Question

The rate law for the following reaction: $$ \mathrm{CO}(g)+\mathrm{NO}_{2}(g) \longrightarrow \mathrm{CO}_{2}(g)+\mathrm{NO}(g) $$ is rate \(=k\left[\mathrm{NO}_{2}\right]^{2}\). Suggest a plausible mechanism for the reaction, given that the unstable species \(\mathrm{NO}_{3}\) is an intermediate.

Short answer

Possible mechanism: 1) \( 2\text{NO}_2 \rightarrow \text{NO}_3 + \text{NO} \); 2) \( \text{NO}_3 + \text{CO} \rightarrow \text{CO}_2 + \text{NO}_2 \).

Step-by-step solution

Understand the Rate Law

The given rate law is \( \text{rate} = k[\text{NO}_2]^2 \). This indicates that the reaction is second order with respect to \( \text{NO}_2 \) and suggests that two \( \text{NO}_2 \) molecules are involved in the rate-determining step.

Identify Possible Intermediates and Products

We know that \( \text{NO}_3 \) is an intermediate. This means that \( \text{NO}_3 \) is not in the initial or final products but forms temporarily. The final products are \( \text{CO}_2 \) and \( \text{NO} \).

Propose the First Step of the Mechanism

In the first step, assume that two \( \text{NO}_2 \) molecules react to form \( \text{NO}_3 \) and \( \text{NO} \). This can be written as: \( \text{NO}_2 + \text{NO}_2 \rightarrow \text{NO}_3 + \text{NO} \). This step involves two \( \text{NO}_2 \) molecules, fitting the rate law.

Propose the Second Step of the Mechanism

In the second step, \( \text{NO}_3 \) reacts with \( \text{CO} \) to form \( \text{CO}_2 \) and \( \text{NO}_2 \). This can be written as: \( \text{NO}_3 + \text{CO} \rightarrow \text{CO}_2 + \text{NO}_2 \). \( \text{NO}_3 \) is used up and \( \text{NO}_2 \) is reformed, consistent with \( \text{NO}_3 \) as an intermediate.

Verify Consistency with Rate Law

The first step is the rate-determining step (slow step), involving two \( \text{NO}_2 \) molecules: \( \text{rate} = k[\text{NO}_2]^2 \). This is directly consistent with the provided rate law. The second step is fast and regenerates \( \text{NO}_2 \), maintaining overall stoichiometry.

Key concepts

Rate Law

When a chemical reaction takes place, its rate can often be described by a rate law. The rate law is an equation that links the reaction rate with the concentrations of reactants. For the reaction involving carbon monoxide (\( \text{CO} \)) and nitrogen dioxide (\( \text{NO}_2 \)), the rate law is given by:
\[ \text{rate} = k[\text{NO}_2]^2 \]
This means the rate of the reaction depends on the concentration of \( \text{NO}_2 \) squared. The coefficient 2 indicates the reaction is second-order with respect to \( \text{NO}_2 \).
There are key aspects to consider about this rate law:

• The constant \( k \) is the rate constant, which varies with temperature and provides insight into the speed of the reaction.
• The second-order dependency implies that for the formation of products, two \( \text{NO}_2 \) molecules are necessary in the rate-determining step.

Understanding this helps in proposing the sequence of steps involved in a reaction mechanism.

Intermediate Species

An intermediate species in a chemical reaction is a molecule that is formed in one of the steps and consumed in another, never appearing in the overall reaction equation. In the given problem, \( \text{NO}_3 \) is identified as an intermediate.Why is \( \text{NO}_3 \) crucial in this context?

• It is formed from the reaction of \( \text{NO}_2 \) molecules, which supports the second-order kinetic expression.
• It does not appear in the final products, indicating it is consumed during the process.
• Its temporary formation allows for the overall mechanism to work, facilitating the transformation of \( \text{CO} \) into \( \text{CO}_2 \).

Understanding intermediates help us piece together how individual steps lead to the overall reaction. They provide a clearer picture of the transition and transformation states involved in the mechanism.

Rate-Determining Step

In reaction mechanisms, the rate-determining step is the slowest step which controls the speed of the entire reaction. It is like a bottleneck that decides how fast the overall process can proceed.For the reaction of \( \text{CO} \) and \( \text{NO}_2 \), the rate-determining step involves two molecules of \( \text{NO}_2 \) forming \( \text{NO}_3 \) and \( \text{NO} \). This is significant because:

• It correlates directly with the rate law \( \text{rate} = k[\text{NO}_2]^2 \), showing that the concentration of \( \text{NO}_2 \) directly impacts the rate.
• The reaction cannot proceed faster than this step, making it critical to understand and analyze this step when modifying or optimizing conditions to speed up the reaction.

The concept of a rate-determining step helps chemists identify which steps in a reaction mechanism need more focus to either accelerate or control the reaction efficiently.