Rudolf Clausius · Entropy
Why does heat flow only one way?
That time, that place
Heat flows from hot to cold, and not the other way by itself. So obvious that nobody wrote it down as a law.
Then in the mid-nineteenth century two results appeared to clash. Carnot had shown in 1824 that a heat engine's efficiency has a ceiling — but his calculation assumed heat flows from high to low while keeping its quantity. Joule showed that heat turns into work, that is, heat is not conserved.
Both looked right, on opposite premises.
Why this question
Clausius reconciled them in 1850.
Energy is conserved — Joule is right that far. But quite apart from the quantity of energy, there is how spread out it is. Carnot's ceiling comes from that second thing.
He raised to a principle that heat never flows of itself from cold to hot. That is the second law of thermodynamics.
In 1865 he gave the spread-out-ness a name. He showed that heat exchanged, divided by the temperature at which it is exchanged, never decreases in an isolated system, and called that quantity entropy — from the Greek for change, deliberately shaped to sound like energy, as he noted.
What was found
At the end of the same paper he set two sentences side by side: the energy of the universe is constant; the entropy of the universe tends to a maximum.
The first law says nothing is lost. The second says the usable form keeps shrinking. When hot coffee cools, its energy has not gone anywhere — it is in the room. It simply cannot be made to do anything now.
This law has a direction. Most physical laws run the same backwards: a film of a thrown ball reversed looks fine. A film of a cup shattering, reversed, does not. Where that wrongness comes from becomes a question here.
Some thirty years later Boltzmann re-explained entropy through probability: scattered arrangements outnumber gathered ones overwhelmingly.
Heat flowing one way was an obvious fact, not something to be written as a law
That heat exchanged divided by temperature never decreases in an isolated system
Boltzmann recast the quantity as probability, and time's direction itself became a problem for physics