Decay constant & exponential decay

A radioisotope thermoelectric generator (RTG) uses radioactive decay to generate electrical energy.

A vehicle that is used to examine the surface of Mars includes an RTG that initially contains 3.23 × 10₂₄ atoms of plutonium-238 238Pu 94 .

Determine the initial mass of plutonium-238 in the RTG.

The plutonium-238 undergoes alpha decay. The energy released by this decay is converted into electrical energy with an efficiency of 4.58%.

The power provided by the RTG decreases over time as the plutonium-238 decays.

The vehicle requires a minimum electrical power of 29.6 W to operate.

Calculate the initial activity of the plutonium-238. Go on to determine, in Earth years, the time for which the RTG can provide sufficient power for the vehicle to operate.

decay constant of plutonium-238 = 2.51 × 10⁻¹⁰ s⁻¹ energy of alpha particle emitted by plutonium-238 = 8.94 × 10⁻¹³ J

The vehicle has a battery-powered laser that is used to vaporise small samples of ice. On Mars, ice changes directly from solid to vapour at a temperature of 0 °C. The ice does not form a liquid.

The energy required to vaporise 1.0 g of solid ice directly into water vapour is the same as the total energy required to melt 1.0 g of ice and then to vaporise the melted ice.

The laser produces an output power of 1.0 kW for a maximum time of 3.0 s before its battery needs to be recharged.

The laser heats the ice from an initial temperature of −25 °C to a temperature of 0.0 °C. The laser then transfers energy to vaporise the ice at 0.0 °C.

Determine whether the laser is able to vaporise 1.0 g of the ice that is initially at −25 °C before the battery needs to be recharged.

specific heat capacity of ice = 2.1 kJ kg⁻¹ K⁻¹ specific latent heat of fusion of ice = 334 kJ kg⁻¹ specific latent heat of vaporisation = 2500 kJ kg⁻¹

A team of students uses 900 dice, each with n sides, to model the decay of a radioactive material. Each dice represents a single undecayed nucleus. A throw of the dice represents a constant time interval.

When the dice are thrown, those that show a 1 represent decayed nuclei and are removed. The students count the number N of ‘undecayed’ dice that remain. The procedure is repeated using the undecayed dice.

Figure 10 shows the students’ data.

Explain why N has been plotted on a logarithmic scale in Figure 10.

In this experiment, a decay constant λ can be defined that models the radioactive decay constant.

Determine λ. Go on to use your value for λ to show that n = 4 for the dice used in this experiment.

A typical radioactive source used in schools has an activity of 100 kBq. A radioactive source used in a hospital has an activity of 370 GBq.

State one safety measure when using a radioactive source in a school laboratory. Go on to discuss how this safety measure needs to be adapted for safe use of the hospital radioactive source.

X-rays are a form of ionising radiation.

A person has check-ups with a dentist every six months. The dentist only takes X-ray images when the person has reported a problem.

Suggest why.

A rock sample is found to contain the stable isotope lead-207. When it was formed, the rock contained uranium-235 but did not contain any lead-207.

Uranium-235 decays by a series of steps into lead-207. The half-life of uranium-235 is 0.71 billion years. The half-lives of the nuclides in the intermediate steps are negligible.

The sample of rock now contains one atom of lead-207 for every four atoms of uranium-235.

How long ago was the rock formed?

Figure 4 shows a spark detector used to detect alpha particles.

The detector consists of a metal mesh placed 5.0 mm above a wire. A potential difference of 4000 V is applied between the mesh and the wire.

Molecules in the air between the mesh and the wire are ionised by an alpha particle and a spark is produced.

Figure 5 shows equipotentials between the mesh and the wire.

Figure 5 shows a dashed line between the mesh and the wire.

Sketch on Figure 6 a graph to show how the magnitude E of the electric field strength varies with the distance d from the mesh along this dashed line. No values are required on the E axis.

An alpha particle passes through the mesh. The alpha particle ionises an argon atom at P on Figure 5, releasing one electron. The electron and the argon ion have no kinetic energy at P. The electron then travels to the wire and the argon ion travels to the mesh.

Assume that the air has no effect on the motion of the electron or on the motion of the argon ion.

mass of argon ion = 6.64 × 10⁻²⁶ kg

In practice, the air does affect the motion of the electron and the motion of the argon ion.

Suggest how the presence of air between the mesh and the wire changes the ratio in Question 04.2. No numerical detail is required.

The alpha source in Figure 4 is moved to different heights h above the mesh.

Figure 7 shows how the number of sparks N produced in 10 minutes varies with h. No sparks are produced when the source is not present.

Student A suggests that the spark rate obeys an inverse-square law. Student B suggests that the spark rate decreases exponentially with h.

Determine whether either student is correct.

The random nature of radioactive decay means that it is never possible to predict

Tritium is a radioactive nuclide used in ‘Exit’ signs. When a sign was manufactured the activity of the tritium in it was 37 MBq. After 10 years the tritium in the sign has an activity of 21 MBq.

What will the activity be 15 years after it was manufactured?

The graph shows the variation of activity with time for a sample of a nuclide X.

What was the initial number of nuclei of X in the sample?

A sample of radioactive material consists of 200 g of nuclide P and 100 g of nuclide Q.

Nuclide P has a half-life of 2 days and nuclide Q has a half-life of 4 days.

What is the total mass of nuclides P and Q after 12 days?

A nuclide has a half-life of 10 ms.

The decay constant for this nuclide lies between

X and Y are two radioactive nuclides. X has a half-life of 3.0 minutes and Y has a half-life of 9.0 minutes.

Two freshly prepared samples of X and Y start decaying at the same time. After 18 minutes the number of radioactive nuclei in both samples is the same. The sample of Y initially contained N radioactive nuclei.

What was the initial number of radioactive nuclei in the sample of X ?

Figure 1 shows apparatus used to investigate the inverse-square law for gamma radiation.

A sealed source that emits gamma radiation is held in a socket attached to clamp B. The vertical distance between the open end of the source and the bench is 138 mm. A radiation detector, positioned vertically above the source, is attached to clamp T.

A student is told not to move the stands closer together.

Describe a procedure for the student to find the value of d, the vertical distance between the open end of the source and the radiation detector.

In your answer, annotate Figure 1 to show how a set-square can be used in this procedure.

Before the source was brought into the room, a background count C_b was recorded.

With the source and detector in the positions shown in Figure 1, d = 530 mm. Separate counts C₁, C₂ and C₃ are recorded.

R_C is the mean count rate corrected for background radiation.

Show that when d = 530 mm, R_C is about 0.3 s⁻¹.

The apparatus is adjusted so that d = 380 mm. Counts are made that show R_C = 0.76 s⁻¹.

The student predicts that:

where k is a constant.

Explain whether the values of R_C in Questions 01.2 and 01.3 support the student’s prediction.

Describe a safe procedure to reduce d. Give a reason for your procedure.

The student determines R_C for further values of d. The values of d change by the same amount ∆ d between each measurement. Figure 2 shows these data.

Determine ∆ d.

Explain how the student could confirm whether Figure 2 supports the prediction:

No calculation is required.

When a gamma photon is detected by the detector, another photon cannot be detected for a time t_d called the dead time.

It can be shown that:

where R₁ is the measured count rate R₂ is the count rate when R₁ is corrected for dead time error.

The distance between the source and the detector is adjusted so that d is very small

On average, two of the gamma photons that enter the detector every second are not detected.

Calculate t_d for this detector.

A student says that if 100 gamma photons enter a detector in one second and t_d is 0.01 s, all the photons should be detected.

Explain, with reference to the nature of radioactive decay, why this idea is not correct.

A thermal nuclear reactor uses enriched uranium as its fuel. This is fuel in which the ratio of U-235 to U-238 has been artificially increased from that found in naturally-occurring ore.

Describe what happens when neutrons interact with U-235 and U-238 nuclei in a thermal nuclear reactor.

The amounts of U-235 and U-238 in the ore decrease due to radioactive decay at different rates.

A sample of uranium ore today contains 993 g of U-238 The mass of U-238 in this sample was greater 2.00 × 10⁹ years ago.

Show that the mass of U-238 in this sample at that time was about 1.4 kg.

decay constant of U-238 = 1.54 × 10⁻¹⁰ year⁻¹

A thermal nuclear reactor requires a minimum of 3.0% of its uranium mass to be U-235

The ratio of U-235 to U-238 in the ore has changed over time. 2.00 × 10⁹ years ago, the sample in Question 06.2 contained 52 g of U-235

Deduce whether the sample had a high enough U-235 content to be used in a reactor

The number of parent nuclei in a sample of a radioactive element is N at time t. The radioactive element has a half-life The rate of decay is proportional to

Figure 11 shows how radioactive decay of one nuclide can be modelled by draining water through a tap from a cylindrical tube.

The water flow-rate is proportional to the pressure of the water. The pressure of the water is proportional to the depth of the water. Therefore the rate at which the depth decreases is proportional to the depth of the water.

Before the tap is opened the depth is 16.0 cm

The tap is opened and the depth is measured at regular intervals. These data are plotted on the graph in Figure 12.

Determine the predicted depth of water when the time is 57 s

Suggest how the apparatus in Figure 11 may be changed to represent a radioactive sample of the same nuclide with a greater number of nuclei.

Suggest how the apparatus in Figure 11 may be changed to represent a radioactive sample of a nuclide with a smaller decay constant.

The age of the Moon has been estimated from rock samples containing rubidium (Rb and strontium (Sr), brought back from Moon landings. ⁸⁷₃₇Rb decays to⁸⁷₃₈Sr with a radioactive decay constant of 1.42 10^–11 year^–1

Calculate, in years, the half-life of⁸⁷₃₇Rb.

A sample of Moon rock contains 1.23 mg of⁸⁷₃₇Rb.

Calculate the mass, in g, of⁸⁷₃₇Rb that the rock sample contained when it was formed 4.47 10⁹ years ago.

Give your answer to an appropriate number of significant figures.

Calculate the activity of a sample of⁸⁷₃₇Rb of mass 1.23 mg

Give an appropriate unit for your answer.

What is the decay constant of nobelium - 259 ?

A pure sample of nuclide Y containing 3 N nuclei has an activity 6 A.

What is the half-life of nuclide Y ?

What is the total activity of 1.0 g of cobalt - 60 ?