Radioactive Decay & Elimination Half-Life Calculator
Solve any half-life problem: find the amount remaining, the half-life, the time elapsed or the starting amount. Convert between half-life, decay constant and mean life. Enter amounts as numbers, percentages or fractions and get the full working, a decay table and a graph.
What Is Half-Life?
Half-life (t½) is the time it takes for half of a substance to decay or disappear. After one half-life, 50% is left. After another, half of that, or 25%, is left. The amount keeps halving at the same steady pace.
The term comes from radioactive decay, where each unstable nucleus has the same chance of decaying in any given second. Across billions of atoms, that fixed chance produces a very regular half-life. Iodine-131 halves every 8.02 days, whether you start with a microgram or a kilogram.
The same maths describes other first-order processes: a drug leaving the bloodstream, caffeine wearing off, a capacitor discharging, or the fizz leaving a drink. The calculator works for all of them, as long as the rate of loss is proportional to the amount left.
How to Use a Half-Life Calculator
- Choose what you want to find: remaining amount, half-life, time elapsed, initial amount, or the decay constant and mean life.
- Fill in the values you know. The field you are solving for is greyed out and marked “calculated”.
- Pick an isotope to fill in its half-life, or type your own value and choose its unit.
- Enter the remaining amount your way: as a number (25), a percentage (25%) or a fraction (1/4).
- Read the result. Under the answer you get the working with your numbers substituted in, a table of the first half-lives and a decay curve with your point marked.
Half-life and time can use different units. For example, a half-life in days and a time in hours are converted automatically.
How to Calculate Half-Life From Initial and Remaining Amount
If you know how much you started with, how much is left and how long it took, the half-life is:
Example: a sample drops from 80 mg to 5 mg in 12 hours.
- Ratio: 80 ÷ 5 = 16.
- Natural log: ln 16 = 2.7726.
- Half-life: 12 × 0.6931 ÷ 2.7726 = 3 hours.
Check: 16 = 2⁴, so four half-lives passed in 12 hours, which is 3 hours each. When the ratio is a power of 2 you can do it in your head. For any other ratio, use the formula or the calculator.
Half-Life Formula Explained
All half-life problems come from one equation, written in three equivalent ways:
- N₀ is the amount at the start (t = 0).
- N is the amount left after time t.
- t ÷ t½ is the number of half-lives that have passed. It does not need to be a whole number.
- (1/2) raised to that power is the fraction left.
Rearranging the first form gives every other formula on this page:
| To find | Formula |
|---|---|
| Remaining amount N | N₀ × (1/2)^(t ÷ t½) |
| Initial amount N₀ | N × 2^(t ÷ t½) |
| Half-life t½ | t × ln 2 ÷ ln(N₀ ÷ N) |
| Time elapsed t | t½ × log₂(N₀ ÷ N) = t½ × ln(N₀ ÷ N) ÷ ln 2 |
| Number of half-lives n | t ÷ t½ = log₂(N₀ ÷ N) |
| Decay constant λ | ln 2 ÷ t½ |
| Mean life τ | t½ ÷ ln 2 ≈ 1.4427 × t½ |
Swipe the table sideways to see every column.
How to Calculate the Amount Remaining After Several Half-Lives
Multiply the starting amount by one half, once for every half-life that passes:
| Half-lives (n) | Fraction left | % left | % decayed |
|---|---|---|---|
| 0 | 1 | 100% | 0% |
| 1 | 1/2 | 50% | 50% |
| 2 | 1/4 | 25% | 75% |
| 3 | 1/8 | 12.5% | 87.5% |
| 4 | 1/16 | 6.25% | 93.75% |
| 5 | 1/32 | 3.125% | 96.875% |
| 7 | 1/128 | 0.78% | 99.22% |
| 10 | 1/1024 | 0.098% | 99.90% |
Swipe the table sideways to see every column.
Example: 50 mg of iodine-131 after 7 days. n = 7 ÷ 8.02 = 0.873 half-lives, so N = 50 × (1/2)^0.873 = 27.3 mg. Less than one half-life has passed, so more than half is left, which is a quick way to sanity-check an answer.
How to Calculate the Number of Half-Lives That Have Passed
There are two ways, depending on what you know:
- 30 years of caesium-137 (t½ = 30.08 years) is 30 ÷ 30.08 = 0.997, just under one half-life.
- If 1/64 of a sample is left, n = log₂ 64 = 6 half-lives.
- If 10% is left, n = log₂ 10 = 3.32 half-lives. Not every answer is a whole number.
Knowing n first makes the rest easy: time is n × t½, and the fraction left is (1/2)ⁿ.
How to Calculate Time From Half-Life and Remaining Amount
This is the question behind radiocarbon dating: how long ago did this start decaying?
Example: a piece of wood has 30% of the carbon-14 found in living trees. The half-life is 5,730 years.
- N₀ ÷ N = 100 ÷ 30 = 3.333.
- ln 3.333 = 1.204.
- t = 5,730 × 1.204 ÷ 0.6931 = about 9,950 years.
Real radiocarbon labs also adjust for past changes in atmospheric carbon-14 (calibration curves), so a lab date is more precise than this simple result. The calculation above is what textbooks and exams expect.
How to Calculate Half-Life From a Decay Constant
The half-life comes out in the time unit of λ. If λ is per second, t½ is in seconds.
Example: λ = 0.0866 per day. t½ = 0.6931 ÷ 0.0866 = 8.00 days, very close to iodine-131.
Going the other way, carbon-14 has λ = 0.6931 ÷ 5,730 = 1.21 × 10⁻⁴ per year, or 3.83 × 10⁻¹² per second. Use the “λ and mean life” tab to convert between the three quantities in any unit.
Half-Life and Decay Constant: What Is the Difference?
| Half-life (t½) | Decay constant (λ) | |
|---|---|---|
| Meaning | Time for half to decay | Fraction decaying per unit time (for small times) |
| Units | Time (s, days, years) | 1/time (s⁻¹, per year) |
| Fast decay | Short half-life | Large λ |
| Used in | Everyday descriptions, dating | Activity A = λN, differential equations |
| Link | t½ × λ = ln 2 ≈ 0.6931 | |
Swipe the table sideways to see every column.
They carry the same information. Half-life is easier to picture. The decay constant is handier in equations, for example the activity of a sample is A = λN, the number of decays per second.
Half-Life vs. Mean Life
Mean life (τ, tau) is the average time an atom survives before decaying. It is longer than the half-life, because a few atoms last a very long time and pull the average up.
- After one half-life, 50% is left.
- After one mean life, 1/e = 36.8% is left.
- Carbon-14: t½ = 5,730 years, τ = 8,267 years.
Physicists often quote τ for particles (the muon’s mean life is about 2.2 microseconds). Chemists and geologists usually quote t½.
Half-Life and Exponential Decay
Half-life only exists because decay is exponential: a fixed fraction is lost in each equal time step, not a fixed amount. That is why the curve drops steeply at first and then flattens out, and why it never quite reaches zero.
A quick comparison over 3 time steps, starting from 100:
| Step | Linear (lose 50 each step) | Exponential (lose half each step) |
|---|---|---|
| 0 | 100 | 100 |
| 1 | 50 | 50 |
| 2 | 0 | 25 |
| 3 | — | 12.5 |
Swipe the table sideways to see every column.
Any process where the rate of loss is proportional to the amount present behaves this way. That includes most drug elimination (first-order kinetics), light passing through an absorbing material and the cooling of a hot object towards room temperature.
Half-Life Calculation Examples
1. Remaining amount: iodine-131
100 g of I-131 (t½ = 8.02 days) after 24.06 days: n = 24.06 ÷ 8.02 = 3, so N = 100 × (1/2)³ = 12.5 g.
2. Initial amount: cobalt-60
6.25 g of Co-60 is left after 21.08 years (t½ = 5.27 years). n = 4, so N₀ = 6.25 × 2⁴ = 100 g.
3. Half-life from data
An unknown sample falls from 80 mg to 5 mg in 12 hours: t½ = 12 × ln 2 ÷ ln 16 = 3 hours.
4. Time elapsed: caesium-137 from Chernobyl
The 1986 accident released Cs-137 (t½ = 30.08 years). By 2026, 40 years later, the fraction left is (1/2)^(40 ÷ 30.08) = 39.8%. About 100 years after the accident it drops to roughly 10%.
5. Medical imaging: technetium-99m
Tc-99m (t½ = 6.01 hours) is used in hospital scans because it fades fast. After 24 hours only (1/2)^(24 ÷ 6.01) = 6.3% is left.
Half-Life Problems With Fractions and Percentages
Many exam questions give a fraction or a percentage instead of amounts. Turn it into N₀ ÷ N and take log₂:
| Question says | N₀ ÷ N | Half-lives n |
|---|---|---|
| “1/8 remains” | 8 | 3 |
| “1/32 remains” | 32 | 5 |
| “75% has decayed” | 4 (25% remains) | 2 |
| “10% remains” | 10 | 3.32 |
| “90% has decayed” | 10 | 3.32 |
| “1% remains” | 100 | 6.64 |
| “99.9% has decayed” | 1000 | 9.97 |
Swipe the table sideways to see every column.
Read carefully whether the question gives the amount remaining or the amount decayed. “75% decayed” means 25% remains. The calculator accepts “25%” or “1/4” directly in the remaining box, so you can type the question as it is written after that one conversion.
Half-Life Calculation for Radioactive Isotopes
Half-lives range from fractions of a second to billions of years. These are built into the calculator:
| Isotope | Half-life | Where it is used or found |
|---|---|---|
| Fluorine-18 | 109.8 minutes | PET scans |
| Technetium-99m | 6.01 hours | Most common medical imaging tracer |
| Radon-222 | 3.82 days | Indoor radon gas from soil |
| Iodine-131 | 8.02 days | Thyroid treatment; nuclear fallout |
| Phosphorus-32 | 14.27 days | Biology labs, DNA labelling |
| Cobalt-60 | 5.27 years | Radiotherapy, sterilising equipment |
| Tritium (H-3) | 12.32 years | Glow-in-the-dark signs and watch dials |
| Strontium-90 | about 28.8 years | Fission product; bone-seeking |
| Caesium-137 | 30.08 years | Fission product; Chernobyl and Fukushima |
| Radium-226 | 1,600 years | Historic luminous paint |
| Carbon-14 | 5,730 years | Radiocarbon dating up to about 50,000 years |
| Plutonium-239 | 24,110 years | Nuclear fuel and weapons |
| Uranium-235 | 704 million years | Nuclear reactor fuel |
| Potassium-40 | 1.25 billion years | Potassium–argon dating of rocks |
| Uranium-238 | 4.468 billion years | Dating the oldest rocks |
Swipe the table sideways to see every column.
Carbon-14’s half-life is quoted as 5,730 years in most textbooks. The latest nuclear data gives about 5,700 years; radiocarbon labs still report ages using the older 5,568-year “Libby” value by convention. Check which value your course uses.
Uranium-238’s half-life is close to the age of the Earth, about 4.54 billion years. So roughly half of the U-238 the planet started with is still here.
Common Half-Life Calculation Mistakes
- Subtracting instead of halving. A sample does not lose 50% of the original each half-life. It loses half of what is left, so after 2 half-lives 25% remains, not 0%.
- Mixing time units. A half-life in days and a time in hours must be converted to the same unit before you divide.
- Confusing decayed and remaining. “80% decayed” means 20% remains.
- Assuming n must be a whole number. 1.5 half-lives leaves 35.4%, not the 37.5% you get by averaging 50% and 25%.
- Getting the ratio upside down. Use ln(N₀ ÷ N), not ln(N ÷ N₀). The second gives a negative number and a negative half-life.
- Rounding λ too early. Rounding 0.000121 to 0.0001 changes a carbon-14 date by about 20%. Keep extra digits until the last step.
- Treating 5 half-lives as “all gone”. 3.1% is still left. That may matter for radiation or medicine.
Half-Life Calculator FAQs
The amount left after time t is N = N₀ × (1/2)^(t / t½). Rearranged to find the half-life: t½ = t × ln 2 ÷ ln(N₀ / N). N₀ is the starting amount, N is the amount left, t is the time that has passed and t½ is the half-life.
Divide the initial amount by the remaining amount, take the natural log, and divide ln 2 by it. Then multiply by the time. Example: 80 mg falls to 5 mg in 12 hours. 80 ÷ 5 = 16, ln 16 = 2.773, and 12 × 0.693 ÷ 2.773 = 3 hours.
One eighth, or 12.5%. Each half-life halves what is left: 100% → 50% → 25% → 12.5%. After 4 half-lives 6.25% is left, after 5 half-lives 3.125%, and after 10 half-lives about 0.1%.
They are inversely related: t½ = ln 2 ÷ λ ≈ 0.693 ÷ λ, and λ = 0.693 ÷ t½. A large decay constant means a short half-life. The decay constant has units of 1/time, such as per second or per year.
Half-life is the time for half the atoms to decay. Mean life (τ) is the average lifetime of one atom, τ = 1/λ = t½ ÷ 0.693, which is about 1.443 times the half-life. After one mean life, 36.8% of the original amount is left.
No. Radioactive half-life is a property of the isotope. 1 gram and 1 kilogram of iodine-131 both lose half their atoms in 8.02 days. Temperature, pressure and chemical form have essentially no effect on it.
Yes. Decay is continuous, so any time can be expressed as a number of half-lives. After 1.5 half-lives, (1/2)^1.5 = 35.4% is left, not 37.5%. The calculator handles any value.
In theory it never reaches exactly zero, because each half-life only halves what is left. In practice people use a cut-off: about 7 half-lives leaves under 1%, and about 10 half-lives leaves about 0.1%.