Question #44276

The diamagnetic complex ion \ce{[Co(C2O4)3]^{3-}} undergoes an acid-catalyzed redox decomposition in aqueous solution to yield the paramagnetic \ce{[Co(H2O)6]^{2+}} ion and \ce{CO2} gas. The reaction proceeds via the following mechanism:

$\ce{[Co(C2O4)3]^{3-} + H+ <=>[K] [Co(C2O4)2(OC2O3H)]^{2-}} \quad \text{(Fast equilibrium)}$ $\ce{[Co(C2O4)2(OC2O3H)]^{2-} ->[k_1] [Co(C2O4)2]^{2-} + C2O4^{\bullet-} + H+} \quad \text{(Slow)}$ $\ce{[Co(C2O4)3]^{3-} + C2O4^{\bullet-} ->[fast] [Co(C2O4)2]^{2-} + 2 CO2 + C2O4^{2-}} \quad \text{(Fast)}$

Given: Equilibrium constant $K = 400 \text{ M}^{-1}$ Rate constant $k_1 = 4.33 \times 10^{-3} \text{ s}^{-1}$

A kinetic experiment is initiated by dissolving a small amount of \ce{K3[Co(C2O4)3]} in a large excess of an aqueous buffer at $pH = 2.0$ and $298 \text{ K}$. Calculate the time required for the total concentration of the diamagnetic Co(III) species to reduce to $12.5\%$ of its initial value. (Assume activity coefficients are unity and $\ln 2 = 0.693$)