Answer
(a) 3 (b) 6 (c) 12
Work Step by Step
-- To compute the number of vibrational modes, we subtract the number of rotational modes and the number of translation modes from the total number of modes.
-- For all molecules, the total number of modes is equal 3N (N is number of atoms in a molecule) because each atom has three degrees of freedoms in x, y and z directions.
-- For all molecules, the number of translation modes is equal 3 because the entire molecule can only move in x, y and z directions.
-- For linear molecules, there are only 2 rotation modes as the molecules can only rotate in yz and xz plants but cannot rotate in xy plane as shown by the image below. In comparison, for non-linear molecules, there are 3 rotation modes as the molecules can rotate in yz, xz and xy plants as shown by the image below.
-- So, the number of vibrational modes for linear molecules is equal (3N-5) and the number of vibrational modes for non-linear molecules is equal (3N-6). In this problem, all three molecules are non-linear molecules with (3N-6) vibrational modes.
