Half-life
From Freepedia
- For other uses, see Half-life (disambiguation).
The half-life of a radioactive substance is the time required for half of a sample to undergo radioactive decay. The term also has pharmaceutical and other uses.
More generally, for a quantity subject to exponential decay, the half-life is the time required for the quantity to fall to half of its initial value. (This article is a narrow discussion of half-life. For phenomena where half-life is applied, see "Related topics" below.)
| After # of Half-lives | Percent of quantity remaining |
|---|---|
| 0 | 100% |
| 1 | 50% |
| 2 | 25% |
| 3 | 12.5% |
| 4 | 6.25% |
| 5 | 3.125% |
| 6 | 1.5625% |
| 7 | 0.78125% |
| ... | ... |
| N | <math>\frac{100%}{2^N}</math> |
| ... | ... |
The table at right shows the reduction of the quantity in terms of the number of half-lives elapsed.
Quantities subject to exponential decay are commonly denoted by the symbol N. (This convention suggests a decaying number of discrete items. This interpretation is valid in many, but not all, cases of exponential decay.) If the quantity is denoted by the symbol N, the value of N at a time t is given by the formula:
- <math>N(t) = N_0 e^{-\lambda t} \,</math>
where
- <math>N_0</math> is the initial value of N (at t=0)
- λ is a positive constant (the decay constant).
When t=0, the exponential is equal to 1, and N(t) is equal to <math>N_0</math>. As t approaches infinity, the exponential approaches zero.
In particular, there is a time <math>t_{1/2} \,</math> such that:
- <math>N(t_{1/2}) = N_0\cdot\frac{1}{2} </math>
Substituting into the formula above, we have:
- <math>N_0\cdot\frac{1}{2} = N_0 e^{-\lambda t_{1/2}} \,</math>
- <math>e^{-\lambda t_{1/2}} = \frac{1}{2} \,</math>
- <math>- \lambda t_{1/2} = \ln \frac{1}{2} = - \ln{2} \,</math>
- <math>t_{1/2} = \frac{\ln 2}{\lambda} \,</math>
Thus the half-life is 69.3% of the mean lifetime.
Decay by two or more processes
A radioactive element may decay via two or more different processes. These processes may have different probabilities of occurring, and thus there is also a different half-life associated with each process.
As an example, for two decay modes, the amount of substance left after time t is given by
- <math>N(t) = N_0 e^{-\lambda _1 t} e^{-\lambda _2 t} = N_0 e^{-(\lambda _1 + \lambda _2) t}</math>
In a fashion similar to the previous section, we can calculate the new total half-life <math>T _{1/2} \,</math> and we'll find it to be
- <math>T_{1/2} = \frac{\ln 2}{\lambda _1 + \lambda _2} \,</math>
or, in terms of the two half-lives
- <math>T_{1/2} = \frac{t _1 t _2}{t _1 + t_2} \,</math>
Where <math>t _1 \,</math> is the half-life of the first process, and <math>t _2 \,</math> is the half life of the second process.
Related topics
- Exponential decay
- Mean lifetime
- Radioactive decay
- Radiometric dating
- Tables of nuclides with color-coding of half-lives:
