Write the balanced chemical equations for the following reactions:
$(1)$ Reaction of $Cr_2O_7^{2-}$ with $Fe^{2+}$ in acidic medium.
$(2)$ Reaction of $MnO_4^-$ with $Fe^{2+}$ in acidic medium.

(N/A) $(1)$ The reaction of dichromate ion with ferrous ion in acidic medium is:
$Cr_2O_7^{2-} + 14H^+ + 6Fe^{2+} \rightarrow 2Cr^{3+} + 6Fe^{3+} + 7H_2O$
$(2)$ The reaction of permanganate ion with ferrous ion in acidic medium is:
$MnO_4^- + 8H^+ + 5Fe^{2+} \rightarrow Mn^{2+} + 5Fe^{3+} + 4H_2O$