For the decomposition reaction: \(\mathrm{N}_{2} \mathrm{O}_{4(\mathrm{~g})} \longrightarrow 2 \mathrm{NO}_{2(\mathrm{~g})} ;\) the initial pressure of \(\mathrm{N}_{2} \mathrm{O}_{4}\) falls from \(0.46\) atm to \(0.28\) atm in 30 minute. What is the rate of appearance of \(\mathrm{NO}_{2}\) ?

Decomposition reactions are a type of chemical reaction where a single compound splits into two or more simpler substances. These reactions often require an input of energy, such as heat, light, or electricity, to occur. The general formula for a decomposition reaction is:

\( AB \to A + B \).

In the context of the textbook problem, dinitrogen tetroxide (\( \mathrm{N_2O_4} \)) decomposes into nitrogen dioxide (\( \mathrm{NO_2} \)), following the stoichiometry of 1:2. This means that each molecule of \( \mathrm{N_2O_4} \) breaks down to form two molecules of \( \mathrm{NO_2} \). Understanding this reaction is crucial, as it lays the foundation for calculating the stoichiometry, and the rate of the reaction.

Stoichiometry is the quantitative relationship between the reactants and products in a chemical reaction. It is based on the conservation of mass and the balanced chemical equation. For the problem at hand, the stoichiometry is derived from the balanced equation:

\( \mathrm{N_2O_4_{(g)}} \rightarrow 2 NO_{2_{(g)}} \).

This indicates that for every mole of \( \mathrm{N_2O_4} \) that decomposes, two moles of \( \mathrm{NO_2} \) are produced. When calculating stoichiometric quantities, it's important to use a consistent set of units—moles in this case—and to understand how these relate to measurable quantities like pressure, especially under ideal gas behavior conditions. When students master stoichiometric calculations, they can predict the amounts of products formed or reactants needed in a chemical reaction, which is essential for laboratory work and industrial processes.

The rate of reaction is a key concept in chemical kinetics, referring to the speed at which a chemical reaction proceeds. It is typically expressed as the change in concentration of a reactant or product per unit time. In the case of gases, where concentration is often related to pressure under the assumption of ideal gas behavior, the rate can be given in terms of pressure change per unit time.

For a decomposition reaction like \( \mathrm{N_2O_4} \rightarrow 2 \mathrm{NO_2} \), the rate of appearance of \( \mathrm{NO_2} \) is linked to the rate of decomposition of \( \mathrm{N_2O_4} \). The rate can be determined by measuring the pressure change of \( \mathrm{NO_2} \) over a specific time interval, given the stoichiometric relationships. In the exercise, the decrease in pressure of \( \mathrm{N_2O_4} \) directly translates to the creation of \( \mathrm{NO_2} \) pressure, allowing students to calculate the rate of appearance with the formula:

\( \text{Rate of appearance} = \frac{\Delta \text{Pressure}_{NO_2}}{\Delta \text{time}} \).

Understanding the rate of reaction not only helps in academic settings but is also fundamental in industries that depend on controlled reaction speeds, such as pharmaceuticals, manufacturing, and materials science.