Carnots Theorem and the Entropy of Clausius

We can redraw the Carnot cycle in a slightly simplified form  

Without loss of generality, this summation could be extended to all four limbs (i = 1. 4) of the Carnot cycle (because adiabatic steps contribute nothing to the sum), 

It was recognized by Carnot that cyclic integrals of the form (4.24c) must actually vanish for all reversible cycles  

The proof of this important theorem is sketched in Sidebar 4.7. 

Given the validity of (4.25b) for any Carnot cycle, we can extend the result to a general cyclic path, as shown in the following diagram  

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