1. Recall the meaning of degrees of freedom

For a molecule, the degrees of freedom are the independent ways in which it can possess energy.

By the equipartition theorem, for each quadratic degree of freedom, the average energy is:

$\frac{1}{2}kT \quad \text{per molecule}$

or

$\frac{1}{2}RT \quad \text{per mole}$

2. Check statement (A)

Statement (A): A molecule with $n$ degrees of freedom has $n^2$ different ways of storing energy.

This is incorrect.

If a molecule has $n$ degrees of freedom, then it has $n$ independent modes/degrees of energy storage, not $n^2$.

So, (A) is false.

3. Check statement (B)

Statement (B): Each degree of freedom is associated with $\frac{1}{2}RT$ average energy per mole.

This is the standard equipartition result.

So, (B) is true.

4. Check statement (C)

Statement (C): A monatomic gas molecule has 1 rotational degree of freedom whereas diatomic molecule has 2 rotational degrees of freedom.

• A monatomic gas molecule is treated as a point particle, so it has no rotational degree of freedom contributing in the usual kinetic theory sense.
• A diatomic molecule has 2 rotational degrees of freedom at ordinary temperatures.

Thus the first part is wrong.

So, (C) is false.

5. Check statement (D)

Statement (D): $\mathrm{CH_4}$ has a total of 6 degrees of freedom.

$\mathrm{CH_4}$ is a non-linear polyatomic molecule.

For a non-linear molecule:

• Translational degrees of freedom $= 3$
• Rotational degrees of freedom $= 3$

Hence total (ignoring vibrational contribution in this basic count):

$3 + 3 = 6$

So, (D) is true.

6. Determine the correct option

True statements are:

$\boxed{(B) \text{ and } (D)}$

This corresponds to Option B.

7. Comparison with stored answer

Stored correct answer: B

Our derived answer: B

Hence, they agree.