the cold side. And a bit of heat multiplies the number of ways far more in a cold body, whose atoms have little energy to share, than in a hot one, as a coin means more to an empty purse than to a full one. So the colder that side, the less heat has to go.
So the best any engine can do is set by its two temperatures alone, counted up from absolute zero. The share of the heat that can become work is the difference between them, divided by the hot one. Between boiling water and a room at 20°C that is about a fifth, however perfect the engine. That is why engines run as hot as their metal can stand.
Only an engine with no friction, whose heat crosses no gap in temperature on its way in or out, reaches that ceiling, and such an engine would run infinitely slowly. Every gap wastes part of the difference, and the bigger the gap, the more. A flame of well over a thousand degrees that heats steam to 600°C throws much of it away before any work is done.
So the best gas-fired power station uses its heat twice. It burns gas in a turbine, then uses the turbine’s exhaust, still at about 670°C, to raise steam for a second one. Together they turn 64 percent of the fuel’s energy into electricity. That figure leaves out the energy the water vapor from the burning could still give up by condensing; counting it, the share is about 58 percent. The best road-car engines turn a little under half into work at their best, and much less in everyday driving.
Not every power source is a heat engine. Dams and wind turbines are not, nor are batteries and muscles, which turn a chemical difference straight into work at one temperature. The heat-engine limit does not bind them. A solar cell is not a heat engine either, but a version of the limit, set by the temperature of the Sun’s surface, still caps it, far above what any cell reaches. What limits a battery is its free energy: the energy left once the count of ways has taken its share. That share can even pull heat in. An instant cold pack chills itself as the salt inside dissolves, because the dissolved salt gains so many ways to arrange itself that it can afford to draw heat from your hand.
On average, the rule holds even for an engine made of one atom. Physicists held a single electrically charged calcium atom in a funnel of electric fields, and warmed and cooled it by turns. Warm, it jiggled wider and the funnel’s slope pushed it toward the open end; cooled, it slid back. Timed with the heating, those pushes built up a swing along the funnel, the way pushes build up a swing in a playground. Of the heat it took in, the atom turned a few tenths of a percent into motion, and it did so only because it had a hot side and a cold side.
Running Heat Uphill
In anything large enough to see, heat never flows from cold to hot on its own. It can be carried there, at a price, and a fridge does it all day: it takes heat out of its cold box and gives it to the kitchen through the warm grille at its back. A heat pump is the same machine turned around to warm a house. It takes heat from the air outdoors, even freezing air, and gives it to the rooms.
The trick is a fluid that boils at very low temperatures, and the lower the pressure, the colder it boils. A narrow valve lets it into a coil where the compressor keeps the pressure low, and there it boils at about minus 15°C, colder than the winter air. So heat flows from the air into it, and that heat is what boils it. A compressor squeezes the vapor, and squeezing warms it, until it is hotter than the room. Then heat flows out of it into the house, and it turns back into liquid.
At each coil, heat flows downhill, from warmer to colder. The compressor’s work lifts it from the outdoor level to the indoor one. Because the work only lifts the heat and does not make it, a heat pump gives out more heat than the electricity it uses. In real homes it gives about three times as much over a winter, where an electric heater gives exactly as much as it uses. A small lift costs less than a big one, so a heat pump does best on mild days, warming radiators that need only lukewarm water.
A heat pump is a heat engine run backward. An engine takes heat from the hot side, turns part of it into work and gives the rest to the cold side. A heat pump takes work and heat from the cold side, and gives both to the hot one.
The lift costs more the colder the cold side. Pulling heat out of a freezer and into the kitchen takes a little work for each bit of heat. Pulling it out of something at a tenth of a degree above absolute zero and into the kitchen takes, even in a perfect machine, nearly three thousand times as much work as the heat it pulls out.
A Floor That Cannot Be Reached
There is a coldest possible temperature, absolute zero, at minus 273.15°C. The negative temperatures that some magnetic systems reach are not below it: they are hotter than any positive one. At absolute zero a body has given up all the energy it can give up. Even then its atoms are not perfectly still: quantum mechanics leaves each one a spread in where it is and how fast it moves, which cannot be taken away.
Every way of making something colder than everything around it works by changing some setting and letting the body adjust. In magnetic cooling, a strong field lines up the tiny magnets of a salt’s atoms while the heat this gives off is carried away. Then the salt is sealed off and the field switched off. As its magnets scatter, they take over part of the jumble from the jiggling atoms, and the salt gets colder.
But as a body nears absolute zero, its entropy falls to the same smallest value whatever the setting: for a perfect crystal, with a single arrangement left, that value is zero. That is the third law of thermodynamics. The settings stop making a difference, so each step cools by less than the last, and reaching zero would take endlessly many steps.
Real cooling goes down in stages, each with its own method. The chips of a quantum computer sit in a refrigerator built as a stack of plates, each colder than the one above it: about 50 degrees above absolute zero, then 4, then about 1, a tenth and a hundredth. Every plate shields the next from heat flowing down from the one above.
The coldest atoms ever measured moved as slowly as those of a gas at 38 trillionths of a degree above absolute zero. Physicists dropped a cloud of atoms, with the machine holding it, 110 meters down a tower, and while it fell, used a brief pulse of magnetic field as a lens that stopped its atoms drifting apart. Even that is a step, not an arrival.
From Stars to Living Things
The same laws hold far from any kitchen, sometimes with a twist. A star is held up by the pressure of its hot gas against its own gravity, and it behaves backward: when it loses energy, it gets hotter. Its gas shrinks under its own weight, and the fall sets free twice the energy the star lost: half makes up the loss, and half warms the gas. This keeps a star steady. If its core burns too fast, the star swells and cools until the burning slows; if too slowly, it shrinks and heats until the burning picks up.
A black hole is backward too. Theory gives it a temperature, and the bigger it is, the colder. One with the mass of the Sun would be about 60 billionths of a degree above absolute zero, far colder than the faint glow that fills space. So it takes in more heat than it gives out, and as it grows, it gets colder still. Its own glow is far too faint to have been seen. One part of the theory has been checked: when two black holes merged, the total area of their surfaces grew, as the theory’s version of the second law demands.
Near the Sun’s surface, the gas itself carries the energy up. Hot gas rises in cells about the size of a country, gives up energy to space as light, and sinks again, cooler, at their darker edges.
Earth sends back into space almost exactly as much energy as it gets from the Sun; the small surplus is what is warming it now. What Earth lives on is a difference in temperature. Sunlight comes from a surface at about 5,500°C. Earth reflects about three tenths of it straight back, and gives the rest back as infrared, mostly from high in its air, at about minus 18°C on average: some twenty particles of light for every one it absorbed. Plants, and everything that eats them, live on that difference, much as the little engine lives on the coffee’s.
Four Short Laws
Put together, the laws are short. Two things that each match a third in temperature match each other. Energy is never made or lost. Differences run down on their own, and work can be had only from a difference. Absolute zero can be neared but never reached.
They hold for coffee cups, power stations and stars, and on average for single atoms. Theory says they hold for black holes too, and one of them has been checked there. Only universe as a whole bends the rule on energy: as it stretches, light crossing it arrives with less energy than it set out with. Across universe the differences are running down: stars burn their fuel, and their light carries the energy off into the cold of space. How far that goes, over times far longer than universe’s present age, is a question cosmology is still working on.
