Problem 81 - Chemistry: The Central Science

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 the species \(\mathrm{CN}^{+}, \mathrm{CN}\), and \(\mathrm{CN}^{-}\) are: 1. \(\mathrm{CN}^{+}\): \(\sigma_{2s}^2 \sigma_{2s}^{*2} \sigma_{2p}^2 \pi_{2p}^4\) 2. \(\mathrm{CN}\): \(\sigma_{2s}^2 \sigma_{2s}^{*2} \sigma_{2p}^2 \pi_{2p}^4 \pi_{2p}^{*1}\) 3. \(\mathrm{CN}^{-}\): \(\sigma_{2s}^2 \sigma_{2s}^{*2} \sigma_{2p}^2 \pi_{2p}^4 \pi_{2p}^{*2}\) (a) \(\mathrm{CN}^{+}\) has the strongest C-N bond with a bond order of 2. (b) Only the neutral molecule, \(\mathrm{CN}\), has unpaired electrons, with one unpaired electron in the \(\pi_{2p}^{*}\) orbital.

Step 1: The atomic orbitals of Carbon and Nitrogen

First, we have to consider the atomic orbitals of the Carbon and Nitrogen atoms that will combine to form the molecular orbitals. The valence orbitals of Carbon are in the 2s and 2p_shell, while Nitrogen also has its valence electrons in the 2s and 2p_shell. Therefore, for both atoms, the relevant atomic orbitals are 2s and 2p.

Step 2: Formation of molecular orbitals

The atomic orbitals can combine constructively or destructively, giving us bonding and antibonding molecular orbitals. For this diatomic molecule, the molecular orbitals formed are as follows: 1. \(\sigma_{2s}\) (bonding) and \(\sigma_{2s}^{*}\) (antibonding) from 2s orbitals 2. \(\sigma_{2p}\) (bonding) from 2p_z orbitals 3. \(\sigma_{2p}^{*}\) (antibonding) from 2p_z orbitals 4. \(\pi_{2p}\) (bonding) and \(\pi_{2p}^{*}\) (antibonding; there are two of these) from 2p_x, and 2p_y orbitals.

Step 3: Determine the electron count for each species

Now we need to find out the total number of electrons in each species to fill the molecular orbitals: 1. \(\mathrm{CN}^{+}\): Carbon has 6 electrons and Nitrogen has 7 electrons. In the positive ion, one electron is removed, so we have a total of 12 electrons. 2. \(\mathrm{CN}\): Carbon has 6 electrons, and Nitrogen has 7 electrons, so we have a total of 13 electrons. 3. \(\mathrm{CN}^{-}\): Carbon has 6 electrons and Nitrogen has 7 electrons. In the negative ion, one electron is added, so we have a total of 14 electrons.

Step 4: Fill the molecular orbitals with electrons

Using the electron count, we will now fill the molecular orbitals in order of increasing energy levels according to the Aufbau principle: 1. \(\mathrm{CN}^{+}\): 12 electrons will fill the molecular orbitals in the following order: - 2 electrons in \(\sigma_{2s}\) - 2 electrons in \(\sigma_{2s}^{*}\) - 2 electrons in \(\sigma_{2p}\) - 4 electrons in \(\pi_{2p}\) (2 in each \(\pi\) orbital) - 2 more electrons can't fill the antibonding orbitals. 2. \(\mathrm{CN}\): 13 electrons will fill the molecular orbitals in the following order: - 2 electrons in \(\sigma_{2s}\) - 2 electrons in \(\sigma_{2s}^{*}\) - 2 electrons in \(\sigma_{2p}\) - 4 electrons in \(\pi_{2p}\) (2 in each \(\pi\) orbital) - 1 electron in \(\pi_{2p}^{*}\) 3. \(\mathrm{CN}^{-}\): 14 electrons will fill the molecular orbitals in the following order: - 2 electrons in \(\sigma_{2s}\) - 2 electrons in \(\sigma_{2s}^{*}\) - 2 electrons in \(\sigma_{2p}\) - 4 electrons in \(\pi_{2p}\) (2 in each \(\pi\) orbital) - 2 electrons in \(\pi_{2p}^{*}\) (1 in each \(\pi^{*}\) orbital) Now we can answer the questions:

Step 5: Answer (a): Which species has the strongest C-N bond?

The species with the strongest C-N bond will have the highest bond order. The bond order can be calculated using the following formula: Bond order = (Number of electrons in bonding orbitals - Number of electrons in antibonding orbitals) / 2 - For \(\mathrm{CN}^{+}\): Bond order = (8 - 4) / 2 = 2 - For \(\mathrm{CN}\): Bond order = (8 - 5) / 2 = 1.5 - For \(\mathrm{CN}^{-}\): Bond order = (8 - 6) / 2 = 1 Thus, \(\mathrm{CN}^{+}\) has the strongest C-N bond.

Step 6: Answer (b): Which species, if any, has unpaired electrons?

A species has unpaired electrons if there are any orbitals with one electron: - \(\mathrm{CN}^{+}\): No unpaired electrons - \(\mathrm{CN}\): One unpaired electron in \(\pi_{2p}^{*}\) orbital - \(\mathrm{CN}^{-}\): No unpaired electrons Hence, only the neutral molecule, \(\mathrm{CN}\), has unpaired electrons.

Key Concepts

Molecular Orbitals

Molecular orbitals (MOs) are a foundational concept in understanding how atoms bond in molecules. They are formed when atomic orbitals from different atoms overlap during the formation of a molecule.

Within a molecule, electrons are not confined to individual atoms but are distributed over these molecular orbitals. These MOs can be bonding or antibonding depending on the phase of the atomic orbitals that combine. Bonding molecular orbitals result from constructive interference, lowering the energy and stabilizing the molecule. On the contrary, antibonding orbitals are formed through destructive interference, raising the energy and potentially destabilizing the molecule.

When filling MOs with electrons, the lower energy bonding orbitals are filled first, followed by higher energy antibonding orbitals. The order and nature of molecular orbital filling directly influence the bond strength and magnetic properties of the molecule.

Bond Order Calculation

The bond order of a molecule is an indicator of the stability and strength of the bond between atoms. It can be calculated using a simple formula:

Bond order = \( \frac{\text{Number of electrons in bonding MOs - Number of electrons in antibonding MOs}}{2} \)

A higher bond order suggests a stronger and more stable bond. In the CN example given, the bond orders are calculated based on how the electrons fill the molecular orbitals. For instance, \(\mathrm{CN}^{+}\) has a bond order of 2, hinting at a double bond that is the strongest among the species compared. In contrast, \(\mathrm{CN}^{−}\) has a bond order of 1, suggestive of a single bond, which is the weakest configuration for the CN molecules analyzed.

Unpaired Electrons

Unpaired electrons are those that occupy an orbital without a partner electron. Their presence can give a molecule paramagnetic properties, making it attracted to magnetic fields.

In the context of molecular orbitals, unpaired electrons result when there's an odd number of electrons to distribute into orbitals, or when there are degenerate orbitals (orbitals at the same energy level) that get half-filled. When filling molecular orbitals, Hund's Rule applies, stating that each degenerate orbital must be singly occupied before any is doubly occupied. In our CN molecule example, \(\mathrm{CN}\) has an unpaired electron occupying one of the antibonding \(\pi_{2p}^{*}\) orbitals, which is responsible for the molecule's paramagnetic nature and an important characteristic in determining its chemical reactivity and bonding properties.