What happens when two systems at different temperatures interact with each other? In Chapter 19 we said that “heat is transferred,” but what does that mean? What is the mechanism by which energy is transferred from a hotter object to a colder object? An atomic-level view will improve our understanding of heat and thermal equilibrium.

FIGURE 20.12 shows a rigid, insulated container divided into two sections by a very thin membrane. The left side, which we’ll call system 1, has $N_1$ atoms at an initial temperature $T_{1\text{i}}$. System 2 on the right has $N_2$ atoms at an initial temperature $T_{2\text{i}}$. The membrane is so thin that atoms can collide at the boundary as if the membrane were not there, yet it is a barrier that prevents atoms from moving from one side to the other. The situation is analogous, on an atomic scale, to basketballs colliding through a shower curtain.

FIGURE 20.12 Two gases can interact thermally through a very thin barrier.

Suppose that system 1 is initially at a higher temperature: $T_{1\text{i}}>T_{2\text{i}}$. This is not an equilibrium situation. The temperatures will change with time until the systems eventually reach a common final temperature $T_{\text{f}}$. If you watch the gases as one warms and the other cools, you see nothing happening. This interaction is quite different from a mechanical interaction in which, for example, you might see a piston move from one side toward the other. The only way in which the gases can interact is via molecular collisions at the boundary. This is a thermal interaction, and our goal is to understand how thermal interactions bring the systems to thermal equilibrium.

System 1 and system 2 begin with thermal energies

We’ve written the energies for monatomic gases; you could do the same calculation if one or both of the gases is diatomic by replacing the $\frac{3}{2}$ with $\frac{5}{2}$. Notice that we’ve omitted the subscript “th” to keep the notation manageable.

The total energy of the combined systems is $E_{\text{tot}}=E_{1\text{i}}+E_{2\text{i}}$. As systems 1 and 2 interact, their individual thermal energies $E_1$ and $E_2$ can change but their sum $E_{\text{tot}}$ remains constant. The system will have reached thermal equilibrium when the individual thermal energies reach final values $E_{1\text{f}}$ and $E_{2\text{f}}$ that no longer change.