Enter the half-life into the calculator to determine the decay constant (decay-rate constant).
Decay Constant Formula
The decay constant λ describes how fast a radioactive isotope decays. It is tied directly to the half-life and the mean lifetime through these formulas:
λ = ln(2) / t½
λ = ln(N₀ / N(t)) / t
τ = 1 / λ
- λ = decay constant (per unit time)
- t½ = half-life (time for the amount to drop by half)
- N₀ = initial amount
- N(t) = amount remaining after time t
- t = elapsed time
- τ = mean lifetime
- ln(2) ≈ 0.693147
The calculator uses each formula in a different mode. The Half-life mode applies λ = ln(2)/t½ and also returns τ. The Amount change mode solves λ = ln(N₀/N(t))/t when you know how much material is left after some elapsed time. The Convert mode takes any one of λ, t½, or τ and returns the other two using τ = 1/λ and t½ = ln(2)/λ.
Reference Values
Use these tables to sanity-check your inputs and to interpret the result.
Half-life and decay constant for common isotopes
| Isotope | Half-life | λ |
|---|---|---|
| Technetium-99m | 6.01 hr | 0.1153 hr⁻¹ |
| Radon-222 | 3.8235 day | 0.1813 day⁻¹ |
| Iodine-131 | 8.02 day | 0.0864 day⁻¹ |
| Cobalt-60 | 5.271 yr | 0.1315 yr⁻¹ |
| Carbon-14 | 5,730 yr | 1.21×10⁻⁴ yr⁻¹ |
| Uranium-238 | 4.468×10⁹ yr | 1.55×10⁻¹⁰ yr⁻¹ |
Fraction remaining after n half-lives
| Half-lives elapsed | Fraction left | Percent left |
|---|---|---|
| 1 | 1/2 | 50% |
| 2 | 1/4 | 25% |
| 3 | 1/8 | 12.5% |
| 4 | 1/16 | 6.25% |
| 5 | 1/32 | 3.125% |
| 7 | 1/128 | ~0.78% |
| 10 | 1/1024 | ~0.10% |
Worked Examples and FAQ
Example 1: From half-life. Carbon-14 has a half-life of 5,730 years. The decay constant is λ = ln(2)/5730 = 0.6931/5730 ≈ 1.21×10⁻⁴ yr⁻¹. The mean lifetime is τ = 1/λ ≈ 8,267 yr.
Example 2: From an amount change. A 100 g sample drops to 25 g over 11,460 years. Using λ = ln(N₀/N)/t = ln(100/25)/11460 = ln(4)/11460 = 1.386/11460 ≈ 1.21×10⁻⁴ yr⁻¹. That matches Carbon-14, which makes sense since 11,460 years is exactly two half-lives.
What units does λ use? Whatever time unit you choose. If t½ is in seconds, λ is per second (s⁻¹). If t½ is in years, λ is per year (yr⁻¹). The calculator returns λ in the same time unit you entered, plus a per-second value for cross-checking.
What is the difference between half-life and mean lifetime? The half-life is when 50% remains. The mean lifetime τ is the average time an atom exists before decaying, and it is longer: τ = t½ / ln(2) ≈ 1.4427 × t½.
Can the decay constant change? For practical purposes, no. λ is a fixed property of the isotope and does not depend on temperature, pressure, or chemical state.
Why must the remaining amount be less than the initial amount? The formula assumes decay, so N(t) < N₀. If the remaining amount is equal or larger, the calculator flags it because ln of a number ≤ 1 would give zero or a negative λ.
