Step 1: Understanding the Concept:
The magnetic behavior of a molecule or ion depends on the arrangement of its electrons in molecular orbitals.
If a species contains one or more unpaired electrons, it is paramagnetic (attracted to a magnetic field).
If all the electrons in the species are paired, it is diamagnetic (repelled slightly by a magnetic field).
To determine the magnetic property of diatomic species like oxygen ions, we use the Molecular Orbital (MO) Theory.
Step 2: Key Formula or Approach:
The peroxide ion is $O_2^{2-}$.
Total number of electrons $= 2 \times 8 (\text{from O atoms}) + 2 (\text{negative charge}) = 18$ electrons.
The MO electronic configuration for an 18-electron diatomic species is:
\[ \sigma 1s^2, \sigma^* 1s^2, \sigma 2s^2, \sigma^* 2s^2, \sigma 2p_z^2, \pi 2p_x^2 = \pi 2p_y^2, \pi^* 2p_x^2 = \pi^* 2p_y^2 \]
Step 3: Detailed Explanation:
Let's populate the molecular orbitals with 18 electrons:
1. The 1s orbitals combine to give $\sigma 1s$ and $\sigma^* 1s$ (4 electrons).
2. The 2s orbitals combine to give $\sigma 2s$ and $\sigma^* 2s$ (4 electrons).
3. The 2p orbitals combine to give $\sigma 2p_z$, $\pi 2p_x$, $\pi 2p_y$, $\pi^* 2p_x$, $\pi^* 2p_y$, and $\sigma^* 2p_z$.
Filling the 2p molecular orbitals:
- $\sigma 2p_z$: 2 electrons.
- $\pi 2p_x$ and $\pi 2p_y$ (degenerate): 4 electrons total.
- $\pi^* 2p_x$ and $\pi^* 2p_y$ (degenerate): In neutral $O_2$ (16e), these contain 1 electron each (unpaired). In $O_2^{2-}$ (18e), two more electrons are added.
- These antibonding $\pi^*$ orbitals now contain 2 electrons each, making them completely filled.
4. Total electron count: $4 (1s) + 4 (2s) + 2 (\sigma 2p) + 4 (\pi 2p) + 4 (\pi^* 2p) = 18$ electrons.
As we can see, every single molecular orbital is completely occupied by two electrons with opposite spins.
There are no unpaired electrons in the peroxide ion. Therefore, it is diamagnetic.
Step 4: Final Answer:
According to Molecular Orbital Theory, $O_2^{2-}$ has all its electrons paired. Thus, it is diamagnetic. The correct option is (C).