# Application of Carnot's theorem to calculate efficiency of thermal engine

I understood Carnot's theorem in theory, but I still have some doubt on the practical use of it.

In particular I would like to know: if I have any reversible cycle followed by a gas, then the thermal efficiency of that cycle will be $$\eta=1-\frac{T_{\mathrm{min}}}{T_{\mathrm{max}}}$$

Where $T_{\mathrm{min}}$ is the lowest and $T_{\mathrm{max}}$ the highest among the temperatures of the states involved in the cycle. Is this use of the theorem correct or is there something wrong?

In this case $T_{\mathrm{min}}=T_D$ and $T_{\mathrm{max}}=T_B$ but the efficiency turns out to be

$$\eta=1-\frac{T_D}{T_A}$$

While, using Carnot's theorem I would have said $$\eta= 1-\frac{T_D}{T_B}$$

So I think that my interpretation of the way to use the theorem is wrong. Any suggestion on where my mistake is is highly appreciated.

• Canot's theorem states that the efficiency obtained by the Carnot engine is the maximum efficiency, not that any reversible process has that efficiency. – Sanya Jul 12 '16 at 20:09
• @Sanya you are correct, just a clarification: you can show that any reversible process between two heat reservoirs have the same efficiency that the Carnot cycle. The point here is that you have to have TWO reservoirs not three or more. – Wolphram jonny Jul 12 '16 at 20:14
• @Wolphramjonny thank you, it has been some years ... But after reading your post, I even remember the proof :D – Sanya Jul 12 '16 at 20:20

In particular I would like to know: if I have any reversible cycle followed by a gas, then the thermal efficiency of that cycle will be $$\eta=1-\frac{T_{\mathrm{min}}}{T_{\mathrm{max}}}.$$