Which of the following relation is correct for a first order reaction? \((k=\) rate constant; \(\mathrm{r}=\) rate of reaction; \(\mathrm{C}=\) conc. of reactant) (a) \(k=\mathrm{r} \times \mathrm{C}^{2}\) (b) \(k=\mathrm{r} \mathrm{x}\) (c) \(k=\mathrm{C} / r\) (d) \(k=\mathrm{r} / \mathrm{C}\)

In the realm of first-order reaction chemistry, understanding the rate constant is pivotal. It symbolizes the speed at which a reaction progresses and is represented by the notation 'k'. The rate constant is unique to each chemical reaction and can be influenced by factors such as temperature and the presence of a catalyst.

In mathematical terms, for a first-order reaction where the reaction rate is directly proportional to the reactant's concentration, the rate constant 'k' is a coefficient that relates the rate of reaction to the concentration of the reactant. If the concentration doubles, for a first-order reaction, so does the reaction rate, as they are directly linked through the constant 'k'. This is fundamental in solving exercises where determining the relationship between the rate, the rate constant, and the concentration of the reactant is required. Furthermore, it's crucial when predicting how long a reaction will take or how its rate will change under different conditions.

The reaction rate is the velocity at which a chemical reaction occurs. In simple terms, it’s a measure of how quickly reactants are converted into products. For a first-order reaction, the reaction rate can be represented by the change in concentration of a reactant over a specific time period.

Using the notation 'r' for reaction rate, the equation for a first-order reaction is expressed as \( r = k \times [Reactant] \) where '[Reactant]' is the concentration of the reactant. As the solution to the original exercise indicated, for a first-order reaction, the rate is directly proportional to the concentration of one reactant, leading to our equation and the ability to solve for either the rate constant 'k' or the reactant concentration. Understanding this concept is key to not only solving textbook exercises but also to grasping how chemical reactions progress in real-world scenarios.

The science of chemical kinetics delves into the rates of chemical processes and the steps by which they occur. It includes studying both the reaction rate and the rate constant among many other parameters. Chemical kinetics gives us insights into the mechanisms of reactions, which are series of steps that lead to the formation of products.

Studies in this field can explain why certain reactions are instantaneous while others take millennia to progress. Chemical kinetics also extends to the influence of external factors on the reaction rate, including temperature, pressure, and the presence of a catalyst. Through the principles of kinetics, we can also understand why similar substances react at varying speeds and why some substances are more reactive than others. For students and professionals in the field of chemistry, a deep understanding of chemical kinetics is instrumental for predicting and controlling chemical reactions in both laboratory and industrial settings.