The activation energy for a reaction is changed from \(184 \mathrm{~kJ} / \mathrm{mol}\) to \(59.0 \mathrm{~kJ} / \mathrm{mol}\) at \(600 . \mathrm{K}\) by the introduction of a catalyst. If the uncatalyzed reaction takes about 2400 years to occur, about how long will the catalyzed reaction take? Assume the frequency factor \(A\) is constant, and assume the initial concentrations are the same.

Understanding the term activation energy is key to unlocking the secrets of chemical reactions. In essence, it's the energy threshold that reactant molecules must overcome to transform into products. Picture a hill that molecules must climb before they can coast down the other side; activation energy is that hill's height. The bigger the hill, the less likely the molecules are to make it over without a little push. That's why a higher activation energy generally means a slower reaction, as fewer molecules have the energy to scale the summit.

Looking at the Arrhenius equation, \( k = Ae^{\frac{-E_a}{RT}} \), it's clear that the variable \(E_a\) symbolizes activation energy. The negative exponent expresses an inverse relationship; as \(E_a\) increases, the reaction rate constant \(k\) decreases, all else held constant. This is why manipulating activation energy is so powerful—it has a direct impact on how swiftly a reaction proceeds. In our textbook problem, a catalyst reduces the activation energy needed, which in course leads to a significant speed-up of the reaction time.

A catalyzed reaction is a bit of molecular magic. It’s when a substance, called a catalyst, joins the party but leaves without being consumed in the reaction. This guest star has one job: to make the reaction easier to happen. It's like a savvy coach who trains a track runner to hurdle a lower bar, effectively decreasing their activation energy.

In technical terms, a catalyst offers an alternative pathway for the reaction to occur with a lower activation energy. This doesn't affect the final destination—the products are the same as they would be without the catalyst—but it alters the journey, making the reaction happen at a much faster rate. In the exercise at hand, the introduction of a catalyst has slashed the activation energy from \(184 \text{kJ/mol}\) to \(59.0 \text{kJ/mol}\). The stunning result? A process that could otherwise take millennia completes in less than a second—a testament to the power of catalysis.

Delving into the reaction rate constant, denoted as \(k\), opens a window to how nimbly a chemical reaction advances under specific conditions. Think of it as a factor that bundles up the temperature, the presence of a catalyst, and the intrinsic properties of the reactants—all rolled into one. Its value determines the speed at which a given number of reactants will transform into products.

The Arrhenius equation beautifully captures the essence of \(k\): \(k = Ae^{\frac{-E_a}{RT}}\). From this perspective, it’s not just a number; it's a dynamic outcome of varying factors, with the temperature (\(T\)) and activation energy (\(E_a\)) playing key roles. A higher \(k\) indicates a faster reaction, which can be influenced by either ramping up the temperature or, as in our exercise, employing a catalyst to pull down the activation energy. Hence, understanding \(k\) is fundamental for predicting and controlling how fast a reaction will take place in the laboratory setting or in industry applications.