Order by bond length? "NO", "NO"^(+), "NO"^(-)

Sep 17, 2017

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

${r}_{{\text{N"-"O")^("NO"^(-)) > r_("N"-"O")^("NO") > r_("N"-"O")^("NO}}^{+}}$

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

$\textcolor{red}{\left[\text{*}\right]}$ - except for the orbital ordering below the ${\pi}^{\text{*}}$ (the $2 {b}_{1} , 2 {b}_{2}$) of these species

To prove this, here is the MO diagram of $\text{NO}$ (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 ${\sigma}_{2 s}^{\text{*}}$ antibonding MO.
• $1 {b}_{1}$ is the ${\pi}_{2 {p}_{x}}$ bonding MO.
• $1 {b}_{2}$ is the ${\pi}_{2 {p}_{y}}$ 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 ${\pi}_{2 {p}_{x}}^{\text{*}}$ antibonding MO.
• $2 {b}_{2}$ is the ${\pi}_{2 {p}_{y}}^{\text{*}}$ antibonding MO.
• $4 {a}_{1}$ is the ${\sigma}_{2 {p}_{z}}^{\text{*}}$ antibonding MO.

Note that for ${\text{O}}_{2}$, the $1 {b}_{1} , 1 {b}_{2}$ and $3 {a}_{1}$ orbitals are switched in energy. From this MO diagram, we can see that:

• ${\text{NO}}^{+}$ has $\boldsymbol{0}$ ${\pi}^{\text{*}}$ antibonding electrons.
• $\text{NO}$ has $\boldsymbol{1}$ ${\pi}^{\text{*}}$ antibonding electron.
• ${\text{NO}}^{-}$ has $\boldsymbol{2}$ ${\pi}^{\text{*}}$ antibonding electrons.

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

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