E.4 Fission

A thermal nuclear reactor contains control rods and a moderator.

Which row gives the main function of each?

In the nuclear power plant shown, a coolant carries thermal energy from the reactor core to a heat exchanger.

What is the main purpose of the heat exchanger?

The core of a nuclear reactor is surrounded by thick shielding made of steel and concrete.

Which statement about this shielding is correct?

Some heavy nuclides, such as uranium-238 and plutonium-240, can undergo spontaneous fission.

Which statement about spontaneous fission is correct?

A university research reactor uses uranium-235 fuel. One possible fission reaction is
²³⁵₉₂U + ¹₀n → ¹⁴⁰₅₄Xe + ⁹⁴₃₈Sr + x ¹₀n
What is x?

A nuclear power station is producing a constant power output. On average, how many of the neutrons released by each fission go on to cause another fission?

Demand on the electricity grid falls at night. The operator of a nuclear power station wants to reduce the rate of fission in the core. What should the operator do?

The fragments produced when a uranium-235 nucleus undergoes fission are almost always radioactive and decay by β⁻ emission. Which statement best explains this?

About 200 MeV of energy is released when a nucleus of uranium-235 undergoes fission. Most of this energy appears as

Spent fuel rods removed from a nuclear reactor are kept in deep pools of water for several years before they are moved to dry storage. What is the main reason for this?

An engineer is checking a briefing sheet for a thermal fission reactor. Which component is correctly matched with its function?

Californium-252 is used in hospitals and research laboratories as a compact neutron source because it undergoes spontaneous fission. Which statement about spontaneous fission is correct?

In a pressurised water reactor, what is the main function of the heat exchanger (steam generator)?

A neutron-induced fission is represented by ²³⁵₉₂U + ¹₀n → ¹⁴⁰₅₅Cs + ^A_ZRb + 2 ¹₀n.

What are A and Z for the rubidium nuclide?

In a sample of uranium-235, k is the average number of neutrons from each fission that go on to cause another fission.

Which row describes the reaction when k = 1.5 and gives the value of k for a reactor working at steady power?

Spent fuel removed from a reactor is first kept for several years in a pool of water and is later sent for long-term storage deep underground.

Which row gives a correct reason for each stage?

The moderator in a thermal fission reactor slows down the fast neutrons released in fission.

Which row describes a good moderator?

Uranium fuel that has been enriched, so that a larger fraction of its nuclei are uranium-235, is stored in small containers that are kept well apart from one another.

What is the main reason for this?

When a uranium-235 nucleus absorbs a neutron, the total mass of the nucleus and the neutron is about 236 u. The fission that follows releases about 200 MeV.

What fraction of the mass of the nucleus and the neutron is converted into the energy released? (1 u = 931.5 MeV c⁻²)

In one neutron-induced fission of uranium-235 the fragments are ¹⁴⁷₅₇La and ⁸⁷₃₅Br, and two neutrons are released. The heaviest stable isotopes of these two elements are ¹³⁹₅₇La and ⁸¹₃₅Br.

Which row compares the number of neutrons in each fragment with the number in the stable isotope, and gives the decay that the fragments are most likely to undergo?

A reactor is operating at a constant fission rate. At time t = 0 the control rods are withdrawn a small distance and then held in their new position, so that the average number of neutrons from each fission that cause a further fission becomes 1.002 and stays at this value.

Which graph shows how the fission rate R varies with t after t = 0?

Just after a uranium-235 nucleus has split, the two fragments ¹⁴⁴₅₅Cs and ⁹⁰₃₇Rb have their centres 1.25 × 10⁻¹⁴ m apart. Treat each fragment as a point charge at its centre.

What is the magnitude of the electric force between the fragments?

Neutrons released in fission have a kinetic energy of about 2.0 MeV. The moderator of a reactor slows them until their kinetic energy is about 0.025 eV.

What is the ratio (speed of a neutron released in fission)/(speed of a slowed neutron)?

Molybdenum-99 is used to make the medical tracer technetium-99m. It can be produced by the fission of uranium-235:
²³⁵₉₂U + ¹₀n → ⁹⁹₄₂Mo + X + 2 ¹₀n
What is nuclide X?

The pressurized-water reactor in a submarine uses ordinary water as its moderator. Which property of the hydrogen nuclei in water makes water an effective moderator?

Why is energy released when a uranium-235 nucleus undergoes fission into two medium-mass nuclei?

In a single fission of uranium-235, about 200 MeV of energy is released. In which form is most of this energy released immediately?

Engineers designing a compact fission reactor for a deep-space probe find that a small sphere of fuel cannot sustain a chain reaction, but a larger sphere of the same material can. What is the best explanation?

A nuclear-powered icebreaker ship has a reactor with a thermal power of 320 MW. Each fission releases 200 MeV, which is 3.2 × 10⁻¹¹ J. What is the rate of fission in the reactor?

The binding energy per nucleon of uranium-235 is about 7.6 MeV. The average binding energy per nucleon of its fission fragments is about 8.5 MeV. What is the best estimate of the energy released when one uranium-235 nucleus undergoes fission?

A compact fission reactor designed to power a lunar base supplies thermal energy at a steady rate of 4.0 kW for 3.2×10⁷ s (about one year). What mass is converted into energy in this time?

In the fuel of a reactor, a nucleus of \(^{238}_{92}\text{U}\) absorbs a neutron. The nucleus formed then undergoes two successive β⁻ decays. Which nucleus is finally formed?

Spent reactor fuel is stored securely for several thousand years. At the end of this time, the remaining activity of the stored waste is mainly due to

A nucleus of \(^{235}_{92}\text{U}\) absorbs a slow neutron and splits into two fragments X and Y. Three neutrons are released. Which pair of fragments is possible?

A fission event releases 200 MeV. The rest energy of a neutron plus a uranium-235 nucleus is approximately 236 × 931.5 MeV. Approximately what percentage of the rest mass of the reactants is converted into energy?

Some reactors run on natural (unenriched) uranium. These reactors use heavy water (D₂O) as the moderator rather than ordinary water (H₂O). What is the main reason for this choice?

A submarine reactor is operating at a steady power. The control rods are then withdrawn slightly from the core. What happens immediately afterwards?

The reactor of a nuclear-powered icebreaker ship produces a thermal power of 160 MW. Each fission of uranium-235 releases about 200 MeV. Take 1 MeV = 1.6×10⁻¹³ J. What is the approximate number of fissions per second in the core?

Why does the fission of a uranium-235 nucleus release energy?

Geologists date volcanic crystals by counting microscopic damage tracks. These tracks are left when uranium-238 nuclei in the crystal undergo spontaneous fission. The half-life of U-238 is 4.5×10⁹ years. Two crystals, A and B, have the same uranium concentration. Crystal A has twice as many tracks per unit area as crystal B. Both crystals are a few million years old. What can be concluded?

A nucleus of californium-252, ²⁵²₉₈Cf, undergoes spontaneous fission. It produces a barium-144 nucleus, ¹⁴⁴₅₆Ba, four neutrons and one other nucleus. What is the other nucleus?

A nucleus X of nucleon number A and binding energy per nucleon B_X absorbs a slow neutron and splits into two nuclei, P and Q, and three neutrons.

P has nucleon number A_P and binding energy per nucleon B_P; Q has nucleon number A_Q and binding energy per nucleon B_Q. What is the energy released?

The fission of one uranium-235 nucleus releases about 200 MeV. The fusion reaction ²₁H + ³₁H → ⁴₂He + ¹₀n releases 17.6 MeV.

Taking the mass of each nucleus to be its nucleon number × 1 u, what is (energy released per kilogram of fusion fuel)/(energy released per kilogram of uranium-235)?

A nuclear power station has an electrical output power P and an overall efficiency η. Each fission in the reactor releases energy E.

How many fissions occur in the reactor each second?

A neutron-induced fission is represented by

²³⁵₉₂U + ¹₀n → X + Y + 3 ¹₀n

The mass of the uranium-235 nucleus is m_U, the mass of a neutron is m_n, and the masses of the nuclei X and Y are m_X and m_Y. What is the energy released? (c is the speed of light.)

A neutron of kinetic energy E_n is absorbed by a stationary uranium-235 nucleus, which then undergoes fission. The products have a total kinetic energy E_k, and gamma photons of total energy E_γ are emitted.

What is the decrease in total rest mass? (c is the speed of light.)

The thermal power of a reactor, found from the heating of its coolant, is 2.00 GW. Each fission of uranium-235 releases a total of 202 MeV, including the energy from the decay of its fission products. Of this, 9 MeV is carried by the antineutrinos emitted in the β⁻ decays of the fission products.

How many fissions occur each second? (1 MeV = 1.60 × 10⁻¹³ J)

Fission fragments from uranium-235 are nearly always radioactive and usually decay by β⁻ emission. Which statement explains this?

Three statements are made about the fission of uranium-235 into two fragments and some neutrons.
I. The total binding energy of the products is greater than the binding energy of the uranium-236 compound nucleus.
II. The total rest mass of the products is less than the total rest mass of the uranium-235 nucleus and the absorbed neutron.
III. The binding energy per nucleon of the fragments is less than that of uranium-235.
Which statements are correct?

In a heavy-water moderator, a fast neutron collides elastically and head-on with a stationary deuterium nucleus. Take the mass of the deuterium nucleus to be twice the mass of the neutron. What fraction of the neutron's initial kinetic energy is transferred to the deuterium nucleus?

In a university laboratory, a sealed source emits fast neutrons. A detector that responds only to slow neutrons is placed a fixed distance from the source. Sheets of polyethylene, a plastic containing many hydrogen atoms, are placed between the source and the detector, and the number of counts N in 300 s is recorded for each total thickness x. The background count is negligible. The uncertainty in a number of counts N is ±√N. The graph shows the data with error bars.

To model how a moderator slows neutrons, a student uses two gliders on a level air track. Glider 1, of mass 0.200 kg, represents a neutron. It moves at speed u towards glider 2, of mass M, which is at rest and represents a nucleus of the moderator. Magnetic bumpers make the collision nearly elastic, and glider 1 rebounds with speed v. A card of length (100.0 ± 0.5) mm on glider 1 passes through a light gate before and after the collision, taking times t₁ and t₂. Each time has an uncertainty of ±0.002 s.

For a perfectly elastic head-on collision, theory predicts v/u = (A − 1)/(A + 1), where A = M/0.200 kg.

After fuel is removed from a nuclear reactor it must be stored in a cooling pond. A fuel-storage company publishes the thermal power P released per tonne of spent fuel at times t after removal, for one type of fuel. Each value of P has an uncertainty of ±4 %.

An engineer suggests the model P = kt⁻ⁿ, where k and n are constants. The graph shows log(P/kW t⁻¹) against log(t/days), with error bars, for six of the times.

A published database lists measurements made on samples of spent fuel from pressurized-water reactors. For each sample it gives the burn-up B, the thermal energy released per tonne of uranium in the fuel, in GW d t⁻¹ (1 GW d is the energy released in one day at a power of 1 GW), and the mass m of caesium-137 per tonne of uranium. A student selects seven samples of one fuel type and plots m against B. The uncertainties are ±3 % in B and ±2 % in m.

Energy released per fission = 200 MeV; 1 MeV = 1.60 × 10⁻¹³ J; 1 u = 1.661 × 10⁻²⁷ kg.

Uranium-238 undergoes spontaneous fission, very rarely. A laboratory uses this to measure the mass of uranium in sealed drums of depleted uranium oxide (almost pure uranium-238) without opening them. Each drum is placed in a neutron counter for 1000 s and the number of counts N is recorded. Six calibration drums of the same size and type, containing known masses M of uranium, give the results below. The uncertainty in a number of counts N is ±√N; the error bars are too small to show.

A student estimates the energy released in the neutron-induced fission

²³⁵₉₂U + ¹₀n → ¹⁴¹₅₆Ba + ⁹²₃₆Kr + 3 ¹₀n

in two ways. First, the student plots the binding energy per nucleon of the long-lived nuclides listed in a nuclear database (graph) and reads values from a curve drawn through them. Second, the student uses the atomic masses in the table. (1 u = 931.5 MeV c⁻²)

The thermal power of a small research reactor fuelled with uranium-235 is found from its cooling circuit. Water flows through the core at a mass flow rate of (42.0 ± 0.5) kg s⁻¹. During a 6.0 hour run at constant power, the operators record the temperatures of the water entering and leaving the core every hour. Each thermometer has a resolution of 0.1 °C.

Specific heat capacity of water = 4180 J kg⁻¹ K⁻¹. Average energy released per fission of uranium-235 = 200 MeV. 1 u = 1.661 × 10⁻²⁷ kg.

A student uses a computer simulation to study the chain reaction in a cube of a solution of a uranium-235 salt in water. For a cube of side L the simulation follows many generations of neutrons and gives the multiplication factor k, the average number of neutrons from each fission that go on to cause another fission. Each value of k has an uncertainty of ±0.010. To save computing time, only cubes with L ≥ 25 cm were simulated.

The student tests the model k = k_∞ − b/L, where k_∞ and b are constants. The model assumes that the fraction of neutrons lost through the surface of the cube is proportional to surface area/volume.

The diagram shows a simplified nuclear power station of the pressurized-water type. Water in the reactor vessel surrounds the uranium fuel rods and is pumped round a closed primary circuit. Four parts of the power station are marked by numbered boxes.

A small fission reactor is designed to supply a base on the Moon. It must deliver an electrical power of 40 kW for 10 years. Its core contains 8.0 kg of uranium-235 and is surrounded by a reflector that scatters escaping neutrons back into the core. The overall efficiency of the conversion of thermal energy into electrical energy is 25 %. The energy that is not converted is radiated into space from radiator panels at a temperature of 400 K, with emissivity 0.90.

(energy released per fission = 200 MeV, 1 MeV = 1.60 × 10⁻¹³ J, 1 u = 1.661 × 10⁻²⁷ kg, σ = 5.67 × 10⁻⁸ W m⁻² K⁻⁴, 1 year = 3.16 × 10⁷ s)

A coal-fired power station has an electrical power output of 1.2 GW and an overall efficiency of 38 %. It is to be replaced by a nuclear power station with the same electrical power output and an overall efficiency of 34 %.

Energy density of the coal = 29 MJ kg⁻¹. Burning 1.0 kg of this coal produces 2.6 kg of carbon dioxide. Energy released per fission of uranium-235 = 200 MeV. 1 MeV = 1.60 × 10⁻¹³ J; 1 u = 1.661 × 10⁻²⁷ kg.

In a pool-type research reactor the core is near the bottom of an open tank of water 8.0 m deep. The top of the core is 1.2 m above the floor of the tank. The water acts as the moderator, the coolant and the shielding. The reactor operates at a thermal power of 2.0 MW; a pump circulates the water through the core and through a heat exchanger.

A design study for a lunar surface power station considers plutonium-239 as the fuel. One possible fission reaction is
²³⁹₉₄Pu + ¹₀n → ¹⁴⁵₅₆Ba + ⁹²₃₈Sr + x ¹₀n

A submarine is powered by a pressurized-water reactor that uses enriched uranium fuel, water as moderator and coolant, boron control rods and thick steel and concrete shielding.

The reactor of a nuclear-powered submarine produces thermal energy at a steady rate of 150 MW. Each fission of a uranium-235 nucleus releases 200 MeV.
Molar mass of uranium-235 = 0.235 kg mol⁻¹.

A small fission reactor is proposed to supply power to a crewed base on the Moon. Its compact core of uranium-235 is surrounded by a beryllium oxide reflector. A single boron carbide control rod runs through the centre of the core. The reactor delivers a thermal power of 40 kW. Each fission of U-235 releases 3.2×10⁻¹¹ J.
Take 1 u = 1.661×10⁻²⁷ kg and 1 year = 3.16×10⁷ s.

A submarine is powered by a pressurised water reactor. The fuel is enriched uranium. Water in the primary circuit is pumped through the core, where it acts as both moderator and coolant. The core produces a thermal power of 150 MW. The primary water rises in temperature by 30 K as it passes through the core. At the operating temperature and pressure, the specific heat capacity of the water is 5.4×10³ J kg⁻¹ K⁻¹.

Spent fuel removed from a nuclear reactor contains many radioactive fission products, among them iodine-131 (half-life 8.0 days) and caesium-137 (half-life 30 years). The spent fuel is first stored under water in a cooling pond.

In a pressurized-water reactor, a typical neutron-induced fission is

²³⁵₉₂U + ¹₀n → ¹⁴⁴₅₆Ba + ⁸⁹₃₆Kr + x ¹₀n

The table gives the binding energy per nucleon of the three nuclides. The diagram shows the reactor core and the heat exchanger of the power station.

Californium-252, ²⁵²₉₈Cf, is used as a neutron source to start up nuclear reactors. Most californium-252 nuclei decay by alpha emission, but about 3 % undergo spontaneous fission, for example

²⁵²₉₈Cf → ¹⁴⁰₅₄Xe + ¹⁰⁸₄₄Ru + x ¹₀n

The table gives some masses. (1 u = 931.5 MeV c⁻²)

A slow neutron is absorbed by a stationary uranium-235 nucleus, which then splits:

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

Immediately after the fission the two fragments have a total kinetic energy of 170 MeV. Take the masses of the barium and krypton nuclei to be 141 u and 92 u, and ignore the momentum of the neutrons. (1 u = 1.661 × 10⁻²⁷ kg, e = 1.60 × 10⁻¹⁹ C)

In a graphite-moderated reactor, neutrons released by fission with kinetic energies of about 2 MeV are slowed down by elastic collisions with carbon-12 nuclei until their kinetic energy is about 0.025 eV. Take the mass of a carbon-12 nucleus to be 12 times the mass m of a neutron. The carbon nuclei may be treated as stationary.

For a head-on elastic collision of a neutron with a stationary nucleus of nucleon number A, the fraction of its kinetic energy that the neutron keeps is f = ((A − 1)/(A + 1))².

High-level waste from reprocessed fuel is melted into glass, sealed in a steel canister and placed 500 m underground in a hole in stable rock. The canister is surrounded by a layer of bentonite clay 0.35 m thick, which swells when wet and keeps groundwater away from the canister. The clay must stay below 100 °C.

The thermal power of the canister comes from the decay of fission products, mainly strontium-90 and caesium-137; take the half-life of both to be 30 years. Forty years after the fuel left the reactor the thermal power of the canister is 1.2 kW. Model the clay as a flat layer of area 16 m² and thermal conductivity 1.0 W m⁻¹ K⁻¹, whose outer surface is kept at 50 °C by the surrounding rock.

Molybdenum-99 is used in hospitals as the source of technetium-99m for medical imaging. It is made by neutron-induced fission: small targets containing uranium-235 are placed in a research reactor for several days, and the molybdenum is then separated chemically from the other fission products. One possible fission is

²³⁵₉₂U + ¹₀n → ⁹⁹₄₂Mo + ^A_ZSn + 3 ¹₀n

About 6.1 % of all fissions of uranium-235 lead to a nucleus of molybdenum-99.

A nuclear-powered icebreaker has two pressurized-water reactors, each with a thermal power of 171 MW. Steam turbines drive generators that supply electric motors, which turn the propellers. The total power delivered to the propellers is 54 MW.

A typical fission in the fuel is

²³⁵₉₂U + ¹₀n → ¹⁴⁴₅₅Cs + ⁹⁰₃₇Rb + x ¹₀n

The table gives some atomic masses. 1 u = 931.5 MeV c⁻²; 1 MeV = 1.60 × 10⁻¹³ J.