Problem 50 - Chemistry and Chemical Reactivity

Question

The reaction of \(\mathrm{NO}_{2}(\mathrm{g})\) and \(\mathrm{CO}(\mathrm{g})\) is thought to occur in two steps: Step 1 Slow \(\quad \mathrm{NO}_{2}(\mathrm{g})+\mathrm{NO}_{2}(\mathrm{g}) \longrightarrow \mathrm{NO}(\mathrm{g})+\mathrm{NO}_{3}(\mathrm{g})\) Step 2 Fast \(\quad \mathrm{NO}_{3}(\mathrm{g})+\mathrm{CO}(\mathrm{g}) \longrightarrow \mathrm{NO}_{2}(\mathrm{g})+\mathrm{CO}_{2}(\mathrm{g})\) (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.

Step-by-Step Solution

Answer

(a) The steps combine to form the overall reaction. (b) Both steps are bimolecular. (c) Rate equation: \( \text{rate} = k[\text{NO}_2]^2 \). (d) Intermediate: \( \text{NO}_3(\text{g}) \).

Step 1: Identify the Intermediates

An intermediate is a species that is formed in one step of the reaction and consumed in another. From Step 1 and Step 2, we note that \( \text{NO}_3(\text{g}) \) is formed in Step 1 and consumed in Step 2. Thus, \( \text{NO}_3(\text{g}) \) is an intermediate.

Step 2: Combine Elementary Steps

Add the reactants and products of both steps together: \[ \text{NO}_2(\text{g}) + \text{NO}_2(\text{g}) + \text{NO}_3(\text{g}) + \text{CO}(\text{g}) \rightarrow \text{NO}(\text{g}) + \text{NO}_3(\text{g}) + \text{NO}_2(\text{g}) + \text{CO}_2(\text{g}) \]. Cancel out the intermediate \( \text{NO}_3(\text{g}) \) to yield the overall reaction: \[ \text{NO}_2(\text{g}) + \text{CO}(\text{g}) \rightarrow \text{NO}(\text{g}) + \text{CO}_2(\text{g}) \].

Step 3: Determine Molecularity of Each Step

Molecularity refers to the number of molecules colliding in an elementary step. For Step 1, there are two molecules of \( \text{NO}_2(\text{g}) \), so it's bimolecular. For Step 2, one molecule each of \( \text{NO}_3(\text{g}) \) and \( \text{CO}(\text{g}) \) react, so it's also bimolecular.

Step 4: Derive Rate Equation for Consistency

The slowest elementary step determines the rate law. From Step 1, which is slow, the rate is determined by \( \text{NO}_2(\text{g}) + \text{NO}_2(\text{g}) \). The rate law for this step is rate \( = k[\text{NO}_2]^2 \), where \( k \) is the rate constant.

Step 5: Verify and Summarize

Summarize the findings: (a) The overall reaction \( \text{NO}_2(\text{g}) + \text{CO}(\text{g}) \rightarrow \text{NO}(\text{g}) + \text{CO}_2(\text{g}) \) is consistent. (b) Each step is bimolecular. (c) The experimental rate equation is \( \text{rate} = k[\text{NO}_2]^2 \). (d) \( \text{NO}_3(\text{g}) \) is the intermediate.

Key Concepts

Elementary Steps

Elementary steps in a reaction mechanism are the individual stages that make up a complex reaction. Each of these steps represents a single molecular event. When you add all the elementary steps together, they give you the overall chemical equation. For instance, in the reaction between \(\text{NO}_2(\text{g})\) and \(\text{CO}(\text{g})\), we have two steps:

• Step 1 involves the collision of two \(\text{NO}_2(\text{g})\) molecules to form \(\text{NO}(\text{g})\) and \(\text{NO}_3(\text{g})\).
• Step 2 involves \(\text{NO}_3(\text{g})\) reacting with \(\text{CO}(\text{g})\) to form \(\text{NO}_2(\text{g})\) and \(\text{CO}_2(\text{g})\).

By combining these steps and eliminating intermediates like \(\text{NO}_3(\text{g})\), you get the overall reaction: \[\text{NO}_2(\text{g}) + \text{CO}(\text{g}) \rightarrow \text{NO}(\text{g}) + \text{CO}_2(\text{g})\] Understanding these steps helps clarify how chemical reactions occur.

Molecularity

Molecularity is the number of molecules that collide during an elementary step. It is a term used to describe the mechanism's stages, not the complete balanced equation.

• A step said to be unimolecular involves only one molecule acting alone.
• Bimolecular steps include two molecules colliding and reacting with one another.
• Termolecular steps, which are rare, involve three molecules colliding simultaneously.

Looking at our specific reaction, both Step 1 and Step 2 are bimolecular. In Step 1, two \(\text{NO}_2(\text{g})\) molecules collide, and in Step 2, one \(\text{NO}_3(\text{g})\) and one \(\text{CO}(\text{g})\) molecule react together. Each molecular interaction is critical for advancing the reaction forward.

Rate Equation

The rate equation (or rate law) for a reaction indicates the relationship between the concentration of reactants and the reaction rate. For complex reactions, the slowest step, also known as the rate-determining step, governs the rate law.In the given reaction mechanism, Step 1 is the slowest step. Therefore, the rate of the overall reaction depends on the collision of \(\text{NO}_2(\text{g})\) molecules in this step. The rate law derived from Step 1 is:\[ \text{rate} = k[\text{NO}_2]^2 \] Where:

• \(k\) is the rate constant.
• \([\text{NO}_2]\) is the concentration of \(\text{NO}_2(\text{g})\).

The squared term \([\text{NO}_2]^2\) reflects that two \(\text{NO}_2\) molecules are involved in the slowest (rate-determining) step. This equation allows scientists to predict how changes in the \(\text{NO}_2\) concentration will affect the reaction rate.

Reaction Intermediates

Reaction intermediates are species that appear in the reaction mechanism but not in the overall stoichiometric equation. They are formed in one step and consumed in another, helping facilitate the progression of the reaction. In the reaction mechanism for the interaction between \(\text{NO}_2(\text{g})\) and \(\text{CO}(\text{g})\), \(\text{NO}_3(\text{g})\) acts as an intermediate.

• It is created in Step 1 when two \(\text{NO}_2(\text{g})\) molecules react to produce \(\text{NO}(\text{g})\) and \(\text{NO}_3(\text{g})\).
• Immediately following its formation, in Step 2, \(\text{NO}_3(\text{g})\) participates in a reaction with \(\text{CO}(\text{g})\), leading to the formation of \(\text{NO}_2(\text{g})\) and \(\text{CO}_2(\text{g})\).

Despite its crucial role, \(\text{NO}_3(\text{g})\) does not appear in the net equation because it is neither seen as a starting material nor a final product.