Half-life and radioactive decay calculator
Solve a half-life problem from three known values and get decayed amount, remaining percentage and number of half-lives.
Calculate remaining quantity, elapsed time, half-life or initial quantity with exponential decay. Leave exactly one quantity blank when reverse solving is supported.
Results
Calculation details
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How does this tool work?
Ideal radioactive decay is exponential: after every half-life, the expected number of undecayed nuclei is halved. This calculator can solve for any one of the four main quantities from the other three.
The governing relation is N(t)=N₀(1/2)^(t/T½). The same form also applies to activity when it is proportional to the number of undecayed nuclei.
When elapsed time or half-life is unknown, the equation is inverted with a base-2 logarithm.
N₀ and N(t) must use the same quantity unit, but that unit can represent mass, number of nuclei or a proportional activity.
How to use this tool
Choose the unknown quantity
Enter three of initial quantity N₀, half-life T½, elapsed time t and remaining quantity N(t). Leave only the quantity to solve for blank.
Keep quantity and time units coherent
N₀ and N(t) must represent the same quantity in the same unit. Time and half-life can use a supported time unit as long as the conversion is consistent.
Apply exponential decay
The engine uses N(t)=N₀(1/2)^(t/T½). If elapsed time or half-life is unknown, the relation is inverted logarithmically.
Interpret remaining fraction and half-lives elapsed
Results also include decayed amount, remaining percentage and t/T½, which shows how many equivalent halvings have passed.
Good to know
Half-life is not the lifetime of one nucleus
Radioactive decay is statistical. T½ is the time in which the expected number of undecayed nuclei in a large population is halved; it cannot predict the exact decay time of an individual nucleus.
Fractional numbers of half-lives are valid
The factor (1/2)^(t/T½) works for any positive ratio t/T½. A duration of 2.5 half-lives produces an intermediate remaining fraction rather than requiring an integer number of halvings.
Activity and number of undecayed nuclei share the same relative law
For a given radionuclide with fixed decay constant, activity is proportional to the number of undecayed nuclei. Their relative decrease therefore follows the same exponential factor when units are kept consistent.
Formula
Exponential decay links the remaining fraction to the number of half-lives elapsed. The decay constant λ and half-life are related by λ = ln(2)/T½.
Decay law
N(t) = N₀(1/2)^(t/T½)
Each half-life halves the expected remaining quantity.
- N₀
- initial quantity
- N(t)
- remaining quantity
- T½
- half-life
Assumes a constant decay probability for the radionuclide and does not model radioactive daughter chains.
Use cases
Radioactivity homework
Check standard calculations involving an initial amount, a known half-life and an elapsed time. Counting half-lives first provides an intuitive estimate before evaluating the exponential expression.
Solving for elapsed time
Given an initial and measured remaining amount, determine how much time corresponds to that reduction when the radionuclide half-life is known.
Comparing radionuclides
Explore how short and long half-lives change the remaining fraction over the same time interval and separate decay rate from the chosen initial amount.
Learning exponential decay
Use half-life as a concrete exponential model: 50% remains after one half-life, 25% after two and 12.5% after three, while the continuous model never reaches exactly zero.
Examples
Three half-lives
15 years equals three half-lives.
Input
N₀ = 100; T½ = 5 years; t = 15 yearsResult
N = 12.5; 12.5% remainsSolve for time
The quantity has halved twice.
Input
N₀ = 80; N = 20; T½ = 10 daysResult
t = 20 daysTips and best practices
- Use the same unit for initial and remaining quantity.
- Half-life is a statistical property of a radionuclide, not the lifetime of one nucleus.
- The model assumes a constant decay rate.
- For extremely large or small quantities, use consistent scientific notation to avoid misplaced zeros.
- Do not mix a half-life in years with an elapsed time interpreted as days unless the time units are converted consistently.
Sources and references
- OpenStax, 22.3 Half Life and Radiometric Dating (opens in a new tab), PhysicsPage in English
- BIPM, The International System of Units (SI Brochure) (opens in a new tab), Bureau international des poids et mesuresPage in English
Frequently asked questions
What remains after one half-life?
On average, half of the initial quantity remains.
What remains after two half-lives?
One quarter, or 25%.
Can the calculator solve for time?
Yes. Leave elapsed time blank and provide N₀, N and T½.
Must the quantity be in grams?
No. Any consistent quantity unit works when N₀ and N use the same unit.
Does the amount ever become exactly zero?
Not in the continuous exponential model; it approaches zero asymptotically. For small actual populations, the discrete statistical nature of nuclei matters.
How are half-life and decay constant related?
They describe the same decay rate in different forms: λ = ln(2)/T½.
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