Half-life of a Reaction

Half-life (t1/2) is the time required for a reactant's concentration to decrease to half its initial value. It helps estimate reactant concentration after an elapsed time and provides insight into reaction rates. Shorter half-lives indicate faster reactions, while longer half-lives indicate slower reactions, making it useful for comparing reaction speeds.

First-order reaction half-life is independent of initial reactant concentration and remains constant throughout the reaction. Second-order reaction half-life is inversely proportional to initial concentration, meaning it increases as the reaction proceeds and concentration decreases. This fundamental difference affects how each reaction type behaves over successive half-life periods.

Half-life expressions are derived from the integrated rate law by substituting specific conditions: time equals t1/2 and concentration equals half the initial value. This mathematical relationship allows chemists to calculate half-life for different reaction orders using the integrated rate law the dependence of concentration on time, connecting kinetic theory to practical predictions.

For zero-order reactions, half-life is directly proportional to initial reactant concentration and inversely proportional to the rate constant. As the reaction proceeds and concentration decreases, the half-life becomes shorter. This contrasts with first-order reactions, where half-life remains constant regardless of concentration changes.

For first-order reactions, half-life and rate constant have an inverse relationship, allowing direct calculation of the rate constant from half-life. However, for second-order reactions, the initial concentration must be known to calculate the rate constant from half-life. This dependency highlights how reaction order determines the relationship between kinetic parameters.

Hydrogen peroxide decomposition exhibits first-order kinetics, so its half-life remains constant. During each 6-hour period, the remaining concentration is halved: 1.000 M to 0.500 M, then 0.500 M to 0.250 M, then 0.250 M to 0.125 M. This predictable pattern occurs because first-order half-life is independent of concentration.

Sodium-24 has a short half-life of 14.7 hours and exhibits a faster decay rate, while cobalt-60 has a longer half-life of 5.3 years and decays more slowly. This inverse relationship between half-life and decay rate illustrates a fundamental principle: shorter half-lives correspond to faster reaction rates, making half-life a useful tool for comparing reaction speeds.