Diatomic molecules are a bigger challenge. How many degrees of freedom does a diatomic molecule have? FIGURE 20.10 shows a diatomic molecule, such as molecular nitrogen $\text{N}{}_2$, oriented along the x-axis. Three degrees of freedom are associated with the molecule’s translational kinetic energy. The molecule can have a dumbbell-like end-over-end rotation about either the y-axis or the z-axis. It can also rotate about its own axis. These are three rotational degrees of freedom. The two atoms can also vibrate back and forth, stretching and compressing the molecular bond. This vibrational motion has both kinetic and potential energy—thus two more degrees of freedom.

FIGURE 20.10 A diatomic molecule can rotate or vibrate.

Altogether, then, a diatomic molecule has eight degrees of freedom, and we would expect the thermal energy of a gas of diatomic molecules to be $E_{\text{th}}=4N{k}_{\text{B}}T$. The analysis we followed for a monatomic gas would then lead to the prediction $C_{\text{V}}=4R=33.2\text{J}\text{/}\text{mol}\text{K}$. As compelling as this reasoning seems to be, this is not the experimental value of $C_{\text{V}}$ that was reported for diatomic gases in Table 19.4. Instead, we found $C_{\text{V}}=20.8\text{J}\text{/}\text{mol}\text{K}$.

Why should a theory that works so well for monatomic gases and solids fail so miserably for diatomic molecules? To see what’s going on, notice that $20.8\text{J/mol K}=\frac{5}{2}R$. A monatomic gas, with three degrees of freedom, has $C_{\text{V}}=\frac{3}{2}R$. A solid, with six degrees of freedom, has $C=3R$. A diatomic gas would have $C_{\text{V}}=\frac{5}{2}R$ if it had five degrees of freedom, not eight.

This discrepancy was a major conundrum as statistical physics developed in the late 19th century. Although it was not recognized as such at the time, we are here seeing our first evidence for the breakdown of classical Newtonian physics. Classically, a diatomic molecule has eight degrees of freedom. The equipartition theorem doesn’t distinguish between them; all eight should have the same energy. But atoms and molecules are not classical particles. It took the development of quantum theory in the 1920s to accurately characterize the behavior of atoms and molecules. We don’t yet have the tools needed to see why, but quantum effects prevent three of the modes—the two vibrational modes and the rotation of the molecule about its own axis—from being active at room temperature.

FIGURE 20.11 shows $C_{\text{V}}$ as a function of temperature for hydrogen gas. $C_{\text{V}}$ is right at $\frac{5}{2}R$ for temperatures from $\approx 200\text{K}$ up to $\approx 800\text{K}$. But at very low temperatures $C_{\text{V}}$ drops to the monatomic-gas value $\frac{3}{2}R$. The two rotational modes become “frozen out” and the nonrotating molecule has only translational kinetic energy. Quantum physics can explain this, but not Newtonian physics. You can also see that the two vibrational modes do become active at very high temperatures, where $C_{\text{V}}$ rises to $\frac{7}{2}R$. Thus the real answer to What’s wrong? is that Newtonian physics is not the right physics for describing atoms and molecules. We are somewhat fortunate that Newtonian physics is adequate to understand monatomic gases and solids, at least at room temperature.

FIGURE 20.11 Hydrogen molar specific heat at constant volume as a function of temperature.

Accepting the quantum result that a diatomic gas has only five degrees of freedom at commonly used temperatures (the translational degrees of freedom and the two end-over-end rotations), we find

A diatomic gas has more thermal energy than a monatomic gas at the same temperature because the molecules have rotational as well as translational kinetic energy.

While the micro/macro connection firmly establishes the atomic structure of matter, it also heralds the need for a new theory of matter at the atomic level. That is a task we will take up in Part VIII. For now, TABLE 20.1 summarizes what we have learned from kinetic theory about thermal energy and molar specific heats.