The translational degrees of freedom (ft ) and rotational degrees of freedom (fr ) of CH4 molecule are :

The correct answer is f_t = 3 and f_r = 3.

To understand why, we need to recall the concept of degrees of freedom in kinetic theory of gases. A degree of freedom is any independent way in which a molecule can store energy — either by moving (translational), rotating (rotational), or vibrating (vibrational).

Step 1: Identify the structure of CH₄ (Methane)

CH₄ is methane. It consists of 1 carbon atom bonded to 4 hydrogen atoms arranged in a tetrahedral (3D) geometry. This means it is a non-linear polyatomic molecule — its atoms do not all lie on a single straight line.

Step 2: Translational Degrees of Freedom (f_t)

Translational degrees of freedom refer to the number of independent directions in which the centre of mass of the molecule can move in space.

Since CH₄ exists in 3-dimensional space, its centre of mass can move independently along:

→ the x-axis
→ the y-axis
→ the z-axis

Therefore, f_t = 3 for all molecules (monoatomic, diatomic, or polyatomic) that exist in 3D space.

Step 3: Rotational Degrees of Freedom (f_r)

Rotational degrees of freedom refer to the number of independent axes about which the molecule can rotate and store appreciable rotational kinetic energy. This depends on the geometry of the molecule:

• Monoatomic molecules (e.g., He, Ar): The molecule is a point mass — rotation about any axis through the atom carries negligible (effectively zero) moment of inertia, so f_r = 0.

• Linear molecules (e.g., O₂, CO₂, HCl): All atoms lie on one axis. Rotation about that bond axis has negligible moment of inertia, leaving only 2 meaningful rotational axes. So f_r = 2.

• Non-linear polyatomic molecules (e.g., H₂O, NH₃, CH₄): Atoms are arranged in 3D space. The molecule has three distinct, mutually perpendicular axes (x, y, z) about which it can rotate with significant moment of inertia. So f_r = 3.

Since CH₄ is a non-linear polyatomic molecule with a 3D tetrahedral shape, it can rotate about all three independent axes in space.

Therefore, f_r = 3.

Step 4: Summary using the Equipartition Theorem Framework

For any molecule in 3D space, the total classical degrees of freedom = 3N (where N = number of atoms). These are distributed among translational, rotational, and vibrational modes. For CH₄ (N = 5):

Total degrees of freedom = 3 × 5 = 15
• Translational: f_t = 3
• Rotational: f_r = 3 (non-linear molecule)
• Vibrational: f_v = 15 - 3 - 3 = 9

Conclusion:

For the CH₄ molecule:

f_t=3 (translational — movement along x, y, z axes)

f_r=3 (rotational — rotation about all three axes, because CH₄ is non-linear/tetrahedral)

This corresponds to the correct option: f_t = 3 and f_r = 3.