A common assumption about engines
A student learns that a power station turns only part of the energy in its fuel into electricity. The rest passes to the surroundings by heating. The loss is large, often more than half. The natural conclusion is that it reflects imperfect engineering. A better engine would waste less, and a perfect one would waste nothing, turning all the energy it takes in into work.
No engine can do that. The energy an engine sends on to a colder place is part of what lets it do any work at all, and it cannot be kept. That an engine must have a cold side was recognised before anyone had a modern account of what heat is.
Carnot’s question
In 1824 a French engineer, Sadi Carnot, published a short book on the motive power of fire, meaning the useful work obtainable from heat. His advance was to change the question. Engine builders asked how much coal a machine burned. Carnot asked how much heat had to flow through it, and what that flow could be made to do.
He reasoned inside the theory of his day, which was caloric, the fluid the previous article overturned. The experiments that overturned it were still twenty years away. Yet his central finding did not depend on the theory being right.
Carnot pictured caloric flowing from the hot furnace down to a colder body, with the fall doing the work, much as falling water drives a mill. The picture was wrong about heat.
It carried a conclusion that was right. An engine needs its heat to fall from hot to cold, as a mill needs its water to fall. Without a difference of temperature to drive it, the flow yields no work at all.
Carnot put it plainly: “The production of heat alone is not sufficient to give birth to the impelling power: it is necessary that there should also be cold; without it, the heat would be useless.”
From the same reasoning he drew a second conclusion. The best an ideal engine can do depends only on the two temperatures it works between. Whether the working substance is steam or air makes no difference.
An apparent contradiction
There is a difficulty here. Joule’s experiments had shown that heating a body and doing work on it are two ways of changing one thing. The energy an engine takes in by heating is energy like any other, the same in kind as the energy it gives out as work.
If that is so, it is not clear why the engine cannot give out the whole of it. Nor is it clear why any must pass to a colder place.
Carnot’s own picture sharpened the difficulty, because in it nothing is converted. A fixed quantity of caloric enters at the top and leaves at the bottom unchanged, having done its work by falling. That picture and Joule’s could not both be right, and reconciling them fell to someone else.
Clausius’s reconciliation
In 1850 Rudolf Clausius found the way through, and it began by discarding the analogy that had served Carnot. Water turning a mill does not become the mill’s work. The work comes from the water’s fall under gravity, and the water reaches the bottom in the quantity it had at the top.
Heat in an engine is not like that. Some of the energy taken in by heating leaves the engine as work, and the rest is passed to the colder body.
The account still balances, but differently. The energy an engine takes in by heating equals the work it does plus the energy it rejects to the cold side. Carnot had been right that an engine needs a hot side and a cold one, and that the working substance does not matter. He had been wrong that all the heat entering also leaves.
So the rejected energy is not caloric leaking from a store, because there is no such fluid. Whether it is mere waste, or something the engine cannot work without, is the next question.
Why a colder body is needed
An engine is a cycle and not a single push. To keep working it must return, over and over, to the state it began in. That requirement is where the cold body becomes necessary.
Follow one turn. Energy is transferred to the gas by heating. As the gas expands, it drives the piston and energy leaves the gas as work. An expansion that leaves the gas in a new state is not yet an engine, though. To work again it must bring the gas back to where it began, which means compressing it.
Compression costs work, and there is the trap. If compression retraces the hot expansion, it takes back all the work just gained. The engine has gained nothing.
The escape is to cool the gas first, by bringing it against a colder body, so that it is at lower pressure and cheaper to compress. Compressing a gas tends to warm it, as the air in a bicycle pump warms under the plunger.
So the gas is pressed back while it rests against the colder body. The energy the compression adds passes into that body by heating, and the gas stays cool. That colder body is the engine’s cold sink.
The last part of the compression is done with the gas insulated, which warms it back to its starting temperature. Only then is the gas returned to its starting state, ready to begin again.
The work an engine yields is what remains after the gas is reset, and it can be reset cheaply only by shedding energy to something cold. Remove the cold body and there is nowhere for that energy to go.
The gas cannot return to its starting state without spending all the work the expansion produced. A colder place to reject energy to is as much a part of the engine as the furnace. Without it there is no engine, only a gas that expands once.
The second law
Clausius’s argument compares engines working between the same two temperatures. An ideal reversible engine can run backwards as a refrigerator when work is supplied. Now take an engine better than the ideal one and set it to drive such a refrigerator.
Over a cycle the pair would move energy from the cold body up to the hot one, while drawing no work from outside. That would be a refrigerator that runs on nothing. No such machine has ever worked, so no engine beats the ideal. That is Carnot’s conclusion again, now resting on energy conservation and one new principle.
The principle, which Clausius stated as a law, is this. Energy does not pass by heating from a colder body to a hotter one of its own accord. Left to itself, energy is transferred by heating from hot to cold, and the reverse takes work. This is the second law of thermodynamics, standing beside the first, which says that energy is conserved.
It also names what the perfect engine of the opening would have to be. Such an engine need not make energy from nothing. It could obey the first law, turning into work just the energy it took in.
Its fault lies elsewhere. Rejecting nothing, it would need no colder body at all. An engine with no colder body is forbidden by the second law, just as an engine making energy from nothing is forbidden by the first.
The machine that would make work from nothing is perpetual motion of the first kind. A machine that takes in energy by heating from one source and turns the whole of it into work is perpetual motion of the second kind. The cold sink is what the second kind lacks, and what the second law requires of every engine.
What the cold sink shows
The first law is silent on direction. It says the energy an engine takes in is fully accounted for, so much as work and the rest passed to the cold body. Nothing in that accounting forbids the rejected energy from making its way back up to the furnace. It is the second law that forbids the return, and so decides which of the energy-balanced processes can run.
A cyclic engine, then, cannot take in energy by heating from a single hot source and leave nothing behind but work. Some must be passed to a colder place, because only then can the working substance be returned to the state it began in.
Energy is conserved, and its usefulness is not. The cold sink is where an engine pays that price for any work at all. It is part of what makes the engine run, and not a flaw in the machine.
Sources
Sadi Carnot, Reflections on the Motive Power of Heat and on Machines Fitted to Develop That Power, translated and edited by R. H. Thurston (New York: John Wiley & Sons, 1897; original French edition 1824).
Paul Sen, Einstein’s Fridge: How the Difference Between Hot and Cold Explains the Universe (Scribner, 2021).