Question

Describe the bonding in the \(\mathrm{C}_{2}^{2-}\) ion in terms of the molecular orbital theory.

Short answer

The \(\mathrm{C}_{2}^{2-}\) ion has a bond order of 3, indicating a triple bond.

Step-by-step solution

Determine the Total Number of Electrons

The \(\mathrm{C}_{2}^{2-}\) ion is derived from two carbon atoms, each contributing 6 electrons, and the 2- charge contributes 2 additional electrons. Therefore, the total number of electrons is \\(6 + 6 + 2 = 14\).

Set Up the Molecular Orbital Diagram

For \(\mathrm{C}_{2}^{2-}\), we will use the molecular orbital configuration that applies to diatomic molecules with 14 electrons. The order for carbon is: \\(\sigma_{1s}, \sigma_{1s}^*, \sigma_{2s}, \sigma_{2s}^*, \pi_{2p_x}, \pi_{2p_y}, \sigma_{2p_z}, \pi_{2p_x}^*, \pi_{2p_y}^*, \sigma_{2p_z}^*\).

Fill the Molecular Orbitals with Electrons

Assign electrons starting from the lowest energy level orbital, following the Pauli exclusion principle and Hund's rule. The filling is: \\[(\sigma_{1s})^2 (\sigma_{1s}^*)^2 (\sigma_{2s})^2 (\sigma_{2s}^*)^2 (\pi_{2p_x})^2 (\pi_{2p_y})^2 (\sigma_{2p_z})^2.\]

Calculate the Bond Order

The bond order can be calculated using the formula: \\[\text{Bond Order} = \frac{\text{(Number of bonding electrons)} - \text{(Number of antibonding electrons)}}{2}.\] \(\mathrm{C}_{2}^{2-}\) has 10 bonding electrons and 4 antibonding electrons. Thus, \(\text{Bond Order} = \frac{10 - 4}{2} = 3.\)

Interpret the Bond Order

A bond order of 3 indicates a triple bond, suggesting that \(\mathrm{C}_{2}^{2-}\) has a stronger and shorter bond compared to a single or double bond, consistent with its electronic configuration in terms of molecular orbitals.

Key concepts

Understanding Bond Order

Bond order is a fundamental concept in chemistry that describes the strength and length of a chemical bond. It gives us insight into the stability of a molecule, which is particularly useful when we're looking at more complicated structures like ions.

The bond order is calculated using the formula: \[\text{Bond Order} = \frac{\text{(Number of bonding electrons)} - \text{(Number of antibonding electrons)}}{2}.\]This formula essentially measures the difference between electrons in bonding and antibonding molecular orbitals.

• A higher bond order indicates a stronger bond. For example, a bond order of 3, as seen in the \(\mathrm{C}_{2}^{2-}\) ion, usually means a triple bond.
• Conversely, a bond order of 1 would signify a single bond.
• A bond order of 0 implies no bond exists, making the molecule highly unstable.

Understanding bond order provides insights into the molecule’s reactivity and properties, and it’s a critical parameter when analyzing molecular interactions.

Exploring Molecular Orbitals

Molecular orbitals (MOs) are a core aspect of molecular orbital theory and are pivotal in explaining the bonding within molecules and ions like \(\mathrm{C}_{2}^{2-}\).

Molecular orbitals are formed by the combination of atomic orbitals from each atom participating in the bond. These MOs can be bonding or antibonding:

• Bonding orbitals result from constructive interference of atomic orbitals. Electrons in these orbitals help stabilize the molecule.
• Antibonding orbitals come from destructive interference. Electrons in these orbitals can destabilize the molecule.

In the \(\mathrm{C}_{2}^{2-}\) ion, the MOs are filled according to the order of their energy levels:\[(\sigma_{1s})^2 (\sigma_{1s}^*)^2 (\sigma_{2s})^2 (\sigma_{2s}^*)^2 (\pi_{2p_x})^2 (\pi_{2p_y})^2 (\sigma_{2p_z})^2,\]which includes 10 electrons in bonding and 4 electrons in antibonding orbitals. This arrangement significantly impacts the properties of the ion, such as its bond length and stability.

Bonding in Ions: The C_{2}^{2-} Example

Bonding in ions, including \(\mathrm{C}_{2}^{2-}\), can be uniquely different compared to neutral molecules. Molecular orbital theory provides a thorough explanation for these differences.

Unlike neutral molecules, ions have an added or subtracted set of electrons, which can change the electron configuration dramatically.

For the \(\mathrm{C}_{2}^{2-}\) ion:

• The extra two electrons due to the 2- charge need to be considered when filling molecular orbitals. This causes changes in bond characteristics as compared to a neutral \(\mathrm{C}_2\) molecule.
• By occupying additional molecular orbitals, these electrons influence both the bond length and bond order of the ion, making bonds generally stronger and shorter.

Molecular orbital diagrams allow us to visualize this process and assess the impact of electron changes on an ion's properties. In \(\mathrm{C}_{2}^{2-}\), this means analyzing how added electrons increase stability through affecting the balance of bonding versus antibonding orbitals. This results in a calculated chemical bond order that reflects the ion's true nature.