[Physics] Why is the efficiency of a carnot engine maximum

Question

I know carnot engine consists of isothermal and adiabatic process. My question is why the efficiency of carnot engine is maximum?

Why not it is maximum with other processes.

In one book I read that it is maximum because it is based on reversible cycle. I could not understand. Please help me out.

Answer 1

Lets look at some of the corollaries of the fact that the Carnot efficiency is $$\eta_{carnot}=1-\frac{T_H}{T_C}$$ where $T_H$ and $T_C$ are the fixed temperatures of the hot and cold reservoirs respectively (as these reservoirs are considered infinitely large).

Consider these $3$ cases:

1. If the heat engine is more efficient than a Carnot engine then the net heat extracted (or put in) of the two reservoirs is $𝑄_{𝑛𝑒𝑡} \lt 0$. This corresponds to the sole effect of the Carnot fridge plus heat engine being a transfer of heat from the cold reservoir to the hot one. This would violate the second law and thus this level of efficiency is impossible.
2. If the efficiency of an arbitrary engine $\eta^{E}$ is such that $\eta^{E}=\eta_{carnot}$ then the net heat transfer is zero and everything is transferred back to its initial state after one cycle implying with this efficiency the process is completely reversible.
3. $\eta^{E}\lt\eta_{carnot}$ then $𝑄_{𝑛𝑒𝑡} \gt 0$ so the cycle involves a net transfer of heat from the hot reservoir to the cold one. This is of course fine but to restore the reservoirs to their original state would require a shifting of the heat back thus the process is irreversible.

So from these $3$ cases we conclude that no heat engine operating between two reservoirs can have a greater efficiency than a Carnot engine.