If a reaction rate has a first-order dependence on a given reactant concentration, what will happen to the rate when the concentration of that reactant is doubled?

Chemical kinetics is the study of the rates at which chemical processes occur and the factors that affect these rates. It's an essential field in chemistry because it helps predict how fast a reaction will proceed under certain conditions, and what we can do to control the speed of these reactions.

The fundamental purpose of studying kinetics is to gain an understanding of the underlying pathways and the mechanism through which a reaction proceeds. It involves analyzing the frequency of collisions between molecules, the energy with which they collide, and the orientation of the molecules during collision. This knowledge allows chemists to design experiments, to optimize conditions for desired reactions, and to prevent unwanted reactions.

For instance, in industrial chemical processes, understanding kinetics can optimize production rates and lead to cost savings. Similarly, in biological systems, enzyme kinetics can explain how biochemical reactions occur within our bodies at incredibly fast rates.

The rate of a chemical reaction often depends on the concentrations of the reactants. In a first-order reaction, each time the concentration of a reactant is doubled, the reaction rate also doubles. This is because the rate law for a first-order reaction can be written as \(\text{Rate} = k[A]\), where \(k\) is the rate constant and \(\mathbf{[A]}\) is the concentration of the reactant A.

A great way to visualize this is to imagine a crowd in a stadium doing 'the wave'. If the number of people participating doubles, the waves around the stadium will occur twice as frequently- similar to how molecules in a solution react. This dependency is not only crucial in the laboratory; it's also applied in pharmaceutical drug development, environmental modeling, and even food science where reaction rates can affect flavor and texture.

The concentration effect on reaction rate is a cornerstone of chemical kinetics. Invariably, as the concentration of a reactant increases, the number of collisions among particles also increases, potentially leading to more frequent effective collisions where products are formed.

Specifically, for a first-order reaction, the rate directly depends on the concentration of the reactant. This can be mathematically represented as \(\text{Rate}=k[A]\), emphasizing that the rate is proportional to the concentration. If \(\text{[A]}\) doubles, so does the rate, as noted in the solution to our exercise. This has practical implications, for instance, in controlling pollution: reducing the concentration of harmful substances decreases the rate of undesirable reactions, and thus the formation of pollutants.

It's also worth noting, the effect of concentration on rate is a vital consideration in the design of chemical reactors, ensuring that reactions remain at optimal speed and also safe from reacting too violently.