For each of the following gas-phase reactions, write the rate expression in terms of the appearance of each product and disappearance of each reactant: (a) \(2 \mathrm{H}_{2} \mathrm{O}(g) \longrightarrow 2 \mathrm{H}_{2}(g)+\mathrm{O}_{2}(g)\) (b) \(2 \mathrm{SO}_{2}(g)+\mathrm{O}_{2}(g) \longrightarrow 2 \mathrm{SO}_{3}(g)\) (c) \(2 \mathrm{NO}(g)+2 \mathrm{H}_{2}(g) \longrightarrow \mathrm{N}_{2}(g)+2 \mathrm{H}_{2} \mathrm{O}(g)\) (d) \(\mathrm{N}_{2}(g)+2 \mathrm{H}_{2}(g) \longrightarrow \mathrm{N}_{2} \mathrm{H}_{4}(g)\)

The study of how reactions occur and the factors that influence them is called chemical kinetics. It examines the reaction rates, which are measures of how quickly products form from reactants. Understanding kinetics involves analyzing factors such as temperature, concentration, surface area, and the presence of catalysts, as all can affect the speed of a reaction.

When writing rate expressions, it is essential to consider the stoichiometry of the reaction, as it helps in quantifying the exact relationship between reactants and products over time. The coefficients of the balanced chemical equation indicate the proportion in which molecules react and form products; these coefficients become integral to the rate expressions. Knowing the rate expressions and how they're derived is crucial in chemical kinetics to predict the rate of a reaction under various conditions.

The reaction rate is a key concept in kinetics, representing the speed at which a chemical reaction proceeds. It is often expressed in terms of the rate of change in concentration of a reactant or product per unit time.

For instance, in a given reaction where hydrogen gas combines with oxygen to form water, the reaction rate can be expressed by the change in concentration of hydrogen or water over time. There's an important aspect of reaction rates to keep in mind: the rate of disappearance of reactants will always be negative because their concentrations decrease over time, while the rate of appearance of products is positive. For balanced chemical equations, the stoichiometric coefficients must be considered when writing rate expressions to ensure the relationship between the rates of reactants and products is accurate.

The term stoichiometry refers to the calculation of the quantities of reactants and products involved in a chemical reaction. It is based on the principle of conservation of mass, which states that matter is neither created nor destroyed in a chemical reaction. Thus, the number of atoms of each element must be the same on both sides of a balanced equation.

When dealing with gases, as in the provided exercises, it's also important to remember Avogadro's law, which states that equal volumes of gases at the same temperature and pressure contain an equal number of molecules. Stoichiometry is not just about balancing equations; it provides a quantitative measure of the relative amounts of substances involved in a reaction, which is used to derive the rate expressions for those substances. A clear understanding of stoichiometry is essential, not only to balance chemical equations but also to connect the coefficients to the rates at which reactants are consumed and products are produced in chemical reactions.