Concept: Molecular Orbital Theory (MOT)

Molecular Orbital Theory describes the electronic structure of molecules in terms of molecular orbitals, which are formed by the linear combination of atomic orbitals. For diatomic molecules, the order of filling molecular orbitals depends on the total number of electrons. For molecules with up to 14 electrons (like N\u2082 and N\u2082\u207a), the energy order of molecular orbitals is: $\sigma 1s<{\sigma }^{\ast }1s<\sigma 2s<{\sigma }^{\ast }2s<\pi 2p_x=\pi 2p_y<\sigma 2p_z<{\pi }^{\ast }2p_x={\pi }^{\ast }2p_y<{\sigma }^{\ast }2p_z$. For molecules with more than 14 electrons, the order of $\sigma 2p_z$ and $\pi 2p$ orbitals is swapped.

Why (A) is correct:

1. Determine total electrons: A nitrogen atom (N) has 7 electrons. So, N\u2082 has $7\times 2=14$ electrons. N\u2082\u207a is formed by removing one electron from N\u2082, so N\u2082\u207a has $14-1=13$ electrons.

2. Apply MOT filling order for \u2264 14 electrons: The correct order of filling molecular orbitals for molecules with 14 or fewer electrons is: $\sigma 1s<{\sigma }^{\ast }1s<\sigma 2s<{\sigma }^{\ast }2s<\pi 2p_x=\pi 2p_y<\sigma 2p_z<{\pi }^{\ast }2p_x={\pi }^{\ast }2p_y<{\sigma }^{\ast }2p_z$

3. Fill 13 electrons according to Hund's rule and Pauli's exclusion principle:

• $(\sigma 1s{)}^2$ (2 electrons)
• $({\sigma }^{\ast }1s{)}^2$ (4 electrons)
• $(\sigma 2s{)}^2$ (6 electrons)
• $({\sigma }^{\ast }2s{)}^2$ (8 electrons)
• $(\pi 2p_x{)}^2=(\pi 2p_y{)}^2$ (8 + 4 = 12 electrons) - The $\pi 2p$ orbitals are degenerate, so they are filled completely before moving to the next higher energy orbital.
• $(\sigma 2p_z{)}^1$ (12 + 1 = 13 electrons) - The last electron goes into the $\sigma 2p_z$ orbital.

Therefore, the electronic configuration for N\u2082\u207a is $(\sigma 1s{)}^2({\sigma }^{\ast }1s{)}^2(\sigma 2s{)}^2({\sigma }^{\ast }2s{)}^2(\pi 2p_x^2=\pi 2p_y^2)(\sigma 2p_z{)}^1$.

Option Analysis:

• A) $(\sigma 1s{)}^2({\sigma }^{\ast }1s{)}^2(\sigma 2s{)}^2({\sigma }^{\ast }2s{)}^2(\pi 2p_x^2=\pi 2p_y^2)(\sigma 2p_z{)}^1$: This matches the correct filling order and electron count for 13 electrons.
• B) $(\sigma 1s{)}^2({\sigma }^{\ast }1s{)}^2(\sigma 2s{)}^2({\sigma }^{\ast }2s{)}^2(\sigma 2p_x{)}^2(\pi 2p_x^2=\pi 2p_z^1)$: This option uses an incorrect orbital notation $(\sigma 2p_x{)}^2$ and an incorrect filling order for $\pi 2p$ orbitals.
• C) $(\sigma 1s{)}^2({\sigma }^{\ast }1s{)}^2(\sigma 2s{)}^2({\sigma }^{\ast }2s{)}^2(\sigma 2p_z{)}^2(\pi 2p_x^2=\pi 2p_y^2)$: This configuration has 14 electrons, not 13. Also, it incorrectly places $\sigma 2p_z$ before $\pi 2p$ for molecules with \u2264 14 electrons.
• D) $(\sigma 1s{)}^2({\sigma }^{\ast }1s{)}^2(\sigma 2s{)}^2({\sigma }^{\ast }2s{)}^2(\sigma 2p_z{)}^2(\pi 2p_x^1=\pi 2p_z^1)$: This configuration has 12 electrons, not 13. It also incorrectly places $\sigma 2p_z$ before $\pi 2p$ and incorrectly fills the $\pi 2p$ orbitals.

Correct Answer: (A)