In a first-order reaction, the concentration of the reactant, decreases from \(0.8 \mathrm{M}\) to \(0.4 \mathrm{M}\) in 15 minutes. The time taken for the concentration to change from \(0.1 \mathrm{M}\) to \(0.025 \mathrm{M}\) is: (a) 30 minutes (b) 60 minutes (c) \(7.5\) minutes (d) 15 minutes

Chemical kinetics is the study of the speed or rate at which chemical reactions occur and the factors that affect this rate. By understanding the kinetics of a chemical reaction, we can predict the concentration of reactants or products at a given time, control the reaction conditions to optimize product yield, and deduce the reaction mechanism.

First-order reactions are a fundamental concept within chemical kinetics. In these reactions, the rate of reaction is directly proportional to the concentration of one reactant. The rate law for a first-order reaction can be expressed as Rate = k[A], where k is the rate constant and [A] is the concentration of the reactant. For first-order reactions, plotting the natural logarithm of the reactant concentration versus time gives a straight line, which indicates a constant rate of reaction over time.

Understanding first-order kinetics is crucial because it allows predictions about how long a reaction will take, which is essential in industrial applications where time and efficiency are critical. Furthermore, medications that follow first-order kinetics can have their dosages more easily adjusted, as their behavior in the body will be consistent and predictable.

The half-life of a reaction is the time it takes for the concentration of a reactant to decrease to half of its initial value. For first-order reactions, the half-life is given by the formula t_(1/2) = ln(2) / k, where ln(2) is the natural logarithm of 2, approximately equal to 0.693, and k is the rate constant. One intriguing feature of first-order reactions is that their half-life is constant and does not depend on the initial concentration of reactants.

In the exercise, we apply this principle to determine the time required for the concentration to fall to a quarter of its initial value. The half-life calculation simplifies this process: since the concentration halves in 15 minutes, and halving the concentration twice (which is what is needed to go from 0.1 M to 0.025 M) would require two half-lives, the total time can be easily computed. This relationship enables us to predict how concentrations will change over time in a systematic, straightforward manner.

The reaction rate and the concentration of reactants are intimately connected in chemical kinetics, especially for first-order reactions. The rate of a reaction is often found by monitoring the concentration of a reactant over time. For first-order reactions, the rate is directly dependent on the concentration of the reactant; as the concentration decreases, the rate of the reaction decreases as well.

The concentration dependency of the reaction rate is crucial when dealing with situations where precise control over the reaction speed is necessary. In pharmaceuticals, for instance, maintaining a specific concentration of a drug in the bloodstream may be critical for its therapeutic effect. By understanding the concentration dependency, pharmacists and chemists can design drug release systems that maintain the optimal concentration over time, ensuring the effectiveness of the medication. In the context of the exercise, knowing that the half-life remains constant regardless of concentration allows for straightforward calculations and reinforces the importance of concentration dependency in predicting how long it will take for a reaction to proceed to a certain point.