Half-Life: formula
Enter a nonnegative initial amount and elapsed time, plus a positive half-life in the same time unit. The amount unit can be mass, activity or count when the chosen constant-rate model applies. A biological drug-elimination estimate needs its own validated context.
Half-Life: worked example
| Starting amount | 100 |
|---|---|
| Half-life | 5 |
| Elapsed time | 12 |
| Amount remaining | 18.9 |
| Share remaining | 18.9% |
| Decay constant (per time unit) | 0.139 |
Real drugs are cleared by more complex processes, and radioactive decay is random for small samples. The formula applies to exponential decay only. Different isotopes have half-lives from fractions of a second to billions of years.
Safety limits are set by regulators, not by this calculation.
Half-Life: changing Starting amount
| Starting amount | Amount remaining | Share remaining | Decay constant (per time unit) |
|---|---|---|---|
| 80 | 15.2 | 18.9% | 0.139 |
| 90 | 17.1 | 18.9% | 0.139 |
| 100 (example) | 18.9 | 18.9% | 0.139 |
| 110 | 20.8 | 18.9% | 0.139 |
| 120 | 22.7 | 18.9% | 0.139 |
Half-Life: compare input ranges
| Input | Tested from | To | Amount remaining: low input | Amount remaining: high input | Swing |
|---|---|---|---|---|---|
| Half-life | 4.5 | 5.5 | 15.7 | 22 | 6.3 (33%) |
| Elapsed time | 11 | 13 | 21.8 | 16.5 | 5.3 (28%) |
| Starting amount | 90 | 110 | 17.1 | 20.8 | 3.8 (20%) |
Half-Life: individual input comparisons
| Input | Value | Amount remaining | Share remaining | Decay constant (per time unit) |
|---|---|---|---|---|
| Starting amount | 80 | 15.2 | 18.9% | 0.139 |
| 90 | 17.1 | 18.9% | 0.139 | |
| 100 | 18.9 | 18.9% | 0.139 | |
| 110 | 20.8 | 18.9% | 0.139 | |
| 120 | 22.7 | 18.9% | 0.139 | |
| Half-life | 4 | 12.5 | 12.5% | 0.173 |
| 4.5 | 15.7 | 15.7% | 0.154 | |
| 5 | 18.9 | 18.9% | 0.139 | |
| 5.5 | 22 | 22.0% | 0.126 | |
| 6 | 25 | 25.0% | 0.116 | |
| Elapsed time | 10 | 25 | 25.0% | 0.139 |
| 11 | 21.8 | 21.8% | 0.139 | |
| 12 | 18.9 | 18.9% | 0.139 | |
| 13 | 16.5 | 16.5% | 0.139 | |
| 14 | 14.4 | 14.4% | 0.139 |
Half-Life: combined input changes
| Starting amount / Half-life | 4 | 5 | 6 |
|---|---|---|---|
| 80 | 10 | 15.2 | 20 |
| 90 | 11.3 | 17.1 | 22.5 |
| 100 | 12.5 | 18.9 | 25 |
| 110 | 13.8 | 20.8 | 27.5 |
| 120 | 15 | 22.7 | 30 |
Half-Life: which assumptions move the result?
Half-Life: values and definitions
| Quantity and definition | Value used |
|---|---|
| Starting amount | 100 |
| Half-life — Any time unit, as long as elapsed time uses the same one. | 5 |
| Elapsed time | 12 |
| Amount remaining | 18.9 |
| Share remaining | 18.9% |
| Decay constant (per time unit) | 0.139 |
Half-Life: half and double Half-life
| Half-life | Amount remaining |
|---|---|
| 2.5 | 3.6 |
| 5 | 18.9 |
| 10 | 43.5 |
Common error: Assuming the quantity is gone after two half-lives. Half of it remains after one, a quarter after two, and it never quite reaches zero.
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Half-life is the time for half of a decaying quantity to disappear, and the same half is lost in each equal interval. With a five year half-life, after twelve years a starting amount of 100 has fallen to about 19. The decay constant output gives the continuous rate, which is what you need when the time is not a clean multiple of the half-life.
Give the half-life and the elapsed time in the same unit. A half-life in years and an elapsed time in months gives a result off by a factor of twelve.
See growth in the opposite direction on the exponential growth calculator, logarithms on the logarithm page, and doubling time on the rule of 72 calculator. The percentage and percent difference pages help with shares, and the scientific notation calculator handles very small amounts.
What is a half-life?
The time it takes for half of a quantity of a decaying substance to be gone, which stays constant however much you start with.
How much remains after 3 half-lives?
One eighth, or 12.5%, since the amount halves three times.
How is half-life used in medicine?
To estimate how long a drug stays active in the body and how often it needs to be dosed.