KAHIBARO
Table of Contents
Meaning of Half-Life
Half life is the time required for half of the radioactive nuclei in a sample to decay. It is one of the most important ways to describe how quickly a radioactive substance changes with time.
If a sample begins with $N_0$ unstable nuclei, then after one half life only half remain undecayed. After a second half life, half of that remaining half is left, and so on. This repeated halving gives radioactive decay its characteristic pattern.
For example, if a sample starts with 1000 radioactive nuclei, then:
| Number of half-lives | Undecayed nuclei remaining |
|---|---|
| 0 | 1000 |
| 1 | 500 |
| 2 | 250 |
| 3 | 125 |
| 4 | 62.5 |
This table may describe actual nuclei on average, or a large sample very well. For small numbers of nuclei, decay is still random, so exact counts may differ.
Half-Life Formula
The decay law can be written in terms of half life. If $T_{1/2}$ is the half life, then the number of undecayed nuclei after time $t$ is
$$
N(t) = N_0 \left(\frac{1}{2}\right)^{t/T_{1/2}}
$$
This equation says that each time the interval $T_{1/2}$ passes, the number is multiplied by $\frac{1}{2}$.
Important half-life equation:
$$
N(t) = N_0 \left(\frac{1}{2}\right)^{t/T_{1/2}}
$$
After one half life, $N = \frac{N_0}{2}$.
After two half-lives, $N = \frac{N_0}{4}$.
After $n$ half-lives, $N = N_0 \left(\frac{1}{2}\right)^n$.
Because activity is proportional to the number of undecayed nuclei, activity also falls by half every half life:
$$
A(t) = A_0 \left(\frac{1}{2}\right)^{t/T_{1/2}}
$$
Relation to the Decay Constant
Half life is closely related to the decay constant $\lambda$. The exponential decay law is
$$
N(t) = N_0 e^{-\lambda t}
$$
Comparing this with the half-life form gives
$$
T_{1/2} = \frac{\ln 2}{\lambda}
$$
Since $\ln 2 \approx 0.693$,
$$
T_{1/2} \approx \frac{0.693}{\lambda}
$$
Key relation between half life and decay constant:
$$
T_{1/2} = \frac{\ln 2}{\lambda}
\qquad\text{and}\qquad
\lambda = \frac{\ln 2}{T_{1/2}}
$$
A large decay constant means rapid decay and therefore a short half life. A small decay constant means slow decay and therefore a long half life.
Visualizing Repeated Halving
A simple way to picture half life is to imagine the sample shrinking by equal factors over equal time intervals.
The curve gets smaller and smaller, but it never suddenly becomes zero. In principle, the amount keeps decreasing continuously.
Examples
Suppose a radioactive isotope has a half life of 10 days and the initial number of undecayed nuclei is 800.
After 10 days,
$$
N = 800 \left(\frac{1}{2}\right)^1 = 400
$$
After 20 days,
$$
N = 800 \left(\frac{1}{2}\right)^2 = 200
$$
After 30 days,
$$
N = 800 \left(\frac{1}{2}\right)^3 = 100
$$
Now suppose 15 days have passed. Then
$$
N = 800\left(\frac{1}{2}\right)^{15/10}
= 800\left(\frac{1}{2}\right)^{1.5}
$$
Since
$$
\left(\frac{1}{2}\right)^{1.5} \approx 0.354
$$
we get
$$
N \approx 800 \times 0.354 \approx 283
$$
So about 283 nuclei remain on average.
How to Find Half-Life from Data
If you know that the number of undecayed nuclei falls from $N_0$ to $N$ in time $t$, you can solve for the half life.
Starting from
$$
N = N_0 \left(\frac{1}{2}\right)^{t/T_{1/2}}
$$
divide by $N_0$:
$$
\frac{N}{N_0} = \left(\frac{1}{2}\right)^{t/T_{1/2}}
$$
This can be rearranged using logarithms to find $T_{1/2}$:
$$
T_{1/2} = \frac{t \ln 2}{\ln(N_0/N)}
$$
This is useful in experiments and in dating methods.
If a quantity drops to one half of its initial value in time $T_{1/2}$, then that time is the half life, regardless of the starting amount.
Half life does not depend on how much material you begin with.
Important Properties of Half-Life
Half life is a constant property of a given radioactive isotope under ordinary conditions. A large sample and a small sample of the same isotope have the same half life. The larger sample has more decays per second at first, but the fraction that decays behaves the same way.
Different isotopes can have very different half-lives. Some decay in tiny fractions of a second, while others take thousands or millions of years.
| Isotope type | Typical half-life behavior |
|---|---|
| Very unstable nuclei | Very short half-life |
| Moderately unstable nuclei | Medium half-life |
| Relatively stable radioactive nuclei | Very long half-life |
Half-Life and Fraction Remaining
Sometimes it is useful to think in terms of the fraction remaining after several half-lives.
| Half-lives elapsed | Fraction remaining | Percentage remaining |
|---|---|---|
| 1 | $\frac{1}{2}$ | 50\% |
| 2 | $\frac{1}{4}$ | 25\% |
| 3 | $\frac{1}{8}$ | 12.5\% |
| 4 | $\frac{1}{16}$ | 6.25\% |
| 5 | $\frac{1}{32}$ | 3.125\% |
This pattern is useful for quick estimates without a calculator.
Common Misunderstandings
Half life does not mean that all nuclei survive until a fixed time and then half decay suddenly. Decay is random for individual nuclei. Half life describes the behavior of a large collection on average.
Half life also does not mean a sample becomes completely gone after two or three half-lives. There is always some amount left, although it may become extremely small.
Half life is an average statistical property of many nuclei.
An individual nucleus does not have a scheduled decay time.
Summary
Half life is the time needed for half of the radioactive nuclei in a sample to decay. It leads to repeated halving of the number of undecayed nuclei and of the activity. The main formulas are
$$
N(t) = N_0 \left(\frac{1}{2}\right)^{t/T_{1/2}}
$$
and
$$
T_{1/2} = \frac{\ln 2}{\lambda}
$$
These formulas make half life one of the most practical tools for describing radioactive change over time.