## Order by bond length?\(\mathrm{NO}, \mathrm{NO}^{+}, \mathrm{NO}^{-}\)

Answer:

The bond strength \(\text{increases}\) going from \(\mathrm{NO}^{-}\) to \(\mathrm{NO}^{+}\), and the bond length consequently shortens going from \(\mathrm{NO}^{-}\) to \(\mathrm{NO}^{+}\)

\(r_{\mathrm{N}-\mathrm{O}}^{\mathrm{NO}^{-}}>r_{\mathrm{N}-\mathrm{O}}^{\mathrm{NO}}>r_{\mathrm{N}-\mathrm{O}}^{\mathrm{NO}^{+}}\).

To start with, \(\mathrm{O}_{2}\) is (roughly) isoelectronic with \(\mathrm{NO}^{-[*]}\), and \(\mathrm{O}_{2}\) has two \(\pi^{*}\) antibonding electrons, as you should know from its MO diagram from your textbook. Thus, so does \(\mathrm{NO}^{-}\).

\(\left[^{*}\right]-\) except for the orbital ordering below the \(\pi^{*}\) (the\(2 b_{1}, 2 b_{2}\)) of these species.

To prove this, here is the MO diagram of NO (Miessler et al., Answer Key):

(The original was this; I added the orbital depictions and symmetry labels.) Quick overview of what the labels correspond to what MOs:

- \(1 a_{1}\) is the \(\sigma_{2 s}\) bonding MO.
- \(2 a_{1}\) is the \(\sigma_{2 s}^{*}\) antibonding MO.
- \(1 b_{1}\) is the \(\pi_{2 p_{x}}\) bonding MO.
- \(1 b_{2}\) is the \(\pi_{2 p_{x}}\) bonding MO.
- \(3 a_{1}\) is the \(\sigma_{2 p_{z}}\) bonding MO, but it’s relatively nonbonding with respect to oxygen.
- \(2 b_{1}\) is the \(\pi_{2 p_{x}}^{*}\) antibonding MO.
- \(2 b_{2}\) is the \(\pi_{2 p_{y}}^{*}\) antibonding MO.
- \(4 a_{1}\) is the \(\pi_{2 p_{z}}^{*}\) antibonding MO.

Note that for \(\mathrm{O}_{2}\), the \(1 b_{1}, 1 b_{2}\) and \(3 a_{1}\) orbitals are switched in energy. From this MO diagram, we can see that:

- \(\mathrm{NO}^{+}\) has \(\mathbf{0} \pi^{*}\) antibonding electrons.
- \(\text { NO }\) has \(\mathbf{1} \pi^{*}\) antibonding electron.
- \(\mathrm{NO}^{-}\) has \(2 \pi^{*}\) antibonding electrons.

As the number of antibonding electrons \(\text{increases}\), the \(N − O\) bond \(\text{weakens}\), having acquired antibonding character (which as the name suggests, goes against making a bond).

Thus, the bond strength \(\text{increases}\) going from \(\mathrm{NO}^{-}\) to \(\mathrm{NO}^{+}\), and the bond length consequently shortens going from \(\mathrm{NO}^{-}\) to \(\mathrm{NO}^{+}\).