Answer first
What this calculator tells you
Calculate the amount remaining, the share remaining and the decay constant after a given time. Track how much of a decaying quantity is left after a time you choose. Formula: Remaining = initial × (½)^(elapsed ÷ half-life); decay constant λ = ln 2 ÷ half-life. At the worked-example inputs, the amount remaining is 18.9. Holding every other input steady, moving starting amount from 80 to 120 moves the result from 15.2 to 22.7.
Transparent method
The formula
Track how much of a decaying quantity is left after a time you choose.
Worked example
Example inputs
How to interpret the result
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.
At the worked-example inputs the amount remaining is 18.9. It rises with half-life and starting amount and falls as elapsed time increases.
These are exact physics and chemistry formulas. Real-world results add tolerances from component quality, temperature and measurement error that this calculator does not model.
Before you rely on it
What to check
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.
The common error
Where people go wrong with half-life calculator
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.
Sensitivity evidence
How starting amount changes the amount remaining
Holding every other input at the worked-example value, moving starting amount from 80 to 120 moves the amount remaining from 15.2 to 22.7: a spread of 7.6, or 40% of the worked-example result.
| 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 |
| 100worked example | 18.9 | 18.9% | 0.139 |
| 110 | 20.8 | 18.9% | 0.139 |
| 120 | 22.7 | 18.9% | 0.139 |
Every input, tested
Which input moves the amount remaining most
Of the 3 inputs, half-life moves the amount remaining most (6.3 across the range tested) and starting amount moves it least (3.8).
| Input | Tested from | To | Amount remaining at each end | Swing |
|---|---|---|---|---|
| Half-life | 4.5 | 5.5 | 15.7 to 22 | 6.3 (33%) |
| Elapsed time | 11 | 13 | 21.8 to 16.5 | 5.3 (28%) |
| Starting amount | 90 | 110 | 17.1 to 20.8 | 3.8 (20%) |
Two variables at once
Amount remaining by starting amount and half-life
Across the grid the amount remaining runs from 10 to 30. Moving starting amount from 80 to 120 shifts it by 7.6 at the middle column, and moving half-life from 4 to 6 shifts it by 12.5 at the middle row, so half-life is the bigger lever here.
| 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 |
The highlighted cell is the worked example: 18.9.
Step by step
The worked example, input by input
| Input | Value used | What it means |
|---|---|---|
| Starting amount | 100 | Enter the starting amount used in this calculation. |
| Half-life | 5 | Any time unit, as long as elapsed time uses the same one. |
| Elapsed time | 12 | Enter the elapsed time used in this calculation. |
| Amount remaining | 18.9 | |
| Share remaining | 18.9% | |
| Decay constant (per time unit) | 0.139 | |
Inputs, definitions and assumptions
Starting amount
Enter the starting amount used in this calculation. The prefilled worked-example value is 100.
Half-life
Any time unit, as long as elapsed time uses the same one. The prefilled worked-example value is 5.
Elapsed time
Enter the elapsed time used in this calculation. The prefilled worked-example value is 12.
How to use this calculator
- 1Verify the inputs. Gather starting amount, half-life and elapsed time from your own documents; the prefilled values are examples.
- 2Save a baseline. The worked example puts the amount remaining at 18.9. Store your own version of it as Scenario A.
- 3Test one change. Start with half-life, the input with the biggest effect here: moving half-life from 4.5 to 5.5 takes the amount remaining from 15.7 to 22, a swing of 33% of the worked-example figure.
- 4Check the extremes. At half the example half-life (2.5) the amount remaining is 3.6; at double (10) it is 43.5.
People also ask
Frequently asked questions
How do you calculate half-life?
Remaining = initial × (½)^(elapsed ÷ half-life); decay constant λ = ln 2 ÷ half-life. At the worked-example inputs the amount remaining is 18.9.
What does the half-life result mean?
Track how much of a decaying quantity is left after a time you choose. At the worked-example inputs the amount remaining is 18.9. It rises with half-life and starting amount and falls as elapsed time increases.
How much does starting amount change the amount remaining?
Holding every other input at the worked-example value, moving starting amount from 80 to 120 moves the amount remaining from 15.2 to 22.7, a spread of 7.6.
What are the limits of this half-life calculator?
These are exact physics and chemistry formulas. Real-world results add tolerances from component quality, temperature and measurement error that this calculator does not model. The tables on this page test starting amount only from 80 to 120; a value outside that range is not tabulated here.
Which input moves the amount remaining most in the half-life calculator?
Ranked by how far each moves the amount remaining across the range tested: half-life (6.3, 33%), elapsed time (5.3, 28%) and starting amount (3.8, 20%).
If I double half-life in the half-life calculator, does the amount remaining double?
Doubling it from 5 to 10 takes the amount remaining from 18.9 to 43.5, which is 2.30 times the worked-example figure. So the result grows faster than the input does. Halving it to 2.5 gives 3.6.
How much does half-life matter in the half-life calculator?
The worked example uses 5. Holding every other input at its worked-example value, moving half-life from 4.5 to 5.5 takes the amount remaining from 15.7 to 22, a swing of 33% of the worked-example figure.
How much does elapsed time matter in the half-life calculator?
The worked example uses 12. With the other inputs left at the worked example, moving elapsed time from 11 to 13 takes the amount remaining from 21.8 to 16.5, a swing of 28% of the worked-example figure.
Which inputs change the share remaining in the half-life calculator?
At the worked-example inputs it is 18.9%. Half-life takes it from 15.7% to 22.0% and elapsed time takes it from 21.8% to 16.5%.
Which inputs change the decay constant (per time unit) in the half-life calculator?
At the worked-example inputs it is 0.139. Half-life takes it from 0.154 to 0.126.
Does Ohm's law apply to every electrical component?
Only to purely resistive, linear components: a plain resistor, for instance. Components like diodes, transistors and many real-world loads have a voltage-current relationship that isn't a straight line, so Ohm's law doesn't directly describe them.
Does the combined gas law work if gas escapes?
No. It assumes a fixed amount of gas. If gas is added or lost, the ideal gas law with the amount of gas as a variable is the right tool.
Sources and evidence
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