Calculator inputs
Calculation history
Recent results are stored only in this browser.
- No saved calculations yet.
Formula used
N = N₀ × (1/2)t/T
N = N₀e−λt
λ = ln(2) ÷ T
t = T × ln(N₀/N) ÷ ln(2)
T = t × ln(2) ÷ ln(N₀/N)
N is the remaining quantity. N₀ is the initial quantity. t is elapsed time, T is half-life, and λ is the decay constant.
How to use this calculator
- Select the unknown value you want calculated.
- Enter the required quantities and choose matching units.
- Select precision, notation, table periods, and graph scale.
- Press Calculate to view results, steps, tables, and graphs.
- Copy, print, share, or download the completed calculation.
Worked example
A sample begins with 100 units. Its half-life is 10 days. After 20 days, two half-lives have elapsed.
The remaining quantity is 25 units. Therefore, 75 units have decayed. The remaining percentage is exactly 25 percent.
| Initial | Half-life | Elapsed | Remaining | Decayed |
|---|---|---|---|---|
| 100 units | 10 days | 20 days | 25 units | 75 units |
Understanding half-life
Half-life and mean lifetime
Half-life measures when half a population remains. Mean lifetime is the average survival duration. For exponential decay, mean lifetime equals 1 divided by λ.
Physical and biological half-life
Physical half-life describes spontaneous nuclear or chemical decay. Biological half-life describes removal by living systems. Effective half-life combines both processes mathematically.
Accuracy and rounding
Displayed answers follow your selected decimal precision. Internal calculations retain greater numerical precision. Preset half-lives are approximate reference values.
This calculator supports education and estimation. It does not replace laboratory, medical, pharmaceutical, or regulatory guidance.
Frequently asked questions
What does half-life mean?
It is the time required for half the current quantity to decay or leave a system.
Can elapsed time use different units?
Yes. The calculator converts compatible time units before applying the formula.
Can I calculate the original quantity?
Yes. Choose the initial quantity mode and enter the remaining quantity, time, and half-life.
What is the decay constant?
The decay constant λ is the proportional decay rate and equals ln(2) divided by half-life.
Why cannot remaining quantity exceed initial quantity?
A simple decay model only decreases quantity. Growth or replenishment requires another mathematical model.
Does half-life depend on starting quantity?
No. In an ideal exponential model, half-life remains constant regardless of the starting amount.
Can the calculator use percentages?
Yes. Select percentage as the quantity unit or use the percentage remaining calculation mode.
What does scientific notation do?
It displays very large or small values using powers of ten for easier reading.
Are months and years exact conversions?
They use average astronomical durations. Calendar-specific dates may produce slightly different results.
Can I create a custom time unit?
Yes. Enter its label and define the number of seconds represented by one custom unit.
Can I save the results?
Yes. Use CSV, PDF, print, copy, or the shareable URL controls.