Now we’ve arrived at the critical step in the reasoning. Suppose I claim to have a heat engine that can operate between temperatures $T_{\text{H}}$ and $T_{\text{C}}$ with more efficiency than a perfectly reversible engine. FIGURE 21.19 shows the output of this heat engine operating the same perfectly reversible refrigerator that we used in Figure 21.18b.

FIGURE 21.19 A heat engine more efficient than a perfectly reversible engine could be used to violate the second law of thermodynamics.

Recall that the thermal efficiency and the work of a heat engine are

If the new heat engine is more efficient than the perfectly reversible engine it replaces, it needs less heat $Q_{\text{H}}$ from the hot reservoir to perform the same work $W_{\text{out}}$. If $Q_{\text{H}}$ is less while $W_{\text{out}}$ is the same, then $Q_{\text{C}}$ must also be less. That is, the new heat engine exhausts less heat to the cold reservoir than does the perfectly reversible heat engine.

When this new heat engine drives the perfectly reversible refrigerator, the heat it exhausts to the cold reservoir is less than the heat extracted from the cold reservoir by the refrigerator. Similarly, this engine extracts less heat from the hot reservoir than the refrigerator exhausts. Thus the net result of using this superefficient heat engine to operate a perfectly reversible refrigerator is that heat is transferred from the cold reservoir to the hot reservoir without outside assistance.

But this can’t happen. It would violate the second law of thermodynamics. Hence we have to conclude that no heat engine operating between reservoirs at temperatures $T_{\text{H}}$ and $T_{\text{C}}$ can be more efficient than a perfectly reversible engine. This very important conclusion is another version of the second law:

The answer to our question “Is there a maximum $\eta$ that cannot be exceeded?” is a clear Yes! The maximum possible efficiency $\eta {}_{\text{max}}$ is that of a perfectly reversible engine. Because the perfectly reversible engine is an idealization, any real engine will have an efficiency less than $\eta {}_{\text{max}}$.

A similar argument shows that no refrigerator can be more efficient than a perfectly reversible refrigerator. If we had such a refrigerator, and if we ran it with the output of a perfectly reversible heat engine, we could transfer heat from cold to hot with no outside assistance. Thus: