91Ó°ÊÓ

The rate constant, denoted as \( k \), plays a crucial role in determining how fast a reaction occurs in chemical kinetics. For a second-order reaction, the rate is proportional to the product of the concentrations of the reactants raised to their stoichiometric coefficients. The equation for a second-order reaction is usually expressed as \( ext{Rate} = k[A][B] \), but in the simpler case where only one reactant repeats, it becomes \( ext{Rate} = k[A]^2 \).
In the given exercise, the second-order reaction involves nitric oxide bromide decomposing, and the rate constant is provided as \( 0.80 \, M^{-1} \, s^{-1} \). This rate constant indicates how strongly the concentration of reactants influences the speed of the reaction at a certain temperature, here specified at \( 10^{\circ} \mathrm{C} \). Understanding the meaning of the rate constant is key, as it allows us to predict the reaction behavior under various experimental conditions.

The half-life of a reaction is the time required for the concentration of a reactant to reduce to half its initial value. For second-order reactions, the half-life is not constant and depends on the initial concentration of the reactant. The formula used to calculate the half-life \( t_{1/2} \) for second-order reactions is \( t_{1/2} = \frac{1}{k[A]_0} \).
This exercise highlights how, for different initial concentrations of nitric oxide bromide (\([\mathrm{NOBr}]_0\)), the half-life changes. For \([\mathrm{NOBr}]_0 = 0.072 \, M\), the calculated half-life is approximately \(1.7 \, s\), while for \([\mathrm{NOBr}]_0 = 0.054 \, M\), it is \(2.3 \, s\). This demonstrates the inverse relationship between concentration and half-life in second-order kinetics.

Chemical kinetics involves the study of reaction rates and the steps that occur in chemical processes. It allows us to understand the factors influencing the speed of a reaction, such as reactant concentration, temperature, and the presence of catalysts.
In the exercise, observing a chemical reaction with nitric oxide bromide offers insights into the kinetics of decomposition. The rate constant already mentioned is central to these kinetic considerations, being sensitive to changes in temperature. Hence, understanding kinetics not only helps in predicting how long a process will take but also assesses the conditions under which it will proceed efficiently with regard to time and resource consumption.

In chemical kinetics, calculating the concentration of reactants or products over time is an essential aspect of understanding a reaction's progress. For second-order reactions, the concentration of a reactant at any time \( t \) can be determined using the integrated rate equation:
\[ \frac{1}{[A]_t} = kt + \frac{1}{[A]_0} \]
This equation allows us to calculate the remaining concentration of \([NOBr]\) in the given exercise after \(22\) seconds, with an initial concentration of \(0.086 \, M\). By plugging into the equation and solving, the concentration of \([NOBr]\) is shown to decrease to \(0.0342 \, M\). This calculation shows how the concentration diminishes as the reaction progresses over time.