Question

Using MO theory predict which of the following species has the shortest bond length? (a) \(\mathrm{O}_{2}^{+}\) (b) \(\mathrm{O}_{2}^{-}\) (c) \(\mathrm{O}_{2}^{2-}\) (d) \(\mathrm{O}_{2}^{2+}\)

Short answer

\( \mathrm{O}_2^{2+} \) has the shortest bond length due to the highest bond order.

Step-by-step solution

Understanding Molecular Orbital Theory

Molecular Orbital (MO) Theory explains how atomic orbitals combine to form molecular orbitals, which can accommodate the electrons of a molecule. In MO theory, the bond order, which is calculated from the electron configuration in these orbitals, helps predict the bond strength and length of a molecule.

Defining Bond Order

Bond order is defined as half the difference between the number of bonding electrons and antibonding electrons: \[ \text{Bond Order} = \frac{1}{2} (N_b - N_a) \] where \( N_b \) is the number of electrons in bonding orbitals and \( N_a \) is the number of electrons in antibonding orbitals. A higher bond order indicates a stronger bond and usually a shorter bond length.

Molecular Orbital Configuration for Dioxygen

The electron configuration for neutral \( \mathrm{O}_2 \) in terms of molecular orbitals is: \( (\sigma_{2s})^2(\sigma_{2s}^*)^2(\sigma_{2p_z})^2(\pi_{2p_x})^2(\pi_{2p_y})^2(\pi^*_{2p_x})^1(\pi^*_{2p_y})^1 \). From this, the bond order is 2: \[ \text{Bond Order of } \mathrm{O}_2 = 3 \text{ (bonding electrons)} - 1 \text{ (antibonding electrons)} = 2 \]

Calculating Bond Orders for Given Species

We modify the electron configuration for each given species and calculate their bond orders: - For \( \mathrm{O}_2^+ \): one less electron leads to formula \( (\pi^*_{2p_x})^1 \) being \( (\pi^*_{2p_x})^0 \), bond order \( = \frac{1}{2}(10-5) = 2.5 \).- For \( \mathrm{O}_2^- \): one more electron leads to formula \( (\pi^*_{2p_y})^1 \) being \( (\pi^*_{2p_y})^2 \), bond order \( = \frac{1}{2}(10-7) = 1.5 \).- For \( \mathrm{O}_2^{2-} \): two more electrons, two in antibonding \( \pi^* \), bond order \( = \frac{1}{2}(10-8) = 1 \).- For \( \mathrm{O}_2^{2+} \): two less electrons, removing two from antibonding, bond order \( = \frac{1}{2}(10-4) = 3 \).

Comparing Bond Orders for Shortest Bond Length

Higher bond order generally means shorter bond length due to stronger bonds. Among the calculated bond orders, \( \mathrm{O}_2^{2+} \) has the highest bond order of 3, suggesting it should have the shortest bond length.

Key concepts

Bond Order

Bond order is a crucial concept in Molecular Orbital Theory used to predict the strength and length of a bond. It is calculated using the formula:\[ \text{Bond Order} = \frac{1}{2}(N_b - N_a) \]where \( N_b \) is the number of electrons in bonding orbitals and \( N_a \) is the number of electrons in antibonding orbitals. The bond order gives insight into the number of bonds between a pair of atoms:

• A bond order of 1 indicates a single bond.
• A bond order of 2 indicates a double bond.
• A bond order of 3 indicates a triple bond.

Higher bond orders correspond to stronger bonds, which are generally also shorter in length. This is because more bonding interactions increase the attractive forces between atoms, pulling them closer together. In essence, by understanding bond order, you're gauging not only the bond's strength but also its expected length.

Molecular Orbital Configuration

Molecular Orbital Configuration details how electrons are distributed in molecular orbitals, resulting from the linear combination of atomic orbitals. In the case of oxygen molecules like \( \mathrm{O}_2 \), we can represent this through a specific sequence of filled molecular orbitals:\[(\sigma_{2s})^2(\sigma_{2s}^*)^2(\sigma_{2p_z})^2(\pi_{2p_x})^2(\pi_{2p_y})^2(\pi^*_{2p_x})^1(\pi^*_{2p_y})^1\]This fundamental breakdown is essential because it helps us determine bond order and predict chemical properties. To adapt this configuration for the ions of oxygen, we add or remove electrons:

• \( \mathrm{O}_2^+ \): One electron is removed, adjusting the antibonding orbital.
• \( \mathrm{O}_2^- \): One electron is added, increasing electrons in the antibonding orbital.
• \( \mathrm{O}_2^{2-} \): Two additional electrons occupy the antibonding orbitals.
• \( \mathrm{O}_2^{2+} \): Two electrons are removed, lowering antibonding occupancy.

Every alteration in the number of electrons subsequently shifts the bond order, showing how intimately the configurations tie to bonding properties.

Bond Length Prediction

Predicting bond length in molecules involves understanding how bond order influences the distance between atoms. Higher bond orders generally indicate stronger and therefore shorter bonds. This is because a higher bond order implies more electrons are bonding rather than antibonding, which increases attraction between the atoms:

• For \( \mathrm{O}_2^+ \) with a bond order of 2.5, the bond is shorter than in neutral \( \mathrm{O}_2 \), with order 2.
• \( \mathrm{O}_2^- \) displays a lower bond order of 1.5, suggesting a longer bond length.
• \( \mathrm{O}_2^{2-} \) with a bond order of 1 has the longest bond length among these species, indicative of weaker bonding.
• \( \mathrm{O}_2^{2+} \), with the highest bond order of 3, has the shortest bond length, due to the strongest bonding interaction.

This principle allows chemists to infer physical properties and reactivity from the molecular orbital configuration. The shorter the bond, the more energy it requires to break, and often, more reactive species are those with lower bond orders.