AkCalculators

Radioactive decay — law, half-life and real-world uses

Radioactive decay describes the spontaneous transformation of unstable atomic nuclei into more stable configurations by emitting radiation: alpha particles, beta particles, gamma rays, or combinations thereof. Despite quantum randomness at the level of individual nuclei, large numbers of identical nuclei follow a predictable exponential decay law. The fundamental relation for single-step decay is:

N(t) = N₀ e^{-λ t}

Here N₀ is the initial number of radioactive nuclei (or initial activity), λ the decay constant with units of inverse time, and t the elapsed time. The exponential form arises because the probability of an individual nucleus decaying is constant per unit time—leading to an ensemble average that decreases exponentially.

Half-life

Half-life T₁/₂ is the time required for half the original nuclei to decay. It is related to the decay constant by:

λ = ln(2) / T₁/₂

Half-life is commonly reported for isotopes (e.g., Carbon-14 ≈ 5730 years, Iodine-131 ≈ 8.02 days, Uranium-238 ≈ 4.468 × 10⁹ years) and provides an intuitive measure of how rapidly an isotope decays.

Activity

Activity A is the decay events per unit time and relates to the number of nuclei by A(t) = λ N(t). Activity is usually reported in becquerel (Bq, decays per second) or curie (Ci; 1 Ci ≈ 3.7×10¹⁰ Bq). When using activity inputs, ensure consistent units with λ.

Solving for different variables

The exponential law can be rearranged to solve for any variable given two others. Common rearrangements:

• N = N₀ e^{-λ t}
• N₀ = N e^{λ t}

Use matching time units for t and λ (e.g., seconds with s⁻¹). The calculator accepts scientific notation for large or small numbers to reduce unit conversion errors.

Decay chains and limitations

Many isotopes decay into other radioactive isotopes forming decay chains. Simple exponential formulas apply to each step but coupling between parents and daughters requires solving Bateman equations for accurate time-dependent compositions. This page focuses on single-step decay for clarity and rapid use.

Applications

Radioactive decay mathematics underpins radiometric dating (Carbon-14 dating), nuclear medicine (dosing, tracer dynamics), nuclear power (fuel burnup and decay heat), and environmental monitoring. In radiometric dating, measured ratios of parent to daughter isotopes, together with known half-lives, allow age estimates for geological and archaeological samples. In medicine, understanding decay informs dosing and safe handling schedules.

Worked examples

Example 1 — Remaining fraction: For λ = 0.693 / 1 hour (half-life 1 hour), after 3 hours N/N₀ = e^{-λ t} = e^{-0.693×3} ≈ 0.125 (12.5%).

Example 2 — Compute half-life: If λ = 1.157×10⁻⁵ s⁻¹, then T₁/₂ = ln(2)/λ ≈ 59930 s ≈ 16.65 hours.

Example 3 — Solve for time: N₀ = 1e6 counts, N = 2.5e5 counts, λ = 0.0001 s⁻¹. Then t = (1/λ) ln(N₀/N) = 10000 × ln(4) ≈ 13863 s ≈ 3.85 hours.

Practical measurement notes

Measured counts are subject to statistical (Poisson) uncertainty—standard deviation √N for N counts. When reporting ages or activities, include measurement uncertainty and calibration errors. For small sample sizes or short counting times, statistical errors become significant and must be considered.

Using this calculator effectively

Enter any two independent values among N₀, N, λ, T₁/₂ and t. If you have activity instead of counts, use A = λ N to interconvert. Enable step-by-step derivations for lab reports and set precision for formatted outputs. Remember to maintain consistent time units across inputs.

Exponential decay is a simple yet powerful model describing many natural and engineered processes. While the underlying mechanism is quantum-mechanical and stochastic, the population-level behaviour is deterministic and easily manipulated algebraically for engineering, medical, and scientific applications.