# Proof of a Carnot engine being the most efficient heat engine

Most of the physics textbooks I’ve read try to prove that the Carnot engine is the most efficient engine possible by showing that if you coupled a superefficient engine to a Carnot refrigerator, you’d get a perfect refrigerator, which violates the second law of thermodynamics.

An example from Physics for Scientists and Engineers: A Strategic Approach is shown below.

Another example is this video by Khan Academy.

What I do not understand is what about this refrigerator identifies it as as Carnot refrigerator. Couldn’t you take any refrigerator, including one whose coefficient of performance is less than that of a Carnot refrigerator, couple it to a heat engine that is more efficient than the refrigerator driven in reverse (i.e. its engine “counterpart”), and get the same result that violates the second law of thermodynamics?

No you cannot. It must be a Carnot refrigerator. This is because you want to prove that the super-efficient engine is impossible even if $<Q_H$ means infinitesimally smaller than $Q_H$ (and $<Q_C$ means infinitesimally smaller than $Q_C$).