Half-Life Calculator
Use the half-life formula to find the amount left, the starting amount, the half-life or the time passed, with the decay constant, mean lifetime and steps.
Results
| Item | Value |
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Steps
| Step | Working |
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Decay by half-lives
| Half-lives | Time | Amount left | Percent left |
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Quick Answer
The half-life formula is N = N₀ × (1/2)^(t / T½). Carbon-14 has a half-life of about 5,730 years, so after 10,000 years, 100 g becomes 100 × (1/2)^(10,000 / 5,730) ≈ 29.83 g. After one half-life, half is left; after two, a quarter.
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Table of Contents
- Half-Life Calculator
- Quick Answer
- Related Calculators
- How This Half-Life Calculator Works
- How to Use the Half-Life Calculator
- Core Formulas
- Key Terms
- Worked Example
- Real-Life Examples
- Half-Life vs Mean Lifetime
- Amount Left After Each Half-Life
- Where Half-Life Is Used
- Half-Life Beyond Radioactivity
- Common Mistakes to Avoid
- Frequently Asked Questions
- References & Sources
How This Half-Life Calculator Works
Choose what to solve for, then enter the other three values: the starting amount, the amount remaining, the half-life and the time passed. The calculator uses the exponential decay formula, rearranged with logarithms when needed, and shows the decay constant, the mean lifetime and a table of how much is left after each half-life.
Quick explanation: each half-life cuts the amount in half.
- 1 half-life: 50% left.
- 2 half-lives: 25% left.
- 10 half-lives: under 0.1% left.
How to Use the Half-Life Calculator
- Choose the value to solve for.
- Enter the other three values.
- Use the same time unit for half-life and time.
- Press Calculate.
- Read the result, decay constant and the half-life table.
Core Formulas
- Amount left: N = N₀ × (1/2)t / T½
- Half-life: T½ = t × ln(1/2) ÷ ln(N / N₀)
- Time passed: t = T½ × ln(N / N₀) ÷ ln(1/2)
- Decay constant: λ = ln(2) ÷ T½
- Mean lifetime: τ = 1 ÷ λ = T½ ÷ ln(2)
Key Terms
- Half-life: the time for half of a substance to decay.
- Decay constant: λ, the rate of decay per unit time.
- Mean lifetime: the average time before an atom decays.
- Radioactive decay: the process of unstable atoms changing into other atoms.
- Exponential decay: a decrease by the same fraction in each equal time step.
Worked Example
A sample has 25% of its original carbon-14. How old is it?
- N / N₀ = 0.25, which is (1/2)²
- So 2 half-lives have passed
- Age ≈ 2 × 5,730 = 11,460 years
Real-Life Examples
- Dating: 40% of carbon-14 left means about 7,575 years old.
- Medicine: a drug with a 6-hour half-life drops to 25% after 12 hours.
- Iodine-131: with a half-life of about 8 days, 100 mg becomes about 7.4 mg after 30 days.
- Homework: check any decay problem step by step.
Half-Life vs Mean Lifetime
- Half-life: time for half of the amount to decay.
- Mean lifetime: the average time a single atom lasts, about 1.443 × the half-life.
- Decay constant: the fraction decaying per unit of time, ln(2) ÷ half-life.
Amount Left After Each Half-Life
| Half-lives | Fraction left | Percent left | Percent decayed |
|---|---|---|---|
| 0 | 1/1 | 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% |
| 6 | 1/64 | 1.5625% | 98.4375% |
| 7 | 1/128 | 0.78125% | 99.2188% |
| 8 | 1/256 | 0.390625% | 99.6094% |
| 9 | 1/512 | 0.195312% | 99.8047% |
| 10 | 1/1024 | 0.0976562% | 99.9023% |
Where Half-Life Is Used
- Radiocarbon dating: estimates the age of once-living things.
- Medicine: drug half-life tells how fast the body clears a medicine.
- Nuclear safety: shows how long radioactive material stays active.
- Chemistry: first-order reactions also follow a half-life pattern.
Half-Life Beyond Radioactivity
The same math describes many things that shrink by a steady fraction over time. In medicine, a drug's elimination half-life is the time for the amount in the blood to fall by half once the body is clearing it. Doctors use it to plan how often a medicine is taken. A common rule of thumb is that a drug is mostly cleared after about 4 to 5 half-lives, when roughly 94% to 97% is gone.
- Caffeine example: if a drink's caffeine had a 5-hour half-life, 200 mg would drop to 100 mg after 5 hours and 50 mg after 10 hours.
- Finance: the time for money to lose half its buying power to inflation follows the same pattern.
Always follow your doctor or pharmacist for real medicines, since half-lives vary from person to person.
Common Mistakes to Avoid
- Mixing time units. Use years with years, or hours with hours.
- Subtracting half each time from the original. Each half-life halves what is left, not the starting amount.
- Assuming it reaches zero. In theory, a tiny amount always remains.
- Using log of the wrong ratio. Use N ÷ N₀, which is less than 1.
- Applying it to small samples exactly. Decay is random for single atoms.
Frequently Asked Questions
N = N₀ × (1/2)^(t / T½).
T½ = t × ln(1/2) ÷ ln(N / N₀). Choose half-life in the calculator.
1/8, or 12.5%.
About 5,730 years.
t = T½ × ln(N / N₀) ÷ ln(1/2).
λ = ln(2) ÷ T½, the fraction that decays per time unit.
Mathematically, no. It just gets very small.
Yes, as long as the half-life and time use the same unit.
Yes. Many medicines are cleared from the body at a roughly constant fraction per hour.
The average time an atom survives: T½ ÷ ln(2).
After 7 half-lives less than 1% remains.
They undo the exponent so you can solve for time or half-life.
References & Sources
| Source | Used For | Link |
|---|---|---|
| Wikipedia, Half-life | Definition, formulas and examples. | https://en.wikipedia.org/wiki/Half-life |
| Wikipedia, Carbon-14 | The half-life of carbon-14 and its use in dating. | https://en.wikipedia.org/wiki/Carbon-14 |