$MnO _{4}^{-}+ H ^{\oplus}+5 e ^{-} \rightarrow Mn ^{+2}+4 H _{2} O$

Nernst equation:

$E _{\text {cell }}= E _{ Cell }^{0}-\frac{0.059}{5} \log \frac{\left[ Mn ^{+2}\right]}{\left[ MnO _{4}^{-}\right]}\left[\frac{1}{ H ^{+}}\right]^{8}$

$(I)$ Given $\left[ H ^{\oplus}\right]=1\, M$

$E _{1}= E ^{0}-\frac{0.059}{5} \log \frac{\left[ Mn ^{+2}\right]}{\left[ MnO _{4}^{-}\right]}$

$(II)$ Now : $\left[ H ^{\oplus}\right]=10^{-4}\, M$

$E _{2}= E ^{0}-\frac{0.059}{5} \log \frac{\left[ Mn ^{+2}\right]}{\left[ MnO _{4}^{-}\right]} \times \frac{1}{\left(10^{-4}\right)^{8}}$

$= E ^{0}-\frac{0.059}{5} \log \frac{ Mn ^{+2}}{\left[ MnO _{4}^{-}\right]}+\frac{0.059}{5} \log 10^{-32}$

therefore : $\left| E _{1}- E _{2}\right|=\frac{0.059}{5} \times 32$

$=0.3776\, V =3776 \times 10^{-4}$

$x =3776$