Calculate the half life of a first order reaction from their rate constants given below : (a) \(200 \mathrm{~s}^{-1}\); (b) \(2 \mathrm{~min}^{-1}\); (c) 4 year \(^{-1}\).

The rate constant, often denoted as k, is a crucial parameter in chemical kinetics that provides the speed at which a chemical reaction proceeds. It's unique to each reaction and is affected by factors like temperature and the presence of a catalyst.
In the context of a first-order reaction, the rate constant has the units of reciprocal seconds (s^-1), minutes (min^-1), or years (year^-1) depending on the time scale of the reaction. The larger the value of k, the faster the reaction goes to completion. Understanding the rate constant is essential for calculating the half-life of a reaction, which indicates how long it takes for half of the reactant to be consumed.

Chemical kinetics is the study of reaction rates, how reaction rates change under varying conditions and by which mechanism the reaction proceeds. It not only helps in determining the speed of a reaction but also provides insights into the steps that constitute the overall process.
A first-order reaction is one such type of reaction where the rate is directly proportional to the concentration of one reactant. This implies that as the reactant concentration decreases, the rate of reaction decreases at the same rate. Chemical kinetics encompasses the mathematical expressions that relate the reaction rate to the concentration of reactants, which is vital when calculating half-lives of reactions.

The half-life of a reaction is the time required for the concentration of a reactant to decrease by half its initial value. For a first-order reaction, the half-life is calculated using the expression t_1/2 = (ln(2)) / k. This formula shows that the half-life is independent of the initial concentration; it only depends on the rate constant k.
When performing the half-life calculation, it is important to ensure that the units of the rate constant and time are consistent. For instance, if k is given in s^-1, the resulting half-life would be in seconds, whereas if k is given in min^-1, the half-life will be in minutes.

The natural logarithm of 2, denoted as ln(2), appears in the half-life formula for first-order reactions due to the nature of exponential decay processes. The value of ln(2) is approximately 0.693. This value is a constant that emerges from the relationship between the decrease in concentration over time in a first-order reaction.
The inclusion of ln(2) in the half-life formula can be thought of as a result of integrating the rate law of a first-order reaction to obtain the relationship between concentration and time. The ln(2) term essentially indicates that at the half-life time point, the concentration of the reactant has reached half of its initial value, a fundamental understanding for students tackling half-life problems.