Write the rate expression for each of the following elementary steps: (a) \(\mathrm{NO}_{3}+\mathrm{CO} \longrightarrow \mathrm{NO}_{2}+\mathrm{CO}_{2}\) (b) \(\mathrm{I}_{2} \longrightarrow 2 \mathrm{I}\) (c) \(\mathrm{NO}+\mathrm{O}_{2} \longrightarrow \mathrm{NO}_{3}\)

An elementary step in chemical reactions is a single reaction stage that describes a change that happens in a single, decisive step. It represents a fundamental building block of complex reactions that occur through multiple steps. Each elementary step is characterized by its molecularity, which refers to the number of reactant molecules involved.

For instance, when we look at the elementary step \(\mathrm{NO}_3+\mathrm{CO}\longrightarrow\mathrm{NO}_2+\mathrm{CO}_2\), we're observing a bimolecular reaction involving two species, \(\mathrm{NO}_3\) and \(\mathrm{CO}\), undergoing a direct transformation without any intermediate stages. In contrast, a unimolecular reaction like \(\mathrm{I}_2\longrightarrow\mathrm{I}\) only involves a single molecule breaking apart.

To ensure students fully comprehend this concept, remember that an elementary step assumes a direct conversion from reactants to products, as described in the problem's solutions.

Chemical kinetics is the branch of chemistry that deals with the rates of chemical reactions and the factors that affect these rates. It involves the study of how different variables such as concentration, temperature, and catalysts impact the speed at which reactants are converted into products.

Understanding the kinetics of a reaction provides insight into the reaction mechanism, which is the step-by-step sequence of elementary steps that describes the pathway from reactants to products. For example, the rate at which \(\mathrm{NO}_3\) and \(\mathrm{CO}\) react to form \(\mathrm{NO}_2\) and \(\mathrm{CO}_2\) is determined by factors such as the frequency of collisions and the energy of the reactant molecules. Kinetics is not only about speed but also about the detailed process of how reactions proceed, which can often involve complex sequences of elementary steps.

To improve understanding, emphasize that kinetics is more than just measuring reaction speed; it's about dissecting the dynamic process of the chemical reaction itself.

The reaction rate law, also simply known as the rate law, expresses the relationship between the rate of a chemical reaction and the concentration of its reactants. For an elementary step, the rate law is derived directly from the reaction stoichiometry, held true for most elementary reactions. The exponents in a rate law correspond to the stoichiometric coefficients of the reactants for an elementary step.

Consider the examples given in the exercise, each rate expression corresponds to the formula 'Rate = k[Reactant1]^[m] [Reactant2]^[n]...', where 'k' is the rate constant, and 'm' and 'n' are the reaction orders with respect to each reactant. For example, the rate law for the reaction of \(\mathrm{NO}\) with \(\mathrm{O}_2\) to form \(\mathrm{NO}_3\) is Rate = k[NO][O2], indicating that the rate is proportional to the concentration of both \(\mathrm{NO}\) and \(\mathrm{O}_2\).

To make this digestible for students, emphasize that the rate law is a mathematical way to predict how changes in reactant concentrations will influence the reaction rate, adhering to the specific molecularity of the elementary steps involved.