91Ó°ÊÓ

In chemical kinetics, the rate constant \(k\) is a crucial parameter that describes the speed of a reaction. For a second-order reaction like \(2 \text{NO}_{2} \longrightarrow 2 \text{NO} + \text{O}_{2}\), the rate constant has the units of \(M^{-1}\cdot s^{-1}\). This indicates how the concentration of reactants decreases over time. The rate constant for this reaction is given as \(0.54 / M \cdot \mathrm{s}\) at \(300^{\circ} \mathrm{C}\). This means that at this specific temperature, the reaction will proceed at a measurable pace determined by this constant.Understanding the rate constant helps us predict how fast reactants will convert to products under given conditions. It is part of the rate law equation, which combines the rate constant with the concentration of the reactants to give the overall rate of reaction.

The integrated rate law for second-order reactions provides insight into the relationship between concentration and time. The formula is given by: \[ \frac{1}{[A]_t} = \frac{1}{[A]_0} + kt \] Where:

• \([A]_t\) is the concentration at time \(t\)
• \([A]_0\) is the initial concentration
• \(k\) is the rate constant

This law allows us to calculate how long it will take for a reactant to reach a certain concentration at a given rate constant. By rearranging and solving this equation, you can determine the time needed for \([\text{NO}_2]\) to decrease from \(0.62 M\) to \(0.28 M\) in the given reaction.The step-by-step substitution and calculation demonstrate the application of the integrated rate law, yielding important insights needed for planning and anticipating reaction progress.

The concept of half-life \(t_{1/2}\) in chemistry tells us how long it takes for the concentration of a reactant to decrease to half of its initial value. For second-order reactions, the half-life is dependent on both the rate constant and the initial concentration. The equation for half-life in a second-order reaction is: \[ t_{1/2} = \frac{1}{k[A]_0} \] Unlike first-order reactions, where half-life remains constant, in a second-order reaction, it changes with the concentration. For instance, when \([\text{NO}_2] = 0.62 \, M\), the half-life is approximately \(3.06\) seconds. However, when \([\text{NO}_2] = 0.28 \, M\), the half-life extends to about \(6.62\) seconds.These calculations illustrate that as the concentration of a reactant decreases, the time required for further reduction to half becomes longer. This illustrates the dynamic nature of reaction rates and how they are influenced by reactant concentrations and serves as a significant tool in reaction analysis.

In the context of reaction kinetics, concentration plays a pivotal role in determining the rate of reaction. In a second-order reaction, the rate at which reactants convert to products is directly proportional to the square of the reactant concentration. As a result, variations in initial concentration significantly alter both the reaction rate and half-life.For example, in the given reaction, starting with a concentration of \(0.62 \, M\) and transitioning to a concentration of \(0.28 \, M\), the progress of the reaction is tracked using the integrated rate law. This gives insight into how concentration shifts influence reaction completion times and half-life durations.Therefore, understanding the initial and changing concentrations is critical for analyzing how fast the reaction proceeds and planning accordingly for industrial or laboratory reactions. This helps predict how much product will be formed or how long it will take under specific concentrations.