Half Life Calculator
Radioactive (and any exponential) decay follows N = N₀ × (1/2)^(t / T½), where every T½ of time halves the amount.
Enter Values
How to use this calculator
- Enter the initial amount N₀ and the half-life T½ (in any time unit — seconds, years, whatever you like).
- Enter the elapsed time t to find how much is left, OR leave t blank and enter the remaining amount N to find how long it took.
- Press Calculate. Keep t and T½ in the same time units.
How it works
Radioactive (and any exponential) decay follows N = N₀ × (1/2)^(t / T½), where every T½ of time halves the amount. Equivalently N = N₀ · e^(−λt) with the decay constant λ = ln 2 ÷ T½.
To find the time for a known drop, invert the formula: t = T½ × log₂(N₀ ÷ N). The mean lifetime is τ = 1 ÷ λ = T½ ÷ ln 2 ≈ 1.4427 × T½, the average time a particle survives before decaying.
Worked example
100 units, half-life 5 years, after 15 years. That's 15 ÷ 5 = 3 half-lives, so N = 100 × (1/2)³ = 12.5 units remaining (12.5% left). The decay constant is λ = ln 2 ÷ 5 = 0.138629 per year, and the mean lifetime is τ = 1 ÷ λ = 7.213475 years.
Common mistakes
- Mixing time units — the elapsed time t and the half-life T½ must be in the same units (both years, or both seconds).
- Confusing the half-life with the mean lifetime τ. The mean lifetime is longer: τ = T½ ÷ ln 2 ≈ 1.44 × T½.
- Assuming a fixed amount decays each period. Decay is exponential, not linear — after two half-lives you have one quarter left, not zero.
Frequently asked questions
What is the half-life formula?
N = N₀ × (1/2)^(t / T½), where N₀ is the starting amount, T½ is the half-life and t is the elapsed time. In decay-constant form it's N = N₀ · e^(−λt) with λ = ln 2 ÷ T½.
How do I find how long decay takes to reach a given amount?
Leave the elapsed time blank and enter the remaining amount N. The tool inverts the formula: t = T½ × log₂(N₀ ÷ N). For example, dropping from 100 to 12.5 with a 5-year half-life takes 5 × log₂(8) = 15 years.
How is the decay constant λ related to the half-life?
λ = ln 2 ÷ T½ ≈ 0.6931 ÷ T½. A larger λ means faster decay and a shorter half-life. The mean lifetime is its reciprocal, τ = 1 ÷ λ.
Does this only work for radioactive isotopes?
No. The same exponential-decay maths applies to drug concentration in the body, capacitor discharge, cooling and any process with a constant fractional decay rate.
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