Problem 112 - Chemistry The Central Science

Question

The following mechanism has been proposed for the reaction of \(\mathrm{NO}\) with \(\mathrm{H}_{2}\) to form \(\mathrm{N}_{2} \mathrm{O}\) and \(\mathrm{H}_{2} \mathrm{O}\) : $$ \begin{aligned} \mathrm{NO}(g)+\mathrm{NO}(g) & \longrightarrow \mathrm{N}_{2} \mathrm{O}_{2}(g) \\ \mathrm{N}_{2} \mathrm{O}_{2}(g)+\mathrm{H}_{2}(g) & \longrightarrow \mathrm{N}_{2} \mathrm{O}(g)+\mathrm{H}_{2} \mathrm{O}(g) \end{aligned} $$ (a) Show that the elementary reactions of the proposed mechanism add to provide a balanced equation for the reaction. (b) Write a rate law for each elementary reaction in the mechanism. (c) Identify any intermediates in the mechanism. (d) The observed rate law is rate \(=k[\mathrm{NO}]^{2}\left[\mathrm{H}_{2}\right]\). If the proposed mechanism is correct, what can we conclude about the relative speeds of the first and second reactions?

Step-by-Step Solution

Answer

The balanced chemical equation for the reaction is \(2\mathrm{NO}(g) + \mathrm{H}_{2}(g) \longrightarrow \mathrm{N}_{2}\mathrm{O}(g) + \mathrm{H}_{2}\mathrm{O}(g)\). The rate laws for the elementary reactions are \(rate_1 = k_1 [\mathrm{NO}]^2\) and \(rate_2 = k_2 [\mathrm{N}_{2}\mathrm{O}_{2}] [\mathrm{H}_{2}]\). The intermediate in the mechanism is \(\mathrm{N}_{2}\mathrm{O}_{2}\). Comparing the observed rate law (\(rate = k[\mathrm{NO}]^2[\mathrm{H}_{2}]\)) to the proposed mechanism, we can conclude that the first reaction is fast and reaches equilibrium, while the second one is slow. This is consistent with the observed rate law.

Step 1: Check the balanced chemical equation

To verify if the proposed mechanism adds up to a balanced chemical equation, we need to add the elementary reactions and check if the sum of the reactants equals the sum of the products: \[\mathrm{NO}(g)+\mathrm{NO}(g) \longrightarrow \mathrm{N}_{2}\mathrm{O}_{2}(g)\] \[\mathrm{N}_{2}\mathrm{O}_{2}(g)+\mathrm{H}_{2}(g) \longrightarrow \mathrm{N}_{2}\mathrm{O}(g)+\mathrm{H}_{2}\mathrm{O}(g)\] Adding the reactions gives: \[2\,\mathrm{NO}(g) + \mathrm{H}_{2}(g) \longrightarrow \mathrm{N}_{2}\mathrm{O}(g) + \mathrm{H}_{2}\mathrm{O}(g)\] The reaction is balanced: \(2\,\mathrm{NO}\) molecules and one \(\mathrm{H}_{2}\) molecule react to form one molecule of \(\mathrm{N}_{2}\mathrm{O}\) and one molecule of \(\mathrm{H}_{2}\mathrm{O}\). Step 2: Write the rate law for each elementary reaction

Step 2: Rate laws for elementary reactions

For each elementary reaction, the rate law can be written as follows: Reaction 1: \[rate_1 = k_1 [\mathrm{NO}]^2\] Reaction 2: \[rate_2 = k_2 [\mathrm{N}_{2}\mathrm{O}_{2}] [\mathrm{H}_{2}]\] Step 3: Identify the Intermediate in the Mechanism

Step 3: Identifying intermediates

An intermediate is a species that is formed and consumed in the mechanism but does not appear in the overall balanced equation. In this case, the intermediate is \(\mathrm{N}_{2}\mathrm{O}_{2}\). It is formed in the first elementary reaction and consumed in the second elementary reaction, but it does not appear in the overall balanced equation. Step 4: Compare the Observed Rate Law to the Proposed Mechanism

Step 4: Comparing observed rate law to proposed mechanism

The observed rate law is given by: \[rate = k[\mathrm{NO}]^2[\mathrm{H}_{2}]\] To relate this observed rate law to the proposed mechanism, we can consider that the first reaction reaches equilibrium and that the second reaction is slow compared to the first one. In such a case, we can assume that \[\frac{rate_1}{rate_2} \approx [\mathrm{N}_{2}\mathrm{O}_{2}]\] Since \(rate_1 = k_1 [\mathrm{NO}]^2\), we can calculate the concentration of \(\mathrm{N}_{2}\mathrm{O}_{2}\) as: \[[\mathrm{N}_{2}\mathrm{O}_{2}] = \frac{rate_1}{rate_2} \approx \frac{k_1[\mathrm{NO}]^2}{k_2 [\mathrm{H}_{2}]}\] Now we can substitute the concentration of \(\mathrm{N}_{2}\mathrm{O}_{2}\) in the rate law for Reaction 2: \[rate_2 = k_2 \times \frac{k_1[\mathrm{NO}]^2}{k_2 [\mathrm{H}_{2}]} \times [\mathrm{H}_{2}]\] \[rate_2 = k_1k_2\frac{[\mathrm{NO}]^2[\mathrm{H}_{2}]}{k_2}\] The observed rate law and the proposed mechanism rate law are equal, which means that the proposed mechanism with Reaction 1 being fast and in equilibrium and Reaction 2 being slow is consistent with the observed rate law.

Key Concepts

Elementary Reactions

An elementary reaction is a single step in a reaction mechanism, involving just a few molecules. These reactions occur exactly as written and give insight into the overall reaction process. For the reaction of \(\mathrm{NO}\) with \(\mathrm{H}_{2}\), we have two elementary steps:

1. \(\mathrm{NO}(g) + \mathrm{NO}(g) \rightarrow \mathrm{N}_{2}\mathrm{O}_{2}(g)\)
2. \(\mathrm{N}_{2}\mathrm{O}_{2}(g) + \mathrm{H}_{2}(g) \rightarrow \mathrm{N}_{2}\mathrm{O}(g) + \mathrm{H}_{2}\mathrm{O}(g)\)

By adding these reactions, we form the balanced overall equation. Each step happens independently, and combining them reveals the complete transformation from reactants to products.
Understanding elementary reactions helps chemists design mechanisms and predict reaction behavior.

Rate Laws

Rate laws express how the rate of a reaction depends on the concentration of its reactants. For elementary reactions, these laws are simple because they mirror the reaction's molecular form. In this mechanism:

- For reaction 1: \(rate_1 = k_1 [\mathrm{NO}]^2\)
- For reaction 2: \(rate_2 = k_2 [\mathrm{N}_{2}\mathrm{O}_{2}][\mathrm{H}_{2}]\)

Every rate law uses a rate constant \(k\), unique for each step. This helps understand how fast or slow a reaction progresses. Mastering rate laws aids in predicting the speed of chemical processes and developing efficient reaction conditions.

Chemical Intermediates

Intermediates are species produced in one step of a reaction mechanism and consumed in another. They don't appear in the overall equation as they exist only briefly. In our reaction, \(\mathrm{N}_{2}\mathrm{O}_{2}\) serves as an intermediate.

It forms during the first step and reacts further in the next step, making it vital for the transition from reactants to products. Recognizing intermediates allows chemists to identify crucial molecules in complex processes and enhance control over reactions.

Observations of Reaction Rates

Observing reaction rates allows us to compare proposed mechanisms with actual behavior. For this mechanism, the observed rate law is \(rate = k[\mathrm{NO}]^2[\mathrm{H}_{2}]\). It suggests that the first reaction is fast and possibly in equilibrium, while the second is slower and rate-determining.

This observation aligns with the proposed mechanism. Monitoring reaction rates gives insights into which steps are essential and informs improvements to mechanisms, ensuring accurate predictions of reaction progress.