Question

A reaction has an activation energy of \(363 \mathrm{~kJ}\) at \(25^{\circ} \mathrm{C}\). If the rate constant has to increase ten-fold, what should the activation energy of the catalyzed reaction be?

Short answer

Answer: The activation energy of the catalyzed reaction should be approximately \(334.7 \mathrm{~kJ \cdot mol^{-1}}\).

Step-by-step solution

Recall the Arrhenius equation

The Arrhenius equation relates the rate constant (k), activation energy (Ea), and temperature (T): \(k = Ae^ {-\frac{Ea}{RT}}\) where A is the pre-exponential factor, R is the universal gas constant (\(8.314 \mathrm{~J \cdot K^{-1} \cdot mol^{-1}}\)), and T is the temperature in Kelvin.

Calculate the initial temperature in Kelvin

Given the initial temperature as \(25^{\circ} \mathrm{C}\), convert it into Kelvin by adding 273.15: \(T = 25 + 273.15 = 298.15 \mathrm{~K}\)

Find the ratio of the catalyzed rate constant to the initial rate constant

According to the problem, the rate constant has to increase ten-fold, meaning the catalyzed rate constant (\(k_2\)) would be 10 times greater than the initial rate constant (\(k_1\)). So, \(\frac{k_2}{k_1} = 10\)

Substitute the Arrhenius equation into the ratio and derive an equation relating initial and final activation energies

Substitute the Arrhenius equation for \(k_2\) and \(k_1\), and then solve for the activation energy of the catalyzed reaction (\(Ea_2\)): \(\frac{Ae^ {-\frac{Ea_2}{RT}}}{Ae^ {-\frac{Ea_1}{RT}}} = 10\) Cancel out the pre-exponential factor (A) and then take the natural logarithm of both sides: \(-\frac{Ea_2}{RT} + \frac{Ea_1}{RT} = \ln{10}\) Now solve for \(Ea_2\): \(Ea_2 = Ea_1 - RT \cdot \ln{10}\)

Plug in the given values and solve for the activation energy of the catalyzed reaction

Using the given values, we can now calculate the activation energy of the catalyzed reaction: \(Ea_2 = 363000 \mathrm{~J\cdot mol^{-1}} - (8.314 \mathrm{~J\cdot K^{-1}\cdot mol^{-1}}) \cdot (298.15 \mathrm{~K}) \cdot \ln{10}\) \(Ea_2 \approx 334727.6 \mathrm{~J\cdot mol^{-1}}\) So, the activation energy of the catalyzed reaction should be approximately \(334.7 \mathrm{~kJ \cdot mol^{-1}}\) for the rate constant to increase ten-fold.

Key concepts

Arrhenius Equation

The Arrhenius equation is a formula that describes how the rate of a chemical reaction depends on temperature and activation energy. It is expressed as
\(k = Ae^{-\frac{Ea}{RT}}\)
where

• \(k\) is the rate constant of the reaction,
• \(A\) is the pre-exponential factor, a constant for each chemical reaction,
• \(Ea\) is the activation energy, or the minimum energy required for a reaction to occur,
• \(R\) is the universal gas constant (\(8.314 \mathrm{~J \cdot K^{-1} \cdot mol^{-1}}\)), and
• \(T\) is the temperature in Kelvin.

This equation is vital for understanding how temperature affects the speed of chemical reactions. The exponential term \(e^{-\frac{Ea}{RT}}\) indicates that even a small decrease in activation energy can significantly increase the rate constant and thereby speed up the reaction.

Catalyzed Reaction

A catalyzed reaction is a chemical reaction that is sped up by the presence of a catalyst. Catalysts are substances that lower the activation energy \(Ea\) required for a reaction to occur, without being consumed in the process.

Catalysts work by providing an alternative pathway for the reaction that has a lower activation energy while not altering the reactants or the final products. As a result, the reaction can occur more rapidly at the same temperature. In the given exercise, the presence of a catalyst is implied to decrease the activation energy from \(363 \mathrm{~kJ \cdot mol^{-1}}\) to \(334.7 \mathrm{~kJ \cdot mol^{-1}}\) which ten-folds the rate constant, illustrating the profound impact a catalyst can have on reaction kinetics.

Rate Constant

The rate constant, denoted as \(k\), is a proportionality constant in the rate equation of a chemical reaction that is specific to the reaction at a given temperature. It reflects how quickly a reaction will proceed. The magnitude of the rate constant depends on various factors, including the presence of a catalyst, the concentration of reactants, and the reaction conditions such as temperature.

The exercise addresses how the rate constant changes when the activation energy is altered by the use of a catalyst. With a ten-fold increase in the rate constant, the reaction would progress much faster, demonstrating that even small changes in the rate constant due to changes in activation energy have significant implications for the rate at which reactions proceed.