Question

Explain the following: (a) The peroxide ion, ${\mathrm{O}}_2^{2-}$, has a longer bond length than the superoxide ion, ${\mathrm{O}}_2^{-}$. (b) The magnetic properties of ${\mathrm{B}}_2$ are consistent with the ${\pi }_{2p}$ MOs being lower in energy than the ${\sigma }_{2p}$ MO. (c) The ${\mathrm{O}}_2^{2+}$ ion has a stronger O $-$ O bond than ${\mathrm{O}}_2$ itself.

Short answer

(a) The peroxide ion, O2^(2-), has a longer bond length than the superoxide ion, O2^(-), due to the presence of an additional electron in the antibonding molecular orbital, which weakens the bond and increases the bond length. (b) B2 exhibits paramagnetic behavior due to unpaired electrons in the π2p molecular orbitals; this supports that π2p orbitals have lower energy than σ2p orbitals in B2. (c) The O2^(2+) ion has a stronger O-O bond than O2, as a result of the removal of an electron from the antibonding orbital (${\pi }_{2p}^{\ast }$), which strengthens the bond.

Step-by-step solution

Understanding Molecular Orbital Theory

Molecular Orbital Theory (MO Theory) is a method for determining molecular structure by describing the electronic structure of a molecule through MOs that are formed from the linear combination of atomic orbitals (AOs). This combines atomic orbitals from individual atoms to generate molecular orbitals, which then helps us understand properties such as bond strength, bond length, and magnetic properties of the molecules.

Peroxide ion and Superoxide ion bond length comparison

Peroxide ion, O2^2-, has an additional electron compared to the superoxide ion, O2^-. Due to the repulsion between these additional electrons, existing in an antibonding molecular orbital, the peroxide ion will have a longer bond length compared to the superoxide ion. In other words, the extra electron in O2^2- weakens the bond, thus increasing the bond length in comparison to O2^-.

Magnetic properties of B2

In B2, there are 6 valence electrons which are filled in the molecular orbitals in the following order: ${\sigma }_{1s}<{\sigma }_{1s}^{\ast }<{\sigma }_{2s}<{\sigma }_{2s}^{\ast }<{\pi }_{2p}={\pi }_{2p}<{\sigma }_{2p}$. As a result, there are unpaired electrons present in the π2p molecular orbitals. The presence of these unpaired electrons means that B2 exhibits paramagnetic behavior, which supports the fact that π2p orbitals have lower energy than σ2p orbitals in B2.

Bond strength comparison between O2 and O2^2+

To compare the bond strength between O2 and O2^2+, we need to analyze how the distribution of electrons in their molecular orbitals affects bond strength. In O2, the molecular orbitals are filled in the following order: ${\sigma }_{1s}<{\sigma }_{1s}^{\ast }<{\sigma }_{2s}<{\sigma }_{2s}^{\ast }<{\sigma }_{2p}<{\pi }_{2p}={\pi }_{2p}<{\pi }_{2p}^{\ast }={\pi }_{2p}^{\ast }<{\sigma }_{2p}^{\ast }$. When another electron is removed (forming O2^2+), an electron will be removed from the antibonding orbital (${\pi }_{2p}^{\ast }$). The removal of this electron will strengthen the bond, ultimately resulting in a stronger O-O bond in O2^2+ compared to O2.

Key concepts

Peroxide ion

The peroxide ion is denoted as ${\text{O}}_2^{2-}$. It has a longer bond length compared to the superoxide ion, ${\text{O}}_2^{-}$. This is because the peroxide ion contains two additional electrons compared to the neutral oxygen molecule, ${\text{O}}_2$. These added electrons occupy antibonding molecular orbitals.
$$\text{Antibonding orbitals are electron orbitals that, when occupied, weaken the bond between atoms.}$$
This occupation reduces the bond order, which is the number of chemical bonds between a pair of atoms.

Lower bond order results in a weaker bond and typically a longer bond length. So, in the case of the peroxide ion, the additional electron in the antibonding orbital stretches the bond compared to the superoxide ion. This makes it a crucial concept in understanding molecular structure via MO theory.

Superoxide ion

The superoxide ion, ${\text{O}}_2^{-}$, has one less electron compared to the peroxide ion which results in a shorter bond length.
This ion forms when an oxygen molecule gains an extra electron.
Since it has an even number of electrons, it typically does not fill antibonding orbitals to the extent that the peroxide ion does.

• The lesser occupation in antibonding orbitals means a higher bond order.
• A higher bond order generally correlates with more stable and stronger bonds.

Although it's negatively charged, the repulsive interactions are not as significant as in the case of ${\text{O}}_2^{2-}$. Therefore, its bond length remains shorter, illustrating how the electron configuration affects molecular properties.

Magnetic properties of B2

Understanding the magnetic properties of ${\text{B}}_2$ helps illustrate how molecular orbital theory explains paramagnetism. Boron, being in Group 13, has three valence electrons per atom, totaling six valence electrons in ${\text{B}}_2$.
These electrons fill the molecular orbitals in a distinct sequence. First are the ${\sigma }_{1s}$, ${\sigma }_{2s}$, and then the ${\pi }_{2p}$ orbitals before the ${\sigma }_{2p}$ manifold. This ordering means:

• Two unpaired electrons remain in the ${\pi }_{2p}$ orbitals.
• This makes ${\text{B}}_2$ itself paramagnetic since it has unpaired electrons.
• Paramagnetism occurs when unpaired electrons react to magnetic fields.

Thus, showcasing why ${\text{B}}_2$, based on electron configuration, exhibits this magnetic behavior and further reaffirms MO theory's predictiveness.

Bond strength comparison

Considering bond strength between ${\text{O}}_2$ and ${\text{O}}_2^{2+}$ involves analyzing their electron configurations.
${\text{O}}_2$ commonly fills its molecular orbitals including the antibonding ${\pi }_{2p}^{\ast }$ orbitals. When it forms ${\text{O}}_2^{2+}$, two electrons are removed, usually affecting the antibonding orbitals.
This removal leads to:

• A reduction in electron repulsion within antibonding orbitals, strengthening the overall bond.
• 
  A higher bond order results, meaning the $\text{O}-\text{O}$ bond is stronger in ${\text{O}}_2^{2+}$.

Such concepts are essential when considering chemical reactivity and stability of substances, explicitly showing how understanding molecular orbital vacancy and occupancy is crucial in predicting molecular behavior.