The second law and the Carnot engine

The first law says energy is conserved, but it does not say which way things happen. Heat always flows from hot to cold, never back by itself; a spinning wheel stopped by friction warms its brake, but warming the brake never makes the wheel spin again. The second law gives the direction.

It has two equivalent statements:

• Kelvin–Planck: no engine working in a cycle can turn all the heat it takes in into work. Some heat must always be thrown out to a colder place. That is why every car or motorbike has hot exhaust gases.
• Clausius: heat cannot by itself flow from a colder body to a hotter one. A refrigerator does move heat out of its cold inside into the warm kitchen, but only because its compressor does work.

A reversible process is one that can be run backwards so that the system and the surroundings both return exactly to where they started. It must be quasi-static (so slow that the system is always in equilibrium) and have no dissipation. It is an ideal limit, never quite reached.

Real, natural processes are irreversible:

• Friction turns work into heat, which cannot all be turned back.
• Free expansion of a gas into a vacuum cannot undo itself.
• Heat flow across a finite temperature difference goes only one way.

A heat engine takes in heat Q₁ from a hot reservoir, does work W, and gives out heat Q₂ to a cold one. Its efficiency is η = W/Q₁ = 1 – Q₂/Q₁.

The best possible engine is the Carnot engine: a reversible cycle of two isothermal and two adiabatic steps between a hot reservoir at T₁ and a cold one at T₂. Its efficiency is η = 1 – T₂/T₁, with temperatures in kelvin, and no engine working between the same two temperatures can beat it.

Example: an engine burns its fuel at about 800 K and dumps heat to air at 300 K. Even a perfect Carnot engine could reach only 1 – 300/800 = 62.5%; real engines manage much less.

So η can be 100% only if T₂ = 0 K, which can never be reached. Every engine must waste some heat.