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According to the Molecular Orbital Theory (MOT),
For $$O_2$$ (16 electrons):
The configuration is $$\sigma1s^2, \sigma^*1s^2, \sigma2s^2, \sigma^*2s^2, \sigma2p_z^2, (\pi2p_x^2 = \pi2p_y^2), (\pi^*2p_x^1 = \pi^*2p_y^1)$$
Oxygen has two unpaired electrons in the degenerate antibonding $$\pi^*$$ orbitals. Thus, it is paramagnetic.
For $$O_2^{2-}$$ (Peroxide ion, 18 electrons):
The configuration is $$\sigma1s^2, \sigma^*1s^2, \sigma2s^2, \sigma^*2s^2, \sigma2p_z^2, (\pi2p_x^2 = \pi2p_y^2), (\pi^*2p_x^2 = \pi^*2p_y^2)$$
All electrons are paired. Thus, it is diamagnetic.
For $$C_2$$ (12 electrons):
The configuration is $$\sigma1s^2, \sigma^*1s^2, \sigma2s^2, \sigma^*2s^2, (\pi2p_x^2 = \pi2p_y^2)$$
All electrons are paired. Thus, it is diamagnetic.
For $$N_2$$ (14 electrons):
The configuration is $$\sigma1s^2, \sigma^*1s^2, \sigma2s^2, \sigma^*2s^2, (\pi2p_x^2 = \pi2p_y^2), \sigma2p_z^2$$
All electrons are paired. Thus, it is diamagnetic.
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