Work out how much of a substance remains after a given time from its half-life, plus the decay constant and mean lifetime.
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With starting quantity (n₀) 100, half-life (t½) 5,730, time elapsed (t) 10,000, the quantity remaining is 29.82924364.
| Starting quantity (N₀) | 100 |
|---|---|
| Half-life (T½) | 5,730 |
| Time elapsed (t) | 10,000 |
| Quantity remaining | 29.82924364 |
|---|---|
| Percentage remaining | 29.82924364% |
| Amount decayed | 70.17075636 |
| Half-lives elapsed | 1.745200698 |
| Decay constant (per time unit) | 0.0001209680943 |
| Mean lifetime | 8,266.642584 |
The half-life is the time it takes for half of a decaying quantity to disappear. After one half-life 50% remains, after two 25%, after three 12.5%, and so on. The same rule describes radioactive isotopes, the elimination of many medicines from the body and other first-order processes. Use the same time unit for the half-life and the elapsed time.
N(t) = N₀ × (1/2)^(t ÷ T½). Decay constant λ = ln 2 ÷ T½. Mean lifetime τ = T½ ÷ ln 2. The half-life is the time it takes for half of a decaying quantity to disappear. After one half-life 50% remains, after two 25%, after three 12.5%, and so on. The same rule describes radioactive isotopes, the elimination of many medicines from the body and other first-order processes. Use the same time unit for the half-life and the elapsed time.
With starting quantity (n₀) 100, half-life (t½) 5,730, time elapsed (t) 10,000, the quantity remaining is 29.82924364. Percentage remaining: 29.82924364%; Amount decayed: 70.17075636; Half-lives elapsed: 1.745200698; Decay constant (per time unit): 0.0001209680943; Mean lifetime: 8,266.642584.
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(1/2)³ = 1/8, or 12.5% of the starting amount.
25% is two half-lives, so about 2 × 5,730 = 11,460 years, assuming the standard carbon-14 half-life and no contamination.
The mean lifetime is the average time an individual atom or molecule survives. It is longer than the half-life by a factor of 1 ÷ ln 2, about 1.443.