Diatomic Gases

The Specific Heat Capacity of Diatomic Gases

The kinetic theory of gases, when applied to diatomic molecules, reveals a richer story than for monatomic gases. A diatomic molecule is not a point particle; it can rotate and vibrate. Each of these additional motions provides a new "store" for energy, and the equipartition theorem tells us exactly how much energy each store holds.

For a diatomic molecule, we consider three distinct types of motion:

1. Translational motion of the centre of mass (3 degrees of freedom).
2. Rotational motion about axes perpendicular to the line joining the two atoms (2 degrees of freedom). Rotation about the line joining the atoms (the molecular axis) is neglected because the moment of inertia about that axis is negligible.
3. Vibrational motion of the two atoms relative to each other (1 degree of freedom for kinetic energy of vibration + 1 degree of freedom for potential energy of vibration = 2 degrees of freedom).

At ordinary temperatures (around room temperature), the vibrational modes are not fully excited. The molecule behaves as a "rigid rotator" — it can translate and rotate, but it does not vibrate. Therefore, at room temperature, a diatomic gas effectively has only 5 active degrees of freedom: 3 translational and 2 rotational.

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Molar Specific Heats for a Diatomic Gas (Rigid Rotator Model)

Using the equipartition theorem, the total internal energy of one mole of a diatomic gas (with $f=5$) is the sum of the energies in each active degree of freedom.

Each translational degree of freedom contributes $\frac{1}{2}RT$ per mole.

Each rotational degree of freedom contributes $\frac{1}{2}RT$ per mole.

Therefore, the total internal energy $U$ per mole is:

From this, we can directly calculate the molar specific heat at constant volume, $C_V$.

The molar specific heat at constant pressure, $C_P$, is then found using the universal relation $C_P - C_V = R$.

Finally, the ratio of specific heats, $\gamma$, is:

These theoretical values — $C_V = \frac{5}{2}R$, $C_P = \frac{7}{2}R$, and $\gamma = 1.40$ — agree remarkably well with experimental data for many diatomic gases like hydrogen ($H_2$), nitrogen ($N_2$), and oxygen ($O_2$) at room temperature.

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The Role of Vibrational Modes at High Temperatures

What happens when the temperature is raised significantly? The vibrational mode of the molecule becomes active. A diatomic molecule's vibration is like a simple harmonic oscillator. It has two degrees of freedom: one for kinetic energy and one for potential energy. According to equipartition, each contributes $\frac{1}{2}RT$, so the vibrational mode contributes a total of $RT$ to the molar internal energy.

When vibration is active, the total number of degrees of freedom becomes $f = 3 + 2 + 2 = 7$. The internal energy per mole is then:

This leads to the following high-temperature predictions:

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The Anomalous Behaviour of Hydrogen at Low Temperatures

Hydrogen ($H_2$) provides a fascinating test case at low temperatures. As the temperature drops, the rotational modes also begin to "freeze out." At very low temperatures (below about 100 K), the hydrogen molecule has insufficient energy to rotate. The only active degrees of freedom are the three translational ones.

In this low-temperature regime, hydrogen behaves like a monatomic gas: