Half-Life

The half-life of a reaction, t_1/2, is the duration of time required for the concentration of a reactant to drop to one-half of its initial concentration.

[latex] [A]{}_{t1/2}{}_{ } [/latex]= [latex] frac{1}{2} [A]{}_{0} [/latex]

Half-life is typically used to describe first-order reactions and serves as a metric to discuss the relative speeds of reactions. A slower reaction will have a longer half-life, while a faster reaction will have a shorter half-life.

To determine the half-life of a first-order reaction, we can manipulate the integrated rate law by substituting t_1/2 for t and [A]_t1/2 =  [A]₀  for [A]_t, then solve for t_1/2:

ln = –kt + ln       (integrated rate law for a first-order reaction)

ln [latex] frac{1}{2} [A]{}_{0} [/latex]= –[latex] extit{k} t{}_{1/2 }+ ln {[A]}_0 [/latex]

ln [latex] frac{frac{1}{2}{ m }{{ m [A]}}_0{ m }}{{[A]}_0} [/latex]= –[latex] extit{k} t{}_{1/2 } [/latex]

ln [latex] frac{1}{2} [/latex]= –[latex] extit{k} t{}_{1/2 } [/latex]

[latex] t{}_{1/2 } [/latex] = –  [latex] frac{{ m ln }frac{1}{2} }{k} [/latex] = [latex] frac{0.693}{k} [/latex]

Since the half-life equation of a first-order reaction does not include a reactant concentration term, it does not rely on the concentration of reactant present. In other words, a half-life is independent of concentration and remains constant throughout the duration of the reaction. Consequently, plots of kinetic data for first-order reactions exhibit a series of regularly spaced t_1/2 intervals (Figure 17.10 “Generic First-Order Reaction Kinetics Plot”).

Figure 17.10. Generic First-Order Reaction Kinetics Plot 

This graph shows repeating half-lives on a kinetics plot of a generic first-order reaction.