Question

Briefly explain how we arrive at the formula 3N-6 for calculating the number of vibrational modes for a non-linear molecule. Why is the formula 3N-5 for linear molecules? (Be sure to talk about degrees of freedom in your explanation).

Answer 1

3n degrees of freedom describe the motion of a molecule in relation to the coordinates (x,y,z). The 3n degrees of freedom also describe the translational, rotational, and vibrational motions of the molecule. There are three degrees of freedom for translational, movement through space, and rotational motion, each for a nonlinear molecule. Therefore, translational and rotational can move and rotate around each of the three Cartesian axes. However, a linear molecule can only rotate around 2 of the Cartesian axes because the rotation about the molecular axis does not represent a change of the nuclear coordinates. If you subtract the translational and rotational degrees of freedom, you obtain the following equations shown below for the degrees of vibrational freedom.

The degrees of vibrational modes for linear molecules = 3n − 5

The degrees of freedom for nonlinear molecules = 3n−6

where, n is equal to the number of atoms within the molecule of interest.