Problem 70 - Chemistry The Molecular Science

Question

The reaction of \(\mathrm{NO}_{2}(\mathrm{~g})\) and \(\mathrm{CO}(\mathrm{g})\) to form \(\mathrm{CO}_{2}(\mathrm{~g})\) and \(\mathrm{NO}(\mathrm{g})\) is thought to occur in two steps: Step 1: \(\mathrm{NO}_{2}(\mathrm{~g})+\mathrm{NO}_{2}(\mathrm{~g}) \longrightarrow \mathrm{NO}(\mathrm{g})+\mathrm{NO}_{3}(\mathrm{~g}) \quad\) slow Step \(2: \mathrm{NO}_{3}(\mathrm{~g})+\mathrm{CO}(\mathrm{g}) \longrightarrow \mathrm{NO}_{2}(\mathrm{~g})+\mathrm{CO}_{2}(\mathrm{~g}) \quad\) fast (a) Show that the elementary steps add up to give the overall, stoichiometric equation. (b) Determine 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.

Step-by-Step Solution

Answer

Overall reaction: \(\mathrm{NO}_{2}(\mathrm{~g}) + \mathrm{CO}(\mathrm{g}) \rightarrow \mathrm{CO}_{2}(\mathrm{~g}) + \mathrm{NO}(\mathrm{g})\). Molecularity: both steps are bimolecular. Rate law: \(\text{Rate} = k[\mathrm{NO}_{2}]^2\). Intermediate: \(\mathrm{NO}_{3}(\mathrm{~g})\).

Step 1: Add the Elementary Steps

To find the overall reaction equation, sum the two elementary steps:From Step 1: \( \mathrm{NO}_{2}(\mathrm{~g})+\mathrm{NO}_{2}(\mathrm{~g}) \rightarrow \mathrm{NO}(\mathrm{g})+\mathrm{NO}_{3}(\mathrm{~g}) \)From Step 2: \( \mathrm{NO}_{3}(\mathrm{~g})+\mathrm{CO}(\mathrm{g}) \rightarrow \mathrm{NO}_{2}(\mathrm{~g})+\mathrm{CO}_{2}(\mathrm{~g}) \)Net equation: Combine and cancel the intermediate \( \mathrm{NO}_{3}(\mathrm{~g}) \).\[ \mathrm{NO}_{2}(\mathrm{~g}) + \mathrm{CO}(\mathrm{g}) \rightarrow \mathrm{CO}_{2}(\mathrm{~g}) + \mathrm{NO}(\mathrm{g}) \]

Step 2: Determine Molecularity

Molecularity is the number of molecules involved in an elementary step.For Step 1: \( \mathrm{NO}_{2}(\mathrm{~g}) + \mathrm{NO}_{2}(\mathrm{~g}) \), the molecularity is 2 (bimolecular).For Step 2: \( \mathrm{NO}_{3}(\mathrm{~g}) + \mathrm{CO}(\mathrm{g}) \), the molecularity is 2 (bimolecular).

Step 3: Derive the Rate Equation

The rate of the overall reaction is determined by the slowest step (rate-determining step) which is Step 1.The experimental rate law is based on Step 1: \[ \text{Rate} = k[\mathrm{NO}_{2}]^2 \]

Step 4: Identify the Reaction Intermediates

An intermediate is a species that is formed and consumed during the reaction mechanism but does not appear in the overall reaction.In this mechanism, \( \mathrm{NO}_{3}(\mathrm{~g}) \) is formed in Step 1 and consumed in Step 2, making it the intermediate.

Key Concepts

Elementary Steps

In a chemical reaction mechanism, understanding each piece or 'elementary step' is essential. An elementary step is a single transformation that happens as the reactants turn into products. In the given reaction mechanism, there are two elementary steps.

• Step 1: Involves the reaction of two molecules of \(\mathrm{NO}_2(\mathrm{~g})\) to form one \(\mathrm{NO}(\mathrm{g})\) and one \(\mathrm{NO}_3(\mathrm{~g})\). This step proceeds slowly.
• Step 2: Consists of the reaction of \(\mathrm{NO}_3(\mathrm{~g})\) with \(\mathrm{CO}(\mathrm{g})\) to produce \(\mathrm{NO}_2(\mathrm{~g})\) and \(\mathrm{CO}_2(\mathrm{~g})\), occurring fast.

You add these steps to get the complete reaction, which involves canceling out intermediates like \(\mathrm{NO}_3(\mathrm{~g})\). Each step needs to correctly align with the overall chemical transformation when added together.

Molecularity

Molecularity refers to how many molecules take part in an elementary step of a reaction. It gives a simple count of molecules or atoms involved in the transformation. Let's examine the molecularity of each step in our example:

• Step 1: Two \(\mathrm{NO}_2(\mathrm{~g})\) molecules come together, making it a bimolecular reaction, as two species are reacting.
• Step 2: Here, one molecule each of \(\mathrm{NO}_3(\mathrm{~g})\) and \(\mathrm{CO}(\mathrm{g})\) means this is also a bimolecular step.

Keep in mind that molecularity is a characteristic of elementary steps only, not overall reactions.

Rate Equation

The rate equation, or rate law, gives insight into how the concentration of reactants affects the reaction rate. In complex reactions, the rate is often measured by the slowest step, known as the rate-determining step. In our mechanism, Step 1 is slow, making it the rate-determining step. Hence, the rate equation reflects this step: \[\text{Rate} = k[\mathrm{NO}_2]^2\] Here, \(k\) is the rate constant, and \([\mathrm{NO}_2]^2\) indicates that two molecules of \(\mathrm{NO}_2\) are involved in the slow step. This equation suggests that doubling the concentration of \(\mathrm{NO}_2\) will quadruple the reaction rate.

Reaction Intermediates

Understanding reaction intermediates is crucial for fully grasping a reaction mechanism. Intermediates are species that are produced during one step and consumed in another, and they do not appear in the overall equation since they are entirely used up before the reaction is complete. In our specific example, the intermediate involved is \(\mathrm{NO}_3(\mathrm{~g})\). It gets created in Step 1 and promptly used up in Step 2. Thus, while \(\mathrm{NO}_3(\mathrm{~g})\) is central to the mechanism, it's absent in the overall balanced equation, reflecting its transient nature. Recognizing intermediates can also illuminate additional pathways or branches in reaction mechanisms.