Radioactive Decay (AQA A Level Physics)

The radioisotope iodine-131  is used in medicine to treat overactive thyroid glands. It has a half-life of 8.02 days. 

Calculate the decay constant of  .

Calculate the number of atoms of   necessary to produce a sample with an activity of 8.0 × 10⁵ disintegrations s^–1.

Calculate the time taken, in hours, for the activity of the same sample of   to fall from 8.5 × 10⁵ disintegrations s^–1 to 8.0 × 10⁵ disintegrations s^–1.

Calculate the time (in days) for a sample of iodine to decay to 1% of its initial activity.

State two reasons why radioisotopes such as  are particularly suitable for use as a treatment for overactive thyroid glands.

The age of an ancient boat may be determined by comparing the radioactive decay of  from living wood with that of wood taken from the ancient boat.

A sample of 4.30 × 10²³ atoms of carbon is removed for investigation from a block of living wood. In living wood one in 10¹² of the carbon atoms is of the radioactive isotope , which has a decay constant of 3.84 × 10^–12 s^–1. 

Calculate the half-life of  in years, giving your answer to an appropriate number of significant figures.

Show that the rate of decay of the  atoms in the living wood sample is 1.65 Bq.

A sample of 4.30 × 10²³ atoms of carbon is removed from a piece of wood taken from the ancient boat. The rate of decay due to the   atoms in this sample is 1.15 Bq. 

Calculate the age of the ancient boat in years.

(i) State two reasons why it is difficult to obtain a reliable age of the ancient boat from the carbon dating described.

(ii) Suggest why the method of carbon dating is likely to be unreliable if the sample is:

• Less than 200 years old.

• More than 60 000 years old.

The radioisotope bismuth () can decay to become an isotope of lead by two different decay methods. These are partially completed in Table 1. 

Complete the decay equations and decay methods in Table 1. 

Table 1

 is also an alpha particle emitter. 

A student carries out an experiment to measure alpha particle activity in a sample of this isotope. They take two readings of the corrected count rate over 24 hours. Their data is as follows: 

• Initial corrected count rate = 1500 counts s^–1­­­­

• Final corrected count rate = 270 counts s^–1­­­­ 

Show that the decay constant of  is about 2.0 × 10^–5 s^­–1.

Calculate the half-life of this sample.

State two major sources of radioactivity that the student would need to account for when measuring the activity of the sample. 

An isotope of technetium is a gamma emitter used in medicine as a tracer in the human body. It is injected into the patient’s bloodstream. Scanners outside the body detect the gamma emissions, enabling internal structures to be imaged.

Figure 1 shows the variation of activity with time, t, for a sample of technetium. 

Figure 1

Using Figure 1, determine the half-life of the isotope of technetium.

Use data from the graph to calculate the number of nuclei of the radioactive technetium isotope present at time t = 0.

Figure 2 shows the variation of the number of nuclei remaining with time, t, for the same sample of the technetium isotope. 

Figure 2

Using Figure 2, show that, after 50 000 s, about 1 × 10¹⁶ nuclei decay every second.

All nuclei of technetium-99 have the same decay probability. 

Explain how this statement accounts for the observation that the number, N, of radioactive technetium atoms in the sample varies with time, t, as shown in Figure 2.

The half-life of lead–214 is 26.8 minutes. 

Show that the decay constant of lead–214 is approximately 4 × 10^–4 s^–1.

Calculate the percentage of the original number of nuclei of lead–214 left in a sample after a period of 120 minutes.

At t = 0, a sample contains 5.63 × 10⁷ nuclei of lead–214. 

Calculate the number of nuclei which decay in the time interval between t = 60 min and t = 120 min.

Each decay releases 2.0 × 10^–10 J of energy. 

Calculate the energy released in the time interval between t = 60 min and t = 120 min.

The radioisotope uranium-238  decays through a decay chain to the radioisotope lead-206 , which is stable. 

During the decay chain, 8 alpha particles are emitted, and X beta particles are emitted. 

Calculate the value of X.

The half-life of uranium-238 is so long in comparison to any of the isotopes in its decay chain that we can assume the number of lead-206 nuclei,  at any time is equal to the number of uranium-238 that have decayed.

The number of uranium-238 nuclei  at time t is given by the equation: 

where  is the number of uranium-238 nuclei at t = 0. 

Show that the ratio  is given by: 

Enriched uranium fuel is a mixture of the fissionable uranium-235 with the more naturally abundant uranium-238. Mixtures of radioactive nuclides such as this are very common in the nuclear power industry.  

Two samples of radioactive nuclides X and Y each have an activity of A₀ at t = 0. They are subsequently mixed together. 

Show that the total activity of the mixture at time t = 48 years is equal to A₀.  

The half-life of X and Y are 16 and 8 years respectively.

Table 1 shows measurements of activity for two imaginary radioactive isotopes. 

Table 1

 The decay constant  gives the probability (per second) that a radioactive isotope will decay. 

Show that the ratio of decay constants , for the two radioactive isotopes in Table 1 is approximately equal to 2.

A physics postgraduate wants to compare the activity of the two isotopes in Table 1 graphically. She decides to sketch a graph as shown in Figure 1. 

Figure 1

Add labels to the spaces provided in Figure 1 to show which line corresponds to isotope X and isotope Y and justify your answer.

Rubidium (Rb) was found in samples of moon rock which were retrieved from the Apollo missions. The isotope  is known to have a half-life of 4.90 × 10¹⁰ years.

One of the moon rock samples contains 1.2 mg of . 

Show that the mass of  that the rock sample contained when the moon was formed 4.47 × 10⁹ years ago was about 1.3 mg.

Calculate the activity of 1.2 mg of .

Radioactive carbon dating is a method for determining the age of samples containing organic material. This is a commonly used method for terrestrial objects. 

Pieces of ancient wood found at an archaeological site can be dated by measuring the activity of carbon-14. One sample contains one carbon-14 atom per 8.0 × 10¹⁰ carbon-12 atoms.

In a sample of living wood, the concentration of carbon-14 atoms is one carbon-14 atom per 3.0 × 10¹⁰ carbon-12 atoms.

The half-life of carbon-14 is 1.8 × 10¹¹ s.

(i) Explain why the concentration of carbon-14 is greater in a sample of living wood.

(ii) Calculate the age, in years, of the sample of ancient wood taken from the archaeological site.

This method of radioactive carbon dating, which uses a ratio of carbon-14 atoms to carbon-12 atoms, can be improved by using all the carbon-14 atoms, rather than only those which happen to decay when dating is carried out.

Suggest why using all the carbon-14 atoms in the sample of ancient wood would give a more reliable value for its age.

The radioactive isotope uranium-238 decays in a decay series to the stable lead-206. 

The half-life of  is 4.5 × 10⁹ years, which is much longer than the half-lives of the rest of the decays in the series. 

Initially, a rock sample contained 6.0 × 10²² atoms of  and no  atoms. 

Assume that at any given time, the majority of the atoms in the sample are  or  and the concentration of other atoms formed in the decay series is negligible. 

On Figure 1, sketch the variation of the number of atoms and the number of  atoms in the rock sample over a period of 1.0 × 10¹⁰ years from its formation.

Label your graphs U and Pb. 

Figure 1

A certain time, t, after its formation the sample contained twice as many  atoms as  atoms. 

Show that the number of atoms in the rock sample at time t was 4.0 × 10²².

The ratio of the number of lead nuclei to the number of uranium nuclei  at some time t is given by: 

Calculate how long it takes, in years, for the number of atoms to be twice that of atoms.

Lead-214 is an unstable isotope of lead-206. It decays by emitting a β particle to form bismuth-214 (Bi)

Bismuth is also unstable and has two decay modes:

• Emitting an α particle to form thallium-210 (Tl) + energy

• Emitting a β particle to form polonium-214 (Po) + energy 

Write decay equations for the decay chain of lead -214 to thallium -210 and to polonium -214 and comment on the nature of the energy released.