Question #45042

The auto-redox (disproportionation) of sodium hypochlorite (\ce{NaClO}) in water proceeds via the intermediate \ce{HClO}, formed by salt hydrolysis. The empirical rate law for this aqueous pathway is $Rate = k_{aq}[\ce{HClO}]^2$. In a $1.0 \text{ M}$ aqueous solution of \ce{NaClO} buffered at $pH = 10.0$, the initial rate is $R_{aq}$. When \ce{NaClO} is instead dissolved to a concentration of $1.0 \text{ M}$ in anhydrous Dimethyl Sulfoxide (DMSO), hydrolysis is completely suppressed. The reaction then proceeds via direct bimolecular collision of \ce{ClO^-} ions with rate law $Rate = k_{DMSO}[\ce{ClO^-}]^2$ and initial rate $R_{DMSO}$. The activation energy for the direct \ce{ClO^-} pathway in DMSO is $57.06 \text{ kJ/mol}$ higher than that of the \ce{HClO} pathway in water. Assuming the pre-exponential factors ($A$) for both rate constants are identical, calculate the ratio $R_{DMSO} / R_{aq}$ at $298 \text{ K}$. (Given: $pK_a$ of \ce{HClO} = $7.5$, $2.303 RT = 5.706 \text{ kJ/mol}$ at $298 \text{ K}$)