Question #44444

D-Glucose undergoes mutarotation in aqueous solution at $298 \text{ K}$ via an open-chain intermediate. The overall process involves the $\alpha$-pyranose ($A$), $\beta$-pyranose ($B$), the open-chain free aldehyde ($I$), and its gem-diol hydrate ($H$). The mutarotation mechanism is strictly mediated by the free aldehyde ($I$):

$\ce{A <=>[k_1][k_{-1}] I <=>[k_2][k_{-2}] B}$

The free aldehyde is in rapid equilibrium with its hydrate:

$\ce{I + H2O(l) <=>[K_h] H}$

where the hydration equilibrium constant is $K_h = \frac{[H]}{[I]} = 3.0$. At equilibrium, the mole fractions of the cyclic species relative to the total dissolved glucose are $[A] = 0.36$ and $[B] = 0.64$. The total mole fraction of the open-chain species ($I + H$) is experimentally determined to be $2.0 \times 10^{-4}$.

Given that the rate constant for the ring closure of the $\alpha$-anomer is $k_{-1} = 14.4 \text{ s}^{-1}$, determine the rate constant for the ring opening of the $\alpha$-anomer ($k_1$).