Write the rate law for each of the following elementary reactions: (a) \(\mathrm{O}_{3} \stackrel{\text { sunlight }}{\longrightarrow} \mathrm{O}_{2}+\mathrm{O}\) (b) \(\mathrm{O}_{3}+\mathrm{Cl} \longrightarrow \mathrm{O}_{2}+\mathrm{ClO}\) (c) \(\mathrm{ClO}+\mathrm{O} \longrightarrow \mathrm{Cl}+\mathrm{O}_{2}\) (d) \(\mathrm{O}_{3}+\mathrm{NO} \longrightarrow \mathrm{NO}_{2}+\mathrm{O}_{2}\) (e) \(\mathrm{NO}_{2}+\mathrm{O} \longrightarrow \mathrm{NO}+\mathrm{O}_{2}\)

Understanding the rate law is essential for predicting how changes in concentration affect the rate of a chemical reaction. The rate law expresses the relationship between the rate of a chemical reaction and the concentration of its reactants. For example, consider an elementary reaction where ozone decomposes to dioxygen and a lone oxygen atom:
\(\mathrm{O_{3}} \stackrel{\text{sunlight}}{\longrightarrow} \mathrm{O_{2}} + \mathrm{O}\).
The rate law for this reaction is written as \[\text{rate} = k[\mathrm{O_{3}}]\],where \(k\) is the rate constant and \([\mathrm{O_{3}}]\) is the concentration of ozone. It's important to note that for elementary reactions, the rate law can often be written directly from the stoichiometry of the reaction, which is conveniently the case in the provided examples. Elemental steps, being the simplest, directly translate their stoichiometry to the rate law, as reflected in the example above.

The reaction order indicates the power to which the concentration of a reactant is raised in the rate law, reflecting how the rate is affected by changes in concentration. For elementary reactions, the reaction order is typically derived from the molecularity, which refers to the number of reactant molecules participating in a single step. If a reaction involves a single molecule, such as the decomposition of ozone, it is classified as unimolecular and has a reaction order of 1. Conversely, if a reaction involves two molecules, such as \(\mathrm{O_{3}} + \mathrm{Cl} \longrightarrow \mathrm{O_{2}} + \mathrm{ClO}\),it is termed bimolecular and holds a reaction order of 2, being first order with respect to each reactant. This means, mathematically, the rate of the reaction is directly proportional to the product of the concentrations of the two reactants, as noted in the rate law \[\text{rate} = k[\mathrm{O_{3}}][\mathrm{Cl}]\].

Elementary reactions are the basic steps that constitute a complex reaction mechanism. Each elementary step proceeds through a single transition state and represents a single molecular event. An important aspect of elementary reactions is that their rate laws reflect the stoichiometric coefficients of the reactants directly. This means each reactant's concentration affects the reaction rate as per its stoichiometric coefficient in the balanced equation for the step. In other words, the reaction order for an elementary step is given by its molecularity. Thus, for the provided examples, the moleculartity of reactions (a), (b), (c), (d), and (e) are all clear from the number of reactant molecules, resulting in rate laws that directly mirror these reaction orders.

The rate constant, symbolized as \(k\), is a proportionality factor in the rate law equation that provides the connection between reactant concentrations and reaction rate. Its value is determined experimentally and varies with temperature and the presence of a catalyst. For instance, in the reaction \(\mathrm{NO_{2}} + \mathrm{O} \longrightarrow \mathrm{NO} + \mathrm{O_{2}}\),the rate constant plays a crucial role in determining the reaction's speed under specific conditions, represented in the rate law as \[\text{rate} = k[\mathrm{NO_{2}}][\mathrm{O}]\].The rate of any elementary reaction is not directly observable but is inferred from the rate at which reactant concentrations change. The rate constant is unique to each reaction and provides a quantitative measure of how quickly a reaction proceeds. It is essential for chemists to understand and calculate the rate constant for accurate predictions of reaction behavior.