Determine the electron configurations for cn+, cn, and cn−. part a which species has the strongest c−n bond?

For Lewis theory, the most stable species will have a complete octet for as many atoms as possible. Construct Lewis dot structures for each species. You should see that CN+ cannot give a complete octet to the C atom unless a quadruple bond - unknown except in transition metals - is formed. CN will have an odd number of electrons, and is thus a free radical and unstable with respect to dimerization (it forms cyanogen). CN-, the familiar cyanide ion, gives both C and N a complete octet with a triple bond, and is thus the most stable. Molecular orbital theory is a bit more complex. Nitrogen and carbon are close enough in electronegativity, so the orbitals from the C atom will mix with the same orbitals from the N atom. The molecular orbitals formed will be sigma2s, sigma*2s, pi2p, sigma2p, pi*2p, and sigma*2p. The * denotes an antibonding orbital; these are higher in energy, and electrons placed into these orbitals weaken the bonding between two atoms. CN+ will completely fill the sigma2s, sigma*2s, and pi2p orbitals. CN will add an electron in the bonding sigma2p orbital, and the atoms are thus more strongly bonded than in CN+. CN- fills the sigma2p orbital, and the addition of another bonding electron means that this species has the strongest bond of the three. I might have the names of some of the filled levels incorrect; the energy levels of the sigma2p and pi2p swap at some point. This concept is hard to explain without a picture; see the link. Thus, both MO and Lewis theory predict CN- as the most stable species, a prediction that matches well with experimental data.