Using molecular orbital theory, explain why the removal of one electron in O2 strengthens bonding, while the removal of one electron in N2 weakens bonding?

First you have a knowledge of bond order which is

B. O = (no. of electrons in bonding orbital - no. of electrons in non-bonding orbital) : 2

Note:

bond strength is directly proportional to bond order.

For oxygen:

B. O = (6-2) / 2 = 2; after the removal of two electrons (removal occur from non-bonding orbital)

B. O = (6-0) / 2 = 3 (As B. O increased bond strength increased)

For Nitrogen:

B. O = (6-0) / 2 = 3; after the removal of two electrons (removal occur from bonding orbital)

B. O = (4-0) / 2 = 2 (As B. O decreased bond strength decreased)