After two half-lives of carbon-14, 11,460 years, a quarter of the original remains: 100 becomes 25. Carbon-14's half-life is 5,730 years, and each half-life halves what is left.
What this calculates
Choose what you want to find, then enter the other two of the initial amount, the amount remaining and the half-life or time. This works out the third, with the number of half-lives, the decay constant and the mean life, a table of the first ten half-lives and a chart of the decay. The half-life and the time can be in different units.
The formula
Remaining
remaining = initial × (1/2)^(time ÷ half-life)Half-life
half-life = time × ln 2 ÷ ln(initial ÷ remaining)Time
time = half-life × ln(initial ÷ remaining) ÷ ln 2Constants
λ = ln 2 ÷ half-life; mean life = half-life ÷ ln 2The amounts can be in any unit - grams, atoms or becquerels - and the answer is in the same one.
Worked example
After one half-life 50 is left, after three (17,190 years) 12.5. Going the other way, taking 100 down to 10 needs log₂(10) = 3.32193 half-lives, or about 19,034.6 years.
When to use it, and the mistakes to avoid
Use it for radiometric dating, nuclear medicine doses, how long a sample stays hazardous, or a chemistry or physics problem on decay.
The mistakes that cost the most:
- Mixing units. A half-life in hours and a time in days must be converted; the calculator does it, but check the units you chose.
- Expecting it to reach zero. The amount halves forever and never quite empties.
- Treating a half-life as half the lifetime. Half is gone after one half-life; a quarter is still there after two.
- Applying it to one atom. Decay is random; the half-life describes a large sample.
- Using the wrong half-life. Different isotopes of one element differ enormously.
FAQ
What is a half-life?
The time it takes for half of a radioactive substance to decay. After one half-life 50% is left, after two 25%, after three 12.5%, and so on, halving each time. It does not depend on how much you start with.
How do I calculate how much is left after some time?
Divide the time by the half-life to get the number of half-lives, then halve the starting amount that many times: remaining = initial × (1/2)^(time ÷ half-life). The number of half-lives need not be a whole number.
How do I find the half-life from a measurement?
Measure the starting and remaining amounts and the time between them. The number of half-lives is ln(initial ÷ remaining) ÷ ln 2, and the half-life is the time divided by that. Switch to the half-life mode and enter the three values.
Can I use different units for the half-life and the time?
Yes. Each has its own unit, from seconds to years, and they are converted to seconds before dividing. A year is the average calendar year of 365.2425 days.
What are the decay constant and the mean life?
The decay constant λ is the chance per unit of time that an atom decays, ln 2 ÷ half-life. The mean life is its reciprocal, half-life ÷ ln 2: the average time an atom survives, about 1.44 times the half-life.