Polyatomic Gases

Polyatomic Gases: Degrees of Freedom and Specific Heats

The kinetic theory we have applied so far assumed molecules are point masses with only translational motion. That works well for monatomic gases like helium or argon. But most gases are polyatomic — molecules with two or more atoms. A diatomic molecule like oxygen ($O_2$) or nitrogen ($N_2$) can rotate and vibrate, not just translate. Each of these additional modes of motion stores energy, and that changes the specific heat capacity.

The central idea is this: each independent way a molecule can store energy is called a degree of freedom. The equipartition theorem then says that each quadratic degree of freedom contributes $\frac{1}{2}k_BT$ to the average energy per molecule (or $\frac{1}{2}RT$ per mole). The total internal energy of the gas is the sum over all active degrees of freedom, and from that we can calculate $C_V$ and $C_P$.

Degrees of Freedom for Different Molecules

Monatomic gas (e.g. He, Ar, Ne): A single atom has only three translational degrees of freedom — motion along the $x$, $y$, and $z$ axes. No rotation (a point mass has negligible moment of inertia) and no vibration. So $f = 3$.

Diatomic gas (e.g. $N_2$, $O_2$, $H_2$, CO): A diatomic molecule has:

• 3 translational degrees of freedom (motion of the centre of mass)
• 2 rotational degrees of freedom (rotation about two axes perpendicular to the bond; rotation about the bond axis has negligible moment of inertia)
• 2 vibrational degrees of freedom (one for kinetic energy of vibration, one for potential energy of vibration)

That gives $f = 3 + 2 + 2 = 7$ in principle. But at ordinary temperatures, the vibrational modes are "frozen out" — they do not get excited because the energy gap between vibrational quantum levels is much larger than $k_BT$. So at room temperature, only translational and rotational modes contribute: $f = 3 + 2 = 5$.

Triatomic (linear) gas (e.g. $CO_2$): Three atoms in a straight line. Degrees of freedom:

• 3 translational
• 2 rotational (same as diatomic — rotation about the two axes perpendicular to the molecular axis)
• 4 vibrational (each atom can vibrate in various normal modes, but the total vibrational contribution is 4 quadratic terms: 2 kinetic + 2 potential, corresponding to 2 normal modes at ordinary temperatures)

At room temperature, only translational and rotational modes are active: $f = 3 + 2 = 5$.

Triatomic (non-linear) gas (e.g. $H_2O$, $SO_2$): Three atoms not in a straight line. Degrees of freedom:

• 3 translational
• 3 rotational (now all three axes have significant moment of inertia)
• 3 vibrational (3 normal modes, each contributing 2 quadratic terms = 6 vibrational degrees of freedom)

At room temperature, only translational and rotational modes are active: $f = 3 + 3 = 6$.

Applying Equipartition to Find $C_V$ and $C_P$

For a gas with $f$ active degrees of freedom per molecule, the average energy per molecule is:

For one mole ($N_A$ molecules), the internal energy is:

The molar specific heat at constant volume is:

Using the relation $C_P - C_V = R$ (which holds for ideal gases regardless of molecular structure), we get:

The ratio of specific heats (adiabatic index) is:

Summary Table of Predicted Values

Comparison with Experimental Values

The predictions match experimental data remarkably well for monatomic gases. For diatomic gases like $O_2$ and $N_2$ at room temperature, $\gamma \approx 1.40$, which agrees with $f = 5$. This confirms that vibrational modes are indeed frozen out at ordinary temperatures.

For triatomic gases, the agreement is less perfect. Water vapour ($H_2O$) at 100°C has $\gamma \approx 1.33$, matching the non-linear prediction. But $CO_2$ at room temperature has $\gamma \approx 1.30$, which is lower than the predicted 1.40. This discrepancy arises because some vibrational modes in $CO_2$ begin to get excited even at room temperature, effectively increasing $f$ beyond 5.

The Temperature Dependence of Specific Heats …