Theorem 1. “No heat engine working in a cyclic process between two fixed temperatures can be more efficient than a reversible heat engine working between the same two fixed temperatures.”

Proof: Refer Fig. 7.3 (a)

Let                                           HE_A =  Any heat engine

RHE_B= Reversible heat engine and both the engine draw Q₁ amount a heat. Suppose the statement is not true, so,

or                                 W_A / Q_A > W_B / Q₁

or                                         W_A > W_B

Now we reverse the RHE8 as it is reversible. So all directions of flow will be reversed.

If we combine HE _A & EHRB, we get work W A – W 8 out of exchange heat (Q2B – Q2A) with single temperature sink T2. Thus it violates Kelvin-Planks law, so ɳ _B >ɳ _A is true

ɳ _B >ɳ _A means that HEA is also a reversible heat engine. This leads to the important

corollary of Carnot’ s theorem.

Corollary of Carnot’s Theorem

All reversible engines working between the same fixed two temperatures have the same efficiency.

Proof : We have          ɳ _B >ɳ _A for B reversed

ɳ _A>ɳ _B  for A reversed

ɳ _A>ɳ _B  proved for both the engines are reversible.