Question

The following system obeys second-order kinetics. $$ \begin{gathered} 2 \mathrm{NO}_2 \rightarrow \mathrm{NO}_3+\mathrm{NO} \text { (slow) } \\ \mathrm{NO}_3+\mathrm{CO} \rightarrow \mathrm{NO}_2+\mathrm{CO}_2 \text { (fast) } \end{gathered} $$ What is the rate law for this reaction? A. Rate \(=\mathrm{k}\left[\mathrm{NO}_2\right][\mathrm{CO}]\). B. Rate \(=\mathrm{k}\left[\mathrm{NO}_2\right]^2[\mathrm{CO}]\) C. Rate \(=\mathrm{k}\left[\mathrm{NO}_2\right]\left[\mathrm{NO}_3\right]\). D. Rate \(=\mathrm{k}\left[\mathrm{NO}_2\right]^2\).

Short answer

D. Rate \(= k[\mathrm{NO}_2]^2\)

Step-by-step solution

Understand the Reaction Mechanism

The given system involves two steps: 1) Slow step: \(2 \mathrm{NO}_2 \rightarrow \mathrm{NO}_3 + \mathrm{NO}\) 2) Fast step: \(\mathrm{NO}_3 + \mathrm{CO} \rightarrow \mathrm{NO}_2 + \mathrm{CO}_2\)

Identify the Rate-Determining Step

The slow step () is the rate-determining step. The rate law is primarily determined by this step.

Write the Rate Law for the Slow Step

The rate law is based on the slow step: \(\text{Rate} = k[\mathrm{NO}_2]^2\).

Compare with Given Options

Now compare the derived rate law with the given options: A, B, C, and D. The rate law we derived matches option D: \(\text{Rate} = k[\mathrm{NO}_2]^2\).

Key concepts

rate law

The rate law expresses the relationship between the rate of a chemical reaction and the concentration of its reactants. For the given reaction system, we note that it follows second-order kinetics. This means the rate law is not as straightforward as being directly proportional to the concentration of a single reactant. Instead, it involves the concentrations of two reactants in a particular manner.
In the slow step:

• We have the reaction: 2 NO₂ → NO₃ + NO
• The rate law for this reaction can be written as:
  Rate = k[NO₂]²

This tells us that the rate of reaction is directly proportional to the square of the concentration of NO₂. This is indicative of a second-order reaction, where doubling the concentration of NO₂ would quadruple the reaction rate.

reaction mechanism

A reaction mechanism outlines the individual steps that make up the overall reaction. Each step in the mechanism involves different molecular interactions and rearrangements that contribute to the final products. For our reaction, there are two steps:

• Slow step: 2 NO₂ → NO₃ + NO (this step controls the rate of the overall reaction)
• Fast step: NO₃ + CO → NO₂ + CO₂

Reaction mechanisms are critical as they provide a detailed understanding of the pathway taken by reactants to form products. This helps in predicting how the reaction will behave under different conditions.
It's important to recognize that the overall reaction rate is not influenced by the fast step because it's quickly completed once the slow step has taken place.

rate-determining step

The rate-determining step (RDS) is the slowest step in a reaction mechanism that determines the overall reaction rate. For the given reaction, the slow step is:

• 2 NO₂ → NO₃ + NO (slow)

Because this step is slow, it acts as a bottleneck for the entire reaction process. The rate law is derived based on this step since it limits the speed at which the reactants convert into products.
This concept is crucial because it simplifies the determination of the rate law. By focusing on the slowest, rate-determining step, we ignore the faster steps that do not significantly impact the overall rate of reaction.

chemical kinetics

Chemical kinetics is the branch of chemistry that studies the speed (rate) of chemical reactions and the factors that influence this speed. Key factors include the concentration of reactants, temperature, presence of a catalyst, and the nature of the reactants and products.
For the given reaction:

• 2 NO₂ → NO₃ + NO (slow)
• NO₃ + CO → NO₂ + CO₂ (fast)

The kinetics focus on how changing the concentration of NO₂ or adding a catalyst affects the reaction rate. By understanding these factors, chemists can control and optimize reactions for industrial and laboratory applications.
Chemical kinetics also involves studying reaction mechanisms to predict how changes to reaction conditions might influence the speed and outcome of the reaction. This foundational knowledge is critical for developing efficient chemical processes in various fields, from pharmaceuticals to energy production.