Calculator Inputs
Choose the value to solve, then enter the other known values. Keep elapsed time, half-life, and decay constant in the same selected time unit.
Example Data Table
| Scenario | Initial Amount | Half-Life | Elapsed Time | Decay Constant | Remaining Amount | Remaining % |
|---|---|---|---|---|---|---|
| Sample A | 1000 grams | 5 years | 12 years | 0.138629 per year | 189.465 grams | 18.9465% |
| Sample B | 500 atoms | 8.6643 days | 20 days | 0.08 per day | 100.948 atoms | 20.1897% |
| Sample C | 300 mg | 5 hours | 10 hours | 0.138629 per hour | 75 mg | 25% |
Formula Used
Core Decay Equation
N(t) = N0 × e−λt
N(t) is the remaining amount, N0 is the initial amount, λ is the decay constant, and t is elapsed time.
Half-Life Relationship
T1/2 = ln(2) / λ
You can convert between half-life and decay constant whenever both use the same time unit.
Solving for Time
t = ln(N0 / N(t)) / λ
This rearrangement finds the elapsed time needed for a quantity to decay from its initial amount to its remaining amount.
Activity and Mean Life
A = λN and Mean Life = 1 / λ
Activity is proportional to the current amount, while mean life gives the average lifetime in the same chosen time unit.
How to Use This Calculator
- Select the variable you want to solve from the Solve For menu.
- Choose one time unit and keep elapsed time, half-life, and decay constant consistent with it.
- Enter the known amount values and add either decay constant or half-life when required.
- Set your preferred precision and graph detail, then press Solve Equation.
- Review the result panel above the form, inspect the formula steps, and study the chart trend.
- Use the CSV or PDF buttons to save the result summary for reports, notes, or classroom work.
FAQs
1) What does this calculator solve?
It solves remaining amount, initial amount, decay constant, elapsed time, or half-life when the other required values are known. It also derives percentages, mean life, and activity values.
2) Do all time values need the same unit?
Yes. Elapsed time, half-life, and decay constant must match the selected time unit. If you choose years, interpret λ as per year and enter every time value in years.
3) Can I use grams, atoms, or moles?
Yes. The amount field is unit-agnostic. Enter any consistent quantity, label it clearly, and the solver applies that same label to initial and remaining amounts.
4) Why can’t remaining amount exceed initial amount?
For ordinary decay with positive time, the remaining amount cannot be larger than the starting amount. A larger value suggests growth, measurement error, or inconsistent inputs.
5) What is the decay constant?
The decay constant, λ, measures how quickly the quantity shrinks per selected time unit. Larger λ values mean faster decay and a shorter half-life.
6) What does an infinite half-life mean?
It appears when the decay constant is zero. Mathematically, the amount stays unchanged over time, so the model behaves like a stable substance.
7) Why does the graph curve downward?
Radioactive decay is exponential. The amount falls quickly at first in absolute terms, then the drop becomes smaller as less material remains to decay.
8) Does the amount ever reach zero exactly?
Not in the continuous exponential model. The amount approaches zero asymptotically, which is why finite inputs cannot solve a time for exactly zero remaining quantity.