Degrees of Freedom

A degree of freedom (DOF) is an independent way in which a molecule can store energy. The number of degrees of freedom directly controls the molar heat capacities $C_v$ and $C_p$ and hence the adiabatic ratio $\gamma$.

Concept

For a single molecule:

• Translation: 3 DOF (motion along $x$, $y$, $z$).
• Rotation: depends on geometry.
• Vibration: each vibrational mode counts as 2 DOF (kinetic + potential).

Monatomic gas (He, Ar, Ne)

• 3 translational DOF.
• Rotation about an axis through the centre carries no significant moment of inertia for a point-like atom.
• No vibration.
• Total $f = 3$.

Diatomic gas (O$_2$, N$_2$, H$_2$) — rigid rotator at moderate $T$

• 3 translational + 2 rotational (perpendicular to bond axis) DOF.
• Rotation about the bond axis is negligible (very small moment of inertia).
• Vibration is frozen out at room temperature.
• Total $f = 5$.

At high temperatures vibration becomes excited: add 2 more DOF, giving $f = 7$.

Polyatomic gas (non-linear: H$_2$O, CH$_4$)

• 3 translational + 3 rotational DOF.
• Plus any active vibrational modes.
• Total at moderate T $f = 6$.

Heat capacities

By equipartition (next sub-topic), each DOF contributes $\tfrac{1}{2} R$ to $C_v$ per mole:

$C_v = \frac{f}{2} R, \quad C_p = C_v + R = \frac{f+2}{2} R, \quad \gamma = \frac{C_p}{C_v} = 1 + \frac{2}{f}$

Worked Example

Q: A gas has $\gamma = 1.40$. Identify the type of gas and compute its $C_v$ and $C_p$ in J/(mol·K).

A: $\gamma = 1 + 2/f = 1.40 \Rightarrow 2/f = 0.40 \Rightarrow f = 5$. Diatomic gas (e.g., N$_2$, O$_2$).

• $C_v = (5/2)R = (5/2)(8.314) \approx 20.8$ J/(mol·K).
• $C_p = (7/2)R \approx 29.1$ J/(mol·K).

Q2: Why is the measured $\gamma$ for H$_2$ at very low temperatures closer to 5/3 than to 7/5?

A: At low $T$, even rotational modes get "frozen out" (quantum mechanics — rotational quanta become comparable to $k_B T$), so only 3 translational DOF remain active. $\gamma \to 1 + 2/3 = 5/3$.

Common Confusions

• Each vibrational mode contributes 2 DOF (one KE, one PE), not 1.
• For a linear molecule, only 2 rotational DOF count, not 3 — rotation about the bond axis has negligible moment of inertia.
• DOF can be temperature-dependent: at low $T$ rotations may freeze; at very high $T$ vibrations switch on.

Key Takeaways

• $C_v = \tfrac{f}{2} R$ and $\gamma = 1 + 2/f$.
• Monatomic: $f = 3$, $\gamma = 5/3$. Diatomic (rigid): $f = 5$, $\gamma = 7/5$. Polyatomic: $f \ge 6$, $\gamma \le 4/3$.
• More DOF $\Rightarrow$ more "energy storage capacity" per molecule $\Rightarrow$ larger $C_v$ and smaller $\gamma$.