AP® Chemistry · Unit 5 · Topic 5.3
Half-Life
Half-life is the time needed for a reactant's concentration to drop by half. For a first order reaction this time is always the same, no matter how much you start with, because it depends only on the rate constant.
A first order reaction has rate = k[A], so the bigger the concentration, the faster it drops, but those two effects cancel out exactly. Solving the first order integrated rate law for the time it takes concentration to fall to half its starting value gives t1/2 = ln(2)/k. Concentration never appears in that formula. That is why going from 100 to 50 takes the same time as going from 50 to 25, or from 25 to 12.5: each step is a halving, and halving always takes the same amount of time when the rate constant is fixed. This is a direct fingerprint of first order behavior. If you plot concentration versus time for a first order reaction, you get a curve that keeps losing half its height over equal time intervals, which looks different from the straight-line drop of a zero order reaction or the ever-slowing decay of a second order one.
Second order reactions break this pattern. There, rate = k[A]2, and the integrated rate law gives t1/2 = 1/(k[A]_0), which depends on the starting concentration. A concentrated second order reaction has a short first half-life, but as concentration drops, each successive half-life takes longer. So constant half-life is not a general kinetics fact, it is a special signature of first order reactions, and checking whether successive half-lives are equal is one way to test reaction order from data.
Half-life connects directly to the integrated rate laws from earlier in the unit: it is just a specific case of solving for time when the concentration ratio is one half. It also sets up later ideas like radioactive decay and catalysis timing, where the same first order math reappears.
Key terms
- Half-life (t1/2)
- The time required for a reactant's concentration to fall to half its previous value.
- First order reaction
- A reaction whose rate depends on concentration raised to the first power, rate = k[A].
- Rate constant (k)
- The proportionality constant linking rate to concentration, with units that depend on reaction order.
- Integrated rate law
- An equation relating concentration directly to time, derived by integrating the rate law.
Watch it explained
- How it worksHow does the half-life of a first order reaction come out independent of concentration?
- Worked exampleFind the half-life of a first order reaction with a rate constant of 0.0231 per second.
- Worked exampleFind how much of a 4.00 g sample remains after 3 half-lives have gone by.
- Common mistakeWhy isn't half-life independent of concentration for a second order reaction?
Check yourself
End of topic.