Radioactive decay, half-life, and decay constant
Radioactive decay is random for an individual nucleus, but a sufficiently large population follows an exponential expectation. Half-life is the time for the expected undecayed population or activity to fall by one-half.
This forward calculator reports remaining and decayed quantities plus λ. The separate Half-Life Calculator is better when the unknown is elapsed time, initial quantity, or half-life.
How to calculate nuclear decay
- Enter initial quantity: Use nuclei count, amount, mass, or activity when proportionality remains valid.
- Match time units: Enter elapsed time and isotope half-life in the same unit.
- Calculate: Review the expected remaining, decayed, and decay-constant values.
- Respect the model: Do not treat an expected fractional nucleus count as a literal small-sample outcome.
Formula and variables
Elapsed time and half-life must use the same unit; λ is returned per that time unit.
N(t) = N₀(1/2)^(t/T½); λ = ln(2)/T½- N₀ — Initial quantity
- Undecayed quantity at time zero (count or proportional quantity)
- N(t) — Remaining quantity
- Expected undecayed quantity after time t (same as N₀)
- t — Elapsed time
- Nonnegative decay interval (time)
- T½ — Half-life
- Time for expected quantity to halve (same as t)
- λ — Decay constant
- Exponential decay probability rate (inverse time)
Carbon-14 after 1,000 years
Start with 100 proportional units, elapsed time 1,000 years, and half-life 5,730 years.
- Initial quantity
- 100
- Time
- 1,000 years
- Half-life
- 5,730 years
- N = 100(1/2)^(1000/5730)
Result: Approximately 88.60 units remain and 11.40 units have decayed.
The decay constant is about 1.2097 × 10⁻⁴ per year.
Understanding your results
Results are population expectations
The model predicts the mean behavior of a population; individual decay events remain stochastic.
- At time zero, remaining equals initial quantity.
- After one half-life, half remains.
- Decay constant has inverse units matching the selected time unit.
- Activity follows the same exponential factor for a single isolated radionuclide.
Assumptions
- A single radionuclide decays with constant λ.
- The system is closed to production or removal except for decay.
- Entered quantity is proportional to undecayed nuclei.
Limitations
- Does not model decay chains, daughter ingrowth, branching, mixtures, background, detection efficiency, or uncertainty.
- Does not perform radiometric-age calibration or dose calculation.
- Not a radiation safety or medical decision tool.
Common mistakes
- Mixing years and seconds.
- Using mean lifetime as half-life.
- Subtracting a fixed quantity per half-life.
- Interpreting fractional expected nuclei as literal counts in a tiny sample.
Practical use cases
Nuclear physics education
Explore the forward decay curve and the relation between λ and half-life.
Ideal single-isotope estimates
Estimate remaining proportional quantity when half-life and elapsed time are established.
Frequently asked questions
What is the difference between this and the Half-Life Calculator?
This page emphasizes forward decay, decayed quantity, and λ; Half-Life can solve inverse problems for time or half-life.
Can I enter activity instead of nuclei count?
For one radionuclide with constant λ, activity decreases by the same exponential factor.
Does radioactive material reach exactly zero?
The continuous expectation approaches zero but never reaches it at a finite time.
Sources and review
- Radioactive Decay — OpenStax University Physics Volume 3. Accessed 2026-07-13.
Reviewed 2026-07-13.