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Carnot’s Theorem

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 BA is true

ɳ BA 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 BA for B reversed

ɳ AB  for A reversed

ɳ AB  proved for both the engines are reversible.