Last updated: July 3, 2026
Radioactive Decay Calculator
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Quick Answer
The radioactive decay calculator computes remaining quantity from N(t) = N0 × e^(−λt), where λ = ln(2)/half-life. Enter initial quantity, half-life, and elapsed time in matching units. It returns the remaining quantity and the decay constant.
Radioactive decay remaining quantity equals the initial quantity times e to the negative decay constant times time. The decay constant equals natural log of two divided by the half-life.
Key Takeaways
- Radioactive decay follows N(t) = N0 × e^(−λt).
- The decay constant is λ = ln(2)/t½.
- One, two, and three half-lives leave 50%, 25%, and 12.5% of the initial quantity.
- Elapsed time and half-life must use the same units.
- The result keeps the same unit as the initial quantity.
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Formula
N(t) = N0 × e^(−λt); λ = ln(2)/t½
Where:
- N(t)=Remaining quantity after elapsed time(same as N0)
- N0=Initial quantity(mass, atoms, activity, or amount)
- t=Elapsed time(same time unit as t½)
- t½=Half-life(time)
- λ=Decay constant(time⁻¹)
Worked Examples
Two half-lives of a 100 g sample
A radioactive sample starts at 100 g, has a 10-year half-life, and decays for 20 years.
- 1Compute λ = ln(2)/10 = 0.0693147 yr⁻¹.
- 2Multiply λt = 0.0693147 × 20 = 1.386294.
- 3Apply N = 100 × e^(−1.386294) = 100 × 0.25.
One half-life of an 80 g sample
An 80 g sample with a 5-year half-life is measured after 5 years.
- 1One half-life has elapsed because t/t½ = 5/5 = 1.
- 2The exponential form is N = 80 × e^(−ln2).
- 3e^(−ln2) = 0.5, so N = 80 × 0.5.
Three half-lives of a 200 g sample
A 200 g sample with an 8-year half-life decays for 24 years.
- 1Compute elapsed half-lives: 24/8 = 3.
- 2Use the equivalent form N = 200 × (1/2)^3.
- 3(1/2)^3 = 0.125, so N = 25 g.
Introduction
The radioactive decay calculator predicts how much of a radioactive sample remains after a chosen time. It uses the first-order nuclear decay law N(t) = N0 × e^(−λt) and computes the decay constant from the half-life. For broader inverse problems, compare it with the half-life calculator; for Carbon-14 age estimates, use the radiocarbon dating calculator.
Radioactive decay formula
Radioactive nuclei decay randomly one atom at a time, but a large sample follows a predictable exponential curve. The remaining quantity is N(t) = N0 × e^(−λt). Here λ is the probability of decay per unit time for the population, and t is the elapsed time in matching units.
N0 can be mass, atoms, activity, or percent at time zero.
N(t) has the same unit as N0.
t and t½ must use the same time unit.
The model assumes a single nuclide with a constant half-life.
Converting half-life to decay constant
The decay constant is λ = ln(2)/t½ because one half-life makes N/N0 = 1/2. For a 10-year half-life, λ = 0.693147/10 = 0.0693147 yr⁻¹. NIST discusses the experimental importance of precise radionuclide half-life values at NIST Physical Measurement Laboratory.
If half-life is entered in days, λ is day⁻¹; if half-life is entered in years, λ is yr⁻¹.
Quick half-life checkpoints
The exponential equation gives the same checkpoints as repeated halving. These values are useful for checking calculator results and lab reports.
| Elapsed half-lives | Fraction remaining | From 100 units |
|---|---|---|
| 0 | 1 | 100 |
| 1 | 1/2 | 50 |
| 2 | 1/4 | 25 |
| 3 | 1/8 | 12.5 |
| 4 | 1/16 | 6.25 |
Chemistry and nuclear context
Radioactive decay is central to nuclear chemistry, tracer experiments, environmental monitoring, and isotope dating. It is often paired with isotope mass calculations from the atomic mass calculator or formula checks from the molar mass calculator. LibreTexts provides a classroom overview of decay rates at Chemistry LibreTexts/Nuclear_Chemistry/Radioactive_Decay_Rates).
Using the result in lab work
Enter the initial quantity measured at a reference time, the isotope half-life, and the elapsed time. The output gives the predicted remaining quantity and λ. Use it to plan count times, estimate storage activity, or sanity-check decay corrections before applying detector efficiency and background corrections.
For measured activity, the same equation applies because activity is proportional to the number of undecayed nuclei for one isotope.
Assumptions and limitations
This calculator handles one ideal first-order radioactive decay step. It does not model daughter ingrowth, branching chains, activation during irradiation, biological clearance, shielding, or calibrated calendar ages. For decay chains, Bateman equations or specialized nuclear data software are needed.
Half-life must be positive.
Elapsed time cannot be negative.
The sample is treated as one radionuclide.
External production or removal is not included.
Quick Reference Card
Radioactive Decay — Quick Reference
Quick reference • Radioactive Decay Calculator
N = N0e^(−λt); λ = ln2/t½; also N = N0(1/2)^(t/t½)Valid range: N0 > 0, t½ > 0, and t ≥ 0 with consistent time units
Common Values
⚠ Watch Out
- •Do not mix years and days between t and t½.
- •Half-life must be positive for λ to be defined.
- •This model does not include daughter product ingrowth.
- •Do not treat an uncalibrated decay age as a radiocarbon calendar date.
Pro Tips
- →Count t/t½ first for quick mental checks.
- →Use the exponential form when λ is already known.
- →Report λ with reciprocal units such as yr⁻¹ or day⁻¹.
- →For activity measurements, subtract background before applying decay corrections.
FAQs
How do I calculate radioactive decay from half-life?
Compute λ = ln(2)/t½, then use N(t) = N0 × e^(−λt). Equivalently, use N(t) = N0 × (1/2)^(t/t½).
What is the decay constant?
The decay constant λ is the first-order rate constant for radioactive decay. It has reciprocal time units and equals ln(2) divided by the half-life.
Why must elapsed time and half-life use the same units?
The exponent λt must be dimensionless. If half-life is in years, elapsed time must also be in years, and λ is reported in yr⁻¹.
Does the initial quantity need to be grams?
No. You can enter grams, atoms, activity, concentration, or percent as long as the remaining result is interpreted in the same unit.
What remains after two half-lives?
Two half-lives leave one quarter of the initial quantity. For 100 g, the remaining amount is 25 g.
Can this model handle decay chains?
No. It assumes a single radionuclide with a constant half-life and no daughter ingrowth. Decay chains require additional coupled equations.