The bond length in a species is inversely proportional to its bond order. A higher bond order indicates a stronger bond and thus a shorter bond length.

Let's calculate the bond order for each species using the Molecular Orbital Theory (MOT):

1. O₂⁺ (Total electrons = 16 - 1 = 15)
Electronic configuration: σ1s² σ*1s² σ2s² σ*2s² σ2pᵢ² π2pᵢ⁴ π*2pᵢ¹
Bond Order (BO) = ½ (Number of bonding electrons - Number of antibonding electrons)
BO = ½ (10 - 5) = ½ (5) = 2.5

2. O₂²⁻ (Total electrons = 16 + 2 = 18)
Electronic configuration: σ1s² σ*1s² σ2s² σ*2s² σ2pᵢ² π2pᵢ⁴ π*2pᵢ⁴
Bond Order (BO) = ½ (10 - 8) = ½ (2) = 1.0

3. O₂⁻ (Total electrons = 16 + 1 = 17)
Electronic configuration: σ1s² σ*1s² σ2s² σ*2s² σ2pᵢ² π2pᵢ⁴ π*2pᵢ³
Bond Order (BO) = ½ (10 - 7) = ½ (3) = 1.5

4. O₂²⁺ (Total electrons = 16 - 2 = 14)
Electronic configuration: σ1s² σ*1s² σ2s² σ*2s² σ2pᵢ² π2pᵢ⁴
Bond Order (BO) = ½ (10 - 4) = ½ (6) = 3.0

Comparing the bond orders:
O₂²⁺ (BO = 3.0) > O₂⁺ (BO = 2.5) > O₂⁻ (BO = 1.5) > O₂²⁻ (BO = 1.0)

Since O₂²⁺ has the highest bond order (3.0), it will have the shortest bond length.

Option Analysis:

• A) O₂⁺: Bond order = 2.5. This is high, but not the highest.
• B) O₂²⁻: Bond order = 1.0. This is the lowest bond order among the options, meaning it has the longest bond length.
• C) O₂⁻: Bond order = 1.5. This is lower than O₂⁺ and O₂²⁺.
• D) O₂²⁺: Bond order = 3.0. This is the highest bond order, hence the shortest bond length.

Correct Answer: (D)