Question

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

Short answer

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

Step-by-step solution

Define the Concept of Bond Order

Bond order is defined as the difference between the number of bonding electrons and the number of antibonding electrons, divided by two. In molecular orbital (MO) theory, a higher bond order typically indicates a shorter bond length.

Write the Electron Configuration for Each Species

Determine the molecular orbital electron configurations for each of the given species:- For \( \mathrm{O}_2^+ \): Remove one electron from the highest occupied molecular orbital (HOMO) of \( \mathrm{O}_2 \), which is \( \pi^*_{2p} \). - For \( \mathrm{O}_2^{2-} \): Add two electrons to the \( \pi^*_{2p} \) orbitals. - For \( \mathrm{O}_2^- \): Add one electron to the \( \pi^*_{2p} \) orbitals. - For \( \mathrm{O}_2^{2+} \): Remove two electrons from the \( \pi^*_{2p} \) orbitals.

Calculate the Bond Order for Each Species

Using the molecular orbital configurations:- For \( \mathrm{O}_2 \), the bond order is 2.- For \( \mathrm{O}_2^+ \): Bond order = \((10 - 5) / 2 = 2.5\).- For \( \mathrm{O}_2^{2-} \): Bond order = \((10 - 8) / 2 = 1\).- For \( \mathrm{O}_2^- \): Bond order = \((10 - 6) / 2 = 2\).- For \( \mathrm{O}_2^{2+} \): Bond order = \((10 - 4) / 2 = 3\).

Compare the Bond Orders to Predict the Shortest Bond Length

The shortest bond length corresponds to the highest bond order, as more electrons in bonding orbitals mean stronger attraction and shorter bonds. \( \mathrm{O}_2^{2+} \) has the highest bond order of 3, indicating it will have the shortest bond length among the given species.

Key concepts

Bond Order

Bond order is a fundamental concept in Molecular Orbital Theory that helps us determine the strength and length of a chemical bond. It is defined as half the difference between the number of electrons in bonding orbitals and antibonding orbitals. The formula for bond order is:

• Bond Order = \((N_b - N_a) / 2\)

where \(N_b\) is the number of electrons in bonding orbitals, and \(N_a\) is the number in antibonding orbitals.
Bond order gives us an idea about the bond's strength:

• A higher bond order means more bonding interactions than antibonding, leading to a stronger and shorter bond.
• A bond order of zero suggests that the molecule does not exist under normal conditions as there is no net bonding.

Higher bond orders correlate with higher bond strengths and are often used to predict chemical stability and reactivity.

Bond Length

Bond length is the distance between the nuclei of two bonded atoms. It is an essential factor in determining the properties of a molecule. Molecular Orbital Theory provides insight into how bond order affects bond length:

• Higher Bond Order: Discovers shorter bonds. With more electrons in bonding orbitals, the atomic nuclei are pulled closer together.
• Lower Bond Order: Results in longer bonds, as there is less electronic attraction between the nuclei.

For oxygen species, the bond length can be predicted by calculating bond order. Typically, if bond order increases, the bond length decreases. Understanding this relationship allows chemists to predict molecular properties and behaviors based on calculations rather than just experimental measurements.

Oxygen Species

The oxygen species we are examining are important because they show differences in bond lengths due to variations in electron configuration. Here's a breakdown:

• Oxygen Molecule \(\mathrm{O}_2\): Standard diatomic form with a bond order of 2.
• \(\mathrm{O}_2^+\): Loses an electron, which increases its bond order to 2.5, leading to a shorter bond than \(\mathrm{O}_2\).
• \(\mathrm{O}_2^-\): Gains an electron, decreasing its bond order to 2, similar to \(\mathrm{O}_2\), thus, similar bond length.
• \(\mathrm{O}_2^{2+}\): Loses two electrons, resulting in a bond order of 3, indicating the shortest bond length among the species.
• \(\mathrm{O}_2^{2-}\): Gains two electrons, reducing bond order to 1, the longest bond length among the oxygen species.

These configurations help us understand the effects of electron removal or addition on bond properties.

Electron Configuration

Electron configuration in Molecular Orbital Theory is crucial for understanding bond order and properties of molecules. In this theory, electrons are placed in molecular orbitals that form when atomic orbitals combine.
For oxygen species:

• Start from the baseline of \(\mathrm{O}_2\) molecular orbitals, which have electron filling in order from lower to higher energy levels.
• Adding or removing electrons adjusts the balance between bonding and antibonding orbital occupancy:
• \(\mathrm{O}_2\): Contains even distribution in bonding and antibonding orbitals, giving a bond order of 2.
  • \(\mathrm{O}_2^+\): Removal from antibonding orbital improves bond order to 2.5.
  • \(\mathrm{O}_2^{2-}\): Additional electrons in antibonding orbitals lower the bond order to 1.
• Each insertion or withdrawal of electrons affects molecular stability and bond length.

Understanding this electron distribution helps predict molecular behavior and interaction implications.