Radioactive Decay (Cambridge (CIE) A Level Physics): Exam Questions

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 

(i) the spontaneous nature of radioactive decay,

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(ii) the random nature of radioactive decay.

[2]

Technetium-101 $\left({}_{43}{}^{101}\mathrm{Tc}\right)$ decays by beta-minus emission to form a stable isotope of Ruthenium $\left(\mathrm{Ru}\right)$.

Complete the equation for this decay.

${}_{43}{}^{101}\mathrm{Tc}\to {}_{......}{}^{......}\mathrm{Ru}+{}_{......}{}^{......}\mathrm{\beta }+......$

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

(i) Use Fig. 1.1 to determine the activity, in Bq, of the sample of technetium-101 at time t = 14.0 minutes.

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(ii) Use your answer in (c)(i) to calculate the decay constant λ of technetium-101.

[2]

(iii) On Fig. 1.1, sketch a line to show the variation with t of the number of ruthenium nuclei in the sample.

[2]

Each decay releases a beta particle with energy 487 keV. 

(i) Calculate, in J, the total amount of energy given to beta particles that are emitted between time t = 10 min and time t = 30 min.

[3]

(ii) Suggest why the total amount of energy released by the decay process between time t = 10 min and time t = 30 min is actually greater than your answer in (d)(i).

[1]

A source of water is found to be contaminated with the radioactive isotope radium-228 $\left({}_{88}{}^{228}\mathrm{Ra}\right)$.

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

(i) the number of radium-228 atoms in 1.0 kg of the contaminated water

[2]

(ii) the ratio 

$\frac{\mathrm{number}\mathrm{of}\mathrm{molecules}\mathrm{of}\mathrm{water}\mathrm{in}1.0\mathrm{kg}\mathrm{of}\mathrm{water}}{\mathrm{number}\mathrm{of}{}^{228}\mathrm{Ra}\mathrm{atoms}\mathrm{in}1.0\mathrm{kg}\mathrm{of}\mathrm{water}}$

[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. 

(i) On Fig. 1.1, sketch the variation with t of the number N of nuclei of the isotope present in the sample from time t = 0 to time t = 3T.

Fig. 1.1

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(ii) State the name of the quantity represented by the gradient of the graph in Fig. 1.1.

[1]

(i) On Fig. 1.2, sketch the variation with N of the activity A of the sample for values of N between N = 0 and N = N₀.

Fig. 1.2

[2]

(ii) State the name of the quantity represented by the gradient in Fig. 1.2.

[1]

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

The radioisotope uranium-238 $\left({}_{92}{}^{238}\mathrm{U}\right)$ decays through a decay chain to the radioisotope lead-206 $\left({}_{82}{}^{206}\mathrm{Pb}\right)$, 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, $N_{Pb}$ at any time is equal to the number of uranium-238 that have decayed. 

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

                  $N_U=N_0{e}^{-\lambda t}$ 

where $N_0$ is the number of uranium-238 nuclei at t = 0. 

Show that the ratio $\frac{N_{Pb}}{N_U}$ is given by: 

                  $\frac{N_{Pb}}{N_U}=e^{\lambda t}-1$

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 $\frac{9}{64}$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 ${}_{37}{}^{87}Rb$ is known to have a half-life of 4.90 × 10¹⁰years.

One of the moon rock samples contains 1.2 mg of ${}_{37}{}^{87}Rb$. 

Show that the mass of ${}_{37}{}^{87}Rb$ 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 ${}_{37}{}^{87}Rb$.

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.