The first-order rate constant for the decomposition of \(\mathrm{N}_{2} \mathrm{O}_{5}\) is \(6.2 \times 10^{-4} \mathrm{~s}^{-1} .\) The half-life for this decomposition is: (a) \(1177.7 \mathrm{~s}\) (b) \(1117.7 \mathrm{~s}\) (c) \(6.077 \mathrm{~s}\) (d) \(110.77 \mathrm{~s}\)

Chemical kinetics is the branch of physical chemistry that studies the rates of chemical reactions and how these rates change under varying conditions. It's crucial for understanding the speed at which reactions occur and what can be done to speed them up or slow them down. In the context of our discussion, when looking at the decomposition of a substance like 2O5, kinetics helps us predict how quickly this will happen.

The rate of a reaction can be affected by various factors, including temperature, concentration of reactants, surface area, the presence of catalysts, and light. During the process of decomposition, reactants are gradually converted into products, and it's through kinetics that we can quantify this change over time. For students tackling problems related to reaction rates, it's essential to understand that the detailed mechanisms by which reactions occur can directly affect the kinetics, and thereby the overall rate of the reactions.

In chemical kinetics, the rate constant is a proportionality constant in the rate law equation, which illustrates the relationship between the reactant concentrations and the reaction rate. For a first-order reaction, the rate law is expressed as Rate = k[A], where [A] stands for the molar concentration of reactant A, and k is the rate constant.

The rate constant, denoted as k, holds critical information about the reaction's speed. It has specific units depending on the order of the reaction and is influenced by temperature and the presence of a catalyst. In the case of first-order reactions, the units are typically s^(-1) (inverse seconds). It's essential to understand that the rate constant does not change with the concentration or the amount of reactants but can change with temperature. In the problem provided, k for the decomposition of 2O5 helps us predict how rapidly the decomposition proceeds by calculating the half-life of the reaction.

Decomposition reactions occur when a compound breaks down into simpler substances. The decomposition of dinitrogen pentoxide (2O5) to nitrogen dioxide (NO2) and oxygen (O2) is commonly studied in chemical kinetics due to its relatively simple reaction mechanism and its occurrence in the Earth's atmosphere. As the reaction proceeds, the concentration of 2O5 decreases, and the amounts of NO2 and O2 increase.

Mathematically, the decomposition of 2O5 can be described by the rate equation for a first-order reaction. This gives us a clear relationship between the half-life (the time it takes for half of the 2O5 to decompose) and the rate constant. In educational materials, it's helpful for students to visualize the reaction using graphs plotting concentration versus time, which shows the exponential decrease of 2O5 and the consistent half-life intervals. By solving the given exercise, students learn how to apply the concept of half-life to real chemical systems, enhancing their ability to predict reaction behavior over time.