Question

The reaction of \(\mathrm{NO}_{2}(\mathrm{g})\) and \(\mathrm{CO}(\mathrm{g})\) is thought to occur in two steps: Step 1: Slow \(\mathrm{NO}_{2}(\mathrm{g})+\mathrm{NO}_{2}(\mathrm{g}) \rightarrow \mathrm{NO}(\mathrm{g})+\mathrm{NO}_{3}(\mathrm{g})\) Step 2: Fast \(\mathrm{NO}_{3}(\mathrm{g})+\mathrm{CO}(\mathrm{g}) \rightarrow \mathrm{NO}_{2}(\mathrm{g})+\mathrm{CO}_{2}(\mathrm{g})\) (a) Show that the elementary steps add up to give the overall, stoichiometric equation. (b) What is the molecularity of each step? (c) For this mechanism to be consistent with kinetic data, what must be the experimental rate equation? (d) Identify any intermediates in this reaction.

Short answer

(a) \( \mathrm{NO}_{2} + \mathrm{CO} \rightarrow \mathrm{NO} + \mathrm{CO}_{2} \); (b) Both steps are bimolecular; (c) Rate = \( k[\mathrm{NO}_{2}]^2 \); (d) Intermediate: \( \mathrm{NO}_{3} \).

Step-by-step solution

Combining Elementary Steps

First, we need to combine the two given steps to verify if they produce the overall reaction. Add together the reactants and products from each step.Step 1: \( \mathrm{NO}_{2} + \mathrm{NO}_{2} \rightarrow \mathrm{NO} + \mathrm{NO}_{3} \)Step 2: \( \mathrm{NO}_{3} + \mathrm{CO} \rightarrow \mathrm{NO}_{2} + \mathrm{CO}_{2} \)Adding both:Reactants: \( \mathrm{NO}_{2} + \mathrm{NO}_{2} + \mathrm{NO}_{3} + \mathrm{CO} \)Products: \( \mathrm{NO} + \mathrm{NO}_{3} + \mathrm{NO}_{2} + \mathrm{CO}_{2} \) Cancel \( \mathrm{NO}_{3} \) from both sides, leaving:\( 2\mathrm{NO}_{2} + \mathrm{CO} \rightarrow \mathrm{NO} + \mathrm{NO}_{2} + \mathrm{CO}_{2} \)Further simplify by canceling \( \mathrm{NO}_{2} \) on both sides:\( \mathrm{NO}_{2} + \mathrm{CO} \rightarrow \mathrm{NO} + \mathrm{CO}_{2} \)So, the simplified equation is \( \mathrm{NO}_{2} + \mathrm{CO} \rightarrow \mathrm{NO} + \mathrm{CO}_{2} \). This verifies the stoichiometric equation.

Determining Molecularity

Molecularity refers to the number of molecules coming together in a step of a reaction. For Step 1: \( \mathrm{NO}_{2} + \mathrm{NO}_{2} \rightarrow \mathrm{NO} + \mathrm{NO}_{3} \), two \( \mathrm{NO}_{2} \) molecules are involved, so it is bimolecular.For Step 2: \( \mathrm{NO}_{3} + \mathrm{CO} \rightarrow \mathrm{NO}_{2} + \mathrm{CO}_{2} \), one \( \mathrm{NO}_{3} \) and one \( \mathrm{CO} \) are involved, making it also bimolecular.

Deriving the Rate Equation

For kinetic data consistency, the rate law must reflect the reaction mechanism and include the slow step. The given mechanism indicates Step 1 is slow, and determines the rate law.For step 1: \( \mathrm{NO}_{2} + \mathrm{NO}_{2} \rightarrow \mathrm{NO} + \mathrm{NO}_{3} \), the rate law is based on the molecularity giving the rate as\( ext{Rate} = k [\mathrm{NO}_{2}]^2 \).

Identifying Reaction Intermediates

A reaction intermediate is a species that is produced in one step of a reaction mechanism and consumed in another. In this reaction, \( \mathrm{NO}_{3} \) is created in Step 1 and consumed in Step 2. Therefore, \( \mathrm{NO}_{3} \) is the intermediate.

Key concepts

Elementary Steps

An elementary step is a single reaction event that occurs as part of a larger reaction mechanism. These steps break down complex reactions into simpler events that show how molecules rearrange to form products. Each elementary step provides insight into the actual molecular interactions during the reaction process.

In the given reaction between \(\mathrm{NO}_{2}(\mathrm{g})\) and \(\mathrm{CO}(\mathrm{g})\), the mechanism is composed of two elementary steps. The first step involves two \(\mathrm{NO}_{2}\) molecules producing \(\mathrm{NO}\) and \(\mathrm{NO}_{3}\). The second step occurs when \(\mathrm{NO}_{3}\) reacts with \(\mathrm{CO}\) to produce \(\mathrm{NO}_{2}\) and \(\mathrm{CO}_{2}\).

Elementary steps must collectively summarize or simplify to represent the overall stoichiometric reaction, which is \(\mathrm{NO}_{2} + \mathrm{CO} \rightarrow \mathrm{NO} + \mathrm{CO}_{2}\).

Molecularity

Molecularity refers to the number of reactant molecules participating in an elementary step. It provides a direct count of molecules that collide to bring about the reaction.

• **Unimolecular:** Involves one molecule decomposing or rearranging.
• **Bimolecular:** Requires two molecules to interact or collide.
• **Termolecular:** Involves three molecules; rare due to the improbability of three molecules colliding simultaneously.

In the problem, both steps are **bimolecular**:
- Step 1: Involves collision between two \(\mathrm{NO}_{2}\) molecules.
- Step 2: Involves interaction between \(\mathrm{NO}_{3}\) and \(\mathrm{CO}\).

Each step's molecularity reflects the complexity and likelihood of the process.

Rate Law

The rate law shows how the rate of reaction depends on the concentrations of the reactants. For reactions involving multiple steps, the rate law is often determined by the slowest step, also known as the rate-determining step.

In this particular mechanism, Step 1 is designated as the slow step. The slow step predominantly affects the overall reaction rate. Thus, the rate equation is derived from this step's reactants.

For Step 1: \(\mathrm{NO}_{2} + \mathrm{NO}_{2} \rightarrow \mathrm{NO} + \mathrm{NO}_{3}\)
The rate law based on the molecularity (bimolecular) is:
\[\text{Rate} = k [\mathrm{NO}_{2}]^2\]

This equation demonstrates that the reaction rate is proportional to the square of the concentration of \(\mathrm{NO}_{2}\). Understanding the rate law helps predict how changing concentrations can affect reaction speed.

Reaction Intermediates

Reaction intermediates are species that are formed in one elementary step and consumed in another during the reaction mechanism. They are transient, meaning they do not appear in the overall reaction equation.

In the given mechanism, \(\mathrm{NO}_{3}\) plays the role of an intermediate. It is created in Step 1 during the reaction \(\mathrm{NO}_{2} + \mathrm{NO}_{2} \rightarrow \mathrm{NO} + \mathrm{NO}_{3}\) and then immediately used up in Step 2, where it reacts with \(\mathrm{CO}\) to form products.

Intermediates are crucial for understanding the step-by-step progression of chemical transformations. Identifying these helps scientists infer the details of the molecular path taken by reactions.