Derive a balanced equation for the reaction occurring in the cell: $$\mathrm{Fe}(\mathrm{s})\left|\mathrm{Fe}^{2+}(\mathrm{aq}) \| \mathrm{Fe}^{3+}(\mathrm{aq}), \mathrm{Fe}^{2+}(\mathrm{aq})\right| \mathrm{Pt}(\mathrm{s})$$ (a) If \(E_{\text {cell }}^{\circ}=1.21 \mathrm{V},\) calculate \(\Delta G^{\circ}\) and the equilibrium constant for the reaction. (b) Use the Nernst equation to determine the potential for the cell: $$\begin{array}{r} \mathrm{Fe}(\mathrm{s}) | \mathrm{Fe}^{2+}\left(\mathrm{aq}, 1.0 \times 10^{-3} \mathrm{M}\right) \| \mathrm{Fe}^{3+}\left(\mathrm{aq}, 1.0 \times 10^{-3} \mathrm{M}\right) \\ \mathrm{Fe}^{2+}(\mathrm{aq}, 0.10 \mathrm{M}) | \mathrm{Pt}(\mathrm{s}) \end{array}$$ (c) In light of (a) and (b), what is the likelihood of being able to observe the disproportionation of \(\mathrm{Fe}^{2+}\) into \(\mathrm{Fe}^{3+}\) and Fe under standard conditions?

Step by step solution

01

Formulation of Chemical Equations

Based on the cell notation, the half-reactions for the given cell are: \(\mathrm{Fe}^{2+} + 2e^- \leftrightarrow \mathrm{Fe}\) and \(\mathrm{Fe}^{2+} \leftrightarrow \mathrm{Fe}^{3+} + e^-\). Hence, the balanced overall cell reaction is \(\mathrm{Fe} + \mathrm{Fe}^{2+} \leftrightarrow \mathrm{Fe} + \mathrm{Fe}^{3+}\). Simplifying, the final balanced cell reaction is \(\mathrm{Fe}^{2+} \leftrightarrow \mathrm{Fe}^{3+}\).

02

Calculation of \(\Delta G^{\circ}\) and Equilibrium Constant

The standard free energy change, \(\Delta G^{\circ}\), can be calculated using the equation: \(\Delta G^{\circ} = -nFE_{\text {cell }}^{\circ}\). Where \(F = 96485 C/mol\) is the Faraday constant, \(n\) is the number of moles of electrons transferred (for this reaction, \(n=1\)), and \(E_{\text {cell }}^{\circ}=1.21 V\) is the standard cell potential. From this, \(\Delta G^{\circ}\) would be \(-1 * 96485 * 1.21\). The equilibrium constant, \(K\) for the reaction can then be calculated using the formula: \(K=e^{(-\Delta G^{\circ} /RT)}\) where \(R = 8.314 J/(mol*K)\) is the universal gas constant and \(T\) is the temperature in Kelvin.

03

Using Nernst Equation

The Nernst equation is used to determine the cell potential under non-standard conditions. It is represented as \(E = E^{\circ} - (RT/nF)lnQ\), where \(E\) is the cell potential, \(Q\) is the reaction quotient, and the other symbols have their usual meanings.

04

Likelihood of Disproportionation of \(\mathrm{Fe}^{2+}\)

The calculated equilibrium constant (\(K\)) gives an indication of which direction the reaction is likely to proceed. If it’s greater than 1, the reaction will proceed to the right, i.e., the disproportionation of \(\mathrm{Fe}^{2+}\) into \(\mathrm{Fe}^{3+}\) and Fe is more likely to occur under standard conditions.

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