According to Molecular Orbital Theory,

Options

A. `C_(2)^(2-)` is expected to be diamagnetic
B. `O_(2)^(2+)` is expected to have a longer bond length than `O_(2)`
C. `N_(2)^(+) and N_(2)^(-)` have the same bond order
D. `He_(2)^(+)` has the same energy as two isolated He atoms

Explanation

To solve the question according to Molecular Orbital Theory, we will analyze each option provided in the question step by step.

### Step 1: Analyze Option 1 - C2 2- is expected to be diamagnetic
1. **Determine the number of electrons**: C2 2- has 14 electrons (12 from C2 and 2 additional electrons).
2. **Write the molecular orbital configuration**:
- σ1s² σ*1s² σ2s² σ*2s² π2px² π2py² σ2pz²
3. **Identify unpaired electrons**: In the configuration, all electrons are paired.
4. **Conclusion**: Since there are no unpaired electrons, C2 2- is diamagnetic.

### Step 2: Analyze Option 2 - O2 2+ is expected to have a longer bond length than O2
1. **Determine the molecular orbital configuration of O2 2+**:
- O2 2+ has 10 electrons: σ1s² σ*1s² σ2s² σ*2s² σ2pz² π2px² π2py²
2. **Calculate bond order**:
- Bond order = 1/2 (number of bonding electrons - number of antibonding electrons)
- Bonding electrons = 10, Antibonding electrons = 4
- Bond order = 1/2 (10 - 4) = 3
3. **Determine the molecular orbital configuration of O2**:
- O2 has 12 electrons: σ1s² σ*1s² σ2s² σ*2s² σ2pz² π2px² π2py² π*2px¹ π*2py¹
4. **Calculate bond order for O2**:
- Bonding electrons = 10, Antibonding electrons = 6
- Bond order = 1/2 (10 - 6) = 2
5. **Conclusion**: Since O2 has a lower bond order than O2 2+, it has a longer bond length. Therefore, this option is incorrect.

### Step 3: Analyze Option 3 - N2+ and N2- have the same bond order
1. **Determine the molecular orbital configuration of N2+**:
- N2+ has 9 electrons: σ1s² σ*1s² σ2s² σ*2s² π2px² π2py² σ2pz¹
2. **Calculate bond order for N2+**:
- Bonding electrons = 9, Antibonding electrons = 4
- Bond order = 1/2 (9 - 4) = 2.5
3. **Determine the molecular orbital configuration of N2-**:
- N2- has 10 electrons: σ1s² σ*1s² σ2s² σ*2s² π2px² π2py² σ2pz² π*2px¹
4. **Calculate bond order for N2-**:
- Bonding electrons = 10, Antibonding electrons = 5
- Bond order = 1/2 (10 - 5) = 2.5
5. **Conclusion**: Both N2+ and N2- have the same bond order of 2.5. Therefore, this option is correct.

### Step 4: Analyze Option 4 - Helium 2+ has the same energy as 2 isolated helium
1. **Consider the molecular orbital configuration of He2**:
- He2 would have 2 electrons in bonding and 2 in antibonding, leading to no net bonding.
2. **Conclusion**: Helium 2+ does not have the same energy as two isolated helium atoms due to the presence of bonding and antibonding interactions. Therefore, this option is incorrect.

### Final Conclusion:
- **Correct Options**: Option 1 (C2 2- is diamagnetic) and Option 3 (N2+ and N2- have the same bond order) are correct.
- **Incorrect Options**: Option 2 (O2 2+ has a longer bond length than O2) and Option 4 (Helium 2+ has the same energy as 2 isolated helium) are incorrect.