The decay law, half-life and mean life

Which nucleus decays next?

Nobody can say when one given uranium nucleus will decay. It may be in the next second or in a billion years. But for a million nuclei we can say exactly how many will be left after a day. Radioactive decay is a matter of chance, as the throw of a coin is. For a thousand coins, you can say that about five hundred will fall as heads.

We make two assumptions, and experiments back them up.

• Each nucleus has a fixed chance of decaying in each second. We call this chance the decay constant, $\lambda$, and measure it in $\text{s}^{-1}$.
• The nucleus has no memory. An old nucleus is as likely to decay as a new one. Heat, pressure or chemical bonding do not change $\lambda$ in any way that matters here.

The law of radioactive decay

If $N$ nuclei are present, the number that decay in a short time $dt$ is $\lambda N\,dt$. So the number falls like this:

The solution is an exponential, and you can check it by differentiating:

Here $N_0$ is the number of nuclei at $t = 0$. The more nuclei there are, the faster they decay. So the number falls quickly at first and slowly later, and it never reaches zero exactly.

Half-life

The half-life $T$ is the time in which half of the nuclei decay. Put $N = N_0/2$ and $t = T$:

After one half-life, $\tfrac{1}{2}$ is left. After two, $\tfrac{1}{4}$. After three, $\tfrac{1}{8}$. In general,

This is the same law as $e^{-\lambda t}$, and it is easier to use when $t$ is a simple number of half-lives. A-Level and IB often write the half-life as $T_{1/2}$ or $t_{1/2}$.

The half-life does not depend on how many nuclei you start with. Half of a big sample and half of a small one both decay in the same time. The half-lives of different nuclei are very different: less than a microsecond for some and many billions of years for others.

Mean life

Some nuclei in the sample decay at once and some last for a very long time. The mean life $\tau$ is the average time that a nucleus lives:

At $t = \tau$ the number left is $N_0 e^{-1} = 0.368\,N_0$. So after one mean life about $37\ \%$ is left. The mean life is longer than the half-life, because the nuclei that last a long time pull the average up.

Worked example: a hospital isotope

Technetium-99m is used in scans. Its half-life is $6.0$ h. Find the fraction left after $18$ h and after $15$ h, and the time in which $90\ \%$ has decayed.

1. After 18 h. That is $\dfrac{18}{6.0} = 3$ half-lives. $\dfrac{N}{N_0} = \left(\dfrac{1}{2}\right)^3 = \dfrac{1}{8} = 12.5\ \%$.
2. After 15 h. That is $2.5$ half-lives. $\dfrac{N}{N_0} = 2^{-2.5} = 0.177$, which is $17.7\ \%$.
3. When $90\ \%$ has decayed. The fraction left is $0.10 = 2^{-t/T}$. Take logarithms: $t = T\,\dfrac{\ln 10}{\ln 2} = 6.0\ \text{h} \times 3.32 = 19.9\ \text{h}$.

Check. After $18$ h, $87.5\ \%$ has decayed. The $90\ \%$ mark comes a little later, at $19.9$ h. It fits.

Worked example: decay constant and mean life

Cobalt-60 has a half-life of $5.27$ years. Find $\lambda$ and $\tau$, and the fraction left after $10$ years.

1. The decay constant. We want $\lambda$, so $\lambda = \dfrac{0.693}{T} = \dfrac{0.693}{5.27\ \text{y}} = 0.1315\ \text{y}^{-1}$. In seconds, $T = 5.27 \times 3.156 \times 10^{7}\ \text{s} = 1.663 \times 10^{8}\ \text{s}$, so $\lambda = 4.17 \times 10^{-9}\ \text{s}^{-1}$.
2. The mean life. $\tau = \dfrac{1}{\lambda} = 7.60$ y.
3. After 10 years. $\dfrac{N}{N_0} = 2^{-10/5.27} = 0.268$.

Two ways to decay

Some nuclei can decay in two ways, for example by alpha and by beta decay. Each way has its own decay constant, $\lambda_1$ and $\lambda_2$. A nucleus has two chances to decay, so they add:

The fraction of the decays that go by the first way is $\dfrac{\lambda_1}{\lambda}$. For example, suppose the first way alone would have a half-life of $20$ min and the second way alone $30$ min. Then $\dfrac{1}{T} = \dfrac{1}{20} + \dfrac{1}{30} = \dfrac{1}{12}$, so $T = 12$ min. The first way takes $\dfrac{12}{20} = 60\ \%$ of the decays.

A short time is not a half-life

If the time $\Delta t$ is very small compared with the half-life, the number that decay is about $\lambda N\,\Delta t$. This is the first term of the exponential. It is wrong for long times. After one half-life it would say that $0.693$ of the nuclei have decayed, when only $0.5$ has.

Set the half-life to $6$ h and the time to $12$ h. What is the fraction left? Now double the half-life. What time gives the same fraction?

Practise with new numbers

Try it in Python

The program starts with $2000$ nuclei. In every step each nucleus that is still there has a fixed chance of decaying. It then compares the count with the law.

Common mistakes

• Thinking that all nuclei decay in two half-lives. After two half-lives one quarter is left, not zero.
• Using $T = 1/\lambda$. That is the mean life. The half-life is $0.693/\lambda$.
• Saying that the half-life is shorter for a smaller sample. It does not depend on the size of the sample.
• Subtracting in the wrong way. After $t$ the fraction that has decayed is $1 - 2^{-t/T}$, and the fraction left is $2^{-t/T}$.
• Mixing time units. $\lambda$ and $t$ must use the same unit, for example both per year and years.
• Using $\lambda N\,\Delta t$ for a long time. It is only true when $\lambda\,\Delta t$ is much smaller than $1$.

Remember these

• $\dfrac{dN}{dt} = -\lambda N$ and $N = N_0e^{-\lambda t} = N_0\,2^{-t/T}$.
• $T = \dfrac{\ln 2}{\lambda} = \dfrac{0.693}{\lambda}$, and the mean life is $\tau = \dfrac{1}{\lambda} = 1.44\,T$.
• After $n$ half-lives the fraction left is $\left(\dfrac{1}{2}\right)^n$. After one mean life it is $\dfrac{1}{e} = 0.368$.
• Two ways to decay: $\lambda = \lambda_1 + \lambda_2$, and $\dfrac{1}{T} = \dfrac{1}{T_1} + \dfrac{1}{T_2}$.
• The half-life does not depend on the number of nuclei or on heat and pressure.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A sample has a half-life of $3$ days. What fraction is left after $12$ days?

2. A nucleus has a decay constant of $0.050\ \text{s}^{-1}$. What are its half-life and its mean life?

3. A student says: in two half-lives all the nuclei will have decayed. What is wrong?

4. Two samples, A and B, have half-lives of $10$ min and $40$ min. They start with the same number of nuclei. After $40$ min, which has more nuclei left, and by what factor?

References

• NCERT Physics Part II (Class 12, rationalised edition), Chapter 13 Nuclei, on the law of radioactive decay.
• OpenStax University Physics Volume 3, Chapter 10 Nuclear Physics, on radioactive decay and half-life.
• OpenStax Chemistry 2e, Chapter 21 Nuclear Chemistry, on rates of radioactive decay.