# Why is the efficiency of a carnot engine maximum?

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.

• Care to elaborate on what you mean by "other processes"? I have already given you an answer to your first question. Please let me know if this is understood and then we can perhaps address the second question you have about other processes. Dec 7, 2016 at 9:49
• @BLAZE I mean if we have isothermal and isochoric process Dec 7, 2016 at 9:58
• I think you meant Isobaric for the first process you mentioned, but you said yourself; "I know carnot engine consists of isothermal and adiabatic processes". A carnot engine only consists of these thermodynamic processes, this is what enables the efficiency to be optimal. So you cannot have isobaric and isochoric process in a carnot cycle as they will not yield maximum efficiency. A carnot cycle can only consist of Isotherms and Adiabats on a pressure vs volume graph. Dec 7, 2016 at 10:11
• @BLAZE by that I meant that if there is another cycle which consists of these different process and have more efficiency Dec 7, 2016 at 10:13
• There are no thermodynamic cycles with different processes that have a higher efficiency than a Carnot cycle. It is a fundamental mathematically proven fact by Sadi Carnot in 1824. Dec 7, 2016 at 10:19

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.