For the sulphate ion, SO_4^2-, a) What is the total number of vibrational modes expected? b) In which irreducible representations of the appropriate point group will vibrational modes be IR active in the free SO_4^2 ion? c) In which irreducible representations of the appropriate point group will vibrational modes be Roman active in the free SO_4^2- ion? d) What symmetry operation is necessary in a point group for there to be no common frequencies (mutual exclusion) between the IR and Raman spectrum of a species? Would the SO_4^2- ion shows this mutual exclusion? e) Give the symmetry point group of a SO_4^2- ion chelated through two O atoms to a metal ion. Should IR spectroscopy be able to distinguish a free SO_4^2- ion clearly from a SO_4^2 ligand coordinated to a metal? Explain.