Animated Solution for Chemistry - Chemical Bonding and Molecular Structure: The bond order and the magnetic characteristics of ${\text{CN}}^{-}$ are

Molecular Orbital Theory (MOT) is one of the most powerful tools in chemistry for predicting the stability and magnetic properties of molecules. In this walkthrough, we will dissect the cyanide ion, ${\text{CN}}^{-}$, step-by-step to uncover its bond order and magnetic nature.

The very first step in any MOT problem is to accurately count the total number of electrons in the species. This dictates which energy level diagram we will use.

For the ${\text{CN}}^{-}$ ion: - Carbon (C) contributes 6 electrons. - Nitrogen (N) contributes 7 electrons. - The negative charge indicates an extra electron, adding 1 more.

Total Electrons = $6+7+1=14$.

This is a crucial number! Any diatomic species with 14 electrons is isoelectronic with the nitrogen molecule (${\text{N}}_2$). Because it has 14 or fewer electrons, it experiences significant s-p mixing, which alters the standard order of molecular orbitals.

Due to s-p mixing, the ${\pi }_{2p}$ orbitals are pushed lower in energy than the ${\sigma }_{2p_z}$ orbital. Let's fill our 14 electrons into the molecular orbitals following the Aufbau principle, Pauli exclusion principle, and Hund's rule:

1. ${\sigma }_{1s}$ takes 2 electrons. 2. ${\sigma }_{1s}^{\ast }$ takes 2 electrons. 3. ${\sigma }_{2s}$ takes 2 electrons. 4. ${\sigma }_{2s}^{\ast }$ takes 2 electrons. 5. ${\pi }_{2p_x}$ and ${\pi }_{2p_y}$ are degenerate (equal energy) and take 4 electrons total. 6. ${\sigma }_{2p_z}$ takes the final 2 electrons.

The complete electronic configuration is:

Bond order gives us a direct measure of the bond strength and stability. It is defined as half the difference between the number of bonding electrons ($N_b$) and antibonding electrons ($N_a$).

Let's count them up: - Bonding Electrons ($N_b$): These are in orbitals without an asterisk. We have 2 in ${\sigma }_{1s}$, 2 in ${\sigma }_{2s}$, 4 in ${\pi }_{2p}$, and 2 in ${\sigma }_{2p_z}$. Total $N_b=10$. - Antibonding Electrons ($N_a$): These are in orbitals with an asterisk. We have 2 in ${\sigma }_{1s}^{\ast }$ and 2 in ${\sigma }_{2s}^{\ast }$. Total $N_a=4$.

Plugging these into our formula:

A bond order of 3 perfectly aligns with the Lewis structure of cyanide, which features a strong triple bond between the carbon and nitrogen atoms.

Paired or Unpaired?

The magnetic nature of a molecule depends entirely on whether it has unpaired electrons. - Paramagnetic: Has one or more unpaired electrons (attracted to magnetic fields). - Diamagnetic: All electrons are paired (weakly repelled by magnetic fields).

Looking back at our filled molecular orbital configuration, every single orbital that contains electrons is completely full. There are zero unpaired electrons. Therefore, the ${\text{CN}}^{-}$ ion is strictly diamagnetic.

We have successfully determined that ${\text{CN}}^{-}$ has a bond order of 3 and is diamagnetic. This makes option (a) the correct answer.

To truly master this concept, ask yourself: What if we had the neutral $\text{CN}$ radical instead? Neutral $\text{CN}$ has 13 electrons. The last electron would be removed from the highest occupied molecular orbital (${\sigma }_{2p_z}$). This would leave 9 bonding electrons and 4 antibonding electrons, dropping the bond order to $2.5$. Furthermore, that single electron in the ${\sigma }_{2p_z}$ orbital would be unpaired, instantly making the molecule paramagnetic! Always practice these variations to solidify your understanding.