Radioactive decay is one of the standard examples of exponential change in physics. A sample containing unstable nuclei
does not lose the same number of nuclei in every equal time interval. Instead, each nucleus has the same probability of
decaying per unit time, so the total rate of decay is proportional to how many undecayed nuclei are still present.
This leads directly to the exponential decay law.
Decay law
If \(N(t)\) is the number of undecayed nuclei at time \(t\), then the basic differential equation is
Rate law for radioactive decay.
\[
\begin{aligned}
\frac{dN}{dt} &= -\lambda N
\end{aligned}
\]
where \(\lambda\) is the decay constant. The negative sign means the amount decreases with time. Solving
this differential equation gives
\[
\begin{aligned}
N(t) &= N_0 e^{-\lambda t}
\end{aligned}
\]
where \(N_0\) is the initial amount at \(t=0\). This is the main formula used in radioactive-decay calculations.
It tells us the remaining amount after any elapsed time, provided the decay constant stays fixed.
Half-life
The half-life is the time required for the amount to drop to half of its initial value. By definition,
\[
\begin{aligned}
N(T_{1/2}) &= \frac{N_0}{2}.
\end{aligned}
\]
Substituting that into the decay law gives
Half-life relation.
\[
\begin{aligned}
\frac{N_0}{2} &= N_0 e^{-\lambda T_{1/2}} \\
\frac{1}{2} &= e^{-\lambda T_{1/2}} \\
\lambda T_{1/2} &= \ln 2 \\
T_{1/2} &= \frac{\ln 2}{\lambda}.
\end{aligned}
\]
This relation is extremely important because some problems give the decay constant while others give the half-life.
You can always convert one into the other using the formula above.
Activity
The activity is the decay rate, meaning the number of decays per unit time. Since the decay law starts
from \(dN/dt = -\lambda N\), the magnitude of the activity is
Activity formula.
\[
\begin{aligned}
A(t) &= \lambda N(t).
\end{aligned}
\]
This shows that activity follows the same exponential trend as the number of remaining nuclei. When the number of
undecayed nuclei becomes smaller, the activity also becomes smaller by the same factor.
Remaining fraction and decayed amount
Two other useful quantities are the surviving fraction and the number that have already decayed:
\[
\begin{aligned}
\frac{N(t)}{N_0} &= e^{-\lambda t}, \\
N_{\mathrm{decayed}} &= N_0 - N(t).
\end{aligned}
\]
The surviving fraction is especially convenient because it does not depend on the absolute sample size. It tells you
directly what proportion of the original sample is still present.
Sample calculation: cobalt-60
Suppose a sample begins with \(N_0 = 1000\) nuclei, has half-life \(T_{1/2} = 5.27\) years, and you want the remaining
amount after \(t = 10\) years. First convert the half-life into the decay constant:
\[
\begin{aligned}
\lambda &= \frac{\ln 2}{5.27}.
\end{aligned}
\]
Numerically, this gives
\[
\begin{aligned}
\lambda &\approx 0.1315\ \mathrm{year^{-1}}.
\end{aligned}
\]
Then substitute into the decay law:
\[
\begin{aligned}
N(10)
&= 1000\,e^{-(0.1315)(10)} \\
&\approx 269.
\end{aligned}
\]
So the sample has about \(26.9\%\) of its original amount left after 10 years. Because activity is proportional to
\(N\), the activity is reduced by the same factor:
\[
\begin{aligned}
\frac{A(10)}{A_0} &= \frac{N(10)}{N_0} \approx 0.269.
\end{aligned}
\]
Why half-life markers are useful
Half-life markers make exponential decay easier to interpret. After one half-life, the sample is reduced to
\(N_0/2\). After two half-lives, it is reduced to \(N_0/4\). After three half-lives, it is reduced to \(N_0/8\), and so on.
This pattern is often more intuitive than using the exponential formula directly.
| Elapsed time |
Remaining amount |
Remaining fraction |
| \(T_{1/2}\) |
\(N_0/2\) |
\(1/2\) |
| \(2T_{1/2}\) |
\(N_0/4\) |
\(1/4\) |
| \(3T_{1/2}\) |
\(N_0/8\) |
\(1/8\) |
| \(4T_{1/2}\) |
\(N_0/16\) |
\(1/16\) |
Advanced note
This calculator treats a single isotope with one decay constant. At a more advanced university level, one may study
branched decay, decay chains, daughter products, and coupled differential equations. Those systems are
more complicated, but the basic exponential law for a single unstable species remains the foundation of the whole topic.
| Concept |
Main relation |
Meaning |
| Decay law |
\(N(t)=N_0 e^{-\lambda t}\) |
Remaining amount after time \(t\) |
| Activity |
\(A(t)=\lambda N(t)\) |
Decay rate at time \(t\) |
| Half-life |
\(T_{1/2}=\ln 2/\lambda\) |
Time for the amount to halve |
| Decay constant |
\(\lambda=\ln 2/T_{1/2}\) |
Inverse-time decay parameter |
| Surviving fraction |
\(N/N_0=e^{-\lambda t}\) |
Fraction still present |
| Decayed amount |
\(N_{\mathrm{decayed}}=N_0-N\) |
Amount lost by decay |