Why is the half-life constant for a first-order reaction?

The half-life is constant for a first-order reaction because it is independent of the initial concentration of the reactants.

In a first-order reaction, the rate of the reaction is directly proportional to the concentration of one reactant. This means that as the reaction progresses and the concentration of the reactant decreases, the rate of the reaction also decreases. However, the half-life of a first-order reaction, which is the time taken for the concentration of the reactant to decrease by half, remains constant throughout the reaction. This is a unique characteristic of first-order reactions.

The mathematical expression for a first-order reaction is: rate = k[A], where 'rate' is the rate of the reaction, 'k' is the rate constant, and '[A]' is the concentration of the reactant. If we rearrange this equation to find the time it takes for the concentration of the reactant to decrease by half (the half-life), we get: t1/2 = 0.693/k. As you can see, the half-life (t1/2) only depends on the rate constant (k), and not on the initial concentration of the reactant. This is why the half-life is constant for a first-order reaction.

This property of first-order reactions is very useful in many areas of chemistry and other sciences. For example, it is used in the field of nuclear chemistry to predict the rate at which radioactive substances decay. It is also used in pharmacology to predict how quickly a drug will be metabolised in the body. Understanding this concept is therefore not only important for your chemistry studies, but also for your broader scientific knowledge.