Interpret Statement I

For a gas in thermal equilibrium, molecular energies are statistically distributed.

• Translational speeds follow the Maxwell distribution.
• Rotational degrees of freedom of molecules in thermal equilibrium also have a thermal distribution governed by Boltzmann statistics.

For a diatomic molecule, rotational motion contributes energy, and over a large ensemble the rotational energies are distributed statistically at a given temperature.

Hence, Statement I is taken as true.

Interpret Statement II

Statement II says:

In a diatomic molecule, the rotational energy at a given temperature equals the translational kinetic energy for each molecule.

This is incorrect.

By the equipartition theorem:

• Average translational kinetic energy per molecule: $\langle E_{\text{trans}} \rangle = \frac{3}{2}kT$
• Average rotational energy per diatomic molecule (ignoring vibration, with 2 active rotational degrees): $\langle E_{\text{rot}} \rangle = \frac{2}{2}kT = kT$

These are not equal, even on average.

Also, the statement says "for each molecule," which is even more wrong, because individual molecules do not all have the same energy.

So, Statement II is false.

Choose the correct option

• Statement I: True
• Statement II: False

Therefore, the correct option is: $\boxed{\text{C}}$

Compare with stored answer

Stored correct answer: C

My derived answer matches the stored answer.