Recall that for a first-order reaction: $$ \ln (\mathrm{R})=\ln (\mathrm{R})_{0}-k t $$ a) When \(t=t_{1 / 2}\), what is the value of \((\mathrm{R})\) in terms of \((\mathrm{R})_{0}\) ? b) Show that \(t_{1 / 2}=\frac{\ln 2}{k}=\frac{0.693}{k}\) for a first-order reaction.

In the fascinating realm of chemical kinetics, the reaction rate is a cornerstone concept. It refers to the speed at which reactants are converted into products in a chemical reaction. In a mathematical sense, it's the change in concentration of a reactant or product per unit time. For a first-order reaction, the rate is directly proportional to the concentration of the single reactant.

Understanding reaction rate is paramount because it dictates how quickly a reaction proceeds under given conditions. It's influenced by various factors such as temperature, pressure, and the presence of a catalyst. Knowing the rate at which a drug is metabolized, for example, can help in designing appropriate dosage regimes in medicine.

The term half-life is often tossed around in discussions on radioactivity, but it's equally important in chemistry. It's defined as the time required for the concentration of a reactant to reach half its initial value. This concept is critical when it comes to first-order reactions, where the half-life is constant regardless of the initial concentration.

To illustrate, consider a radioactive isotope decaying or a medication being metabolized in the body; the half-life gives a measure of how long it takes for half of the material to vanish or be rendered inactive. This understanding of half-life can assist in determining the longevity and dosage intervals for pharmaceuticals, or in estimating the age of archaeological artifacts through carbon dating.

The natural logarithm, represented as \( \ln \), is a logarithm to the base \( e \) where \( e \) is an irrational and transcendental constant approximately equal to 2.71828. This particular logarithm is pivotal in growth and decay problems, compounding interest calculations, and of course, in expressing the kinetics of a first-order reaction. The natural logarithm allows us to linearize the exponential decay in such reactions, linking the concentration of reactants to time in an easily understandable way.

The beauty of the natural logarithm lies in its inherent relationship with exponential functions, making it an indispensable tool in not only chemistry but in various branches of mathematics and physics as well.

In any chemical reaction, the rate constant is the proportionality factor that relates the reaction rate to the reactant concentrations as expressed by the rate equation. For a first-order reaction, the rate constant \( k \) ties together the time, \( t \), and the concentration of the reactant. It's a measure of how quickly a reaction occurs, and its value is determined by the particular reaction's mechanism and its temperature dependence, often described by the Arrhenius equation.

The rate constant holds the key to unlocking the mysteries of a chemical reaction's timing. In the given exercise, the inversely proportional relationship between the half-life and the rate constant elegantly demonstrates how a higher rate constant signifies a faster reaction, leading to a shorter half-life. Such information is crucial for chemists in the design and prediction of reaction behaviors.