Question

Use MO diagram to compare bond length and bond order of following species: \(\mathrm{O}_{2}^{+}, \mathrm{O}_{2}, \mathrm{O}_{2}^{-}, \mathrm{O}_{2}^{2-}\).

Short answer

The bond orders for the species are \(O_2^+\) (2.5), \(O_2\) (2), \(O_2^-\) (1.5) and \(O_2^{2-}\) (1). Hence, the order of decreasing bond length is \(O_2^{2-}\), \(O_2^-\), \(O_2\), \(O_2^+\).

Step-by-step solution

Understanding Molecular Orbital Theory

Understand that according to Molecular Orbital theory, atomic orbitals of the same energy levels can combine to form molecular orbitals. Construct an MO diagram for each species. For oxygen and its ions, the order of energy levels is: 1s < 2s < 2p < 2p* < 1s* < 2s*, where * denotes antibonding orbitals.

Determining Electron Configuration

Count the number of valence electrons in each species. Neutral oxygen molecule \(O_2\) has 12 valence electrons (6 from each atom), \(O_2^+\) has 11 electrons, \(O_2^-\) has 13 electrons, and \(O_2^{2-}\) has 14 electrons. Fill these electrons in the MO diagram starting from the lowest energy level.

Calculate the Bond Order

Calculate the bond order for each species using the formula: Bond Order = 0.5 * (number of bonding electrons - number of antibonding electrons). For all four species, the bond orders are: \(O_2^+\) is 2.5, \(O_2\) is 2, \(O_2^-\) is 1.5, \(O_2^{2-}\) is 1.

Infer about Bond Length

Remember that the higher the bond order, the shorter the bond length. Thus, the order of decreasing bond length should be: \(O_2^{2-}\), \(O_2^-\), \(O_2\), \(O_2^+\).

Key concepts

Molecular Orbital Theory

Molecular Orbital Theory (MOT) provides an in-depth explanation of the bond formation between atoms. It reveals how atomic orbitals merge to form molecular orbitals (MOs), each with its own energy level. In simple terms, MOT suggests that when atoms combine, their individual orbitals overlap to create new orbitals that belong to the molecule as a whole, rather than to any single atom.

MOT offers two types of orbitals: bonding and antibonding. Bonding orbitals stabilize the molecule as they are lower in energy compared to the original atomic orbitals. On the flip side, antibonding orbitals, which are higher in energy and denoted with an asterisk (*), can destabilize the molecule when populated with electrons.

When constructing an MO diagram, orbitals are filled according to the Aufbau principle, with electrons added starting from the lowest energy level. For molecules like diatomic oxygen and its ions, the MO diagram includes both sigma (σ) and pi (π) bonds formed from the combination of s and p atomic orbitals. Understanding these concepts is crucial when comparing different molecular species, as it allows for insights into their stability, reactivity, and physical properties.

Bond Order Calculation

The bond order of a molecule is an indication of the stability and strength of the chemical bond. Calculating the bond order is straightforward: You take half the difference between the number of electrons in bonding MOs and the number of electrons in antibonding MOs. The formula is written mathematically as: \( \text{Bond Order} = \frac{{\text{Number of Bonding Electrons} - \text{Number of Antibonding Electrons}}}{2} \).

A higher bond order implies a stronger, shorter bond, because there are more electrons in bonding orbitals relative to antibonding orbitals, thus the attractive force between the nuclei is greater. Conversely, a lower bond order suggests a weaker, longer bond. When comparing species like \( \mathrm{O}_{2}^{+} \), \( \mathrm{O}_{2} \), \( \mathrm{O}_{2}^{-} \), and \( \mathrm{O}_{2}^{2-} \), their bond orders provide insights into their comparative stabilities and strengths of the oxygen-oxygen bond within each ion.

Bond Length Comparison

Bond length is the physical distance between the nuclei of two bonded atoms in a molecule. In general, the bond length is inversely related to the bond order: higher bond orders translate to shorter bonds. This is due to the increased electron density in the space between the two nuclei, which pulls them closer together.

In the context of \( \mathrm{O}_{2}^{+} \), \( \mathrm{O}_{2} \), \( \mathrm{O}_{2}^{-} \), and \( \mathrm{O}_{2}^{2-} \), we can use their bond orders to predict relative bond lengths. Since \( \mathrm{O}_{2}^{+} \) has the highest bond order (2.5), it will have the shortest bond length. On the other hand, \( \mathrm{O}_{2}^{2-} \) with the lowest bond order (1) is expected to have the longest bond length. Bond lengths can be topically significant when considering the physical properties of a substance, such as boiling and melting points, and are also an important factor in the determination of the molecular geometry.