## What is the Carnot theorem?

Sadi Carnot, a French engineer, was the first to use cyclic process in thermodynamic reasoning. According to Carnot theorem, no cycle can be more efficient than a reversible cycle operating between the same temperature limits;

Also, any two reversible cycles will have the same thermal efficiency when they operate between the same heat source and heat sink temperatures.

From the theorem, we can conclude that the efficiency of the reversible cycle engine is greater than the irreversible cycle engine and this statement is also a corollary of the second law of thermodynamics.

I guess many of you may feel it is hard to accept this theorem without the mathematical proof if you are reading this theorem for the first time right? but it is proved brilliantly by the logical analysis of the combined operation of the reversible and the irreversible cycle engine!!

## Proof for Carnot theorem

It is proved brilliantly by a logical analysis of the system which uses both reversible and irreversible cycle engines for the operation.

Let’s assume we have two heat engines that produce work using heat. One is Reversible heat engine “R” and the other one is irreversible engine “I”

Now assume the heat engine “I” has higher efficiency than reversible engine “R” for analysing purposes.

Then for a heat input “Qk”, work done by the irreversible engine(Wi) is higher than that of the reversible engine (Wr) and also heat rejected by the Reversible engine (Qo,R) is higher than the irreversible engine(Qo,I)

If now, the reversible heat engine is made to work as a refrigeration machine and the irreversible heat engine continues to work as a heat engine as shown in Figure, the resultant combined system will work as a perpetual motion machine of the second kind.

It means by taking heat equal to “Qo,R – Qo,I” from the cold reservoir and converting it completely into work, thus violating the Kelvin-Planck statement of the Second Law applicable to heat engines as shown in Figure!!

Since the heat rejected from the reversible engine “R” is used by the irreversible heat engine “I” hence the hot reservoir doesn’t require this system to function.

Hence we can logically conclude that the efficiency of the irreversible engine can’t be higher than the reversible engine.

To know more about the efficiency of the Carnot cycle then click/tap here

To know why the Carnot cycle is not used in the engine even though efficiency is high then click/tap here

## References

Refrigeration and Air Conditioning by C P Arora

Carnot’s theorem (thermodynamics) – Wikipedia

Kelvin-Planck Statement Example And Its Applications – BYJU’S (byjus.com)