Question

P35.32 The reaction of atomic chlorine with ozone is the first step in the catalytic decomposition of stratospheric ozone by \(\mathrm{Cl} \bullet\):$$\mathrm{Cl} \cdot(g)+\mathrm{O}_{3}(g) \rightarrow \mathrm{ClO} \cdot(g)+\mathrm{O}_{2}(g)$$ At \(298 \mathrm{K}\) the rate constant for this reaction is \(6.7 \times 10^{9} \mathrm{M}^{-1} \mathrm{s}^{-1}\) Experimentally, the Arrhenius pre-exponential factor was determined to be \(1.4 \times 10^{10} \mathrm{M}^{-1} \mathrm{s}^{-1} .\) Using this information determine the activation energy for this reaction.

Short answer

The activation energy for the reaction between atomic chlorine and ozone at 298 K is approximately \(7283\ \text{J mol}^{-1}\).

Step-by-step solution

Identify the given variables

We are given the following data: - Rate constant (\(k\)): \(6.7 \times 10^{9}\ \mathrm{M}^{-1} \mathrm{s}^{-1}\) - Arrhenius pre-exponential factor (\(A\)): \(1.4 \times 10^{10} \mathrm{M}^{-1} \mathrm{s}^{-1}\) - Temperature (\(T\)): 298 K

Rearrange the Arrhenius equation for activation energy

The Arrhenius equation is: \(k = Ae^{\dfrac{-E_{a}}{RT}}\) We want to solve for activation energy (\(E_{a}\)). First, divide both sides of the equation by \(A\): \(\dfrac{k}{A} = e^{\dfrac{-E_{a}}{RT}}\) Next, take the natural logarithm of both sides to isolate the exponent: \(\ln(\dfrac{k}{A}) = \dfrac{-E_{a}}{RT}\) Finally, multiply both sides by -RT: \(E_{a} = -RT \ln(\dfrac{k}{A})\)

Substitute the given values and solve for activation energy

Using the given values, we can now calculate the activation energy: \(E_{a} = -(8.314\ \text{J mol}^{-1} \text{K}^{-1})(298\ \text{K}) \ln(\dfrac{6.7 \times 10^{9}\ \mathrm{M}^{-1} \mathrm{s}^{-1}}{1.4 \times 10^{10}\ \mathrm{M}^{-1} \mathrm{s}^{-1}})\) \(E_{a} \approx -24624\ \text{J mol}^{-1} \ln(0.4786)\) \(E_{a} \approx 7283\ \text{J mol}^{-1}\) Thus, the activation energy for this reaction is approximately \(7283\ \text{J mol}^{-1}\).

Key concepts

Arrhenius Equation

The Arrhenius Equation is a fundamental formula used to calculate the rate constant of a chemical reaction, and it helps us understand how temperature influences reaction rates. The equation is given by:\[ k = Ae^{\dfrac{-E_{a}}{RT}} \]where:

• \(k\) is the rate constant.
• \(A\) is the pre-exponential factor, often called the frequency factor.
• \(E_a\) is the activation energy.
• \(R\) is the gas constant (8.314 J mol\(^{-1}\) K\(^{-1}\)).
• \(T\) is the temperature in Kelvin.

The equation shows that as the temperature increases, the rate constant \(k\) increases, assuming the activation energy \(E_a\) remains the same. Taking the natural log of both sides simplifies calculations involving the equation, especially when we need to solve for the activation energy, like in your exercise.

Catalytic Decomposition

Catalytic decomposition involves the breakdown of molecules into simpler chemical species using a catalyst, which accelerates the reaction without being consumed. In the context of the reaction given in the exercise, the molecule ozone \( (O_3) \) is decomposed in the presence of atomic chlorine \(( \mathrm{Cl} \bullet)\). This reaction is the first step in the catalytic cycle that leads to the destruction of ozone in the stratosphere. Chlorine atoms act as catalysts in this process because they enable reactions to occur at a lower activation energy and can participate in many reaction cycles repeatedly. These reactions are significant in atmospheric chemistry as they contribute to ozone layer depletion, highlighting the importance of understanding and controlling catalytic decomposition processes in nature.

Rate Constant

The rate constant, symbolized as \(k\), is a measure of the speed of a chemical reaction. In chemical kinetics, it is pivotal in defining how quickly reactants are transformed into products under given conditions. From the Arrhenius equation, we see the rate constant depends on temperature and the activation energy. In the exercise example, the rate constant is given as \(6.7 \times 10^{9} \ \mathrm{M}^{-1} \mathrm{s}^{-1}\) at 298 K. This value tells us that the reaction is quite fast, as indicated by the large magnitude of the rate constant. Understanding the rate constant helps scientists predict how changes in temperature or catalysts can affect the speed of a reaction, making it critical for everything from drug design to industrial synthesis.

Pre-exponential Factor

The pre-exponential factor, denoted as \(A\) in the Arrhenius equation, provides insight into the frequency of collisions and the orientation of reactant molecules. It is also known as the frequency factor or the steric factor. This term indicates how many times reactant molecules approach each other in the correct orientation to react per unit time. In your exercise, the Arrhenius pre-exponential factor is given as \(1.4 \times 10^{10} \ \mathrm{M}^{-1} \mathrm{s}^{-1}\), indicating a high frequency which is typical for reactions involving gas-phase molecules like those in the stratosphere. Understanding \(A\) helps in analyzing the molecular dynamics of reactions, providing further insights beyond what temperature and activation energy alone can deliver.