Radioactive Decay (HL IB Physics)

The table shows some of the isotopes of phosphorus and, where they are unstable, the type of decay.

Isotope
Type of decay			stable

State whether the isotope is stable or not. If not, determine, with a reason, the type of decay it experiences.

The isotope of phosphorus decays into an isotope of silicon, .

Write a decay equation for this decay, finding the values of A and Z, and explain why each emission product occurs.

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 larger than all the other half−lives of the decays in the series.

A rock sample, when formed originally, contained 6.0 × 10²² atoms of and no atoms. At any given time, most of the atoms are either or with a negligible number of atoms in other forms in the decay series.

Sketch on the axes below the variation of number of atoms and the number of atoms in the rock sample as they vary over a period of 1.0 × 10¹⁰ years from its formation. Label your graphs U and Pb.

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: 

λ is the decay constant and has a value of 1.54 × 10⁻¹⁰ years.

Calculate the time taken (in years) for there to be twice as many atoms as atoms.

Lead−214 is an unstable isotope of lead−206. It decays by emitting a  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. Comment on the nature of the energy released. 

Xenon−140 is one of the waste products from the fission of uranium-235. Xenon−140 is radioactive and decays through decay.

The graph shows the variation with time of the mass of 1kg of xenon−140 remaining in the sample.

An alternative nuclear fuel to the traditionally used uranium-235 is thorium-232. When thorium-232 is exposed to neutrons, it will undergo a series of nuclear reactions until it eventually emerges as an isotope of uranium-233, which will readily split and release energy the next time it absorbs a neutron.

Part of the thorium fuel cycle is shown below.

Once the uranium-233 nucleus absorbs a neutron, it undergoes fission, releasing energy and two neutrons and forming the fission products Xenon and Strontium as in parts a-c. Any isotopes of uranium-233 which do not undergo fission decay through a chain ending with a stable nucleus of thallium-205 . 

Show that 12 particles, not including neutrons, are emitted during this combination of decay chains. Explain your reasoning.

Show that the decay constant is related to the half-life by the expression 

Uranium-238 has a half-life of 4.47 × 10⁹ years and decays to thorium-234. The thorium decays (by a series of further nuclear processes with short half-lives) to lead.

Assuming that a rock was originally entirely uranium and that at present, 1.5% of the nuclei are now lead, calculate the age of the rock. Give your answer in years to 2 significant figures.

The ionisation current I produced by α-particles emitted in the decay of radon can be measured experimentally. The logarithmic graph shows how current, ln I, varies with time, t.

Using the graph, determine the half-life of radon.

Unstable uranium-238 has various nuclear decay modes to become stable thorium-234. The total amount of energy released when it decays is measured to be 210 keV. 

Outline, without calculation, the intermediate decay modes between the unstable uranium-238 to the stable thorium-234. 

A possible decay chain for uranium-238 is: 

Calculate the total amount of energy, in joules, carried away as gamma radiation in this decay chain. 

Deduce an alternative decay chain from unstable uranium-238 to stable thorium-234 which releases the same amount of energy in the form of gamma radiation as in part (b). 

Justify your answer with a calculation. 

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 of to  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. 

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

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

A radioactive source is used to measure the thickness of paper. A Geiger counter is used to measure the count rate on the opposite side of the paper to the radioactive source.  The radioactive source used must be chosen carefully.

The arrangement below is used to maintain a constant 0.10 m thickness of aluminium sheets. Alpha, beta or gamma sources are available to be used.

Outline the most suitable radioactive source for this arrangement and explain why the other sources may not be appropriate.

The source used in part (b) has a half-life of 14 days and it has an initial count rate of 240 counts per minute when first used in the apparatus.

Giving your answer in weeks, calculate the length of time it takes for the Geiger counter to detect a count rate of 0.25 s^–1.

Once the source has reached an activity of 0.25 s^–1, it is replaced as the count rate of the source is comparable with that of background radiation.

State two natural sources of background radiation and two man-made sources of background radiation.

A sample’s count rate in counts per minute (cpm) is measured using a  ray detector. This data is plotted on a graph.

[2]

[2]

The scientist wonders how the experiment in part (a) would have changed if the sample was twice the size.

Assuming the experiment from part (a) was repeated with a sample the exact same age but twice the mass, calculate the length of time it would have taken to reach a count rate of 22.5 cpm.

In reality the detector will measure a count rate of more than 5 cpm long after the length of time in part (b) has passed.

[2]

[2]

The scientist can measure the count rate of the source but is unable to directly measure the activity of the source using their detector. Activity is the total number of particles emitted from the sample per unit time.

Explain why this is not possible.

Fluorescent tubes operate by exciting the electrons of mercury atoms.

The energy levels of a mercury atom are shown below (not to scale):

An electron is excited to n = 4. On the diagram, draw all the possible de-excitation routes from n = 4 to the ground state. 

State and explain which energy transition will emit the photon with the lowest frequency.

An unstable isotope of mercury, Hg-203, is tested for its radioactive emissions in a laboratory that has a background rate of 0.3 s^–1.

A source is placed a fixed distance from a Geiger-Muller tube. Various materials are placed in between the detector and the source while the count rate is recorded. The results are shown below.

State and explain what types of radiation are being emitted by the Hg-203 source.

A student notices that the count rate recorded actually increases when 0.5 mm thick paper was placed between the Geiger-Muller tube and the source.

The image below shows how the binding energy per nucleon varies with nucleon number. 

Fission and fusion are two nuclear processes in which energy can be released. 

[1]

Explain with reference to the figure in part (a), why the energy released per nucleon from fusion is greater than that from fission.

Explain how the binding energy of an oxygen  nucleus can be calculated with information obtained in the figure from part (a).

The mass of an  nucleus is 15.991 u. 

Calculate: 

The mass difference, in kg, of the nucleus.

 [2] 

The binding energy, in MeV, of an oxygen nucleus.

[1]

Bismuth-214 () decays into Polonium-214 () by beta minus decay. 

The binding energy per nucleon of Bismuth-214 is 7.774 MeV and the binding energy per nucleon of Polonium-214 is 7.785 MeV.

Beta-minus decay is described by the following equation:

Show that the energy released in the decay of bismuth is about 2.35 MeV and state where the energy comes from.

If an additional neutron is accelerated into the Polonium-214 () to produce the isotope Polonium-215 (), use the following information to deduce the binding energy per nucleon of this new isotope.

         Mass of   nucleus = 3.571140 × 10⁻²⁵ kg

Polonium-215 () is radioactive and decays by the producing alpha radiation, which is known to be a particularly stable. 

Determine the binding energy of alpha radiation. 

The following information is available:

• Mass of a Helium-4 nucleus: 4.001265 u

A student claims that the amount of matter within a marble directly converted into energy would be enough to provide 1 year of current human energy consumption globally which is estimated to be 5.80 × 10¹⁸ J.

 If the matter within marble is approximately 6.02 × 10²³ u, determine if this statement is true, using the mass-energy equivalence.

Explain why the mass of an alpha-particle (α) is less than the total mass of two individual protons and two individual neutrons.

Show that the energy equivalence of 1.0 u is 931.5 MeV.

Data for the masses of some nuclei are given below 

Use the data to determine the binding energy of deuterium in MeV.

Using the data given in part (c), determine the binding energy per nucleon of zirconium in MeV.

Iodine-131  has a half-life of 8.02 days.

Calculate the decay constant of .

The initial activity of the sample of Iodine−131 is 6.5 × 10⁴ Bq.

Determine the activity after 16 days.

Determine the mass of the iodine−131 in the sample after 16 days.

Iodine−131 decays through a number of decay modes. Two of these are decay of 606 keV and gamma emission of 364 keV. The product of the  decay is . 

Sketch a nuclear energy level diagram to represent these decays. 

A nucleus of sodium−24 decays into a stable nucleus of magnesium−24. It decays by β^– emission followed by the emission of γ-radiation as the magnesium−24 nucleus de-excites into its ground state.

The sodium−24 nucleus can decay to one of three excited states of the magnesium−24 nucleus. This is shown in the diagram below:

The energies of the excited states are shown relative to the ground state.

Calculate the maximum possible speed of the emitted beta particle in MeV.

The excited magnesium nucleus de-excites through production of gamma radiation of discrete wavelengths.

Calculate the shortest wavelength of emitted radiation.

The graph shows the activity of a sample of sodium−24 with time.

Use the graph to calculate the decay constant of sodium−24.

The detector in this experiment measures 4% of the activity from the sample.

Determine the activity of sample after 27 hours from the start of the recording,

Americium-241 has a half-life of 432 years. A small sample is held in a school for use in experiments.

The teacher uses a Geiger-Müller counter to measure the count rate at close range. The relationship between activity and count rate is a ratio of 6:1. Over 5 minutes, the count is 13 600. 

Determine the activity of the sample.

Determine the activity of the americium sample after 748 years.