Question

One mechanism for the destruction of ozone in the upper atmosphere is $$ \begin{array}{ll} \mathrm{O}_{3}(g)+\mathrm{NO}(g) \longrightarrow \mathrm{NO}_{2}(g)+\mathrm{O}_{2}(g) & \text { Slov } \\ \mathrm{NO}_{2}(g)+\mathrm{O}(g) \longrightarrow \mathrm{NO}(g)+\mathrm{O}_{2}(g) & \text { Fast } \\ \hline \end{array} $$ Overall reaction \(\mathrm{O}_{3}(\mathrm{~g})+\mathrm{O}(\mathrm{g}) \longrightarrow 2 \mathrm{O}_{2}(\mathrm{~g})\) a. Which species is a catalyst? b. Which species is an intermediate? c. \(E_{\mathrm{a}}\) for the uncatalyzed reaction $$ \mathrm{O}_{3}(g)+\mathrm{O}(g) \longrightarrow 2 \mathrm{O}_{2} $$ is \(14.0 \mathrm{~kJ} . E_{\mathrm{a}}\) for the same reaction when catalyzed is \(11.9 \mathrm{~kJ}\). What is the ratio of the rate constant for the catalyzed reaction to that for the uncatalyzed reaction at \(25^{\circ} \mathrm{C}\) ? Assume that the frequency factor \(A\) is the same for each reaction.

Short answer

a. The catalyst is NO, as it is involved in both elementary reactions but doesn't appear in the overall reaction. b. The intermediate species is NO2, as it is formed and consumed in the reaction mechanism but doesn't appear in the overall reaction. c. Using the Arrhenius equation, the ratio of the rate constant for the catalyzed reaction to that for the uncatalyzed reaction at 25°C is approximately 4.04.

Step-by-step solution

Identifying the catalyst

A catalyst is a substance that increases the rate of a chemical reaction without being consumed in the overall reaction. In the given mechanism, we observe that NO is present in both elementary reactions as both a reactant and a product, and doesn't appear in the overall reaction. Thus, NO is the catalyst for this mechanism.

Identifying the intermediate

An intermediate species is formed and consumed in the course of a reaction mechanism, and does not appear in the overall reaction. In the given mechanism, we observe that NO2 is formed in the first elementary reaction and consumed in the second elementary reaction, and it doesn't appear in the overall reaction. Thus, NO2 is an intermediate species.

Applying the Arrhenius equation to find the rate constant ratio

The Arrhenius equation is used to relate the rate constant k of a reaction to the activation energy Ea, the temperature T, and the frequency factor A: \(k=A \cdot e^{-\frac{Ea}{RT}}\) where R is the gas constant (8.314 J/mol·K in this case). We are given the activation energies Ea for both the catalyzed (11.9 kJ/mol) and uncatalyzed (14.0 kJ/mol) reactions, and the temperature T = 298 K = 25°C. We need to determine the ratio of the rate constants for the catalyzed (k1) and uncatalyzed (k2) reactions. The ratio can be calculated as: \(\frac{k1}{k2} = \frac{A \cdot e^{-\frac{Ea_{1}}{RT}}}{A \cdot e^{-\frac{Ea_{2}}{RT}}}\) Since the frequency factors A are the same for both reactions, they will cancel out, leaving only the exponential terms: \(\frac{k1}{k2} = \frac{e^{-\frac{Ea_{1}}{RT}}}{e^{-\frac{Ea_{2}}{RT}}}\) Now, we can plug in the given values for Ea and T, and calculate the rate constant ratio.

Calculate the rate constant ratio

With the values given: \(Ea_{1} = 11.9 \times 10^{3} J/mol\) \(Ea_{2} = 14.0 \times 10^{3} J/mol\) \(T = 298 K\) \(R = 8.314 J/mol·K\) Applying these into the equation: \(\frac{k1}{k2} = \frac{e^{-\frac{11.9 \times 10^{3}}{8.314 \times 298}}}{e^{-\frac{14.0 \times 10^{3}}{8.314 \times 298}}}\) Computing this gives: \(\frac{k1}{k2} \approx 4.04\) So, the rate constant for the catalyzed reaction is roughly 4.04 times greater than that for the uncatalyzed reaction at 25°C.

Key concepts

Catalysis

Catalysis plays a vital role in many chemical reactions by speeding them up without being part of the final product. In the case of ozone destruction in the atmosphere, a catalyst is a magical helper that aids the reaction process. In this specific problem, nitric oxide (NO) acts as the catalyst.
It reappears in both steps of the reaction mechanism and remains unchanged by the end of the reaction. This amazing property allows it to continuously facilitate the reaction, making the process more efficient while not being used up itself.

• No permanent transformation: Catalysts are present by the end of the reaction.
• Catalyst cycle: Appears in reactants and products at different stages.
• Speeds up the reaction: Reduces the overall activation energy, making reactions happen faster.

Reaction Mechanism

Understanding the reaction mechanism is like having a roadmap for the chemical reaction journey. It dissects the overall reaction into simpler steps called elementary reactions. For the breakdown of ozone, two key reactions take place.
1. The first step involves ozone reacting with NO to form NO2 and O2. 2. The second step, which is faster, sees NO2 react with an oxygen atom to regenerate NO and produce another O2 molecule. By piecing these together, we realize how molecules are changed at each stage, allowing the understanding of their interaction and transformation.

• Elementary Steps: The reaction is broken down into smaller, sequential actions.
• Intermediates: Temporary species, like NO2, formed and consumed during the process.
• Overall Reaction: Result of combining steps to show initial and final states.

Arrhenius Equation

The Arrhenius equation is like the mathematical recipe for figuring how fast a reaction happens. It links the rate constant (k) to important factors like activation energy (Ea), temperature (T), and the frequency factor (A). The equation is given by:
\[ k = A \cdot e^{-\frac{Ea}{RT}} \]
Here, R is the universal gas constant. In this scenario, both catalyzed and uncatalyzed reactions share the same frequency factor A. By carefully comparing them, we use the Arrhenius equation to find how much faster the catalyzed reaction occurs. This is achieved by examining the ratio of their rate constants (\( \frac{k_1}{k_2} \)), which indicates just how effective the catalyst is. The ratio signifies how much more swift the catalyzed pathway is compared to its uncatalyzed counterpart.

• Rate Prediction: A formula to predict how fast reactions proceed.
• Exponent Component: Shows the influence of temperature and activation energy.

Activation Energy

Activation energy is like the energy hurdle that reactants need to jump to transform into products. In catalysis, one of the great achievements is lowering this barrier, making the path smoother for the reaction to proceed. For the destruction of ozone, the activation energy for the uncatalyzed reaction is 14.0 kJ/mol, which is relatively high.
By introducing a catalyst, the activation energy drops to 11.9 kJ/mol. This decrease reflects how the catalyst facilitates the reaction process, enabling reactants to transform more easily and effectively.

• Energy Requirement: Minimum energy needed to start a reaction.
• Catalyst Effect: Reduction in the energy needed, aiding reaction progress.

The smaller the energy barrier, the greater the number of molecules that can participate in the reaction, thus speeding up the rate.