Problem 81 - Chemistry: The Central Science in SI Units, Global Edition

Question

Determine the electron configurations for \(\mathrm{CN}^{+}, \mathrm{CN}\), and \(\mathrm{CN}^{-}\). (a) Which species has the strongest \(\mathrm{C}-\mathrm{N}\) bond? (b) Which species, if any, has unpaired electrons?

Step-by-Step Solution

Answer

The electron configurations for CN+, CN, and CN- are as follows: CN+: \(\sigma_{1s}^2\sigma^*_{1s}^2\sigma_{2s}^2\sigma^*_{2s}^2\pi_{2p}^4\sigma_{2p}^0\pi^*_{2p}^0\sigma^*_{2p}^0\) CN: \(\sigma_{1s}^2\sigma^*_{1s}^2\sigma_{2s}^2\sigma^*_{2s}^2\pi_{2p}^4\sigma_{2p}^1\pi^*_{2p}^0\sigma^*_{2p}^0\) CN-: \(\sigma_{1s}^2\sigma^*_{1s}^2\sigma_{2s}^2\sigma^*_{2s}^2\pi_{2p}^4\sigma_{2p}^2\pi^*_{2p}^0\sigma^*_{2p}^0\) (a) The species with the strongest C-N bond is CN- because it has the highest bond order of 3. (b) CN is the only species with an unpaired electron, which is in the \(\sigma_{2p}\) orbital.

Step 1: Determine the total number of electrons in each species

For each molecule, find the atomic numbers of carbon (C) and nitrogen (N) and add or subtract the charge. Carbon has an atomic number of 6 and nitrogen has an atomic number of 7. For CN+: \(6 + 7 - 1 = 12\) electrons For CN: \(6 + 7 = 13\) electrons For CN-: \(6 + 7 + 1 = 14\) electrons

Step 2: Identify the molecular orbital diagram

Since the number of atomic orbitals for C and N is closer to that of oxygen rather than to that of hydrogen, we will use the oxygen molecular orbital diagram.

Step 3: Determine the electron configurations

Fill in the molecular orbitals according to the diagram. For species with fewer than 14 electrons (CN+ and CN), fill in the electrons from the lowest to the highest energy orbitals, abiding to the Aufbau Principle and Hund's Rule. For species with 14 electrons (CN-), all orbitals will be full. CN+: \(\sigma_{1s}^2\sigma^*_{1s}^2\sigma_{2s}^2\sigma^*_{2s}^2\pi_{2p}^4\sigma_{2p}^0\pi^*_{2p}^0\sigma^*_{2p}^0\) CN: \(\sigma_{1s}^2\sigma^*_{1s}^2\sigma_{2s}^2\sigma^*_{2s}^2\pi_{2p}^4\sigma_{2p}^1\pi^*_{2p}^0\sigma^*_{2p}^0\) CN-: \(\sigma_{1s}^2\sigma^*_{1s}^2\sigma_{2s}^2\sigma^*_{2s}^2\pi_{2p}^4\sigma_{2p}^2\pi^*_{2p}^0\sigma^*_{2p}^0\) Now we can use the electron configurations to answer parts (a) and (b).

Step 4: Identify the species with the strongest C-N bond

The species with the highest bond order will have the strongest bond. Bond order is determined by subtracting the number of electrons in antibonding orbitals from the number of electrons in bonding orbitals and dividing by 2. CN+ \(Bond Order = (2-2 + 2-2 + 4 + 0)/2 = 4/2 = 2\) CN \(Bond Order = (2-2 + 2-2 + 4 + 1)/2 = 5/2 = 2.5\) CN- \(Bond Order = (2-2 + 2-2 + 4 + 2)/2 = 6/2 = 3\) The species with the strongest bond is CN- because it has the highest bond order of 3. (a)

Step 5: Determine if any species has unpaired electrons

Observe the electron configurations and look for any orbitals with an unpaired electron. CN+ has paired electrons in all orbitals, so it has no unpaired electrons. CN has one unpaired electron in the \(\sigma_{2p}\) orbital. CN- has paired electrons in all orbitals, so it has no unpaired electrons. Therefore, CN is the only species with an unpaired electron. (b)

Key Concepts

Molecular Orbital Theory

Molecular orbital theory is a foundational concept in chemistry that helps in understanding the behavior of electrons in molecules. It differs from valence bond theory by considering orbitals to belong to the entire molecule, rather than being localized between atoms. Here's an easy breakdown of how it works:

• Atoms, such as carbon and nitrogen, have atomic orbitals that combine to form molecular orbitals when these atoms bond.
• Molecular orbitals are categorized into bonding, antibonding, and non-bonding orbitals. In bonding orbitals, electrons stabilize the molecule, while antibonding orbitals, indicated with an asterisk (*), can destabilize it if occupied.
• Electrons fill these orbitals following principles such as the Aufbau principle (lowest energy first), Pauli exclusion principle (no two electrons can have identical quantum numbers), and Hund's rule (maximizing unpaired electrons in separate orbitals).

Utilizing molecular orbital diagrams, as in the case of the CN species, helps us visualize electron placement and predict molecular properties such as magnetism, stability, and bond order.

Bond Order

Bond order is a useful metric for determining the stability and strength of bonds within a molecule. It is summarized by the formula:\[\text{Bond Order} = \frac{\text{Number of electrons in bonding orbitals} - \text{Number of electrons in antibonding orbitals}}{2}\]
The higher the bond order, the stronger and more stable the bond. Here's how you can think about bond order:

• A bond order of 0 implies no bond exists.
• Higher bond orders indicate stronger bonds, typically leading to shorter bond lengths and higher bond energies.
• For the CN species review, CN- has the largest bond order of 3, revealing the strongest C-N bond compared to CN and CN+.

Thus, determining bond order aids in comparing molecule stability, helping to understand which molecular structures are more robust.

Unpaired Electrons

Unpaired electrons are electrons that occupy an orbital singly, without a spin-paired partner. These electrons play a significant role in the magnetic properties of molecules. Molecules or species with unpaired electrons are often paramagnetic, meaning they are attracted to magnetic fields. In contrast, molecules with all electrons paired are diamagnetic and not attracted.
Here's how to identify them:

• Review the molecular orbital configuration. Electrons in partially filled orbitals can be paired or unpaired.
• Look for orbitals with a single electron; this indicates an unpaired electron is present.

In the exercise given, CN is the only one among CN+, CN, and CN- that has an unpaired electron, specifically within the \(\sigma_{2p}\) orbital. Knowing which species have unpaired electrons provides insight into their potential reactivity and magnetic behavior.