It seems that the only condition used in proving that the Carnot engine is the most efficient is that it is reversible. More specifically, the Carnot engine can be run in reverse as a refrigerator. Furthermore, it is asserted that all reversible engines will have the same efficiency.
However, isn't any closed loop on a PV diagram reversible? The arrows can simply be drawn in the reverse way to create a refrigerator. If any closed loop is reversible then why does the specific Carnot engine (a specific loop) have the highest efficiency?
Answer
However, isn't any closed loop on a PV diagram reversible? The arrows can simply be drawn in the reverse way to create a refrigerator. If any closed loop is reversible then why does the specific Carnot engine (a specific loop) have the highest efficiency?
This was exactly the question I asked myself ten years ago :-) The problem is that often students do not appreciate the whole statement: Carnot's engine is operating between two temperatures (heat sources). Any circle on the PV-plane is reversible if you have many heat sources. In the case of many heat sources, you may also know that you do not talk about the efficiency of the engine, but you talk about the Clausius' equality: $$\sum_{i} \frac{Q_i}{T_i}= 0.$$ Note that $T_i$ is the temperature of the $i$th heat source (this is a very important point often missed!), which equals the temperatures of the system when they are in reversible contact. This is not true if the process is irreversible: you have heat flow from hot sources to the (colder) engine. Then one has the Clausius' inequality: $$\sum_{i}\frac{Q_i}{T_i}<0.$$
So, in short: Carnot's engine is the only reversible engine operating between two temperatures.
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