Question #44768

During the discharge cycle of a secondary battery, a localized pH gradient alters the redox behavior of the electrolyte. To investigate this thermodynamic shift, a technician constructs the following concentration-dependent electrochemical cell at $298 \text{ K}$:

$Pt | M^{2+} (0.1 \text{ M}), M^{3+} (0.01 \text{ M}) || MnO_4^- (0.01 \text{ M}), Mn^{2+} (0.1 \text{ M}), H^+ (pH = x) | Pt$

The technician observes that the cell potential is exactly $0 \text{ V}$, indicating that the $KMnO_4$ indicator fails to spontaneously oxidize $M^{2+}$ under these specific conditions.

Given: $E^\circ_{M^{3+}/M^{2+}} = 1.27 \text{ V}$ $E^\circ_{MnO_4^-/Mn^{2+}} = 1.51 \text{ V}$ $\frac{2.303 RT}{F} = 0.06 \text{ V}$

Calculate the value of $x$ (the pH of the cathodic compartment).