Chapter 12, Problem 65E

The hydrolysis of the sugar sucrose to the sugars glucose and fructose,

${\text{C}}_{12}{\text{H}}_{22}{\text{O}}_{11}+{\text{H}}_2\text{O}\to {\text{C}}_6{\text{H}}_{12}{\text{O}}_6+{\text{C}}_6{\text{H}}_{12}{\text{O}}_6$

follows a first-order rate equation for the disappearance of sucrose: Rate $=k\left[{\text{C}}_{12}{\text{H}}_{22}{\text{O}}_{11}\right]$ (The products of the reaction, glucose and fructose, have the same molecular formulas but differ in the arrangement of the atoms in their molecules.)

(a) In neutral solution, $k=2.1\times {10}^{-11}{\text{s}}^{-1}$ at 27 °C and $8.5\times {10}^{-11}{\text{s}}^{-1}$ at 37 °C. Determine the activation energy, the frequency factor, and the rate constant for this equation at 47 °C (assuming the kinetics remain consistent with the Arrhenius equation at this temperature).

(b) When a solution of sucrose with an initial concentration of 0.150 M reaches equilibrium, the concentration of sucrose is $1.65\times {10}^{-7}$M . How long will it take the solution to reach equilibrium at 27 °C in the absence of a catalyst? Because the concentration of sucrose at equilibrium is so low, assume that the reaction is irreversible.

(c) Why does assuming that the reaction is irreversible simplify the calculation in pan (b)?