Half-Life Calculator
Solve radioactive decay and first-order kinetics problems with example presets, a live decay visual, and step-by-step rearrangements.
What can you enter?
Enter any three values and leave exactly one blank. The calculator can solve the missing initial quantity, remaining quantity, elapsed time, or half-life, then it also reports the decay constant k.
| Input | Meaning | What you can type |
|---|---|---|
| N0 | Initial quantity before decay starts | 100 g, 1.0 mol/L, 5000 counts, 80 Bq, or 100% |
| N | Remaining quantity after time passes | 12.5 g, 0.125 mol/L, 625 counts, 10 Bq, or 25% |
| t | Elapsed time | Years, days, hours, minutes, or seconds |
| t_half | Time needed for half the current amount to remain | Use the same time unit as t |
| Time unit | Label for t, t_half, and k | k is shown as unit^-1, such as days^-1 |
Formula used
k = ln(2) / t_half
N(t) = N0 * e^(-k*t)
The key idea is fraction decay. After one half-life, 50% remains; after two half-lives, 25% remains; after three half-lives, 12.5% remains. The sample loses half of what is currently present, not the same fixed mass each time.
Worked example
Problem: A 100 g sample has a half-life of 5 years. How much remains after 15 years?
Number of half-lives = t / t_half = 15 / 5 = 3.
N = 100 * (1/2)^3 = 100 * 0.125 = 12.5 g. The decay constant is k = ln(2) / 5 = 0.1386 years^-1.
Built-in example data
Use the quick example dropdown to load common classroom setups. These examples are meant for calculation practice, so always use the exact half-life value your textbook, lab, or teacher provides.
| Example | N0 | N | t | t_half | What it teaches |
|---|---|---|---|---|---|
| 100 g sample | 100 | blank | 15 years | 5 years | Solves remaining amount after 3 half-lives |
| Carbon-14 style | 100 | blank | 11460 years | 5730 years | Shows two half-lives, so 25% remains |
| Iodine-131 style | 80 | 10 | blank | 8.02 days | Solves elapsed time from measured decay |
| First-order reaction | 1 | 0.125 | 36 hours | blank | Finds half-life from concentration change |
| Original sample | blank | 12.5 | 15 years | 5 years | Works backward to the initial amount |
Why this calculator is useful
| Use case | How half-life helps |
|---|---|
| Nuclear chemistry homework | Find remaining isotope mass, activity, or count rate after a known time. |
| Radiocarbon-style practice | Estimate elapsed time from the fraction of carbon-14 remaining. |
| First-order kinetics | Connect concentration data, half-life, and the rate constant k. |
| Lab data checking | See whether measured decay follows a constant half-life model. |
| Exam revision | Practice rearranging the same equation for different unknowns. |
How to read the output
- Remaining fraction is N / N0. A value of 0.125 means one-eighth remains.
- Percent remaining is the remaining fraction multiplied by 100.
- Half-lives elapsed is t / t_half. Whole numbers give familiar fractions: 1, 2, 3, 4 half-lives become 50%, 25%, 12.5%, 6.25%.
- Decay constant k is ln(2) / t_half. If t_half is in days, k is in days^-1.
- Decay visual shows where your time lands on the exponential curve and half-life strip.
Radioactive decay and first-order reactions
Radioactive decay is statistical: each nucleus has the same probability of decaying per unit time, so the sample follows an exponential curve. First-order chemical reactions use the same mathematical form because the reaction rate depends on the amount or concentration of one reactant.
For a first-order reaction, half-life is independent of starting concentration. That is why a concentration can drop from 1.00 M to 0.500 M in one half-life, then from 0.500 M to 0.250 M in the next half-life. The change in amount gets smaller, but the fraction lost stays the same.
Common mistakes
- Using t_half / t instead of t / t_half for the exponent.
- Mixing time units, such as entering t in hours and t_half in days.
- Subtracting the same quantity every half-life instead of halving the amount left.
- Using log base 10 in the decay constant equation instead of natural log.
- Trying to apply this first-order model to zero-order decay or multi-path biological clearance.
Rounding and checks
Keep extra digits for k when calculating over many half-lives. The remaining amount should be positive and should not exceed the initial amount for ordinary decay. If a setup gives negative time, a negative half-life, or a remaining amount larger than the starting amount, the problem is not describing simple decay with the entered values.
Related Chemistry Tools
Half-Life Calculator FAQs
How to calculate half-life in chemistry?
Half-life (t½) is the time required for half the atoms of a radioactive sample (or half the concentration of a reactant in a first-order reaction) to disappear. For a first-order process, t½ = 0.693/k, where k is the rate constant. The half-life is independent of the starting amount — a crucial property of first-order kinetics. After n half-lives, the fraction remaining is (½)n. For instance, after 3 half-lives, only 12.5% of the original sample remains. t½ = 0.693 / k (first-order)
What is the formula for calculating half-life?
For first-order radioactive decay or reactions, the formulae are: (i) N = N0 · (½)t/t½ for the amount remaining after time t; (ii) t½ = ln 2 / λ = 0.693/λ, where λ is the decay constant (same as k); and (iii) λ = 2.303/t × log(N0/N) from integrated rate law. Use whichever formula suits the variables you have, and remember to be consistent with units of time. N = N0 · (½)t/t½
How to calculate half-life of radioactive elements?
Take readings of activity (A, counts per minute) at different times. Plot ln A vs t — the slope gives −λ, then t½ = 0.693/λ. Alternatively, find the time taken for A to fall to A0/2 directly from a linear graph of A vs t. For very long-lived isotopes, like U-238 (t½ = 4.5 × 109 y), we measure the small present rate and use the equation A = λN to back-calculate λ, then t½.
How to calculate half-life of Uranium-235?
U-235 has an experimentally determined half-life of about 7.04 × 108 years (704 million years). It cannot be directly observed in laboratory time; instead, we measure its activity (A = λN) using a sensitive detector, knowing the number of atoms N from mass and Avogadro's number. From λ = A/N, we then compute t½ = 0.693/λ. Alternatively, geochronologists use isotope ratios in old rocks (Pb-207/U-235 method) to confirm this half-life — vital for radiometric dating of rocks and meteorites.
How to calculate the half-life of a radioactive isotope?
Measure the activity (A) of a known amount (N atoms) of the isotope. Compute the decay constant: λ = A/N (in s-1). Then t½ = 0.693/λ. Alternatively, follow the activity over time and use t½ = (t × 0.693)/(ln(A0/A)). For very short-lived isotopes, monitor activity decline directly; for long-lived isotopes, count present-day activity and use known atom count. The principle is always the same, just adapted to the isotope's lifetime.