How is half-life related to the rate constant in second-order reactions?

The half-life in second-order reactions is inversely proportional to the initial concentration and directly proportional to the rate constant.

In second-order reactions, the half-life (t½) is dependent on both the rate constant (k) and the initial concentration of the reactant ([A]₀). The relationship between these variables is given by the equation t½ = 1 / k[A]₀. This means that the half-life is inversely proportional to the initial concentration of the reactant - as the initial concentration increases, the half-life decreases. Conversely, the half-life is directly proportional to the rate constant - as the rate constant increases, so does the half-life.

This relationship is unique to second-order reactions. In first-order reactions, the half-life is independent of the initial concentration and is directly proportional to the reciprocal of the rate constant. However, in second-order reactions, the rate of reaction is dependent on the concentration of the reactants. This means that as the reaction progresses and the concentration of the reactants decreases, the rate of reaction also decreases. Therefore, the half-life, which is the time taken for the concentration of a reactant to decrease by half, increases as the reaction progresses.

The relationship between half-life, rate constant, and initial concentration in second-order reactions can be used to determine the order of a reaction experimentally. By measuring the half-life at different initial concentrations, one can determine whether the half-life changes with concentration (indicating a second-order reaction) or remains constant (indicating a first-order reaction). Similarly, by measuring the rate constant at different concentrations, one can determine whether the rate of reaction changes with concentration.

In summary, the half-life in second-order reactions is a function of both the rate constant and the initial concentration of the reactant. This relationship provides valuable information about the kinetics of the reaction and can be used to identify the order of the reaction.