Degrees of Freedom — Kinetic Theory of Gases

1. What Are Degrees of Freedom?

Degrees of freedom refer to the independent ways in which a molecule can move or store energy. These include motion along different axes and, for some molecules, rotation and vibration.

The more complex the molecule, the more degrees of freedom it usually has.

2. Translational Degrees of Freedom

All gas molecules—whether atoms or complex molecules—can move freely in three dimensions. This gives every gas molecule:

3 \text{ translational degrees of freedom}

The molecule can move along:

x-axis
y-axis
z-axis

2.1. Meaning

These motions contribute directly to the kinetic energy and pressure of the gas.

3. Rotational Degrees of Freedom

Depending on the structure of the molecule, rotation contributes additional degrees of freedom.

3.1. Monatomic Gases

Monatomic gases such as He, Ne, Ar cannot rotate about axes that change their kinetic energy meaningfully. So they have:

0 \text{ rotational degrees of freedom}

3.2. Diatomic Gases

Diatomic molecules (like O₂, N₂, H₂) can rotate about two perpendicular axes. So they have:

2 \text{ rotational degrees of freedom}

Rotation about the molecular axis is usually negligible at normal temperatures due to small moment of inertia.

3.3. Polyatomic Gases

Non-linear polyatomic molecules (like H₂O and CO₂) can rotate about all three axes. So they have:

3 \text{ rotational degrees of freedom}

4. Vibrational Degrees of Freedom

In vibrational motion, atoms in a molecule move back and forth relative to each other. Each vibrational mode contributes:

2 \text{ degrees of freedom per vibrational mode}

However, vibrational modes become active only at high temperatures because they require more energy.

4.1. Example

At room temperature, diatomic gases usually do not show vibrational motion, but at high temperatures, these modes become significant.

5. Total Degrees of Freedom for Different Gases

The total degrees of freedom (f) depend on the type of molecule:

5.1. Monatomic Gas

f = 3

(Only translational motion)

5.2. Diatomic Gas (Normal Temperatures)

f = 3 + 2 = 5

(Translational + rotational)

5.3. Polyatomic Gas (Non-Linear)

f = 3 + 3 = 6