The absolute activation energy of an uncatalyzed reaction is determined to be exactly . The addition — Chemical Kinetics Chemistry Question
Question
The absolute activation energy of an uncatalyzed reaction is determined to be exactly $75 \text{ kJ mol}^{-1}$. The addition of a specific catalyst lowers the effective activation energy to $50 \text{ kJ mol}^{-1}$. At what specific elevated temperature (in Kelvin) will the reaction rate of the uncatalyzed reaction be exactly equal to the accelerated rate of the catalyzed reaction operating at $300 \text{ K}$? (Assume the pre-exponential factor $A$ remains strictly identical for both reaction pathways).
💡 Solution & Explanation
Since the rates (and thus rate constants) are strictly equal, $k_{uncat} = k_{cat}$. Given that the Arrhenius constant $A$ is the same, we can equate the exponential terms: $A e^{-E_{uncat} / R T_{uncat}} = A e^{-E_{cat} / R T_{cat}}$. Canceling $A$ and taking the natural log yields: $E_{uncat} / T_{uncat} = E_{cat} / T_{cat}$. Substituting the provided numerical values: $75 / T_{uncat} = 50 / 300$. This simplifies to $75 / T_{uncat} = 1 / 6$. Solving for the unknown temperature gives $T_{uncat} = 75 \times 6 = 450 \text{ K}$.