To determine the increasing order of bond length, we need to analyze the bond order of each species. Bond length is inversely proportional to bond order; higher the bond order, shorter the bond length.

Step 1: Determine the total number of electrons for each species.

• NO: Nitrogen (7) + Oxygen (8) = 15 electrons
• NO${}^{+}$: Nitrogen (7) + Oxygen (8) - 1 = 14 electrons
• NO${}^{-}$: Nitrogen (7) + Oxygen (8) + 1 = 16 electrons

Step 2: Write the molecular orbital (MO) configuration and calculate the bond order.

For species with up to 14 electrons, the MO configuration is ${\sigma }_{1s},{\sigma }_{1s}^{\ast },{\sigma }_{2s},{\sigma }_{2s}^{\ast },{\pi }_{2p_x}={\pi }_{2p_y},{\sigma }_{2p_z},{\pi }_{2p_x}^{\ast }={\pi }_{2p_y}^{\ast },{\sigma }_{2p_z}^{\ast }$.

For species with more than 14 electrons, the MO configuration is ${\sigma }_{1s},{\sigma }_{1s}^{\ast },{\sigma }_{2s},{\sigma }_{2s}^{\ast },{\sigma }_{2p_z},{\pi }_{2p_x}={\pi }_{2p_y},{\pi }_{2p_x}^{\ast }={\pi }_{2p_y}^{\ast },{\sigma }_{2p_z}^{\ast }$.

Bond Order (BO) = $\frac{1}{2}$ (Number of electrons in bonding MOs - Number of electrons in antibonding MOs)

• NO${}^{+}$ (14 electrons):
  ${\sigma }_{1s}^2{\sigma }_{1s}^{\ast 2}{\sigma }_{2s}^2{\sigma }_{2s}^{\ast 2}{\pi }_{2p_x}^2{\pi }_{2p_y}^2{\sigma }_{2p_z}^2$
  BO = $\frac{1}{2}(10-4)=3.0$
• NO (15 electrons):
  ${\sigma }_{1s}^2{\sigma }_{1s}^{\ast 2}{\sigma }_{2s}^2{\sigma }_{2s}^{\ast 2}{\sigma }_{2p_z}^2{\pi }_{2p_x}^2{\pi }_{2p_y}^2{\pi }_{2p_x}^{\ast 1}$
  BO = $\frac{1}{2}(10-5)=2.5$
• NO${}^{-}$ (16 electrons):
  ${\sigma }_{1s}^2{\sigma }_{1s}^{\ast 2}{\sigma }_{2s}^2{\sigma }_{2s}^{\ast 2}{\sigma }_{2p_z}^2{\pi }_{2p_x}^2{\pi }_{2p_y}^2{\pi }_{2p_x}^{\ast 1}{\pi }_{2p_y}^{\ast 1}$
  BO = $\frac{1}{2}(10-6)=2.0$

Step 3: Compare bond orders and determine bond lengths.

Bond orders: NO${}^{+}$ (3.0) > NO (2.5) > NO${}^{-}$ (2.0)

Since bond length is inversely proportional to bond order, the order of bond length will be the reverse:

Bond lengths: NO${}^{+}$ < NO < NO${}^{-}$

Option Analysis:

• A) $\mathrm{NO}>{\mathrm{NO}}^{-}>{\mathrm{NO}}^{+}$: Incorrect. This order implies NO has the longest bond, which is false.
• B) ${\mathrm{NO}}^{+}<\mathrm{NO}<{\mathrm{NO}}^{-}$: Correct. This matches our derived order of bond lengths.
• C) $\mathrm{NO}<{\mathrm{NO}}^{+}<{\mathrm{NO}}^{-}$: Incorrect. This implies NO has the shortest bond, which is false.
• D) $\mathrm{NO}<{\mathrm{NO}}^{+}={\mathrm{NO}}^{-}$: Incorrect. The bond lengths are not equal, and the order is wrong.

Correct Answer: (B)