In a first-order reaction \(\mathrm{A} \longrightarrow \mathrm{P}\), the ratio of \(\mathrm{a} /(\mathrm{a}-\mathrm{x})\) was found to be 8 after 60 minutes. If the concentration is \(0.1 \mathrm{M}\) then the rate of reaction in moles of A reacted per minutes is: (a) \(2.226 \times 10^{-3}\) mol litre \(^{-1} \min ^{-1}\) (b) \(3.466 \times 10^{-3}\) mol litre \(^{-1} \min ^{-1}\) (c) \(4.455 \times 10^{-3}\) mol litre \(^{-1} \min ^{-1}\) (d) \(5.532 \times 10^{-3}\) mol litre \(^{-1} \mathrm{~min}^{-1}\)

Chemical kinetics, at its core, is the study of the speed at which chemical reactions occur. It's a fundamental topic in physical chemistry that investigates the factors affecting rates of reactions and the mechanisms by which they occur. Understanding kinetics can help predict how long it will take for a reaction to reach a certain point, which is crucial in fields such as pharmaceuticals, environmental science, and materials engineering.

When examining reaction rates, we look at the change in concentration of reactants or products over time. This is expressed mathematically as the reaction rate. For a first-order reaction, one of the simplest forms, the rate is directly proportional to the concentration of the reactant. This means that as the reactant is consumed, the rate decreases linearly over time.

An equation that describes first-order kinetics is the integrated rate law, which relates the concentrations of reactants to time. The tool of half-life is often used in kinetics, which in a first-order reaction remains constant regardless of initial concentration. This consistent half-life characteristic and straightforward rate laws make first-order kinetics a favorite for preliminary studies in reaction mechanics.

The rate of reaction tells us how fast or slow a chemical reaction proceeds. It's usually quantified by the change in concentration of a reactant or product over a period of time. Different reactions have different rates, and this is dependent on various factors such as the nature of the reactants, temperature, pressure, the presence of a catalyst, and surface area in heterogeneous reactions.

In the context of a first-order reaction, the rate of reaction can be determined using the rate constant and the concentration of the reactant. For the example from the exercise where we have a first-order reaction of the type \( A \longrightarrow P \) and are given that the ratio \( \frac{a}{a-x} \) is 8, we can interpret this as the concentration of reactant 'A' is reduced to one-eighth its original value over a given time period, which in this case is 60 minutes.

To compute the rate, we first determine the concentration of A that has reacted, and then divide this by the time interval to find the rate at which A is converted to P. This calculation is critical for chemists and other scientists who need to optimize reaction conditions for industrial processes, drug synthesis, or even just simple lab experiments.

NEET chemistry is a section within the National Eligibility cum Entrance Test, an examination that is pivotal for students in India aspiring to embark on a career in medicine. It covers a vast syllabus, including the physical, organic, and inorganic branches of chemistry.

Regarding chemical kinetics, and more specifically the rate of reactions, NEET candidates must have a solid grasp of the concepts and mathematical skills to solve related problems. As illustrated in the exercise, being able to calculate the rate of a first-order reaction from given data is a fundamental skill that could be tested in the examination. Problems typically require understanding of the integrated rate laws and half-life equations, along with the ability to perform calculations like those in the exercise efficiently and accurately.

Acing these kinds of questions often involves practice in problem-solving and familiarity with steps usually taken to solve kinetic problems, which are crucial not only for the NEET examination but also for practical laboratory and industrial applications where these principles are applied.