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?

Short answer

The elementary reactions add up to give the balanced equation: $$\mathrm{2NO}(g)+\mathrm{H}_{2}(g) \longrightarrow \mathrm{N}_{2}\mathrm{O}(g)+\mathrm{H}_{2}\mathrm{O}(g)$$. The rate laws for each elementary reaction are: $$\mathrm{rate}_{1} = k_{1}[\mathrm{NO}]^{2}$$ and $$\mathrm{rate}_{2} = k_{2}[\mathrm{N}_{2}\mathrm{O}_{2}][\mathrm{H}_{2}]$$. In this mechanism, the intermediate is \(\mathrm{N}_{2}\mathrm{O}_{2}\). Since the observed rate law matches the rate law for the first reaction, it suggests that the first reaction is the rate-determining step and is slower than the second reaction.

Step-by-step solution

(a) Balanced Equation

First, we need to add the elementary reactions to verify if they provide a balanced equation for the overall reaction. 1. $$\mathrm{NO}(g)+\mathrm{NO}(g) \longrightarrow \mathrm{N}_{2}\mathrm{O}_{2}(g)$$ 2. $$\mathrm{N}_{2}\mathrm{O}_{2}(g)+\mathrm{H}_{2}(g) \longrightarrow \mathrm{N}_{2}\mathrm{O}(g)+\mathrm{H}_{2}\mathrm{O}(g)$$ Now, let's add these reactions together: $$\mathrm{2NO}(g)+\mathrm{H}_{2}(g) \longrightarrow \mathrm{N}_{2}\mathrm{O}(g)+\mathrm{H}_{2}\mathrm{O}(g)$$ The elementary reactions add up to give the balanced equation for the reaction.

(b) Rate Laws for Elementary Reactions

Now let's write the rate laws for each elementary reaction: 1. For the first reaction, $$\mathrm{rate}_{1} = k_{1}[\mathrm{NO}]^{2}$$ 2. For the second reaction, $$\mathrm{rate}_{2} = k_{2}[\mathrm{N}_{2}\mathrm{O}_{2}][\mathrm{H}_{2}]$$

(c) Intermediates in the Mechanism

Now, we must identify any intermediates in the mechanism. Intermediates are species that are produced in one step of the reaction and consumed in the following step. In this mechanism, the intermediate is \(\mathrm{N}_{2}\mathrm{O}_{2}\), which is produced in the first reaction and consumed in the second reaction.

(d) Relative Speeds of Reactions

If the proposed mechanism is correct, we can compare the relative speeds of the first and second reactions using the observed rate law: $$\mathrm{rate}=k[\mathrm{NO}]^{2}\left[\mathrm{H}_{2}\right]$$. Since the observed rate law matches the rate law for the first reaction, it suggests that the first reaction is the rate-determining step. Therefore, the first reaction is slower than the second reaction. If the second reaction were slower, the observed rate law would depend on the intermediate \(\mathrm{N}_{2}\mathrm{O}_{2}\) instead of just \(\mathrm{NO}\) and \(\mathrm{H}_{2}\).

Key concepts

Reaction Mechanism

In chemical kinetics, a reaction mechanism is a detailed step-by-step description of how a reaction occurs at the molecular level. It breaks down the overall reaction into a series of basic steps called elementary reactions. Each of these steps involves the making and breaking of chemical bonds. For example, in the reaction mechanism provided, two elementary steps describe how nitric oxide (NO) reacts with hydrogen (H₂) to ultimately form nitrous oxide (N₂O) and water (H₂O). These elementary reactions help us understand the molecular process behind the overall reaction. By analyzing each step, chemists can predict the behavior and speed of a reaction under different conditions, providing insight into new ways to control chemical reactions.

Rate Laws

Rate laws are mathematical expressions that describe the speed of a chemical reaction in relation to the concentrations of the reactants. The rate of a reaction tells us how quickly reactants are converted into products. For elementary reactions, the rate law reflects the stoichiometry of the reaction—that is, the concentration of the reactants raised to the power of their coefficients in the balanced equation.
For the given mechanism, the rate law for the first elementary step is given by \( ext{rate}_1 = k_1[ ext{NO}]^2 \), showing its dependence on the concentration of NO. The rate law for the second step is \( ext{rate}_2 = k_2[ ext{N}_2 ext{O}_2][ ext{H}_2] \). These rate laws indicate how each step contributes to the overall speed of the reaction.

• Rate constants \( k_1 \) and \( k_2 \) are unique to each step and affected by factors like temperature and catalyst presence.

Reaction Intermediates

In a reaction mechanism, intermediates are species that appear in the steps of a reaction but not in the overall balanced equation. They are formed in one step and consumed in another. Therefore, they are not seen among the initial reactants or final products. In the provided exercise, N₂O₂ is an intermediate. It is produced in the first elementary step and immediately consumed in the second. Understanding intermediates is crucial because they often determine the pathway of a reaction.
If an intermediate accumulates, it can sometimes impede the progress of the reaction or indicate that one of the steps is much slower than others. Therefore, identifying reaction intermediates helps chemists better understand and potentially manipulate the reaction mechanisms for desired outcomes.

Rate-Determining Step

The rate-determining step is essentially the bottleneck of the reaction. It is the slowest step in a mechanism and thus dictates the rate of the overall reaction. When evaluating the proposed mechanism, the observed rate law helps pinpoint the rate-determining step. In this scenario, the rate law matches that of the first reaction step: \( ext{rate} = k[ ext{NO}]^2[ ext{H}_2] \).
This shows that the first step is the slowest and controls the rate because it's consistent with the observed kinetics. Understanding the rate-determining step is critical for controlling reactions. By accelerating this step using catalysts or changing conditions, chemists can enhance the efficiency and speed of industrial and laboratory reactions.