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 $$\begin{aligned}&\text { Slow } \quad \mathrm{NO}_{2}+\mathrm{NO}_{2} \rightarrow\mathrm{NO}_{3}+\mathrm{NO}\\\&\text { Fast } \quad \mathrm{NO}_{3}+\mathrm{CO} \rightarrow \mathrm{NO}_{2}+\mathrm{CO}_{2}\end{aligned}$$ Mechanism \(3 \quad\) Two steps $$\begin{array}{ll}\text { Slow } & \mathrm{NO}_{2} \rightarrow \mathrm{NO}+\mathrm{O} \\\\\text { Fast } & \mathrm{CO}+\mathrm{O} \rightarrow \mathrm{CO}_{2}\end{array}$$

Short answer

Mechanism 2 best matches the experimental rate equation.

Step-by-step solution

Understand the Experimental Rate Law

The experimentally observed rate equation is given as \( \text{Rate} = k[\text{NO}_2]^2 \). This indicates that the rate-determining step should involve two \( \text{NO}_2 \) molecules in the reaction process.

Analyze Mechanism 1

Mechanism 1 suggests a single elementary step: \( \text{NO}_2 + \text{CO} \rightarrow \text{CO}_2 + \text{NO} \). As an elementary reaction, the rate law from this step should be \( k[\text{NO}_2][\text{CO}] \), which does not match the experimental rate law \( k[\text{NO}_2]^2 \). Therefore, Mechanism 1 does not agree with the experiment.

Analyze Mechanism 2

Mechanism 2 consists of two steps: a slow step \( \text{NO}_2 + \text{NO}_2 \rightarrow \text{NO}_3 + \text{NO} \) and a fast step \( \text{NO}_3 + \text{CO} \rightarrow \text{NO}_2 + \text{CO}_2 \). The slow step is rate-determining and involves \( \text{NO}_2 + \text{NO}_2 \), which provides a rate law of \( k[\text{NO}_2]^2 \). This matches the experimental rate law, making Mechanism 2 consistent with the experimental data.

Analyze Mechanism 3

Mechanism 3 involves a slow step \( \text{NO}_2 \rightarrow \text{NO} + \text{O} \) and a fast step \( \text{CO} + \text{O} \rightarrow \text{CO}_2 \). The slow step has a rate law of \( k[\text{NO}_2] \), which does not fit the experimental rate law of \( k[\text{NO}_2]^2 \). Therefore, Mechanism 3 does not match the experimental observations.

Conclusion

After comparing all mechanisms, only Mechanism 2 provides a rate law that matches the experimentally observed rate equation \( \text{Rate} = k[\text{NO}_2]^2 \). Thus, Mechanism 2 is the best fit for the experiment.

Key concepts

Rate Law

The rate law of a chemical reaction provides important insights into the reaction mechanism. It expresses the rate of the reaction as a function of the concentration of reactants. For the CO and NO₂ reaction, the experimental rate law is \[\text{Rate} = k[\text{NO}_2]^2\]This tells us that the reaction rate is directly proportional to the square of the concentration of NO₂, indicating that two NO₂ molecules are involved in the critical step of the reaction. Understanding why this rate law is observed requires examining the proposed mechanisms and identifying which one involves these two NO₂ molecules in a slow, rate-determining step.
In summary:

• Rate laws are deduced from experimental data.
• The exponents in the rate law reflect the number of each reactant molecule involved in the rate-determining step.
• Matching a rate law to a mechanism helps confirm or refute proposed reaction pathways.

Elementary Reaction

An elementary reaction is a single step process that occurs in a chemical reaction, representing an individual collision between reactant molecules. Each elementary step has its own rate law derived from its molecularity, which describes the number of molecules involved. This is crucial because the overall reaction rate often hinges on identifying the correct sequence of elementary steps that match the experimental rate law.
Let's consider the proposed mechanisms:

• Mechanism 1 suggests an elementary step: \[\text{NO}_2 + \text{CO} \rightarrow \text{CO}_2 + \text{NO}\]However, its rate law \[k[\text{NO}_2][\text{CO}]\]does not match the experimental \[k[\text{NO}_2]^2\]
• Mechanism 2 suggests two elementary steps, and the initial step aligns with the observed rate law, making it plausible.
• Mechanism 3 also suggests two steps, but fails to fit the experimental rate, thus is less likely.

Recognizing the difference between these steps is key to understanding how reactions proceed from reactants to products.

Rate-Determining Step

The rate-determining step is the slowest step in a reaction mechanism and acts as a bottleneck for the entire process. It's crucial because it dictates the overall reaction rate and thus, is reflected in the experimentally observed rate law. In mechanism analysis, identifying the rate-determining step is essential to see if experimental data supports the proposed reaction pathway.
For our reaction:

• Mechanism 2's initial step \[\text{NO}_2 + \text{NO}_2 \rightarrow \text{NO}_3 + \text{NO}\]is the rate-determining step and matches the observed rate law \[k[\text{NO}_2]^2\]
• This shows that the slow formation of \( \text{NO}_3 \) and \( \text{NO} \) governs the entire reaction rate.
• Mechanisms 1 and 3's rate-determining steps do not align with the experimental rate law, hence they aren't viable.

In conclusion, the rate-determining step is invaluable in validating correct mechanisms and understanding reaction dynamics.

Chemical Kinetics

Chemical kinetics is the branch of chemistry that studies the speed of chemical reactions and provides insight into reaction mechanisms. It connects quantitative measures like rate laws with qualitative reaction paths, offering explanations for how reactions occur and what factors influence their rates.
Key aspects of chemical kinetics include:

• The role of concentration, temperature, and catalysts in influencing reaction rates.
• Understanding mechanisms through stepwise reactions and identifying the rate-determining step.
• Using rate laws to derive mechanistic insights that are consistent with experimental observations.

In our CO and NO₂ reaction example, chemical kinetics analysis allows us to conclude that Mechanism 2 best aligns with the observed data, as it explains not only how the reaction proceeds but also why the rate law takes its specific form. This understanding is crucial for predicting and controlling reaction behavior in both scientific and industrial contexts.