**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 Q_{1 }amount a heat. Suppose the statement is not true, so,

or W_{A} / Q_{A} > W_{B} / Q_{1}

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.