Which of the following have same bond order and are paramagnetic?

The correct option is O₂⁺, N₂⁻.

To determine which pair has the same bond order and is paramagnetic, we can analyze their total number of electrons and use Molecular Orbital (MO) theory.

1. Concept of Bond Order:
According to Molecular Orbital theory, the bond order (BO) is calculated using the formula:

$\mathrm{BO}=\frac{N_b-N_a}{2}$

where $N_b$ is the number of bonding electrons and $N_a$ is the number of antibonding electrons.

Alternatively, we can use a quick method based on the total number of electrons for diatomic species:
- A species with 14 electrons (like N₂) has a bond order of 3.0.
- For every electron added or removed from 14, the bond order decreases by 0.5:
  • 13 electrons: BO = 2.5
  • 15 electrons: BO = 2.5
  • 16 electrons (like O₂): BO = 2.0
  • 17 electrons: BO = 1.5

2. Calculations for O₂⁺:
- A neutral oxygen atom (O) has 8 electrons, so O₂ has 16 electrons.
- O₂⁺ has lost 1 electron, so it has 15 electrons.
- With 15 electrons, its bond order is:

$\mathrm{BO}=2.5$

- Magnetic Behavior: The molecular orbital configuration for O₂⁺ (15 electrons) is:
$\sigma 1s^2,{\sigma }^{*}1s^2,\sigma 2s^2,{\sigma }^{*}2s^2,\sigma 2p_z^2,(\pi 2p_x^2=\pi 2p_y^2),({\pi }^{*}2p_x^1={\pi }^{*}2p_y^0)$.
Since there is 1 unpaired electron in the antibonding ${\pi }^{*}$ orbital, O₂⁺ is paramagnetic.

3. Calculations for N₂⁻:
- A neutral nitrogen atom (N) has 7 electrons, so N₂ has 14 electrons.
- N₂⁻ has gained 1 electron, so it has 15 electrons.
- With 15 electrons, its bond order is:

$\mathrm{BO}=2.5$

- Magnetic Behavior: The molecular orbital configuration for N₂⁻ (15 electrons) is:
$\sigma 1s^2,{\sigma }^{*}1s^2,\sigma 2s^2,{\sigma }^{*}2s^2,(\pi 2p_x^2=\pi 2p_y^2),\sigma 2p_z^2,({\pi }^{*}2p_x^1={\pi }^{*}2p_y^0)$.
Since it also contains 1 unpaired electron in the antibonding ${\pi }^{*}$ orbital, N₂⁻ is paramagnetic.

Conclusion:
Both O₂⁺ and N₂⁻ have a bond order of 2.5 and are paramagnetic due to the presence of an unpaired electron.