We hypothesized above that there are no perfect heat engines—that is, no heat engines like the one shown in Figure 21.4 with $Q_{\text{C}}=0$ and $\eta =1$. Now we’re ready to prove this hypothesis. FIGURE 21.11 shows a hot reservoir at temperature $T_{\text{H}}$ and a cold reservoir at temperature $T_{\text{C}}$. An ordinary refrigerator, one that obeys all the laws of physics, is operating between these two reservoirs.

FIGURE 21.11 A perfect engine driving an ordinary refrigerator would be able to violate the second law of thermodynamics.

Suppose we had a perfect heat engine, one that takes in heat $Q_{\text{H}}$ from the high-temperature reservoir and transforms that energy entirely into work $W_{\text{out}}$. If we had such a heat engine, we could use its output to provide the work input to the refrigerator. The two devices combined have no connection to the external world. That is, there’s no net input or net output of work.

If we built a box around the heat engine and refrigerator, so that you couldn’t see what was inside, the only thing you would observe is heat being transferred with no outside assistance from the cold reservoir to the hot reservoir. But a spontaneous or unassisted transfer of heat from a colder to a hotter object is exactly what the second law of thermodynamics forbids. Consequently, our assumption of a perfect heat engine must be wrong. Hence another statement of the second law of thermodynamics is:

Any real heat engine must exhaust waste heat $Q_{\text{C}}$ to a cold reservoir.