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

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

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

\(x =3776\)