Decay Law and Half-Life
How radioactive decay changes with time
Exponential decay
Radioactive decay is random for individual nuclei, but for a large sample the number of undecayed nuclei decreases exponentially: .
Half-life
The half-life is the time it takes for half of the nuclei in a sample to decay. It is related to the decay constant by .
A useful pattern
After one half-life, remains. After two half-lives, remains. After three half-lives, remains.
Solving half-life problems
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1
Identify the half-life and the elapsed time.
-
2
Count how many half-lives have passed: .
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3
Use the fraction remaining .
Sample decay fractions
| Number of half-lives | Fraction remaining |
|---|---|
| 1 | |
| 2 | |
| 3 | |
| 4 |
If a sample has a half-life of 10 days, what fraction remains after 20 days?
Twenty days is two half-lives, so the remaining fraction is .
Correct answer:
Write the exponential decay law for the number of undecayed nuclei.
This is the standard model for radioactive decay in a large collection of nuclei.
Correct answer: N(t)=N0 e^{-lambda t}
If the half-life is very short, the sample decays
A short half-life means the quantity halves in a short time, so decay is rapid.
Correct answer: very quickly