A rechargeable cell employs the redox couple MnO _ 4 ^ - ( aq ) + 8 , H ^ + ( aq ) +…

CHEMISTRY · Multi-part · CBSE · Chemistry · Electrochemistry · Electrochemistry

Problem

A rechargeable cell employs the redox couple

\mathrm{MnO}_{4}^{-}(\mathrm{aq}) + 8\,\mathrm{H}^{+}(\mathrm{aq}) + 5\,e^{-} \to \mathrm{Mn}^{2+}(\mathrm{aq}) + 4\,\mathrm{H}_{2}\mathrm{O}(\mathrm{l}) \quad E^{\circ} = +1.51\,\text{V}

along with the half-reaction

\mathrm{Al}^{3+}(\mathrm{aq}) + 3\,e^{-} \to \mathrm{Al}(\mathrm{s}) \quad E^{\circ} = -1.66\,\text{V}

The cell is assembled at

298\,\text{K}

with the following initial ion concentrations:

[\mathrm{MnO}_{4}^{-}] = 0.010\,\text{M} \quad [\mathrm{Mn}^{2+}] = 0.10\,\text{M} \quad [\mathrm{H}^{+}] = 1.0\,\text{M} \quad [\mathrm{Al}^{3+}] = 0.0010\,\text{M}

During operation, protons are consumed in the reduction half-reaction shown above. (a) Write the balanced overall reaction for the cell when it operates galvanically. (b) Derive an expression for the instantaneous cell potential in terms of the ratio

\dfrac{[\mathrm{Mn}^{2+}]^{5}[\mathrm{Al}^{3+}]^{3}}{[\mathrm{MnO}_{4}^{-}]^{5}[\mathrm{H}^{+}]^{40}}

, omitting numerical constants. Clearly state any assumptions you make about activities. (c) Calculate the numerical value of the cell potential with the initial ion concentrations provided. (d) As the cell discharges,

[\mathrm{H}^{+}]

falls to

0.010\,\text{M}

while the other concentrations remain effectively unchanged. Determine the cell potential at this point and explain whether the cell still operates spontaneously as a galvanic cell or now requires external energy input to run in the forward direction. (e) Using your expression from part (b), determine the minimum value of

[\mathrm{H}^{+}]

at which the cell potential becomes zero, and interpret this condition in terms of equilibrium and the feasibility of recharging the cell.

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