Question #44846

A $1.0\text{ L}$ aqueous solution at $298\text{ K}$ is continuously fed with a hypothetical radioactive isotope \ce{^{240}M^{3+}(aq)} at a constant rate of $6.0 \times 10^{-4}\text{ mol s}^{-1}$. \ce{^{240}M^{3+}} undergoes radioactive decay to \ce{^{240}Z^{3+}(aq)} with a decay constant $\lambda_M = 2.0 \times 10^{-3}\text{ s}^{-1}$. The daughter isotope \ce{^{240}Z^{3+}} further decays to a stable element \ce{Y(aq)} with a decay constant $\lambda_Z = 1.0 \times 10^{-3}\text{ s}^{-1}$. Simultaneously, a constant direct current of $19.30\text{ A}$ is passed through the solution using inert electrodes.

Given the standard reduction potentials: $E^\circ_{\ce{M^{3+}/M^{2+}}} = +0.77\text{ V}$ $E^\circ_{\ce{Z^{3+}/Z^{2+}}} = +1.20\text{ V}$

Assume that the decay constants of the $+2$ oxidation states are identical to their $+3$ counterparts (i.e., $\lambda_{\ce{M^{2+}}} = \lambda_{\ce{M^{3+}}}$ and $\lambda_{\ce{Z^{2+}}} = \lambda_{\ce{Z^{3+}}}$), and that all species remain perfectly soluble. What is the steady-state reduction potential of the cathode?

(Given: $F = 96500\text{ C mol}^{-1}$, $\frac{2.303 RT}{F} = 0.06\text{ V}$, $\log 2 = 0.30$, $\log 3 = 0.48$)