How to use the half-life calculator
- Choose what to solve for: the amount remaining, the elapsed time, the half-life or the initial amount.
- Enter the other three values. The unit of the amount does not matter, but time and half-life must use the same unit.
- Pick the time unit so the labels read correctly.
- Read the answer, the decay constant, the mean lifetime and the table of amounts after whole half-lives.
Formula
N(t) = N0 x (1/2)^(t / T)
t = T x log2(N0 / N(t))
T = t / log2(N0 / N(t))
Decay constant = ln 2 / T, mean lifetime = T / ln 2
N0 is the starting amount, N(t) the amount left after time t, and T the half-life. All the formulas come from the same exponential decay law.
Worked example
Carbon-14 has a half-life of about 5,730 years. A sample that starts at 100 and ages 11,460 years has gone through exactly two half-lives, so 25 remains.
Run backward, going from 100 to 25 with a half-life of 5,730 years takes 5,730 x log2(4) = 11,460 years. With the same numbers the decay constant is 0.693147 / 5,730 = 0.00012097 per year, and the mean lifetime is about 8,266.6 years.
What half-life means
The half-life is the time for half of a quantity to decay. After one half-life, half remains, after two a quarter, after three an eighth. The rate of decay is proportional to the amount present, which gives an exponential curve. The half-life stays the same no matter how much you start with.
Other uses of the same math
The decay law describes radioactive isotopes, and it is also used for the elimination of some drugs from the body and for the discharge of capacitors, where the time constant plays the role of the mean lifetime. The tool treats the amount as a plain number, so any unit works. For percent changes between two values see the percent change calculator.
Limits
Real decay is random, so small samples fluctuate around the smooth curve. A substance may also have several decay paths. For very large or small numbers the scientific notation converter can help read results.
Assumptions and limits
- Decay is smooth exponential decay with a single half-life.
- Time and half-life use the same unit, which you choose.
- The amount remaining cannot exceed the initial amount.
- The 5,730 year figure for carbon-14 in the example is a widely published value, used only to illustrate the math.
Frequently asked questions
How do you calculate half-life?
Divide the elapsed time by log base 2 of the ratio of the starting amount to the amount left. This tool does it for you.
How much is left after 3 half-lives?
One eighth, or 12.5 percent. Each half-life cuts the amount in half.
What is the decay constant?
It is the fraction that decays per unit of time in the continuous model. It equals the natural log of 2 divided by the half-life.
Can the half-life change over time?
For first order decay, no. The half-life is constant, which is what makes it useful for dating and for planning.