Question #44667

Consider the acid-catalyzed hydrolysis of an ester \ce{A} in a highly concentrated aqueous solution at $298\text{ K}$: $\ce{A(aq) + H2O(l) ->[H^+] B(aq) + C(aq)}$ The initial reaction mixture contains $1.0\text{ mol}$ of \ce{A} and $4.0\text{ mol}$ of \ce{H2O}. The partial molar volumes of the components, which are assumed to remain constant over the course of the reaction, are given as: $\bar{V}_{\ce{A}} = 128\text{ mL/mol}$, $\bar{V}_{\ce{H2O}} = 18\text{ mL/mol}$, $\bar{V}_{\ce{B}} = 26\text{ mL/mol}$, and $\bar{V}_{\ce{C}} = 20\text{ mL/mol}$.

The true rate law for the reaction is $\text{Rate} = k[\ce{A}][\ce{H2O}]$. In standard dilute solutions, this is approximated as a pseudo-first-order reaction with an apparent rate constant $k_{\text{obs}} = k[\ce{H2O}]$. Due to the high concentration and significant volume contraction upon reaction, $[\ce{H2O}]$ is not constant. Calculate the exact ratio of the instantaneous pseudo-first-order rate constant at $50\%$ completion of \ce{A} to its initial value, i.e., $\frac{k_{\text{obs}}(50\%)}{k_{\text{obs}}(0\%)}$.