What half-life means for a first-order reaction
Half-life is the time it takes for the concentration of a reactant to fall to half of its starting value. For a first-order process, such as radioactive decay or the breakdown of many drugs and reagents, this time is a constant: it takes the same number of seconds to go from 100% to 50% as from 50% to 25%. That property makes half-life a convenient way to describe how fast something disappears without quoting a concentration.
Use this calculator when you already know the first-order rate constant k, for example from a fitted concentration-versus-time plot or from a data table, and you want the half-life in seconds. It is handy for planning sampling times in kinetics experiments, checking lab-report answers, or judging how long a reagent solution will stay usable. It does not apply to zero-order or second-order reactions, whose half-lives depend on the starting concentration.
The formula
For a first-order reaction the concentration follows [A] = [A]0 e^(-kt). Setting [A] = [A]0 / 2 and solving gives:
t1/2 = ln 2 / k = 0.693 / k
- t1/2 is the half-life, in the same time unit as 1/k. The field is labeled s-1, so the answer is in seconds.
- k is the first-order rate constant. It must be greater than zero.
- 0.693 is the natural log of 2 (0.6931...) rounded to three digits. The calculator uses this rounded value, so results differ from the exact figure by about 0.02%.
Worked example
A first-order decomposition has k = 0.0231 s-1. The half-life is 0.693 / 0.0231 = 30.0 s, and the calculator displays 30.0000 s. To check it, note that after 30 s the fraction remaining is e^(-0.0231 x 30) = e^(-0.693), which is 0.500. After 60 s (two half-lives) one quarter remains, and after 90 s one eighth.
Common mistakes and how to interpret the result
- Applying the formula to the wrong order. For second-order reactions t1/2 = 1/(k[A]0), and for zero-order t1/2 = [A]0/(2k). Using 0.693/k for those gives wrong answers.
- Mixing time units. If your k is per minute or per hour, the half-life comes out in minutes or hours even though the label says seconds. Convert first if you need seconds.
- Entering the rate constant of the wrong step. Complex mechanisms have several constants; only the overall first-order k gives a true half-life.
- Expecting the reaction to finish after two half-lives. After n half-lives the fraction left is (1/2)^n, so it never reaches exactly zero; after 10 half-lives about 0.1% remains.