How many vibrational modes does a linear molecule of N atoms have?

To determine the number of vibrational modes in a linear molecule with N atoms, we can use the general formula for vibrational modes of a system in terms of the number of atoms (N) and the degrees of freedom.

For any molecule, the total degrees of freedom (in three-dimensional space) is given by 3 times the number of atoms (3N). However, when considering molecular vibrations, we need to subtract the translational and rotational degrees of freedom.

In the case of a linear molecule, it has 3 translational degrees of freedom (movement along the x, y, and z axes) and 2 rotational degrees of freedom (rotation about two axes perpendicular to the molecular axis). Therefore, we subtract 5 degrees of freedom:

• 3 translational for moving in space.

• 2 rotational for the motion around two perpendicular axes.

This gives us the formula:

Total vibrational modes = Total degrees of freedom - (Translational + Rotational)

= 3N - (3 + 2)

= 3N - 5.

Therefore, the number of vibrational modes for a linear molecule comprised of N atoms is represented by 3N - 5. This is why the correct answer aligns