For a first-order reaction, how long will it take for the concentration of reactant to fall to one-eighth its original value? Express your answer in terms of the half-life \(\left(t_{1}\right)\) and in terms of the rate constant \(k\).

Reaction kinetics is the study of the rates at which chemical reactions proceed and the factors that affect these rates. In the context of a first-order reaction, kinetics involves understanding how the reaction rate depends on the concentration of the reactant. A first-order reaction has the characteristic that the rate of reaction is directly proportional to the concentration of one reactant only. This means that if you were to double the concentration of the reactant, the reaction rate would also double.

Understanding the kinetics of such reactions is crucial for predicting how the concentration of reactants will change over time, which is fundamental to both theoretical study and practical applications such as pharmaceuticals, industrial processes, and environmental modeling. For students, grasping reaction kinetics is often achieved by focusing on the relationship between concentration and time, as represented by the equation \( A = A_{0}e^{-kt} \).

The rate constant, symbolized by \( k \), is a numerical value that represents the speed at which a chemical reaction proceeds. For a first-order reaction, it helps determine how quickly the concentration of the reactant decreases over time. It's important to understand that the rate constant is not affected by the reactant's concentration; it remains constant for a given reaction at a particular temperature.

A higher rate constant indicates a faster reaction. The magnitude of \( k \) depends on various factors, such as temperature, the presence of a catalyst, and the molecular nature of the reactants. In our exercise, the rate constant is critical to finding the time at which the reactant's concentration falls to one-eighth of its original value. By manipulating the first-order reaction equation, we can solve for \( t \), thus connecting the rate constant to an observable quantity—the time it takes for the reactant concentration to change.

The half-life of a reaction, denoted as \( t_{1/2} \), is the time required for the concentration of a reactant to decrease to half of its initial value. In the realm of first-order reactions, the half-life is a constant, which signifies a distinctive feature—it doesn't change as the reaction progresses. This is unlike other reaction orders where the half-life may vary with concentration.

For students delving into reaction kinetics, the half-life concept serves as a useful tool for simplifying calculations. As shown in the exercise solution, we can express the time for a reactant to reach one-eighth of its initial concentration as three times the half-life \( (t = 3t_{1/2}) \). This relation showcases the power of using the half-life concept for predictions and highlights the exponential nature of first-order kinetics. Understanding the half-life helps in visualizing the decay of reactants and is also a stepping stone to more complex kinetic analysis.