Sure, let's solve this step-by-step.
First, it's important to understand a few basics about molecular orbital (MO) theory:
1. Bond Order: It is calculated as the difference between the number of bonding and antibonding electrons, divided by two. The bond order gives us an idea of the strength and length of the bond; higher bond orders indicate stronger, shorter bonds.
2. Magnetic Properties: Molecules can be paramagnetic (with unpaired electrons) or diamagnetic (with all paired electrons).
Given these basics, let's determine the bond order and magnetic properties of the peroxide ion [tex]\(O _2^{2-}\)[/tex].
1. Molecular Orbital Diagram for [tex]\(O_2\)[/tex]:
- For [tex]\(O_2\)[/tex], the molecular orbital filling is as follows (from lowest to highest energy orbital):
- [tex]\( \sigma_{2s}^2 \)[/tex]
- [tex]\( \sigma_{2s}^ \ ^2 \)[/tex]
- [tex]\( \sigma_{2p_z}^2 \)[/tex]
- [tex]\( \pi_{2p_x}^2 \)[/tex] and [tex]\( \pi_{2p_y}^2 \)[/tex]
- [tex]\( \pi_{2p_x}^ \)[/tex] and [tex]\( \pi_{2p_y}^ \)[/tex]
- For [tex]\(O_2\)[/tex], the total number of electrons is 16. The filling of molecular orbitals according to these 16 electrons is as follows:
- [tex]\( \sigma_{2s}^2 \)[/tex]
- [tex]\( \sigma_{2s}^ \ ^2 \)[/tex]
- [tex]\( \sigma_{2p_z}^2 \)[/tex]
- [tex]\( \pi_{2p_x}^2 \)[/tex] and [tex]\( \pi_{2p_y}^2 \)[/tex]
- [tex]\( \pi_{2p_x}^ \ ^1 \)[/tex] and [tex]\( \pi_{2p_y}^ \ ^1 \)[/tex]
2. Addition of [tex]\(2-\)[/tex] charge (peroxide ion):
- By adding two additional electrons to [tex]\(O_2^{2-}\)[/tex], these electrons will occupy the next available lowest energy molecular orbitals, which are the [tex]\( \pi_{2p_x}^ \)[/tex] and [tex]\( \pi_{2p_y}^ \)[/tex] orbitals.
- Therefore, the electron configuration for [tex]\(O_2^{2-}\)[/tex] will be:
- [tex]\( \sigma_{2s}^2 \)[/tex]
- [tex]\( \sigma_{2s}^ \ ^2 \)[/tex]
- [tex]\( \sigma_{2p_z}^2 \)[/tex]
- [tex]\( \pi_{2p_x}^2 \)[/tex] and [tex]\( \pi_{2p_y}^2 \)[/tex]
- [tex]\( \pi_{2p_x}^ \ ^2 \)[/tex] and [tex]\( \pi_{2p_y}^ \ ^2 \)[/tex]
- The total number of electrons is now 18.
3. Calculate Bond Order:
- Bond order = (Number of electrons in bonding MO - Number of electrons in antibonding MO) / 2
- Number of bonding electrons = ( [tex]\(\sigma_{2s}^2\)[/tex] + [tex]\( \sigma_{2p_z}^2 \)[/tex]) + 2[tex]\( \pi_{2p_x}^2\)[/tex] = 4 ( [tex]\(\sigma\)[/tex] + 4 ( [tex]\(\pi\)[/tex] = 8 electrons
- Number of antibonding electrons = ([tex]\(\sigma_{2s}^ \ ^2\)[/tex] + [tex]\( \pi_{2p_x}^ \ ^2 \)[/tex] + [tex]\( \pi_{2p_y}^ \ ^2 \)[/tex]) = 8 electrons
- Bond order = (8 - 8) / 2 = 0
4. Determine Magnetic Properties:
- Since there are no unpaired electrons, the molecule is diamagnetic.
Thus, the bond order for [tex]\(O_2^{2-}\)[/tex] is 0, and it is diamagnetic. Therefore, the answer is:
0 , diamagnetic
