Question #44232

Consider the standard Gibbs free energy changes ($\Delta G^\circ$) for the following processes at 298 K:

For a transition metal $M$: Atomization: $M(s) \rightarrow M(g)$, $\Delta G^\circ = 330 \text{ kJ mol}^{-1}$ Ionization: $M(g) \rightarrow M^{2+}(g) + 2e^-$, $\Delta G^\circ = 2700 \text{ kJ mol}^{-1}$ Hydration: $M^{2+}(g) \rightarrow M^{2+}(aq)$, $\Delta G^\circ = -1998.8 \text{ kJ mol}^{-1}$

For a transition metal $N$: Atomization: $N(s) \rightarrow N(g)$, $\Delta G^\circ = 130 \text{ kJ mol}^{-1}$ Ionization: $N(g) \rightarrow N^{2+}(g) + 2e^-$, $\Delta G^\circ = 2640 \text{ kJ mol}^{-1}$ Hydration: $N^{2+}(g) \rightarrow N^{2+}(aq)$, $\Delta G^\circ = -1950 \text{ kJ mol}^{-1}$

The absolute standard Gibbs free energy change for the oxidation half-reaction $H_2(g) \rightarrow 2H^+(aq) + 2e^-$ is $965.6 \text{ kJ mol}^{-1}$.

An aqueous solution containing $1.0 \text{ M}$ of $M^{2+}$ and $1.0 \text{ M}$ of $N^{2+}$ at pH = 0 is electrolyzed using inert electrodes at 298 K and 1 bar pressure. The overpotential for $H_2$ evolution at the cathode is $0.40 \text{ V}$. Given $F = 96500 \text{ C mol}^{-1}$.

What are the standard reduction potentials $E^\circ_{M^{2+}/M}$ and $E^\circ_{N^{2+}/N}$ respectively, and which species is discharged first at the cathode?