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The Nearst Equation: A galvanic cell is ...

The Nearst Equation: A galvanic cell is constructed at `25^(@)C` as follows. Once half-cell consists of the `Fe^(3+)//Fe^(2+)` couple in which `C_(Fe^(3+)) = 1.00 M` and `C_(Fe^(2+)) = 0.100 M`, the other involves the `MnO_(4)^(-)//Mn^(2+)` couple in which `C_(MnO_(4)^(-)) = 1.00 xx 10^(-2)M, C_(Mn^(2+)) = 1.00 xx 10^(-4) M`, and `C_Mn^(2+) = 1.00 xx 10^(-3) M`. (a) Find the elecrode potential for each half-cell with these concentrations, and (b) calculate the overall cell potential.
Strategy: (a) Apply the Nernst equation to find the electrode potential of each half-cell with the given concentrations.
Write the half-reaction with the more positive (or less negative) potentikal (after correction) as the cathode reaction along with its potential. Reverse the other half-reaction (to make it anode reaction) and change the sign of its `E` value. Finally balance the electron transfer and then add the half-reactions and their electrode potentials to find the overall cell potentail.

Text Solution

Verified by Experts

(a) For the `Fe^(3+)//Fe^(2+)` half-cell as a reduction.
`Fe^(3+)(aq.)+e^(-)hArrFe^(2+)(aq.)E_(Fe^(3+)//Fe^(2+))^(2+)=+0.771V`
`E_(Fe^(3+)//Fe^(2+))^(@) - (0.0592 V)/(n) log (C_Fe^(2+))/(C_Fe^(3+))`
` = +0.77 V - (0.0592 V)/(1) log (0.100)/(1.00)`
`= +0.77 V - (0.0592 V) (-1.00)`
`= +0.829 V`
For the `MnO_(4)^(-) + 8H6(+) + 5e^(-) rarr Mn^(2+) + H_(2)O E_(MnO_(4)//Mn^(2+))^(@) = +1.507`
`E_(MnO_(4)//Mn^(2+)) = E_(MnO_(4)//Mn^(2+)) - (0.0592 V)/(n)log (C_(Mn^(2+)))/(C_(MnO_(4)^(-))C_(H+)^(8)`
`= +0.77 V - (0.0592 V)/(5)`
`"log" (1.00 xx 10^(-4))/(1.00 xx 10^(-2))(1.00 xx 10^(-3))^(8)`
`= +1.507 V - (0.0592 V)/(5) "log" (1.00 xx 10^(22))`
`= +1.507 V - (0.0592 V)/(5) (22.0) = +1.246 V`
(b) Since the corrected electrode potential for the `MnO_(4)^(-)//Mn^(2+)` half-cell is greater than that for the `Fe^(3+)//Fe^(2+)` half-cell, we reaverse the half-cell reaction for the `Fe^(3+)//Fe^(2+)` couple, balance the electron trasfer, and add.
`{:(MnO_(4)^(-)(aq.)+8H^(+)(aq.)5e^(-)hArr Mn^(2+)(aq.)4H_(2)P(1)+1.246 V-0.0829V),(5xx[Fe^(2+)(aq)hArr Fe^(3+)(aq)+e^(-)]),(bar(MnO_(4)^(-)(aq,)+8H^(+)(aq.)+5Fe^(2+)(aq.)rarr Mn^(2+)(aq.)+4H_(2)O(1)5Fe^(3+)(aq.)E_("cell")=0.417V)):}`
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