A substance reacts according to I order kinetics and rate constant for the reaction is \(1 \times 10^{-2} \mathrm{sec}^{-1}\). If its initial concentration is \(1 M\) (a) What is initial rate? (b) What is rate after 1 minute?

Understanding the reaction rate law is crucial to studying the dynamics of how chemicals transform over time. Simply put, the rate law expresses the speed of a chemical reaction in terms of the concentration of its reactants. For a first-order reaction, the rate law has a direct relationship with the concentration of a single reactant. Mathematically, this relationship is represented as the rate equals the rate constant multiplied by the reactant's concentration, or in symbols, \( \text{rate} = k \times [A] \).

In practice, if you know the rate law for a reaction, you can predict how quickly a reactant will be consumed at any given concentration, as shown in the original exercise where the initial rate was determined using the given concentration and the rate constant.

The rate constant is a proportionality factor in the rate law that influences how quickly a reaction proceeds. It is represented by the symbol \( k \) and can change with temperature, but is independent of reactant concentrations for a given reaction under constant conditions. For the exercise provided, the rate constant \( k \) is given as \(1 \times 10^{-2} \mathrm{sec}^{-1}\).

The rate constant gives us insight into the intrinsic speed of the reaction, and when coupled with reactant concentration, it allows us to calculate the initial reaction rate. The higher the value of \( k \) for a reaction at a given temperature, the faster the reaction takes place.

Moving beyond the initial rate, we need to understand the integrated rate equation. This equation gives the concentration of a reactant at any time \( t \) during a first-order reaction, allowing us to see how the concentration decreases exponentially over time. The equation for a first-order reaction is \( [A] = [A_{\text{initial}}] \times e^{-kt} \), where \( [A] \) is the concentration at time \( t \) and \( [A_{\text{initial}}] \) is the initial concentration.

As shown in the exercise, this equation is used to determine the concentration of a reactant after a certain period has elapsed, which then helps to find the rate at that later time. Using this approach provides a powerful tool for predicting the future concentration of a reactant.

The concept of half-life in the context of reaction kinetics is a measure of how long it takes for half of the reactant in a chemical reaction to be consumed. For a first-order reaction, the half-life is a constant that does not depend on the initial concentration and is represented by the formula \( t_{1/2} = \frac{\ln(2)}{k} \).

The half-life provides a convenient way to understand the timescale of a reaction. For example, if the half-life is short, the reactant concentration decreases rapidly, indicating a fast reaction. Conversely, a long half-life means the reaction is slower. Being able to calculate half-life for a first-order reaction is particularly useful because it remains consistent throughout the reaction, thus serving as a straightforward and reliable measure of reaction speed.