Radioactive Decay (CIE A Level Physics)

A student sets up a radioactive source and a Geiger counter to investigate how the count rate of the source varies with time.

State and explain how the student could demonstrate

Technetium-101 decays by beta-minus emission to form a stable isotope of Ruthenium .

Complete the equation for this decay.

The variation with time t of the number N of technetium-101 nuclei in a sample of radioactive material is shown in Fig. 1.1.

Fig. 1.1

Each decay releases a beta particle with energy 487 keV. 

A source of water is found to be contaminated with the radioactive isotope radium-228 .

This isotope of radium has a half-life of 5.75 years.

Explain the meaning of half-life in this context.

Calculate the decay constant, in s^–1, of radium-228.

A sample is taken of the contaminated water. The activity of radium-228 in a sample of 1.0 kg of water is found to be 20 mBq.

The mass of water in 1.0 mol is 18 g.

Calculate

[2]

The maximum safe limit for the activity of radium-228 in water has been set as 18.5 mBq kg⁻¹. 

Calculate the time, in days, for the activity of the contaminated water to be reduced to the safe limit.

State what is meant by radioactive decay.

A radioactive sample consists of an isotope with a half-life T that decays to form a stable product. Only the isotope and the stable product are present in the sample. 

At time t = 0, the sample has an activity of A₀ and contains N₀ nuclei of the isotope. 

Fig. 1.1

Fig. 1.2

Calculate the fraction of undecayed nuclei N remaining relative to N₀ at time t = 1.25T.

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.

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 × 10¹⁰years.

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 in a fireplace at an archaeological site can be dated by measuring the activity of carbon-14. A sample contains one carbon-14 atom per 8.0 × 10¹⁰ carbon-12 atoms. In living wood, the concentration of carbon-14 atoms is greater at one carbon-14 atom per 3.0 × 10¹⁰ carbon-12 atoms.

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

Explain why the concentration of carbon-14 is greater in living wood and calculate the age, in years, of the sample taken from the archaeological site.