Why heat flows downhill

The second law

The first law of thermodynamics says energy is conserved. The second law is harder to state and more surprising. In the words of the German physicist Rudolf Clausius: heat cannot pass from a colder body to a warmer one without some other change happening at the same time.

Picture an ice pack on a sore knee. Heat flows from the warm knee into the cold ice. That happens on its own; nobody has to do anything. Have you ever seen the reverse, heat flowing out of the ice pack into your knee, making the ice colder and the knee warmer? Never. Yet an air conditioner seems to do exactly that. It moves heat out of an 18°C room into 40°C air outside. But it only works because a compressor runs on electricity. That is the “other change” the law requires: work has to be done, and energy has to be spent.

A word for transformation

The second law led to one of the most important words in science: entropy. Anyone who studies physics should look at where the word came from, because the great scientists chose their words with care. Clausius coined it in 1865 from the Greek tropē, “transformation,” adding a beginning that echoes energy. Entropy concerns the transformation of energy.

The Chinese translation is just as apt. The character 熵 combines “fire” with 商, which in mathematics means a quotient, the result of a division. That is the classical definition: when heat flows into or out of a body, its entropy changes by the heat divided by the body’s temperature. We can’t measure a body’s total entropy, only how much it changes.

The water wheel of heat

A heat engine is hard to picture, so start with a water wheel. Water pours from a high reservoir to a low one, turns the wheel on the way down, and the wheel does work. A heat engine works the same way, except that what falls is heat. It flows from a hot reservoir to a cold one, and the engine captures some of it as work. The French engineer Sadi Carnot used the same comparison in 1824, in the book that founded the science of heat engines.

Here is a worked example. A hot reservoir at 1,000 kelvin gives up 4,000 joules of heat, and its entropy falls by 4,000 ÷ 1,000 = 4 units. The second law says that entropy can’t decrease overall. So the cold reservoir, at 100 kelvin, must take in at least enough heat to raise its entropy by 4, and 4 × 100 = 400 joules. The rest can be work.

So the best possible engine turns 3,600 of the 4,000 joules into work, an efficiency of 90 percent, and it can do this only if total entropy doesn’t increase at all. In the worst case, all 4,000 joules simply leak into the cold reservoir. The cold side’s entropy rises by 4,000 ÷ 100 = 40, the total rises by 36, and no work is done. Energy is still conserved, since all 4,000 joules are still there, but now they sit in the cold reservoir where they can’t turn a wheel. In plain language: entropy measures energy that has become useless for work.

Counting arrangements

There is a second, microscopic definition of entropy, and it explains why the first one works. A balloon holds something like 10²³ gas molecules, rushing about and colliding at random. Nobody can track them with Newton’s laws. The founders of statistical physics realized that they didn’t need to.

Think of a Galton board, a board of pegs where each dropped ball bounces left or right at random. You can’t predict where any one ball will land. Drop thousands and they pile into a predictable bell-shaped heap. Coins make it exact. Toss four coins: there are 16 equally likely ways they can land, which physicists call microstates. But if you only count how many heads, the macrostate, the outcomes aren’t equal. Four heads can happen only one way. Two heads can happen six ways. So two heads is six times as likely as four.

Replace heads and tails with gas molecules on the left or right side of a box. Start with all four on the left, then open a partition. A little later, the chance of finding all four back on the left is 1 in 16. With more molecules, the odds pile up ever more steeply around an even split.

The Austrian physicist Ludwig Boltzmann turned this into a definition. Entropy is proportional to the logarithm of the number of microstates that make up a macrostate. A state that can happen in more ways has higher entropy. Left alone, a system drifts from less probable states to more probable ones, and its entropy rises. With 10²³ molecules, the even state is so overwhelmingly probable that the gas will, in practice, never gather back into one corner.

That is the second law in its general form: the entropy of an isolated system, one that exchanges neither heat nor matter with anything outside, never decreases. In a reversible process it stays the same; in any real, irreversible process it increases. English writers describe rising entropy as rising disorder, the loss of order and structure. Melt ice into water and the neat crystal loosens into a liquid. Boil the water and the molecules scatter everywhere. Each step takes in heat, raises entropy, and lowers order.