The bond length of ` H_(2)^(+), H_(2)^(-) " and "H_(2)` are in the order

Options

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

Explanation

To determine the bond lengths of \( H_2^+, H_2^-, \) and \( H_2 \), we will follow these steps:

### Step 1: Understand the Concept of Bond Order
Bond order is defined as the number of bonding pairs of electrons between two atoms. It can be calculated using the formula:
\[ \text{Bond Order} = \frac{1}{2} \times (\text{Number of electrons in bonding MOs} - \text{Number of electrons in anti-bonding MOs}) \]
Higher bond order indicates a stronger bond and a shorter bond length.

### Step 2: Determine the Molecular Orbital Configurations
1. **For \( H_2^+ \)**:
- Total electrons = 1 (since \( H_2 \) has 2 electrons, and we remove one for \( H_2^+ \)).
- Molecular orbital configuration: \( \sigma_{1s}^1 \)

2. **For \( H_2^- \)**:
- Total electrons = 3 (adding one electron to \( H_2 \)).
- Molecular orbital configuration: \( \sigma_{1s}^2 \sigma_{1s}^1 \)

3. **For \( H_2 \)**:
- Total electrons = 2.
- Molecular orbital configuration: \( \sigma_{1s}^2 \)

### Step 3: Calculate the Bond Order for Each Species
1. **For \( H_2^+ \)**:
\[ \text{Bond Order} = \frac{1}{2} \times (1 - 0) = \frac{1}{2} \]

2. **For \( H_2^- \)**:
\[ \text{Bond Order} = \frac{1}{2} \times (2 - 1) = \frac{1}{2} \]

3. **For \( H_2 \)**:
\[ \text{Bond Order} = \frac{1}{2} \times (2 - 0) = 1 \]

### Step 4: Analyze the Bond Lengths Based on Bond Order
- **Bond order and bond length relationship**: As bond order increases, bond length decreases.
- \( H_2 \) has the highest bond order of 1, meaning it has the shortest bond length.
- Both \( H_2^+ \) and \( H_2^- \) have a bond order of \( \frac{1}{2} \). However, \( H_2^- \) has an additional electron which leads to increased electron-electron repulsion, resulting in a longer bond length compared to \( H_2^+ \).

### Conclusion: Order of Bond Lengths
Thus, the order of bond lengths from longest to shortest is:
\[ H_2^- > H_2^+ > H_2 \]