Question #44777

A cylindrical corrosion pit of depth $L = 1.0 \times 10^{-2}$ m and cross-sectional area $A = 1.0 \times 10^{-4}$ m$^2$ is formed on an iron surface. The pit is filled with an aqueous solution of a weak monoprotic acid \ce{HA} at a uniform concentration of $C = 0.1$ M. Due to a localized gradient, the degree of dissociation of \ce{HA} varies with depth $z$ (where $z=0$ at the pit opening and $z=L$ at the iron base) according to the relation $\alpha(z) = \frac{\alpha_0 L}{L + z}$, with $\alpha_0 = 0.025$. The limiting molar conductivity of \ce{HA} is $\Lambda_m^\infty = 4.0 \times 10^{-2} \ \text{S m}^2 \text{mol}^{-1}$. A constant potential difference of $V = 1.5$ V drives the anodic dissolution of iron (\ce{Fe -> Fe^{2+} + 2e^-}) at the base of the pit. Assuming the resistance of the electrolyte in the pit is the only resistance in the circuit, calculate the mass of iron lost due to corrosion after $t = 96500$ s. \n(Given: Atomic mass of Fe = 56 g/mol, $1 F = 96500$ C/mol, assume $\Lambda_m \approx \Lambda_m^\infty$ for the weak acid).