Question #44843

A $1.0 \text{ \mu mol}$ sample of a pure radioisotope $\ce{^{A}_{Z}X}$ ($t_{1/2} = 100 \text{ min}$) is placed in a specialized reactor. It undergoes $\alpha$-decay with a mass defect of $0.005 \text{ amu}$ per disintegration. The energy released is captured with $40\%$ efficiency and used to drive the electrolysis of water ($\ce{H2O(l) -> H2(g) + 1/2 O2(g)}$) at a constant applied potential of $1.5 \text{ V}$.

Assume the nuclear decay can be modeled as a thermodynamic process. Calculate the volume of $\ce{H2(g)}$ collected at $300 \text{ K}$ and $1 \text{ atm}$ after $200 \text{ min}$, and determine the equivalent standard cell potential ($E^\circ_{eq}$) for the decay of 1 mole of $X$, assuming it acts as a 2-electron spontaneous process ($\Delta G^\circ_{decay} = -n F E^\circ_{eq}$).

(Given: $1 \text{ amu} \equiv 1000 \text{ MeV}$, $1 \text{ eV} = 1.6 \times 10^{-19} \text{ J}$, $N_A = 6.0 \times 10^{23} \text{ mol}^{-1}$, $F = 96000 \text{ C mol}^{-1}$, $R = 0.082 \text{ L atm K}^{-1} \text{ mol}^{-1}$. Assume $\Delta G^\circ_{decay} \approx \Delta H^\circ_{decay}$)