Half-Life Calculator for Decay, Time and Activity
Calculate the quantity remaining after a known time, solve elapsed time or half-life from two amounts, recover the initial amount, convert between half-life and decay constant, or estimate activity from a radioactive-nuclei count. The tool keeps time units explicit and evaluates sensitive ratios in the logarithmic domain.
- Six calculation modes
- Nanoseconds to billion years
- Decay constant and mean lifetime
- Becquerel and curie conversions
Online Half-Life and Exponential Decay Calculator
Select one solve mode and enter only the active values. Amounts must use the same unit. Time values can use different units because the calculator normalizes them before applying the decay law.
The result, decay measures, unit conversions and calculation steps will appear here.
Decay Results and Conversions
| Quantity | Unit | Value |
|---|---|---|
| Result | — | — |
Normalized Decay Curve
Calculation Steps
- Select a solve mode and enter its active values.
- The calculator will normalize time units and apply the exponential decay relation.
How to Use This Half-Life Calculator
- Choose the quantity to solve. Select remaining amount, elapsed time, half-life, initial amount, decay constants or activity.
- Enter only positive quantities. Elapsed time can be zero in the remaining-amount and initial-amount modes, but half-life and decay constants must be positive.
- Keep amount units identical. Initial and remaining values must describe the same quantity in the same unit before they are compared.
- Select every time unit. Nanoseconds through billion years are converted to seconds using the stated unit definitions.
- Choose result units and precision. The selection changes display and conversion only; the calculator keeps unrounded working values.
- Press Calculate Half-Life Result. Review the main answer, remaining and decayed fractions, decay constant, mean lifetime, curve and substitution steps.
- Check the model assumptions. Confirm one constant exponential-decay process applies and obtain qualified review for laboratory, medical or radiation-safety work.
Half-Life and Exponential Decay Formula
Half-life is the time required for the expected number of undecayed nuclei, or another quantity following the same exponential law, to fall to one-half of its starting value. The remaining quantity is halved again during every equal half-life interval. The loss is therefore multiplicative, not a fixed subtraction.
N0 is the initial quantity, N(t) is the quantity remaining after elapsed time t, T1/2 is the half-life and λ is the decay constant. The dimensionless ratio t/T1/2 is the number of half-lives elapsed. After one, two and three half-lives, the ideal remaining fractions are 1/2, 1/4 and 1/8.
The same calculation applies to activity because activity is proportional to the number of radioactive nuclei when the decay constant is fixed. It can also describe a concentration, signal or population only when that subject genuinely follows one first-order exponential-decay process.
Six Half-Life Calculation Modes
| Find | Formula | Required values |
|---|---|---|
| Remaining quantity | N = N02−t/T1/2 | Initial quantity, elapsed time and half-life |
| Elapsed time | t = T1/2 ln(N0/N) / ln(2) | Initial quantity, remaining quantity and half-life |
| Half-life | T1/2 = t ln(2) / ln(N0/N) | Initial quantity, remaining quantity and elapsed time |
| Initial quantity | N0 = N2t/T1/2 | Remaining quantity, elapsed time and half-life |
| Decay measures | λ = ln(2)/T1/2, τ = 1/λ | One of half-life, decay constant or mean lifetime |
| Activity | A = λN | Radioactive-nuclei count and half-life |
The inverse modes require a positive remaining amount. In the ideal exponential model, the calculated amount approaches zero but never reaches exactly zero at a finite time. An entered zero therefore cannot produce a finite elapsed-time answer.
Decay Constant, Mean Lifetime and Half-Life
The decay constant λ is the probability rate per unit time for one radioactive nucleus. Its unit is reciprocal time, such as s−1 or d−1. A larger decay constant means faster decay and a shorter half-life.
Mean lifetime τ is the expected lifetime of a nucleus under the model. It is about 1.442695 times the half-life. Half-life is the median lifetime in the sense that half the original population is expected to have decayed by that time. These measures are related but answer different questions.
Always attach the reciprocal unit to λ. A value of 0.1 d−1 is not the same as 0.1 s−1. The calculator converts a rate stated per selected unit into s−1 before producing other values.
Worked Half-Life Examples
Amount remaining after three half-lives
Suppose 160 mg follows a six-hour half-life and 18 hours pass. The interval contains 18/6 = 3 half-lives.
The remaining fraction is 0.125, or 12.5%. The decayed fraction is 0.875, or 87.5%. This calculation does not say what products were created or whether any daughter product is also radioactive.
Time until 1% remains
For N/N0 = 0.01, the number of half-lives is log2(100) = 6.643856. If the half-life is 12 years, the ideal elapsed time is about 79.7263 years.
Activity from nuclei and half-life
For 1.00 × 1012 radioactive nuclei with an eight-day half-life, convert eight days to 691,200 seconds. The decay constant is 0.693147/691,200 = 1.002817 × 10−6 s−1.
This is the expected nuclear-transformation rate. A detector count rate is not the same as activity. Geometry, efficiency, dead time, emission probability and absorption often reduce detected events, while background can add unrelated gross counts.
Time and Activity Units Used
| Unit | Calculator definition | Use |
|---|---|---|
| Second, minute and hour | 1 min = 60 s; 1 h = 3,600 s | Short laboratory and process intervals |
| Day and week | 1 d = 86,400 s; 1 wk = 7 d | Fixed-duration decay calculations |
| Julian year | 365.25 d = 31,557,600 s | Consistent scientific conversion, not calendar scheduling |
| Becquerel | 1 Bq = 1 nuclear transformation per second | SI unit of radionuclide activity |
| Curie | 1 Ci = 3.7 × 1010 Bq | Non-SI activity unit retained in some contexts |
Calendar years vary between 365 and 366 days. This calculator uses the fixed Julian year so a unit conversion remains reproducible. If your source defines a year differently, convert the source interval to days or seconds before calculating.
Amount, Activity, Counts and Radiation Dose Are Different
Amount measures how much radioactive material or how many radioactive nuclei are present. Activity measures the expected number of nuclear transformations per second. Detector counts measure recorded events and depend on the instrument and measurement geometry. Absorbed dose measures energy imparted per unit mass. Equivalent dose applies radiation weighting, and effective dose also applies tissue weighting. A half-life calculation does not convert activity into dose.
Mass alone is not enough to calculate activity. You also need the radionuclide identity, molar mass, isotopic abundance or purity and a defensible number of radioactive nuclei. The activity mode therefore accepts N directly. It does not guess isotope composition from grams.
Single-Component Decay, Effective Half-Life and Decay Chains
The calculator assumes one independent component with one constant decay probability. It does not model production during the interval, daughter ingrowth, branching chains, mixtures of radionuclides, changing environmental removal or detector response. A sum of several exponentials does not generally have one constant half-life.
Physical half-life describes nuclear decay. Biological half-life describes removal from an organism or compartment. Effective half-life combines independent physical and biological removal rates under a specific model. Do not substitute one definition for another without confirming the subject and equation.
For a decay chain, the amount of a daughter nuclide can rise before falling because it is produced by the parent while also decaying. Bateman equations or validated numerical software are required. This page reports only the selected parent or single decaying quantity.
Common Half-Life Calculation Mistakes
- Subtracting one-half of the original amount every interval instead of halving the amount currently present.
- Mixing hours and days without conversion or attaching the wrong reciprocal unit to the decay constant.
- Entering a zero remaining amount and expecting a finite time from an ideal exponential model.
- Using a measured detector count rate as though it were the source activity in becquerels.
- Converting mass to activity without the radionuclide molar mass and isotopic fraction.
- Treating half-life as the exact decay time of each individual nucleus instead of a population statistic.
- Applying a single half-life to a mixture, decay chain or process with a changing rate.
- Rounding the remaining fraction before solving an inverse logarithmic formula.
Accuracy, Validation and Numeric Limits
- Initial and remaining quantities must be finite and positive. The remaining value cannot exceed the initial value in a decay-only inverse calculation.
- Half-life, decay constant and mean lifetime must be finite and positive. Selected elapsed-time inputs allow zero only where the equation remains defined.
- The calculator evaluates amount ratios and exponentials in logarithmic form. This preserves many results that would otherwise overflow, underflow or lose precision.
- Extremely small or large answers can be displayed as powers of ten even when the direct decimal does not fit in browser floating-point range.
- Displayed significant digits control presentation only. They do not represent measurement uncertainty or improve the quality of source data.
- The normalized curve illustrates the mathematical model. It is not measured data and it does not show uncertainty, background or daughter activity.
For scientific reporting, use an evaluated half-life with its uncertainty, retain source precision, propagate uncertainty through the model and document the time reference. Confirm the result with validated software when it affects research, health, safety or compliance.
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Half-Life Calculator FAQs
What is the half-life formula?
The remaining quantity is N(t) = N0 × 2^(−t/T1/2). The equivalent decay-constant form is N(t) = N0 × e^(−λt), where λ = ln(2)/T1/2.
How do I calculate the amount remaining after several half-lives?
Divide elapsed time by half-life to find n, then multiply the initial amount by 2^(−n). After three half-lives, 2^(−3) = 1/8, so 12.5% remains.
How many half-lives pass before 1% remains?
Use n = log2(100) because the initial-to-remaining ratio is 100. The result is about 6.643856 half-lives. Multiply that value by the half-life to obtain elapsed time.
Does an exponentially decaying amount ever reach exactly zero?
No finite time gives exactly zero in the ideal continuous equation. The value approaches zero. A real sample can contain zero radioactive nuclei after a random final decay, but that discrete outcome is outside the continuous expected-population model.
What is the difference between half-life and mean lifetime?
Half-life is the time by which half the original population is expected to decay. Mean lifetime is the expected lifetime of one nucleus under the model and equals T1/2/ln(2), about 1.442695 times the half-life.
How is the decay constant related to half-life?
The relationship is λ = ln(2)/T1/2. If half-life is expressed in seconds, λ is in s^(−1). A larger λ means a shorter half-life and faster exponential decay.
Can I calculate activity directly from radioactive mass?
Not from mass alone. You need the number of radioactive nuclei, which depends on radionuclide molar mass and isotopic abundance or purity. This calculator accepts nuclei count directly and applies A = λN.
What is the difference between a becquerel and a curie?
One becquerel is one nuclear transformation per second. One curie is exactly 3.7 × 10^10 Bq. Activity is not the same as detector counts, absorbed dose or biological risk.
Is physical half-life the same as biological or effective half-life?
No. Physical half-life describes nuclear decay. Biological half-life describes removal by a biological system. Effective half-life combines independent removal rates under a stated model and should not be substituted without context.
Can I use this result for laboratory, medical or radiation-safety decisions?
Use it only as an educational estimate or independent arithmetic check. Consequential work requires evaluated nuclear data, calibrated instruments, uncertainty analysis, approved procedures, current regulations and qualified professional review.
Formula, Unit and Safety Sources
- IAEA Nuclear Physics and Reactor Theory, exponential decay, half-life, decay constant, mean lifetime, activity and curie conversion.
- NIST Technical Note 2073, exponential decay correction and the relation between decay constant and half-life.
- NIST SP 330, SI Units, the becquerel as the SI derived unit for radionuclide activity.
- NIST Radioactivity SRM General Information, activity definitions and the exact 1 Ci = 3.7 × 1010 Bq relation.
- NIST Time and Frequency Glossary, the Julian year defined as exactly 365.25 days of 86,400 seconds.
Disclaimer: This calculator provides educational estimates for one ideal exponential-decay component. It does not calculate radiation dose, shielding, exposure risk, detector response, regulatory release, patient dosage or decay chains. Verify consequential work with authoritative radionuclide data, calibrated measurements, approved procedures and qualified professionals.