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Step-by-Step Solution
Step 1: Identify the Given Half-Reactions and Their Potentials
We have two relevant half-cell potentials in the question:
Reduction of Mn2+ to Mn(s):
$ \text{Mn}^{2+} + 2e^- \to \text{Mn(s)}, \quad E^\circ = -1.18 \text{ V} $
Oxidation of Mn2+ to Mn3+:
$ \text{Mn}^{2+} \to \text{Mn}^{3+} + e^-, \quad E^\circ = -1.51 \text{ V} $
Note that the second potential is given directly in the oxidation direction (Mn2+ ā Mn3+ + eā).
Step 2: Write the Desired Overall Reaction
The target reaction is:
$ 3\,\text{Mn}^{2+} \to \text{Mn(s)} + 2\,\text{Mn}^{3+}. $
To form this overall reaction, we combine:
1 Ć (Reduction) Mn2+ + 2eā ā Mn(s)
2 Ć (Oxidation) Mn2+ ā Mn3+ + eā
Step 3: Combine the Half-Reactions and Cancel Electrons
When we multiply the oxidation half-reaction by 2, it becomes:
$ 2 \times \bigl(\text{Mn}^{2+} \to \text{Mn}^{3+} + e^-\bigr)\,:\quad 2\,\text{Mn}^{2+} \to 2\,\text{Mn}^{3+} + 2\,e^- $
Adding this to the reduction half-reaction:
$ \text{Mn}^{2+} + 2\,e^- \to \text{Mn(s)} $
we see that 2eā from oxidation and 2eā from reduction cancel each other. Hence, the net overall reaction is:
$ 3\,\text{Mn}^{2+} \to \text{Mn(s)} + 2\,\text{Mn}^{3+}. $
Step 4: Calculate the Overall Standard Potential
The overall standard potential $E^\circ_\text{net}$ is the sum of the potentials for (1) the reduction step and (2) the oxidation step. Since the second reaction is given in the oxidation direction with $E^\circ = -1.51\text{ V}$, we can directly add them:
$ E^\circ_\text{net} = (-1.18\text{ V}) + (-1.51\text{ V}) = -2.69\text{ V}. $
Step 5: Assess the Feasibility of the Forward Reaction
Because the overall cell potential $E^\circ_\text{net}$ is negative ($-2.69\text{ V}$), the forward reaction is not feasible as written. A negative $E^\circ$ implies a positive $ \Delta G^\circ $, meaning the process is non-spontaneous in the forward direction.
Final Answer
The value of $E^\circ$ for the reaction $3\,\text{Mn}^{2+} \to \text{Mn(s)} + 2\,\text{Mn}^{3+}$ is $-2.69\text{ V}$, and the forward reaction does not occur spontaneously.
