E.4 Fission: IB Physics HL exam-style questions

A uranium-236 nucleus (X on the graph) undergoes fission into two nuclei, each with nucleon number 117 (Y on the graph), and two free neutrons.

What is the best estimate of the energy released?

Each fission of a uranium-235 nucleus releases about 200 MeV. (N_A = 6.0 × 10²³ mol⁻¹, 1 eV = 1.6 × 10⁻¹⁹ J)

What is the approximate energy released by the complete fission of 1 kg of uranium-235?

In a nuclear reactor fuelled by uranium-235, each fission releases on average k neutrons. The reactor is operating at a constant power.

What fraction of all the neutrons released must, on average, be absorbed without causing fission or escape from the core?

Uranium-235 has a neutron-to-proton ratio of 1.55. Its fission produces two nuclei with nucleon numbers roughly between 90 and 145. Stable nuclides in this range have neutron-to-proton ratios between about 1.2 and 1.45.

Which row gives the reason why the fission products are radioactive and the decay they usually undergo?

Three statements about a thermal nuclear fission reactor are:

I. Pushing the control rods further into the core reduces the average number of neutrons per fission that go on to cause another fission.
II. The moderator slows the neutrons down, which makes them more likely to cause fission of uranium-235 nuclei.
III. The thick shielding around the core is needed mainly to absorb the alpha particles emitted by the fuel.

Which statements are correct?

In a simple model of a sphere of fissile uranium, neutrons are produced by fission at a rate proportional to the volume of the sphere, and neutrons escape through its surface at a rate proportional to its surface area. The sphere is replaced by a sphere of the same material with twice the radius.

By what factor does (rate at which neutrons escape)/(rate at which neutrons are produced) change?

A sample of waste from a nuclear reactor contains two radioactive nuclides: X, of half-life 30 years, and Y, of half-life 2.4 × 10⁴ years. At a certain time the sample contains equal numbers of nuclei of X and of Y.

Which row identifies the nuclide with the greater activity at that time and the nuclide with the greater activity 600 years later?

A plutonium-239 nucleus absorbs a slow neutron and undergoes fission:

¹₀n + ²³⁹₉₄Pu → ¹³⁴₅₄Xe + ^A_ZX + 3¹₀n

Which row gives A and Z for nuclide X?

A uranium-235 nucleus of mass m_U absorbs a slow neutron of mass m_n. It splits into nuclei X and Y, of masses m_X and m_Y, and three neutrons. The kinetic energy of the slow neutron is negligible.

What is the energy released in this fission?

In a pressurized water reactor, the thermal power produced in the core is 3.0 GW and the overall efficiency of the power station is 33 %. All of this thermal power is transferred in the heat exchanger to the water of the secondary circuit, which enters the heat exchanger at its boiling point and leaves it as steam at the same temperature. The specific latent heat of vaporization of this water is 1.6 MJ kg⁻¹.

What mass of steam is produced each second?

Which property makes a material suitable for use as the moderator in a thermal fission reactor?

In a reactor core, neutrons are slowed by the water moderator until they are in thermal equilibrium with it. The temperature of the water is 290 °C. The mean kinetic energy of the neutrons is then (3/2)k_BT.

Which is the best estimate of the speed of a neutron with this kinetic energy?

Two nuclear power stations X and Y have the same electrical power output. The data below give their overall efficiency and the percentage by mass of uranium-235 in their fuel. In both, all the uranium-235 undergoes fission and the uranium-238 releases no energy.

Power station X: efficiency 30 %, uranium-235 4.0 %
Power station Y: efficiency 40 %, uranium-235 3.0 %

What is (mass of fuel used by X)/(mass of fuel used by Y) in the same time?

Three statements about the fission of nuclei in the fuel of a reactor are:

I. Spontaneous fission can occur without a neutron being absorbed by the nucleus.
II. Spontaneous fission of nuclei in fresh fuel can provide the first neutrons for a chain reaction.
III. The rate of spontaneous fission in the fuel is reduced when the control rods are pushed further into the core.

Which statements are correct?

Spent fuel removed from a reactor is first stored under water for several years. It is later sealed in containers and placed in a deep geological repository.

Which row gives a main reason for each stage?

Uranium-235 nuclei undergo fission after absorbing slow neutrons. The graph shows the distribution of the kinetic energies E_k of the fragments. The two peaks correspond to the most probable light fragment and the most probable heavy fragment.

Assume that the uranium-236 nucleus is at rest when it splits, that the momentum of the released neutrons is negligible, and that the total nucleon number of the two fragments is 234.

What is the nucleon number of the most probable light fragment?

The chart shows how the average energy released per fission of uranium-235 is shared between different forms. The decay energies are those released later by the radioactive fission products.

A reactor has operated at constant power for a long time. Which is the best estimate of the percentage of the energy released per fission that is transferred to internal energy of the reactor (core, moderator, coolant and shielding)?

The diagram shows the main parts of a nuclear power station with a pressurized water reactor. Four parts are labelled P, Q, R and S.

Which part transfers thermal energy from the water that has passed through the core to the water that drives the turbine, so that no radioactive material reaches the turbine?

A reactor is operating at constant power. The control rods are withdrawn slightly, and the power starts to rise. When the power has doubled, the control rods are returned to their original positions. Ignore any effects of the changes in temperature and in the amount of fission products.

Three statements are:

I. While the control rods are withdrawn, on average more than one neutron from each fission causes a further fission.
II. After the control rods are returned, the power falls back to its original value.
III. After the control rods are returned, the rate of fission stays at about twice its original value.

Which statements are correct?

In a model of a chain reaction, each fission causes on average k further fissions in the next generation. The first generation contains N fissions.

How many fissions are there in generation n?

In a pool-type research reactor, the thermal energy released in the core is removed by water pumped through it. With the reactor held at constant power, the operators vary the mass flow rate ṁ of the cooling water and record the rise ΔT in water temperature between the inlet and the outlet of the core. Energy losses from the core other than to the cooling water are negligible, and each fission of a uranium-235 nucleus releases 200 MeV.

The graph shows ΔT against 1/ṁ with a line of best fit through the origin. (Specific heat capacity of water = 4180 J kg⁻¹ K⁻¹; 1 MeV = 1.60 × 10⁻¹³ J; 1 u = 1.66 × 10⁻²⁷ kg)

A student models the heat exchanger of a nuclear power station. Hot water, representing the primary circuit, flows at a constant mass flow rate of (2.00 ± 0.04) g s⁻¹ through a copper coil immersed in 2.00 kg of cold water, representing the secondary circuit, in an insulated beaker. The secondary water is stirred and its temperature θ is recorded every 60 s.

During the 600 s of the experiment, the mean temperatures of the hot water entering and leaving the coil are 75.0 °C and 51.0 °C. Each thermometer reading has an uncertainty of ±0.5 °C. The specific heat capacity of water is 4180 J kg⁻¹ K⁻¹.

A new research reactor is loaded with fuel for the first time. A neutron source is placed at the centre of the core and fuel elements are added in batches. After each batch a neutron detector counts for 200 s. The count rate due to background is negligible.

Let k be the average number of neutrons from each fission that cause a further fission. A model predicts that the count rate C is proportional to 1/(1 − k), as long as k < 1. The table and the graph show the results, where n is the number of fuel elements in the core.

Water is used as a shield around research reactors. To test its effect, a student places tanks of water of total thickness x between a source of fast neutrons and a detector of slow neutrons. For x ≥ 10 cm, the model predicts that the count rate due to the source is C − B = C₀e^−μx, where C is the measured count rate and B is the background count rate.

With the source removed, the student measures B = 0.60 s⁻¹. The table shows the results; the graph shows ln((C − B)/s⁻¹) against x for all six thicknesses, with the line of best fit.

A pressurized-water reactor delivers 1.10 GW of electrical power with an overall efficiency of 33 %. Each fission of uranium-235 releases on average 200 MeV, and the fuel contains 4.0 % uranium-235 by mass; the rest is uranium-238.

Water pumped through the core at high pressure enters at 290 °C and leaves at 325 °C. Its specific heat capacity at these temperatures is 5.5 kJ kg⁻¹ K⁻¹.

(1 u = 1.661 × 10⁻²⁷ kg, 1 MeV = 1.60 × 10⁻¹³ J, 1 year = 3.16 × 10⁷ s)

In a thermal reactor, fission neutrons of kinetic energy about 2.0 MeV are slowed down by collisions with the carbon-12 nuclei of a graphite moderator.

In a head-on elastic collision between a neutron of mass m, moving with speed v, and a stationary nucleus of mass M, the neutron rebounds with speed v(M − m)/(M + m). Take M = 12m for carbon.

After removal from a reactor, a spent fuel assembly is stored under water. The graph shows the thermal power P produced in the assembly by radioactive decay against the time t after its removal.

Californium-252 (²⁵²₉₈Cf) has a half-life of 2.65 years. Of its nuclei that decay, 96.9 % emit an alpha particle and 3.1 % undergo spontaneous fission, which releases on average 3.8 neutrons. Small californium-252 sources are used as neutron sources to start up nuclear reactors.

Atomic masses: ²⁵²Cf 252.081 627 u, ¹⁴⁰Xe 139.921 645 u, ¹⁰⁸Ru 107.910 186 u; neutron 1.008 665 u. (1 year = 3.16 × 10⁷ s)

When a uranium-235 nucleus absorbs a slow neutron, the resulting uranium-236 nucleus splits into two fragments and a few neutrons. The graph shows the percentage of fissions, the fission yield, that produce a fragment of each nucleon number A.

(1 u = 1.661 × 10⁻²⁷ kg, 1 MeV = 1.60 × 10⁻¹³ J)

About 2.0 × 10⁹ years ago, in a uranium ore deposit in Gabon, groundwater allowed natural chain reactions of neutron-induced fission of uranium-235 to run for a long period. Today, uranium-235 makes up 0.720 % of the uranium atoms in natural uranium; almost all the rest is uranium-238. Modern reactors that use ordinary water need fuel in which at least about 3 % of the uranium atoms are uranium-235.

Half-lives: uranium-235 7.04 × 10⁸ years; uranium-238 4.47 × 10⁹ years. (1 u = 1.661 × 10⁻²⁷ kg, 1 MeV = 1.60 × 10⁻¹³ J, 1 year = 3.16 × 10⁷ s)

A spacecraft travelling to the outer planets is powered by a small uranium-235 fission reactor. The thermal energy from the core is used by heat engines, which deliver an electrical power of 10.0 kW with an overall efficiency of 25.0 %. The hot end of the engines is at 1070 K and the cold end at 420 K. All the energy not converted to electrical energy is transferred to a radiator at 420 K, which emits it into space.

Each fission releases 200 MeV. (1 u = 1.661 × 10⁻²⁷ kg, 1 MeV = 1.60 × 10⁻¹³ J, 1 year = 3.16 × 10⁷ s)

A fission chamber monitors the neutron flux in a reactor. It contains argon gas between two parallel metal plates 5.0 mm apart, with a potential difference of 250 V between them. One plate is coated with a thin layer of uranium-235. When a slow neutron causes fission of a uranium-235 nucleus in the coating, one of the two fragments enters the gas and loses all its kinetic energy there by ionizing argon atoms. The energy needed to produce one ion pair in argon is 26 eV.

(1 u = 1.661 × 10⁻²⁷ kg, 1 year = 3.16 × 10⁷ s)

A reactor has a thermal power of 3.2 GW. Each fission of uranium-235 releases 200 MeV. One possible fission is

¹₀n + ²³⁵₉₂U → ¹⁴¹₅₆Ba + ⁹²₃₆Kr + 3¹₀n

The fragments are radioactive and decay by a series of β⁻ decays: krypton-92 decays to the stable nuclide zirconium-92 (Z = 40) and barium-141 to the stable nuclide praseodymium-141 (Z = 59). On average, the products of one fission emit 6.0 antineutrinos as they decay. (1 MeV = 1.60 × 10⁻¹³ J)

Thorium-232 does not undergo fission when it absorbs a slow neutron. Instead it forms thorium-233, which decays by β⁻ emission (half-life 22 minutes) to protactinium-233. Protactinium-233 decays by β⁻ emission (half-life 27.0 days) to uranium-233, which undergoes fission with slow neutrons. In a thorium reactor some protactinium-233 is removed from the core and stored while it decays.

Atomic masses: uranium-233 233.039634 u; xenon-139 138.918792 u; strontium-93 92.914024 u; neutron 1.008665 u. (1 u = 1.661 × 10⁻²⁷ kg = 931.5 MeV c⁻², 1 MeV = 1.60 × 10⁻¹³ J, 1 year = 3.16 × 10⁷ s)

In a pulsed research reactor the uranium-235 fuel is mixed with the moderator. A control rod is suddenly fired out of the core, and the chain reaction grows very rapidly. The fuel heats up, and when its temperature rises the neutrons are slowed down less effectively. The graph shows the power P of the reactor against time t during one pulse.

Each fission releases 200 MeV. (1 MeV = 1.60 × 10⁻¹³ J)