8.3 Radioactive decay and half-life — Learn

Prerequisites: exponential decay from Section 5.4; the graph of an exponential from Chapter 3; reading a gradient; natural logarithms; SI prefixes.

By the end of this section you can: define activity and half-life; use $A = \lambda N$; find a decay constant from a half-life; use $N = N_0 e^{-\lambda t}$ and $A = A_0 e^{-\lambda t}$; read a decay curve; and write a balanced nuclear equation for alpha, beta and gamma decay.

1. The nucleus and its notation

An atom has a small, dense nucleus. The nucleus contains protons and neutrons, which are together called nucleons. The number of protons is the proton number $Z$, and it decides which element the atom is. The number of nucleons is the nucleon number $A$. Atoms of the same element with different numbers of neutrons are isotopes.

A nuclide is written with these numbers:

For example, $^{18}_{9}\text{F}$ has 9 protons and 18 nucleons, so it has $18 - 9 = 9$ neutrons.

Some nuclei are unstable. They change into a more stable nucleus and release radiation. This is radioactive decay. The decay is random: you cannot predict which nucleus will decay next, or when. It is also spontaneous: nothing outside the nucleus has to set it off.

2. Activity

The activity of a source is the number of nuclei that decay each second.

The unit of activity is the becquerel, $\text{Bq}$. One becquerel is one decay per second.

A detector measures a count rate, not the activity. Not every decay sends a particle into the detector, so the count rate is smaller than the activity. The detector also records a background count rate from nearby rocks, building materials and cosmic rays. Subtract the background before you use the count rate:

3. Half-life

The half-life $T_{1/2}$ of a source is the average time for half of the unstable nuclei in a sample to decay. It is also the time for the activity to fall to half of its starting value.

After $n$ half-lives, the fraction that remains is $\left(\tfrac{1}{2}\right)^n$:

Worked example 1. The hospital tracer starts the morning with an activity of $8.0 \times 10^{8}\,\text{Bq}$. Its half-life is 110 minutes. Find the activity after 330 minutes.

The time is three half-lives, because $\dfrac{330}{110} = 3$.

After 330 minutes the activity is $1.0 \times 10^{8}\,\text{Bq}$.

4. The decay constant and the exponential law

Half-lives describe whole steps. For any time, the decay follows an exponential curve. The decay constant $\lambda$ is the probability that a given nucleus decays in one second. It has the unit $\text{s}^{-1}$.

The activity is proportional to the number of undecayed nuclei, which gives

The decay constant and the half-life are linked:

Here $\ln 2 = 0.693$. Because the decay rate depends only on the nucleus, the half-life does not change with temperature or pressure. It is a property of the nuclide.

Worked example 2. Find the decay constant of fluorine-18, whose half-life is 110 minutes.

First change the half-life to seconds.

Worked example 3. The tracer has an activity of $8.0 \times 10^{8}\,\text{Bq}$ at the start. Find the number of fluorine-18 nuclei that are still undecayed at that moment.

Rearrange $A = \lambda N$.

About $7.6 \times 10^{12}$ nuclei are undecayed.

5. Nuclear equations

The nucleon number and the proton number are conserved in every decay. That is how a decay equation is balanced.

• Alpha decay emits a helium nucleus, $^{4}_{2}\text{He}$. The nucleon number falls by 4 and the proton number falls by 2.
• Beta-minus decay emits an electron, $^{0}_{-1}\text{e}$, and an antineutrino. A neutron turns into a proton, so the proton number rises by 1.
• Beta-plus decay emits a positron, $^{0}_{+1}\text{e}$, and a neutrino. A proton turns into a neutron, so the proton number falls by 1.
• Gamma emission releases a photon. The nucleon number and the proton number do not change.

Worked example 4. Write the balanced equation for the beta-plus decay of fluorine-18, $^{18}_{9}\text{F}$.

Beta-plus decay emits a positron. To balance the nucleon numbers, the product nucleus also has 18 nucleons. To balance the proton numbers, it has $9 - 1 = 8$ protons, which is oxygen.

The neutrino $\nu$ carries away some energy and momentum. It is why the emitted positron has a range of kinetic energies rather than one fixed value.

6. The decay curve

A graph of activity against time for a radioactive source is a falling exponential curve. It is steep at the start, when many nuclei remain, and it flattens as the sample runs out.

You can read the half-life from the curve. Find the starting activity, halve it, and read the time at which the activity reaches that value. Repeat for the next halving to check.

A straight-line graph is easier to use. Taking logarithms of $A = A_0 e^{-\lambda t}$ gives

A graph of $\ln A$ against $t$ is a straight line with gradient $-\lambda$. The half-life follows from $T_{1/2} = \dfrac{\ln 2}{\lambda}$.

Watch out. Half-life does not mean "the time for all the atoms to decay". After one half-life, half remain. After two, a quarter remain. After ten, about one thousandth remain, and the sample is still slightly radioactive. A second trap is to use different units for the time and the half-life. If the half-life is in minutes, convert the time to minutes, or convert both to seconds. A third trap is to think that a temperature change alters the half-life. It does not: the decay is a property of the nucleus.

7. Try it: the decay curve

The figure draws the activity of a source against time. Two sliders set the half-life and the starting activity. The readout gives the activity after the chosen time and the number of half-lives that have passed.

Predict first. Do not move the sliders yet. Write down your answers.

1. After three half-lives, what fraction of the starting activity remains?
2. If the half-life is halved, does the curve fall faster or more slowly?
3. Does doubling the starting activity change the half-life?

Test. Set a half-life and a starting activity, then read the activity at one, two and three half-lives. Change one slider at a time and check each prediction.

Explain. Why is the curve steepest at the start? Write two or three sentences. Use the idea that the activity is proportional to the number of undecayed nuclei, and that many nuclei remain at the start.

Progress check

Return to the hospital tracer. It starts the morning with an activity of $8.0 \times 10^{8}\,\text{Bq}$ and a half-life of 110 minutes.

1. Find the activity after 330 minutes, and after 550 minutes.
2. Find the decay constant in $\text{s}^{-1}$.
3. Find the time at which the activity has fallen to $2.0 \times 10^{8}\,\text{Bq}$.
4. Explain why the team cannot predict which nucleus will decay next, even though they can predict the activity of the whole sample well.

If question 1 goes wrong, count the half-lives and divide by $2^n$. If question 2 goes wrong, change 110 minutes to seconds and use $\lambda = \dfrac{\ln 2}{T_{1/2}}$. If question 3 goes wrong, find how many halvings take the activity from $8.0 \times 10^{8}$ to $2.0 \times 10^{8}$. If question 4 goes wrong, think about the size of the sample and the meaning of a probability.

You should now be able to

• describe the nucleus, the proton number, the nucleon number and an isotope;
• define activity and use $A = \dfrac{\Delta N}{\Delta t}$;
• correct a count rate for the background;
• define half-life and use $A = \dfrac{A_0}{2^n}$;
• use $A = \lambda N$ and $N = N_0 e^{-\lambda t}$;
• find the decay constant from the half-life, and the half-life from the decay constant;
• write balanced equations for alpha, beta-minus, beta-plus and gamma decay;
• read a half-life from a decay curve, and use a $\ln A$ against $t$ graph;
• explain why radioactive decay is random and spontaneous.