Carnot Cycle

Nicholas Sadi Carnot devised a reversible cycle in 1824 called the Carnot cycle for an engine working between two reservoirs at different temperatures. It consists of two reversible isothermal and two reversible adiabatic processes. For a cycle 1-2-3-4, the working material

1. Undergoes isothermal expansion in 1-2 while absorbing heat from high temperature reservoir
2. Undergoes adiabatic expansion in 2-3
3. Undergoes isothermal compression in 3-4, and
4. Undergoes adiabatic compression in 4-1.

Heat is transferred to the working material during 1-2 (Q₁) and heat is rejected during 3-4 (Q₂). The thermal efficiency is thus η_th = W/Q₁. Applying first law, we have, W = Q₁ − Q₂, so that η_th = 1 − Q₂/Q₁.

Carnot's principle states that

1. No heat engine working between two thermal reservoirs is more efficient than the Carnot engine, and
2. All Carnot engines working between reservoirs of the same temperature have the same efficiency.

The proof by contradiction of the above statements come from the second law, by considering cases where they are violated. For instance, if you had a Carnot engine which was more efficient than another one, we could use that as a heat pump (since processes in a Carnot cycle are reversible) and combine with the other engine to produce work without heat rejection, to violate the second law. A corollary of the Carnot principle is that Q₂/Q₁ is purely a function of t₂ and t₁, the reservoir temperatures. Or,