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 HEA = Any heat engine
RHEB= Reversible heat engine and both the engine draw Q1 amount a heat. Suppose the statement is not true, so,
or WA / QA > WB / Q1
or WA > WB
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.