Why is manganese more stable in the `+2` state than the `+3` state and the reverse is true for iron?

Explanation

(1) The electronic configuration of Mn is `._(25)Mn[Ar]3d^(5)4s^(2)`.
(2) In `+2` and `+3` oxidation states, the electronic configuration of Mn is, `Mn^(2+)[Ar]3d^(5) and Mn^(3+)[Ar]3d^(4)`.
Since half-filled d-orbital `(3d^(5))` has more stability and lower energy than `3d^(4),Mn^(2+)` is more stable than `Mn^(3+)`.
(4) The electronic configuration of Fe is `._(26)Fe[Ar]3d^(6)4s^(2)`.
In `+2` and `+3` oxidation states of Fe, the electronic configuration is, `Fe^(2+)[Ar]3d^(6)` and `Fe^(3+)[Ar]3d^(5)`.
Since half-filled orbital is more stable, `+3` state of Fe is more stable than `+2` state.