Rate of a reaction: \(\mathrm{A}+2 \mathrm{~B} \rightarrow \mathrm{P}\) is \(2 \times 10^{-2} \mathrm{M} / \mathrm{min}\), when concentrations of each \(A\) and \(B\) are \(1.0 \mathrm{M}\). If the rate of reaction, \(r=K[\mathrm{~A}]^{2}[\mathrm{~B}]\), the rate of reaction when half of the \(\mathrm{B}\) has reacted should be (a) \(5.625 \times 10^{-3} \mathrm{M} / \mathrm{min}\) (b) \(3.75 \times 10^{-3} \mathrm{M} / \mathrm{min}\) (c) \(9.375 \mathrm{M} / \mathrm{min}\) (d) \(2.5 \times 10^{-3} \mathrm{M} / \mathrm{min}\)

Understanding the rate law of a chemical reaction unlocks the relationship between reactant concentrations and the rate at which products form. The rate law is an equation that mathematically describes this relationship. Specifically, it indicates how the rate depends on the concentration of each reactant. For instance, in the provided exercise, the rate law is expressed as r = K[A]^2[B], where r represents the rate, K is the rate constant, and the exponents of [A] and [B] signify their respective effects on the rate.

In this scenario, the rate is directly proportional to the square of the concentration of A and the first power of B. These exponents are known as reaction orders and play a critical role in the kinetics of the reaction. When working with rate laws, a step-by-step approach involves first identifying the rate law from experimental data, then using known concentrations to solve for the rate constant.

The rate constant, often denoted as K, is a proportionality factor that directly impacts the rate of a chemical reaction. It is a unique value for every chemical reaction at a given temperature. In simple terms, it quantifies the rate at which reactants transform into products.

To calculate the rate constant from the rate law, one must know the rate of the reaction and the concentrations of the reactants under specific conditions, as seen in step 2 of the provided solution. After obtaining the rate constant, it can be used to predict the rate of the reaction under different concentrations of reactants. This is vital as it remains constant at a constant temperature and provides insights into the reactivity and mechanism of the reaction.

Reactant concentration, represented by [A], [B], etc., is a measure of the amount of a reactant within a given volume. In chemical kinetics, it's crucial to understand how changes in reactant concentrations influence the reaction rate. As exemplified in the exercise, if the concentration of B is reduced by half, from 1.0 M to 0.5 M, the rate of the reaction will also change.

This is because the rate law equation includes the concentrations raised to their respective reaction orders. When a reactant concentration changes, the rate can be recalculated by substituting the new concentration value into the rate law, while keeping the rate constant, K, the same (unless the temperature changes). The concentration of reactants typically decreases over time as they form products, and monitoring these changes can help control the reaction rate for desired outcomes.

Chemical kinetics is a branch of physical chemistry that deals with the speed or rate at which chemical reactions occur. It examines the factors that influence rates of reactions and establishes the mechanisms by which reactions proceed. Some of these factors include reactant concentrations, temperature, and the presence of catalysts.

The study of kinetics is fundamental in predicting how long a reaction will take and how it can be optimized for industrial or laboratory settings. Understanding kinetics can help in the design of chemical reactors and in the safe storage and handling of reactive materials. It's pivotal for students to grasp the concepts of chemical kinetics as they can then apply these principles to various situations, including the exercise given, which focuses on manipulating reactant concentrations to determine a reaction's rate.