Question #44772

A modified \ce{H2-O2} fuel cell operates at $298 \text{ K}$ using a dual-membrane system that creates an internal concentration gradient. The anode compartment interfaces with a weak polymeric acid membrane (effective monomeric concentration $C = 0.1 \text{ M}$, acid dissociation constant $K_a = 10^{-5}$). The cathode compartment interfaces with a fully dissociated strong acid membrane ($C = 0.1 \text{ M}$). Both \ce{H2} and \ce{O2} gases are supplied at $1 \text{ bar}$. Given the standard cell potential $E^\circ = 1.23 \text{ V}$, the standard enthalpy of combustion of \ce{H2(g)} to \ce{H2O(l)} as $\Delta H^\circ = -289.5 \text{ kJ/mol}$, and Faraday's constant $F = 96500 \text{ C/mol}$. Assuming $\frac{2.303 RT}{F} = 0.059 \text{ V}$ at $298 \text{ K}$, what is the theoretical thermodynamic efficiency ($\eta = \frac{\Delta G}{\Delta H} \times 100$) of this modified fuel cell?