Problem 78 - Chemistry A Molecular Approach

Question

Consider this two-step mechanism for a reaction: $$ \mathrm{NO}_{2}(g)+\mathrm{Cl}_{2}(g) \longrightarrow \mathrm{ClNO}_{2}(g)+\mathrm{Cl}(g) \quad \text { Slow } $$ $$ \mathrm{NO}_{2}(g)+\mathrm{Cl}(g) \longrightarrow \mathrm{ClNO}_{2}(g) $$ Fast a. What is the overall reaction? b. Identify the intermediates in the mechanism. c. What is the predicted rate law?

Step-by-Step Solution

Answer

Overall reaction: \(\mathrm{2NO}_2(g) + \mathrm{Cl}_2(g) \longrightarrow 2\mathrm{ClNO}_2(g)\). Intermediates: \(\mathrm{Cl}(g)\). Predicted rate law: \(\text{rate} = k[\mathrm{NO}_2][\mathrm{Cl}_2]\).

Step 1: Determine the Overall Reaction

To find the overall reaction, we need to add the two steps of the mechanism together. We must pay attention to species that appear on both sides of the reaction arrows because they may cancel out.

Step 2: Cancel Intermediates and Derive Overall Equation

After adding the two steps, cancel any species that appear on both sides of the reaction arrows. These are intermediates. The species that do not cancel out will be part of the overall reaction.

Step 3: Write the Overall Reaction

Once the intermediates have been canceled, we construct the overall reaction by combining the remaining reactants and products from both steps.

Step 4: Identify the Intermediates

Intermediates are species that are produced in one step of a mechanism and consumed in a subsequent step. They do not appear in the overall reaction.

Step 5: Predict the Rate Law

The rate law is based on the slowest step in the mechanism, known as the rate-determining step. The rate law includes only the reactants that are part of the slow step and is based on their stoichiometry in the rate-determining step.

Key Concepts

Overall Reaction

Understanding the overall reaction in a chemical process means identifying the initial reactants and final products after the completion of the reaction, disregarding the individual steps that take place throughout the mechanism.

For example, consider the provided two-step reaction mechanism. The two steps involved are:

• \text{Step 1: } \text{(Slow)} \[\mathrm{NO}_2(g) + \mathrm{Cl}_2(g) \rightarrow \mathrm{ClNO}_2(g) + \mathrm{Cl}(g)\]
• \text{Step 2: } \text{(Fast)} \[\mathrm{NO}_2(g) + \mathrm{Cl}(g) \rightarrow \mathrm{ClNO}_2(g)\]

To find the overall chemical equation, we simply add the two steps together and cancel out the intermediates. The intermediates are species that appear on both sides of the equation and thus do not show up in the final overall reaction. In this case, when we add Step 1 and Step 2, the \( \mathrm{Cl}(g) \) molecules produced in Step 1 and consumed in Step 2 cancel out, leaving the overall reaction as: \[\mathrm{NO}_2(g) + \mathrm{Cl}_2(g) \rightarrow 2\mathrm{ClNO}_2(g)\].

Thus, the overall reaction tells us that one molecule of nitrogen dioxide and one molecule of chlorine react to produce two molecules of chloronitrite.

Reaction Intermediates

A reaction intermediate is a transient species within a reaction mechanism that is formed in one step and used up in another. These species are crucial for the progression of the overall reaction but are not present in the initial reactants or final products.In the given reaction mechanism, the intermediate can be identified through the cancellation process. As we look at the steps, \(\mathrm{Cl}(g)\) is produced in the first (slow) step and then used up in the second (fast) step. Because it is not present in the overall reaction, \(\mathrm{Cl}(g)\) is the intermediate in this case. It's important for students to note that intermediates can help us understand the pathway a reaction takes, but since they are not part of the overall reaction, they won't appear in the final equation we write for the reaction.

Rate Law

The rate law is an expression that relates the rate of a reaction to the concentration of the reactants. It can't be derived simply from the overall balanced equation; instead, it depends on the detailed sequence of steps that constitute the reaction mechanism, with special emphasis on the rate-determining step.For the mechanism provided, the rate-determining step is the slow initial reaction: \[\mathrm{NO}_2(g) + \mathrm{Cl}_2(g) \rightarrow \mathrm{ClNO}_2(g) + \mathrm{Cl}(g)\]. This step is generally the bottleneck of the reaction, meaning that the overall reaction can't proceed any faster than this slowest step allows.Hence, the predicted rate law would be proportional to the concentrations of the reactants in this slowest step. The rate law can therefore be written as \[\text{Rate} = k[\mathrm{NO}_2][\mathrm{Cl}_2]\] where \(k\) is the rate constant and the brackets represent the concentration of each reactant. This indicates that the reaction rate doubles if the concentration of either \(\mathrm{NO}_2\) or \(\mathrm{Cl}_2\) is doubled.

Rate-Determining Step

The rate-determining step is the slowest step in a reaction mechanism, effectively setting the pace for the entire reaction because all other steps are much faster in comparison. It is often equated to the bottleneck in a process.In our exercise, the first step of the mechanism is the slow one, and thus, the rate-determining step. This slow step's reactants are directly involved in determining the reaction's rate law. Students must focus on this concept as it not only determines the rate law but also allows chemists to devise ways to optimize reactions. In real-world applications, finding a way to speed up the rate-determining step can lead to more efficient processes in industries such as pharmaceuticals, manufacturing, and environmental chemistry.In summary, understanding the rate-determining step is fundamental to controlling the kinetics of a reaction and predicting how changes in conditions will affect the reaction rate.