Suppose you drop a hot rock into the ocean. Heat energy is transferred from the rock to the ocean until the rock and ocean are at the same temperature. Although the ocean warms up ever so slightly, $\text{Δ}{T}_{\text{ocean}}$ is so small as to be completely insignificant. For all practical purposes, the ocean is infinite and unchangeable.

An energy reservoir is an object or a part of the environment so large that its temperature does not change when heat is transferred between the system and the reservoir. A reservoir at a higher temperature than the system is called a hot reservoir. A vigorously burning flame is a hot reservoir for small objects placed in the flame. A reservoir at a lower temperature than the system is called a cold reservoir. The ocean is a cold reservoir for the hot rock. We will use $T_{\text{H}}$ and $T_{\text{C}}$ to designate the temperatures of the hot and cold reservoirs.

Hot and cold reservoirs are idealizations, in the same category as frictionless surfaces and massless strings. No real object can maintain a perfectly constant temperature as heat is transferred in or out. Even so, an object can be modeled as a reservoir if it is much larger than the system that thermally interacts with it.

Heat energy is transferred between a system and a reservoir if they have different temperatures. We will define

By definition, $Q_{\text{H}}$ and $Q_{\text{C}}$ are positive quantities. The direction of heat transfer, which determines the sign of Q in the first law, will always be clear as we deal with thermodynamic devices.

FIGURE 21.2a shows a heavy copper bar between a hot reservoir (at temperature $T_{\text{H}}$) and a cold reservoir (at temperature $T_{\text{C}})$. Heat $Q_{\text{H}}$ is transferred from the hot reservoir into the copper and heat $Q_{\text{C}}$ is transferred from the copper to the cold reservoir. FIGURE 21.2b is an energy-transfer diagram for this process. The hot reservoir is always drawn at the top, the cold reservoir at the bottom, and the system—the copper bar in this case—between them. Figure 21.2b shows heat $Q_{\text{H}}$ being transferred into the system and $Q_{\text{C}}$ being transferred out.

FIGURE 21.2 Energy-transfer diagrams.

Full Alternative Text

Full Alternative Text

The first law of thermodynamics $Q=W_{\text{s}}+\text{Δ}{E}_{\text{th}}$ refers to the system. Q is the net heat to the system, which, in this case, is $Q=Q_{\text{H}}-Q_{\text{C}}$. The copper bar does no work, so $W_{\text{s}}=0$. The bar warms up when first placed between the two reservoirs, but it soon comes to a steady state where its temperature no longer changes. Then $\text{Δ}{E}_{\text{th}}=0$. Thus the first law tells us that $Q=Q_{\text{H}}-Q_{\text{C}}=0$, from which we conclude that $Q_{\text{C}}=Q_{\text{H}}$.

In other words, all of the heat transferred into the hot end of the rod is subsequently transferred out of the cold end. This isn’t surprising. After all, we know that heat is transferred spontaneously from a hotter object to a colder object. Even so, there has to be some means by which the heat energy gets from the hotter object to the colder. The copper bar provides a route for the energy transfer, and $Q_{\text{C}}=Q_{\text{H}}$ is the statement that energy is conserved as it moves through the bar.

Contrast Figure 21.2b with FIGURE 21.2c. Figure 21.2c shows a system in which heat is being transferred from the cold reservoir to the hot reservoir. The first law of thermodynamics is not violated, because $Q_{\text{H}}=Q_{\text{C}}$, but the second law is. If there were such a system, it would allow the spontaneous (i.e., with no outside input or assistance) transfer of heat from a colder object to a hotter object. The process of Figure 21.2c is forbidden by the second law of thermodynamics.