As I was reading through my physics book I came across a proof of Carnot's Theorem.

Consider two reversible engines A and B working between same temperatures limits T_{1 }and T_{2}. Source being T1 and sink T2 are coupled.

eff1(heat engine) = (Q1-Q2) / Q1 = W / Q1

eff2(refrigerator) = (Q1'-Q2') / Q1' = W / Q1'

( I )

if eff1 > eff2,

Q1' > Q1

Also Q1 - Q2 = Q1' - Q2'

i.e. Q2' > Q2

Here both Q2' - Q2 and Q1' - Q1 are positive quantity. Which means heat flows from colder body to hotter body with any external work. hence it violates 2nd law of thermodynamics .

But in ( II )

if eff2 > eff1 ,

Q1 > Q1'

also Q1 - Q2 = Q1' - Q2'

i.e. Q2' > Q2

Here both Q2 - Q2' and Q1 - Q1' are positive quantity . Which means heat flows from hotter body to colder body.

**How does this violate any law of thermodynamics to prove Carnot's Theorem**

** 1 ) No engine working between two given temperature can be more efficient than a reversible ( Carnot's ) engine working between the same limits of temperature (i.e between the same source and sink).**

** 2 ) All the reversible engine working between same limits of temperature have the same efficiency whatever be the working substance.**