How to Calculate Radioactive Half-Life: Exponential Decay Formula
Learn how to calculate radioactive half-life, remaining isotope quantities, and decay constants (lambda) for radiocarbon dating and nuclear physics.
How to Calculate Radioactive Half-Life: Exponential Decay Formula
Radioactive decay is a stochastic quantum process where unstable atomic nuclei spontaneously release ionizing radiation (alpha particles, beta particles, gamma rays) to reach a lower-energy stable nuclear state. The half-life (t½) is the characteristic time required for exactly 50% of the radioactive parent atoms in a sample to decay.
This guide explains the exponential decay equations, radiocarbon dating mathematics, and connects to the CalculatorAll Half-Life Calculator.
The Radioactive Decay Formulas
1. Remaining Quantity Formula
N(t) = N₀ × (0.5)^(t / t½) = N₀ × e^(-λ·t)
Where:
- N(t): Quantity of radioactive material remaining at time
t. - N₀: Initial quantity at time zero (
t = 0). - t½: Half-life of the radioactive isotope.
- t: Total elapsed time.
- λ (Lambda): Radioactive decay constant, defined as
λ = ln(2) / t½ ≈ 0.693147 / t½.
Worked Example: Carbon-14 Archeological Dating
Carbon-14 (¹⁴C) has a validated half-life of 5,730 years. Suppose an ancient wooden artifact contains 25.0% (0.25) of its original ¹⁴C concentration:
- Calculate Number of Elapsed Half-Lives (n):
0.25 = (0.5)^n → n = 2 half-lives. - Calculate Artifact Age:
Age = 2 × 5,730 years = 11,460 years old.
Sample Decay Progression Table
| Number of Half-Lives | % Remaining | % Decayed | Fraction Remaining |
|---|---|---|---|
| 0 Half-Lives | 100.0% | 0.0% | 1 (N₀) |
| 1 Half-Life | 50.0% | 50.0% | 1/2 |
| 2 Half-Lives | 25.0% | 75.0% | 1/4 |
| 3 Half-Lives | 12.5% | 87.5% | 1/8 |
| 4 Half-Lives | 6.25% | 93.75% | 1/16 |
| 5 Half-Lives | 3.125% | 96.875% | 1/32 |
| 10 Half-Lives | ~0.098% | ~99.90% | 1/1,024 |
Try the numbers with our calculator
Use your own assumptions instead of relying on a generic example.
Calculate Half-Life & Radioactive Decay
