I have this homework question:
"Show that any reversible engine operating between T1 and T2 is a carnot engine."
I think I have a solution, but it feels very hand-wavy. We know that any process that can be represented as a loop in the PV plane is reversible as the net entropy change will be zero. We must operate between two specifice temperatures, so the loop must comprise of two isotherms a T1 and T2. So the question is what curves join the isotherms. As a heat engine comprises of energy input at constant temperature, there will be no energy change between the isotherms. So the curves connecting the isotherms must be adiabatic curves. So we have a carnot cycle.
Is this sufficient? I don't know why, but I doubt it.
$$W_{total}=nR\left ( T_h\ln \frac{V_2}{V_1}-T_c\ln\frac{V_3}{V_4} \right )$$
Also for adiabatic processes we have :
$$T_h V_2^{\gamma-1 }=T_c V_3^{\gamma-1}\tag{2}$$
$$T_hV_1^{\gamma-1}=T_cV_4^{\gamma-1}\tag{3}$$
Substituting for $V_2$ and $V_4$ from $(2)$ and $(3)$ in the first equation, we can find the ratio $V_3\over V_1$. (Find the final result yourself!)
Best Answer
The Carnot cycle is the unique reversible cycle working between two reservoirs of temperature $T_1$ and $T_2$, such that when the engine is in contact with reservoir 1, evolution is isothermal at temperature $T_1$, isothermal at $T_2$ when the engine is in contact with reservoir 2 and adiabatic (isentropic) elsewhere. There are no other possibilities with these "boundary conditions" i.e. interface to the outside World.
However, there are many other possibilities for the interface to the outside World. The engine could make contact with many more than two different temperature reservoirs during a cycle, or heat could be added at constant volume (e.g. from detonation of a chemical reaction inside a rigid vessel, as approximately happens during a Diesel cycle).