For a first-order reaction, the ratio of time for \(99.9 \%\) of the reaction to complete and half of the reaction to complete is (a) 1 (b) 2 (c) 4 (d) 10

Understanding the rate law for first-order reactions is essential for predicting how the concentration of reactants will change over time. In these reactions, the rate at which the reaction proceeds is directly proportional to the concentration of a single reactant. Formally, this can be expressed as the rate \( \frac{d[A]}{dt} \) being equal to \(-k[A]\), where \( [A] \) is the concentration of the reactant, and \( k \) is the rate constant specific to the reaction under given conditions. The negative sign indicates that the concentration of the reactant decreases over time.
This relationship is simple yet powerful, as it tells us that no matter how much reactant we start with, the rate at which it reacts is always proportional to its current concentration. Because of this proportionality, we can use calculus to derive the integrated rate equation, which provides a direct link between the concentration of the reactant at any given time and the elapsed time.

Importance of Rate Constant \( k \)

One key aspect of the first-order rate law is the rate constant \( k \)—a unique value for each reaction that can be influenced by factors such as temperature and the presence of catalysts. Knowing \( k \) is essential for determining the speed of the reaction and for calculating the half-life, which is the time it takes for half of the reactant to be consumed.

Reaction Order Determination

To confirm that a reaction is first-order, one would conduct experiments to measure the concentration of reactants over time and ensure that the rate of reaction is consistent with the expected behavior dictated by the first-order rate law. This process often involves plotting data in a way that should yield a straight line if the reaction indeed follows first-order kinetics.

The concept of half-life is particularly important in the study of first-order reactions, as it is a constant value that indicates the time taken for the concentration of a reactant to decrease by half its initial value. For first-order reactions, the half-life is described by the equation \( t_{1/2} = \frac{\ln(2)}{k} \), where \( k \) is the rate constant.
This equation reveals a unique feature of first-order reactions: the half-life remains constant regardless of the starting concentration of the reactant. This is distinct from other reaction orders, where the half-life can change as the reaction progresses and the concentration of reactants changes. For students and chemists alike, the half-life is a convenient and quick way to assess the rate of a reaction without having to measure concentrations at multiple time points.

Practical Applications of Half-Life

In pharmacology, the half-life of drugs is critical for determining dosing schedules. Similarly, in environmental science, the half-life of pollutants dictates how long they remain in the environment. In each case, the underlying principles of half-life in first-order kinetics enable precise calculations and predictions for real-world applications. By mastering this concept, students can bridge the gap between theoretical chemistry and its practical uses in various fields.

The integrated rate equation for a first-order reaction is a fundamental tool in chemical kinetics, linking the concentration of a reactant to the time elapsed during the reaction. It is derived from the rate law and takes the form \( \ln(\frac{[A]_0}{[A]}) = kt \), where \( [A]_0 \) is the initial concentration, \( [A] \) is the concentration at time \( t \), and \( k \) is the rate constant. By rearranging the equation, we can solve for the time \( t \) that corresponds to any given concentration of the reactant.
This equation provides a direct method for calculating not only the half-life of the reaction, but also the time required for any percentage of the reactant to be consumed. For example, it can be used to determine the time it takes for 99.9% of a reactant to react, by substituting the appropriate concentration values into the equation.

Visualizing the Reaction Progress

Plotting data from the integrated rate equation typically yields a straight line on a semi-logarithmic graph, with the slope equal to \( -k \). This graphical approach is helpful for analyzing reaction kinetics and confirming that a reaction follows first-order kinetics. Understanding how to utilize the integrated rate equation gives students a powerful analytic method to interpret kinetic data, predict future concentrations, and calculate significant parameters such as the rate constant and reaction time for specific completion percentages.