Degrees of freedom and equipartition

Where does the energy of a gas go?

Heat a balloon of neon and a balloon of carbon dioxide by the same amount. The neon gets hotter. Why? The carbon dioxide molecule can do more things with the energy. It can move, and it can also turn. So less of the energy goes into moving, and moving is what we read as temperature.

In this lesson we count the ways a molecule can store energy. Then we share the energy out among them.

Degrees of freedom

A degree of freedom is one independent way in which a molecule can store energy. Each one adds a squared term to the energy.

Moving (translation). A molecule can move along $x$, along $y$ and along $z$. Its kinetic energy of motion is

That is three squared terms. Every molecule has these 3 translational degrees of freedom.

Turning (rotation). A molecule of two atoms, like a tiny dumbbell, can also turn. It turns about two axes at right angles to the line joining the atoms. Each turn adds a term $\tfrac12 I\omega^2$, where $I$ is the moment of inertia. So a linear molecule has 2 rotational degrees of freedom.

What about turning about the line through both atoms? The atoms are almost points on that line. The moment of inertia about it is tiny, and so is its energy. We do not count it.

A bent molecule, like water or methane, can turn about all three axes. It has 3 rotational degrees of freedom.

Vibrating. At high temperatures the bond between two atoms can stretch and squeeze, like a spring. One vibration has kinetic energy $\tfrac12 m u^2$ and potential energy $\tfrac12 k y^2$. That is two squared terms for one vibration.

So the count $f$ for a few molecules is:

• Monatomic (neon, argon): $f = 3$.
• Rigid diatomic or linear (carbon dioxide at room temperature, treated as rigid): $f = 3 + 2 = 5$.
• Rigid non-linear (methane, steam): $f = 3 + 3 = 6$.
• Diatomic that also vibrates: $f = 3 + 2 + 2 = 7$.

The law of equipartition

How is the energy shared? The answer is a law of classical physics.

Law of equipartition of energy. In thermal equilibrium at temperature $T$, every squared term in the energy of a molecule has, on average, an energy $\tfrac12 k_BT$.

Here $k_B = 1.38 \times 10^{-23}\ \text{J/K}$ is the Boltzmann constant. So one molecule with $f$ degrees of freedom has an average energy

One mole has $N_A$ molecules, and $N_A k_B = R$. So the internal energy of $n$ moles of an ideal gas is

There is one important result here. Every molecule, of any gas, has $\tfrac32 k_BT$ of translational energy. Neon and carbon dioxide at the same temperature move with the same average kinetic energy. The extra energy of carbon dioxide sits in its turning.

Set the turns to 0 and the vibrations to 0. That is neon. Now set the turns to 2. The bar for moving does not change. Only a new bar appears. Raise the temperature: do all the bars grow by the same factor?

A worked example

Take $0.50\ \text{mol}$ of neon and $0.50\ \text{mol}$ of carbon dioxide, both at $400\ \text{K}$. Treat carbon dioxide as rigid. Which has more internal energy?

1. Neon. $f = 3$, so $U = \tfrac32 \times 0.50\ \text{mol} \times 8.31\ \text{J/(mol K)} \times 400\ \text{K} = 2493\ \text{J}$.
2. Carbon dioxide. $f = 5$, so $U = \tfrac52 \times 0.50\ \text{mol} \times 8.31\ \text{J/(mol K)} \times 400\ \text{K} = 4155\ \text{J}$.
3. Compare. Both have the same translational energy, $2493\ \text{J}$. Carbon dioxide has another $1662\ \text{J}$ in turning.

Practise with new numbers

Try it in Python

The program counts the degrees of freedom of four gases. Then it shares the energy at $300\ \text{K}$.

Common mistakes

• Giving each degree of freedom $k_BT$. Each squared term gets half of that, $\tfrac12 k_BT$.
• Counting a vibration once. A vibration has kinetic and potential energy, so it counts as two terms.
• Counting three turns for a linear molecule. The turn about its own axis stores almost no energy.
• Thinking a heavier molecule moves with more energy. At the same temperature, every molecule has the same translational energy, $\tfrac32 k_BT$.

Remember these

• Translation: 3 for every molecule. Rotation: 2 for a linear molecule, 3 for a bent one. Each active vibration: 2.
• Equipartition: $\tfrac12 k_BT$ for each squared term, on average.
• One molecule: $E = \tfrac{f}{2}k_BT$. One gas: $U = \tfrac{f}{2}nRT$.
• Translational energy is $\tfrac32 k_BT$ for every gas.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. How many degrees of freedom does a molecule of methane have, if it does not vibrate?

2. What is the average energy of one argon atom at $500\ \text{K}$?

3. Neon and carbon dioxide are at the same temperature. Which has the larger average translational kinetic energy per molecule?

4. A diatomic molecule starts to vibrate when the gas is very hot. By how much does $f$ grow?