Using `MO` theory predict which of the following sepcies has the shortest bond length ?

Options

A. `O_(2)^(2+)`
B. `O_(2)^(+)`
C. `O_(2)^(-)`
D. `O_(2)^(2-)`

Answer

To determine which species has the shortest bond length using Molecular Orbital (MO) theory, we will follow these steps:

### Step 1: Identify the species and their electron counts
The species given are:
1. O₂²⁺
2. O₂⁺
3. O₂⁻
4. O₂²⁻

We know that the neutral O₂ molecule has 16 electrons. We will adjust the electron count based on the charge of each species:
- O₂²⁺: 16 - 2 = 14 electrons
- O₂⁺: 16 - 1 = 15 electrons
- O₂⁻: 16 + 1 = 17 electrons
- O₂²⁻: 16 + 2 = 18 electrons

### Step 2: Calculate the bond order for each species
The bond order can be calculated using the formula:
\[ \text{Bond Order} = \frac{(\text{Number of bonding electrons} - \text{Number of antibonding electrons})}{2} \]

We can use a known bond order for specific electron counts:
- 10 electrons: Bond order = 5
- 11 electrons: Bond order = 4.5
- 12 electrons: Bond order = 4
- 13 electrons: Bond order = 3.5
- 14 electrons: Bond order = 3
- 15 electrons: Bond order = 2.5
- 16 electrons: Bond order = 2
- 17 electrons: Bond order = 1.5
- 18 electrons: Bond order = 1

From the above, we can summarize the bond orders:
- O₂²⁺ (14 electrons): Bond order = 3
- O₂⁺ (15 electrons): Bond order = 2.5
- O₂⁻ (17 electrons): Bond order = 1.5
- O₂²⁻ (18 electrons): Bond order = 1

### Step 3: Compare bond orders to determine the shortest bond length
The bond length is inversely related to the bond order; the higher the bond order, the shorter the bond length.

From our calculations:
- O₂²⁺: Bond order = 3
- O₂⁺: Bond order = 2.5
- O₂⁻: Bond order = 1.5
- O₂²⁻: Bond order = 1

### Conclusion
The species with the highest bond order is O₂²⁺ with a bond order of 3. Therefore, O₂²⁺ has the shortest bond length among the given species.

**Final Answer: O₂²⁺**

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