Question

The rate law for this reaction $$ \mathrm{CO}(g)+\mathrm{NO}_{2}(g) \longrightarrow \mathrm{CO}_{2}(g)+\mathrm{NO}(g) $$ is rate \(=k\left[\mathrm{NO}_{2}\right]^{2}\). Suggest a plausible mechanism for the reaction, given that the unstable species \(\mathrm{NO}_{3}\) is an intermediate.

Short answer

The suggested mechanism for the reaction is as follows: \n1. Slow: \(\mathrm{NO}_{2}(g) + \(\mathrm{NO}_{2}(g) \rightarrow \(\mathrm{NO}_{3}(g) + \(\mathrm{NO}(g)\) \n2. Fast: \(\mathrm{CO}(g) + \(\mathrm{NO}_{3}(g) \rightarrow \(\mathrm{CO}_{2}(g) + \(\mathrm{NO}_{2}(g)\)

Step-by-step solution

Identifying Elementary Steps

In the first step, it is inferred that two molecules of \(\mathrm{NO}_{2}(g)\) must come together to form \(\mathrm{NO}_{3}\) and \(\mathrm{NO}(g)\). This makes sense as \(\mathrm{NO}_{3}\) is an intermediate species and is not found in the final products. The elementary reaction for the first step can be represented as: \(\mathrm{NO}_{2}(g) + \(\mathrm{NO}_{2}(g) \rightarrow \(\mathrm{NO}_{3}(g) + \(\mathrm{NO}(g)

Completing the Reaction

The next step is needed to generate the final products. In this step, \(\mathrm{CO}(g)\) reacts with the \(\mathrm{NO}_{3}(g)\) intermediate to produce \(\mathrm{CO}_{2}(g)\) and \(\mathrm{NO}_{2}(g)\). The elementary reaction for the second step can be represented as: \(\mathrm{CO}(g) + \(\mathrm{NO}_{3}(g) \rightarrow \(\mathrm{CO}_{2}(g) + \(\mathrm{NO}_{2}(g)

Confirming the Mechanism

Now that both steps of the reaction mechanism have been identified, it is necessary to ensure they align with the given rate law, rate \(=k\left[\mathrm{NO}_{2}\right]^{2}\). In order for this to be true, the first step must be the rate-determining step, meaning it is the slowest and thus controls the rate of the overall reaction. As the rate law is second order with respect to \(\mathrm{NO}_{2}\), and the reaction consumes two molecules of \(\mathrm{NO}_{2}\), this is indeed the rate-determining step.

Key concepts

Rate Law

Understanding the rate law is fundamental when studying the kinetics of a chemical reaction. A rate law expresses the relationship between the rate of a chemical reaction and the concentration of its reactants. Mathematically, it is represented by an equation showing the rate (usually denoted by the variable 'rate' or 'v') as proportional to the product of constant 'k' (the rate constant) and the concentrations of the reactants raised to some powers (the reaction orders).

For example, consider the rate law for the reaction between carbon monoxide, CO(g), and nitrogen dioxide, NO₂(g), which is: rate = k[NO₂]². This indicates that the reaction rate is directly proportional to the square of the concentration of NO₂, making it a second-order reaction with respect to NO₂.

Here, 'k' is specific to the reaction and temperature-dependent, the square of [NO₂] implies that if you double the concentration of NO₂, the rate of the reaction will increase by a factor of four. This is key to predicting how changes in concentration affect the rate, which can be tested through experiments and used to confirm the proposed reaction mechanism.

Elementary Steps

Elementary steps are the basic, individual processes that occur during a chemical reaction. Each of these steps involves a direct interaction between molecules or atoms resulting in the formation or breaking of bonds. The overall reaction is the sum of these elementary steps. The mechanism that comprises the elementary steps gives us insight into the 'molecular-level' pathway of a complex reaction.

In our example, where CO(g) and NO₂(g) react to form CO₂(g) and NO(g), the mechanism involves two elementary steps. The first elementary step proposes the formation of an unstable intermediate, NO₃, by the dimerization of two NO₂ molecules. The occurrence of such an intermediate is consistent with the rate law, as the rate-determining step involves NO₂ molecules interacting with each other. The second step involves CO(g) reacting with NO₃, the intermediate, to form the final products, CO₂ and NO. Since each step is considered elementary, all of them occur in a single-stage rather than through a series of events.

Rate-Determining Step

The rate-determining step in a reaction mechanism is the slowest step, which therefore controls the overall rate of the reaction. Think of it as the 'bottleneck' in the process. No matter how fast the other steps are, the overall reaction cannot proceed any faster than this slowest step.

In our reaction, the formation of NO₃ from two NO₂ molecules is the rate-determining step since it must be the slowest step to be consistent with the rate law given. It poignantly demonstrates that only the rate-determining step's reactant concentrations appear in the rate law. The subsequent steps are faster and do not influence the rate law directly. Identifying the rate-determining step is imperative when deducing the mechanism of a reaction. It aids in optimizing and controlling the conditions to improve reaction rates for practical applications, such as in industrial chemical processes.