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 to give NO and \(\mathrm{CO}_{2}\) : 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.

Step-by-Step Solution

Answer

The overall reaction is \(\mathrm{NO}_{2} + \mathrm{CO} \rightarrow \mathrm{NO} + \mathrm{CO}_{2}\). Both steps are bimolecular. The rate law is \(\text{Rate} = k[\mathrm{NO}_{2}]^2\), and \(\mathrm{NO}_{3}\) is an intermediate.

Step 1: Express Overall Reaction

First, combine the two given steps to find the overall reaction. In Step 1, we have: \(\mathrm{NO}_{2} + \mathrm{NO}_{2} \rightarrow \mathrm{NO} + \mathrm{NO}_{3} \) and in Step 2: \(\mathrm{NO}_{3} + \mathrm{CO} \rightarrow \mathrm{NO}_{2} + \mathrm{CO}_{2}\). Adding the reactions: \(\mathrm{NO}_{2} + \mathrm{NO}_{2} + \mathrm{NO}_{3} + \mathrm{CO} \rightarrow \mathrm{NO} + \mathrm{NO}_{3} + \mathrm{NO}_{2} + \mathrm{CO}_{2}\), which simplifies to: \(\mathrm{NO}_{2} + \mathrm{CO} \rightarrow \mathrm{NO} + \mathrm{CO}_{2}\).

Step 2: Molecularity of Each Step

Molecularity refers to the number of molecules involved in an elementary step. Step 1 involves \(\mathrm{NO}_{2} + \mathrm{NO}_{2}\), thus it has a molecularity of two and is bimolecular. Step 2 involves \(\mathrm{NO}_{3} + \mathrm{CO}\), which is also bimolecular as it involves two reactant molecules.

Step 3: Determine Experimental Rate Equation

The slow step determines the rate of the overall reaction. Writing the rate law for Step 1: rate = \(k[\mathrm{NO}_{2}]^2\). Therefore, the experimental rate equation should be based on this step: \(\text{Rate} = k[\mathrm{NO}_{2}]^2\).

Step 4: Identify Intermediates

Intermediates are species that are produced in one step and consumed in another, not appearing in the overall stoichiometric equation. In this mechanism, \(\mathrm{NO}_{3}(g)\) is produced in Step 1 and consumed in Step 2, so it is an intermediate.

Key Concepts

Elementary Steps

A reaction mechanism is a detailed sequence of elementary steps that describes how reactants transform into products. Each elementary step in a reaction involves a specific molecular event. In the case of the reaction between \(\text{NO}_2(\text{g})\) and \(\text{CO}(\text{g})\), the mechanism consists of two such elementary steps. The first step involves two \(\text{NO}_2\) molecules reacting to form \(\text{NO}\) and \(\text{NO}_3\). The second step involves \(\text{NO}_3\) interacting with \(\text{CO}\) to form \(\text{NO}_2\) and \(\text{CO}_2\).

It is essential to understand that each elementary step is a hypothetical construct. This means it is a proposed simplification that allows chemists to rationalize and predict the behavior of chemical reactions. When describing reactions in terms of elementary steps, the overall reaction can be obtained by adding all the individual steps together. In our reaction, when we add the two steps, the \(\text{NO}_3\) produced and consumed gets canceled out, resulting in the overall reaction \(\text{NO}_2 + \text{CO} \rightarrow \text{NO} + \text{CO}_2\).

Elementary steps are crucial for understanding reaction mechanisms and predicting how changes in conditions can affect reaction rates.

Molecularity

Molecularity is the term used to describe the number of reactant molecules involved in an elementary step in a reaction mechanism. Unlike reaction order, which is determined experimentally, molecularity is a theoretical concept based directly on the specific reaction step.

In our reaction mechanism:

• Step 1 involves two \(\text{NO}_2\) molecules colliding to form products, making it bimolecular.
• Similarly, Step 2 involves a \(\text{NO}_3\) molecule and a \(\text{CO}\) molecule coming together, which is also bimolecular.

Both steps are classified as bimolecular because they involve the interaction between two molecules. It is important to note that molecularity only applies to elementary steps, not to the overall reaction. Understanding the molecularity of steps is vital as it gives insight into the probability of collisions between molecules changing into products.

Rate Equation

The rate equation is a mathematical expression that describes the speed of a chemical reaction in terms of the concentration of reactants. For reactions with multiple steps, the slowest step, known as the rate-determining step, governs the overall rate. It acts as a bottleneck, and the rate law for the entire reaction is based on this step.

In the reaction between \(\text{NO}_2\) and \(\text{CO}\), Step 1 is identified as the slow step. Thus, the rate equation is derived from this elementary step. For Step 1, the rate can be expressed as:
\[\text{Rate} = k[\text{NO}_2]^2\]
where \(k\) is the rate constant, and \(\text{NO}_2\) is raised to the power of 2, consistent with its bimolecular nature. This equation implies that the reaction rate depends on the concentration of \(\text{NO}_2\) squared. Understanding the rate equation is crucial for predicting how changes in reactant concentrations can impact speed and efficiency of chemical reactions.

Reaction Intermediates

Reaction intermediates are transient species that appear in the steps of a reaction mechanism but do not appear in the final stoichiometry of the overall reaction. They are produced in one elementary step and consumed in a subsequent step.

In the reaction sequence of \(\text{NO}_2\) reacting with \(\text{CO}\), \(\text{NO}_3\) is produced during Step 1 and consumed in Step 2 before it can appear as a final product. This makes \(\text{NO}_3\) an intermediate. It is important because intermediates often play critical roles in facilitating the transformation of reactants to products even if they are not present in the final balanced chemical equation.

Identifying intermediates helps chemists understand the detailed progress of reactions and can aid in the development of catalysts or optimized reaction pathways.