2.3 Heat capacity and equipartition of energy  (Page 2/6)

We would like to generalize our results to ideal gases with more than one atom per molecule. In such systems, the molecules can have other forms of energy beside translational kinetic energy, such as rotational kinetic energy and vibrational kinetic and potential energies. We will see that a simple rule lets us determine the average energies present in these forms and solve problems in much the same way as we have for monatomic gases.

Degrees of freedom

In the previous section, we found that $\frac{1}{2}m\stackrel{\text{–}}{v^2}=\frac{3}{2}{k}_{\text{B}}T$ and $\stackrel{\text{–}}{v^2}=3\stackrel{\text{–}}{v_x^2}$ , from which it follows that $\frac{1}{2}m\stackrel{\text{–}}{v_x^2}=\frac{1}{2}{k}_{\text{B}}T$ . The same equation holds for $\stackrel{\text{–}}{v_y^2}$ and for $\stackrel{\text{–}}{v_z^2}$ . Thus, we can look at our energy of $\frac{3}{2}{k}_{\text{B}}T$ as the sum of contributions of $\frac{1}{2}{k}_{\text{B}}T$ from each of the three dimensions of translational motion. Shifting to the gas as a whole, we see that the 3 in the formula $C_V=\frac{3}{2}R$ also reflects those three dimensions. We define a degree of freedom    as an independent possible motion of a molecule, such as each of the three dimensions of translation. Then, letting d represent the number of degrees of freedom, the molar heat capacity at constant volume of a monatomic ideal gas is $C_V=\frac{d}{2}R,$ where $d=3$ .

The branch of physics called statistical mechanics tells us, and experiment confirms, that $C_V$ of any ideal gas is given by this equation, regardless of the number of degrees of freedom. This fact follows from a more general result, the equipartition theorem    , which holds in classical (non-quantum) thermodynamics for systems in thermal equilibrium under technical conditions that are beyond our scope. Here, we mention only that in a system, the energy is shared among the degrees of freedom by collisions.

This result is due to the Scottish physicist James Clerk Maxwell (1831−1871), whose name will appear several more times in this book.

For example, consider a diatomic ideal gas (a good model for nitrogen, ${\text{N}}_2,$ and oxygen, ${\text{O}}_2).$ Such a gas has more degrees of freedom than a monatomic gas. In addition to the three degrees of freedom for translation, it has two degrees of freedom for rotation perpendicular to its axis. Furthermore, the molecule can vibrate along its axis. This motion is often modeled by imagining a spring connecting the two atoms, and we know from simple harmonic motion that such motion has both kinetic and potential energy. Each of these forms of energy corresponds to a degree of freedom, giving two more.

We might expect that for a diatomic gas, we should use 7 as the number of degrees of freedom; classically, if the molecules of a gas had only translational kinetic energy, collisions between molecules would soon make them rotate and vibrate. However, as explained in the previous module, quantum mechanics controls which degrees of freedom are active. The result is shown in [link] . Both rotational and vibrational energies are limited to discrete values. For temperatures below about 60 K, the energies of hydrogen molecules are too low for a collision to bring the rotational state or vibrational state of a molecule from the lowest energy to the second lowest, so the only form of energy is translational kinetic energy, and $d=3$ or $C_V=3R\text{/}2$ as in a monatomic gas. Above that temperature, the two rotational degrees of freedom begin to contribute, that is, some molecules are excited to the rotational state with the second-lowest energy. (This temperature is much lower than that where rotations of monatomic gases contribute, because diatomic molecules have much higher rotational inertias and hence much lower rotational energies.) From about room temperature (a bit less than 300 K) to about 600 K, the rotational degrees of freedom are fully active, but the vibrational ones are not, and $d=5$ . Then, finally, above about 3000 K, the vibrational degrees of freedom are fully active, and $d=7$ as the classical theory predicted.