The internuclear distances in O-O bonds for $$O_2^+$$, $$O_2$$, $$O_2^-$$ and $$O_2^{2-}$$ respectively are :

To determine the internuclear distances in the O-O bonds for $$O_2^+$$, $$O_2$$, $$O_2^-$$, and $$O_2^{2-}$$, we need to understand how bond length relates to bond order. Bond length is inversely proportional to bond order: higher bond order means a stronger bond and shorter bond length. Therefore, we must calculate the bond order for each species using molecular orbital theory.

First, recall the molecular orbital configuration for oxygen molecules. Oxygen has an atomic number of 8, so $$O_2$$ has 16 electrons. However, we focus on valence electrons (12 electrons) since core electrons (1s orbitals) cancel out. The molecular orbital energy order for $$O_2$$ is: $$\sigma(2s) \lt \sigma^*(2s) \lt \sigma(2p_z) \lt \pi(2p_x) = \pi(2p_y) \lt \pi^*(2p_x) = \pi^*(2p_y) \lt \sigma^*(2p_z)$$.

For $$O_2$$ (16 electrons, 12 valence electrons):

The electron configuration is: $$\sigma(2s)^2 \sigma^*(2s)^2 \sigma(2p_z)^2 \pi(2p_x)^2 \pi(2p_y)^2 \pi^*(2p_x)^1 \pi^*(2p_y)^1$$.

Bonding electrons: $$\sigma(2s)^2$$ (2 electrons), $$\sigma(2p_z)^2$$ (2 electrons), $$\pi(2p_x)^2 \pi(2p_y)^2$$ (4 electrons) → total bonding electrons = 8.

Antibonding electrons: $$\sigma^*(2s)^2$$ (2 electrons), $$\pi^*(2p_x)^1 \pi^*(2p_y)^1$$ (2 electrons) → total antibonding electrons = 4.

Bond order = $$\frac{$$ number of bonding electrons $$-$$ number of antibonding electrons $$}{2} = \frac{8 - 4}{2} = \frac{4}{2} = 2$$.

For $$O_2^+$$ (15 electrons, 11 valence electrons):

Remove one electron from the highest occupied molecular orbital (HOMO), which is $$\pi^*$$. Configuration: $$\sigma(2s)^2 \sigma^*(2s)^2 \sigma(2p_z)^2 \pi(2p_x)^2 \pi(2p_y)^2 \pi^*(2p_x)^1 \pi^*(2p_y)^0$$ (or equivalent).

Bonding electrons: same as $$O_2$$, 8 electrons.

Antibonding electrons: $$\sigma^*(2s)^2$$ (2 electrons), $$\pi^*$$ orbitals have 1 electron → total antibonding electrons = 3.

Bond order = $$\frac{8 - 3}{2} = \frac{5}{2} = 2.5$$.

For $$O_2^-$$ (17 electrons, 13 valence electrons):

Add one electron to the $$\pi^*$$ orbital. Configuration: $$\sigma(2s)^2 \sigma^*(2s)^2 \sigma(2p_z)^2 \pi(2p_x)^2 \pi(2p_y)^2 \pi^*(2p_x)^2 \pi^*(2p_y)^1$$ (or equivalent).

Bonding electrons: same as $$O_2$$, 8 electrons.

Antibonding electrons: $$\sigma^*(2s)^2$$ (2 electrons), $$\pi^*$$ orbitals have 3 electrons → total antibonding electrons = 5.

Bond order = $$\frac{8 - 5}{2} = \frac{3}{2} = 1.5$$.

For $$O_2^{2-}$$ (18 electrons, 14 valence electrons):

Add two electrons to the $$\pi^*$$ orbitals. Configuration: $$\sigma(2s)^2 \sigma^*(2s)^2 \sigma(2p_z)^2 \pi(2p_x)^2 \pi(2p_y)^2 \pi^*(2p_x)^2 \pi^*(2p_y)^2$$.

Bonding electrons: same as $$O_2$$, 8 electrons.

Antibonding electrons: $$\sigma^*(2s)^2$$ (2 electrons), $$\pi^*$$ orbitals have 4 electrons → total antibonding electrons = 6.

Bond order = $$\frac{8 - 6}{2} = \frac{2}{2} = 1$$.

Summary of bond orders:

• $$O_2^+$$: 2.5
• $$O_2$$: 2
• $$O_2^-$$: 1.5
• $$O_2^{2-}$$: 1

Since bond length is inversely proportional to bond order, the order of increasing bond length (shortest to longest) is: $$O_2^+ \lt O_2 \lt O_2^- \lt O_2^{2-}$$.

Now, comparing with the given options:

• Option A: 1.30 Å ($$O_2^+$$), 1.49 Å ($$O_2$$), 1.12 Å ($$O_2^-$$), 1.21 Å ($$O_2^{2-}$$) → Incorrect, as $$O_2^+$$ should be shortest, but 1.30 Å is longer than 1.12 Å.
• Option B: 1.49 Å ($$O_2^+$$), 1.21 Å ($$O_2$$), 1.12 Å ($$O_2^-$$), 1.30 Å ($$O_2^{2-}$$) → Incorrect, as $$O_2^+$$ bond length (1.49 Å) is longer than $$O_2$$ (1.21 Å), but it should be shorter.
• Option C: 1.21 Å ($$O_2^+$$), 1.12 Å ($$O_2$$), 1.49 Å ($$O_2^-$$), 1.30 Å ($$O_2^{2-}$$) → Incorrect, as $$O_2^+$$ bond length (1.21 Å) is longer than $$O_2$$ (1.12 Å), but it should be shorter.
• Option D: 1.12 Å ($$O_2^+$$), 1.21 Å ($$O_2$$), 1.30 Å ($$O_2^-$$), 1.49 Å ($$O_2^{2-}$$) → Correct, matches the order: shortest for $$O_2^+$$, then $$O_2$$, then $$O_2^-$$, longest for $$O_2^{2-}$$.

Standard bond length values confirm this: $$O_2^+$$ ≈ 1.12 Å, $$O_2$$ ≈ 1.21 Å, $$O_2^-$$ ≈ 1.30 Å, $$O_2^{2-}$$ ≈ 1.49 Å.

Hence, the correct answer is Option D.