The reduction potential ( E ^ 0 ), in V) of MnO _ 4 ^ - (aq) /...

Given

(1) ${MnO}_{4}^{-}\left(aq\right) + 4{H}^{+} + 3e ⟶ {MnO}_{2}\left(s\right) + 2{H}_{2}O$; $E ° = 1 . 68V$

${\Delta G}_{1} ° = - 3F\left(1 . 68\right) = - 5 . 04F$

(2) ${MnO}_{2}\left(s\right) + 4{H}^{+} + 2e ⟶ {Mn}^{2 +}( aq ) + 2{H}_{2}O$; $E ° = 1 . 21V$

${\Delta G}_{2} ° = - 2F\left(1 . 21\right) = - 2 . 42F$

(3) ${Mn}^{2 +}\left(aq\right) + 2e ⟶ Mn\left(s\right)$; $E ° = - 1 . 03V$

${\Delta G}_{3} ° = - 2F\left(- 1 . 03\right) = + 2 . 06F$

Adding \left(1\right), \left(2\right)$and \left(3\right)$,

${MnO}_{4}^{-}\left(aq\right) + 8{H}^{+} + 7e ⟶ Mn\left(s\right) + 4{H}_{2}O$

${\Delta G}^{o} = {\Delta G}_{1} ° + {\Delta G}_{2} ° + {\Delta G}_{3} °$

$= \left(- 5 . 04 - 2 . 42 + 2 . 06\right)F$

$- 7FE ° = - 5 . 40F$

$E ° = 0 . 77V$