The \(k_{\text {cat }}\) for alkaline phosphatase-catalyzed hydrolysis of methylphosphate is approximately \(14 / \sec\) at \(\mathrm{pH} 8\) and \(25^{\circ} \mathrm{C}\). The rate constant for the uncatalyzed hydrolysis of methylphosphate under the same conditions is approximately \(10^{-15} /\) sec. What is the difference in the free energies of activation of these two reactions?

A catalyzed reaction is one that is accelerated by the addition of a catalyst, a substance that increases the rate of the reaction without undergoing any permanent chemical change itself. In biological systems, enzymes are natural catalysts that facilitate various biochemical reactions necessary for life. In the given exercise, alkaline phosphatase accelerates the hydrolysis of methylphosphate. Enzymes like alkaline phosphatase operate by providing an alternative reaction pathway with a lower activation energy compared to the uncatalyzed pathway, hence significantly enhancing the reaction rate. This concept is crucial as it explains the mechanism through which living organisms achieve rapid and highly regulated metabolic processes.

In contrast to a catalyzed reaction, an uncatalyzed reaction proceeds without the assistance of a catalyst. It relies solely on the kinetic energy of the reactants and environmental conditions, such as temperature and pressure, to reach the activation energy threshold. As a result, uncatalyzed reactions typically occur at a much slower rate because the probability of reactant molecules colliding with sufficient energy to surpass the activation energy barrier is lower. The vast difference in rate constants of the catalyzed and uncatalyzed hydrolysis of methylphosphate highlights the significant role enzymes play in increasing reaction rates, which the exercise solution illustrates by calculating the immense difference in activation energies.