When the rate of the reaction \(2 \mathrm{NO}(\mathrm{g})+\mathrm{O}_{2}(\mathrm{~g}) \rightarrow\) \(2 \mathrm{NO}_{2}(\mathrm{~g})\) was studied, the rate was found to double when the \(\mathrm{O}_{2}\) concentration alone was doubled but to quadruple when the NO concentration alone was doubled. Which of the following mechanisms accounts for these observations? Explain your reasoning. (a) Step \(1 \mathrm{NO}+\mathrm{O}_{2} \longrightarrow \mathrm{NO}_{3}\) and its reverse (both fast, equilibrium) Step \(2 \mathrm{NO}+\mathrm{NO}_{3} \rightarrow \mathrm{NO}_{2}+\mathrm{NO}_{2}\) (slow) (b) Step \(1 \mathrm{NO}+\mathrm{NO} \rightarrow \mathrm{N}_{2} \mathrm{O}_{2}\) (slow) Step \(2 \mathrm{O}_{2}+\mathrm{N}_{2} \mathrm{O}_{2} \rightarrow \mathrm{N}_{2} \mathrm{O}_{4}\) (fast) Step \(3 \mathrm{~N}_{2} \mathrm{O}_{4} \rightarrow \mathrm{NO}_{2}+\mathrm{NO}_{2}\) (fast)

Understanding how a reaction rate is influenced by the concentration of reactants is fundamental in the study of chemical kinetics. A reaction's rate often depends on the concentration of its reactants in a predictable way. In the provided example, the rate of the formation of \( \mathrm{NO}_{2} \) gas doubles when the concentration of \( \mathrm{O}_{2} \) is doubled, suggesting a direct, first-order relationship for \( \mathrm{O}_{2} \) in the reaction. Conversely, when the concentration of \( \mathrm{NO} \) gas is doubled, the rate of the reaction quadruples, indicating a second-order relationship for \( \mathrm{NO} \) in the reaction.

This information is particularly valuable for predicting how changes in concentrations will affect the speed at which the reaction proceeds. It's important to note that the overall reaction rate can be determined by the rates of the individual steps in a multi-step reaction, which leads us to consider the concept of the rate-determining step.

In a multi-step chemical reaction, one step typically limits the overall rate of the reaction. This is known as the rate-determining step (RDS). The RDS has a higher activation energy than the other steps, and therefore, it proceeds more slowly. As a result, the overall reaction cannot proceed faster than this slowest step.

In the exercise at hand, the rate dependence on \( \mathrm{NO} \) and \( \mathrm{O}_{2} \) suggests that the mechanism we choose must include a slow step that is consistent with the observed order of reaction. Only the mechanism that accurately reflects the dependence of the reaction rate on the concentrations of the reactants can be correct. Hence, the RDS for the correct mechanism should show a second-order dependence on \( \mathrm{NO} \) and a first-order dependence on \( \mathrm{O}_{2} \) to match the experimental observations.

Reaction order is a term used to express how the rate of a chemical reaction depends on the concentration of one or more reactants. It is determined by summing the powers of the concentration terms in the rate law of the reaction. For example, a reaction that is first-order with respect to \( \mathrm{NO} \) would imply that if the concentration of \( \mathrm{NO} \) is doubled, the rate of the reaction would double as well.

In the provided exercise, we observed that the reaction is of the first-order in \( \mathrm{O}_{2} \) and second-order in \( \mathrm{NO} \) because the rate of reaction changes in direct proportion to the concentration changes of these reactants. The overall reaction order is the sum of the individual orders with respect to each reactant involved in the rate-determining step.

Chemical kinetics is the branch of chemistry that concerns the rates of chemical reactions and the mechanisms by which they occur. By studying the factors that affect reaction rates, such as reactant concentrations, temperature, and catalysts, chemists can understand the sequence of steps, or the reaction mechanism, that leads from reactants to products.

Understanding kinetics is crucial in a number of practical settings, including the development of new reactions in industrial processes, understanding metabolic pathways in biochemistry, and even predicting how quickly a medication will be metabolized in the body. The principles of kinetics allow scientists and engineers to design and optimize reactions for a wide variety of applications.

Equilibrium reactions are chemical reactions that proceed in both the forward and reverse directions. At equilibrium, the rate of the forward reaction is equal to the rate of the reverse reaction, resulting in no net change in the concentration of reactants and products over time. Unlike reactions that go to completion, equilibrium reactions are characterized by constant, stable concentrations of all the reactants and products involved.

Understanding equilibrium is important when examining reaction mechanisms, as some mechanisms involve fast, reversible steps that establish an equilibrium before the rate-determining step. In the exercise, mechanism (a) proposes such an equilibrium in the first step. When analyzing such mechanisms, the concentrations of intermediates at equilibrium can significantly influence the overall reaction rate, as they may be involved in the slowest step of the reaction.