2.3 Carnot cycles

Take-home message: The efficiency of a Carnot engine is independent of its construction, and no irreversible engine can beat it.

A Carnot engine is, simply, a reversible engine acting between only two heat reservoirs. That means that all processes are either isothermal (heat transfer at a constant temperature) or adiabatic (no heat transfer). By contrast the Otto cycle, which has heating at constant volume, would need a whole series of heat reservoirs at incrementally higher temperatures to carry out the heating reversibly.

Carnot's theorem says that a reversible engine is the most efficient engine which can operate between two reservoirs. If you want to see the proof, see here. An equally important corollory is that any reversible engine working between two heat reservoirs has the same efficiency as any other, irrespective of the details of the engine.

Much is made of the fact that the Carnot engine is the most efficient engine. Actually, this is not mysterious. First, if we specify only two reservoirs, then all it says is that a reversible engine is more efficient than an irreversible engine, which isn't surprising (no friction...) Second, we will see that the efficiency of a Carnot engine increases with the temperature difference between the reservoirs. So it makes sense to use only the hottest and coldest heat baths you have available, rather than a whole series of them at intermediate temperatures. But the independence of the details of the engine is rather deeper, and has far-reaching consequences.

As a result, if we can calculate the efficiency for one Carnot engine, we know it for all. We can calculate it for an ideal gas Carnot cycle, and find

$\mbox{\large\colorbox{yellow}{\rule[-3mm]{0mm}{10mm} \ $\displaystyle \eta_{\rm carnot}=1-{T_C\over T_H},$ }}$

details here. Hence this is true for all Carnot engines.

By comparing with the definition of the efficiency of any heat engine, $\eta=W/Q_H=1-Q_C/Q_H$, we get the even more useful relation:

$\mbox{\large\colorbox{yellow}{\rule[-3mm]{0mm}{10mm} \ $\displaystyle {Q_C\over Q_H}={T_C\over T_H}. $ }}$

See here for examples of questions about Carnot heat engines.

Carnot developed the concept of reversibility and showed that no engine could be more efficient than a reversible one before either the first or second law of thermodynamics had been formulated! See here for a link to his essay, and to a good description of his contribution to thermodynamics.

References

• Mandl 5.2
• Bowley and Sánchez 2.3
• Adkins 4.2, 4.5
• Zemansky 7.1-4