How to Use This Half-Life Calculator
- Half-life and decay: choose what to solve for, then fill in the other three values. You can give the half-life or the rate constant k, and the result shows both, the number of half-lives, the percent remaining, and a decay curve.
- Integrated rate law: choose the reaction order and the unknown, then fill in the rest. The result includes the half-life at that starting concentration and the plot that would be a straight line.
Answers are rounded to the fewest significant figures in your inputs, and each step shows the numbers you entered.
The Half-Life Formula for First-Order Decay
The half-life, t1/2, is the time it takes for half of a reactant or radioactive sample to be used up. For a first-order process it never changes, so the amount left after n half-lives is
N = N0 × (1/2)n, where n = t ÷ t1/2
The half-life is tied to the rate constant by t1/2 = 0.693 ÷ k, the equation printed on the AP Chemistry equations sheet. The 0.693 is ln 2: setting [A]t = ½[A]0 in the first-order integrated rate law, ln[A]t − ln[A]0 = −kt, gives ln 2 = kt1/2.
Example: iodine-131 has a half-life of 8.02 days. After 24.06 days, n = 24.06 ÷ 8.02 = 3.00 half-lives, so an 80.0 mg sample has 80.0 × (1/2)3 = 10.0 mg left, or 12.5%.
How to Find a Half-Life From Two Measurements
- Take the natural log of the ratio of the two amounts: ln(N0 ÷ N).
- Divide by the time between them to get k.
- Divide 0.693 by k.
If a first-order reactant falls from 0.800 M to 0.100 M in 90.0 s: ln(0.800 ÷ 0.100) = 2.079, k = 2.079 ÷ 90.0 s = 0.0231 s−1, and t1/2 = 0.693 ÷ 0.0231 = 30.0 s. A quick check: 0.800 to 0.100 is three halvings (0.400, 0.200, 0.100), and 90.0 s ÷ 3 = 30.0 s. The same data set, with the plots that prove it is first order, is worked in AP Chemistry rate law.
Half-Life for Zero-, First-, and Second-Order Reactions
| Order | Integrated rate law | Straight-line plot | Half-life | Units of k |
|---|---|---|---|---|
| Zero | [A]t − [A]0 = −kt | [A] vs. t, slope −k | [A]0 ÷ 2k | M s−1 |
| First | ln[A]t − ln[A]0 = −kt | ln[A] vs. t, slope −k | 0.693 ÷ k | s−1 |
| Second | 1/[A]t − 1/[A]0 = kt | 1/[A] vs. t, slope +k | 1 ÷ (k[A]0) | M−1 s−1 |
The equations sheet prints all three integrated rate laws and the first-order half-life. Only the first-order half-life is constant: a zero-order half-life shrinks as the reactant runs down, and a second-order half-life grows. That is why the AP course treats half-life as the key quantity for first-order reactions. How to pick the order from a data table, and what each slope tells you, is covered step by step in AP Chemistry integrated rate laws.
Radioactive Decay and Carbon-14 Dating
Every radioactive isotope decays by first-order kinetics, which is why the AP course uses radioactive decay as its main illustration of a constant half-life. Carbon-14 has a half-life of about 5,730 years. A sample of old wood with 25.0% of the carbon-14 found in living wood has gone through two half-lives (100% → 50% → 25%), so it is about 2 × 5,730 = 11,460 years old, which rounds to 1.15 × 104 years at three significant figures. The calculator handles cases that are not whole numbers of half-lives, such as 30.0% remaining, using n = ln(N0 ÷ N) ÷ 0.693.
Kinetics is one unit of nine on the exam. To see how your practice scores across all of them translate into a 1–5 estimate, use the AP Chem Score Calculator.
Related Resources
- AP Chem Score Calculator
- AP Chemistry Integrated Rate Laws
- AP Chemistry Rate Law
- Unit 5 Review: Kinetics
- AP Chemistry Activation Energy
- Scientific Notation Calculator
- Sig Fig Calculator
Frequently Asked Questions
How do you calculate half-life?
For a first-order process, divide 0.693 by the rate constant: t1/2 = 0.693 / k. If you have two measurements instead, find k = ln(N0 / N) / t first. For example, 0.800 M falling to 0.100 M in 90.0 s gives k = 2.079 / 90.0 = 0.0231 per second and a half-life of 30.0 s.
What is the formula for the amount remaining after a number of half-lives?
N = N0 × (1/2)^n, where n = t / t1/2 is the number of half-lives that have passed. After 3 half-lives, (1/2)^3 = 1/8 of the sample remains, so 80.0 mg becomes 10.0 mg.
Is the half-life constant for every reaction?
No. Only first-order reactions, including radioactive decay, have a half-life that stays the same no matter how much reactant is left. For a zero-order reaction the half-life is [A]0 / 2k and gets shorter as the reactant is used up; for a second-order reaction it is 1 / (k[A]0) and gets longer.
How many half-lives until less than 1% of a sample is left?
Seven. After 6 half-lives, (1/2)^6 = 1.56% remains; after 7, 0.78% remains. Solving exactly, n = ln(100) / 0.693 = 6.64 half-lives brings the sample to 1.00%.
Is the half-life equation on the AP Chemistry equations sheet?
Yes. The kinetics section of the AP Chemistry equations sheet prints t1/2 = 0.693 / k along with the zero-, first-, and second-order integrated rate laws. The half-life expressions for zero- and second-order reactions are not printed, but they follow from setting [A]t equal to half of [A]0.
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