Problem 63 - Chemistry and Chemical Reactivity

Question

At temperatures below \(500 \mathrm{K}\), the reaction between carbon monoxide and nitrogen dioxide $$ \mathrm{CO}(\mathrm{g})+\mathrm{NO}_{2}(\mathrm{g}) \rightarrow \mathrm{CO}_{2}(\mathrm{g})+\mathrm{NO}(\mathrm{g}) $$ has the following rate equation: Rate \(=k\left[\mathrm{NO}_{2}\right]^{2}\) Which of the three mechanisms suggested here best agrees with the experimentally observed rate equation? Mechanism 1 \(\quad\) single, elementary step $$ \mathrm{NO}_{2}+\mathrm{CO} \rightarrow \mathrm{CO}_{2}+\mathrm{NO} $$ Mechanism \(2 \quad\) Two steps Slow $$ \mathrm{NO}_{2}+\mathrm{NO}_{2} \rightarrow \mathrm{NO}_{3}+\mathrm{NO} $$ Fast $$ \mathrm{NO}_{3}+\mathrm{CO} \rightarrow \mathrm{NO}_{2}+\mathrm{CO}_{2} $$ Mechanism 3 Two steps Slow $$ \mathrm{NO}_{2} \rightarrow \mathrm{NO}+\mathrm{O} $$ Fast $$ \mathrm{CO}+\mathrm{O} \rightarrow \mathrm{CO}_{2} $$

Step-by-Step Solution

Answer

Mechanism 2 is the correct mechanism.

Step 1: Identify the given rate equation

The problem states that the rate of the reaction is given by the equation \( \text{Rate} = k[\text{NO}_2]^2 \). This implies that the reaction rate is dependent on the concentration of \( \text{NO}_2 \) squared, suggesting that two molecules of \( \text{NO}_2 \) are involved in the rate-determining step.

Step 2: Analyze Mechanism 1

Mechanism 1 proposes a single, elementary step: \( \text{NO}_2 + \text{CO} \rightarrow \text{CO}_2 + \text{NO} \). In this mechanism, the rate law would be \( \text{Rate} = k[\text{NO}_2][\text{CO}] \), which doesn't match the given rate law of \( \text{Rate} = k[\text{NO}_2]^2 \). Therefore, Mechanism 1 cannot be the correct mechanism.

Step 3: Analyze Mechanism 2

Mechanism 2 consists of two steps:1. Slow: \( \text{NO}_2 + \text{NO}_2 \rightarrow \text{NO}_3 + \text{NO} \)2. Fast: \( \text{NO}_3 + \text{CO} \rightarrow \text{NO}_2 + \text{CO}_2 \)In the slow step, two molecules of \( \text{NO}_2 \) are involved, which aligns with the given rate law \( \text{Rate} = k[\text{NO}_2]^2 \). This mechanism matches the observed rate equation, indicating that the first step is the rate-determining step.

Step 4: Analyze Mechanism 3

Mechanism 3 also consists of two steps:1. Slow: \( \text{NO}_2 \rightarrow \text{NO} + \text{O} \)2. Fast: \( \text{CO} + \text{O} \rightarrow \text{CO}_2 \)The slow step involves only one \( \text{NO}_2 \) molecule, leading to a rate law of \( \text{Rate} = k[\text{NO}_2] \), which does not match the observed rate law \( \text{Rate} = k[\text{NO}_2]^2 \). Thus, Mechanism 3 isn't suitable.

Step 5: Conclusion: Determine the suitable mechanism

Based on the analysis, Mechanism 2 is the only one that matches the experimental rate law \( \text{Rate} = k[\text{NO}_2]^2 \) because the rate-determining step involves two molecules of \( \text{NO}_2 \) reacting.

Key Concepts

Rate-Determining Step

In a multi-step chemical reaction, not all steps occur at the same speed. The rate-determining step is the slowest step in the sequence and effectively limits the rate at which the overall reaction proceeds. When analyzing reaction mechanisms, identifying the rate-determining step is crucial because it controls the kinetics of the reaction.

Consider the reaction between carbon monoxide and nitrogen dioxide where we have to choose from three proposed mechanisms. Each mechanism offers a different pathway for the reaction, and the rate-determining step would dictate which mechanism matches the observed rate law. A slow step involving two NO₂ molecules aligns with the experimental observation, as it matches the rate law of Rate = k[NO₂]², indicating that two molecules must collide in the rate-determining step. Therefore, Mechanism 2, with a slow step involving NO₂ + NO₂, is consistent with the given data.

Rate Law

The rate law for a chemical reaction expresses the relationship between the reaction rate and the concentration of its reactants. It takes the form Rate = k[ ext{Reactant1}][ ext{Reactant2}], where k is the rate constant, reflecting factors like temperature and catalyst presence.

For understanding our reaction, we consider the given rate law Rate = k[NO₂]². This indicates that the rate is second-order with respect to NO₂ and independent of CO concentration under the conditions specified. Such a rate law implies that any plausible mechanism must involve two NO₂ molecules in the rate-determining step. When comparing the potential mechanisms, the rate law points us towards Mechanism 2, which involves two NO₂ molecules in its slow step, conforming to the experimentally determined rate expression.

Chemical Kinetics

Chemical kinetics is the study of reaction rates and the steps by which they occur. It helps us understand how different variables such as concentration, temperature, and catalysts affect reaction progress. This vital area of chemistry extends beyond merely predicting how fast a reaction will occur by dissecting the mechanism—or the series of steps—leading to the product formation.

Examining the reaction of CO with NO₂ offers a classic application of principles from chemical kinetics. The process involves analyzing each proposed mechanism step-by-step, considering the slowest (rate-determining) step's contribution to the reaction kinetics. Mechanism 2 contains a slow initial step involving two NO₂ molecules, which chemically validates the observed rate law according to theories of chemical kinetics, making it the most suitable mechanism for explaining the reaction's behavior at temperatures below 500 K.