Decay constant & exponential decay

Some rocks contain polonium. The polonium is radioactive, and decays to a stable isotope of lead.

Which of the following graphs shows how the number N of lead atoms produced from the decay of polonium varies with time?

Glaciers are massive bodies of slowly moving ice. Over time, a fine sediment forms on the surface of a glacier. The sediment contains radioactive isotopes.

Caesium‑137,¹³⁷Cs, is a common radioisotope found in the sediment.

The concentration of¹³⁷Cs on a Norwegian glacier was investigated.

half‑life of¹³⁷Cs = 9.510⁸s

The activity of¹³⁷Cs was determined to be 2.510⁴Bq per kg of sediment.

Calculate the number of atoms of¹³⁷Cs in 1 kg of sediment.

Number of atoms of¹³⁷Cs =_........

Calculate the time taken for 10% of the¹³⁷Cs atoms to have decayed.

Nuclear decay is described as being spontaneous and random.

State what is meant by spontaneous and random in this context.

Explain why the decay constant of an isotope can be determined even though nuclear decay is random.

A radioactive source used in a school laboratory emits alpha and beta radiation.

Describe how the percentage of the activity due to beta radiation may be determined using a Geiger–Müller tube and ratemeter.

Americium‑241 is used in schools as a source of alpha radiation. A pure americium‑241 source was bought 34 years ago by a school.

Determine the percentage of the initial activity that would be expected today for the americium‑241 source.

half-life of americium-241 = 432 years

The decay products of americium are unstable and undergo a series of further decays.

The table shows the first three decays in this sequence.

A student states, “Protactinium‑233 emits beta particles when it decays, so by now the americium‑241 source bought 34 years ago will be emitting a significant amount of beta radiation.”

Discuss the student’s statement.

Radon is a radioactive gas. One isotope of radon,²²⁰₈₆Rn, decays to polonium, Po, by emitting an alpha particle.

Complete the nuclear equation for the decay of radon.

Radon is produced in the ground and escapes into the atmosphere.

Explain why this is a safety hazard.

The diagram shows apparatus for monitoring the decay of radon in the laboratory. Radon gas is produced in the plastic bottle from the decay of radium. A small amount of radon is then inserted into the decay chamber by squeezing the plastic bottle. A current is produced between two electrodes inside the chamber. This current is amplified and recorded by the ammeter.

Explain why a current is produced in the decay chamber.

A teacher is demonstrating the operation of the decay chamber to her class. She squeezes the bottle to introduce radon into the chamber.

She claims that within 450 s the activity of the radon in the chamber will be less than 1% of its initial value.

Assess whether her claim is correct.

half-life of radon = 55.6 s

Carbon‑14 is a radioactive isotope with a decay constant of 1.2×10^–4 year^–1.

The fossil of a plant contains 24% of the amount of carbon‑14 that would have been present when the plant was alive.

Which of the following expressions gives the age of the fossil in years?

Actinium-225 and bismuth-210 are radioactive isotopes. A sample of each isotope is prepared so that each sample has the same number of nuclei initially.

Explain why the activity of each sample would be the same after 10 days.

half-life of actinium-225 = 10 days half-life of bismuth-210 = 5 days

A hundred years ago, a method to determine the age of certain rocks was developed. An unstable isotope of rubidium is present in some rocks when they form. Over time the rubidium decays to a stable isotope of strontium.

Rubidium decays to strontium via β⁻ decay. Complete the nuclear equation representing the decay.

A sample of Moon rock from the Apollo 11 mission was analysed to determine the age of the rock. When the sample was analysed the number of rubidium atoms was N_R and the number of strontium atoms was N_S. As strontium atoms have all been produced from the decay of rubidium, the original number of rubidium atoms in the sample was (N_R + N_S). From the analysis of the sample, it was determined that = 0.0532

Deduce whether this ratio is consistent with the Earth and the Moon forming at the same time.

age of Earth = 4.5 10⁹ years half-life of rubidium isotope = 4.88 10¹⁰ years

Give a reason why the half-life of the rubidium isotope is hard to determine.

Recent investigations suggest that the half-life of the rubidium isotope may be larger than the traditionally accepted value.

Explain how this would affect the ages obtained by this dating method.

In 2011, a tsunami was caused by a massive earthquake centred some distance off the coast of Japan. The tsunami caused a cooling system failure at the Fukushima Nuclear Power Plant. This resulted in a nuclear meltdown and radioactive materials were released into the surroundings.

A reservoir beside one of the reactor buildings contained a large volume of water. In 2013, this water was found to have an extremely high concentration of caesium-137. Caesium-137 is a radioactive isotope of caesium.

Complete the nuclear equation for the decay of caesium-137.

An activity of 2.35 × 10¹² Bq per m³ of water in the reservoir was measured. It is suggested that a safe level for the activity of all water in the reservoir would be 100 Bq.

Calculate the time in years for the caesium-137 to decay to a safe level.

volume of water in reservoir = 5000 m³ half-life of caesium-137 = 30 years

The most common radionuclide amongst the fission products in the fuel was iodine-131, which decays with a half-life of 8.0 days to form a stable isotope of the gas xenon.

Deduce whether enough xenon would have collected in 32 days to exert a pressure of 1.0 10⁵ Pa in a volume of 450 m³. Assume that no gas escapes.

temperature = 20 °C initial number of iodine nuclei = 1.25 × 10²⁸

Buildings in nearby Tohoku University suffered structural damage during the 2011 earthquake.

The graph shows how the acceleration of one of the buildings, measured on the 9th floor, varied with time during the earthquake.

(Source: https://www.sciencedirect.com/science/article/pii/S0038080612001035)

At the time it was reported that during the earthquake the 9th floor of the building displaced by more than 30 cm from its normal position.

Assess the accuracy of this report.

The photograph shows a vase made of uranium glass. Uranium glass is radioactive.

Uranium glass usually contains a maximum of 2% uranium. Uranium glass made in the early part of the 20th century can contain up to 25% uranium.

A student carried out an investigation to determine the percentage of uranium in the glass.

The student measured the count rate by placing a Geiger Muller (GM) tube against the vase at a single position. This value was used to calculate the decay rate for the whole vase.

Show that the decay constant for uranium is about 5 10⁻¹⁸s⁻¹

half-life of uranium = 1.41 10¹⁷s

Calculate the percentage of uranium, by mass, in the glass.

area of GM tube window = 6.36 10⁻⁵m² surface area of vase = 0.0177 m² background count rate = 525 counts in 10 minutes count rate when GM tube next to vase = 3623 counts in 5 minutes mass of vase = 149 g mass of uranium atom = 238 u

The uranium decays by emitting alpha particles.

Criticise the method used to determine the percentage of uranium in the vase.

A uranium nucleus decays to thorium by emission of an alpha particle.

It can be assumed that all the energy of the decay is transferred to kinetic energy of the alpha particle.

Calculate the speed of the emitted alpha particle.

mass of uranium nucleus = 238.0003 u mass of thorium nucleus = 233.9942 u mass of alpha particle = 4.0015 u

Radioactive decay is often described in textbooks as a spontaneous, random process.

State what is meant by spontaneous decay.

*(b) Explain why there is an exponential decrease in the rate of decay for a sample containing a large number of unstable nuclei.

Astronauts on the 1971 Apollo 14 mission to the Moon brought back many rock samples. It is now believed that one of these contains a piece of rock that originated on Earth about 4 billion years (4 × 10⁹ years) ago.

The piece of rock is believed to have been launched into space when an asteroid struck the Earth.

The rock sample contains uranium. The radioactive decay of uranium allows it to be used to determine the time since the rock was formed on the Earth.

The uranium isotope²³⁸₉₂U becomes the lead isotope²⁰⁶₈₂Pb through a series of radioactive decays.

Calculate the number of α particles and the number of β particles emitted for one nucleus of²³⁸₉₂U to decay to become a nucleus of²⁰⁶₈₂Pb.

The half-life of²³⁸₉₂U is 4.47 10⁹ years. The half-lives of the other stages in the decay to²⁰⁶₈₂Pb are relatively so short that they can be ignored.

There was no lead in the rock when it formed, so all the²⁰⁶₈₂Pb in the sample is a product of²³⁸₉₂U decay. In the sample, for every 103 uranium nuclei present at the start, 50 are now lead nuclei.

Show that the age of the sample is about 4 × 10⁹ years.

The gravitational potential between the Earth and the Moon due to the combined effect of their gravitational fields increases to a maximum value of −1.28 MJ kg⁻¹ at a point between them.

Calculate the minimum speed at which a rock would have to leave the Earth in order to reach the Moon.

In your calculation, you may assume the rock has zero kinetic energy when it has maximum potential energy.

mass of Earth = 5.97 10²⁴kg radius of Earth = 6370 km

Four billion years ago, the Moon had a different orbital period, because it was closer to the Earth than it is today.

Calculate the period of the Moon’s orbit four billion years ago, when the radius of its orbit was 1.34 10⁸m.

mass of Earth = 5.97 × 10²⁴kg

Astronauts on the 1971 Apollo 14 mission to the Moon brought back many rock samples. It is now believed that one of these contains a piece of rock that originated on Earth about 4 billion years (4 × 10⁹ years) ago.

The piece of rock is believed to have been launched into space when an asteroid struck the Earth.

The rock sample contains uranium. The radioactive decay of uranium allows it to be used to determine the time since the rock was formed on the Earth.

The uranium isotope²³⁸₉₂U becomes the lead isotope²⁰⁶₈₂Pb through a series of radioactive decays.

Calculate the number of α particles and the number of β particles emitted for one nucleus of²³⁸₉₂U to decay to become a nucleus of²⁰⁶₈₂Pb.

The half-life of²³⁸₉₂U is 4.47 10⁹ years. The half-lives of the other stages in the decay to²⁰⁶₈₂Pb are relatively so short that they can be ignored.

There was no lead in the rock when it formed, so all the²⁰⁶₈₂Pb in the sample is a product of²³⁸₉₂U decay. In the sample, for every 103 uranium nuclei present at the start, 50 are now lead nuclei.

Show that the age of the sample is about 4 × 10⁹ years.

The gravitational potential between the Earth and the Moon due to the combined effect of their gravitational fields increases to a maximum value of −1.28 MJ kg⁻¹ at a point between them.

Calculate the minimum speed at which a rock would have to leave the Earth in order to reach the Moon.

In your calculation, you may assume the rock has zero kinetic energy when it has maximum potential energy.

mass of Earth = 5.97 10²⁴kg radius of Earth = 6370 km

Four billion years ago, the Moon had a different orbital period, because it was closer to the Earth than it is today.

Calculate the period of the Moon’s orbit four billion years ago, when the radius of its orbit was 1.34 10⁸m.

mass of Earth = 5.97 × 10²⁴kg

*4 Radium is a radioactive element. The most common isotope of radium has a half-life of almost two thousand years. A sample of radium can remain at a higher temperature than its surroundings for a long period of time.

Explain how a sample of radium is able to release significant amounts of energy over a long period of time.

*4 Radium is a radioactive element. The most common isotope of radium has a half-life of almost two thousand years. A sample of radium can remain at a higher temperature than its surroundings for a long period of time.

Explain how a sample of radium is able to release significant amounts of energy over a long period of time.

A school science department keeps a sample of potassium chloride to use as a test source for Geiger‑Müller tubes.

Potassium contains 0.012% of the unstable isotope potassium‑40.

Potassium‑40 undergoes β^– decay, producing a stable isotope of calcium.

Complete the nuclear equation for this decay.

A teacher makes some measurements using the potassium chloride test source to determine whether a Geiger‑Müller tube is sufficiently efficient at detecting β radiation.

The potassium chloride sample has a mass of 300 mg. Show that the number of nuclei of potassium‑40 in the sample is about 310¹⁷.

number of potassium nuclei in 1 g of potassium chloride = 8.110²¹

Show that the activity of this sample is about 5 Bq.

half‑life of potassium‑40 = 1.2510⁹ years

With no sample in front of the Geiger‑Müller tube, a count rate of 15 counts per minute is recorded. When the potassium chloride test sample is placed next to the Geiger‑Müller tube 176 counts are recorded in a period of 10 minutes.

A detector is considered efficient if it detects at least 7.5% of beta emissions from the source.

Determine whether this Geiger‑Müller tube can be considered efficient.

Explain a possible reason why only a low proportion of the decays are detected.

The science department also has a sample of strontium‑90. This undergoes beta decay with a half‑life of 29 years.

State why the half‑life of potassium-40 makes the potassium chloride a more suitable material than strontium-90 for the test.