Question

The number of antibonding electron pairs in $ {\text{O}}_2^{2 - } $ molecular ion on the basis of molecular orbital theory is:
(A) $ 2 $
(B) $ 3 $
(C) $ 4 $
(D) $ 5 $

Answer 1

Hint: On the basis of molecular orbital theory, the molecular orbital configuration is different for upto $ 14 $ electrons and that for more than $ 14 $ electrons. Once we know the correct energy order for more than $ 14 $ electrons, we can put two electrons in each orbital and then count the number of antibonding electron pairs.

Complete step by step solution:
The number of electrons in $ {\text{O}} $ atom $ = 8 $ .
Therefore, the number of electrons in $ {{\text{O}}_2} $ molecule $ = 2 \times 8 = 16 $ .
Also, the number of electrons in $ {\text{O}}_2^{2 - } $ molecular ion $ = 16 + 2 = 18 $ .
The bonding molecular orbitals are lower in energy than their corresponding anti-bonding molecular orbitals. The energy of various molecular orbitals for $ {{\text{O}}_2} $ molecule follows the following increasing order:
 $ \sigma 1s $ < $ \sigma *1s $ < $ \sigma 2s $ < $ \sigma *2s $ < $ \sigma 2{p_z} $ < $ \pi 2{p_x} $ = $ \pi 2{p_y} $ < $ \pi *2{p_x} $ = $ \pi *2{p_y} $ < $ \sigma *2{p_z} $ .
Here, $ * $ represents anti-bonding orbitals.
Now, filling all the $ 18 $ electrons in this order, we can get the molecular orbital configuration as:
 $ {(\sigma 1s)^2}{(\sigma *1s)^2}{(\sigma 2s)^2}{(\sigma *2s)^2}{(\sigma 2{p_z})^2}{(\pi 2{p_x})^2}{(\pi 2{p_y})^2}{(\pi *2{p_x})^2}{(\pi *2{p_y})^2} $
The number of antibonding electron pairs $ = 4 $
Hence, the correct option is (C).

Additional Information
The energy of various molecular orbitals for molecules/ ion having less than or equal to $ 14 $ electrons follows the following increasing order:
 $ \sigma 1s $ < $ \sigma *1s $ < $ \sigma 2s $ < $ \sigma *2s $ < $ \pi 2{p_x} $ = $ \pi 2{p_y} $ < $ \sigma 2{p_z} $ < $ \pi *2{p_x} $ = $ \pi *2{p_y} $ < $ \sigma *2{p_z} $ .
Note the difference of energy order of $ \sigma 2{p_z} $ orbital.

Note:
If we remember that there are six electrons in antibonding orbitals in an $ {{\text{O}}_2} $ molecule, we can calculate that there will be eight electrons in antibonding orbitals of $ {\text{O}}_2^{2 - } $ molecule ion. This is another short method to know the number of antibonding electron pairs.