Animated Solution for Physics - Thermodynamics: A vessel contains a mixture of one mole of oxygen and two moles of nitrogen at 300 K. The ratio of the average rotational kinetic energy per ${\text{O}}_2$ molecule to per ${\text{N}}_2$ molecule is

This problem is a beautiful demonstration of the Law of Equipartition of Energy, a cornerstone of the Kinetic Theory of Gases. It shows us that in the microscopic world of thermal equilibrium, energy is a great equalizer.

Imagine a vessel containing a mixture of oxygen (${\text{O}}_2$) and nitrogen (${\text{N}}_2$) gases. The entire mixture is sitting at a comfortable room temperature of $300\text{ K}$. The question asks us to find the ratio of the average rotational kinetic energy per molecule for these two different gases.

At first glance, you might think, "Wait, oxygen and nitrogen have different masses and different moments of inertia. Shouldn't their rotational energies be different?" Let's see why that intuition is actually incorrect in the realm of thermodynamics.

The key to unlocking this problem is the Law of Equipartition of Energy. This fundamental principle states that for a system in thermal equilibrium at temperature $T$, the average kinetic energy associated with each active degree of freedom is exactly $\frac{1}{2}kT$, where $k$ is the Boltzmann constant.

Now, let's look at our specific molecules. Both ${\text{O}}_2$ and ${\text{N}}_2$ are diatomic molecules. Geometrically, they resemble tiny dumbbells. Because of this linear shape, they can rotate significantly along two independent axes that are perpendicular to the bond connecting the two atoms. (Rotation along the bond axis itself has a negligible moment of inertia and doesn't store thermal energy at normal temperatures).

Therefore, both oxygen and nitrogen molecules possess exactly $2$ rotational degrees of freedom ($f_{rot}=2$).

Let's calculate the average rotational kinetic energy ($E_{rot}$) for a single molecule of either gas. We simply multiply the number of rotational degrees of freedom by the energy per degree of freedom:

Substituting $f_{rot}=2$:

Notice the elegance of this result! The average rotational kinetic energy depends only on the absolute temperature ($T$). It does not depend on the mass of the molecule, the number of moles present in the vessel, or the moment of inertia.

Since both the oxygen and nitrogen gases are in the same vessel, they are in thermal equilibrium and share the exact same temperature ($T=300\text{ K}$).

For an ${\text{O}}_2$ molecule: $E_{rot}({\text{O}}_2)=kT$ For an ${\text{N}}_2$ molecule: $E_{rot}({\text{N}}_2)=kT$

Taking the ratio of the two:

The energy is distributed perfectly equally. This is the magic of equipartition!