Here’s a explanation without any equations:
D-block elements show variable oxidation states because:
- Both outermost s and d electrons can participate in bonding.
- In transition metals, the electrons in the outermost shell (s) and the penultimate d shell are close in energy.
- This allows the metal to lose different numbers of electrons in different situations.
- Similar energy levels of s and d orbitals.
- Since the energy difference between these orbitals is small, electrons from either orbital can be removed.
- Stability of certain electron arrangements.
- Some oxidation states are more stable because having a half-filled or fully-filled d orbital gives extra stability.
- Ability to form multiple bonds and complexes.
- Transition metals can form different compounds in which they use different numbers of electrons, resulting in multiple oxidation states.
In short:
D-block elements can show more than one oxidation state because their outer s and d electrons are both available for bonding, and different arrangements of these electrons can be stable in different compounds.