Thermodynamics has its historical roots in the development of the steam engine and other machines of the early industrial revolution. Early steam engines, built on the basis of experience rather than scientific understanding, were not very efficient at converting fuel energy into work. The first major theoretical analysis of heat engines was published by the French engineer Sadi Carnot in 1824. The question that Carnot raised was: Can we make a heat engine whose thermal efficiency $\eta$ approaches 1, or is there an upper limit $\eta {}_{\text{max}}$ that cannot be exceeded? To frame the question more clearly, imagine we have a hot reservoir at temperature $T_{\text{H}}$ and a cold reservoir at $T_{\text{C}}$. What is the most efficient heat engine (maximum $\eta$) that can operate between these two energy reservoirs? Similarly, what is the most efficient refrigerator (maximum K) that can operate between the two reservoirs?

We just saw that a refrigerator is, in some sense, a heat engine running backward. We might thus suspect that the most efficient heat engine is related to the most efficient refrigerator. Suppose we have a heat engine that we can turn into a refrigerator by reversing the direction of operation, thus changing the direction of the energy transfers, and with no other changes. In particular, the heat engine and the refrigerator operate between the same two energy reservoirs at temperatures $T_{\text{H}}$ and $T_{\text{C}}$.

FIGURE 21.18a shows such a heat engine and its corresponding refrigerator. Notice that the refrigerator has exactly the same work and heat transfer as the heat engine, only in the opposite directions. A device that can be operated as either a heat engine or a refrigerator between the same two energy reservoirs and with the same energy transfers, with only their direction changed, is called a perfectly reversible engine. A perfectly reversible engine is an idealization, as was the concept of a perfectly elastic collision. Nonetheless, it will allow us to establish limits that no real engine can exceed.

FIGURE 21.18 If a perfectly reversible heat engine is used to operate a perfectly reversible refrigerator, the two devices exactly cancel each other.

Suppose we have a perfectly reversible heat engine and a perfectly reversible refrigerator (the same device running backward) operating between a hot reservoir at temperature $T_{\text{H}}$ and a cold reservoir at temperature $T_{\text{C}}$. Because the work $W_{\text{in}}$ needed to operate the refrigerator is exactly the same as the useful work $W_{\text{out}}$ done by the heat engine, we can use the heat engine, as shown in FIGURE 21.18b, to drive the refrigerator. The heat $Q_{\text{C}}$ the engine exhausts to the cold reservoir is exactly the same as the heat $Q_{\text{C}}$ the refrigerator extracts from the cold reservoir. Similarly, the heat $Q_{\text{H}}$ the engine extracts from the hot reservoir matches the heat $Q_{\text{H}}$ the refrigerator exhausts to the hot reservoir. Consequently, there is no net heat transfer in either direction. The refrigerator exactly replaces all the heat energy that had been transferred out of the hot reservoir by the heat engine.

You may want to compare the reasoning used here with the reasoning we used with Figure 21.11. There we tried to use the output of a “perfect” heat engine to run a refrigerator but did not succeed.